1Introduction fl owwithgeomechanicaleffect Theoreticalstabilityanalysisofmixed fi niteelementmodelofshale-gas

Theoretical stability analysis of mixed fi nite element model of shale-gas fl ow with geomechanical effect

Mohamed F. El-Amin^1,2,*, Jisheng Kou³, and Shuyu Sun⁴

1Energy Research Laboratory, College of Engineering, Effat University, 21478 Jeddah, Kingdom of Saudi Arabia

2Department of Mathematics, Faculty of Science, Aswan University, 81528 Aswan, Egypt

3School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000 Hubei, PR China

4Division of Physical Sciences and Engineering (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900 Jeddah, Kingdom of Saudi Arabia

Received: 6 February 2020 / Accepted: 6 April 2020

Abstract.In this work, we introduce a theoretical foundation of the stability analysis of the mixed finite element solution to the problem of shale-gas transport in fractured porous media with geomechanical effects.

The differential system was solved numerically by the Mixed Finite Element Method (MFEM). The results include seven lemmas and a theorem with rigorous mathematical proofs. The stability analysis presents the boundedness condition of the MFE solution.

Nomenclature

bd Factor of slippage k0;d Intrinsic permeability /m Porosity of the matrix blocks /f Porosity of the fractures

qm Mass density of the gas in the matrix blocks qf Mass density of the gas in the fractures qs Mass density of the core

M_w Gas molar weight P_L Langmuir pressure

p_m Gas pressure in matrix blocks p_f Gas pressure in fractures p_c Critical pressure

p_fe Average fractures pressure around the well VL Langmuir volume

Vstd Mole volume under standard conditions R General constant of gases

T Temperature

Tc Critical temperature Z Gas compressibility factor rw Well-radius

r_e Drainage-radius n Set of normal fractures rm Mean stress in matrix blocks rf Mean stress in fractures

r⁰m Effective stress in matrix blocks r⁰f Effective stress in fractures ad Biot’s effective parameter Lx Fracture spacing inx Ly Fracture spacing iny Lz Fracture spacing inz Superscripts and subscripts

f Fractures

m Matrix

d mor f

1 Introduction

Finite Element Methods (FEMs) are effective numerical techniques for solving the complex engineering problems.

FEMs appeared in the middle of the last century and were widely used in solid mechanical applications [1]. The FEM has been extensively investigated and developed in the last five decades [2], and now is used for highly nonlinear problems [3]. However, the standard FEM faced certain instabilities for solving elliptic problem that describe the flow in porous media [4,5]. On the other hand, the Mixed Finite Element Methods (MFEMs) have succeeded in eliminating such instabilities [6, 7] as it may be extended to higher-order approximations as well as it is a locally

Advanced modeling and simulation of flow in subsurface reservoirs with fractures and wells for a sustainable industry S. Sun, M.G. Edwards, F. Frank, J.F. Li, A. Salama, B. Yu (Guest editors)

