SPE-162516-PA (Analytical Model for Unconventional Multifractured Composite Systems)

Analytical Model forUnconventional Multifractured

Composite SystemsEkaterina Stalgorova and Louis Mattar, IHS

Summary

This paper presents an analytical model for unconventional reser-voirs with horizontal wells with multiple fractures. The model isan extension of the trilinear flow solution, but it subdivides thereservoir into five regions instead of three. This enables it to beused for more-complex reservoirs. Accordingly, the model cansimulate a fracture that is surrounded by a stimulated region oflimited extent (fracture branching), whereas the remaining reser-voir is nonstimulated. In addition to modeling flow within thefracture and flow within the stimulated region, the model takesinto account flow from the surrounding nonstimulated region, bothparallel to and perpendicular to the fracture.

The model can be used to simulate the flow in tight reservoirswith multifractured horizontal wells. In many cases, the fracturesdo not have a simple biwing shape, but are branched. This effec-tively creates regions of higher permeability around each fracture,which obviously affect the production performance significantly.However, in many tight reservoirs, in spite of their low permeability,the surrounding nonstimulated region can also be a significant con-tributor to long-term production. The five-region model accounts forthis contribution. Thus, it is particularly valuable when generatingproduction forecasts for reserves evaluation.

The model was validated by comparing its results with numeri-cal simulation. We found that analytical and numerical results arein good agreement only when the geometry of the system fallswithin certain limitations. However, these limitations are met inmost cases of interest. Therefore, the model is useful for practicalengineering purposes.

Introduction

A common way to produce hydrocarbons from tight reservoirs isto use horizontal wells with multiple hydraulic fractures. It hasbeen postulated that, in many cases, the propagation of a fracturemay create a branch pattern (Daneshy 2003). Fracture branchingcreates a stimulated region around each hydraulic fracture, whichcan be modeled by introducing a region of higher permeability(Fig. 1).

Brown et al. (2009) suggested an analytical solution todescribe the pressure distribution for a system in which the en-hanced region occupies all the space between fractures (Fig. 2a).They considered that the model consists of three regions: fracture,high-permeability region, and low-permeability region. A 1D lin-ear flow was assumed within each region. Brohi et al. (2011) pro-vided another solution for the same configuration; the maindifference between these two solutions is the way in which thethree regions are coupled.

In our earlier study (Stalgorova and Mattar 2012), we sug-gested a solution for the case when the enhanced region occupiesonly part of the space between the fractures, but the regionbeyond the fractures was neglected (Fig. 2b) Similar to Brownet al. (2009), we considered three linear-flow regions (fracture,

high-permeability region, and low-permeability region), but thelocation of the regions and their interaction were different.

In this paper, we present a solution for the case shown inFig. 2c. To provide the pressure distribution for such a system, weconsider five regions of linear flow; therefore, the suggestedmodel is referred as a five-region model. This case is a generaliza-tion that encompasses both the trilinear-model solution suggestedby Brown et al. (2009) and the enhanced-fracture-region modelpresented by Stalgorova and Mattar (2012).

Five-Region Model

The mathematical formulation of the proposed model is anextension of the one presented in our earlier study (Stalgorovaand Mattar 2012). We present an outline of the solution in thissection, whereas the detailed derivation is given in AppendicesA and B.

To model a reservoir with a multifractured horizontal well, weassume that all fractures have the same length and conductivityand are spaced uniformly along the horizontal well. This assump-tion is realistic because it is common field practice to designequally spaced hydraulic fractures with similar properties. Weintroduce a region of higher permeability around each fracture torepresent fracture branching. Because of the symmetry of the sys-tem, we can perform calculations on one-quarter of the spacebetween the fractures. Flow in the model is treated as a combina-tion of five linear flows within contiguous regions, as shown inFig. 3. We formulate the 1D flow solutions for each region andthen couple them by imposing flux and pressure continuity acrossthe boundaries between regions.

A schematic for the suggested five-region model is presented inFig. 3. The model consists of Regions 1 through 4 and the FractureRegion. To keep the solution general, we allow different properties(permeability, porosity, and total compressibility) in each region.However, for practical purposes, properties for Regions 2, 3, and 4can be considered identical because they represent the original res-ervoir rock, whereas Region 1 represents the region of enhancedpermeability immediately near the fracture.

The pressure distribution in each region is governed by the

flow equation r q klrq

@

@t/q (Dake 1978). For the case

of liquid flow, this equation can be rewritten as

r2p /lck

@p

@t1

Note: for the case of gas flow, the equation should be rewritten interms of pseudopressure, as discussed later in the Possible ModelModifications section.

To simplify the solution, we rewrite this equation for eachregion in dimensionless terms and convert it to the Laplace do-main. Definitions of dimensionless variables are given in Appen-dix A; derivations of equations and solutions are given inAppendix B.

