Rate Decline Behavior of Selectively Completed Horizontal Wells … · 2019. 7. 30. · 51 %0˝eD 71 %0˝eD 70 %0D +2 ∞ 3 = *50 %nD + 51 % ˝eD 71 % ˝eD 70 %n ()D . ⋅cos - D

Research ArticleRate Decline Behavior of Selectively Completed HorizontalWells in Naturally Fractured Oil Reservoirs

Qi-guo Liu 1 You-jie Xu 1 Long-xin Li2 and An-zhao Ji3

1State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum University Chengdu 610500 China2Research Institute of Exploration Development PetroChina Southwest Oil amp Gasfield Company Chengdu 610041 China3School of Energy Engineer Longdong University Qingyang 745000 China

Correspondence should be addressed to Qi-guo Liu liuqiguoswpueducn and You-jie Xu xuyoujie920309163com

Received 10 September 2018 Revised 20 November 2018 Accepted 9 December 2018 Published 18 April 2019

Guest Editor Gerhard-WilhelmWeber

Copyright copy 2019 Qi-guo Liu 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

Selectively completed horizontal wells (SCHWs) can significantly reduce cost of completingwells and delaywater breakthrough andprevent wellbore collapse in weak formations Thus SCHWs have been widely used in petroleum development industry SCHWscan shorten the effective length of horizontal wells and thus have a vital effect on production It is significant for SCHWs to studytheir ratedecline and flux distribution in naturally fractured reservoirs In this paper by employingmotion equation state equationandmass conservation equation three-dimension seepage differential equation is established and corresponding analytical solutionis obtained by Laplace transform and finite cosine Fourier transform According to the relationship of constant production andwellbore pressure in Laplace domain dimensionless rate solution is gotten under constant wellbore pressure in Laplace domainDimensionless pressure and pressure derivate curves and rate decline curves are drawn in log-log plot and seven flow regimes areidentified by Stehfest numerical inversionWe compared the simplified results of this paper with the results calculated by Saphir forhorizontal wells in naturally fractured reservoirsThe results showed excellent agreement Some parameters such as outer boundaryradius storativity ratio cross-flow coefficient number and length of open segments can obviously affect the rate integral and rateintegral derivative log-log curves of the SCHWs The proposed model in this paper can help better understand the flow regimecharacteristics of the SCHWs and provide more accurate rate decline analysis of the SCHWs data to evaluate formation

1 Introduction

In the early age since horizontal wells can increase produc-tivity they became a popular method to develop oil and gasCompared with vertical wells horizontal wells can controlsevere water or gas coning problems increase the connectingarea with the reservoir and reduce wellbore turbulence [1]

Although horizontal wells show a number of advantagesincreasing wellbore length may lead to production imbalancealong the wellbore which can lead to water coning anddecreased production Certainly the uneven rate distributioncan lead to bottom-water break through [2] However hori-zontal open holes may be completed by employing prepackedscreens because of their low cost At the same time prepackedscreens can effectively minimize sand production With theproduction of oil or gas the sand will accumulate around thehorizontal wellbore which makes fluids of formation flow

into wellbore by a section of wellbores opened and causesproductivity decline In order to reduce cost of completinghorizontal well delay water breakthrough and prevent well-bore collapse in weak formations SCHWs were used widelyand actual production also proves the effectiveness of themethod An important feature of SCHWswas that only somesegments of the wellbore are open to the formation It is alsoobserved that even if the entire length of the horizontal wellis open or perforated only some segments produce fluid [3]

To analyze the wellbore pressure and rate response ofSCHWs some engineers tend to use an effective horizontalwell length to replace the open length of the horizontalwell This treatment assumes that the open length of thehorizontal well is continuous instead of interval distributionAn analytical model was developed in real domain to predictthe inflow performance of SCHWs and selectively completedvertical wells (SCVWs) [4 5] Their model considers the

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 7281090 13 pageshttpsdoiorg10115520197281090

2 Mathematical Problems in Engineering

O x

y

z

h

Re

Lh

LNciLci

z=h

z=0

zw

Top boundary

Bottom boundary

Completed segmentOpen segment

Figure 1 Schematic of the SCHWs in naturally fractured oil reservoirs

distribution of the open intervals Kamal et al [6] presentedan analytical model of SCHWs by employing instantaneous-point-source solution and the superposition principle Theyused this model to analyze actual cases where pressuretransient is available Yildiz [7ndash9] and Seyide [10] presenteda model of SCHWs and SCVWs and derived asymptoticapproximations of the model in Laplace space Pressure andpressure-derivate log-log curves were plotted flow charac-teristics were discussed and each segment rate of SCHWswas analyzed A new semianalytical model for predictingthe performance of horizontal wells which were completedby inflow control devices in bottom-water reservoirs waspresented The coupled solution is developed for predictingthe performance of horizontal wells in a box-shaped reservoirwith bottom-water drive [11]

In order to analyze the rate decline curve of SCHWsa mathematical model considering difference between hor-izontal and vertical permeability of SCHWs is establishedin naturally fractured reservoirs Based on point source andthe superposition principle pressure analytical solution ofthe SCHWs under the condition of constant production inimpermeable top and bottom boundary and lateral imperme-able boundary by Laplace transform and finite cosine Fouriertransform Log-log curves of pressure and pressure-derivateand rate decline are drawn in naturally fractured reservoirs byemploying Stehfest numerical inversion Seven flow regimesaccording to the characteristic of pressure-derivate curve areidentified and every flow regime characteristic is described indetail This paper discusses that relevant parameters (stora-tivity ratio flow coefficient number and length completedhorizontal sections etc) have effect on pressure and ratedecline curves Corresponding solutions can be useful incompletion design and rate decline in field practice

2 Physical Model of SCHWs and Assumption

Horizontal wells are located in naturally fractured reservoirswith impermeable top and bottom boundary and lateralimpermeable boundary and parallel to the upper and lower

1Fig22


Matrix system Fr

actu

re sy

stem

Horizontal wellbore

Figure 2 Schematic of fluid flowing path for SCHW in naturallyfractured oil reservoirs

impermeable boundary Horizontal well consists of119873Nc opensegments and 119873c completed segments (shown in Figure 1)The naturally fractured reservoir is structured by matrixsystem and natural fracture system The dual-porosity mediaare assumed as the Warren-Root model and pseudo-steadycross flow exits between the matrix and fractures Fluidsflow into natural fractures from matrix firstly and flow intohorizontal wellbore from natural fractures secondly by opensegments (shown in Figure 2) In order tomakemathematicalmodel more reliable and accurate some assumptions arelisted as follows

(a) The fluid flow in the reservoir obeys Darcyrsquos law andlaw of isothermal percolation

(b) Flow is single phase and the fluid has constant andsmall compressibility and constant viscosity

(c) Formation permeability is anisotropic with threemajor directional permeability 119896119909 119896119910 119896119911

(d) Formation initial pressure is 119901e and horizontal pro-duced at a constant surface flow rate 119902sc