* Corresponding author:momousa@effatuniversity.edu.sa

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

https://doi.org/10.2516/ogst/2020025

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conservative method [8]. Different physical variables can be treated differently in the MFEM from other variables. For example, the enhanced linear Brezzi-Douglas-Marini (BDM1) space is used for speed approximation, which requires more precision, while the piece-wise constant space is used for pressure approximation. The models of shale-gas transport were mainly developed by adapting the tradi- tional models of flow in fractured porous media. Warren and Root [9] have developed an idealized geometric model to investigate the behavioral flow in fractured porous media. The dual porosity model including stress of fracture was used to investigate theflow of gas in the shale [10,11], while, the dual-continuum model was used to describe gas flow Kerogen shale [12, 13]. Several versions of the dual- continuum model have been developed to include, for instance, adsorption, heat, deformation and Knudsen diffu- sion [14–17]. Several authors presented research work in shale gas reservoirs with geomechanical effects, such as Wanget al.[18], Linet al.[19], and Yanget al.[20]. Wang et al. [18] developed a general model including embedded discrete fractures, multiple interacting continua and geome- chanics in shale gas reservoirs with multi-scale fractures. In [19], authors provide insights on fracture propagation using reservoirflow simulation integrated with geomechanics in unconventional tight reservoirs. Effects of multicomponent adsorption and geomechanics are investigated in a shale condensate reservoir [20]. Giraultet al.[21] have introduced a priorierror estimates for a discretized poro-elastic-elastic system, in which theflow pressure equation is discretized by either a continuous Galerkin scheme or a mixed scheme, while, elastic displacement equations are discretized by a continuous Galerkin scheme. El-Aminet al.[22] have used the MFEM with stability analysis to simulate the problem of natural gas transport in a low-permeability reservoir without considering fractures. The authors [23] extended their work to cover fractured porous media and the rock stress-sensitivity with considered stability analysis of the MFEM. In this work, we present a theoretical basis with proofs of the stability analysis of the MFEM (in Ref. [23]) including the necessary lemmas and theorem.

2 Modeling and formulation

In this section, the mathematical model of the problem under consideration is developed. The Dual Porosity Dual Permeability (DPDP) model is employed to describe the gas transport in fractured porous media which consists of matrix blocks and fractures. The absorbed gas and the free gas coexist in the matrix blocks, however, the free gas exists in fractures only. The mass of free gas accumulation is /mqm, and the adsorbed gas accumulation (Langmuir isotherm model) is [24,25],

ð1/mÞqsM_wV_Lp_m

V_stdðPLþp_mÞ : ð1Þ The mass density of the gas can be written as,

qg;d¼p_dM_w

ZRT; d¼m;f: ð2Þ

The gas compressibility factor, Z which is given by Peng- Robinson equation of state [26],

Z³ ð1BÞZ²þ ðA3B³2BÞZ ðABB²B³Þ ¼0; ð3Þ

A¼ a_Tp

R²T²; B¼b_Tp

RT; ð4Þ

a_T ¼0:45724R²T²_c

p_c ; b_T ¼0:0778RTc

p_c : ð5Þ In low-permeability formations, Klinkenberg effect, which is described by the apparent permeability, takes place and written as,

k_app;d¼k_0;d 1þb_d p_d

; d¼m;f: ð6Þ

The DPDP model of gas transport in fracture shale strata is represented as,

f1ðp_mÞop_m

ot r qg;mðp_mÞ

l k_0;m 1þbm

pm

rp_m

¼ Sðp_m;p_fÞ; ð7Þ and

f₂ðp_fÞop_f

ot r q_g;fðp_fÞ

l k_0;f 1þb_f p_f

rp_f

¼Sðp_m;p_fÞ Qðp_fÞ; ð8Þ where

f₁ð Þ ¼p_m M_w

ZRT/mð Þ þp_m /⁰_mð Þpp_{m m} þM_wV_Lqs

Vstd

P_Lð1/mð Þp_m Þ PLp_m

ð Þ² /⁰_mð Þp_m PLþp_m

!

; ð9Þ

and

f₂ðp_fÞ ¼ M_w

ZRTh/fðp_fÞ þ/⁰_fðp_fÞp_fi

; ð10Þ

such that/mand/f is the matrix and fractures porosities, respectively,

/⁰dð Þ ¼p_d d/d

dp_d; d¼m;f: ð11Þ Moreover, based on the following definitions,b_mandb_f are treated as constants [17],

b_m ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi 8pRT M_w

r 1

r 2

a0:995

lg; ð12Þ and

b_f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pRT/_0;f

M_wk_0;f s

lg: ð13Þ

The Knudsen diffusion coefficient is defined as, D_kf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pRTk_0;f/_0;f

M_w s

: ð14Þ

Here,Sðp_m;p_fÞis the transfer parameter which is connect- ing of the matrix-fracture domains, and defined as,