Region 4

The diffusivity equation becomes

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CopyrightVC 2013 Society of Petroleum Engineers

This paper (SPE 162516) was accepted for presentation at the SPE CanadianUnconventional Resources Conference, Calgary, 30 October1 November 2012, andrevised for publication. Original manuscript received for review 7 December 2012. Revisedmanuscript received for review 21 March 2013. Paper peer approved 30 May 2013.

246 August 2013 SPE Reservoir Evaluation & Engineering

@2p4D@y2D

sg4D

p4D 0 2 where p4D is the pressure in Region 4 in the Laplace domain. Boundary Condition 1: No-flow condition at the outer reser-voir boundary (y y2)

@p4D@yD

yDy2D

0 3

Boundary Condition 2: Pressure continuity between Regions4 and 2 (at y y1)

p4Dy1D p2Dy1D 4

Region 3


@2p3D@y2D

sg3D

p3D 0 5

Boundary Condition 1: No-flow condition at the outer reser-voir boundary (y y2)

@p3D@yD

yDy2D

0 6


p3Dy1D p1Dy1D 7

Region 2


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a b c

k1 k1 k1

k2

k2k2

Fig. 2(a) Trilinear-flow model (Brown et al. 2009), (b) enhanced-fracture-region model (Stalgorova and Mattar 2012), and (c) five-region model (this study).

Stimulatedregion width (x1)y

Region 3: Region 4:

k3, 3, c3

Region 1:

k1, 1, c1

kf, f, cf

Region 2:

k2, 2, c2

k4, 4, c4

w/2

y 1 =

Fr

act

ure

half

leng

th ((

x f)y)

Fracture:

Fracture half width = w/2

Half distance between fractures (x2)

x

Res

ervo

ir ha

lf wi

dth

(y 2)

Fig. 3Schematic and dimensions for five-region model (one-quarter of a fracture).

a b

k1

k2

Fig. 1Horizontal well with multiple branch fractures (a) and its representation by a model (k1> k2) (b).

August 2013 SPE Reservoir Evaluation & Engineering 247

@2p2D@x2D

k4k2y1D

@p4D@yD

y1D

sg2D

p2D 0 8

Boundary Condition 1: No-flow boundary midway betweenthe fractures

@p2D@xD

x2D

0 9

Boundary Condition 2: Pressure continuity between Regions1 and 2

p2Dx1D p1Dx1D 10

Region 1


@2p1D@x2D

k3k1y1D

@p3D@yD

y1D

sg1D

p1D 0 11

Boundary Condition 1: Flux continuity between Regions 2and 1

k2l

@p2D@xD

x1D

k1l

@p1D@xD

x1D

12

Boundary Condition 2: Continuity between Region 1 andFracture Region

p1DwD=2 pFDwD=2 13

Fracture Region


@2pFD@y2D

2FCD

@p1D@xD

wD=2

sgFD

pFD 0 14

Boundary Condition 1: No flow through the fracture tip(yD y1D)

@pFD@yD

y1D

0 15

Boundary Condition 2: Obtained by applying Darcys law atthe wellbore (yD 0). (A correction to account for flowline con-vergence in the fracture, caused by the well being horizontalrather than vertical, is introduced later in Eq. 20). In dimension-less terms, in the Laplace domain, it becomes

@pFD@yD

0

pFCD s 16

Flow directions and boundary conditions for each region aresummarized in Fig. 4.

It can be shown (see detailed derivation in Appendix B) thatthe solution of the preceding equations with boundary conditions(Eqs. 2 through 16) is:

pFDyD pcoshyD 1

c6s

p FCD s

c6s

psinh c6sp 17

where c6(s) is calculated as shown in Appendix B. The solution atthe wellbore is given by

pwD pFD0 p

FCD s c6s

ptanh c6sp 18

This solution can be inverted from the Laplace to the time domainwith the numerical algorithm given by Stehfest (1970).

It is worth noting that, in the case when the stimulated region(Region 1) occupies all the space between fractures (x1 x2), thefive-region model reduces to the trilinear model of Brown et al. 2009(Fig. 2a). In the case when the contribution of Regions 3 and 4 isneglected (y1 y2), the five-region model reduces to the enhancedfracture region model of Stalgorova and Mattar (2012) (Fig. 2b).Therefore, the five-region model is a generalization that covers boththe trilinear model and the enhanced fracture region model.

Possible Model Modifications

This section follows Stalgorova and Mattar (2012).

Gas Flow

The solution was derived for a liquid system. To use the solutionfor gas flow, the dimensionless pressure should be expressed interms of real-gas pseudopressure (Eq. A-1).

It is, however, important to keep in mind that, for the case of

gas, the diffusivity term g k/lc

is not constant but changes with

pressure, so the diffusivity equation becomes nonlinear. One ofthe ways to deal with this problem is to use pseudotime (Andersonand Mattar 2005).