Mathematical Problems in Engineering 3

(e) Horizontal well consists of119873Nc open segments and119873ccompleted segments and fluids flow into wellbore byonly open segments

(f) The length of the open segments and completedsegments may be unequal and each open segmentmay have a different skin effect and production rate

In this paper we follow the point source theory adoptedby Gringarten and Ramey [12] and Ozkan and Raghavan[13 14] in order to obtain wellbore pressure under constant-rate production and rate distribution under constant-pressure production in naturally fractured reservoirs (seeNotations section)

3 Mathematical Model of SCHWs

31 Point Source Model

311 Governing Equation

(A) Fracture System The 3D governing equation describingtransient fluid flow in natural fracture system can be writtenas follows

1205972119901f1205971199032 + 1119903 120597119901f120597119903 + 119896fv119896fh1205972119901f1205971199112

= 120601f119862ft120583119896fh120597119901f120597119905 + 120601m119862mt120583119896fh

120597119901m120597119905(1)

where 119903 = radic(119909 minus 119909w)2 + (119910 minus 119910w)2(B) Matrix System The 3D governing equation describingtransient fluid flow in natural fracture system can be writtenas follows 120572119896m119896fh (119901m minus 119901f) + 120601m119862m120583119896fh

120597119901m120597119905 = 0 (2)

312 Initial Conditions The initial pressure is assumed to beequal and is represented by original formation pressure innaturally fractured oil reservoirs thus

119901f (119903 119911 119905 = 0) = 119901e (3)

119901m (119903 119911 119905 = 0) = 119901e (4)

313 Inner and Outer Boundary Conditions It is assumedthat production rate of point source is 119902(119905) thus innerboundary condition can be written as

lim120576997888rarr0

int119911w+1205762119911wminus1205762

[ lim120575997888rarr0

2120587119896fh120583 (119903120597119901f120597119903 )119903=120576] 119889119911 = 119902 (119905) (5)

Corresponding outer boundary conditions can beexpressed as for a laterally impermeable boundary top andbottom boundaries being

120597119901f (119903 119911 = 0 119905)120597119911 = 0 (6)

120597119901f (119903 119911 = ℎ 119905)120597119911 = 0 (7)

120597119901f (119903 = 119877e 119911 119905)120597119903 = 0 (8)

32 Dimensionless Point Source Model in Laplace DomainAccording to dimensionless variables definition in Table 1(1)ndash(8) can be transformed into

1205972119901fD1205971199032D + 1119903D120597119901fD120597119903D + 1205972119901fD1205971199112D

= (120596120597119901fD120597119905D + (1 minus 120596) 120597119901mD120597119905D )120582 (119901mD minus 119901fD) + (1 minus 120596) 120597119901mD120597119905D = 0119901fD (119903D 119911D 119905D = 0) = 0119901mD (119903D 119911D 119905D = 0) = 0120597119901fD (119903D 119911D = 0 119905D)120597119911D = 0120597119901fD (119903D 119911D = ℎD 119905D)120597119911D = 0120597119901fD (119903D = 119877eD 119911D 119905D)120597119903D = 0lim120576D997888rarr0

int119911wD+120576D2119911wDminus120576D2

[ lim120575D997888rarr0

(119903D 120597119901fD120597119903D )119903D=120576D

]119889119911D= minusℎD119902D (119905D)

(9)

For convenience in derivation by adopting Laplace trans-form with respect to 119905D and solving pressure of matrixrespectively through (9) thus (9) can be expressed as followsin Laplace domain

1205972119901fD1205971199032D + 1119903D120597119901fD120597119903D + 1205972119901fD1205971199112D = 119906119901fD

120597119901fD (119903D 119911D = 0 119904)120597119911D = 0120597119901fD (119903D 119911D = ℎD 119904)120597119911D = 0120597119901fD (119903D = 119877eD 119911D 119904)120597119903D = 0lim120576119863997888rarr0

int119911wD+120576D2119911wDminus120576D2

[ lim120575D997888rarr0

(119903D 120597119901fD120597119903D )119903D=120576D

]119889119911D= minusℎD119902D (119904)

(10)

where 119906 = ((120582 + 119904120596(1 minus 120596))(120582 + 119904(1 minus 120596)))119904By finite cosine transform with respect to 119911D (10) can be

written as


Table 1 Dimensionless variables definition

Variables Dimensionless definition

Dimensionless pressure of fracture system 119901fD = 2120587119896fhℎ119902sc120583 (119901119890 minus 119901f )Dimensionless pressure of matrix system 119901mD = 2120587119896fhℎ119902sc120583 (119901119890 minus 119901m)Dimensionless wellbore pressure 119901wD = 2120587119896fhℎ119902sc120583 (119901119890 minus 119901w)Dimensionless production time 119905D = 119896fh119905120583(120601119862t)f+m1198712refDimensionless distance 119903D = 119903119871 ref

Dimensionless reservoir thickness ℎD = ℎ119871 ref

Dimensionless radius of impermeable circle boundary 119877eD = 119877e119871 ref

Dimensionless coordinate 119909D = 119909119871 ref119910D = 119910119871 ref

119911D = 119911119871 refradic119896fh119896fv

Dimensionless x-y-z coordinate of point source 119909wD = 119909w119871 ref119910wD = 119910w119871 ref

119911wD = 119911w119871 refradic119896fh119896fv

Dimensionless length of open segment 119871NcD = 119871119873119888119871 ref

Dimensionless wellbore radius 119903wD = 119903w119871 ref

Dimensionless mid-point of ith open segment 119909mD = 119909m119871 ref

Dimensionless continuous production 119902D = 119902119902scDimensionless infinitesimal vertical distance 120576D = 120576119871 ref

Dimensionless infinitesimal radial distance 120575D = 120575119871 ref

11988921006704119901fD1198891199032D + 1119903D119889 1006704119901fD119889119903D = (119906 + 11989921205872ℎ2D ) 1006704119901fD

119889 1006704119901fD (119903D = 119877eD 119899 119904)120597119903D = 0lim120575119863997888rarr0

(119903D 119889 1006704119901fD119889119903D )119903119863=120576119863

= minus119902119863 (119904) ℎD cos(119899120587119911wDℎD )(11)

In deriving (10) we have used the following finite cosinetransform and inverse finite cosine transform

1006704119901fD (119903D 119899 119904) = intℎD0

119901fD (119903D 119911D 119904) cos(119899120587119911DℎD )119889119911D (12)

119901fD (119903D 119911D 119904) = 1ℎD 1006704119901fD (119903D 0 119904)+ 2ℎD

infinsum119899=1

1006704119901fD (119903D 119899 119904) cos(119899120587119911DℎD )(13)

33 Model Solution of Point Source Equation (10) is zero-order Bessel equation and inner and outer boundary condi-tions general solution of zero-order Bessel equation can bewritten as