Sðp_m;p_fÞ ¼rqg;mk_m

l p_mp_f

: ð15Þ

One may provide the definition of the source of the produc- tion well by,

Qðp_fÞ ¼hqg;fk_f lln_r^re

w

p_fep_w

ð Þ; ð16Þ

h¼ h¼2p if the production well at the center of the reservoir h¼^p₂ if the production well at the corner of the reservoir:

Given the constant, r_c, the drainage-radius is represented as,

r_e¼0:14

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxÞ²þ ðyÞ² q

: ð17Þ

The coefficient of crossflow between the matrix and fracture domains is defined as [9],

r¼2nðnþ1Þ

l² ; ð18Þ

such thatl is given by,

l¼ 3L_xL_yL_z

L_xL_yþL_yL_zþL_xL_z: ð19Þ 2.1 Geomechanical effects

Invoking the rock stress-sensitivity [27], the porosity can be expressed as a function of the mean effective-stress,

/dðr⁰dðp_dÞÞ ¼/r;dþ ð/0;d/r;dÞexpðar⁰_dÞ; d¼m;f; ð20Þ r⁰d ¼rdþadp_d; d¼m;f: ð21Þ From(20)and(21), we can obtain,

/dðp_dÞ ¼/r;dþC_1;dexpðap_dÞ;C_1;d

¼ ð/_0;d/_r;dÞexpðardÞ; d¼m;f: ð22Þ As the porosity increases the difference ð/0;d/r;dÞ becomes positive andvice versa. Also, the coefficient C_1;d becomes small based on the change in the porosity, and it becomes positive if porosity increases and negative if poros- ity decreases.

2.2 Initial and boundary conditions

The initial conditions can be represented by,

p_mð;0Þ ¼p_fð;0Þ ¼p₀ in Xm[Xf: ð23Þ The pressure on the production well (boundary condition) is given as,

p_fð Þ ¼;t p_w on C^f_Dð0;TÞ: ð24Þ The no-flow boundary conditions on matrix are given as,

u_mn¼0 on C^mN[C^fN ð0;TÞ; ð25Þ while the no-flow boundary conditions for fracture domain are,

u_f n¼0 on C^f_N ð0;TÞ: ð26Þ

3 MFEM spaces

The MFEM contains two spaces for a scalar variable and it’s flux. The MFEM was developed to approximate both variables simultaneously and to give a higher order approx- imation for the flux. A compatibility condition must hold for the two spaces to insure stability, consistency and convergence of the mixed method and by considering more constraints to the numerical discretization.

Now, let us define the inner product inXas, ðf;gÞ_X¼

Z

X

fðxÞgðxÞdV 8f;g:X!R; ð27Þ and the inner product onoXas,

hf;gi_oX¼ Z

oX

fgdS 8f;g:oX!R: ð28Þ Given,

L²ðXÞ ¼ ff :X!R: Z

X

f²dx<þ1g; ð29Þ is the largest Hilbert space, such that ðD¹u;wÞ and ðp;r wÞ are well defined, therefore, p;r w2L²ðXÞ. It is needed that p;u2L²ðXÞ and u;w2Hðdiv;XÞ. The Hilbert space with norm given by,

Hðdiv;XÞ ¼ fu:r u2L²ðXÞg; ð30Þ jjujj²_Hðdiv;XÞ¼ jjujj² þ jjr ujj²: ð31Þ The seeking solution is,

ðp;uÞ 2L²ðXÞ Hðdiv;XÞ; ð32Þ such that p and u are smooth. For the approximate numerical solution, the two spaces become,

W_hL²ðXÞ; dimðWhÞ<þ1 ð33Þ

and,

V_hHðdiv;XÞ; dimðVhÞ<þ1: ð34Þ Thus, the normal componentunis continuous across the inter-element boundaries.

Moreover, it is useful to present theRT_r elements that are designed to approximateHðdiv;XÞ[7], which satisfies,

W_h¼ w2L²ð ÞX :wj_E2P0ð Þ;E E 2Eh

; ð35Þ

and,

V_h¼ u2Hðdiv;XÞ:uj_E2P1ð Þ;E E2Eh

; ð36Þ

wherewis discontinuous piecewise constant andu is pie- cewise linear.