Dual Porosity

If desired, a dual-porosity modification can be introduced to themodel: The inner region can be treated as a dual continuum. Theessence of the modification consists of replacing s with u sf(s)in Eq. 11, in which f(s) is a function defined by the dual-porositycharacteristics of the system. A detailed description of the dual-porosity modification is given in Brown et al. (2009).

Skin Because of Convergence (Choking Skin). We assumed 1Dlinear flow within the hydraulic fracture. This would be appropri-ate if the well intersecting the fracture were vertical. However, inour present scenario, the well is horizontal, and the flowlines inthe fracture itself are not linear all the way, but as they approachthe horizontal well, they converge toward the wellbore (Fig. 5).This convergence results in an additional pressure drop, which isreferred to as a skin because of convergence (choking skin).Mukherjee and Economides (1991) provided a formulation (sc) toaccount for this additional pressure drop. Eq. 18 is modified as

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. . . . . .

No-flow

No-flow

No-

flow

No-flow

Region 3 Region 4

Region 1 Region 2

p1 = p3

p F =

p 1

p 1 =

p 2

p2 = p4y1 = (xf)y

y2

y

w/2 x1 x2pFy

qF

x

k 1p

1y

p2

x=

k 2

Fig. 4Five-region model: flow directions and boundaryconditions.


pwD p

FCD s c6s

ptanh c6sp

scs

19

where sc is the skin because of convergence, which can be cal-culated from

sc k1hkf w

lnh

2rw

p

2

20

Wellbore Storage

The wellbore-storage effect can also be incorporated as suggestedby Brown et al. (2009). To account for wellbore storage, modifythe wellbore pressure given in Eq. 19 to

pwD;storage pwD

1 CDs2pwD21

where CD is the dimensionless wellbore-storage coefficient.

Model Validation

We compared the results of the five-region model with the resultsof numerical simulation. We used a single-phase liquid for thisexercise. For numerical simulation, we used a 2D finite-differencemodel. To ensure the accuracy of the model, we defined a finegrid throughout the reservoir, and enforced additional refinementnear the fracture and near the boundaries of the regions. One-quarter of a fracture stage was modeled with 33 91 grid cells.Overall, the cell size was approximately 2 3 ft, and the cellsnear the fracture and near the region boundaries were 0.6 0.6 ft.Other model parameters are presented in Table 1. For these pa-rameters, the five-region-model results showed excellent agree-ment with the numerical simulation, as can be seen in Fig. 6.

Alternative Model (for Reservoirs Elongated inx-Direction)

In the original five-region model, it was assumed that flow can beapproximated by a combination of linear flows in each of the fiveregions, as was shown in Fig. 4. This assumption is valid for some

cases. However, there are other cases when such an approxima-tion produces inaccurate results. For example, if x2 is much largerthan y2, we can expect the flowlines in Region 4 to be aligned inthe direction of the pressure sink (the fracture). To account forsuch reservoir geometries, we developed an alternative five-regionmodel that directs the flow in Region 4 toward Region 3 ratherthan Region 2. This is depicted in Fig. 7. The modifications to therelevant equations and boundary conditions were straightforward.

Figs. 8 and 9 compare the original five-region model with thealternative five-region model for two different geometries. For thecase in which y2>>x2 (Fig. 8a), the original five-region model is inclose agreement with the numerical model, whereas the alternativemodel produces inaccurate results (Fig. 8b). On the other hand, forthe cases in which x2>>y2, the alternative five-region model givesmore accurate results than the original one (Figs. 9a and 9b).

For a typical field case of a horizontal well with multiple frac-tures, x2 100 ft (0.5 times the horizontal-well length/number offractures) and y2 500 ft (half the distance between two neigh-boring horizontal wells). Therefore, the use of the original five-region model (shown in Fig. 4) is more appropriate for most casesof interest.

It was shown previously that, under certain reservoir configura-tions, the original five-region model is applicable, whereas otherconfigurations require the use of the alternative five-region model.Unfortunately, there are cases when both the original and the alter-native models are not accurate enough. One such case is whenboth x2 and y2 are much larger than the fracture half-length (xf)y(e.g., a single fracture in an infinite reservoir). Fig. 10 shows thatfor such a configuration, the numerical solution is significantly dif-ferent from that of either model. The explanation for this inaccur-acy is that, at distances far away from the fracture, the flowlinesshould be radial not linear; however, the five-region models do notallow for this. The five-region model was developed to deal withmultiple-fracture horizontal wells. The geometry of these systemsis not likely to lead to radial flow. Therefore, rather than resolvingthis problem, we have focused instead on defining the conditionsof the applicability of the original five-region model.

