production-performance analysis of ... - swansea university

Production-Performance Analysis ofComposite Shale-Gas Reservoirs by the

Boundary-Element MethodMinglu Wu, Mingcai Ding, and Jun Yao, China University of Petroleum, QingDao;

Chenfeng Li, Swansea University; and Zhaoqin Huang and Sinan Xu, China University of Petroleum, QingDao

Summary

A shale-gas reservoir with a multiple-fractured horizontal well (MFHW) is divided into two regions: The inner region is defined asstimulated reservoir volume (SRV), which is interconnected by the fracture network after fracturing, while the outer region is calledunstimulated reservoir volume (USRV), which has not been stimulated by fracturing. Considering an arbitrary interface boundarybetween SRV and USRV, a composite model is presented for MFHWs in shale-gas reservoirs, which is based on multiple flow mecha-nisms, including adsorption/desorption, viscous flow, diffusive flow, and stress sensitivity of natural fractures. The boundary-elementmethod (BEM) is applied to solve the production of MFHWs in shale-gas reservoirs. The accuracy of this model is validated by com-paring its production solution with the result derived from an analytical method and the reservoir simulator. Furthermore, the practic-ability of this model is validated by matching the production history of the MFHW in a shale-gas reservoir. The result shows that themodel in this work is reliable and practicable. The effects of relevant parameters on production curves are analyzed, includingLangmuir volume, Langmuir pressure, hydraulic-fracture width, hydraulic-fracture permeability, natural-fracture permeability, matrixpermeability, diffusion coefficient, stress-sensitivity coefficient, and the shape of the SRV. The model presented here can be used forproduction analysis for shale-gas-reservoir development.

Introduction

Shale gas, an unconventional resource, has attracted wide attention around the world in recent years (Wu et al. 2013). Because of theextremely low permeability and porosity of shale-gas reservoirs, shale gas is difficult to be exploited from the shale formation. TheMFHW has been proved as an efficient technology for enhancing the production and recovery of tight gas/oil reservoirs in the US (Xuet al. 2015). Hydraulic fracturing not only creates several high-conductivity hydraulic fractures, but also generates an interconnectedlarge fracture network (Warpinski et al. 2009; Clarkson 2013; Xu et al. 2015; Zeng et al. 2015). The volumetric extent of the reservoirthat contains hydraulic fractures and a moderate-conductivity fracture network is defined as the SRV (Mayerhofer et al. 2010). Theremaining zone that has not been affected by hydraulic fracturing is called the USRV (Ozkan et al. 2009, 2011).

Previously, various analytical and semianalytical methods have been applied to study the pressure/production performance of theMFHW in composite reservoirs. The common methods, including Green’s function, line-source solution, and integral transformation,can solve efficiently the production/pressure solutions of the MFHW in models with circular boundaries (Wang 2014; Xu et al. 2015;Zeng et al. 2015) and the models with rectangular boundaries (Zerzar and Bettam 2003; Zerzar et al. 2004). Because the fluid-flowperiod of most reservoirs with MFHWs is concentrated mainly on the linear-flow period, many researchers presented linear models thatare used to describe the linear-flow period of the MFHW in tight gas/oil reservoirs with regular boundaries (Ozkan et al. 2011; Nobakhtand Clarkson 2012; Xu et al. 2013). Although these methods can analyze the production/pressure performance of the MFHW in reser-voirs with regular-shaped boundaries, they rarely deal with the situation with a well in reservoirs with arbitrarily shaped boundaries.

Recently, three kinds of numerical methods including the finite-element method (FEM) (Fan et al. 2015), the finite-differencemethod (FDM) (Moinfar et al. 2013), and the BEM (Kikani and Horne 1992; Zhao et al. 2016; Idorenyin and Shirif 2017) have beenused to solve the production/pressure solution of the MFHW in composite reservoirs. The BEM, which is based on the fundamentalsolution that derives from Green’s function and satisfies the governing equation, is effective in solving the production/pressure of thewell in complex-boundary reservoirs (Cheng 2003). In comparison with the FEM and the FDM, the BEM can solve more efficiently theproduction/pressure performance of the MFHW in composite reservoirs with irregular boundaries. The reason is that the BEM needsonly to discretize the boundaries into a small number of boundary elements, and solve the production/pressure solution of the boundaryelements, but FEM and FDM need to divide the reservoir into numerous grids and perform an extensive calculation to solve the pressureof all grids at each timestep.

The BEM has been successfully used to analyze the production/pressure performance of vertical wells, horizontal wells, vertical-fractured wells, and MFHWs in reservoirs with arbitrary boundaries (Kikani and Horne 1992; Sato and Horne 1993a,b; Wang andZhang 2009; Zhao et al. 2016). Zhao et al. (2016) used the BEM to analyze the pressure transient of MFHWs in tight gas reservoirswith an arbitrary outer boundary, and the solutions fit well with the solutions derived from the semianalytical method and well-testingsoftware. However, these models have not considered the SRV, which exists in most reservoirs with MFHWs. Idorenyin and Shirif(2017) applied the BEM to solve the production/pressure of multiwell reservoirs with complex boundary geometries, but the multipleflow mechanisms of the tight gas/oil were neglected.

Many types of pores are in shale-gas reservoirs (Wang and Reed 2009; He et al. 2016), and these pores have a complex structureand are multiscale. It is the special pores in shale-gas reservoirs that lead to multiple flow mechanisms, including absorption/desorption,diffusive flow, and viscous flow in natural fractures (Wang 2014; Xu et al. 2015; Zeng et al. 2015). In addition, the stress sensitivityof natural fractures has a significant effect on the productivity of a well in tight gas/oil reservoirs (Pedrosa Jr. 1986; Archer 2008;Wang 2014).

In view of the previous discussion, a model for an MFHW in a dual-region and dual-porosity composite reservoir with SRV is pre-sented. This model considered multiple flow mechanisms of shale gas, including adsorption/desorption and diffusive flow, viscous

Copyright VC 2018 Society of Petroleum Engineers

Original SPE manuscript received for review 22 May 2017. Revised manuscript received for review 15 March 2018. Paper (SPE 191362) peer approved 4 April 2018.

REE191362 DOI: 10.2118/191362-PA Date: 16-August-18 Stage: Page: 1 Total Pages: 15

ID: jaganm Time: 12:37 I Path: S:/REE#/Vol00000/180048/Comp/APPFile/SA-REE#180048

2018 SPE Reservoir Evaluation & Engineering 1

flow, and stress sensitivity of natural fractures. In addition, the arbitrarily shaped interface boundary and the finite conductivity ofhydraulic fractures were considered. Furthermore, the BEM was introduced to solve the production performance of the MFHW of thismodel. Then, the solution of this model was validated by the analytical method and the reservoir simulator; furthermore, this model wasused to match the producing history of the MFHW in a shale-gas reservoir. Finally, the sensitivity of relevant parameters was analyzed.

Physical Model

The shale-gas reservoir is divided into SRV and USRV. SRV, interconnected by the fracture network, can be simulated by a dual-porosity-medium model, and USRV, which has not been stimulated by fracturing, can be simulated by a single-porosity-medium model(Zeng et al. 2015). According to some previous studies (Mayerhofer et al. 2010; Xu et al. 2015; Idorenyin and Shirif 2017), the interfacebetween SRV and USRV is usually arbitrarily shaped in reservoirs with the MFHW. Microseismic events, which mark the locationwhere the fracture exists, can be used to determine the range of SRV (Xu et al. 2015), because microseismic data cannot differentiatebetween the natural fracture and the induced fracture, which might lead to an overestimation of the SRV. Therefore, more-advancedmethods are needed to determine the scope of the SRV.

