Research ArticleNonlinear Flow Characteristics and Horizontal Well PressureTransient Analysis for Low-Permeability Offshore Reservoirs

Jianchun Xu Ruizhong Jiang and Wenchao Teng

China University of Petroleum (East China) Qingdao 266580 China

Correspondence should be addressed to Jianchun Xu illeyupcgmailcom

Received 21 October 2014 Revised 29 January 2015 Accepted 13 February 2015

Academic Editor Rosana Rodriguez-Lopez

Copyright copy 2015 Jianchun Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Threshold pressure gradient (TPG) and stress sensitivity which cause the nonlinear flow in low permeability reservoirs werecarried out by experiments Firstly the investigation of existing conditions of TPG for oil flow in irreducible water saturationlow-permeability reservoirs was conducted and discussed using the cores from a real offshore oilfield in China The existenceof TPG was proven The relationship between TPG and absolute permeability was obtained by laboratory tests TPG increaseswith decreasing absolute permeability Then stress sensitivity experiment was carried out through depressurizing experimentand step-up pressure experiment Permeability modulus which characterizes stress sensitivity increases with decreasing absolutepermeability Consequently a horizontal well pressure transient analysis mathematical model considering threshold pressuregradient and stress sensitivity was established on the basis of mass and momentum conservation equations The finite elementmethod (FEM) was presented to solve the model Influencing factors such as TPG permeability modulus skin factor wellborestorage horizontal length horizontal position and boundary effect on pressure and pressure derivative curves were also discussedResults analysis demonstrates that the pressure transient curves are different from Darcyrsquos model when considering the nonlinearflow characteristics Both TPG and permeability modulus lead to more energy consumption and the reservoir pressure decreasesmore than Darcyrsquos model

1 Introduction

With the growing tension of oil and gas resources low per-meability reservoirs have become the most important sourceof oil exploration and development in Chinarsquos petroleumindustry They are taking up nearly half reserves of oil andgas respectively

For the lowpermeability formation fluid flow in reservoirhas some special characteristics which can be summarized asfollows (1) flow departures from Darcyrsquos law at low-velocityfluid flow Many authors have noted the departures fromDarcyrsquos law in porous media through experiment methods[1ndash6] and they found that the flow curve is a combinationof a straight line and a concave curve Non-Darcyrsquos flowexists in low permeability porous media in which the TPGalways can be observed Some scholars expounded the non-Darcy phenomenon and thought that it has significanteffect on petroleum development In 2004 Gavin [7] gavea detailed discussion on the non-Darcy phenomenon under

low-velocity conditions in porous media Basak [8] identifiedlow-velocity fluid flow as ldquopre-Darcy flowrdquo where the increaseof fluid flow velocity can be greater than that proportional tothe increase of fluid pressure gradient For tight gas reservoirthe non-Darcy flow was also observed in a water-bearingreservoir and the TPG exists [9 10] Currently the non-Darcyflowhas been discussed inmanyfields in petroleum includingnumerical simulation enhanced oil recovery productivityevaluation and well test [11ndash14] So it is significant to researchon the non-Darcy phenomenon thoroughly

(2) Stress sensitivity is obvious which cannot be ignoredStress sensitivity phenomenon is common in many kinds ofreservoirs which is defined as reservoir rocks with fluid-flow characteristics (permeability) that are highly sensitiveto the effective stress changes andor if they are of weakmechanical strength causing large rock deformation andare considered to be stress-sensitive [15ndash19] For the lowpermeability formation the deformational characteristics ofrocks are always observed and have a great effect on well

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 387149 13 pageshttpdxdoiorg1011552015387149

2 Mathematical Problems in Engineering

Table 1 Property of natural cores in displacement experiment

Sample number Porosity Length Diameter Area Permeability() 119871 (cm) 119863 (cm) 119860 (cm2) 119870 (times10minus3 120583m2)

1 177 3118 2548 5098893 2682 168 6642 2538 5058949 1343 153 2568 2544 5082897 8044 170 4772 2544 5082897 675 155 4254 2566 5171189 5366 158 5864 2538 5058949 402

Table 2 Property of natural cores in stress sensitivity experiment

Sample number Porosity Length Diameter Area Permeability() 119871 (cm) 119863 (cm) 119860 (cm2) 119870 (times10minus3 120583m2)

7 156 4085 2552 5112473 22188 159 3065 2553 511648 40979 158 5215 257 5184847 466610 168 7042 2567 5172749 867011 172 7124 2582 5233378 16200

productivity [20 21] The study on stress sensitivity for lowpermeability formation is meaningful

For the offshore low permeability oil field the horizontalwell is always used to achieve an efficient development Toobtain the formation parameters pressure transient analysisis often used Many scholars had done study on interpreta-tions of formation parameters using pressure data [22ndash27]But for low permeability reservoirs this work has not beendone by considering threshold pressure gradient and stresssensitivity which cause the nonlinear flow

In this paper firstly threshold pressure gradient and stresssensitivity experiments in low permeability oil reservoirswere made and discussed using the cores from a real offshoreoilfield in China Then the pressure transient analysis math-ematical model was derived for the horizontal well FEMwasproposed to solve themodelThe sensitivity of the type curvesand derivative type curves on monitoring TPG permeabilitymodulus well location well length wellbore storage and skinfactor was analyzed finally

2 Experimental Measurement and Analysis

21 TPG Experiment Six natural cores were taken froma low permeability reservoir in the a real offshore oil-field in China with the diameter of 25 cm length rangeof 2568 cmndash6642 cm permeability range of (402ndash268) times

10minus3 120583m2 and porosity range of 153ndash177 (shown in Table 1)The simulated oil was a mixture of degassing crude oiland kerosene with viscosity of 1332mPasdots and density of084 gmL The salinity of formation water was 70mgmL(NaCl CaCl

2

MgCl2

sdot6H2

O = 7 06 04) and the viscosityis 115mPasdots

In order to investigate TPG in low permeability coresthe experimental equipment designed by Wang et al [3]was used which includes five parts power system buffersystem core holder system confining pressure system and

themeasurement system InWangrsquo experiment water was thedisplaced fluid in our experiment the experiment procedurewas adjusted in which the displaced fluid was simulatedoil and the cores were in irreducible water saturation state(Figure 1)

During the experiments the following data needed tobe recorded inlet pressure of the core holder fluid volumeand residual pressure After one displacement the curve ofvelocity-pressure gradient was plotted By epitaxial methodwe can get the intersection between straight line and pressureaxis which is defined as pseudothreshold pressure gradient(TPG)

22 Stress Sensitivity Experiment Five natural cores weretaken from a low permeability reservoir in the South Chinasea oilfield with the diameter of 25 cm length range of3065 cmndash7124 cm permeability range of (2218ndash16200) times

10minus3 120583m2 and porosity range of 156ndash172 (shown inTable 2)The simulated oil was a mixture of degassing crude oiland kerosene with viscosity of 1332mPasdots and density of084 gmL The salinity of formation water is 70mgmL(NaCl CaCl

2

MgCl2

sdot6H2

O = 7 06 04) and the viscositywas 115mPasdots The experimental equipment in Section 21was used

The overburden pressure of the study formation is about22MPa so the confining pressure was set as 22MPa Twokinds of experimentswere conductedmdashdepressurizing exper-iment and step-up pressure experiment During the experi-ment the confining pressure is maintained and inletoutletpressure difference is kept as 05MPa The back-pressureand inlet pressure were adjusted to a given value The flowrate inlet and outlet pressure and pressure differential wererecorded and the core permeability was calculated at acertain pressurewhen the flowwas in steady state Inletoutletpressures are then gradually decreased (depressurizing exper-iment) or increased (step-up pressure experiment) and the

Mathematical Problems in Engineering 3

000000

000001

000002

000003

000004

000005

Velo

city

(ms

)

Number 1Number 2Number 3

Number 4Number 5Number 6

0 1 2 3 4 5 6 7 8 9 10

Pressure gradient (106 Pam)

Figure 1 Experiment results of simulated oil displacement

000

001

002

003

004

005

TPG

Permeability (10minus15 m2)

y = 017xminus107

R2 = 09689

0 5 10 15 20 25 30

Thre

shol

d pr

essu

re g

radi

ent (10

6Pa

m)

Figure 2 Relationship between TPG and permeability

displacement experiment was repeated The core permeabil-ity was calculated at each pressure

23 Analysis of Nonlinear Experiment The TPG exists in lowpermeability reservoirs From Figure 2 we can see that TPGversus permeability presents power function and the TPGwill decrease

Wang et al [20] described the stress sensitivity with thepermeability modulus 120572 which is defined as follows

120572 =1

119870

119889119870

119889119901 (1)

So the permeability can be modified as

119870 = 119870119894

119890minus120572(119901119894minus119901) (2)

2

4

6

8

10

12

14

16

18

Number 7-DNumber 7-SNumber 8-DNumber 8-SNumber 9-D

Number 9-SNumber 10-DNumber 10-SNumber 11-DNumber 11-S

0 2 4 6 8 10 12 14 16

Perm

eabi

lity

(10minus15

m2)

Pore fluid pressure (106 Pa)

Figure 3 Relationship between pore fluid pressure and permeabil-ity

From Figure 3 we can see that the stress sensitivity existswidely in low permeability reservoirs for offshore oilfieldcores both in depressurizing experiment and step-up pres-sure experiment The dimensionless permeability (defined as119870119870max) and permeability modulus were calculated in theexperiment as shown in Figure 4 and Table 3 For the 5 coresas the permeability increases the permeability modulusincreases and the permeability modulus in depressurizingexperiment is larger than step-up pressure experiment

It can be seen in the low permeability reservoir thatthe TPG and stress sensitivity always exist The TPG and

4 Mathematical Problems in Engineering

06

07

08

09

10D

imen

sionl

ess p

erm

eabi

lity

frac

tion

Number 7Number 8Number 9

Number 10Number 11

0 2 4 6 8 10 12 14

Pore fluid pressure (106 Pa)

(a) Depressurizing experiment

070

075

080

085

090

095

100

Dim

ensio

nles

s per

mea

bilit

y fr

actio

n

Number 7Number 8Number 9

Number 10Number 11

0 2 4 6 8 10 12 14

Pore fluid pressure (106 Pa)

(b) Step-up pressure experiment

Figure 4 Relationship between pore fluid pressure and dimensionless permeability

Horizontal well

x

z

y

hL

o

zw

Figure 5 Schematic diagram of a horizontal well in a low perme-ability reservoir

Table 3 Permeability modulus for different core samples

Sample Permeability modulus(depressurizing)MPaminus1

Permeabilitymodulus (step-uppressure)MPaminus1

7 0049 00378 0043 00209 0034 001910 0030 002111 0023 0014

permeability modulus become larger in absolute value whenthe core has a smaller permeability

3 A Horizontal Well in an Anisotropic LowPermeability Reservoir

The nonlinear experiment which contains TPG experimentand stress sensitivity experiment indicates that in low perme-ability reservoir TPG and stress sensitivity cannot be ignoredso when conducting pressure transient analysis for wells theyshould be considered In this section the horizontal wellpressure transient analysis model was established and thesolution method was discussed considering nonlinear flowcharacteristics

31 Mathematical Model Figure 5 is a schematic diagram ofthe horizontal well in a reservoir The119909-119910-axes are in thehorizontal directions and the 119911-axes in the vertical directionTheorigin is at the bottomof the reservoir Some assumptionsare made as follows

(1) The outer boundary of a circular reservoir is closed orwith constant pressure reservoir radius is 119903

119890

(2) The reservoir is horizontal with uniform thickness of

ℎ and original pressure 119901119894

(3) Reservoir permeability anisotropy is considered with

horizontal permeability 119896ℎ

and vertical permeability119896V

(4) The well is located in the 119909119900119911 plane with perforatedlength of 2119871 this well is produced at constant produc-tion rate of 119876

(5) The reservoir fluid is slightly compressible withcompressibility 119862

119905

and viscosity of crude oil 120583(6) The reservoir media is deformational with perme-

ability modulus 120572 and TPG 120582

Mathematical Problems in Engineering 5

(7) The influence of gravity and capillary forces can beignored

(8) Wellbore storage effect and formation damage aretaken into account

32 Mathematical Model The mathematical model in lowpermeability reservoir is

120597

120597119909(119870119909

120583120575119870119909

120597119901

120597119909) +

120597

120597119910(119870119910

120583120575119870119910

120597119901

120597119910)

+120597

120597119911(119870119911

120583120575119870119911

120597119901

120597119911) = 120601119862

119905

120597119901

120597119905+ 119902

(3)

Assuming 119870119909

= 119870119910

= 119870ℎ

119870119911

= 119870V and ignoring source andsink term the equation will be changed to

119870ℎ119894

120583[

120597

120597119909(119890minus120572(119901119894minus119901)120575

119870119909

120597119901

120597119909) +

120597

120597119910(119890minus120572(119901119894minus119901)120575

119870119910

120597119901

120597119910)

+120597

120597119911(119890minus120572(119901119894minus119901)120575

119870119911

120573120597119901

120597119911)] = 120601119862

119905

120597119901

120597119905

(4)

Initial condition is

119901 (119909 119910 119911 119905 = 0) = 119901119894

(5)

Closed outer boundary is

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119890=

120597119901

120597119911

10038161003816100381610038161003816100381610038161003816119911=119911119890= 0 (6)

Constant-pressure outer boundary is

119901 (119903 = 119903119890

) = 119901 (119911 = 119911119890

) = 119901119894

(7)

Some dimensionless terms are defined as Appendix A showsSo the dimensionless mathematical model will be

120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)

= (ℎ119863

119871119863

)2

120597119901119863

120597119905119863

119901119863

(119909119863

119910119863

119911119863

119905119863

= 0) = 0

120597119901119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

=120597119901119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=01

= 0

119901119863

(119903119863

= 119903119890119863

) = 119901119891119863

(119911119863

= 0 1) = 0

(8)

For this model the inner boundary condition is constantrate production The equation of inner boundary conditionis shown in Appendix B2

Analytical solutionNumerical solution

Horizontal line

05 horizontal line

Slope = 05

Slope = 1

101

10minus1

100

10minus2

tDCD

10010minus2 102 104

pwD

(dpwDdt D)lowastt D

Figure 6 Comparison of analytical solution and numerical solution(119862119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 0 120582119863

= 0)

4 Computational Issues

The FEM is used to solve this model The detailed solutionprocedure is shown in Appendix BThe GMRESmethod wasused to solve the equation system [28]

5 Results and Discussions

51 Pressure Transient Behavior To verify the accuracy ofthe model comparisons are made with the solution by Liet al [29] The comparison was made without consideringthe TPG and permeability modulus As shown in Figure 6the type curves and derivative type curves from our modelare in good agreement with those from Li et al [29] FromFigure 6 we can see that considering the TPG and stresssensitivity four stages are observed (1) early pure wellborestorage stage which is characterized by a 1 slope in pressurederivative curve (2) early vertical radial flow stage whichis characterized by a horizontal line in pressure derivativecurve (3) linear flow stage which is characterized by a 12slope in pressure derivative curve (4) late radial flow stagewhich is characterized by a 05 horizontal line in pressurederivative curve

From Figure 7 we can see that the pressure and pressurederivative curveswhen considering stress sensitivity andTPGare different from those based on Darcyrsquos model The effectsof stress sensitivity and TPG cause the pressure and pressurederivative curves to shift upwards in the late stage resultingin the disappearance of pressure derivative horizon with thevalue of 05 in radial flow stage Because of the upward of thecurves late radial flow stage tends to be like linear flow Thereason is that in the formation there is additional pressureloss when considering TPG and stress sensitivity The flowequation never obeys Darcyrsquos model