Applicability of the original five-region model

To evaluate when the Five-Region Model is applicable, we per-formed comparisons with numerical models for different reservoirconfigurations. Agreement between the five-region model and thenumerical model depends on many parameters such as x1, x2,(xf)y, y2, k1, k2, and qF. It may seem difficult to test all possible

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TABLE 1MODEL PARAMETERS

Initial pressure pi 10,000 psi

Viscosity l 1 cpFormation volume factor B 1 bbl/STB

Compressibility c 1106 psi1Horizontal-well length Le 2,000 ft

Reservoir width Ye 1,000 ft

Fracture half-length

(in y-direction)

(xf)y 100 ft

Dimensionless fracture

conductivity

FCD 20.8

Distance from well to

permeability boundary

x1 25 ft

Number of fractures nf 20

Net pay h 100 ft

Permeability of

stimulated region

k1 1 md

Permeability of

nonstimulated regions

k2 k3k4 0.01 md

Porosity / 0.1

Saturation Sw 1

Production rate q 400 STB/D

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

Pwf,

psi

0 200 400 600 800 1,000time, days

Pwf (Five-Region Model)Pwf (numerical simulation)

Fig. 6Comparison of five-region-model results with numeri-cal-model results.

Fracture

Horizontal wellbore

Fig. 5Convergence of flowlines toward the wellbore withinthe fracture.


combinations of parameters. However, looking at the solution,one can notice that the number of parameters that actually affectthe result can be minimized, if we compare pD vs. tD plots (dimen-sionless) instead of comparing p vs. t plots. Fig. 11 illustrates thecomparison of the dimensionless type curves obtained from theoriginal five-region model and the numerical simulator for onereservoir geometry.

We ran a series of such comparisons with different combina-tions of dimensionless parameters, and we observed that the nu-merical and five-region-model type curves show reasonableagreement if all the following conditions are met:

x1 0.1 x2.In other words, the stimulated zone occupies at least 10% of

the space between fractures. (xf)y 0.1 y2.This condition ensures that the fracture length is not too small

compared with the reservoir size. If the fracture length is verysmall, radial flow is more likely to occur, and this is not handledby the five-region model. The (xf)y is usually at least 100 ft; y2 is

the half-distance between neighboring horizontal wells, whichcan be estimated at approximately 1,000 ft; therefore, this condi-tion is usually met.

y2 2 x2. The area drained by each fracture is elongated inthe y-direction (if the geometry were more elongated in the x-direction, the alternative model would be more appropriate). Thex2 is the half-distance between adjacent fractures, so it can beexpressed as x2 Le/(2nf), in which Le is the horizontal-welllength (approximately 4,000 ft) and nf is the number of fractures(at least 10). Therefore, we can safely assume that x2 is less than400; thus, y2 2 x2, and the condition y2 2 x2 is typically met.

Conclusions

We have presented an analytical model to simulate flow through ahorizontal well with branch fractures surrounded by a nonstimu-lated zone. The following conclusions can be made: The proposed five-region model is a generalization for both the tri-

linear model (Brown et al. 2009) and the enhanced-fracture-region

y2

y No-flow

Region 3

Region 1

Region 4

Region 2

No-

flow

No-

flow

No-flow

y1 = (xf)y

pFy

qF

w/2 x1 x2x

p1 = p3p F

=

p 1

p 3 =

p 4

p 1 = p 2

k1p1y

p2x

= k2

Fig. 7Alternative five-region model: flow directions and boundary conditions.

Original

Alternative

a b

x1 = 25 ft

k2 = 0.1 md

x2 = 250 ft

(xf) y

= 10

0 ft

y 2 =

10

00 ft

k1 = 1 md

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

00 2,000 4,000 6,000 8,000 10,000

time, days

pwf,

psi

Original Five-Region Model

Alternative Five-Region Model

Numerical Simulation

Fig. 8y2x2: Model schematic (a) and comparison of the original and alternative five-region models with numerical simulation(b).


model (Stalgorova and Mattar 2012). Equations describing themodel have been solved analytically in the Laplace domain.

The flow through a horizontal well with multiple branch frac-tures can be represented by five regions of linear flow.

We compared the results obtained from the five-region modelwith the results of numerical simulation. The results showedclose agreement for typical well and reservoir configurations.

Depending on the reservoir geometry, the five-region model isnot always applicable. We suggested the alternative five-regionmodel for cases when x2y2. However, this condition does notapply for most of the practical cases of multiple-fractured hori-zontal wells.

We found that the five-region model gives accurate (close tonumerical) results when the stimulated zone is not too smallcompared with the whole reservoir (in the x- and y-direction),and the model is elongated in the y-direction. Formally, it canbe expressed as x1 0.1 x2, (xf)y 0.1 y2, and y2 2 x2. It wasshown that these conditions are met in most practical cases.

Nomenclature

B liquid formation volume factor, RB/STBc compressibility, 1/psi

CD dimensionless wellbore storagect total compressibility, 1/psi

FCD dimensionless fracture conductivityh net pay, ftk permeability, md

OriginalAlternative

k2 = 0.1 md

k1 = 1 md

x1 = 50 ft

(xf) y

= 10

0 ft

y 2 =

10

00 ft

x2 = 1000 ft

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

0

pwf,

psi

b

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000time, days

aOriginal Five-Region Model



Fig. 10x2 and y2(xf)y: Model schematic (a) and comparison of the original and alternative five-region models with numericalsimulation (b).