1006704119901fD = 1198601198700 (120576n119903D) + 1198611198680 (120576n119903D) (14)

where 120576n = radic119906 + 11989921205872ℎ2D (n=0123 )According to the properties ofmodified Besselrsquos functions

and outer boundary condition the coefficient B can beexpressed by

119861 = 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 119860 (15)

Hence (14) also can be written as

1006704119901fD = 119860[1198700 (120576n119903D) + 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 1198680 (120576n119903D)] (16)

Combining with inner boundary condition the coeffi-cient A in (16) can be determined as follows


xmDi

LcDi LNcDi

xmDi+1

Figure 3 Schematic of geometric relationship mid-point of 119894th open segment

119860 = 119902DℎD cos(119899120587119911wDℎD ) (17)

Substituting (17) into (16) and employing finite cosineinverse transform finally (16) can be written as follows

119901fD = 119902D [1198700 (1205760119903D) + 1198701 (1205760119877eD)1198681 (1205760119877eD) 1198680 (1205760119903D)+ 2infinsum119899=1

(1198700 (120576n119903D) + 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 1198680 (120576n119903D))sdot cos(119899120587119911DℎD ) cos(119899120587119911wDℎD )]

(18)

Equation (18) is the point source solution in naturallyfractured oil reservoirs

34 Model Solution of Line Source for SCHWs Taking theSCHWs shown in Figure 1 for example although rate of thehorizontal well in different location open segments can beseen as a uniform rate horizontal line source for SCHWsThus we take the ith open segment as our research objecttakingmid-point of ith open segment as origin coordinate Sowe can get line source of 119894th open segment by integrating withrespect to 119909D from 119909mDi-119871NcDi2 to 119909mDi-119871NcDi2 for pointsource in (18) Finally

119901fD119894

= 119902D119894119871NcD119894int119909mD119894+119871NcD1198942

119909mD119894minus119871NcD1198942[1198700 (1205760radic(119909D minus 120572)2)

+ 1198701 (1205760119877eD)1198681 (1205760119877eD) 1198680 (1205760radic(119909D minus 120572)2) + 2infinsum

119899=1

(1198700 (120576nradic(119909D minus 120572)2)+ 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 1198680 (120576nradic(119909D minus 120572)

2)) cos(119899120587119911DℎD ) cos(119899120587119911wDℎD )]

(19)

It is noted that (19) is only valid to compute the pressureanywhere We use zD=zwD+rwD to calculate wellbore surfacepressure Dimensionless wellbore radius for an anisotropicreservoir is given as follows

119903wD = 119903w119871 refradic( 119896119896119911)

025 + ( 119896119896119911)minus025

(20)

According to geometric relations shown in Figure 3 mid-point of ith open segment can be determined as follows

119909mD119894 = 119871cD119894 + 119871NcD119894 + 119871cD12 (21)

With (19) and by applying the superposition principlethe pressure responses at point (119909D 119910D 119911D) caused by allsegments can be obtained as follows

119901fD (119909D 0 119911D) = 119873sum119894=1

119901fD119894 (120573D 120573D119894)

= 119873sum119894=1

119902D119894119865D119894 (120573D 120573D119894)(22)

where 120573D = (119909D 0 119911D) 120573D119894 = (119909D119894 0 119911D119894)However it is also required that the sum of the flow rates

for each open segment be equal to the total flow rate that is119873sum119894=1

119902D119894 = 1119904 (23)

Combining with (19) and (22) then the matrix form canbe formulated as follows

[[[[[[[[[[

119865D1 (120573D1 120573D1) 119865D2 (120573D1 120573D2) 119865D119873 (120573D1 120573D119873) minus1119865D1 (120573D2 120573D1) 119865D2 (120573D2 120573D2) 119865D119873 (120573D2 120573D119873) minus1

119865D1 (120573DM 120573D1) 119865D2 (120573D2 120573DM) 119865D119873 (120573D119873 120573D119873) minus1

1 1 1 0

]]]]]]]]]]

times[[[[[[[[[

119902D1119902D2 119902D119873119901wD

]]]]]]]]]=[[[[[[[[[[[

00 01119904

]]]]]]]]]]]

(24)

The dimensionless wellbore flow rate for the constant-pressure production in naturally fractured reservoirs can bedetermined by dimensionless pressure with the constant-rateproduction in the Laplace domain [15]

119902D = 11199042119901wD(25)

In order to be consistent with current literature we usethe Fetkovich [16] definitions of the dimensionless declinevariables (119905Dd and 119902Dd) which are given below The 119905Ddfunction is given in terms of dimensionless variables as

119905Dd = 21198772eD119905D

ln119877eD minus 05 (26)

In a similar fashion the 119902Dd function is given in terms ofdimensionless variables as


10minus6 10minus5 10minus4 10minus3 10minus2 10minus1 100 101 102 103 104 105 106 10710minus3

10minus2

10minus1

100

101

102

Result of SaphirResult of this paper

wDampwD

D

Figure 4 Comparison of the results of this paper with that of well-test simulator

119902Dd = [ln119877eD minus 05] 119902D (27)

The rate integral and rate integral derivative functionsintroduced by McCray [17] are given in dimensionless formbelow The dimensionless rate integral function 119902Di is givenas

119902Ddi = 119873pDd119905Dd = 1119905Dd int119905Dd

0119902Dd (119909) 119889119909 (28)

And the dimensionless rate integral derivative function119902Ddid is given as

119902Ddid = 119902Ddid minus 119902Dd (29)

4 Model Verification

To verify the model and solutions derived in the abovesection a relatively particular case is considered and pressureand pressure-derivate curves generated by our solution arecompared to well-test stimulator Saphir Fluid flow into well-bore is treated as infinite conductivity but rate distributionin wellbore is no-uniform Therefore based on differentdimensionless variable definition between this paper andwell-test stimulator we can set 120596=02 for this paper and well-test stimulator 120582=001 for this paper 120582=625 times 10minus8 with119871 ref=40 for well-test stimulator Other parameters can beset as 120596=02 119871h=400 119871Nc=40 119871c=0 119873c=0 119873Nc=10 Thecomparisons presented in Figure 4 suggest that the resultscalculated by our model are consistent with that obtainedby the well-test simulator which verifies the credibility ofthe model presented in this article At the same time byanalyzing each segment rate distribution in different time wecan confirm imbalanced fluid inflow along the wellbore (seeFigure 5)

5 Discussions and Analysis

51 Flow Regimes In order to study the flow regimes ofSCHWs in naturally fractured oil reservoirs more graphi-cally type curves of pressure response and production rate

performance are illustrated in Figures 6 and 7 by Stehfest[18] numerical inversion According to the dimensionlesspressure-derivate characteristic pressure response curves ofSCHWs in naturally fractured oil reservoirs are divided intoseven flow regimes and the important basic data is shown inTable 2

Period I is the first radial (FR) flow period During thisperiod the flow regime is radial flow around open segmentin vertical direction (see Figure 8(a))The pressure derivativecurve is a horizontal line with a value of ldquo1(4119871hD)rdquo Curves ofrate rate integral and integral derivative exhibit a downwardline Because each length of open segment is different firstradial flow may not occur If open segment is very smallspherical flow can appear during this period