During producing, free gas flows through natural fractures and hydraulic fractures into the wellbore first. Then, free gas in the shale-matrix pore transfer flows into natural fractures; matrix/natural-fracture transfer flow is assumed to be pseudosteady state, which can bedescribed by the Warren and Root model (Warren and Root 1963); next, gas in the USRV transports into the SRV to supply fluid.According to previous studies (Wu et al. 2013; Zeng et al. 2015), the amount of adsorption gas varies from 20 to 85% of the total natu-ral gas in the shale-gas reservoir and cannot be neglected in production-performance analysis. When the pressure of the shale formationfalls less than the adsorption pressure, adsorption gas desorbs from the surface of matrix particles, and diffuses into the matrix pore,which was presented in the model proposed by Wang (2014). However, not all shale-gas reservoirs produce through desorption mecha-nisms; when this situation arises, the desorption assumption will no longer be appropriate.

Other assumptions are presented as follows:• The shale-gas reservoirs with uniform thickness are horizontal, homogeneous, and isotropic, and their upper and lower boundaries

are impermeable. The interface between the inner and outer regions is arbitrarily shaped. The properties of shale matrix are thesame in the whole reservoir. The permeability of natural fractures in the inner region is stress-dependent. The initial pressure ofthe whole reservoir is uniform and is equal to pi.

• All induced fractures by fracturing are filled with proppant, which results in the enhanced conductivity of a fracture. When thecase arises in which the induced fracture is fully healed or nonconductive, the dual-porosity media will not be applicable.

• The horizontal well is intersected by Nf transverse hydraulic fractures, which fully penetrate the shale formation. The horizontalwell produces at a constant pressure pwf, and shale gas enters the horizontal wellbore only through hydraulic fractures from theperforation location. In the inner region, shale gas transports mainly in natural fractures, whereas matrix supplies fluid only to nat-ural fractures, and the fluid flow between the matrix pores is neglected. In the outer region, gas flows mainly in shale matrix.

• The adsorption/desorption of shale gas follows the Langmuir isotherm equation. Diffusion from the surface of the matrix particleto the matrix pores is pseudosteady and follows Fick’s law. Gas transport in matrix pores and in natural fractures follows Darcy’slaw. The gravitational and frictional effects are neglected.

Mathematical Model

The SRV can be simulated by the dual-porosity-medium model including shale matrix and natural fractures. Resembling a previousstudy (Zerzar and Bettam 2004), each hydraulic fracture is divided into nf elements. In combining the equation of motion, equation ofstate (EOS), continuity equation, and point source, the governing equations in natural fractures are expressed by Eq. 1; more details canbe found in some previous studies (Wang and Zhang 2009; Wang 2014; Xu et al. 2015),

kfie�bðwi�wf Þ r2wf þ bðrwf Þ

2h i

þ 2akmðw1m � wf Þ �2pscT

hTsc

XNf

j¼1

Xnf

i¼1

qðQi; j; tÞdðP;Qi; jÞ ¼ lCg/f

@wf

@t; ð1Þ

where wf and w1m represent pseudopressure in natural fractures and inner-region matrix, respectively, Pa�s; wi is initial pseudopressureof the reservoir, Pa�s; psc is pressure under the standard condition, Pa; T and Tsc represent temperature under the reservoir condition andthe standard condition, respectively, K; kfi represents the initial permeability of natural fractures, m2; km is the permeability of shalematrix, m2; l is shale-gas viscosity, Pa�s; Cg is gas compressibility, Pa�1; a is the shape factor of the shale-matrix element, m�2; /f isthe porosity of natural fractures, dimensionless; and b is a parameter related to permeability modulus, and is defined by Eq. 2,

b¼ lZg

2pc; ð2Þ

where c is the stress-sensitivity coefficient of natural fractures, Pa�1; Zg is the Z-factor of shale gas, dimensionless; and d() is the Diracdelta function (Wang and Zhang 2009; Zhao et al. 2016), given as

dðP;QjÞ ¼0; P 6¼ Qj

þ1 P ¼ Qj;

�� ð3Þ

which satisfies the following property,ðX

f ðPÞdðP;Qi; jÞdX ¼ f ðQi; jÞ: ð4Þ

When matrix/natural-fracture transfer flow is pseudosteady, the governing equations in the matrix system can be expressed as

�2kmaðw1m � wf Þ ¼ lCg/m

@w1m

@tþ 2pscT

Tsc

@V1

@t; ð5Þ

where V1 is the average volumetric gas concentration in the inner-region matrix, std m3/m3, and /m is the porosity of the matrixsystem, dimensionless.

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2 2018 SPE Reservoir Evaluation & Engineering

The USRV can be simulated by a single-porosity-medium model. By combining the equation of motion, EOS, and continuity equa-tion, the governing equation of the USRV can be derived, given as

r2w2m ¼/mlCg

km

@w2m

@tþ 2pscT

Tsckm

@V2

@t; ð6Þ

where w2m represents the pseudopressure in the outer-region matrix, Pa�s; V2 is the average volumetric gas concentration in the outer-region matrix, std m3/m3.

In considering the well production at a fixed pressure, the inner-boundary condition is defined as

wf jx¼xwi;y¼ywi¼ wwf i ¼ 1; 2; 3…Nf ; ð7Þ

where wwf represents the pseudopressure of the wellbore, Pa�s; (xwi, ywi) is the ith perforation location.In assuming that the outer boundary is closed, the outer-boundary condition can be expressed as

@w2m

@r

��r¼Re

¼ 0: ð8Þ

The pseudopressure and production rate of inner and outer regions are equal at the interface, given as

wf jC1¼ w2mjC1

ð9Þ

kfie�cðwi�wf Þ

l

@wf

@n

��C1

¼ � km

l@w2m

@n

��C1

: ð10Þ

The initial pseudopressure throughout the reservoir is uniform and equal to wi. Thus, the initial condition can be presented as

w1mjt¼0 ¼ w2mjt¼0 ¼ wf jt¼0 ¼ wi: ð11Þ

With the dimensionless-variables definition in Appendix A, the dimensionless forms of Eq. 1 and Eqs. 5 through 11 can be obtained. It isobvious that the model is strongly nonlinear caused by Eq. 1; thus, perturbation technology and Pedrosa’s transformation, presented in previ-ous studies (Pedrosa Jr. 1986; Wang 2014; Xu et al. 2015; Zeng et al. 2015), are applied to linearize the model. Then, by taking the Laplacetransformation with respect to dimensionless time tD, the dimensionless composite model in the Laplace domain can be obtained as

2pXNf

i¼1

Xnf

j¼1

qDðQDi; jÞdðPD;QDi; jÞ þ r2nD0 ¼ Mxf snD0 � k1ðw1mD � nD0Þ ð12Þ

�k1ðw1mD � nD0Þ ¼ sxmMw1mD þ sð1� xm � xf ÞMV1D ð13Þ

r2w2mD ¼ sxmw2mD þ sð1� xm � xf ÞV2D ð14Þ

nD0 jxD¼xwDi;yD¼ywDi¼ 1� e�cD

scD

ð15Þ

@w2mD

@rD

��rD¼ReD

¼ 0 ð16Þ

nD0 jC1¼ w2mD jC1

ð17Þ

M@nD0

@n

��C1

¼ � @w2mD

@n

��C1

: ð18Þ

According to the theory presented by Wang (2014), based on Fick’s first law and the Langmuir isotherm equation, dimensionlessgas concentration of the matrix system in the Laplace domain can be described as