100000

10000

1000

100

10

1

0.1

p D

0 1 10 100 1000 10000 100000 1000000tD

pD (Five-Region Model)pD (Numerical Simulation)

x1D = 1 x2D = 5 y2D = 20 k2 = 0.01

Fig. 11Comparison of dimensionless type curves obtainedfrom the five-region model and numerical simulation.

OriginalAlternative

k2 = 0.1 md

x2 = 3000 ft

(xf) y

= 10

0 ft

k1 = 1 md y 2 =

30

0 ft

x1 = 50 ft

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

0

pwf,

psi

Original Five-Region Model



0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000time, days

a

b

Fig. 9x2y2: Model schematic (a) and comparison of the original and alternative five-region models with numerical simulation(b).


kf fracture permeability, mdki permeability of the ith region, mdLe horizontal-well length, ftnf number of fracturesp pressure, psi

pD dimensionless pressurepFD dimensionless pressure in the fracturepi initial reservoir pressure, psi

piD dimensionless pressure in the ith regionpp pseudopressure, psi2/cppwf wellbore flowing pressure, psiq well-production rate, SDB/D, Mscf/DqF production rate through one fracture, SDB/D, Mscf/Drw wellbore radius, fts Laplace-transform parametersc skin because of convergenceSw water saturation, fractiont time, hoursT reservoir temperature, RtD dimensionless timew fracture width, ft

wD dimensionless fracture widthx coordinate along the horizontal well, ftx1 distance from fracture to permeability boundary, ftx2 half-distance between fractures, ftxD dimensionless coordinate along the horizontal wellx1D dimensionless distance from fracture to permeability

boundaryx2D dimensionless half-distance between fractures

(xf)y half fracture length, fty coordinate perpendicular to the horizontal well, fty1 fracture half-length, fty2 half-distance between horizontal wells, ftyD dimensionless coordinate perpendicular to the horizontal

wellYe reservoir size in y-direction (distance between horizontal

wells), ftz gas-compressibility factorg diffusivity, ft2/hr

gFD dimensionless diffusivity in the fracturegi diffusivity in the ith region, ft2/hrgiD dimensionless diffusivity in the ith regionl viscosity, cp/ porosity, fraction/i porosity of the ith region, fraction/f fracture porosity, fraction

Acknowledgments

The authors would like to thank Karel Zaoral for his advice, valu-able and stimulating discussions, and contribution to the softwareimplementation.

References

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Reservoirs With Significant Transient Flow. Paper 2005-114 presented

at the Canadian International Petroleum Conference, Calgary, Alberta,

79 June. http:/dx.doi.org/10.2118/2005-114.

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Horizontal Wells as Dual Porosity Composite ReservoirsApplica-

tion to Tight Gas, Shale Gas and Tight Oil Cases. Paper SPE 144057

presented at the SPE Western North American Region Meeting, An-

chorage, Alaska, 711 May. http:/dx.doi.org/10.2118/144057-MS.

Brown, M., Ozkan, E., Raghavan, R. et al. 2009. Practical Solutions for

Pressure Transient Responses of Fractured Horizontal Wells in Uncon-

ventional Reservoirs. Paper SPE 125043 presented at the SPE Annual

Technical Conference and Exhibition, New Orleans, Louisiana, 47

October. http:/dx.doi.org/10.2118/125043-MS.

Dake, L.P. 1978. Fundamentals of Reservoir Engineering. Amsterdam:Elsevier.

Daneshy, A.A. 2003. Off-Balance Growth: A New Concept in Hydraulic

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Mukherjee, H. and Economides, M. 1991. A Parametric Comparison of

Horizontal and Vertical Well Performance. SPE Form Eval 6 (2):

209216. http:/dx.doi.org/10.2118/18303-PA.

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late Production of Horizontal Wells With Branch Fractures. Paper SPE

162515 presented at the SPE Canadian Unconventional Resources

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361969.

Appendix ADimensionless Variables

Dimensionless pressure:

pD k1h

141:2qFBlpi p

k1h

1422qFTpppi ppp

for liquid

for gas

8>>>: A-1

where qF q/(number of fractures) is the flow rate through onefracture and pp(p) is the real-gas pseudopressure defined by

ppp 2ppref

p

lpzp dp A-2

Dimensionless time:

tD g1txf 2yA-3

Dimensionless distances:

xD x=xf yx1D x1=xf yx2D x2=xf ywD w=xf yyD y=xf yy1D y1=xf y xf y=xf y 1y2D y2=xf y

A-4

Dimensionless diffusivities for five regions:

g1D g1=g1 1g2D g2=g1g3D g3=g1g4D g4=g1gFD gF=g1

A-5

where gF 2:637 104kF

/ctFland gi 2:637 104

ki/ctil

(i 1,2, 3, or 4 indicates region).Dimensionless fracture conductivity:

FCD kf wk1xf y

kf wDk1

A-6

Appendix BProblem Derivation and Solution

The suggested solution is based on the assumption that the flowcan be represented by a combination of flows within contiguousregions, and that the flow within each region is 1D, as shown inFig. B-1. Formally speaking, this assumption is inconsistent. Forexample, assuming 1D linear flow in Region 4 means that pres-sures at A and B in Fig. B-1 are equal. At the same time, pressure

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at C can be considered equal to pressure at B, as well as pressureat D to pressure at A (a result of pressure continuity); therefore,pressure at C should be equal to pressure at Dbut, at the sametime, we assume flow from C to D in Region 3.