Comparedwith the stress sensitivity TPG can causemoredeviation to the pressure and pressure derivative curvesThisis because that horizontal well has a large contact with thereservoir pressure gradient near the wellbore is not as large

6 Mathematical Problems in Engineering

tDCD

pwD

(dpwDdt D

)lowastt D

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0 120572D = 0

120582D = 0 120572D = 01120582D = 01 120572D = 0

120582D = 01 120572D = 01

Figure 7 Comparison of dimensionless type curves with consider-ing TPG and stress sensitivity (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200)

as in a vertical well but is the main section of TPG effect andthus the effect of stress sensitivity is not as obvious as TPGWhen considering the two factors together the degree of theupward for the curve is the supposition of the two factors tobe considered separately

52 Sensitivity Analysis In this section the sensitivity anal-ysis was obtained which includes the TPG permeabilitymodulus skin factor wellbore storage horizontal lengthhorizontal position and boundary effect

For each sensitivity parameter the pressure and pressurederivative curves were presented In each figure the dimen-sionless parameters were given based on the definition inAppendix A

(1) Effect of 119862119863

Figures 8 and 9 show the effect of wellborestorage coefficient on well test curves The basic parametersare ℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V= 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0016 times

10minus6m3Pa 0032 times 10minus6m3Pa 016 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1 120582 = 0002 times

106 Pam and 119885119908

= 45m Large wellbore storage coefficientcauses the pressure curve to shift upward and the larger thewellbore storage coefficient is the higher the ldquohumprdquo of thepressure derivative curve is and the earlier linear flow occursWhen the wellbore storage coefficient is large enough theearly radial flow stage will be concealed by linear flow Whenthe horizontal axis is 119905

119863

it can be seen that the variationof wellbore storage coefficient causes different duration timeof the wellbore storage stage resulting in horizontal shift ofthe well test curve The pressure and pressure derivate curvescoincide together at later time

(2) Effect of 119878 Figure 10 shows the effect of the skin factoron well test curves The basic parameters are ℎ = 9m 119903

119908

=01m 119871 = 100m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 1 5 10 119862 = 0032 times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1

CD = 50

CD = 100

CD = 500

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 8 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

tD

CD = 50

CD = 100

CD = 500

100

101

10minus1

100

10minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 9 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

120582 = 0002 times 106 Pam and 119885119908

= 45m The larger skin factorrepresents heavier pollution and greater additional pressuredrop it also causes higher ldquohumprdquo on well test curve Largeskin factor may shorten the early radial flow segment or evenmake it disappear

(3) Effect of 120572119863

Figure 11 shows the effect of the permeabilitymodulus on well test curves The basic parameters are ℎ =9m 119903

119908

= 01m 119871 = 200m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0119862 = 0032 times 10minus6m3Pa 120593 =025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0 0023 times 10minus6Paminus1 0049 times

10minus6 Paminus1 120582 = 0003 times 106 Pam and 119885119908

= 45m Accordingto the core experiment the largest permeability modulusin absolute value is 0049 times 10minus6 Paminus1 So we set the largest

Mathematical Problems in Engineering 7

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

S = 1

S = 5

S = 10

pwD

(dpwDdt D)lowastt D

Figure 10 Comparison of dimensionless type curves with differentskin factors (119871

119863

= 5 ℎ119863

= 200 120572119863

= 01 120582119863

= 01)

120572D = 0

120572D = 0047

120572D = 01

tDCD

101

10minus1

100

100

10minus210minus2

102 104

pwD

(dpwDdt D)lowastt D

Figure 11 Comparison of dimensionless type curves with differentpermeability moduli (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120582119863

= 015)

permeability modulus equal to the 0049 times 10minus6 Paminus1 Largepermeability modulus makes the reservoir more sensitive tothe stress causing the pressure and pressure derivate curvesto increase seriously Effect of stress sensitivity on the curvemainly concentrates on the stages after the linear flow

(4) Effect of 120582119863

Figure 12 shows the effect of TPG onhorizontal well test curves The basic parameters are ℎ = 9m119903119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6 Paminus1 120582 = 0 times

106 Pam 0006times 106 Pam 0047times 106 Pam and119885119908

=45mAccording to the core experiment the largest permeabilitymodulus in absolute value is 0047 times 106 Pam So we setthe largest TPG equal to the 0047 times 106 Pam The largeTPG represents strong non-Darcy flow resulting in large

tDCD

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0120582D = 031120582D = 243

pwD

(dpwDdt D)lowastt D

Figure 12 Comparison of dimensionless type curves with differentTPGs (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 005)

tDCD

101

10minus1

100

100

10minus210minus2

102 104

LD = 4

LD = 6

LD = 8

pwD

(dpwDdt D)lowastt D

Figure 13 Comparison of dimensionless type curves with differenthorizontal well lengths (119862

119863

= 100 119878 = 0 ℎ119863

= 200 120572119863

= 005 120582119863

=005)

flow resistance The larger the TPG is the more upturnedthe pressure and pressure derivate curves are The amountof upturning grows with time The 05 value of pressurederivative curve disappears and boundary between the linearflow and late radial flow is not obvious

(5) Effect of 119871119863

Figure 13 shows the effect of the horizontalwell length The basic parameters are ℎ = 9m 119903

119908

= 01m 119871= 80m 120m 160m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam and 119885119908

= 45m It can be seen from the figure thatthe longer the horizontal well is the lower the pressure andpressure derivate curves are and the more obvious the earlyradius flow is Considering the effect of stress sensitivity and

8 Mathematical Problems in Engineering

tDCD

101

10minus1

100

100

10minus210minus2

102 104

hD = 100

hD = 200

hD = 300

pwD

(dpwDdt D)lowastt D

Figure 14 Comparison of dimensionless type curves with differentreservoir thicknesses (119862

119863

= 100 119878 = 0 119871119863

= 5 120572119863

= 005 120582119863

= 005)

nonlinear flow the well test curves of different horizontal welllengths distributed parallel instead of coinciding together

(6) Effect of ℎ119863

Figure 14 shows the effect of reservoirthickness on well test curves The basic parameters are ℎ =45m 9m 135m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2119870V = 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times

10minus6m3Pa 120593 = 025 119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times

10minus6Paminus1 and120582=0001times 106 PamWe set119885119908

equal to 225m45m and 675m respectively which makes the horizontalwell locates in the center in vertical direction It can be seenthat the thicker the reservoir is the longer the early radialflow stage isWhen the reservoir thickness is very small earlyradial flowwill be concealed by the effects of wellbore storage

For traditional horizontal well test with the increase ofreservoir thickness the duration of early vertical radial flowlasts longer and early radial flow stage (the slope of pressureand pressure derivative curves are 05) shifts backward Inthe late radial flow period the pressure derivative curves ofdifferent reservoir thickness coincide together to a horizontalline of 05 that is the speed of pressure drop tends to beuniform

After the introduction of permeability modulus andTPG the duration of early vertical radial flow lasts longerwith increasing reservoir thickness and the effect of stresssensitivity andpseudolinear flowweakens resulting in overalldecrease of pure wellbore storage stage and backwardnessof pressure and pressure derivative curves in linear flow Inthe late radial flow period pressure derivative curve is not astraight line at 05 the pressure and pressure derivative curvesunder different reservoir thicknesses upturn parallely

(7) Effect of 119911119908119863

Figure 15 shows the effect of the horizontalwell position on the well test curve The basic parameters areℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2120583= 1times 0001 Pasdots 119878= 0 5 10119862= 0032times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1

tDCD

101

10minus1

100

100

10minus210minus2

102 104

ZwD = 01

ZwD = 03

ZwD = 05

pwD

(dpwDdt D)lowastt D

Figure 15 Comparison of dimensionless type curves with differenthorizontal well positions (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

102

101

10minus1

100

10minus2

tDCD

10010minus2 102 104 106 108

ReD = 5

ReD = 10

ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 16 Comparison of dimensionless type curves with differentclosed outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

120582 = 0001 times 106 Pam and 119885119908

= 09m 27m 45m It can beseen that when other parameters are kept constant the closerto 05 119911

119908119863

is that is horizontal well is closer to the middleof the reservoir the longer early radical flow lasts Horizontalwell position does not affect thewellbore storage segment thelinear flow and late-linear flow segment

(8) Effect ofOuter Boundary Figures 16 and 17 showboundaryeffect The basic parameters are ℎ = 9m 119903

119908

= 01m 119871 =

100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2 120583 = 1 times

0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025 119888119905

=00022 times 10minus6 Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam 119885119908

= 45m and 119877119890

= 1000m 2000m 3000mFigure 16 shows that the larger the distance of boundary tothe well is the longer it requires to see the reflections and

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Table 1 Property of natural cores in displacement experiment

Sample number Porosity Length Diameter Area Permeability() 119871 (cm) 119863 (cm) 119860 (cm2) 119870 (times10minus3 120583m2)

1 177 3118 2548 5098893 2682 168 6642 2538 5058949 1343 153 2568 2544 5082897 8044 170 4772 2544 5082897 675 155 4254 2566 5171189 5366 158 5864 2538 5058949 402

Table 2 Property of natural cores in stress sensitivity experiment

Sample number Porosity Length Diameter Area Permeability() 119871 (cm) 119863 (cm) 119860 (cm2) 119870 (times10minus3 120583m2)

7 156 4085 2552 5112473 22188 159 3065 2553 511648 40979 158 5215 257 5184847 466610 168 7042 2567 5172749 867011 172 7124 2582 5233378 16200

productivity [20 21] The study on stress sensitivity for lowpermeability formation is meaningful

For the offshore low permeability oil field the horizontalwell is always used to achieve an efficient development Toobtain the formation parameters pressure transient analysisis often used Many scholars had done study on interpreta-tions of formation parameters using pressure data [22ndash27]But for low permeability reservoirs this work has not beendone by considering threshold pressure gradient and stresssensitivity which cause the nonlinear flow

In this paper firstly threshold pressure gradient and stresssensitivity experiments in low permeability oil reservoirswere made and discussed using the cores from a real offshoreoilfield in China Then the pressure transient analysis math-ematical model was derived for the horizontal well FEMwasproposed to solve themodelThe sensitivity of the type curvesand derivative type curves on monitoring TPG permeabilitymodulus well location well length wellbore storage and skinfactor was analyzed finally

2 Experimental Measurement and Analysis

21 TPG Experiment Six natural cores were taken froma low permeability reservoir in the a real offshore oil-field in China with the diameter of 25 cm length rangeof 2568 cmndash6642 cm permeability range of (402ndash268) times

10minus3 120583m2 and porosity range of 153ndash177 (shown in Table 1)The simulated oil was a mixture of degassing crude oiland kerosene with viscosity of 1332mPasdots and density of084 gmL The salinity of formation water was 70mgmL(NaCl CaCl

2

MgCl2

sdot6H2

O = 7 06 04) and the viscosityis 115mPasdots

In order to investigate TPG in low permeability coresthe experimental equipment designed by Wang et al [3]was used which includes five parts power system buffersystem core holder system confining pressure system and

themeasurement system InWangrsquo experiment water was thedisplaced fluid in our experiment the experiment procedurewas adjusted in which the displaced fluid was simulatedoil and the cores were in irreducible water saturation state(Figure 1)

During the experiments the following data needed tobe recorded inlet pressure of the core holder fluid volumeand residual pressure After one displacement the curve ofvelocity-pressure gradient was plotted By epitaxial methodwe can get the intersection between straight line and pressureaxis which is defined as pseudothreshold pressure gradient(TPG)

22 Stress Sensitivity Experiment Five natural cores weretaken from a low permeability reservoir in the South Chinasea oilfield with the diameter of 25 cm length range of3065 cmndash7124 cm permeability range of (2218ndash16200) times

10minus3 120583m2 and porosity range of 156ndash172 (shown inTable 2)The simulated oil was a mixture of degassing crude oiland kerosene with viscosity of 1332mPasdots and density of084 gmL The salinity of formation water is 70mgmL(NaCl CaCl

2

MgCl2

sdot6H2

O = 7 06 04) and the viscositywas 115mPasdots The experimental equipment in Section 21was used

The overburden pressure of the study formation is about22MPa so the confining pressure was set as 22MPa Twokinds of experimentswere conductedmdashdepressurizing exper-iment and step-up pressure experiment During the experi-ment the confining pressure is maintained and inletoutletpressure difference is kept as 05MPa The back-pressureand inlet pressure were adjusted to a given value The flowrate inlet and outlet pressure and pressure differential wererecorded and the core permeability was calculated at acertain pressurewhen the flowwas in steady state Inletoutletpressures are then gradually decreased (depressurizing exper-iment) or increased (step-up pressure experiment) and the

Mathematical Problems in Engineering 3

000000

000001

000002

000003

000004

000005

Velo

city

(ms

)

Number 1Number 2Number 3

Number 4Number 5Number 6

0 1 2 3 4 5 6 7 8 9 10

Pressure gradient (106 Pam)

Figure 1 Experiment results of simulated oil displacement

000

001

002

003

004

005

TPG

Permeability (10minus15 m2)

y = 017xminus107

R2 = 09689

0 5 10 15 20 25 30

Thre

shol

d pr

essu

re g

radi

ent (10

6Pa

m)

Figure 2 Relationship between TPG and permeability

displacement experiment was repeated The core permeabil-ity was calculated at each pressure

23 Analysis of Nonlinear Experiment The TPG exists in lowpermeability reservoirs From Figure 2 we can see that TPGversus permeability presents power function and the TPGwill decrease

Wang et al [20] described the stress sensitivity with thepermeability modulus 120572 which is defined as follows

120572 =1

119870

119889119870

119889119901 (1)

So the permeability can be modified as

119870 = 119870119894

119890minus120572(119901119894minus119901) (2)

2

4

6

8

10

12

14

16

18

Number 7-DNumber 7-SNumber 8-DNumber 8-SNumber 9-D

Number 9-SNumber 10-DNumber 10-SNumber 11-DNumber 11-S

0 2 4 6 8 10 12 14 16

Perm

eabi

lity

(10minus15

m2)

Pore fluid pressure (106 Pa)

Figure 3 Relationship between pore fluid pressure and permeabil-ity

From Figure 3 we can see that the stress sensitivity existswidely in low permeability reservoirs for offshore oilfieldcores both in depressurizing experiment and step-up pres-sure experiment The dimensionless permeability (defined as119870119870max) and permeability modulus were calculated in theexperiment as shown in Figure 4 and Table 3 For the 5 coresas the permeability increases the permeability modulusincreases and the permeability modulus in depressurizingexperiment is larger than step-up pressure experiment

It can be seen in the low permeability reservoir thatthe TPG and stress sensitivity always exist The TPG and

4 Mathematical Problems in Engineering

06

07

08

09

10D

imen

sionl

ess p

erm

eabi

lity

frac

tion

Number 7Number 8Number 9

Number 10Number 11

0 2 4 6 8 10 12 14

Pore fluid pressure (106 Pa)

(a) Depressurizing experiment

070

075

080

085

090

095

100

Dim

ensio

nles

s per

mea

bilit

y fr

actio

n

Number 7Number 8Number 9

Number 10Number 11

0 2 4 6 8 10 12 14

Pore fluid pressure (106 Pa)

(b) Step-up pressure experiment

Figure 4 Relationship between pore fluid pressure and dimensionless permeability

Horizontal well

x

z

y

hL

o

zw

Figure 5 Schematic diagram of a horizontal well in a low perme-ability reservoir

Table 3 Permeability modulus for different core samples

Sample Permeability modulus(depressurizing)MPaminus1

Permeabilitymodulus (step-uppressure)MPaminus1

7 0049 00378 0043 00209 0034 001910 0030 002111 0023 0014

permeability modulus become larger in absolute value whenthe core has a smaller permeability