VjD ¼k2r

sþ k2

wjmD ; j ¼ 1; 2: ð19Þ

By combining Eqs. 12 through 14 and Eq. 19, we can obtain the governing equations in the inner and outer regions, respectively,described as

2pXNf

i¼1

Xnf

j¼1

qDðQDi; jÞdðPD;QDi; jÞ þ r2nD0 ¼ sf1ðsÞnD0 ð20Þ

r2w2mD ¼ sf2ðsÞw2mD ; ð21Þ

where

f1ðsÞ ¼ Mxf þ k1

Mxm þMð1� xf � xmÞk2r

sþ k2

k1 þ sMxm þ sMð1� xf � xmÞk2r

sþ k2

ð22Þ

f2ðsÞ ¼ xm þ ð1� xm � xf Þk2r

sþ k2

: ð23Þ

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The BEM

Boundary Integral Equations. To derive the solution of Eq. 20, the fundamental solution G1 must be imposed, which satisfies Eq. 24(Wang and Zhang 2009; Zhao et al. 2016), given as

r2G1ðPD;RD; sÞ � u1G1ðPD;RD; sÞ þ 2pdðPD;RDÞ ¼ 0; ð24Þ

where PD and RD are arbitrary dimensionless points in the reservoir; u1 can be expressed as

u1 ¼ sf1ðsÞ; ð25Þ

where f1 (s) is defined in Eq. 22. The fundamental solution in the inner region can be expressed as Eq. 26 (Idorenyin and Shirif 2017),

G1ðPD;RD; sÞ ¼ K0½rDðPD;RDÞffiffiffiffiffiu1

p �; ð26Þ

where rD (PD, RD) is the distance between PD and RD, dimensionless; K0(x) represents the modified Bessel function of the second kind,zero order. Eq. 20 can be described as

2pXNf

i¼1

Xnf

j¼1

qDðQDi; jÞdðPD;QDi; jÞ þ r2nD0 ðPD; sÞ ¼ u1nD0ðPD; sÞ: ð27Þ

Multiply Eq. 24 by nD0 ; then Eq. 28 is obtained, given as

r2G1ðPD;RD; sÞnD0 ðPD; sÞ þ 2pdðPD;RDÞnD0ðPD; sÞ ¼ u1G1ðPD;RD; sÞnD0ðPD;RD; sÞ: ð28Þ

Multiply Eq. 27 by G1; then Eq. 29 is obtained, given as

2pXNf

i¼1

Xnf

j¼1

qDðQDi; jÞdðPD;QDi; jÞG1ðPD;RD; sÞ þ r2nD0ðPD; sÞG1ðPD;RD; sÞ ¼ u1nD0ðPD;RD; sÞG1ðPD;RD; sÞ: ð29Þ

Subtract Eq. 28 from Eq. 29, and then integrate the solution with respect to PD over the inner region that is based on the property ofthe Dirac delta function d(); then the following integral equation can be obtained:

ðX

r2nD0ðPD; sÞG1ðPD;RD; sÞ � r2G1ðPD;RD; sÞnD0ðPD; sÞ� �

dXþ 2pXNf

i¼1

Xnf

j¼1

qDðQDi; jÞG1ðRD;QDi; j; sÞ � 2pnD0ðRD; sÞ ¼ 0:

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ð30Þ

According to Green’s second theorem, the boundary integral equation of the governing equation in the inner region can be obtained:

nD0 ðRD; sÞ¼1

2p

ðC1

G1ðPD;RD; sÞ@nD0ðPD; sÞ

@n� nD0ðPD; sÞ

@G1ðPD;RD; sÞ@n

dC þXNf

i¼1

Xnf

j¼1

qDðQDi; jÞG1ðRD;QDi; j; sÞ; ð31Þ

where C1 is the interface boundary between the inner region and the outer region; n represents the normal vector of theboundary element.

As in the inner region, the boundary integral equation of the outer region can be deduced:

w2mD ðRD; sÞ¼1

2p

ðC1[C2

G2ðPD;RD; sÞ@w2mDðPD; sÞ

@n� w2mDðPDÞ

@G2ðPD;RD; sÞ@n

dC; ð32Þ

where G2 is the fundamental solution in the outer region and can be expressed as

G2ðPD;RD; sÞ ¼ K0½rDðPD;RDÞffiffiffiffiffiu2

p �; ð33Þ

where u2 can be given as

u2 ¼ sf2ðsÞ; ð34Þ

where f2 (s) is defined in Eq. 23.

The Solution of Boundary Integral Equations. Interface boundary C1 and outer boundary C2 are divided into Nb1 and Nb2 elements,respectively. Linear elements on both the interface and outer boundaries are used. Integral equations on all boundary elements can beobtained, and the details can be found in some previous studies (Wang and Zhang 2009; Zhao et al. 2016; Idorenyin and Shirif 2017).

Finite-conductivity hydraulic fractures are considered, presented in many previous studies (Zerzar and Bettam 2003; Liu et al. 2015;Xu et al. 2015); this can be expressed as

nwD0 � nD0 þp

CfD

ðxD

0

ðx0

0

qDðx00Þdx00dx0 ¼ pxD

CfDs; ð35Þ

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where nD0 can be expressed by Eq. 31; CfD denotes hydraulic-fracture conductivity, dimensionless. The integral in Eq. 35 was presentedin Eq. 36,

ðxD; j

0

ðx0

0

qDðx00Þdx00dx0 ¼Xj�1

i¼1

qD i

D x2D

2þ DxDðxD; j � iDxDÞ

� �þ Dx2

D

2qD j: ð36Þ

On the basis of the previous discussion, the production rate of all hydraulic-fracture elements can be solved; more details can befound in Zhao et al. (2016); then, the total rate of the gas well can be expressed as

QtD ¼XNf

i¼1

Xnf

j¼1

qDðQDi; jÞ; ð37Þ

where QtD is the total rate of the gas well in the Laplace domain, dimensionless. The Stehfest numerical-inversion algorithm (Stehfest1970) is applied to obtain the production solution in the real-time domain.

Discussion

Validation. Composite Model With Circular Boundaries—Validated by the Analytical Method. This case is the MFHW in a shale-gas reservoir with a circular outer boundary and an interface boundary, in which multiple flow mechanisms and the stress sensitivity ofnatural fractures are considered. The hydraulic fracture is regarded as finite conductivity. A schematic of the physical model is shownin Fig. 1. Based on the physical model, the solution solved by the BEM will be validated by comparing it with the results derived fromthe point-source method. Resembling the method proposed by Wang (2014), Xu et al. (2015), and Zeng et al. (2015), the point-sourcesolution in the inner region of the composite shale-gas reservoir with a closed outer boundary can be given by

nD0 ¼ qD ½ACI0ðg1rDÞ þ K0ðg1rDÞ� ð38Þ

AC ¼ qDg1K1ðg1rmDÞ þ g2MK0ðg1rmDÞBC

g1I1ðg1rmDÞ � g2MI0ðg1rmDÞBC

; g1¼ffiffiffiffiffiu1

p; g2¼

ffiffiffiffiffiu2

p

BC ¼K1ðg2reDÞI1ðg2rmDÞ � K1ðg2rmDÞI1ðg2reDÞK1ðg2reDÞI0ðg2rmDÞ þ K0ðg2rmDÞI1ðg2reDÞ

;

where u1 and u2 are defined in Eq. 25 and Eq. 34, respectively; K1(x) denotes the modified Bessel function of the second kind, first order.