Though this simplified formulation has some inconsistenciesfrom a formal point of view, it gives a solution that is close to theexact solution for the cases of interest (as was demonstrated in theApplicability of the Original Five-Region Model section); there-fore, the suggested solution can be used for practical purposes.

In the general case, for liquid pressure, distribution in eachregion is described by the diffusivity equation

r2p g @p@t

B-1

where g k/lc

; therefore, g is different for different regions.

In dimensionless terms defined in Appendix A, Eq. B-1 can berewritten as

r2piD 1giD@piD@tD

0 B-2

where index i indicates one of the regions1, 2, 3, 4, or fracture(F).

After conversion into the Laplace domain, Eq. B-2 becomes

r2piD s

giDpiD 0 B-3

Here, the bar above the pressure symbol indicates the Laplace do-main, and s is the Laplace-transform parameter with respect to tD.

Equation and Boundary Conditions for Region 4

We assume that flow in Region 4 is a 1D linear flow and p4D does

not depend on xD. In this case, r2p4D @2p4D@y2D

; so, Eq. B-3

becomes

@2p4D@y2D

sg4D

p4D 0 B-4

Boundary Condition 1: No-flow condition at the outer reser-voir boundary (y y2)

@p4D@yD

yDy2D

0 B-5


p4Dy1D p2Dy1D B-6

The general form of the solution for Eq. B-4 can be given as

p4DyD A4cosh yD y2Ds

g4D

r

B4sinh yD y2Ds

g4D

r B-7

By applying the boundary condition given by Eq. B-5, weobtain B4 0. By applying the boundary condition given by Eq.B-6, we obtain A4 p2Dy1D

cosh y1D y2Ds

g4D

r" #

Therefore, the pressure for Region 4 can be rewritten in termsof the pressure for Region 2 at the boundary between Regions 2and 4:

p4DyD p2Dy1D cosh yD y2D

s

g4D

r" #

cosh y1D y2Ds

g4D

r" # B-8

Flux between Regions 2 and 4 is proportional to

@p4D@yD

y1D

p2Dy1Ds

g4D

r tanh y1D y2D

s

g4D

r

B-9


We assume that flow in Region 3 is a 1D linear flow and p3D does

not depend on xD. In this case, r2p3D @2p3D@y2D

; so, Eq. B-3

becomes

@2p3D@y2D

sg3D

p3D 0 B-10

Boundary Condition 1: No-flow condition at the outer-reser-voir boundary (y y2)

@p3D@yD

yDy2D

0 B-11


p3Dy1D p1Dy1D B-12


p3DyD A3cosh yD y2Ds

g3D

r

B3sinh yD y2Ds

g3D

r B-13

By applying the boundary condition given by Eq. B-11, weobtain B3 0. By applying the boundary condition given byEq. B-12, we obtain A3 p1Dy1D

cosh y1D y2Ds

g3D

r" #

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Region 3 Region 4

Region 1 Region 2

A B

C D

Fracture x

y

Fig. B-1Flow represented by a combination of 1D linear flows.


Therefore, the pressure for Region 3 can be rewritten in termsof the pressure for Region 1 at the boundary between Regions 1and 3:

p3DyD p1Dy1D cosh yD y2D

s

g3D

r" #

cosh y1D y2Ds

g3D

r" # B-14

Flux between Regions 1 and 3 is proportional to

@p3D@yD

y1D

p1Dy1Ds

g3D

r tanh y1D y2D

s

g3D

r

B-15


Eq. B-3 for Region 2 is

@2p2D@x2D

@2p2D@y2D

sg2D

p2D 0 B-16

To convert Eq. B-16 into a 1D form, we integrate each of itsterms from 0 to y1D with respect to yD.

Assuming that pressure in the second region does not dependon yD, integrating the first term of Eq. B-16 givesy1D

0

@2p2D@x2D

dyD y1D @2p2D@x2D

B-17

Integrating the second term of Eq. B-16 gives

y1D0

@2p2D@y2D

dyD @p2D@yD

y1D

@p2D@yD

0

B-18

There is a no-flow boundary along the horizontal well; there-

fore,@p2D@yD

0

0:Flux across the boundary between Regions 2 and 4 is continu-

ous; therefore, k2@p2D@yD

y1D

k4 @p4D@yD

y1D

.