3 A Horizontal Well in an Anisotropic LowPermeability Reservoir

The nonlinear experiment which contains TPG experimentand stress sensitivity experiment indicates that in low perme-ability reservoir TPG and stress sensitivity cannot be ignoredso when conducting pressure transient analysis for wells theyshould be considered In this section the horizontal wellpressure transient analysis model was established and thesolution method was discussed considering nonlinear flowcharacteristics

31 Mathematical Model Figure 5 is a schematic diagram ofthe horizontal well in a reservoir The119909-119910-axes are in thehorizontal directions and the 119911-axes in the vertical directionTheorigin is at the bottomof the reservoir Some assumptionsare made as follows

(1) The outer boundary of a circular reservoir is closed orwith constant pressure reservoir radius is 119903

119890

(2) The reservoir is horizontal with uniform thickness of

ℎ and original pressure 119901119894

(3) Reservoir permeability anisotropy is considered with

horizontal permeability 119896ℎ

and vertical permeability119896V

(4) The well is located in the 119909119900119911 plane with perforatedlength of 2119871 this well is produced at constant produc-tion rate of 119876

(5) The reservoir fluid is slightly compressible withcompressibility 119862

119905

and viscosity of crude oil 120583(6) The reservoir media is deformational with perme-

ability modulus 120572 and TPG 120582

Mathematical Problems in Engineering 5

(7) The influence of gravity and capillary forces can beignored

(8) Wellbore storage effect and formation damage aretaken into account

32 Mathematical Model The mathematical model in lowpermeability reservoir is

120597

120597119909(119870119909

120583120575119870119909

120597119901

120597119909) +

120597

120597119910(119870119910

120583120575119870119910

120597119901

120597119910)

+120597

120597119911(119870119911

120583120575119870119911

120597119901

120597119911) = 120601119862

119905

120597119901

120597119905+ 119902

(3)

Assuming 119870119909

= 119870119910

= 119870ℎ

119870119911

= 119870V and ignoring source andsink term the equation will be changed to

119870ℎ119894

120583[

120597

120597119909(119890minus120572(119901119894minus119901)120575

119870119909

120597119901

120597119909) +

120597

120597119910(119890minus120572(119901119894minus119901)120575

119870119910

120597119901

120597119910)

+120597

120597119911(119890minus120572(119901119894minus119901)120575

119870119911

120573120597119901

120597119911)] = 120601119862

119905

120597119901

120597119905

(4)

Initial condition is

119901 (119909 119910 119911 119905 = 0) = 119901119894

(5)

Closed outer boundary is

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119890=

120597119901

120597119911

10038161003816100381610038161003816100381610038161003816119911=119911119890= 0 (6)

Constant-pressure outer boundary is

119901 (119903 = 119903119890

) = 119901 (119911 = 119911119890

) = 119901119894

(7)

Some dimensionless terms are defined as Appendix A showsSo the dimensionless mathematical model will be

120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)

= (ℎ119863

119871119863

)2

120597119901119863

120597119905119863

119901119863

(119909119863

119910119863

119911119863

119905119863

= 0) = 0

120597119901119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

=120597119901119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=01

= 0

119901119863

(119903119863

= 119903119890119863

) = 119901119891119863

(119911119863

= 0 1) = 0

(8)

For this model the inner boundary condition is constantrate production The equation of inner boundary conditionis shown in Appendix B2

Analytical solutionNumerical solution

Horizontal line

05 horizontal line

Slope = 05

Slope = 1

101

10minus1

100

10minus2

tDCD

10010minus2 102 104

pwD

(dpwDdt D)lowastt D

Figure 6 Comparison of analytical solution and numerical solution(119862119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 0 120582119863

= 0)

4 Computational Issues

The FEM is used to solve this model The detailed solutionprocedure is shown in Appendix BThe GMRESmethod wasused to solve the equation system [28]

5 Results and Discussions

51 Pressure Transient Behavior To verify the accuracy ofthe model comparisons are made with the solution by Liet al [29] The comparison was made without consideringthe TPG and permeability modulus As shown in Figure 6the type curves and derivative type curves from our modelare in good agreement with those from Li et al [29] FromFigure 6 we can see that considering the TPG and stresssensitivity four stages are observed (1) early pure wellborestorage stage which is characterized by a 1 slope in pressurederivative curve (2) early vertical radial flow stage whichis characterized by a horizontal line in pressure derivativecurve (3) linear flow stage which is characterized by a 12slope in pressure derivative curve (4) late radial flow stagewhich is characterized by a 05 horizontal line in pressurederivative curve

From Figure 7 we can see that the pressure and pressurederivative curveswhen considering stress sensitivity andTPGare different from those based on Darcyrsquos model The effectsof stress sensitivity and TPG cause the pressure and pressurederivative curves to shift upwards in the late stage resultingin the disappearance of pressure derivative horizon with thevalue of 05 in radial flow stage Because of the upward of thecurves late radial flow stage tends to be like linear flow Thereason is that in the formation there is additional pressureloss when considering TPG and stress sensitivity The flowequation never obeys Darcyrsquos model

Comparedwith the stress sensitivity TPG can causemoredeviation to the pressure and pressure derivative curvesThisis because that horizontal well has a large contact with thereservoir pressure gradient near the wellbore is not as large

6 Mathematical Problems in Engineering

tDCD

pwD

(dpwDdt D

)lowastt D

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0 120572D = 0

120582D = 0 120572D = 01120582D = 01 120572D = 0

120582D = 01 120572D = 01

Figure 7 Comparison of dimensionless type curves with consider-ing TPG and stress sensitivity (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200)

as in a vertical well but is the main section of TPG effect andthus the effect of stress sensitivity is not as obvious as TPGWhen considering the two factors together the degree of theupward for the curve is the supposition of the two factors tobe considered separately

52 Sensitivity Analysis In this section the sensitivity anal-ysis was obtained which includes the TPG permeabilitymodulus skin factor wellbore storage horizontal lengthhorizontal position and boundary effect

For each sensitivity parameter the pressure and pressurederivative curves were presented In each figure the dimen-sionless parameters were given based on the definition inAppendix A

(1) Effect of 119862119863

Figures 8 and 9 show the effect of wellborestorage coefficient on well test curves The basic parametersare ℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V= 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0016 times

10minus6m3Pa 0032 times 10minus6m3Pa 016 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1 120582 = 0002 times

106 Pam and 119885119908

= 45m Large wellbore storage coefficientcauses the pressure curve to shift upward and the larger thewellbore storage coefficient is the higher the ldquohumprdquo of thepressure derivative curve is and the earlier linear flow occursWhen the wellbore storage coefficient is large enough theearly radial flow stage will be concealed by linear flow Whenthe horizontal axis is 119905

119863

it can be seen that the variationof wellbore storage coefficient causes different duration timeof the wellbore storage stage resulting in horizontal shift ofthe well test curve The pressure and pressure derivate curvescoincide together at later time

(2) Effect of 119878 Figure 10 shows the effect of the skin factoron well test curves The basic parameters are ℎ = 9m 119903

119908

=01m 119871 = 100m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 1 5 10 119862 = 0032 times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1

CD = 50

CD = 100

CD = 500

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 8 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

tD

CD = 50

CD = 100

CD = 500

100

101

10minus1

100

10minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 9 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

120582 = 0002 times 106 Pam and 119885119908

= 45m The larger skin factorrepresents heavier pollution and greater additional pressuredrop it also causes higher ldquohumprdquo on well test curve Largeskin factor may shorten the early radial flow segment or evenmake it disappear

(3) Effect of 120572119863

Figure 11 shows the effect of the permeabilitymodulus on well test curves The basic parameters are ℎ =9m 119903

119908

= 01m 119871 = 200m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0119862 = 0032 times 10minus6m3Pa 120593 =025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0 0023 times 10minus6Paminus1 0049 times

10minus6 Paminus1 120582 = 0003 times 106 Pam and 119885119908

= 45m Accordingto the core experiment the largest permeability modulusin absolute value is 0049 times 10minus6 Paminus1 So we set the largest

Mathematical Problems in Engineering 7

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

S = 1

S = 5

S = 10

pwD

(dpwDdt D)lowastt D

Figure 10 Comparison of dimensionless type curves with differentskin factors (119871

119863

= 5 ℎ119863

= 200 120572119863

= 01 120582119863

= 01)

120572D = 0

120572D = 0047

120572D = 01

tDCD

101

10minus1

100

100

10minus210minus2

102 104

pwD

(dpwDdt D)lowastt D

Figure 11 Comparison of dimensionless type curves with differentpermeability moduli (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120582119863

= 015)

permeability modulus equal to the 0049 times 10minus6 Paminus1 Largepermeability modulus makes the reservoir more sensitive tothe stress causing the pressure and pressure derivate curvesto increase seriously Effect of stress sensitivity on the curvemainly concentrates on the stages after the linear flow

(4) Effect of 120582119863

Figure 12 shows the effect of TPG onhorizontal well test curves The basic parameters are ℎ = 9m119903119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6 Paminus1 120582 = 0 times

106 Pam 0006times 106 Pam 0047times 106 Pam and119885119908

=45mAccording to the core experiment the largest permeabilitymodulus in absolute value is 0047 times 106 Pam So we setthe largest TPG equal to the 0047 times 106 Pam The largeTPG represents strong non-Darcy flow resulting in large

tDCD

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0120582D = 031120582D = 243

pwD

(dpwDdt D)lowastt D

Figure 12 Comparison of dimensionless type curves with differentTPGs (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 005)

tDCD

101

10minus1

100

100

10minus210minus2

102 104

LD = 4

LD = 6

LD = 8

pwD

(dpwDdt D)lowastt D

Figure 13 Comparison of dimensionless type curves with differenthorizontal well lengths (119862

119863

= 100 119878 = 0 ℎ119863

= 200 120572119863

= 005 120582119863

=005)

flow resistance The larger the TPG is the more upturnedthe pressure and pressure derivate curves are The amountof upturning grows with time The 05 value of pressurederivative curve disappears and boundary between the linearflow and late radial flow is not obvious

(5) Effect of 119871119863

Figure 13 shows the effect of the horizontalwell length The basic parameters are ℎ = 9m 119903

119908

= 01m 119871= 80m 120m 160m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam and 119885119908

= 45m It can be seen from the figure thatthe longer the horizontal well is the lower the pressure andpressure derivate curves are and the more obvious the earlyradius flow is Considering the effect of stress sensitivity and

8 Mathematical Problems in Engineering

tDCD

101

10minus1

100

100

10minus210minus2

102 104

hD = 100

hD = 200

hD = 300

pwD

(dpwDdt D)lowastt D

Figure 14 Comparison of dimensionless type curves with differentreservoir thicknesses (119862

119863

= 100 119878 = 0 119871119863

= 5 120572119863

= 005 120582119863

= 005)

nonlinear flow the well test curves of different horizontal welllengths distributed parallel instead of coinciding together

(6) Effect of ℎ119863

Figure 14 shows the effect of reservoirthickness on well test curves The basic parameters are ℎ =45m 9m 135m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2119870V = 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times

10minus6m3Pa 120593 = 025 119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times

10minus6Paminus1 and120582=0001times 106 PamWe set119885119908

equal to 225m45m and 675m respectively which makes the horizontalwell locates in the center in vertical direction It can be seenthat the thicker the reservoir is the longer the early radialflow stage isWhen the reservoir thickness is very small earlyradial flowwill be concealed by the effects of wellbore storage

For traditional horizontal well test with the increase ofreservoir thickness the duration of early vertical radial flowlasts longer and early radial flow stage (the slope of pressureand pressure derivative curves are 05) shifts backward Inthe late radial flow period the pressure derivative curves ofdifferent reservoir thickness coincide together to a horizontalline of 05 that is the speed of pressure drop tends to beuniform

After the introduction of permeability modulus andTPG the duration of early vertical radial flow lasts longerwith increasing reservoir thickness and the effect of stresssensitivity andpseudolinear flowweakens resulting in overalldecrease of pure wellbore storage stage and backwardnessof pressure and pressure derivative curves in linear flow Inthe late radial flow period pressure derivative curve is not astraight line at 05 the pressure and pressure derivative curvesunder different reservoir thicknesses upturn parallely

(7) Effect of 119911119908119863

Figure 15 shows the effect of the horizontalwell position on the well test curve The basic parameters areℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2120583= 1times 0001 Pasdots 119878= 0 5 10119862= 0032times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1

tDCD

101

10minus1

100

100

10minus210minus2

102 104

ZwD = 01

ZwD = 03

ZwD = 05

pwD

(dpwDdt D)lowastt D

Figure 15 Comparison of dimensionless type curves with differenthorizontal well positions (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

102

101

10minus1

100

10minus2

tDCD

10010minus2 102 104 106 108

ReD = 5

ReD = 10

ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 16 Comparison of dimensionless type curves with differentclosed outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

120582 = 0001 times 106 Pam and 119885119908

= 09m 27m 45m It can beseen that when other parameters are kept constant the closerto 05 119911

119908119863

is that is horizontal well is closer to the middleof the reservoir the longer early radical flow lasts Horizontalwell position does not affect thewellbore storage segment thelinear flow and late-linear flow segment

(8) Effect ofOuter Boundary Figures 16 and 17 showboundaryeffect The basic parameters are ℎ = 9m 119903

119908

= 01m 119871 =

100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2 120583 = 1 times

0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025 119888119905

=00022 times 10minus6 Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam 119885119908

= 45m and 119877119890

= 1000m 2000m 3000mFigure 16 shows that the larger the distance of boundary tothe well is the longer it requires to see the reflections and

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

000000

000001

000002

000003

000004

000005

Velo

city

(ms

)

Number 1Number 2Number 3

Number 4Number 5Number 6

0 1 2 3 4 5 6 7 8 9 10

Pressure gradient (106 Pam)

Figure 1 Experiment results of simulated oil displacement

000

001

002

003

004

005

TPG

Permeability (10minus15 m2)

y = 017xminus107

R2 = 09689

0 5 10 15 20 25 30

Thre

shol

d pr

essu

re g

radi

ent (10

6Pa

m)

Figure 2 Relationship between TPG and permeability

displacement experiment was repeated The core permeabil-ity was calculated at each pressure

23 Analysis of Nonlinear Experiment The TPG exists in lowpermeability reservoirs From Figure 2 we can see that TPGversus permeability presents power function and the TPGwill decrease

Wang et al [20] described the stress sensitivity with thepermeability modulus 120572 which is defined as follows

120572 =1

119870

119889119870

119889119901 (1)

So the permeability can be modified as

119870 = 119870119894

119890minus120572(119901119894minus119901) (2)

2

4

6

8

10

12

14

16

18

Number 7-DNumber 7-SNumber 8-DNumber 8-SNumber 9-D

Number 9-SNumber 10-DNumber 10-SNumber 11-DNumber 11-S

0 2 4 6 8 10 12 14 16

Perm

eabi

lity

(10minus15

m2)

Pore fluid pressure (106 Pa)

Figure 3 Relationship between pore fluid pressure and permeabil-ity

From Figure 3 we can see that the stress sensitivity existswidely in low permeability reservoirs for offshore oilfieldcores both in depressurizing experiment and step-up pres-sure experiment The dimensionless permeability (defined as119870119870max) and permeability modulus were calculated in theexperiment as shown in Figure 4 and Table 3 For the 5 coresas the permeability increases the permeability modulusincreases and the permeability modulus in depressurizingexperiment is larger than step-up pressure experiment

It can be seen in the low permeability reservoir thatthe TPG and stress sensitivity always exist The TPG and