Each hydraulic fracture is divided into nf elements; then, the production rate of each fracture element will be solved, and the solutionis expressed in Eq. 38. According to the superposition principle, the pseudopressure of the arbitrary fracture element (xDa,yDa) can beexpressed as the superposition of all fracture elements, presented as

nD0a ¼XNf

i¼1

Xnf

j¼1

qD ;i; j AcI0 g1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxD;i; j � xDaÞ2 þ ðyD;i; j � yDaÞ2

q� �� : ð39Þ

By combining Eqs. 35 through 39, the production rate of each fracture element can be obtained as well as the total productionrate of the gas well. Set basic parameters as follows: rm¼ 600 m, Re¼ 1000 m, pi¼ 30 MPa, pwf¼ 20 MPa, kfi¼ 10�5 md,k1m¼ k2m¼ 10�6 md, kh¼ 10 md, wf¼ 10�3 m, c¼ 10�9 MPa�1, Lref¼ 50 m, VL¼ 8 std m3/m3; pL¼ 8 MPa, D¼ 10�8 m2, Nf¼ 3,nf¼ 20, Nb1¼Nb2¼ 20.

As shown in Fig. 2, the line data represent the results from the BEM; the point-data represent the production rate and accumulatedproduction rate solved by the analytical method. It can be seen from Fig. 2 that the solutions from the new model fit well with the solu-tions derived from the analytical method; thus, the correctness of the new model is validated.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

y L

r = rm

re

2Lf

Fig. 1—Schematic of the MFHW in a composite reservoir.




Composite Model With Rectangular Boundaries—Validated by Commercial Software. This case is the MFHW in a shale-gas res-ervoir with rectangular boundaries; a schematic of the physical model is shown in Fig. 3. In this case, both the outer boundary and theinterface are rectangular-shaped. Therefore, it is difficult to analyze the pressure/rate performance of the MFHW by the analytical orsemianalytical method. On the basis of the physical model, the results derived from the BEM can be validated by comparing them withthe results derived from a commercial reservoir simulator (CMG 2010). The reservoir simulator is a useful and effective tool to analyzethe production performance of different kinds of reservoirs, and has been used widely around the world. As shown in Fig. 3, the struc-tured grid was used to discretize the composite model in the reservoir simulator (see Fig. 3a). Each grid in the SRV was refined into severalsubgrids. Set basic parameters as follows: reservoir area: 2,000�2000 ft, pi¼ 2,500 psi, pwf¼ 500 psi, kfi¼ 10�4 md, k1m¼ 5�10�5 md,k2m¼ 10�5 md, kh¼ 100 md, wf¼ 10�3 ft, D¼ 10�6 m2, Nf¼ 4. For the BEM, the outer boundary and interface were divided into 20 and14 linear elements, respectively (see Fig. 3b).

On the basis of the physical model and fundamental parameters, the gas rate can be derived by the reservoir simulator and the BEM,respectively. As shown in Fig. 4, the blue-line data represent the results from the BEM; the red-point data represent the gas rate solvedby the reservoir simulator. It can be seen from Fig. 4 that the solutions by the BEM fit well with the solutions derived from the reservoirsimulator; thus, the availability and correctness of the model in this work are validated.

Qsc

(10

4 m

3 /d)

0

0.4

0.8

1.2

1.6

2

Time (days)

0 2,000

Qsc Gp, by BEMGp, by analytical methodQsc

4,000 6,000 8,000 10,000

Gp

(104

m3 )

1×103

2×103

3×103

4×103

5×103

0

Fig. 2—Comparison of the analytical solution with our model with the BEM.

y (f

t)

x (ft)(a) Composite model in reservoir simulator (b) Composite model by BEM

x (ft)

0

02,0001,0000

2,0001,0000

–1,0

00

–1,0

00

500

1,000

1,500

2,000

500 1,000

USRV

SRV

ElementNode

1,500 2,000

y (f

t)

Fig. 3—Composite model with SRV in reservoir simulator and the BEM.

Qsc

(10

4 m

3 /d)

5

4

3

2

1

00 100 200 300 400 500

Gas rate of this work

Gas rate by reservoir simulator

600 700 800

Time (days)

Fig. 4—Comparison of the results of reservoir simulator with our model with the BEM.




Field-Data Match. To validate the practicality of the model in this paper, published shale-gas reservoir production-data microseis-mic events of Well-1 from Xu et al. (2015) were used, as shown in Fig. 5a; there are three stages of microseismic events around Well-1,and each stage includes approximately 300 microseismic events. According to Xu et al. (2015), the microseismic events can indicate therange of the SRV; thus, the schematic of the SRV can be derived from the microseismic map. Then, the SRV can be taken as the innerregion of the composite model (see Fig. 5b).

Based on the physical model, the model in this paper was used to match the reservoir production history and the detailed reservoirparameters of Well-1 listed in Table 1.

As shown in Fig. 6, red-line data are the results of the model in this paper, and blue-point data represent the actual production datafrom Well-1. The results of the model of this work can match well with the actual field data. They indicate that our model is reasonableand practical. Also, in this case, the interface is arbitrarily shaped, meaning that the model of this work is effective to analyze the pro-duction performance of the reservoirs with complex boundaries.

Flow-Regimes Analysis. Fig. 7a shows the microseismic map of Well-2 that indicates the range of the SRV around the horizontalwell. There are four stages of microseismic events and approximately 500 microseismic data in each stage. Fig. 7b shows a schematicof a composite model with the SRV derived from a microseismic map. Set basic parameters as follows: Re¼ 1000 m, pi¼ 30 MPa,

1200

1000

800

600

Stage 1

Stage 2

Stage 3

400

200

0

1200

1000

800

600

400

200

0

0 200 400 600 800 1000 1200X (m)

(a) Microseismic events of Well-1 (b) Composite model with arbitrary-shaped SRV

0 200 400 600 800 1000 1200 1400X (m)

Y (

m)

Y (

m)

Fig. 5—Composite model derived from the microseismic map of Well-1.

Parameters Value Parameters Value

Initial pressure, pi (MPa) 33.6 Hydraulic-fracture half-length (m) 200

Wellbore pressure, pwf (MPa) 16 Sensitivity coefficient, γ (1/MPa) 0.001

Reservoir temperature, T (K) 360 Gas-compressibility factor, Zg (dimensionless) 0.89

Production period, time (days) 600 Fracture spacing (m) 100

Porosity of fracture, φf 0.001 Reservoir thickness, h (m) 49

Permeability of matrix, km (md) 0.000083 Hydraulic-fracture number, Nf 3

Compressibility of shale, Cm (MPa–1) 0.025 Gas viscosity, µg (cp) 0.022

Table 1—Basic reservoir parameters of Well-1.

Qsc

(10

4 m

3 /d)

Time (days)0

0

1

2

3

Composite model of this work

Field data for Xu et al. (2015)4

5

100 200 300 400 500 600

Fig. 6—Comparison of the field data with our model with the BEM.




pwf¼ 20 MPa, kfi¼ 10�5 md, k1m¼ k2m¼ 10�6 md, kh¼ 10 md, wf¼ 10�3 m, c¼ 10�9 MPa�1, Lref¼ 50 m, VL¼ 8 s m3/m3;pL¼ 8 MPa, D¼ 10�8 m2, Zg¼ 0.89, l¼ 0.036 mPa�s, Nf¼ 4, Nb1¼ 18, Nb2¼ 20.

On the basis of the schematic of the physical model and basic parameters, the production rate of the MFHW in this model can besolved by the BEM. Fig. 8 shows the production-rate curves of the composite shale-gas reservoir in a log-log coordinate system.According to a previous study (Zeng et al. 2015), the derivative of the production rate was introduced to identify the flow regions.