So, integrating the second term of Eq. B-16 gives

y1D0

@2p2D@y2D

dyD k4k2

@p4D@yD

y1D

B-19

Assuming that the pressure in Region 2 does not depend onyD, integrating the third term of Eq. B-16 givesy1D

0

s

g2Dp2DdyD y1D

s

g2Dp2D B-20

Therefore, after integration, Eq. B-16 becomes

y1D@2p2D@x2D

k4k2

@p4D@yD

y1D

y1D sg2Dp2D 0 B-21

Dividing Eq. B-21 by y1D, we obtain

@2p2D@x2D

k4k2y1D

@p4D@yD

y1D

sg2D

p2D 0 B-22

Boundary Condition 1: No-flow boundary midway betweenthe fractures

@p2D@xD

x2D

0 B-23

Boundary Condition 2: Pressure continuity between Regions1 and 2

p2Dx1D p1Dx1D B-24

Using Eq. B-9 and assuming that p2D does not depend on yD,Eq. B-22 can be rewritten as

@2p2D@x2D

c1sp2D 0 B-25

where

c1s sg2D k4

k2y1D

s

g4D

rtanh y1D y2D

s

g4D

r B-26


p2DxD A2coshxD x2Dc1s

p

B2sinhxD x2Dc1s

p B-27

By applying the boundary condition given by Eq. B-23, weobtain B2 0. By applying the boundary condition given byEq. B-24, we obtain A2 p1Dx1D

coshx1D x2Dc1s

p Therefore, pressure for Region 2 (Eq. B-27) can be rewritten

in terms of pressure for Region 1 at the boundary betweenRegions 1 and 2:

p2DxD p1Dx1D coshxD x2D

c1s

p coshx1D x2D

c1s

p B-28Flux between Regions 1 and 2 is proportional to

@p2D@xD

jx1D p1Dx1Dc1s

p tanhx1D x2D

c1s

p

B-29


Eq. B-3 for Region 1 is

@2p1D@x2D

@2p1D@y2D

sg1D

p1D 0 B-30

To convert Eq. B-30 to a 1D form, we integrate each of itsterms from 0 to y1D with respect to yD. In the same way that itwas performed for Region 2, we can obtain that after integrationEq. B-30 becomes

@2p1D@x2D

k3k1y1D

@p3D@yD

y1D

sg1D

p1D 0 B-31

Boundary Condition 1: Flux continuity between Regions 1and 2

k2l

@p2D@xD

x1D

k1l

@p1D@xD

x1D

B-32

Boundary Condition 2: Continuity between Region 1 andFracture Region

p1DwD=2 pFDwD=2 B-33

Using Eq. B-15 and assuming that p1D does not depend on yD,we can rewrite Eq. B-31 as

@2p1D@x2D

c2sp1D 0 B-34

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where

c2s sg1D k3

k1y1D

s

g3D

rtanh y1D y2D

s

g3D

r B-35


p1DxD A1coshxD x1Dc2s

p

B1sinhxD x1Dc2s

p B-36

By substituting xD x1D, we obtain that A1 p1D(x1D). If wesubstitute Eq. B-29 in the left side of the boundary conditiongiven by Eq. B-32 and use Eq. B-36 to calculate the right side ofthe boundary condition given by Eq. B-32, we can express B1 as

B1 p1Dx1D k2k1

c1sc2s

stanhx1D x2D

c1s

p p1Dx1D c3s B-37

where

c3s k2k1

c1sc2s

stanhx1D x2D

c1s

pB-38

Therefore, Eq. B-36 can be rewritten as

p1DxD p1DxD1 coshxD x1Dc2s

p

n o c3ssinhxD x1D

c2s

p

n o B-39

Applying the boundary condition given by Eq. B-33, weobtain

pFDwD=2 p1DxD1 coshwD=2 x1Dc2s

p

n o c3ssinhwD=2 x1D

c2s

p

n o B-40

Therefore,

p1DxD1 pFDwD=2

coshwD=2 x1Dc2s

p c3ssinhwD=2 x1D c2sp pFDwD=2

c4s B-41

where

c4s coshwD=2 x1Dc2s

p

c3ssinhwD=2 x1Dc2s

p B-42

So, the expression for the pressure in Region 1 (Eq. B-39) can berewritten in terms of the pressure in the fracture at the boundarybetween Region 1 and fracture:

p1DxD pFDwD=2

coshxD x1Dc2s

p c3ssinhxD x1D c2sp c4s

B-43Flux from Region 1 to the fracture is proportional to

@p1D@xD

wD=2

pFDwD=2c2s

p

sinhwD=2 x1Dc2s

p c3scoshwD=2 x1D c2sp c4s

pFDwD=2 c5s B-44

where

c5s c2s

p sinhwD=2 x1D

c2s


B-45

Equation and Boundary Conditions for FractureRegion (F)

Eq. B-3 for the Fracture Region is

@2pFD@x2D

@2pFD@y2D

sgFD

pFD 0 B-46

To convert Eq. B-46 to a 1D form, we integrate each of itsterms from 0 to wD/2 with respect to xD.