Period II is first linear (FL) flow period in which fluidflow in the reservoir is parallel to the upper and lower bound-ary of the reservoir and each open segment is independentduring first linear flow stage (see Figure 8(b)) The maincharacteristic of pressure-derivative is a line with a half slopein this stage Corresponding rate integral derivative curvebecomes gentle in this stage Similarly when open segmentis shorter compared with complete segment first linear flowcannot be appearing

Period III is second pseudo-radial (SPR) flow in whichthe pressure derivative curve is horizontal line of ldquo05119873NcrdquoThis flow period is exhibited when the lateral distancebetween open segments is relatively large Before and duringthis flow period each open segment has its own drainage areaand behaves independently without interference from opensegment (see Figure 8(c))

Period IV is second linear (SL) flow Pressure wavepropagates to drainage area controlled by each open segmentand interference between open segments occurs After thesuperposition of the pressure waves pressure waves prop-agate continually as time goes The second linear flow canbe formed in natural fracture reservoir (see Figure 8(d))The pressure derivative curve during this period is exhibitedas a one-half slope straight line again and characteristicsof rate integral derivative curve do not appear in thisstage


Table 2 Important basic data for SCHWs

Parameters (unit) ValueWellbore radius (m) 01Outer boundary radius (m) 10000Horizontal well length (m) 400Length of each open segment (m) 25Length of each completed segment (m) 100Number of open segments (dimensionless) 4Number of completed segments (dimensionless) 3Reference length (m) 40Storativity ratio (dimensionless) 02Flow coefficient (dimensionless) 001

q＄1 q＄10q＄2 q＄9q＄3 q＄8q＄4 q＄7q＄5 q＄6

10minus4 10minus3 10minus2 10minus1 100 101 102 103 104 105 106 107 108002004006008010012014016018020

q D

tD

Figure 5 Rate distribution along the wellbore in different time

Period V is the cross-flow stage in which fluid flows intonatural fracture from matrix firstly when the SCHWs areput into production The pressure of natural fracture systemwill gradually decrease causing pressure difference betweennatural fracture system and matrix system Because existenceof pressure drop between natural fracture system and matrixsystem led to cross-flow from natural fracture system tomatrix system the characteristic of pressure derivative duringstage is ldquodiprdquo Corresponding rate integral derivative curvealso exhibits a ldquodiprdquo in this stage

Period VI is late pseudo-radial (LPR) flow stage Aftercross-flow flow stage the pressures in natural fracture systemand matrix system gradually incline to equilibrium Pseudo-radial flow around SCHWs is formed in naturally fracturedreservoirs (see Figure 8(e)) Pressure derivative exhibits ahorizontal line of ldquo05rdquo during pseudo-radial flow in log-logplot Corresponding rate integral derivative curve is also aslanted line

Period VII is characteristic of closed boundary Pressurewaves propagate to circular impermeable outer boundaryduring this stage Curves of pressure derivative exhibit unite-slope line and corresponding rate integral and derivativecurve coincide and exhibit negative unite-slope line

52 Sensitivity of Parameters Figure 9 shows the effect ofouter boundary radius on dimensionless rate integral 119902Ddiand rate integral derivative 119902Ddid We can know that outerboundary radius has effect on whole flow regime Withincrease of outer boundary radius the value of dimensionlessrate integral and rate integral derivative curves is smallerwhich indicates that larger outer boundary radius can leadto the smaller rate decline curves in whole flow regime

Figure 10 shows the effect of storativity ratio on dimen-sionless rate integral 119902Ddi and rate integral derivative 119902DdidIt is obvious that storativity ratio mainly has significant effecton FR FL SPR and cross-flow regimeThe smaller storativityratio represents that storativity ability of naturally fracturedreservoirs is smaller It can be clearly observed that thesmaller the value of storativity ratio the deeper and widerthe ldquodiprdquo in rate integral derivative curve during cross-flowIn addition smaller value of storativity ratio leads to smallerrate integral of FR FL and SPR

Figure 11 shows the effect of cross-flow coefficient ondimensionless rate integral 119902Ddi and rate integral derivative119902Ddid It is obvious that flow coefficient mainly has significanteffect on SPR cross-flow and LPR regime The larger cross-flow coefficient represents that cross-flow ability fromnatural


10minus5 10minus4 10minus3 10minus2 10minus1 100 101 102 103 104 105 106 10710minus2

10minus1

100

101

102

III III

IVV

VIVII

wDampwD

D

D

pＱ＄

pＱ＄t＄

Figure 6 Pressure and pressure derivative responses of SCHWs with 4 open segments

10minus7 10minus6 10minus5 10minus4 10minus3 10minus2 10minus1 100 101 102 10310minus2

10minus1

100

101

102

Closed boundary

LPRCrossflow

SPRFL

tDd

FR

Ddamp

DdiampDdid

Figure 7 Rate rate integral and rate integral derivative responses of SCHWs with 4 open segments

(a) First radial flow (b) First linear flow

(c) Second pseudo-radial flow (d) Second linear flow

(e) Late pseudo-radial flow

Figure 8 Schematic of flow stage for SCHWs with 4 open segments


10minus6 10minus5 10minus4 10minus3 10minus2 10minus1 100 101 102 10310minus2

10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75

Figure 9 The effect of outer boundary radius on 119902Ddi and 119902Ddid

10minus6 10minus5 10minus4 10minus3 10minus2 10minus1 100 101 102 10310minus710minus2

10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02

Figure 10 The effect of storativity ratio on 119902Ddi and 119902Ddid

fracture to matrix is larger It can be clearly observed that thesmaller the value of cross-flow coefficient is the later the ldquodiprdquoin rate integral derivative curve during cross-flow appearsDuration of LPR regime is shorter

Figures 12 and 13 show the effect of number of open(or completed) segments on dimensionless rate integral 119902Ddiand rate integral derivative 119902Ddid Though number of open(or completed) segments is different it is assumed thatlength of SCHWs is equal Number of completed segmentshas an effect on SPR cross-flow and LR regimes mainlyWith increase of completed segment rate of fluid flowinto wellbore decreases under constant wellbore pressureTherefore higher number of completed segments can leadto larger value of rate integral derivative curves in log-logplot (see Figure 12) At the same time increasing numberof completed segments can delay water breakthrough andprevent wellbore collapse it can also lead to small rate forevery open segment which makes total rate decrease underconstant wellbore pressure (see Figure 13)

Figures 14 and 15 show the effect of length of completedsegment on dimensionless rate integral 119902Ddi and rate integral

derivative 119902Ddid It is assumed that lengths of SCHWs andnumbers of completed segments are equal while completedsegment length is different It is obvious that length ofcompleted segment has an effect on each flow regimeCompared with LR regime length of completed segmenthas an obvious influence on rate integral derivative curveduring FR FL SPR and cross-flow regime Longer lengthof completed segment leads to low rate integral derivativewhich is caused by larger pressure loss from formation towellbore (see Figure 14) With the increase of length ofcompleted segment open segment becomes more and moreshort As the time of production continues to increase longercompleted segment makes each open segment rate smaller(see Figure 15)