There are mainly five flow regimes including early radial flow in the SRV: transfer flow, early pseudosteady flow, diffusive flow,and late-time pseudosteady-flow period. Regime I represents radial flow in the SRV; in this period, fluid in natural fractures flowsaround the fractured horizontal well in the form of radial flow in the SRV (Zeng et al. 2015). Regime II is transfer flow between naturalfracture and matrix, caused by pressure difference between fractures and matrix pores; the free gas of matrix pores transfers flow intothe natural fractures that can be identified by an obvious cavity in the derivative curve. Regime III represents early pseudosteady flow;after the pressure wave arrives at the interface, caused by much lower permeability in the USRV, there is not sufficient fluid to maintainpressure drop in the SRV; thus, production rate decreases rapidly and reaches pseudosteady-state flow. Regime IV represents the gas-diffusive flow period, in the process of depressurization: When the matrix pressure decreases to the desorbed pressure, adsorbed gasdesorbs from the shale-matrix particles and diffuses into matrix pores. This flow regime also can be identified by a cavity in the deriva-tive curves. Regime V represents the late-time pseudosteady-flow period; the shale-gas reservoir in this work is assumed to be a closedboundary. Thus, the late-time pseudosteady-flow period in the composite reservoir occurs. After the pressure wave arrives at the outerboundary, there is not sufficient energy to maintain the previous pressure drop; thus, the production rate decreases rapidly and reaches apseudosteady-state flow.

Sensitivity Analysis. On the basis of the physical model and fundamental parameters in the Flow-Regimes Analysis subsection, thesensitivity of nine major parameters of the shale-gas reservoir with the MFHW can be analyzed, as discussed in the next subsections.

1500

1500

2000

1000

500

0

1000

Stage 1

Stage 2

Stage 3

Stage 4

X (m)

(a) Microseismic eventsof Well-2 USRV

SRV

(b) Composite model inthis model

Y (

m)

Y (

m)

X (m)

500

00 500 1000 1500 1500 200010005000

Fig. 7—Composite model derived from the microseismic map of Well-2.

102

101

100

10–1

10–3 10–2 10–1 100

Production rate

Derivative of rate

ll

lll

lV

V

101 102 103 104

Qsc

(10

4 m

3 /d)

Time (days)

Fig. 8—Production rate and derivative curves in log-log coordinates.




Effect of Langmuir Volume. Fig. 9 shows the effect of Langmuir volume VL on the production curves of the MFHW. VL has nota-ble impact on the gas-well production rate and the accumulative production rate—with larger values of VL, more shale gas will beobtained. This is because a larger value of Langmuir volume leads to a larger adsorption index r, which means more shale gas isadsorbed on the surface of the matrix particles; thus, more adsorbed gas will desorb and diffuse into the matrix system when the pres-sure of shale formation drops. As shown in Fig. 9, the gas production with the Langmuir volume of 12 std m3/m3 is approximately1.3 times as large as the production with the Langmuir volume of 4 std m3/m3. Thus, the effect of Langmuir pressure is clear.

Effect of Langmuir Pressure. Fig. 10 shows the effect of Langmuir pressure pL on the production curves of the MFHW. It is clearthat the Langmuir pressure has significant impact, with all other parameters kept fixed, on the gas-well productivity. With larger valuesof Langmuir pressure, more shale gas will be obtained. Resembling the effect of Langmuir volume, a larger value of Langmuir pressureleads to a larger adsorption index, which means that the larger the value of Langmuir pressure is, the more adsorbed gas the shale for-mation stores and the larger the desorption rate the reservoir has. As shown in Fig. 10, the production rate with the Langmuir pressureof 12 MPa is approximately 1.7 times as large as the rate with the Langmuir pressure of 4 MPa. Thus, the effect of Langmuir pressureis remarkable.

Effect of Hydraulic-Fracture Width. Fig. 11 shows the impact of hydraulic-fracture width wf on the production curves of theMFHW. Hydraulic-fracture width has a significant effect on gas production. With the other parameters kept constant, as hydraulic-fracture width increases, the gas-production rate becomes higher; when hydraulic-fracture width exceeds 5�10�4 m, the production ofthe gas well does not change very much. The reason is that wider hydraulic fractures cause better fracture conductivity, which leads tohigher gas production; however, when hydraulic-fracture width reaches the limit, the production rate of the gas well is limited byparameters other than hydraulic-fracture conductivity.

Time (days)

Gp

(104

m3 )

Qsc

(10

4 m

3 /d)

0

0.4

0.8

1.2

1.6

2

VL = 4 std m3/m3, Qsc

Gp

Gp

Gp



0 2,000 4,000 6,000 8,000 10,000

0.6×103

1.2×103

1.8×103

2.4×103

3×103

0

Fig. 9—Production curves of the Langmuir-volume effect.

pL = 4 MPa, Qsc

pL = 8 MPa, Qsc

pL = 12 MPa, Qsc

Qsc

(10

4 m

3 /d)

0

0.4

0.8

1.2

1.6

2

Time (days)0 2,000 4,000 6,000 8,000 10,000

Gp

(104

m3 )

0.6×103

1.2×103

1.8×103

2.4×103

3×103

0

Gp

Gp

Gp

Fig. 10—Production curves of Langmuir-pressure effect.

wf = 1×10–4 m, Qsc

wf = 5×10–4 m, Qsc

wf = 1×10–3 m, Qsc

Qsc

(10

4 m

3 /d)

0

0.4

0.8

1.2

1.6

2

Time (days)

0 2,000 4,000 6,000 8,000 10,000

Gp

(104

m3 )

0.6×103

1.2×103

1.8×103

2.4×103

3×103

0

Gp

Gp

Gp

Fig. 11—Production curves of hydraulic-fracture-width effect.




Effect of Hydraulic-Fracture Permeability. Fig. 12 shows the impact of hydraulic-fracture permeability kh on the productioncurves of the MFHW. Resembling the hydraulic-fracture width, hydraulic-fracture permeability has a major effect on gas productivity:As it increases, the gas-production rate and accumulative production become higher; when hydraulic-fracture permeability exceeds10 md, the production of the gas well does not change very much. Resembling the effect of the hydraulic-fracture width, a largerhydraulic-fracture permeability causes better fracture conductivity, which leads to a higher gas-production rate; however, whenhydraulic-fracture permeability reaches the limit, the production rate of the gas well is limited by factors other than fracture conductiv-ity. Thus, production increases slowly.

Effect of Matrix Permeability. Fig. 13 shows the effect of matrix permeability km on the production curves of the MFHW. Whenmatrix permeability ranges from 5�10�7 to 5�10�6 md, the production rate of the gas well increases smoothly. A larger matrix perme-ability means that gas in the matrix system has greater capability of migrating into natural fractures. Thus, more gas flows into thewellbore. As shown in Fig. 13, when matrix permeability increases from 5�10�7 to 5�10�6 md, the accumulative gas productionincreases by 34% in 10,000 days. Thus, improving the matrix permeability can increase the gas-well productivity effectively.