Integrating the first term of Eq. B-46 gives

wD=20

@2pFD@x2D

dxD @pFD@xD

wD=2

@pFD@xD

0

B-47

There is a no-flow boundary in the middle of the fracture (in

the y-direction); therefore,@pFD@xD

0

0:Flux across the boundary between Region 1 and the fracture is

continuous; therefore, kf@pFD@xD

wD=2

k1 @p1D@xD

wD=2

.

So, integrating the first term of Eq. B-46 gives

wD=20

@2pFD@x2D

dxD k1kf

@p1D@xD

jwD=2 B-48

Assuming that pressure in the fracture does not depend on xD,integration of the second and third terms of Eq. B-46 gives

wD=20

@2pFD@y2D

sgFD

pFD

dxD wD

2

@2pFD@y2D

sgFD

pFD

B-49Therefore, after integration, Eq. B-46 becomes

k1kf

@p1D@xD

wD=2

wD2

@2pFD@y2D

sgFD

pFD

0 B-50

Dividing Eq. B-50 by wD/2 and using the definition of FCD(Eq. A-6), we obtain

@2pFD@y2D

2FCD

@p1D@xD

wD=2

sgFD

pFD 0 B-51

Boundary Condition 1: No flow through the fracture tip(yD y1D)

@pFD@yD

y1D

0 B-52

Boundary Condition 2: Obtained by applying Darcys Lawat the wellbore (yD 0), as shown on Fig. B-2. In Field unitsDarcys Law through one half of the fracture is given by:

qF2 0:001127 kf h w

l B@p

@yB-53a

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If we convert@p

@yto

@pD@yD

by use of the chain rule and definitions

given by Eqs. A-1 and A-4 and then simplify the obtained equa-tion by use of Eq. A-6 and convert to the Laplace domain, theboundary condition becomes

@pFD@yD

0

pFCD s B-53b

Because this formulation ignores the radial convergence to-ward the horizontal well within the fracture, it will need to be cor-rected later on by incorporating a convergence skinsee thePossible Model Modifications section.

Using Eq. B-44 and assuming that pFD does not depend on xD,we can rewrite Eq. B-51 as

@2pFD@y2D

c6spFD 0 B-54

where

c6s sgFD 2

FCD c5s B-55

The general solution for Eq. B-54 is

pFDyD AFcoshyD y1Dc6s

p

BFsinhyD y1Dc6s

p B-56

Using the boundary condition given by Eq. B-52, we obtainBF 0, and using the boundary condition given by Eq. B-53b, weobtain

AF pFCD s

c6s

psinh c6sp y1D B-57

Therefore, the solution for the Fracture Region is

pFDyD pcoshyD y1D

c6s

p FCD s

c6s

psinh c6sp y1D B-58

Using that y1D 1 (Eq. A-4), we can calculate the solution atthe wellbore:

pwD pFD0 p

FCD s c6s

ptanh c6sp B-59

Calculation Summary

To summarize, wellbore pressure can be obtained with the follow-ing sequence of calculations:

c1s sg2D k4

k2y1D

s

g4D

rtanh y1D y2D

s

g4D

r

c2s sg1D k3

k1y1D

s

g3D

rtanh y1D y2D

s

g3D

r

c3s k2k1

c1sc2s

stanhx1D x2D

c1s

pc4s coshwD=2 x1D

c2s

p

c3ssinhwD=2 x1Dc2s

p

c5s c2s

p sinhwD=2 x1D

c2s


c6s sgFD 2

FCD c5s

pwD p

FCD s c6s

ptanh c6sp

As discussed in the body of the paper (the Possible ModelModifications section), modifications to this calculation procedurecan readily be made to account for the gas case, dual-porosityeffects, skin because of convergence of flow lines within the frac-ture, and wellbore storage.

Louis Mattar is with Fekete Associates, recently acquired byIHS. He specializes in well testing and production-data analy-sis, and has authored 70 technical publications. In 2003, hewas an SPE Distinguished Lecturer in well testing. In 2006, Mat-tar received the SPE International Reservoir Description andDynamics Award. In 2012 he was a Distinguished Member ofSPE. He holds MSc and PEng degrees.

Ekaterina Stalgorova is with Fekete Associates, recentlyacquired by IHS. She holds a degree in in mathematics fromMoscowState University andanMScdegree in petroleumengi-neering from the University of Alberta. Stalgorova specializes inanalytical modeling for well testing and production-data anal-ysis. Her research interests also include classical and nonclassi-cal numerical simulation.

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. . . . .qF/2 qF/2

w

h

Fig. B-2Flow within the fracture toward the wellbore.


Documents

SPE-162516-PA (Analytical Model for Unconventional Multifractured Composite Systems)