4 Mathematical Problems in Engineering

06

07

08

09

10D

imen

sionl

ess p

erm

eabi

lity

frac

tion

Number 7Number 8Number 9

Number 10Number 11

0 2 4 6 8 10 12 14

Pore fluid pressure (106 Pa)

(a) Depressurizing experiment

070

075

080

085

090

095

100

Dim

ensio

nles

s per

mea

bilit

y fr

actio

n

Number 7Number 8Number 9

Number 10Number 11

0 2 4 6 8 10 12 14

Pore fluid pressure (106 Pa)

(b) Step-up pressure experiment

Figure 4 Relationship between pore fluid pressure and dimensionless permeability

Horizontal well

x

z

y

hL

o

zw

Figure 5 Schematic diagram of a horizontal well in a low perme-ability reservoir

Table 3 Permeability modulus for different core samples

Sample Permeability modulus(depressurizing)MPaminus1

Permeabilitymodulus (step-uppressure)MPaminus1

7 0049 00378 0043 00209 0034 001910 0030 002111 0023 0014

permeability modulus become larger in absolute value whenthe core has a smaller permeability

3 A Horizontal Well in an Anisotropic LowPermeability Reservoir

The nonlinear experiment which contains TPG experimentand stress sensitivity experiment indicates that in low perme-ability reservoir TPG and stress sensitivity cannot be ignoredso when conducting pressure transient analysis for wells theyshould be considered In this section the horizontal wellpressure transient analysis model was established and thesolution method was discussed considering nonlinear flowcharacteristics

31 Mathematical Model Figure 5 is a schematic diagram ofthe horizontal well in a reservoir The119909-119910-axes are in thehorizontal directions and the 119911-axes in the vertical directionTheorigin is at the bottomof the reservoir Some assumptionsare made as follows

(1) The outer boundary of a circular reservoir is closed orwith constant pressure reservoir radius is 119903

119890

(2) The reservoir is horizontal with uniform thickness of

ℎ and original pressure 119901119894

(3) Reservoir permeability anisotropy is considered with

horizontal permeability 119896ℎ

and vertical permeability119896V

(4) The well is located in the 119909119900119911 plane with perforatedlength of 2119871 this well is produced at constant produc-tion rate of 119876

(5) The reservoir fluid is slightly compressible withcompressibility 119862

119905

and viscosity of crude oil 120583(6) The reservoir media is deformational with perme-

ability modulus 120572 and TPG 120582

Mathematical Problems in Engineering 5

(7) The influence of gravity and capillary forces can beignored

(8) Wellbore storage effect and formation damage aretaken into account

32 Mathematical Model The mathematical model in lowpermeability reservoir is

120597

120597119909(119870119909

120583120575119870119909

120597119901

120597119909) +

120597

120597119910(119870119910

120583120575119870119910

120597119901

120597119910)

+120597

120597119911(119870119911

120583120575119870119911

120597119901

120597119911) = 120601119862

119905

120597119901

120597119905+ 119902

(3)

Assuming 119870119909

= 119870119910

= 119870ℎ

119870119911

= 119870V and ignoring source andsink term the equation will be changed to

119870ℎ119894

120583[

120597

120597119909(119890minus120572(119901119894minus119901)120575

119870119909

120597119901

120597119909) +

120597

120597119910(119890minus120572(119901119894minus119901)120575

119870119910

120597119901

120597119910)

+120597

120597119911(119890minus120572(119901119894minus119901)120575

119870119911

120573120597119901

120597119911)] = 120601119862

119905

120597119901

120597119905

(4)

Initial condition is

119901 (119909 119910 119911 119905 = 0) = 119901119894

(5)

Closed outer boundary is

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119890=

120597119901

120597119911

10038161003816100381610038161003816100381610038161003816119911=119911119890= 0 (6)

Constant-pressure outer boundary is

119901 (119903 = 119903119890

) = 119901 (119911 = 119911119890

) = 119901119894

(7)

Some dimensionless terms are defined as Appendix A showsSo the dimensionless mathematical model will be

120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)

= (ℎ119863

119871119863

)2

120597119901119863

120597119905119863

119901119863

(119909119863

119910119863

119911119863

119905119863

= 0) = 0

120597119901119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

=120597119901119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=01

= 0

119901119863

(119903119863

= 119903119890119863

) = 119901119891119863

(119911119863

= 0 1) = 0

(8)

For this model the inner boundary condition is constantrate production The equation of inner boundary conditionis shown in Appendix B2

Analytical solutionNumerical solution

Horizontal line

05 horizontal line

Slope = 05

Slope = 1

101

10minus1

100

10minus2

tDCD

10010minus2 102 104

pwD

(dpwDdt D)lowastt D

Figure 6 Comparison of analytical solution and numerical solution(119862119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 0 120582119863

= 0)

4 Computational Issues

The FEM is used to solve this model The detailed solutionprocedure is shown in Appendix BThe GMRESmethod wasused to solve the equation system [28]

5 Results and Discussions

51 Pressure Transient Behavior To verify the accuracy ofthe model comparisons are made with the solution by Liet al [29] The comparison was made without consideringthe TPG and permeability modulus As shown in Figure 6the type curves and derivative type curves from our modelare in good agreement with those from Li et al [29] FromFigure 6 we can see that considering the TPG and stresssensitivity four stages are observed (1) early pure wellborestorage stage which is characterized by a 1 slope in pressurederivative curve (2) early vertical radial flow stage whichis characterized by a horizontal line in pressure derivativecurve (3) linear flow stage which is characterized by a 12slope in pressure derivative curve (4) late radial flow stagewhich is characterized by a 05 horizontal line in pressurederivative curve

From Figure 7 we can see that the pressure and pressurederivative curveswhen considering stress sensitivity andTPGare different from those based on Darcyrsquos model The effectsof stress sensitivity and TPG cause the pressure and pressurederivative curves to shift upwards in the late stage resultingin the disappearance of pressure derivative horizon with thevalue of 05 in radial flow stage Because of the upward of thecurves late radial flow stage tends to be like linear flow Thereason is that in the formation there is additional pressureloss when considering TPG and stress sensitivity The flowequation never obeys Darcyrsquos model

Comparedwith the stress sensitivity TPG can causemoredeviation to the pressure and pressure derivative curvesThisis because that horizontal well has a large contact with thereservoir pressure gradient near the wellbore is not as large

6 Mathematical Problems in Engineering

tDCD

pwD

(dpwDdt D

)lowastt D

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0 120572D = 0

120582D = 0 120572D = 01120582D = 01 120572D = 0

120582D = 01 120572D = 01

Figure 7 Comparison of dimensionless type curves with consider-ing TPG and stress sensitivity (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200)

as in a vertical well but is the main section of TPG effect andthus the effect of stress sensitivity is not as obvious as TPGWhen considering the two factors together the degree of theupward for the curve is the supposition of the two factors tobe considered separately

52 Sensitivity Analysis In this section the sensitivity anal-ysis was obtained which includes the TPG permeabilitymodulus skin factor wellbore storage horizontal lengthhorizontal position and boundary effect

For each sensitivity parameter the pressure and pressurederivative curves were presented In each figure the dimen-sionless parameters were given based on the definition inAppendix A

(1) Effect of 119862119863

Figures 8 and 9 show the effect of wellborestorage coefficient on well test curves The basic parametersare ℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V= 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0016 times

10minus6m3Pa 0032 times 10minus6m3Pa 016 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1 120582 = 0002 times

106 Pam and 119885119908

= 45m Large wellbore storage coefficientcauses the pressure curve to shift upward and the larger thewellbore storage coefficient is the higher the ldquohumprdquo of thepressure derivative curve is and the earlier linear flow occursWhen the wellbore storage coefficient is large enough theearly radial flow stage will be concealed by linear flow Whenthe horizontal axis is 119905

119863

it can be seen that the variationof wellbore storage coefficient causes different duration timeof the wellbore storage stage resulting in horizontal shift ofthe well test curve The pressure and pressure derivate curvescoincide together at later time

(2) Effect of 119878 Figure 10 shows the effect of the skin factoron well test curves The basic parameters are ℎ = 9m 119903

119908

=01m 119871 = 100m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 1 5 10 119862 = 0032 times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1

CD = 50

CD = 100

CD = 500

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 8 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

tD

CD = 50

CD = 100

CD = 500

100

101

10minus1

100

10minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 9 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

120582 = 0002 times 106 Pam and 119885119908

= 45m The larger skin factorrepresents heavier pollution and greater additional pressuredrop it also causes higher ldquohumprdquo on well test curve Largeskin factor may shorten the early radial flow segment or evenmake it disappear

(3) Effect of 120572119863

Figure 11 shows the effect of the permeabilitymodulus on well test curves The basic parameters are ℎ =9m 119903

119908

= 01m 119871 = 200m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0119862 = 0032 times 10minus6m3Pa 120593 =025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0 0023 times 10minus6Paminus1 0049 times

10minus6 Paminus1 120582 = 0003 times 106 Pam and 119885119908

= 45m Accordingto the core experiment the largest permeability modulusin absolute value is 0049 times 10minus6 Paminus1 So we set the largest

Mathematical Problems in Engineering 7

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

S = 1

S = 5

S = 10

pwD

(dpwDdt D)lowastt D

Figure 10 Comparison of dimensionless type curves with differentskin factors (119871

119863

= 5 ℎ119863

= 200 120572119863

= 01 120582119863

= 01)

120572D = 0

120572D = 0047

120572D = 01

tDCD

101

10minus1

100

100

10minus210minus2

102 104

pwD

(dpwDdt D)lowastt D

Figure 11 Comparison of dimensionless type curves with differentpermeability moduli (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120582119863

= 015)

permeability modulus equal to the 0049 times 10minus6 Paminus1 Largepermeability modulus makes the reservoir more sensitive tothe stress causing the pressure and pressure derivate curvesto increase seriously Effect of stress sensitivity on the curvemainly concentrates on the stages after the linear flow

(4) Effect of 120582119863

Figure 12 shows the effect of TPG onhorizontal well test curves The basic parameters are ℎ = 9m119903119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6 Paminus1 120582 = 0 times

106 Pam 0006times 106 Pam 0047times 106 Pam and119885119908

=45mAccording to the core experiment the largest permeabilitymodulus in absolute value is 0047 times 106 Pam So we setthe largest TPG equal to the 0047 times 106 Pam The largeTPG represents strong non-Darcy flow resulting in large

tDCD

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0120582D = 031120582D = 243

pwD

(dpwDdt D)lowastt D

Figure 12 Comparison of dimensionless type curves with differentTPGs (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 005)

tDCD

101

10minus1

100

100

10minus210minus2

102 104

LD = 4

LD = 6

LD = 8

pwD

(dpwDdt D)lowastt D

Figure 13 Comparison of dimensionless type curves with differenthorizontal well lengths (119862

119863

= 100 119878 = 0 ℎ119863

= 200 120572119863

= 005 120582119863

=005)

flow resistance The larger the TPG is the more upturnedthe pressure and pressure derivate curves are The amountof upturning grows with time The 05 value of pressurederivative curve disappears and boundary between the linearflow and late radial flow is not obvious

(5) Effect of 119871119863

Figure 13 shows the effect of the horizontalwell length The basic parameters are ℎ = 9m 119903

119908

= 01m 119871= 80m 120m 160m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam and 119885119908

= 45m It can be seen from the figure thatthe longer the horizontal well is the lower the pressure andpressure derivate curves are and the more obvious the earlyradius flow is Considering the effect of stress sensitivity and

8 Mathematical Problems in Engineering

tDCD

101

10minus1

100

100

10minus210minus2

102 104

hD = 100

hD = 200

hD = 300

pwD

(dpwDdt D)lowastt D

Figure 14 Comparison of dimensionless type curves with differentreservoir thicknesses (119862

119863

= 100 119878 = 0 119871119863

= 5 120572119863

= 005 120582119863

= 005)

nonlinear flow the well test curves of different horizontal welllengths distributed parallel instead of coinciding together

(6) Effect of ℎ119863

Figure 14 shows the effect of reservoirthickness on well test curves The basic parameters are ℎ =45m 9m 135m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2119870V = 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times

10minus6m3Pa 120593 = 025 119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times

10minus6Paminus1 and120582=0001times 106 PamWe set119885119908

equal to 225m45m and 675m respectively which makes the horizontalwell locates in the center in vertical direction It can be seenthat the thicker the reservoir is the longer the early radialflow stage isWhen the reservoir thickness is very small earlyradial flowwill be concealed by the effects of wellbore storage

For traditional horizontal well test with the increase ofreservoir thickness the duration of early vertical radial flowlasts longer and early radial flow stage (the slope of pressureand pressure derivative curves are 05) shifts backward Inthe late radial flow period the pressure derivative curves ofdifferent reservoir thickness coincide together to a horizontalline of 05 that is the speed of pressure drop tends to beuniform

After the introduction of permeability modulus andTPG the duration of early vertical radial flow lasts longerwith increasing reservoir thickness and the effect of stresssensitivity andpseudolinear flowweakens resulting in overalldecrease of pure wellbore storage stage and backwardnessof pressure and pressure derivative curves in linear flow Inthe late radial flow period pressure derivative curve is not astraight line at 05 the pressure and pressure derivative curvesunder different reservoir thicknesses upturn parallely

(7) Effect of 119911119908119863

Figure 15 shows the effect of the horizontalwell position on the well test curve The basic parameters areℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2120583= 1times 0001 Pasdots 119878= 0 5 10119862= 0032times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1

tDCD

101

10minus1

100

100

10minus210minus2

102 104

ZwD = 01

ZwD = 03

ZwD = 05

pwD

(dpwDdt D)lowastt D

Figure 15 Comparison of dimensionless type curves with differenthorizontal well positions (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

102

101

10minus1

100

10minus2

tDCD

10010minus2 102 104 106 108

ReD = 5

ReD = 10

ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 16 Comparison of dimensionless type curves with differentclosed outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

120582 = 0001 times 106 Pam and 119885119908

= 09m 27m 45m It can beseen that when other parameters are kept constant the closerto 05 119911

119908119863

is that is horizontal well is closer to the middleof the reservoir the longer early radical flow lasts Horizontalwell position does not affect thewellbore storage segment thelinear flow and late-linear flow segment

(8) Effect ofOuter Boundary Figures 16 and 17 showboundaryeffect The basic parameters are ℎ = 9m 119903

119908

= 01m 119871 =

100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2 120583 = 1 times

0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025 119888119905

=00022 times 10minus6 Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam 119885119908

= 45m and 119877119890

= 1000m 2000m 3000mFigure 16 shows that the larger the distance of boundary tothe well is the longer it requires to see the reflections and

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

06

07

08

09

10D

imen

sionl

ess p

erm

eabi

lity

frac

tion

Number 7Number 8Number 9

Number 10Number 11

0 2 4 6 8 10 12 14

Pore fluid pressure (106 Pa)

(a) Depressurizing experiment

070

075

080

085

090

095

100

Dim

ensio

nles

s per

mea

bilit

y fr

actio

n

Number 7Number 8Number 9

Number 10Number 11

0 2 4 6 8 10 12 14

Pore fluid pressure (106 Pa)

(b) Step-up pressure experiment

Figure 4 Relationship between pore fluid pressure and dimensionless permeability

Horizontal well

x

z

y

hL

o

zw

Figure 5 Schematic diagram of a horizontal well in a low perme-ability reservoir

Table 3 Permeability modulus for different core samples

Sample Permeability modulus(depressurizing)MPaminus1

Permeabilitymodulus (step-uppressure)MPaminus1

7 0049 00378 0043 00209 0034 001910 0030 002111 0023 0014

permeability modulus become larger in absolute value whenthe core has a smaller permeability