Effect of Natural-Fracture Permeability. Fig. 14 shows the effect of natural-fracture permeability kfi on gas-production curves ofthe MFHW. When natural-fracture permeability ranges from 1�10�6 to 1�10�5 md, the trend of the production change is decreasing.The reason is that a larger natural-fracture permeability leads to greater flow capability of gas migrating into the wellbore; however,when the value of the natural-fracture permeability is sufficiently large, other factors will limit the productivity of the gas well signifi-cantly. As shown in Fig. 14, when natural-fracture permeability increases from 1�10�6 to 1�10�5 md, the accumulative gas productionincreases by 68% in 10,000 days. Thus, the effect of natural-fracture permeability on gas-well productivity is notable.

kh = 1 md, Qsc

kh = 10 md, Qsc

kh = 100 md, Qsc

Qsc

(10

4 m

3 /d)

0

0.4

0.8

1.2

1.6

2

Time (days)0 2,000 4,000 6,000 8,000 10,000

Gp

(104

m3 )

0.6×103

1.2×103

1.8×103

2.4×103

3×103

0

Gp

Gp

Gp

Fig. 12—Production curves of hydraulic-fracture-permeability effect.

km = 5×10–7 md, Qsc

km = 1×10–6 md, Qsc

km = 5×10–6 md, Qsc

Qsc

(10

4 m

3 /d)

0

0.4

0.8

1.2

1.6

2

Time (days)0 2,000 4,000 6,000 8,000 10,000

Gp

(104

m3 )

0.6×103

1.2×103

1.8×103

2.4×103

3×103

0

Gp

Gp

Gp

Fig. 13—Production curves of the matrix-permeability effect.

kfi = 1×10–6 md, Qsc



Qsc

(10

4 m

3 /d)

0

0.4

0.8

1.2

1.6

2

Time (days)0 2,000 4,000 6,000 8,000 10,000

Gp

(104

m3 )

0.6×103

1.2×103

1.8×103

2.4×103

3×103

0

Gp

Gp

Gp

Fig. 14—Production curves of the natural-fracture-permeability effect.




Effect of Diffusion Coefficient. Fig. 15 shows the effect of diffusion coefficient D on the production curves of the MFHW. Clearly,the diffusion coefficient has a significant impact, with other parameters kept constant, on gas-well productivity. With the increase of thediffusion coefficient, the production rate of the gas well increases significantly because a larger diffusion coefficient means that moregas diffuses into the shale matrix, and then more gas flows into the wellbore. As shown in Fig. 15, when the diffusion coefficient rangesfrom 5�10�9 m2 to 5�10�8 m2, the accumulative gas production increases by 106% in 10,000 days. Thus, a high diffusion coefficientcan improve gas-well productivity significantly.

Effect of Stress-Sensitivity Coefficient. Fig. 16 shows the effect of the stress-sensitivity coefficient of natural fracture c on the pro-duction curves of the MFHW. As the stress-sensitivity coefficient c increases, the production rate of the gas well decreases. This isbecause a larger stress-sensitivity coefficient leads to a smaller natural-fracture permeability with the formation pressure decreasing.Thus, a larger stress-sensitivity coefficient makes it difficult for shale gas in natural fractures to flow into the wellbore, and accumula-tive production decreases 37% when the stress-sensitivity coefficient ranges from 0.001 to 0.1 MPa�1. In addition, the gas production atearly time is free gas stored in natural fractures; thus, the production rate decreases quickly at early time, and the high stress-sensitivitycoefficient reduces gas-well production significantly.

Effect of the Shape of SRV. Fig. 17 shows a schematic of a composite model with a different-shaped SRV. When the product ofthe major axis and minor axis of SRV is 1.6�105, the control volume of the model is a constant. For this case, as the value of the majoraxis ranges from 400 to 800 (a¼ 400, 400

ffiffiffi2p

, 800), the value of the eccentricity of the SRV changes from zero toffiffiffiffiffi15p

/4. As shown inFig. 17, the larger the a is, the closer the horizontal well is to the interface.

Time (days)0

0

0.4

0.8

1.2

1.6

2

2,000 4,000 6,000 8,000 10,0000

1×103

2×103

Gp

(104

m3 )

Qsc

(10

4 m

3 /d)

3×103

4×103

5×103

Gp

Gp

Gp

D = 5×10–8 m2/s, Qsc

D = 1×10–8 m2/s, Qsc

D = 5×10–9 m2/s, Qsc

Fig. 15—Production curves of the diffusion-coefficient effect.

Time (days)0 2,000 4,000 6,000 8,000 10,000

0.6×103

1.2×103

1.8×103

2.4×103

3×103

0

Gp

(104

m3 )

0

0.4

0.8

1.2

1.6

2

Qsc

(10

4 m

3 /d)

Gp

Gp

Gp

γ = 0.1 MPa–1, Qsc

γ = 0.01 MPa–1, Qsc

γ = 0.001 MPa–1, Qsc

Fig. 16—Production curves of the stress-sensitivity-coefficient effect.

X (m)

Y (

m)

00

500

1000

1500

2000

a = 400 m, interface

a = 800 m, interface

Outer boundary

a = 400√2 m, interface

2500

500 1000 1500 2000 2500 3000 3500 4000

Fig. 17—Schematic of composite model with different-shaped SRV (a3b 5 1.63105 m2).




Fig. 18 shows the effect of the shape of the SRV on the production curves of the MFHW in shale-gas reservoirs. During depressuri-zation, the pressure wave will reach the closer boundary sooner immediately after the early radial-flow period. As shown in Fig. 18, thelarger the value of the major axis of the SRV, the sooner the production rate drops in the early flow period. However, the model with alarger-eccentricity SRV has a smaller swept area that relates to the cumulative production. The smaller the swept area of the reservoir,the smaller the amount of shale gas mined from formation; thus, the model with a circular SRV has the largest cumulativeproduction rate.

Conclusions

This paper introduces the BEM to solve the production of MFHWs in shale-gas reservoirs. According to the results presented previ-ously, the following conclusions are obtained:• The mathematical model in the SRV and the USRV was derived for the production performance of the MFHW in shale-gas reservoirs

with multiple flow mechanisms including desorption, diffusion, and viscous flow and stress sensitivity of natural fractures.• When analyzing the well-production/pressure performance, it is very difficult for analytical and semianalytical models to deal with

the situation of a well in a complex reservoir, such as reservoirs with impermeable zones, faults, mixed boundaries, or arbitrary-shaped boundaries, whereas the BEM can solve these problems efficiently.

• The accuracy of this model is validated by comparing its solution with the result derived from the analytical method and reservoirsimulator. Furthermore, this model can match well with the production history of the MFHW of a shale-gas reservoir. It indicates thatthe model in this work is reliable and practical.

• The sensitivity analysis of nine uncertain parameters was given previously, which offers a meaningful theoretical reference for theproduction-performance analysis of MFHWs in shale-gas reservoirs.

Nomenclature

Cg ¼ gas compressibility, Pa�1

G1, G2 ¼ fundamental solution in inner and outer regions, respectivelyh ¼ reservoir thickness, m

I0(x) ¼ modified Bessel function of first kind, zero orderkfi ¼ natural-fracture permeability at initial condition, m2

kh ¼ hydraulic-fracture permeability, mdK0(x), K1(x) ¼ modified Bessel function of second kind, zero order, and first order, respectively

Lref ¼ reference length, mM ¼ mobility ratio between inner region and outer region

Nb1, Nb2 ¼ the quantity of elements of interface C1 and outer boundary C2, respectivelyNf ¼ quantity of hydraulic fracturesnf ¼ quantity of elements of each hydraulic fracturepi ¼ pressure under the initial reservoir condition, Pa

psc ¼ pressure under the surface condition, PaP, R ¼ arbitrary point in the region

q ¼ flow rate from the point source, m3/sQi,j ¼ location of ith element of hydraulic fracture j

r ¼ radial distance, r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p, m

s ¼ variable of Laplace transformation, dimensionlesst ¼ time, seconds

V1 ¼ average gas concentration in matrix pores in SRV, std m3/m3

V2 ¼ average gas concentration in matrix pores in the USRV, std m3/m3

VE ¼ equilibrium volumetric gas concentration, std m3/m3

Vi ¼ average volumetric gas concentration in shale-matrix pores at initial condition, std m3/m3

wf ¼ hydraulic-fracture width, mx, y ¼ x- and y-coordinate, m/f ¼ porosity of natural-fracture system, dimensionlessa ¼ shape factor of shale-matrix element, m�2