6 Conclusion

In this work we have developed a solution to computethe rate decline of SCHWs with constant wellbore pressureAccording to characteristic of pressure-derivative curvesunder constant production and rate integral derivative curve


10minus7 10minus6 10minus5 10minus4 10minus3 10minus2 10minus110minus2

10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3

Figure 11 The effect of cross-flow coefficient on 119902Ddi and 119902Ddid


10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd

Figure 12 The effect of number of open (or completed) segments on 119902Ddi and 119902Ddid


024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4

Figure 13 The flux distribution of each open segment



10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

Figure 14 The effect of length of open (or completed) segment on 119902Ddi and 119902Ddid

014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D

tD10minus4 10minus3 10minus2 10minus1 100 101 102 103 104 105 106 107 108

q＄1=q＄3

q＄2


under constantwellbore pressure flow regimes of SCHWsareidentified Specific conclusions can be drawn as follows(1) An analytical model is proposed in this paper toobtain rate decline response and flux distribution of SCHWsin naturally fractured reservoirs under constant wellborepressure Pressure transient responses and Blasingame ratedecline curve are generated and discussed(2) In addition the seven flow periods observed forSCHWs mainly include first radial flow first linear flow sec-ond pseudo-radial flow second pseudo-radial flow secondlinear flow late pseudo-radial flow stage and characteristicof closed boundary(3)By comparing results of simplified model in this paperwith the results calculated by Saphir for horizontal well innaturally fractured reservoirs the results showed excellentagreement Imbalanced flux distribution along the wellboreis verified again(4) The model illustrated how the dimensionless rateintegral and rate integral derivative log-log curves are influ-enced by some parameters (such as outer boundary radius

storativity ratio cross-flow coefficient and number andlength of open segments)(5) The proposed model in this paper can be used tointerpret rate decline signals more accurately for SCHWs innaturally fractured oil reservoirs and provide more accuratedynamic parameters which are important for efficient reser-voir development

Notations

119862ft Total compressibility of natural fracturesystem and oil atmminus1119862mt Total compressibility of matrix system andoil atmminus1ℎ Reservoir thickness cm119896 Equivalent permeability 120583m2119896 = 3radic119896119909119896119910119896119911119896fh Horizontal permeability of naturalfracture system 120583m2


119896fv Vertical permeability of natural fracturesystem 120583m2119896m Permeability of matrix system 120583m2119896z Vertical permeability 120583m2119871c119894 Length of 119894th completed segment cm119871Nc119894 Length of 119894th open segment cm119871 ref Reference length cm119873 Open segment number dimensionless119873p Cumulative production cm3119901e Initial reservoirs pressure atm119901f Pressure of natural fracture system atm119901m Pressure of natural matrix system atm119901w Wellbore pressure of natural matrixsystem atm119902 Production under constant wellborepressure cm3119902d Decline rate function as defined byFetkovich cm3119902di Decline rate integral as defined byMcCray cm3119902did Decline rate integral derivative function asdefined by McCray119902sc Production rate under the standardconditions cm3s119902(119905) Surface production rate of a point sourcecm3s119903 Radial distance cm119877e Radius of impermeable circle boundarycm119903w Wellbore radius cm119904 Laplace variables119905 [Production time s119905d Decline time s119909 x-coordinates cm119909m Mid-point of 119894th open segment cm119909w x-coordinates of a point source cm119910 y-coordinates cm119910w y-coordinates of a point source cm119911 z-coordinates cm119911w z-coordinates of a point source cm120572 Shape factor of dual-porosity systemcmminus2120575 Infinitesimal radial distance cm120576 Infinitesimal vertical distance cm120582 Cross-flow coefficient of dual-porosityreservoirs dimensionless120583 Viscosity at current reservoir pressure cp120593 Reservoir porosity dimensionless120596 Storativity ratio of dual-porosityreservoirs dimensionless1198680(119909) The first kind modified Bessel functionzero order1198700(119909) The second kind modified Besselfunction zero order1198681(119909) The first kind modified Bessel functionfirst order1198702(119909) The second kind modified Besselfunction first order

Subscripts

D Dimensionlessf Natural fracture systemm Matrix system

Superscripts

Laplace domain1006704 Finite cosine transform

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Authorsrsquo Contributions

You-jie Xu and Qi-guo Liu contributed equally to this work(co-first authors)

Acknowledgments

This article was supported by the National Major ResearchProgramme for Science and Technology of China (Grant No2017ZX05009-004 and No 2016ZX05015-003)

References

[1] R Leon-Ventura GGonzalez-G andH Leyva-G ldquoEvaluationof Horizontal Well Productionrdquo in Proceedings of the SPEInternational Petroleum Conference and Exhibition in MexicoVillahermosa Mexico

[2] M M Saggaf ldquoA vision for future upstream technologiesrdquoJournal of Petroleum Technology vol 60 no 3 pp 54ndash98 2008

[3] F Brons and V Marting ldquoThe Effect of Restricted Fluid Entryon Well Productivityrdquo Journal of Petroleum Technology vol 13no 02 pp 172ndash174 2013

[4] P A Goode and D J Wilkinson ldquoInflow performance of par-tially open horizontal wellsrdquo Journal of Petroleum Technologyvol 43 no 8 pp 983ndash987 1991

[5] L Larsen ldquoThe Pressure-Transient Behavior of Vertical WellsWith Multiple Flow Entriesrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Houston Texas

[6] M Kamal I Buhidma S Smith and W Jones ldquoPressure-transient analysis for a well with multiple horizontal sectionsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition Houston Texas 1993

[7] T Yildiz and E Ozkan ldquoTransient pressure behaviour ofselectively completed horizontal wellsrdquo inProceedings of the SPEAnnual Technical Conference and Exhibition New Orleans LaUSA 1994

[8] T Yildiz and Y Cinar ldquoInflow Performance and TransientPressure Behavior of Selectively Completed VerticalWellsrdquo SPEReservoir Engineering vol 1 no 5 pp 467ndash473 1998


[9] T Yildiz ldquoProductivity of horizontal wells completed withscreensrdquo SPE Reservoir Evaluation and Engineering vol 7 no5 pp 342ndash350 2004

[10] H O Seyide ldquoPerformance Analysis of a Selectively CompletedHorizontal Wellrdquo in Proceedings of the SPE Nigeria AnnualInternational Conference and Exhibition Lagos Nigeria

[11] W Luo H-T Li Y-Q Wang and J-C Wang ldquoA new semi-analytical model for predicting the performance of horizontalwells completed by inflow control devices in bottom-waterreservoirsrdquo Journal of Natural Gas Science and Engineering vol27 pp 1328ndash1339 2015