3 A Horizontal Well in an Anisotropic LowPermeability Reservoir

The nonlinear experiment which contains TPG experimentand stress sensitivity experiment indicates that in low perme-ability reservoir TPG and stress sensitivity cannot be ignoredso when conducting pressure transient analysis for wells theyshould be considered In this section the horizontal wellpressure transient analysis model was established and thesolution method was discussed considering nonlinear flowcharacteristics

31 Mathematical Model Figure 5 is a schematic diagram ofthe horizontal well in a reservoir The119909-119910-axes are in thehorizontal directions and the 119911-axes in the vertical directionTheorigin is at the bottomof the reservoir Some assumptionsare made as follows

(1) The outer boundary of a circular reservoir is closed orwith constant pressure reservoir radius is 119903

119890

(2) The reservoir is horizontal with uniform thickness of

ℎ and original pressure 119901119894

(3) Reservoir permeability anisotropy is considered with

horizontal permeability 119896ℎ

and vertical permeability119896V

(4) The well is located in the 119909119900119911 plane with perforatedlength of 2119871 this well is produced at constant produc-tion rate of 119876

(5) The reservoir fluid is slightly compressible withcompressibility 119862

119905

and viscosity of crude oil 120583(6) The reservoir media is deformational with perme-

ability modulus 120572 and TPG 120582

Mathematical Problems in Engineering 5

(7) The influence of gravity and capillary forces can beignored

(8) Wellbore storage effect and formation damage aretaken into account

32 Mathematical Model The mathematical model in lowpermeability reservoir is

120597

120597119909(119870119909

120583120575119870119909

120597119901

120597119909) +

120597

120597119910(119870119910

120583120575119870119910

120597119901

120597119910)

+120597

120597119911(119870119911

120583120575119870119911

120597119901

120597119911) = 120601119862

119905

120597119901

120597119905+ 119902

(3)

Assuming 119870119909

= 119870119910

= 119870ℎ

119870119911

= 119870V and ignoring source andsink term the equation will be changed to

119870ℎ119894

120583[

120597

120597119909(119890minus120572(119901119894minus119901)120575

119870119909

120597119901

120597119909) +

120597

120597119910(119890minus120572(119901119894minus119901)120575

119870119910

120597119901

120597119910)

+120597

120597119911(119890minus120572(119901119894minus119901)120575

119870119911

120573120597119901

120597119911)] = 120601119862

119905

120597119901

120597119905

(4)

Initial condition is

119901 (119909 119910 119911 119905 = 0) = 119901119894

(5)

Closed outer boundary is

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119890=

120597119901

120597119911

10038161003816100381610038161003816100381610038161003816119911=119911119890= 0 (6)

Constant-pressure outer boundary is

119901 (119903 = 119903119890

) = 119901 (119911 = 119911119890

) = 119901119894

(7)

Some dimensionless terms are defined as Appendix A showsSo the dimensionless mathematical model will be

120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)

= (ℎ119863

119871119863

)2

120597119901119863

120597119905119863

119901119863

(119909119863

119910119863

119911119863

119905119863

= 0) = 0

120597119901119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

=120597119901119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=01

= 0

119901119863

(119903119863

= 119903119890119863

) = 119901119891119863

(119911119863

= 0 1) = 0

(8)

For this model the inner boundary condition is constantrate production The equation of inner boundary conditionis shown in Appendix B2

Analytical solutionNumerical solution

Horizontal line

05 horizontal line

Slope = 05

Slope = 1

101

10minus1

100

10minus2

tDCD

10010minus2 102 104

pwD

(dpwDdt D)lowastt D

Figure 6 Comparison of analytical solution and numerical solution(119862119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 0 120582119863

= 0)

4 Computational Issues

The FEM is used to solve this model The detailed solutionprocedure is shown in Appendix BThe GMRESmethod wasused to solve the equation system [28]

5 Results and Discussions

51 Pressure Transient Behavior To verify the accuracy ofthe model comparisons are made with the solution by Liet al [29] The comparison was made without consideringthe TPG and permeability modulus As shown in Figure 6the type curves and derivative type curves from our modelare in good agreement with those from Li et al [29] FromFigure 6 we can see that considering the TPG and stresssensitivity four stages are observed (1) early pure wellborestorage stage which is characterized by a 1 slope in pressurederivative curve (2) early vertical radial flow stage whichis characterized by a horizontal line in pressure derivativecurve (3) linear flow stage which is characterized by a 12slope in pressure derivative curve (4) late radial flow stagewhich is characterized by a 05 horizontal line in pressurederivative curve

From Figure 7 we can see that the pressure and pressurederivative curveswhen considering stress sensitivity andTPGare different from those based on Darcyrsquos model The effectsof stress sensitivity and TPG cause the pressure and pressurederivative curves to shift upwards in the late stage resultingin the disappearance of pressure derivative horizon with thevalue of 05 in radial flow stage Because of the upward of thecurves late radial flow stage tends to be like linear flow Thereason is that in the formation there is additional pressureloss when considering TPG and stress sensitivity The flowequation never obeys Darcyrsquos model

Comparedwith the stress sensitivity TPG can causemoredeviation to the pressure and pressure derivative curvesThisis because that horizontal well has a large contact with thereservoir pressure gradient near the wellbore is not as large

6 Mathematical Problems in Engineering

tDCD

pwD

(dpwDdt D

)lowastt D

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0 120572D = 0

120582D = 0 120572D = 01120582D = 01 120572D = 0

120582D = 01 120572D = 01

Figure 7 Comparison of dimensionless type curves with consider-ing TPG and stress sensitivity (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200)

as in a vertical well but is the main section of TPG effect andthus the effect of stress sensitivity is not as obvious as TPGWhen considering the two factors together the degree of theupward for the curve is the supposition of the two factors tobe considered separately

52 Sensitivity Analysis In this section the sensitivity anal-ysis was obtained which includes the TPG permeabilitymodulus skin factor wellbore storage horizontal lengthhorizontal position and boundary effect

For each sensitivity parameter the pressure and pressurederivative curves were presented In each figure the dimen-sionless parameters were given based on the definition inAppendix A

(1) Effect of 119862119863

Figures 8 and 9 show the effect of wellborestorage coefficient on well test curves The basic parametersare ℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V= 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0016 times

10minus6m3Pa 0032 times 10minus6m3Pa 016 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1 120582 = 0002 times

106 Pam and 119885119908

= 45m Large wellbore storage coefficientcauses the pressure curve to shift upward and the larger thewellbore storage coefficient is the higher the ldquohumprdquo of thepressure derivative curve is and the earlier linear flow occursWhen the wellbore storage coefficient is large enough theearly radial flow stage will be concealed by linear flow Whenthe horizontal axis is 119905

119863

it can be seen that the variationof wellbore storage coefficient causes different duration timeof the wellbore storage stage resulting in horizontal shift ofthe well test curve The pressure and pressure derivate curvescoincide together at later time

(2) Effect of 119878 Figure 10 shows the effect of the skin factoron well test curves The basic parameters are ℎ = 9m 119903

119908

=01m 119871 = 100m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 1 5 10 119862 = 0032 times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1

CD = 50

CD = 100

CD = 500

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 8 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

tD

CD = 50

CD = 100

CD = 500

100

101

10minus1

100

10minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 9 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

120582 = 0002 times 106 Pam and 119885119908

= 45m The larger skin factorrepresents heavier pollution and greater additional pressuredrop it also causes higher ldquohumprdquo on well test curve Largeskin factor may shorten the early radial flow segment or evenmake it disappear

(3) Effect of 120572119863

Figure 11 shows the effect of the permeabilitymodulus on well test curves The basic parameters are ℎ =9m 119903

119908

= 01m 119871 = 200m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0119862 = 0032 times 10minus6m3Pa 120593 =025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0 0023 times 10minus6Paminus1 0049 times

10minus6 Paminus1 120582 = 0003 times 106 Pam and 119885119908

= 45m Accordingto the core experiment the largest permeability modulusin absolute value is 0049 times 10minus6 Paminus1 So we set the largest

Mathematical Problems in Engineering 7

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

S = 1

S = 5

S = 10

pwD

(dpwDdt D)lowastt D

Figure 10 Comparison of dimensionless type curves with differentskin factors (119871

119863

= 5 ℎ119863

= 200 120572119863

= 01 120582119863

= 01)

120572D = 0

120572D = 0047

120572D = 01

tDCD

101

10minus1

100

100

10minus210minus2

102 104

pwD

(dpwDdt D)lowastt D

Figure 11 Comparison of dimensionless type curves with differentpermeability moduli (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120582119863

= 015)

permeability modulus equal to the 0049 times 10minus6 Paminus1 Largepermeability modulus makes the reservoir more sensitive tothe stress causing the pressure and pressure derivate curvesto increase seriously Effect of stress sensitivity on the curvemainly concentrates on the stages after the linear flow

(4) Effect of 120582119863

Figure 12 shows the effect of TPG onhorizontal well test curves The basic parameters are ℎ = 9m119903119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6 Paminus1 120582 = 0 times

106 Pam 0006times 106 Pam 0047times 106 Pam and119885119908

=45mAccording to the core experiment the largest permeabilitymodulus in absolute value is 0047 times 106 Pam So we setthe largest TPG equal to the 0047 times 106 Pam The largeTPG represents strong non-Darcy flow resulting in large

tDCD

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0120582D = 031120582D = 243

pwD

(dpwDdt D)lowastt D

Figure 12 Comparison of dimensionless type curves with differentTPGs (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 005)

tDCD

101

10minus1

100

100

10minus210minus2

102 104

LD = 4

LD = 6

LD = 8

pwD

(dpwDdt D)lowastt D

Figure 13 Comparison of dimensionless type curves with differenthorizontal well lengths (119862

119863

= 100 119878 = 0 ℎ119863

= 200 120572119863

= 005 120582119863

=005)

flow resistance The larger the TPG is the more upturnedthe pressure and pressure derivate curves are The amountof upturning grows with time The 05 value of pressurederivative curve disappears and boundary between the linearflow and late radial flow is not obvious

(5) Effect of 119871119863

Figure 13 shows the effect of the horizontalwell length The basic parameters are ℎ = 9m 119903

119908

= 01m 119871= 80m 120m 160m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam and 119885119908

= 45m It can be seen from the figure thatthe longer the horizontal well is the lower the pressure andpressure derivate curves are and the more obvious the earlyradius flow is Considering the effect of stress sensitivity and

8 Mathematical Problems in Engineering

tDCD

101

10minus1

100

100

10minus210minus2

102 104

hD = 100

hD = 200

hD = 300

pwD

(dpwDdt D)lowastt D

Figure 14 Comparison of dimensionless type curves with differentreservoir thicknesses (119862

119863

= 100 119878 = 0 119871119863

= 5 120572119863

= 005 120582119863

= 005)

nonlinear flow the well test curves of different horizontal welllengths distributed parallel instead of coinciding together

(6) Effect of ℎ119863

Figure 14 shows the effect of reservoirthickness on well test curves The basic parameters are ℎ =45m 9m 135m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2119870V = 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times

10minus6m3Pa 120593 = 025 119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times

10minus6Paminus1 and120582=0001times 106 PamWe set119885119908

equal to 225m45m and 675m respectively which makes the horizontalwell locates in the center in vertical direction It can be seenthat the thicker the reservoir is the longer the early radialflow stage isWhen the reservoir thickness is very small earlyradial flowwill be concealed by the effects of wellbore storage

For traditional horizontal well test with the increase ofreservoir thickness the duration of early vertical radial flowlasts longer and early radial flow stage (the slope of pressureand pressure derivative curves are 05) shifts backward Inthe late radial flow period the pressure derivative curves ofdifferent reservoir thickness coincide together to a horizontalline of 05 that is the speed of pressure drop tends to beuniform

After the introduction of permeability modulus andTPG the duration of early vertical radial flow lasts longerwith increasing reservoir thickness and the effect of stresssensitivity andpseudolinear flowweakens resulting in overalldecrease of pure wellbore storage stage and backwardnessof pressure and pressure derivative curves in linear flow Inthe late radial flow period pressure derivative curve is not astraight line at 05 the pressure and pressure derivative curvesunder different reservoir thicknesses upturn parallely

(7) Effect of 119911119908119863

Figure 15 shows the effect of the horizontalwell position on the well test curve The basic parameters areℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2120583= 1times 0001 Pasdots 119878= 0 5 10119862= 0032times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1

tDCD

101

10minus1

100

100

10minus210minus2

102 104

ZwD = 01

ZwD = 03

ZwD = 05

pwD

(dpwDdt D)lowastt D

Figure 15 Comparison of dimensionless type curves with differenthorizontal well positions (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

102

101

10minus1

100

10minus2

tDCD

10010minus2 102 104 106 108

ReD = 5

ReD = 10

ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 16 Comparison of dimensionless type curves with differentclosed outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

120582 = 0001 times 106 Pam and 119885119908

= 09m 27m 45m It can beseen that when other parameters are kept constant the closerto 05 119911

119908119863

is that is horizontal well is closer to the middleof the reservoir the longer early radical flow lasts Horizontalwell position does not affect thewellbore storage segment thelinear flow and late-linear flow segment

(8) Effect ofOuter Boundary Figures 16 and 17 showboundaryeffect The basic parameters are ℎ = 9m 119903

119908

= 01m 119871 =

100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2 120583 = 1 times

0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025 119888119905

=00022 times 10minus6 Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam 119885119908

= 45m and 119877119890

= 1000m 2000m 3000mFigure 16 shows that the larger the distance of boundary tothe well is the longer it requires to see the reflections and

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

(7) The influence of gravity and capillary forces can beignored

(8) Wellbore storage effect and formation damage aretaken into account

32 Mathematical Model The mathematical model in lowpermeability reservoir is

120597

120597119909(119870119909

120583120575119870119909

120597119901

120597119909) +

120597

120597119910(119870119910

120583120575119870119910

120597119901

120597119910)

+120597

120597119911(119870119911

120583120575119870119911

120597119901

120597119911) = 120601119862

119905

120597119901

120597119905+ 119902

(3)

Assuming 119870119909

= 119870119910

= 119870ℎ

119870119911

= 119870V and ignoring source andsink term the equation will be changed to

119870ℎ119894

120583[

120597

120597119909(119890minus120572(119901119894minus119901)120575

119870119909

120597119901

120597119909) +

120597

120597119910(119890minus120572(119901119894minus119901)120575

119870119910

120597119901

120597119910)

+120597

120597119911(119890minus120572(119901119894minus119901)120575

119870119911

120573120597119901

120597119911)] = 120601119862

119905

120597119901

120597119905

(4)

Initial condition is

119901 (119909 119910 119911 119905 = 0) = 119901119894

(5)

Closed outer boundary is

120597119901

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119890=

120597119901

120597119911

10038161003816100381610038161003816100381610038161003816119911=119911119890= 0 (6)

Constant-pressure outer boundary is

119901 (119903 = 119903119890

) = 119901 (119911 = 119911119890

) = 119901119894

(7)

Some dimensionless terms are defined as Appendix A showsSo the dimensionless mathematical model will be

120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)

= (ℎ119863

119871119863

)2

120597119901119863

120597119905119863

119901119863

(119909119863

119910119863

119911119863

119905119863

= 0) = 0

120597119901119863

120597119903119863

10038161003816100381610038161003816100381610038161003816119903119863=119903119890119863

=120597119901119863

120597119911119863

10038161003816100381610038161003816100381610038161003816119911119863=01

= 0

119901119863

(119903119863

= 119903119890119863

) = 119901119891119863

(119911119863

= 0 1) = 0

(8)

For this model the inner boundary condition is constantrate production The equation of inner boundary conditionis shown in Appendix B2