/m ¼ porosity of shale-matrix system, dimensionlessd() ¼ Dirac delta function

C1, C2 ¼ interface and outer boundary, respectively

Time (days)0 2,000 4,000 6,000 8,000 10,000

0

0.4

0.8

1.2

1.6

2

Qsc

(10

4 m

3 /d)

a = 400 m, Qsc

a = 800 m, Qsc

Gp

Gp

Gp

a = 400√2 m, Qsc

1×103

0

2×103

3×103

4×103

5×103

Gp

(104

m3 )

Fig. 18—Production curves of the shape-of-the-SRV effect.




r ¼ adsorption index, dimensionlesshk ¼ a constant related to the geometrical shape at point RDk

xm, xf ¼ storativity ratio of shale matrix and natural fractures, respectively, dimensionlessK ¼ total storage capacity, Pa�1

k1 ¼ interporosity coefficient, dimensionlessk2 ¼ diffusivity coefficient, dimensionlesswi ¼ pseudopressure at initial condition, Pa�swL ¼ Langmuir pseudopressure, Pa�sl ¼ shale-gas viscosity, Pa�sc ¼ permeability modulus, Pa�1

/f ¼ porosity of natural-fracture system, dimensionless/m ¼ porosity of shale-matrix system, dimensionless

Subscripts

D ¼ dimensionless1m, 2m ¼ matrix system of inner region and outer region, respectively

f ¼ fracture system

Superscript

�¼ Laplace transform

Acknowledgments

This work was supported by the Changjiang Scholars and Innovative Research Team Development Program “Theory and Technologyin Complex Reservoir Development and EOR Theory” (IRT1294) and the National High Technology Research and Development Pro-gram (863 Program) “Key Technologies in Deep Water Oil Fields Intelligent Completion” (2013AA09A215).

References

Archer, R. A. 2008. Impact of Stress Sensitive Permeability on Production Data Analysis. Presented at the SPE Unconventional Reservoirs Conference,

Keystone, Colorado, 10–12 February. SPE-114166-MS. https://doi.org/10.2118/114166-MS.

Cheng, Y. 2003. Pressure Transient Testing and Productivity Analysis for Horizontal Wells. PhD dissertation, Texas A&M University, College Station, Texas.

Clarkson, C. R. 2013. Production Data Analysis of Unconventional Gas Wells: Review of Theory and Best Practices. International Journal of Coal Geol-ogy 109–110: 101–146. https://doi.org/10.1016/j.coal.2013.01.002.

Fan, D., Yao, J., Sun, H. et al. 2015. A Composite Model of Hydraulic Fractured Horizontal Well With Stimulated Reservoir Volume in Tight Oil & Gas

Reservoir. Journal of Natural Gas Science and Engineering 24: 115–123. https://doi.org/10.1016/j.jngse.2015.03.002.

He, J., Teng, W., Xu, J. et al. 2016. A Quadruple-Porosity Model for Shale Gas Reservoirs With Multiple Migration Mechanisms. Journal of Natural

Gas Science and Engineering 33: 918–933. https://doi.org/10.1016/j.jngse.2016.03.059.

Idorenyin, E. H. and Shirif, E. 2017. Transient Response in Arbitrary-Shaped Composite Reservoirs. SPE Res Eval & Eng. 20 (3): 752–764. SPE-

184387-PA. https://doi.org/10.2118/184387-PA.

Kikani, J. and Horne, R. N. 1992. Pressure-Transient Analysis of Arbitrarily Shaped Reservoirs With the Boundary-Element Method. SPE Form Eval7 (1): 53–60. SPE-18159-PA. https://doi.org/10.2118/18159-PA.

Liu, M., Xiao, C., Wang, Y. et al. 2015. Sensitivity Analysis of Geometry for Multi-Stage Fractured Horizontal Wells With Consideration of Finite-

Conductivity Fractures in Shale Gas Reservoirs. Journal of Natural Gas Science and Engineering 22: 182–195. https://doi.org/10.1016/

j.jngse.2014.11.027.

Mayerhofer, M. J., Lolon, E., Warpinski, N. R. et al. 2010. What Is Stimulated Reservoir Volume? SPE Prod & Oper 25 (1): 89–98. SPE-119890-PA.

https://doi.org/10.2118/119890-PA.

Moinfar, A., Varavei, A., Sepehrnoori, K. et al. 2013. Development of a Coupled Dual Continuum and Discrete Fracture Model for the Simulation of

Unconventional Reservoirs. Presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18–20 February. SPE-163647-MS.

https://doi.org/10.2118/163647-MS.

Nobakht, M. and Clarkson, C. R. 2012. A New Analytical Method for Analyzing Linear Flow in Tight/Shale Gas Reservoirs: Constant-Flowing-Pressure

Boundary Condition. SPE Res Eval & Eng 15 (1): 51–59. SPE-143990-PA. https://doi.org/10.2118/143990-PA.

Ozkan, E., Brown, M. L., Raghavan, R. S. et al. 2009. Comparison of Fractured Horizontal-Well Performance in Conventional and Unconventional

Reservoirs. Presented at the SPE Western Regional Meeting, San Jose, California, 24–26 March. SPE-121290-MS. https://doi.org/10.2118/

121290-MS.

Ozkan, E., Brown, M. L., Raghavan, R. et al. 2011. Comparison of Fractured-Horizontal-Well Performance in Tight Sand and Shale Reservoirs. SPERes Eval & Eng 14 (2): 248–259. SPE-121290-PA. https://doi.org/10.2118/121290-PA.

Pedrosa Jr., O. A. 1986. Pressure Transient Response in Stress-Sensitive Formations. Presented at the SPE California Regional Meeting, Oakland, Cali-

fornia, 2–4 April. SPE-15115-MS. https://doi.org/10.2118/15115-MS.

Sato, K. and Horne, R. N. 1993a. Perturbation Boundary Element Method for Heterogeneous Reservoirs: Part 1—Steady-State Flow Problems. SPE

Form Eval 8 (4): 306–314. SPE-25299-PA. https://doi.org/10.2118/25299-PA.

Sato, K. and Horne, R. N. 1993b. Perturbation Boundary Element Method for Heterogeneous Reservoirs: Part 2—Transient-Flow Problems. SPE FormEval 8 (4): 315–322. SPE-25300-PA. https://doi.org/10.2118/25300-PA.

Stehfest, H. 1970. Algorithm 368: Numerical Inversion of Laplace Transforms [D5]. Communications of the ACM 13 (1): 47–49. https://doi.org/10.1145/

361953.361969.

Wang, F. P. and Reed, R. M. 2009. Pore Networks and Fluid Flow in Gas Shales. Presented at the SPE Annual Technical Conference and Exhibition,

New Orleans, 4–7 October. SPE-124253-MS. https://doi.org/10.2118/124253-MS.