[12] A C Gringarten and H J Ramey Jr ldquoThe use of source andgreenrsquos function in solving unsteady-flow problem in reservoirrdquo SPE Journal vol 13 no 5 pp 285ndash296 1973

[13] 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

[14] E Ozkan and R Raghavan ldquoNew solutions for well-test-analysis problems Part 2 Computational considerations andapplicationsrdquo SPE Formation Evaluation vol 6 no 3 pp 369ndash378 1991

[15] A F van Everdingen and W Hurst ldquoThe application of theLaplace transformation to flow problems in reservoirsrdquo Journalof Petroleum Technology vol 1 no 12 pp 305ndash324 1949

[16] M Fetkovich ldquoDecline curve analysis using type curvesrdquoJournal of Petroleum Technology vol 32 no 6 pp 1065ndash10772013

[17] T L Mccray Reservoir analysis using production decline dataand adjusted time TexasAampMUniversity College Station 1990

[18] H Stehfest ldquoNumerical inversion of Laplace transformrdquo Com-munications of the ACM vol 13 no 1 pp 47ndash49 1970

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Dierential EquationsInternational Journal of

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Submit your manuscripts atwwwhindawicom


O x

y

z

h

Re

Lh

LNciLci

z=h

z=0

zw

Top boundary

Bottom boundary


Figure 1 Schematic of the SCHWs in naturally fractured oil reservoirs

distribution of the open intervals Kamal et al [6] presentedan analytical model of SCHWs by employing instantaneous-point-source solution and the superposition principle Theyused this model to analyze actual cases where pressuretransient is available Yildiz [7ndash9] and Seyide [10] presenteda model of SCHWs and SCVWs and derived asymptoticapproximations of the model in Laplace space Pressure andpressure-derivate log-log curves were plotted flow charac-teristics were discussed and each segment rate of SCHWswas analyzed A new semianalytical model for predictingthe performance of horizontal wells which were completedby inflow control devices in bottom-water reservoirs waspresented The coupled solution is developed for predictingthe performance of horizontal wells in a box-shaped reservoirwith bottom-water drive [11]

In order to analyze the rate decline curve of SCHWsa mathematical model considering difference between hor-izontal and vertical permeability of SCHWs is establishedin naturally fractured reservoirs Based on point source andthe superposition principle pressure analytical solution ofthe SCHWs under the condition of constant production inimpermeable top and bottom boundary and lateral imperme-able boundary by Laplace transform and finite cosine Fouriertransform Log-log curves of pressure and pressure-derivateand rate decline are drawn in naturally fractured reservoirs byemploying Stehfest numerical inversion Seven flow regimesaccording to the characteristic of pressure-derivate curve areidentified and every flow regime characteristic is described indetail This paper discusses that relevant parameters (stora-tivity ratio flow coefficient number and length completedhorizontal sections etc) have effect on pressure and ratedecline curves Corresponding solutions can be useful incompletion design and rate decline in field practice

2 Physical Model of SCHWs and Assumption

Horizontal wells are located in naturally fractured reservoirswith impermeable top and bottom boundary and lateralimpermeable boundary and parallel to the upper and lower

1Fig22


Matrix system Fr

actu

re sy

stem

Horizontal wellbore

Figure 2 Schematic of fluid flowing path for SCHW in naturallyfractured oil reservoirs

impermeable boundary Horizontal well consists of119873Nc opensegments and 119873c completed segments (shown in Figure 1)The naturally fractured reservoir is structured by matrixsystem and natural fracture system The dual-porosity mediaare assumed as the Warren-Root model and pseudo-steadycross flow exits between the matrix and fractures Fluidsflow into natural fractures from matrix firstly and flow intohorizontal wellbore from natural fractures secondly by opensegments (shown in Figure 2) In order tomakemathematicalmodel more reliable and accurate some assumptions arelisted as follows

(a) The fluid flow in the reservoir obeys Darcyrsquos law andlaw of isothermal percolation

(b) Flow is single phase and the fluid has constant andsmall compressibility and constant viscosity

(c) Formation permeability is anisotropic with threemajor directional permeability 119896119909 119896119910 119896119911

(d) Formation initial pressure is 119901e and horizontal pro-duced at a constant surface flow rate 119902sc









1205972119901f1205971199032 + 1119903 120597119901f120597119903 + 119896fv119896fh1205972119901f1205971199112

= 120601f119862ft120583119896fh120597119901f120597119905 + 120601m119862mt120583119896fh

120597119901m120597119905(1)


120597119901m120597119905 = 0 (2)


119901f (119903 119911 119905 = 0) = 119901e (3)

119901m (119903 119911 119905 = 0) = 119901e (4)


lim120576997888rarr0

int119911w+1205762119911wminus1205762

[ lim120575997888rarr0

2120587119896fh120583 (119903120597119901f120597119903 )119903=120576] 119889119911 = 119902 (119905) (5)


120597119901f (119903 119911 = 0 119905)120597119911 = 0 (6)

120597119901f (119903 119911 = ℎ 119905)120597119911 = 0 (7)

120597119901f (119903 = 119877e 119911 119905)120597119903 = 0 (8)


1205972119901fD1205971199032D + 1119903D120597119901fD120597119903D + 1205972119901fD1205971199112D


int119911wD+120576D2119911wDminus120576D2

[ lim120575D997888rarr0

(119903D 120597119901fD120597119903D )119903D=120576D

]119889119911D= minusℎD119902D (119905D)

(9)


1205972119901fD1205971199032D + 1119903D120597119901fD120597119903D + 1205972119901fD1205971199112D = 119906119901fD


int119911wD+120576D2119911wDminus120576D2

[ lim120575D997888rarr0

(119903D 120597119901fD120597119903D )119903D=120576D

]119889119911D= minusℎD119902D (119904)

(10)


written as








119911D = 119911119871 refradic119896fh119896fv








11988921006704119901fD1198891199032D + 1119903D119889 1006704119901fD119889119903D = (119906 + 11989921205872ℎ2D ) 1006704119901fD

119889 1006704119901fD (119903D = 119877eD 119899 119904)120597119903D = 0lim120575119863997888rarr0

(119903D 119889 1006704119901fD119889119903D )119903119863=120576119863

= minus119902119863 (119904) ℎD cos(119899120587119911wDℎD )(11)


1006704119901fD (119903D 119899 119904) = intℎD0

119901fD (119903D 119911D 119904) cos(119899120587119911DℎD )119889119911D (12)

119901fD (119903D 119911D 119904) = 1ℎD 1006704119901fD (119903D 0 119904)+ 2ℎD

infinsum119899=1

1006704119901fD (119903D 119899 119904) cos(119899120587119911DℎD )(13)


1006704119901fD = 1198601198700 (120576n119903D) + 1198611198680 (120576n119903D) (14)



119861 = 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 119860 (15)


1006704119901fD = 119860[1198700 (120576n119903D) + 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 1198680 (120576n119903D)] (16)



xmDi

LcDi LNcDi

xmDi+1


119860 = 119902DℎD cos(119899120587119911wDℎD ) (17)