Analytical solutionNumerical solution

Horizontal line

05 horizontal line

Slope = 05

Slope = 1

101

10minus1

100

10minus2

tDCD

10010minus2 102 104

pwD

(dpwDdt D)lowastt D

Figure 6 Comparison of analytical solution and numerical solution(119862119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 0 120582119863

= 0)

4 Computational Issues

The FEM is used to solve this model The detailed solutionprocedure is shown in Appendix BThe GMRESmethod wasused to solve the equation system [28]

5 Results and Discussions

51 Pressure Transient Behavior To verify the accuracy ofthe model comparisons are made with the solution by Liet al [29] The comparison was made without consideringthe TPG and permeability modulus As shown in Figure 6the type curves and derivative type curves from our modelare in good agreement with those from Li et al [29] FromFigure 6 we can see that considering the TPG and stresssensitivity four stages are observed (1) early pure wellborestorage stage which is characterized by a 1 slope in pressurederivative curve (2) early vertical radial flow stage whichis characterized by a horizontal line in pressure derivativecurve (3) linear flow stage which is characterized by a 12slope in pressure derivative curve (4) late radial flow stagewhich is characterized by a 05 horizontal line in pressurederivative curve

From Figure 7 we can see that the pressure and pressurederivative curveswhen considering stress sensitivity andTPGare different from those based on Darcyrsquos model The effectsof stress sensitivity and TPG cause the pressure and pressurederivative curves to shift upwards in the late stage resultingin the disappearance of pressure derivative horizon with thevalue of 05 in radial flow stage Because of the upward of thecurves late radial flow stage tends to be like linear flow Thereason is that in the formation there is additional pressureloss when considering TPG and stress sensitivity The flowequation never obeys Darcyrsquos model

Comparedwith the stress sensitivity TPG can causemoredeviation to the pressure and pressure derivative curvesThisis because that horizontal well has a large contact with thereservoir pressure gradient near the wellbore is not as large

6 Mathematical Problems in Engineering

tDCD

pwD

(dpwDdt D

)lowastt D

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0 120572D = 0

120582D = 0 120572D = 01120582D = 01 120572D = 0

120582D = 01 120572D = 01

Figure 7 Comparison of dimensionless type curves with consider-ing TPG and stress sensitivity (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200)

as in a vertical well but is the main section of TPG effect andthus the effect of stress sensitivity is not as obvious as TPGWhen considering the two factors together the degree of theupward for the curve is the supposition of the two factors tobe considered separately

52 Sensitivity Analysis In this section the sensitivity anal-ysis was obtained which includes the TPG permeabilitymodulus skin factor wellbore storage horizontal lengthhorizontal position and boundary effect

For each sensitivity parameter the pressure and pressurederivative curves were presented In each figure the dimen-sionless parameters were given based on the definition inAppendix A

(1) Effect of 119862119863

Figures 8 and 9 show the effect of wellborestorage coefficient on well test curves The basic parametersare ℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V= 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0016 times

10minus6m3Pa 0032 times 10minus6m3Pa 016 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1 120582 = 0002 times

106 Pam and 119885119908

= 45m Large wellbore storage coefficientcauses the pressure curve to shift upward and the larger thewellbore storage coefficient is the higher the ldquohumprdquo of thepressure derivative curve is and the earlier linear flow occursWhen the wellbore storage coefficient is large enough theearly radial flow stage will be concealed by linear flow Whenthe horizontal axis is 119905

119863

it can be seen that the variationof wellbore storage coefficient causes different duration timeof the wellbore storage stage resulting in horizontal shift ofthe well test curve The pressure and pressure derivate curvescoincide together at later time

(2) Effect of 119878 Figure 10 shows the effect of the skin factoron well test curves The basic parameters are ℎ = 9m 119903

119908

=01m 119871 = 100m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 1 5 10 119862 = 0032 times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1

CD = 50

CD = 100

CD = 500

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 8 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

tD

CD = 50

CD = 100

CD = 500

100

101

10minus1

100

10minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 9 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

120582 = 0002 times 106 Pam and 119885119908

= 45m The larger skin factorrepresents heavier pollution and greater additional pressuredrop it also causes higher ldquohumprdquo on well test curve Largeskin factor may shorten the early radial flow segment or evenmake it disappear

(3) Effect of 120572119863

Figure 11 shows the effect of the permeabilitymodulus on well test curves The basic parameters are ℎ =9m 119903

119908

= 01m 119871 = 200m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0119862 = 0032 times 10minus6m3Pa 120593 =025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0 0023 times 10minus6Paminus1 0049 times

10minus6 Paminus1 120582 = 0003 times 106 Pam and 119885119908

= 45m Accordingto the core experiment the largest permeability modulusin absolute value is 0049 times 10minus6 Paminus1 So we set the largest

Mathematical Problems in Engineering 7

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

S = 1

S = 5

S = 10

pwD

(dpwDdt D)lowastt D

Figure 10 Comparison of dimensionless type curves with differentskin factors (119871

119863

= 5 ℎ119863

= 200 120572119863

= 01 120582119863

= 01)

120572D = 0

120572D = 0047

120572D = 01

tDCD

101

10minus1

100

100

10minus210minus2

102 104

pwD

(dpwDdt D)lowastt D

Figure 11 Comparison of dimensionless type curves with differentpermeability moduli (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120582119863

= 015)

permeability modulus equal to the 0049 times 10minus6 Paminus1 Largepermeability modulus makes the reservoir more sensitive tothe stress causing the pressure and pressure derivate curvesto increase seriously Effect of stress sensitivity on the curvemainly concentrates on the stages after the linear flow

(4) Effect of 120582119863

Figure 12 shows the effect of TPG onhorizontal well test curves The basic parameters are ℎ = 9m119903119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6 Paminus1 120582 = 0 times

106 Pam 0006times 106 Pam 0047times 106 Pam and119885119908

=45mAccording to the core experiment the largest permeabilitymodulus in absolute value is 0047 times 106 Pam So we setthe largest TPG equal to the 0047 times 106 Pam The largeTPG represents strong non-Darcy flow resulting in large

tDCD

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0120582D = 031120582D = 243

pwD

(dpwDdt D)lowastt D

Figure 12 Comparison of dimensionless type curves with differentTPGs (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 005)

tDCD

101

10minus1

100

100

10minus210minus2

102 104

LD = 4

LD = 6

LD = 8

pwD

(dpwDdt D)lowastt D

Figure 13 Comparison of dimensionless type curves with differenthorizontal well lengths (119862

119863

= 100 119878 = 0 ℎ119863

= 200 120572119863

= 005 120582119863

=005)

flow resistance The larger the TPG is the more upturnedthe pressure and pressure derivate curves are The amountof upturning grows with time The 05 value of pressurederivative curve disappears and boundary between the linearflow and late radial flow is not obvious

(5) Effect of 119871119863

Figure 13 shows the effect of the horizontalwell length The basic parameters are ℎ = 9m 119903

119908

= 01m 119871= 80m 120m 160m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam and 119885119908

= 45m It can be seen from the figure thatthe longer the horizontal well is the lower the pressure andpressure derivate curves are and the more obvious the earlyradius flow is Considering the effect of stress sensitivity and

8 Mathematical Problems in Engineering

tDCD

101

10minus1

100

100

10minus210minus2

102 104

hD = 100

hD = 200

hD = 300

pwD

(dpwDdt D)lowastt D

Figure 14 Comparison of dimensionless type curves with differentreservoir thicknesses (119862

119863

= 100 119878 = 0 119871119863

= 5 120572119863

= 005 120582119863

= 005)

nonlinear flow the well test curves of different horizontal welllengths distributed parallel instead of coinciding together

(6) Effect of ℎ119863

Figure 14 shows the effect of reservoirthickness on well test curves The basic parameters are ℎ =45m 9m 135m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2119870V = 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times

10minus6m3Pa 120593 = 025 119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times

10minus6Paminus1 and120582=0001times 106 PamWe set119885119908

equal to 225m45m and 675m respectively which makes the horizontalwell locates in the center in vertical direction It can be seenthat the thicker the reservoir is the longer the early radialflow stage isWhen the reservoir thickness is very small earlyradial flowwill be concealed by the effects of wellbore storage

For traditional horizontal well test with the increase ofreservoir thickness the duration of early vertical radial flowlasts longer and early radial flow stage (the slope of pressureand pressure derivative curves are 05) shifts backward Inthe late radial flow period the pressure derivative curves ofdifferent reservoir thickness coincide together to a horizontalline of 05 that is the speed of pressure drop tends to beuniform

After the introduction of permeability modulus andTPG the duration of early vertical radial flow lasts longerwith increasing reservoir thickness and the effect of stresssensitivity andpseudolinear flowweakens resulting in overalldecrease of pure wellbore storage stage and backwardnessof pressure and pressure derivative curves in linear flow Inthe late radial flow period pressure derivative curve is not astraight line at 05 the pressure and pressure derivative curvesunder different reservoir thicknesses upturn parallely

(7) Effect of 119911119908119863

Figure 15 shows the effect of the horizontalwell position on the well test curve The basic parameters areℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2120583= 1times 0001 Pasdots 119878= 0 5 10119862= 0032times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1

tDCD

101

10minus1

100

100

10minus210minus2

102 104

ZwD = 01

ZwD = 03

ZwD = 05

pwD

(dpwDdt D)lowastt D

Figure 15 Comparison of dimensionless type curves with differenthorizontal well positions (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

102

101

10minus1

100

10minus2

tDCD

10010minus2 102 104 106 108

ReD = 5

ReD = 10

ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 16 Comparison of dimensionless type curves with differentclosed outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

120582 = 0001 times 106 Pam and 119885119908

= 09m 27m 45m It can beseen that when other parameters are kept constant the closerto 05 119911

119908119863

is that is horizontal well is closer to the middleof the reservoir the longer early radical flow lasts Horizontalwell position does not affect thewellbore storage segment thelinear flow and late-linear flow segment

(8) Effect ofOuter Boundary Figures 16 and 17 showboundaryeffect The basic parameters are ℎ = 9m 119903

119908

= 01m 119871 =

100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2 120583 = 1 times

0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025 119888119905

=00022 times 10minus6 Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam 119885119908

= 45m and 119877119890

= 1000m 2000m 3000mFigure 16 shows that the larger the distance of boundary tothe well is the longer it requires to see the reflections and

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

tDCD

pwD

(dpwDdt D

)lowastt D

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0 120572D = 0

120582D = 0 120572D = 01120582D = 01 120572D = 0

120582D = 01 120572D = 01

Figure 7 Comparison of dimensionless type curves with consider-ing TPG and stress sensitivity (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200)

as in a vertical well but is the main section of TPG effect andthus the effect of stress sensitivity is not as obvious as TPGWhen considering the two factors together the degree of theupward for the curve is the supposition of the two factors tobe considered separately

52 Sensitivity Analysis In this section the sensitivity anal-ysis was obtained which includes the TPG permeabilitymodulus skin factor wellbore storage horizontal lengthhorizontal position and boundary effect

For each sensitivity parameter the pressure and pressurederivative curves were presented In each figure the dimen-sionless parameters were given based on the definition inAppendix A

(1) Effect of 119862119863

Figures 8 and 9 show the effect of wellborestorage coefficient on well test curves The basic parametersare ℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V= 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0016 times

10minus6m3Pa 0032 times 10minus6m3Pa 016 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1 120582 = 0002 times

106 Pam and 119885119908

= 45m Large wellbore storage coefficientcauses the pressure curve to shift upward and the larger thewellbore storage coefficient is the higher the ldquohumprdquo of thepressure derivative curve is and the earlier linear flow occursWhen the wellbore storage coefficient is large enough theearly radial flow stage will be concealed by linear flow Whenthe horizontal axis is 119905

119863

it can be seen that the variationof wellbore storage coefficient causes different duration timeof the wellbore storage stage resulting in horizontal shift ofthe well test curve The pressure and pressure derivate curvescoincide together at later time

(2) Effect of 119878 Figure 10 shows the effect of the skin factoron well test curves The basic parameters are ℎ = 9m 119903

119908

=01m 119871 = 100m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 1 5 10 119862 = 0032 times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0049 times 10minus6Paminus1

CD = 50

CD = 100

CD = 500

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 8 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

tD

CD = 50

CD = 100

CD = 500

100

101

10minus1

100

10minus2

102 106104

pwD

(dpwDdt D)lowastt D

Figure 9 Comparison of dimensionless type curves with differentdimensionless wellbore storage coefficients (119878 = 0 119871

119863

= 5 ℎ119863

= 200120572119863

= 01 120582119863

= 01)

120582 = 0002 times 106 Pam and 119885119908

= 45m The larger skin factorrepresents heavier pollution and greater additional pressuredrop it also causes higher ldquohumprdquo on well test curve Largeskin factor may shorten the early radial flow segment or evenmake it disappear

(3) Effect of 120572119863

Figure 11 shows the effect of the permeabilitymodulus on well test curves The basic parameters are ℎ =9m 119903

119908

= 01m 119871 = 200m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0119862 = 0032 times 10minus6m3Pa 120593 =025 119888

119905

= 00022 times 10minus6 Paminus1 120572 = 0 0023 times 10minus6Paminus1 0049 times

10minus6 Paminus1 120582 = 0003 times 106 Pam and 119885119908

= 45m Accordingto the core experiment the largest permeability modulusin absolute value is 0049 times 10minus6 Paminus1 So we set the largest

Mathematical Problems in Engineering 7

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

S = 1

S = 5

S = 10

pwD

(dpwDdt D)lowastt D

Figure 10 Comparison of dimensionless type curves with differentskin factors (119871

119863

= 5 ℎ119863

= 200 120572119863

= 01 120582119863

= 01)

120572D = 0

120572D = 0047

120572D = 01

tDCD

101

10minus1

100

100

10minus210minus2

102 104

pwD

(dpwDdt D)lowastt D

Figure 11 Comparison of dimensionless type curves with differentpermeability moduli (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120582119863

= 015)

permeability modulus equal to the 0049 times 10minus6 Paminus1 Largepermeability modulus makes the reservoir more sensitive tothe stress causing the pressure and pressure derivate curvesto increase seriously Effect of stress sensitivity on the curvemainly concentrates on the stages after the linear flow

(4) Effect of 120582119863

Figure 12 shows the effect of TPG onhorizontal well test curves The basic parameters are ℎ = 9m119903119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6 Paminus1 120582 = 0 times

106 Pam 0006times 106 Pam 0047times 106 Pam and119885119908

=45mAccording to the core experiment the largest permeabilitymodulus in absolute value is 0047 times 106 Pam So we setthe largest TPG equal to the 0047 times 106 Pam The largeTPG represents strong non-Darcy flow resulting in large

tDCD

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0120582D = 031120582D = 243

pwD

(dpwDdt D)lowastt D

Figure 12 Comparison of dimensionless type curves with differentTPGs (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 005)

tDCD

101

10minus1

100

100

10minus210minus2

102 104

LD = 4

LD = 6

LD = 8

pwD

(dpwDdt D)lowastt D

Figure 13 Comparison of dimensionless type curves with differenthorizontal well lengths (119862

119863

= 100 119878 = 0 ℎ119863

= 200 120572119863

= 005 120582119863

=005)

flow resistance The larger the TPG is the more upturnedthe pressure and pressure derivate curves are The amountof upturning grows with time The 05 value of pressurederivative curve disappears and boundary between the linearflow and late radial flow is not obvious

(5) Effect of 119871119863

Figure 13 shows the effect of the horizontalwell length The basic parameters are ℎ = 9m 119903

119908

= 01m 119871= 80m 120m 160m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam and 119885119908

= 45m It can be seen from the figure thatthe longer the horizontal well is the lower the pressure andpressure derivate curves are and the more obvious the earlyradius flow is Considering the effect of stress sensitivity and