Wang, H. and Zhang, L. 2009. A Boundary Element Method Applied to Pressure Transient Analysis of Geometrically Complex Gas Reservoirs. Pre-

sented at the Latin American and Caribbean Petroleum Engineering Conference, Cartagena de Indias, Colombia, 31 May–3 June. SPE-122055-MS.

https://doi.org/10.2118/122055-MS.




https://doi.org/10.2118/114166-MS

https://doi.org/10.1016/j.coal.2013.01.002

https://doi.org/10.1016/j.jngse.2015.03.002


https://doi.org/10.2118/184387-PA

https://doi.org/10.2118/18159-PA



https://doi.org/10.2118/119890-PA

https://doi.org/10.2118/163647-MS

https://doi.org/10.2118/143990-PA

https://doi.org/10.2118/121290-MS

https://doi.org/10.2118/121290-MS

https://doi.org/10.2118/121290-PA

https://doi.org/10.2118/15115-MS



https://doi.org/10.1145/361953.361969

https://doi.org/10.1145/361953.361969

https://doi.org/10.2118/124253-MS

https://doi.org/10.2118/122055-MS

Wang, H.-T. 2014. Performance of Multiple Fractured Horizontal Wells in Shale Gas Reservoirs With Consideration of Multiple Mechanisms. Journalof Hydrology 510: 299–312. https://doi.org/10.1016/j.jhydrol.2013.12.019.

Warpinski, N. R., Mayerhofer, M. J., Vincent, M. C. et al. 2009. Stimulating Unconventional Reservoirs: Maximizing Network Growth While Optimiz-

ing Fracture Conductivity. J Can Pet Technol 48 (10): 39–51. SPE-114173-PA. https://doi.org/10.2118/114173-PA.

Warren, J. and Root, P. J. 1963. The Behavior of Naturally Fractured Reservoirs. SPE J. 3 (3): 245–255. SPE-426-PA. https://doi.org/10.2118/426-PA.

Wu, Y.-S., Li, N., Wang, C. et al. 2013. A Multiple-Continuum Model for Simulation of Gas Production From Shale Gas Reservoirs. Presented at the

SPE Reservoir Characterization and Simulation Conference and Exhibition, Abu Dhabi, 16–18 September. SPE-165991-MS. https://doi.org/10.2118/

165991-MS.

Xu, B., Haghighi, M., Li, X. et al. 2013. Development of New Type Curves for Production Analysis in Naturally Fractured Shale Gas/Tight Gas Reser-

voirs. Journal of Petroleum Science and Engineering 105: 107–115. https://doi.org/10.1016/j.petrol.2013.03.011.

Xu, J., Guo, C., Wei, M. et al. 2015. Production Performance Analysis for Composite Shale Gas Reservoir Considering Multiple Transport Mechanisms.

Journal of Natural Gas Science and Engineering 26: 382–395. https://doi.org/10.1016/j.jngse.2015.05.033.

Zeng, H., Fan, D., Yao, J. et al. 2015. Pressure and Rate Transient Analysis of Composite Shale Gas Reservoirs Considering Multiple Mechanisms. Jour-

nal of Natural Gas Science and Engineering. 27 (Part 2): 914–925. https://doi.org/10.1016/j.jngse.2015.09.039.

Zerzar, A. and Bettam, Y. 2003. Interpretation of Multiple Hydraulically Fractured Horizontal Wells in Closed Systems. Presented at the SPE Interna-

tional Improved Oil Recovery Conference in Asia Pacific, Kuala Lumpur, 20–21 October. SPE-84888-MS. https://doi.org/10.2118/84888-MS.

Zerzar, A., Tiab, D., and Bettam, Y. 2004. Interpretation of Multiple Hydraulically Fractured Horizontal Wells. Abu Dhabi International Conference and

Exhibition, Abu Dhabi, 10–13 October. SPE-88707-MS. https://doi.org/10.2118/88707-MS.

Zhao, Y.-l., Xie, S.-c., Peng, X.-l. et al. 2016. Transient Pressure Response of Fractured Horizontal Wells in Tight Gas Reservoirs With Arbitrary Shapes

by the Boundary Element Method. Environmental Earth Sciences 75: 1220. https://doi.org/10.1007/s12665-016-6013-7.

Appendix A: Definitions of Dimensionless Variables

The dimensionless variables are defined in the following.• Dimensionless pseudopressure,

wjD ¼ ðwi � wjÞ=ðwi � wwf Þ; j ¼ 1m; 2m; f : ðA-1Þ

• Dimensionless distance,

rD ¼ r=Lref ; xD ¼ x=Lref ; yD ¼ y=Lref ;ReD ¼ Re=Lref : ðA-2Þ

• Dimensionless gas concentration,

VD ¼ Vi � V: ðA-3Þ

• Dimensionless time,

tD ¼ kmt.

lKL2ref

�;K ¼ /f Cg þ /mCg þ 2pscT=½ðwi � wwf ÞlTsc�: ðA-4Þ

• Storativity ratio,

xf ¼ /f Cg=K;xm ¼ /mCg=K ðA-5Þ

• Dimensionless interporosity coefficient,

k1 ¼ 2akmL2ref=kfi: ðA-6Þ

• Dimensionless diffusivity coefficient,

k2 ¼ lKL2ref=ðkmsÞ; s ¼ R2

m=ð6Dp2Þ ðA-7Þ

• Dimensionless permeability modulus,

cD ¼ bðwi � wwf Þ: ðA-8Þ

• Dimensionless production rate,

qD ¼ pscqT=½pkf ihTscðwi � wwf Þ�: ðA-9Þ

• Mobility ratio between inner region and outer region,

M ¼ ðkm=lÞ=ðkfi=lÞ: ðA-10Þ

• Adsorption index,

r ¼ ðwi � wwf Þ � VLwL=ðwi þ wLÞ2: ðA-11Þ

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• Hydraulic-fracture conductivity,

CfD ¼ khwf =kmLref : ðA-12Þ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



https://doi.org/10.1016/j.jhydrol.2013.12.019

https://doi.org/10.2118/114173-PA


https://doi.org/10.2118/165991-MS

https://doi.org/10.2118/165991-MS

https://doi.org/10.1016/j.petrol.2013.03.011





https://doi.org/10.1007/s12665-016-6013-7

Minglu Wu is an associate professor in the College of Petroleum Engineering at China University of Petroleum, QingDao. Also, heis a member of the Research Center of Multiphase Flow in Porous Media. Wu has worked at this university for more than 10 years.His current interests include well testing, reservoir numerical simulation, carbon dioxide (CO2) storage, unconventional oil andgas development, hydrate mining, and history matching. Wu holds a PhD degree in petroleum engineering from the ChinaUniversity of Petroleum, and has been a year-long visiting scholar at Swansea University.

Mingcai Ding is a master’s degree student of oil and gas field development engineering at China University of Petroleum, Qing-Dao. His research interests include analysis of pressure/production performance of shale-gas reservoir and CO2 storage.

Jun Yao is a vice president at China University of Petroleum, QingDao. He also is one of the founders of the Research Center ofMultiphase Flow in Porous Media. Yao’s major interests include multiphase flow in porous media, multiscale flow, digital core,numerical well test, development of unconventional oil and gas reservoirs, development of carbonate reservoirs, and the intelli-gent oil reservoir. He holds a PhD degree in petroleum engineering from the China University of Petroleum.

Chenfeng Li is an associate professor in the College of Engineering at Swansea University. His current interests include computersimulation of deformation, fractures, and energy and mass transport; physically based modeling with visual applications; anduncertainty quantification, reliability, and risk assessment.

Zhaoqin Huang is an associate professor in the College of Petroleum Engineering at China University of Petroleum, QingDao. Healso is one of the members of Research Center of Multiphase Flow in Porous Media. Huang’s interests include oil and gas seep-age theory, reservoir numerical simulation, multifield coupling, and multiscale problems. He holds a PhD degree in petroleumengineering from the China University of Petroleum.

Sinan Xu is a master’s degree student of oil and gas engineering at China University of Petroleum, QingDao. His major researcharea is numerical simulation of depressurization mining of hydrate.




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