(18)



119901fD119894

= 119902D119894119871NcD119894int119909mD119894+119871NcD1198942



119899=1


2)) cos(119899120587119911DℎD ) cos(119899120587119911wDℎD )]

(19)


119903wD = 119903w119871 refradic( 119896119896119911)

025 + ( 119896119896119911)minus025

(20)


119909mD119894 = 119871cD119894 + 119871NcD119894 + 119871cD12 (21)


119901fD (119909D 0 119911D) = 119873sum119894=1

119901fD119894 (120573D 120573D119894)

= 119873sum119894=1

119902D119894119865D119894 (120573D 120573D119894)(22)



119902D119894 = 1119904 (23)


[[[[[[[[[[



1 1 1 0

]]]]]]]]]]

times[[[[[[[[[

119902D1119902D2 119902D119873119901wD

]]]]]]]]]=[[[[[[[[[[[

00 01119904

]]]]]]]]]]]

(24)


119902D = 11199042119901wD(25)


119905Dd = 21198772eD119905D

ln119877eD minus 05 (26)




10minus2

10minus1

100

101

102


wDampwD

D


119902Dd = [ln119877eD minus 05] 119902D (27)



0119902Dd (119909) 119889119909 (28)















q＄1 q＄10q＄2 q＄9q＄3 q＄8q＄4 q＄7q＄5 q＄6


q D

tD










10minus1

100

101

102

III III

IVV

VIVII

wDampwD

D

D

pＱ＄

pＱ＄t＄



10minus1

100

101

102

Closed boundary

LPRCrossflow

SPRFL

tDd

FR

Ddamp

DdiampDdid








10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75



10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02






6 Conclusion




10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018
















1205972119901f1205971199032 + 1119903 120597119901f120597119903 + 119896fv119896fh1205972119901f1205971199112

= 120601f119862ft120583119896fh120597119901f120597119905 + 120601m119862mt120583119896fh

120597119901m120597119905(1)


120597119901m120597119905 = 0 (2)


119901f (119903 119911 119905 = 0) = 119901e (3)

119901m (119903 119911 119905 = 0) = 119901e (4)


lim120576997888rarr0

int119911w+1205762119911wminus1205762

[ lim120575997888rarr0

2120587119896fh120583 (119903120597119901f120597119903 )119903=120576] 119889119911 = 119902 (119905) (5)


120597119901f (119903 119911 = 0 119905)120597119911 = 0 (6)

120597119901f (119903 119911 = ℎ 119905)120597119911 = 0 (7)

120597119901f (119903 = 119877e 119911 119905)120597119903 = 0 (8)


1205972119901fD1205971199032D + 1119903D120597119901fD120597119903D + 1205972119901fD1205971199112D


int119911wD+120576D2119911wDminus120576D2

[ lim120575D997888rarr0

(119903D 120597119901fD120597119903D )119903D=120576D

]119889119911D= minusℎD119902D (119905D)

(9)


1205972119901fD1205971199032D + 1119903D120597119901fD120597119903D + 1205972119901fD1205971199112D = 119906119901fD


int119911wD+120576D2119911wDminus120576D2

[ lim120575D997888rarr0

(119903D 120597119901fD120597119903D )119903D=120576D

]119889119911D= minusℎD119902D (119904)

(10)


written as








119911D = 119911119871 refradic119896fh119896fv








11988921006704119901fD1198891199032D + 1119903D119889 1006704119901fD119889119903D = (119906 + 11989921205872ℎ2D ) 1006704119901fD

119889 1006704119901fD (119903D = 119877eD 119899 119904)120597119903D = 0lim120575119863997888rarr0

(119903D 119889 1006704119901fD119889119903D )119903119863=120576119863

= minus119902119863 (119904) ℎD cos(119899120587119911wDℎD )(11)


1006704119901fD (119903D 119899 119904) = intℎD0

119901fD (119903D 119911D 119904) cos(119899120587119911DℎD )119889119911D (12)

119901fD (119903D 119911D 119904) = 1ℎD 1006704119901fD (119903D 0 119904)+ 2ℎD

infinsum119899=1

1006704119901fD (119903D 119899 119904) cos(119899120587119911DℎD )(13)


1006704119901fD = 1198601198700 (120576n119903D) + 1198611198680 (120576n119903D) (14)



119861 = 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 119860 (15)


1006704119901fD = 119860[1198700 (120576n119903D) + 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 1198680 (120576n119903D)] (16)



xmDi

LcDi LNcDi

xmDi+1


119860 = 119902DℎD cos(119899120587119911wDℎD ) (17)




(18)



119901fD119894

= 119902D119894119871NcD119894int119909mD119894+119871NcD1198942



119899=1


2)) cos(119899120587119911DℎD ) cos(119899120587119911wDℎD )]

(19)


119903wD = 119903w119871 refradic( 119896119896119911)

025 + ( 119896119896119911)minus025

(20)


119909mD119894 = 119871cD119894 + 119871NcD119894 + 119871cD12 (21)


119901fD (119909D 0 119911D) = 119873sum119894=1

119901fD119894 (120573D 120573D119894)

= 119873sum119894=1

119902D119894119865D119894 (120573D 120573D119894)(22)



119902D119894 = 1119904 (23)


[[[[[[[[[[



1 1 1 0

]]]]]]]]]]

times[[[[[[[[[

119902D1119902D2 119902D119873119901wD

]]]]]]]]]=[[[[[[[[[[[

00 01119904

]]]]]]]]]]]

(24)


119902D = 11199042119901wD(25)


119905Dd = 21198772eD119905D

ln119877eD minus 05 (26)




10minus2

10minus1

100

101

102


wDampwD

D


119902Dd = [ln119877eD minus 05] 119902D (27)



0119902Dd (119909) 119889119909 (28)















q＄1 q＄10q＄2 q＄9q＄3 q＄8q＄4 q＄7q＄5 q＄6


q D

tD










10minus1

100

101

102

III III

IVV

VIVII

wDampwD

D

D

pＱ＄

pＱ＄t＄



10minus1

100

101

102

Closed boundary

LPRCrossflow

SPRFL

tDd

FR

Ddamp

DdiampDdid








10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75



10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02






6 Conclusion




10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018















119911D = 119911119871 refradic119896fh119896fv








11988921006704119901fD1198891199032D + 1119903D119889 1006704119901fD119889119903D = (119906 + 11989921205872ℎ2D ) 1006704119901fD

119889 1006704119901fD (119903D = 119877eD 119899 119904)120597119903D = 0lim120575119863997888rarr0

(119903D 119889 1006704119901fD119889119903D )119903119863=120576119863

= minus119902119863 (119904) ℎD cos(119899120587119911wDℎD )(11)


1006704119901fD (119903D 119899 119904) = intℎD0

119901fD (119903D 119911D 119904) cos(119899120587119911DℎD )119889119911D (12)