8 Mathematical Problems in Engineering

tDCD

101

10minus1

100

100

10minus210minus2

102 104

hD = 100

hD = 200

hD = 300

pwD

(dpwDdt D)lowastt D

Figure 14 Comparison of dimensionless type curves with differentreservoir thicknesses (119862

119863

= 100 119878 = 0 119871119863

= 5 120572119863

= 005 120582119863

= 005)

nonlinear flow the well test curves of different horizontal welllengths distributed parallel instead of coinciding together

(6) Effect of ℎ119863

Figure 14 shows the effect of reservoirthickness on well test curves The basic parameters are ℎ =45m 9m 135m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2119870V = 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times

10minus6m3Pa 120593 = 025 119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times

10minus6Paminus1 and120582=0001times 106 PamWe set119885119908

equal to 225m45m and 675m respectively which makes the horizontalwell locates in the center in vertical direction It can be seenthat the thicker the reservoir is the longer the early radialflow stage isWhen the reservoir thickness is very small earlyradial flowwill be concealed by the effects of wellbore storage

For traditional horizontal well test with the increase ofreservoir thickness the duration of early vertical radial flowlasts longer and early radial flow stage (the slope of pressureand pressure derivative curves are 05) shifts backward Inthe late radial flow period the pressure derivative curves ofdifferent reservoir thickness coincide together to a horizontalline of 05 that is the speed of pressure drop tends to beuniform

After the introduction of permeability modulus andTPG the duration of early vertical radial flow lasts longerwith increasing reservoir thickness and the effect of stresssensitivity andpseudolinear flowweakens resulting in overalldecrease of pure wellbore storage stage and backwardnessof pressure and pressure derivative curves in linear flow Inthe late radial flow period pressure derivative curve is not astraight line at 05 the pressure and pressure derivative curvesunder different reservoir thicknesses upturn parallely

(7) Effect of 119911119908119863

Figure 15 shows the effect of the horizontalwell position on the well test curve The basic parameters areℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2120583= 1times 0001 Pasdots 119878= 0 5 10119862= 0032times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1

tDCD

101

10minus1

100

100

10minus210minus2

102 104

ZwD = 01

ZwD = 03

ZwD = 05

pwD

(dpwDdt D)lowastt D

Figure 15 Comparison of dimensionless type curves with differenthorizontal well positions (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

102

101

10minus1

100

10minus2

tDCD

10010minus2 102 104 106 108

ReD = 5

ReD = 10

ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 16 Comparison of dimensionless type curves with differentclosed outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

120582 = 0001 times 106 Pam and 119885119908

= 09m 27m 45m It can beseen that when other parameters are kept constant the closerto 05 119911

119908119863

is that is horizontal well is closer to the middleof the reservoir the longer early radical flow lasts Horizontalwell position does not affect thewellbore storage segment thelinear flow and late-linear flow segment

(8) Effect ofOuter Boundary Figures 16 and 17 showboundaryeffect The basic parameters are ℎ = 9m 119903

119908

= 01m 119871 =

100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2 120583 = 1 times

0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025 119888119905

=00022 times 10minus6 Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam 119885119908

= 45m and 119877119890

= 1000m 2000m 3000mFigure 16 shows that the larger the distance of boundary tothe well is the longer it requires to see the reflections and

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

tDCD

101

10minus1

100

100

10minus210minus2

102 106104

S = 1

S = 5

S = 10

pwD

(dpwDdt D)lowastt D

Figure 10 Comparison of dimensionless type curves with differentskin factors (119871

119863

= 5 ℎ119863

= 200 120572119863

= 01 120582119863

= 01)

120572D = 0

120572D = 0047

120572D = 01

tDCD

101

10minus1

100

100

10minus210minus2

102 104

pwD

(dpwDdt D)lowastt D

Figure 11 Comparison of dimensionless type curves with differentpermeability moduli (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120582119863

= 015)

permeability modulus equal to the 0049 times 10minus6 Paminus1 Largepermeability modulus makes the reservoir more sensitive tothe stress causing the pressure and pressure derivate curvesto increase seriously Effect of stress sensitivity on the curvemainly concentrates on the stages after the linear flow

(4) Effect of 120582119863

Figure 12 shows the effect of TPG onhorizontal well test curves The basic parameters are ℎ = 9m119903119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6 Paminus1 120582 = 0 times

106 Pam 0006times 106 Pam 0047times 106 Pam and119885119908

=45mAccording to the core experiment the largest permeabilitymodulus in absolute value is 0047 times 106 Pam So we setthe largest TPG equal to the 0047 times 106 Pam The largeTPG represents strong non-Darcy flow resulting in large

tDCD

101

10minus1

100

100

10minus210minus2

102 104

120582D = 0120582D = 031120582D = 243

pwD

(dpwDdt D)lowastt D

Figure 12 Comparison of dimensionless type curves with differentTPGs (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

= 005)

tDCD

101

10minus1

100

100

10minus210minus2

102 104

LD = 4

LD = 6

LD = 8

pwD

(dpwDdt D)lowastt D

Figure 13 Comparison of dimensionless type curves with differenthorizontal well lengths (119862

119863

= 100 119878 = 0 ℎ119863

= 200 120572119863

= 005 120582119863

=005)

flow resistance The larger the TPG is the more upturnedthe pressure and pressure derivate curves are The amountof upturning grows with time The 05 value of pressurederivative curve disappears and boundary between the linearflow and late radial flow is not obvious

(5) Effect of 119871119863

Figure 13 shows the effect of the horizontalwell length The basic parameters are ℎ = 9m 119903

119908

= 01m 119871= 80m 120m 160m 119870

ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam and 119885119908

= 45m It can be seen from the figure thatthe longer the horizontal well is the lower the pressure andpressure derivate curves are and the more obvious the earlyradius flow is Considering the effect of stress sensitivity and

8 Mathematical Problems in Engineering

tDCD

101

10minus1

100

100

10minus210minus2

102 104

hD = 100

hD = 200

hD = 300

pwD

(dpwDdt D)lowastt D

Figure 14 Comparison of dimensionless type curves with differentreservoir thicknesses (119862

119863

= 100 119878 = 0 119871119863

= 5 120572119863

= 005 120582119863

= 005)

nonlinear flow the well test curves of different horizontal welllengths distributed parallel instead of coinciding together

(6) Effect of ℎ119863

Figure 14 shows the effect of reservoirthickness on well test curves The basic parameters are ℎ =45m 9m 135m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2119870V = 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times

10minus6m3Pa 120593 = 025 119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times

10minus6Paminus1 and120582=0001times 106 PamWe set119885119908

equal to 225m45m and 675m respectively which makes the horizontalwell locates in the center in vertical direction It can be seenthat the thicker the reservoir is the longer the early radialflow stage isWhen the reservoir thickness is very small earlyradial flowwill be concealed by the effects of wellbore storage

For traditional horizontal well test with the increase ofreservoir thickness the duration of early vertical radial flowlasts longer and early radial flow stage (the slope of pressureand pressure derivative curves are 05) shifts backward Inthe late radial flow period the pressure derivative curves ofdifferent reservoir thickness coincide together to a horizontalline of 05 that is the speed of pressure drop tends to beuniform

After the introduction of permeability modulus andTPG the duration of early vertical radial flow lasts longerwith increasing reservoir thickness and the effect of stresssensitivity andpseudolinear flowweakens resulting in overalldecrease of pure wellbore storage stage and backwardnessof pressure and pressure derivative curves in linear flow Inthe late radial flow period pressure derivative curve is not astraight line at 05 the pressure and pressure derivative curvesunder different reservoir thicknesses upturn parallely

(7) Effect of 119911119908119863

Figure 15 shows the effect of the horizontalwell position on the well test curve The basic parameters areℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2120583= 1times 0001 Pasdots 119878= 0 5 10119862= 0032times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1

tDCD

101

10minus1

100

100

10minus210minus2

102 104

ZwD = 01

ZwD = 03

ZwD = 05

pwD

(dpwDdt D)lowastt D

Figure 15 Comparison of dimensionless type curves with differenthorizontal well positions (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

102

101

10minus1

100

10minus2

tDCD

10010minus2 102 104 106 108

ReD = 5

ReD = 10

ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 16 Comparison of dimensionless type curves with differentclosed outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

120582 = 0001 times 106 Pam and 119885119908

= 09m 27m 45m It can beseen that when other parameters are kept constant the closerto 05 119911

119908119863

is that is horizontal well is closer to the middleof the reservoir the longer early radical flow lasts Horizontalwell position does not affect thewellbore storage segment thelinear flow and late-linear flow segment

(8) Effect ofOuter Boundary Figures 16 and 17 showboundaryeffect The basic parameters are ℎ = 9m 119903

119908

= 01m 119871 =

100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2 120583 = 1 times

0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025 119888119905

=00022 times 10minus6 Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam 119885119908

= 45m and 119877119890

= 1000m 2000m 3000mFigure 16 shows that the larger the distance of boundary tothe well is the longer it requires to see the reflections and

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

tDCD

101

10minus1

100

100

10minus210minus2

102 104

hD = 100

hD = 200

hD = 300

pwD

(dpwDdt D)lowastt D

Figure 14 Comparison of dimensionless type curves with differentreservoir thicknesses (119862

119863

= 100 119878 = 0 119871119863

= 5 120572119863

= 005 120582119863

= 005)

nonlinear flow the well test curves of different horizontal welllengths distributed parallel instead of coinciding together

(6) Effect of ℎ119863

Figure 14 shows the effect of reservoirthickness on well test curves The basic parameters are ℎ =45m 9m 135m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2119870V = 18 times 10minus15m2 120583 = 1 times 0001 Pasdots 119878 = 0 119862 = 0032 times

10minus6m3Pa 120593 = 025 119888119905

= 00022 times 10minus6Paminus1 120572 = 00245 times

10minus6Paminus1 and120582=0001times 106 PamWe set119885119908

equal to 225m45m and 675m respectively which makes the horizontalwell locates in the center in vertical direction It can be seenthat the thicker the reservoir is the longer the early radialflow stage isWhen the reservoir thickness is very small earlyradial flowwill be concealed by the effects of wellbore storage

For traditional horizontal well test with the increase ofreservoir thickness the duration of early vertical radial flowlasts longer and early radial flow stage (the slope of pressureand pressure derivative curves are 05) shifts backward Inthe late radial flow period the pressure derivative curves ofdifferent reservoir thickness coincide together to a horizontalline of 05 that is the speed of pressure drop tends to beuniform

After the introduction of permeability modulus andTPG the duration of early vertical radial flow lasts longerwith increasing reservoir thickness and the effect of stresssensitivity andpseudolinear flowweakens resulting in overalldecrease of pure wellbore storage stage and backwardnessof pressure and pressure derivative curves in linear flow Inthe late radial flow period pressure derivative curve is not astraight line at 05 the pressure and pressure derivative curvesunder different reservoir thicknesses upturn parallely

(7) Effect of 119911119908119863

Figure 15 shows the effect of the horizontalwell position on the well test curve The basic parameters areℎ = 9m 119903

119908

= 01m 119871 = 100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times

10minus15m2120583= 1times 0001 Pasdots 119878= 0 5 10119862= 0032times 10minus6m3Pa120593 = 025 119888

119905

= 00022 times 10minus6Paminus1 120572 = 00245 times 10minus6Paminus1

tDCD

101

10minus1

100

100

10minus210minus2

102 104

ZwD = 01

ZwD = 03

ZwD = 05

pwD

(dpwDdt D)lowastt D

Figure 15 Comparison of dimensionless type curves with differenthorizontal well positions (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

102

101

10minus1

100

10minus2

tDCD

10010minus2 102 104 106 108

ReD = 5

ReD = 10

ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 16 Comparison of dimensionless type curves with differentclosed outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

= 200 120572119863

=005 120582

119863

= 005)

120582 = 0001 times 106 Pam and 119885119908

= 09m 27m 45m It can beseen that when other parameters are kept constant the closerto 05 119911

119908119863

is that is horizontal well is closer to the middleof the reservoir the longer early radical flow lasts Horizontalwell position does not affect thewellbore storage segment thelinear flow and late-linear flow segment

(8) Effect ofOuter Boundary Figures 16 and 17 showboundaryeffect The basic parameters are ℎ = 9m 119903

119908

= 01m 119871 =

100m 119870ℎ

= 9 times 10minus15m2 119870V = 18 times 10minus15m2 120583 = 1 times

0001 Pasdots 119878 = 0 119862 = 0032 times 10minus6m3Pa 120593 = 025 119888119905

=00022 times 10minus6 Paminus1 120572 = 00245 times 10minus6Paminus1 120582 = 0001 times

106 Pam 119885119908

= 45m and 119877119890

= 1000m 2000m 3000mFigure 16 shows that the larger the distance of boundary tothe well is the longer it requires to see the reflections and

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

tDCD

101

10minus1

100

100

10minus210minus2

102 104 106 108

ReD = 5ReD = 10ReD = 15

pwD

(dpwDdt D)lowastt D

Figure 17 Comparison of dimensionless type curves with differentconstant-pressure outer boundaries (119862

119863

= 100 119878 = 0 119871119863

= 5 ℎ119863

=200 120572

119863

= 005 120582119863

= 005)

the stages of pressure reflection in the well test curves moveparallel to the right When the distance of the boundary tothe well is small the upward characteristic caused by stresssensitivity and TPG is not obvious concealed by boundariesreflect stage Only when the distance is large enough can theupward characteristic exist obviously Figure 17 reflects pres-sure transient in constant-pressure outer boundary Whenthe pressure reaches the constant-pressure outer boundarypressure curve tends to be horizontal pressure derivativecurve drops down Meanwhile it requires longer time tosee the boundary reflections because of the effect of stresssensitivity and TPG

6 Field Application

Pressure test was performed on a horizontal well for theoffshore oilfield where the experimental cores come fromThe oil production rate is 806m3d The length of horizontalwell is 498mThe effective thickness is 108mThe horizontalwell is 81m far from the bottom boundary The porosity is246 The viscosity of crude oil is 1332mPsdots The volumecoefficient is 1048 The compressibility coefficient is 2215 times

10minus3MPaminus1Using the model in this paper the matching is carried out

to obtain the reservoir parameters as shown in Figure 18Theinterpretation results are as follows horizontal permeabilityis 145mD vertical permeability and horizontal ratio is 0093wellbore storage is 005m3MPa skin factor is 0011 TPG is0005MPam and permeability modulus is 0025MPaminus1 Ascan be seen from Figure 2 and Table 3 when the formationpermeability is 145mD the TPG is close to 0007MPamand the permeabilitymodulus is nearly 0024MPaminus1Thewelltesting interpretation results are very close to the experimen-tal data

0001

001

01

1

10

100

00001 0001 001 01 1 10 100 1000

Fitting pressure curveFitting pressure derivative curveField pressure dataField pressure derivative data

(Δp

Δp998400 ) (

MPa

)

Δt (hours)

Figure 18 Matching curves of well test interpretation

7 Conclusion

(1) With the cores from the a real offshore oilfield in Chinathe TPG and stress sensitivity were tested We verified theexistence of TPG in irreducible water saturation conditionand the permeability stress sensitivity by changing the fluidpressure directly It shows that TPG and permeability stresssensitivity are relative to the absolute permeability of thecoresThe TPG and permeabilitymodulus both increase withthe decrease of permeability

(2) Considering the TPG and permeability modulus thehorizontal well pressure transient mathematical model inan anisotropic low permeability reservoir was establishedThis model is a nonlinear partial differential equation FEMis chosen to solve the problem and the detailed solutionprocedures are discussed Through comparison with theanalytical solution FEM is verified

(3) The type curves and derivative type curves of thehorizontal well depend on TPG permeability modulus welllocation well length wellbore storage skin factor and outerboundary The pressure and pressure derivative curves whenconsidering the stress sensitivity and TPG are different fromthose based onDarcyrsquos modelThe type curves and derivativetype curves at early times are not so sensitive to the stresssensitivity and TPG but in the late stage they cause thepressure and pressure derivative curves to shift upwardsresulting in the disappearance of pressure derivative horizonwith the value of 05 in radial flow stage In the formationthere is additional pressure loss when considering TPG andstress sensitivity

Appendices

A Dimensionless Terms

Consider

119909119863

=119909

119871 119910

119863

=119910

119871 119911

119863

=119911

ℎ

119911119908119863

=119911119908

ℎ 119903

119908119863

=119903119908

ℎ

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

119905119863

=119870ℎ119894

119905

120601120583119862119905

1199032119908

119901119863

=119870ℎ119894

ℎ (119901119894

minus 119901)

119876120583119861

119871119863

=119871

ℎradic

119870V

119870ℎ

ℎ119863

=ℎ

119903119908

radic119870ℎ

119870V

119862119863

=119862

2120587ℎ120601119888119905

1199032119908

120572119863

=119876120583119861

2120587119870ℎ119894

ℎ120572

120582119863

=2120587119870ℎ119894

ℎ119871

119876120583119861120582 120575

119870119863119897

= 1 minus120582119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

120575119870119863119911

= 1 minus120582119863

119871119863

1003816100381610038161003816nabla119901119863

1003816100381610038161003816

(A1)

where 119897 = 119909 119910 and 120582119863119911

= radic119870ℎ

119870V 120582119863ℎ

B Solution for Horizontal Well PressureAnalysis Model

B1 Finite Element Method According to Galerkin method[30] we assume the shape function or basic function

119873 = (1198731

1198732

119873119899

) (B1)

The displacement function is

119901119863

=

119899

sum119895=1

119873119895

119901119863119895

(B2)

where 119899 is the number of nodes and 119901119863119895

is the dimensionlesspressure at the node 119895

By integrating over the volume of the element Ω119890

wehave

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881

(B3)

With the Green function we can get

∭Ω119890

119873119895

[120597

120597119909119863

(119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

)

+120597

120597119910119863

(119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

)

+120597

120597119911119863

(119890minus120572119863119901119863120575

119870119863119911

1198712

119863

120597119901119863

120597119911119863

)]119889119881

= minus∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∬Γ119890

119873119895

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

(B4)

where

∬Γ119890

119890minus120572119863119901119863120575

119870119863119899

120597119901119863

120597119899119863

119889119860

= ∬Γ119890

119899119909

119890minus120572119863119901119863120575

119870119863119909

120597119901119863

120597119909119863

119889119860

+ ∬Γ119890

119899119910

119890minus120572119863119901119863120575

119870119863119910

120597119901119863

120597119910119863

119889119860

+ ∬Γ119890

119899119911

1198712

119863

119890minus120572119863119901119863120575

119870119863119911

120597119901119863

120597119911119863

119889119860

(B5)

For the inner element we can get

∭Ω119890

119890minus120572119863119901119863 (120575

119870119863119909

120597119873119895

120597119909119863

120597119901119863

120597119909119863

+ 120575119870119863119910

120597119873119895

120597119910119863

120597119901119863

120597119910119863

+ 120575119870119863119911

1198712

119863

120597119873119895

120597119911119863

120597119901119863

120597119911119863

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

119873119895

120597119901119863

120597119905119889119881 = 0

(B6)

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

The matrix form is

∭Ω119890

(

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

sdot

(

1205751198701198631199091

1205751198701198631199092

d

120575119870119863119909119899

)

sdot

(((((((

(

1205971198731

120597119909119863

1205971198732

120597119909119863

120597119873119899

120597119909119863

)))))))

)

(1205971198731

120597119909119863

1205971198732

120597119909119863

sdot sdot sdot120597119873119899

120597119909119863

)

+ (

1205751198701198631199101

1205751198701198631199102

d

120575119870119863119910119899

)

sdot

(((((((

(

1205971198731

120597119910119863

1205971198732

120597119910119863

120597119873119899

120597119910119863

)))))))

)

(1205971198731

120597119910119863

1205971198732

120597119910119863

sdot sdot sdot120597119873119899

120597119910119863

)

+1198712

119863

(

1205751198701198631199111

1205751198701198631199112

d

120575119870119863119911119899

)

sdot

(((((((

(

1205971198731

120597119911119863

1205971198732

120597119911119863

120597119873119899

120597119911119863

)))))))

)

(1205971198731

120597119911119863

1205971198732

120597119911119863

sdot sdot sdot120597119873119899

120597119911119863

)

sdot (

1199011198631

1199011198632

119901119863119899

)119889119881

+ ∭Ω119890

(ℎ119863

119871119863

)2

(

1198731

1198732

119873119899

)(1198731

1198732

sdot sdot sdot 119873119899

)

sdot

((((((((

(

119901119898

1198631

minus 119901119898minus11198631

Δ119905119863

1199011198981198632

minus 119901119898minus11198632

Δ119905119863

119901119898119863119899

minus 119901119898minus1119863119899

Δ119905119863

))))))))

)

119889119881 = 0

(B7)

Define the following parameters

B119897

= (1205971198731

120597119897119863

1205971198732

120597119897119863

sdot sdot sdot120597119873119899

120597119897119863

)

N = (

1198731

1198732

119873119899

)

P = (

1199011198631

1199011198632

119901119863119899

)

C = (

119890minus1205721198631199011198631

119890minus1205721198631199011198632

d

119890minus120572119863119901119863119899

)

D119897

= (

1205751198701198631198971

1205751198701198631198972

d

120575119870119863119897119899

)

(B8)

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

The equation will be

∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881P119898

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1

(B9)

A simple form will be

K119890

P119898 = F119890

(B10)

where

K119890

= ∭Ω119890

[C (D119909

B119879119909

B119909

+ D119910

B119879119910

B119910

+ 1198712

119863

D119911

B119879119911

B119911

)

+ (ℎ119863

119871119863

)2

NN119879

Δ119905119863

]119889119881

(B11)

F119890

= ∭Ω119890

(ℎ119863

119871119863

)2

NN119879

Δ119905119863

119889119881P119898minus1 (B12)

B2 Boundary Condition The uniform flux inner boundaryis used to describe the inflow performance of the horizontalwell The wellbore pressure can be considered as the pressureat the position 07 L [22]

For the element including well nodes the F119890

in (B12)needs to be modified to add a sourcesink

f119890

= ∭Ω119890

119873119895

119902119889119881 (B13)

Use the Delta function to describe the sourcesink

119902 =119876

119896120575 (119909 minus 119909

0

) 120575 (119910 minus 1199100

) 120575 (119911 minus 1199110

) (B14)

where 119896 is the element number containing the horizontal well(1199090

1199100

1199110

) is the horizontal well position coordinateThe dimensionless form is

f119890

=2120587

119896∭Ω119890

119873119895

(1199091198630

1199101198630

1199111198630

) 119889119881 (B15)

By solving the equation system the well bore pressure canbe obtained in real space without considering the wellborestorage and skin factor The wellbore storage and skin factorwere considered by using discrete numerical Laplace trans-form method [31] in which the real space well bore pressurecan be converted into Laplace space Then the response of awell withwellbore storage and skin can be obtained using [29]

119901119908119863

=119904119901119863

+ 119878119871119863

119904 (1 + 119862119863

119904 (119904119901119863

+ 119878119871119863

)) (B16)

where 119901119863

is the dimensionless pressure without wellborestorage and skin effects (in Laplace space) With the Stehfest-Laplace numerical inversion method [32] the 119901

119908119863

can beachieved in real space

Nomenclature

nabla119901 Pressure gradient Pam119870 Permeability m2] Velocity ms120583 Fluid viscosity Pasdots119903119908

Wellbore radius m119877119890

Reservoir outer boundary radius mℎ Reservoir thickness m119871 Horizontal well half-length m119911119908

Vertical distance from the formation lowerboundary to wellbore m

119901119894

Initial pressure Pa119876 Production rate m3s119862 Wellbore storage m3Pa119878 Skin factor (dimensionless)119904 Laplace-transform variable with respect to

119905119863

(dimensionless)119905 Time s120572 permeability modulus 1Pa120582 Threshold pressure gradient Pam119909 119910 119911 Cartesian coordinates120588 Density kgm3120593 Porosity 119888119905

Total compressibility Paminus1119901 Pressure PaΩ Volume domainΓ Area domain119881 Volume m3119860 Area m2

Subscripts and Superscripts

119863 Dimensionless119894 Initialℎ Horizontal directionV Vertical direction Laplace

119890 Element119898 Time step

Conflict of Interests

Jianchun Xu Ruizhong Jiang and TengWenchao declare thatthere is no conflict of interests regarding the publication ofthis paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation ofChina (Grants nos are 51174223 and 51374227)the Fundamental Research Funds for the Central Universityunder 14CX06087A the Graduate Innovation Fund of ChinaUniversity of Petroleum (East China) under CX-1210 andYCX2014017 and the National 12th Five-Year Plan under2011ZX05013-006 and 2011ZX05051

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

References

[1] B Zeng L Cheng and C Li ldquoLow velocity non-linear flow inultra-low permeability reservoirrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 1ndash6 2011

[2] F Hao L S Cheng O Hassan J Hou C Z Liu and J DFeng ldquoThreshold pressure gradient in ultra-low permeabilityreservoirsrdquo Petroleum Science and Technology vol 26 no 9 pp1024ndash1035 2008

[3] XWWangM Z Yang P Y Sun andW X Liu ldquoExperimentaland theoretical investigation of nonlinear flow in low perme-ability reservoirrdquo Procedia Environmental Sciences vol 11 pp1392ndash1399 2011

[4] Q Lei W Xiong and J Yuan ldquoBehavior of flow through low-permeability reservoirsrdquo in Proceedings of the EuropecEAGEConference and Exhibition Society of Petroleum Engineers2008

[5] X Wei L Qun G Shusheng H Zhiming and X Hui ldquoPseudothreshold pressure gradient to flow for low permeability reser-voirsrdquo Petroleum Exploration and Development vol 36 no 2pp 232ndash236 2009

[6] A Prada and F Civan ldquoModification of Darcyrsquos law for thethreshold pressure gradientrdquo Journal of Petroleum Science andEngineering vol 22 no 4 pp 237ndash240 1999

[7] L Gavin ldquoPre-Darcy flow a missing piece of the improved oilrecovery puzzlerdquo in Proceedings of the SPEDOE Symposiumon Improved Oil Recovery SPE-89433-MS Society of PetroleumEngineers Tulsa Oklahoma April 2004

[8] P Basak ldquoNon-Darcy flow and its implications to seepageproblemsrdquo Journal of the Irrigation and Drainage Division vol103 no 4 pp 459ndash473 1977

[9] W ZhuH Song XHuang X Liu DHe andQ Ran ldquoPressurecharacteristics and effective deployment in a water-bearingtight gas reservoir with low-velocity non-darcy flowrdquo Energy ampFuels vol 25 no 3 pp 1111ndash1117 2011

[10] Y Hu X Li Y Wan et al ldquoPhysical simulation on gaspercolation in tight sandstonesrdquo Petroleum Exploration andDevelopment vol 40 no 5 pp 621ndash626 2013

[11] Q Xu X Liu Z Yang and J Wang ldquoThe model and algorithmof a new numerical simulation software for low permeabilityreservoirsrdquo Journal of Petroleum Science and Engineering vol78 no 2 pp 239ndash242 2011

[12] D-Y Yin andH Pu ldquoNumerical simulation study on surfactantflooding for low permeability oilfield in the condition ofthreshold pressurerdquo Journal of Hydrodynamics vol 20 no 4 pp492ndash498 2008

[13] H Liu H Zhang and Z Wang ldquoDoubling model for lowerpermeability formation with hydraulic fracturerdquo PetroleumScience and Technology vol 29 no 9 pp 898ndash905 2011

[14] J-J Guo S Zhang L-H Zhang H Qing and Q-G LiuldquoWell testing analysis for horizontal well with consideration ofthreshold pressure gradient in tight gas reservoirsrdquo Journal ofHydrodynamics vol 24 no 4 pp 561ndash568 2012

[15] I Palmer and JMansoori ldquoHowpermeability depends on stressand pore pressure in coalbeds a new modelrdquo SPE ReservoirEvaluation amp Engineering vol 1 no 6 1998

[16] H Ruistuen L W Teufel and D Rhett ldquoInfluence of reser-voir stress path on deformation and permeability of weaklycemented sandstone reservoirsrdquo SPE Reservoir Evaluation andEngineering vol 2 no 3 pp 266ndash272 1999

[17] J C Lorenz ldquoStress-sensitive reservoirsrdquo Journal of PetroleumTechnology vol 51 no 1 pp 61ndash63 1999

[18] S Chen H Li Q Zhang and D Yang ldquoA new technique forproduction prediction in stress-sensitive reservoirsrdquo Journal ofCanadian Petroleum Technology vol 47 no 3 pp 49ndash54 2008

[19] S A Sonnenberg and A Pramudito ldquoPetroleum geology of thegiant Elm Coulee fieldWilliston BasinrdquoAAPG Bulletin vol 93no 9 pp 1127ndash1153 2009

[20] H-XWangGWang Z-X J Chen andRCKWong ldquoDefor-mational characteristics of rock in low permeable reservoir andtheir effect on permeabilityrdquo Journal of Petroleum Science andEngineering vol 75 no 1-2 pp 240ndash243 2010

[21] F Ma S He H Zhu Q Xie and C Jiao ldquoThe effect of stressand pore pressure on formation permeability of ultra-low-permeability reservoirrdquo Petroleum Science and Technology vol30 no 12 pp 1221ndash1231 2012

[22] A J Rosa and R D S Carvalho ldquoA mathematical model forpressure evaluation in an infinite-conductivity horizontal wellrdquoSPE Formation Evaluation vol 4 no 4 pp 559ndash15967 1989

[23] S Knut S C Lien and B T Haug ldquoTroll horizontal well testsdemonstrate large production potential from thin oil zonesrdquoSPE Reservoir Engineering vol 9 no 2 pp 133ndash139 1994

[24] M C Ng and R Aguilera ldquoWell test analysis of horizontal wellsin bounded naturally fractured reservoirsrdquo in Proceedings of theAnnual Technical Meeting Petroleum Society of Canada 1994

[25] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 1 Analytical considerationsrdquo SPE For-mation Evaluation vol 6 no 3 pp 359ndash368 1991

[26] H Zhan L V Wang and E Park ldquoOn the horizontal-wellpumping tests in anisotropic confined aquifersrdquo Journal ofHydrology vol 252 no 1ndash4 pp 37ndash50 2001

[27] E Park andH Zhan ldquoHydraulics of a finite-diameter horizontalwell with wellbore storage and skin effectrdquo Advances in WaterResources vol 25 no 4 pp 389ndash400 2002

[28] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2003

[29] X P Li L H Zhang and Q G LiuWell Test Analysis MethodPetroleum Industry Press 2009

[30] O C Zienkiewicz The Finite Element Method Set McGraw-Hill London UK 1977

[31] J Yao and Z S Wang Theory and Method for Well TestInterpretation in Fractured-Vuggy Carbonate Reservoirs ChinaUniversity of Petroleum Press Dongying China 2008

[32] D K Tong and Q L Chen ldquoA note on the Stehfest Laplacenumerical inversion methodrdquo Acta Petrolei Sinica vol 22 no6 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of