119901fD (119903D 119911D 119904) = 1ℎD 1006704119901fD (119903D 0 119904)+ 2ℎD

infinsum119899=1

1006704119901fD (119903D 119899 119904) cos(119899120587119911DℎD )(13)


1006704119901fD = 1198601198700 (120576n119903D) + 1198611198680 (120576n119903D) (14)



119861 = 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 119860 (15)


1006704119901fD = 119860[1198700 (120576n119903D) + 1198701 (120576119899119877eD)1198681 (120576119899119877eD) 1198680 (120576n119903D)] (16)



xmDi

LcDi LNcDi

xmDi+1


119860 = 119902DℎD cos(119899120587119911wDℎD ) (17)




(18)



119901fD119894

= 119902D119894119871NcD119894int119909mD119894+119871NcD1198942



119899=1


2)) cos(119899120587119911DℎD ) cos(119899120587119911wDℎD )]

(19)


119903wD = 119903w119871 refradic( 119896119896119911)

025 + ( 119896119896119911)minus025

(20)


119909mD119894 = 119871cD119894 + 119871NcD119894 + 119871cD12 (21)


119901fD (119909D 0 119911D) = 119873sum119894=1

119901fD119894 (120573D 120573D119894)

= 119873sum119894=1

119902D119894119865D119894 (120573D 120573D119894)(22)



119902D119894 = 1119904 (23)


[[[[[[[[[[



1 1 1 0

]]]]]]]]]]

times[[[[[[[[[

119902D1119902D2 119902D119873119901wD

]]]]]]]]]=[[[[[[[[[[[

00 01119904

]]]]]]]]]]]

(24)


119902D = 11199042119901wD(25)


119905Dd = 21198772eD119905D

ln119877eD minus 05 (26)




10minus2

10minus1

100

101

102


wDampwD

D


119902Dd = [ln119877eD minus 05] 119902D (27)



0119902Dd (119909) 119889119909 (28)















q＄1 q＄10q＄2 q＄9q＄3 q＄8q＄4 q＄7q＄5 q＄6


q D

tD










10minus1

100

101

102

III III

IVV

VIVII

wDampwD

D

D

pＱ＄

pＱ＄t＄



10minus1

100

101

102

Closed boundary

LPRCrossflow

SPRFL

tDd

FR

Ddamp

DdiampDdid








10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75



10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02






6 Conclusion




10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018









xmDi

LcDi LNcDi

xmDi+1


119860 = 119902DℎD cos(119899120587119911wDℎD ) (17)




(18)



119901fD119894

= 119902D119894119871NcD119894int119909mD119894+119871NcD1198942



119899=1


2)) cos(119899120587119911DℎD ) cos(119899120587119911wDℎD )]

(19)


119903wD = 119903w119871 refradic( 119896119896119911)

025 + ( 119896119896119911)minus025

(20)


119909mD119894 = 119871cD119894 + 119871NcD119894 + 119871cD12 (21)


119901fD (119909D 0 119911D) = 119873sum119894=1

119901fD119894 (120573D 120573D119894)

= 119873sum119894=1

119902D119894119865D119894 (120573D 120573D119894)(22)



119902D119894 = 1119904 (23)


[[[[[[[[[[



1 1 1 0

]]]]]]]]]]

times[[[[[[[[[

119902D1119902D2 119902D119873119901wD

]]]]]]]]]=[[[[[[[[[[[

00 01119904

]]]]]]]]]]]

(24)


119902D = 11199042119901wD(25)


119905Dd = 21198772eD119905D

ln119877eD minus 05 (26)




10minus2

10minus1

100

101

102


wDampwD

D


119902Dd = [ln119877eD minus 05] 119902D (27)



0119902Dd (119909) 119889119909 (28)















q＄1 q＄10q＄2 q＄9q＄3 q＄8q＄4 q＄7q＄5 q＄6


q D

tD










10minus1

100

101

102

III III

IVV

VIVII

wDampwD

D

D

pＱ＄

pＱ＄t＄



10minus1

100

101

102

Closed boundary

LPRCrossflow

SPRFL

tDd

FR

Ddamp

DdiampDdid








10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75



10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02






6 Conclusion




10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018










10minus2

10minus1

100

101

102


wDampwD

D


119902Dd = [ln119877eD minus 05] 119902D (27)



0119902Dd (119909) 119889119909 (28)















q＄1 q＄10q＄2 q＄9q＄3 q＄8q＄4 q＄7q＄5 q＄6


q D

tD










10minus1

100

101

102

III III

IVV

VIVII

wDampwD

D

D

pＱ＄

pＱ＄t＄



10minus1

100

101

102

Closed boundary

LPRCrossflow

SPRFL

tDd

FR

Ddamp

DdiampDdid








10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75



10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02






6 Conclusion




10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018











q＄1 q＄10q＄2 q＄9q＄3 q＄8q＄4 q＄7q＄5 q＄6


q D

tD










10minus1

100

101

102

III III

IVV

VIVII

wDampwD

D

D

pＱ＄

pＱ＄t＄



10minus1

100

101

102

Closed boundary

LPRCrossflow

SPRFL

tDd

FR

Ddamp

DdiampDdid








10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75



10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02






6 Conclusion




10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018










10minus1

100

101

102

III III

IVV

VIVII

wDampwD

D

D

pＱ＄

pＱ＄t＄



10minus1

100

101

102

Closed boundary

LPRCrossflow

SPRFL

tDd

FR

Ddamp

DdiampDdid








10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75



10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02






6 Conclusion




10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018










10minus1

100

101

102

DdiampDdid

tDd

R＄ = 250

R＄ = 125

R＄ = 75



10minus1

100

101

102

DdiampDdid

tDd

= 001= 007= 02






6 Conclusion




10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018










10minus1

100

100

101

101 102 103

102

tDd

DdiampDdid

= 1 times 10-1

= 1 times 10-2

= 1 times 10-3



10minus1

100

101

102

DdiampDdid

N＝=1 N＝=2

N＝=2 N＝=3N＝=3 N＝=4

tDd



024

028

032

036

040

044

048

052

D

tD

q＄1=q＄2

q＄1=q＄3

q＄2

q＄2=q＄3

q＄1=q＄4

N＝=1 N＝=2

N＝=2 N＝=3

N＝=3 N＝=4




10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018










10minus1

100

101

102

DdiampDdid

tDd

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55


014

016

018

020

022

024

026

028

L＝=100 L＝=25

L＝=80 L＝=40

L＝=60 L＝=55

D


q＄1=q＄3

q＄2




Notations




Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018










Subscripts


Superscripts


Data Availability






Acknowledgments


References



























Journal of












Journal of







Volume 2018






Volume 2018


























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Rate Decline Behavior of Selectively Completed Horizontal Wells … · 2019. 7. 30. · 51 %0˝eD 71 %0˝eD 70 %0D +2 ∞ 3 = *50 %nD + 51 % ˝eD 71 % ˝eD 70 %n ()D . ⋅cos - D