4 Mixed finite element approximation

LetXmandXf are, respectively, the matrix and fracture in a polygonal/polyhedral Lipschitz domain XR^d; d2 f1;2;3g(on which we define the standardL²ðXÞspace such thatL²ðXÞ L²ðXÞd

), with the boundaries,oXm¼ C^mD[C^mN and oXf ¼C^fD[C^fN. One may write the above dual-porosity model in the following general form,

f₁ð Þp_m op_m

ot þ r u_m ¼ S p_m;p_f

in Xm ð0;TÞ;

ð37Þ D_mðp_mÞ¹u_m¼ rp_m in Xm ð0;TÞ; ð38Þ

f₂ðp_fÞop_f

ot þ r u_f ¼Sðp_m;p_fÞ Qðp_fÞ in Xf ð0;TÞ;

ð39Þ D_fðp_fÞ¹u_f ¼ rp_f in Xf ð0;TÞ; ð40Þ where

D_mðp_mÞ ¼qðpmÞ

l k_0;m 1þbm

p_m

; ð41Þ

and

D_fðp_fÞ ¼qðpfÞ

l k_0;f 1þb_f p_f

: ð42Þ

The functionsD_mðp_mÞ¹andD_fðp_fÞ¹are moved to the left hand side to avoid discontinuity when we integrate rp_m andrp_f by parts. Selecting anyu2W_h andx2V_h, the mixed finite element weak formulation can be written in the following form,

f₁ðp_mÞop_m ot ;u

þ ðr u_m;uÞ þ ðSðp_m;p_fÞ;uÞ ¼0; ð43Þ

ðDmðp_mÞ¹u_m;xÞ ¼ ðpm;r xÞ; ð44Þ

f2ðp_fÞopf

ot ;u

þ ðr uf;uÞ ðSðpm;p_fÞ;uÞ ¼ ðQðpfÞ;uÞ;

ð45Þ ðDfðp_fÞ¹u_f;xÞ ¼ ðpf;r xÞ hp_w;xi_CD

f : ð46Þ

Now, let the approximating subspace duality V_hHðX;divÞ and W_hL²ðXÞ be the r-th order (r0) Raviart–Thomas space (RT_r) on the partitionTh. The mixedfinite element formulations are stated as below:

findp^h_m;p^h_f 2W_h andu^h_m;u^h_f 2V_h such that, f₁ðp^h_mÞop^h_m

ot ;u

þ ðr u^h_m;uÞ þ ðSðp^h_m;p^h_fÞ;uÞ ¼0; ð47Þ ðDmðp^h_mÞ¹u^h_m;xÞ ¼ ðp^h_m;r xÞ; ð48Þ

f₂ðp^h_fÞop^h_f ot ;u

!

þ ðr u^h_f;uÞ ðSðp^h_m;p^h_fÞ;uÞ ¼ ðQðp^h_fÞ;uÞ;

ð49Þ ðDfðp^h_fÞ¹u^h_f;xÞ ¼ ðp^h_f;r xÞ hp_w;xi_C^D

f; ð50Þ

for anyu2W_h andx2V_h.

In order to obtain an explicit formulation for theflux, we employ a quadrature rule along with the MFE method to [8]. The total time interval½0;Tis divided intoN_T time steps with length tⁿ¼t^nþ1tⁿ. The superscript nþ1 denotes for the current time step, while n denotes for the previous one. We use a backward Euler semi-implicit discre- tization for the time derivative terms. The following scheme has been developed,

f₁ðp^h_m^;nÞp^h;nþ1_m p^h;n_m t ;u

þ ðr u^h_m^;nþ1;uÞ

þðSðp^h;nþ1_m ;p^h_f^;nÞ;uÞ ¼0; ð51Þ

ðDmðp^h;n_m Þ¹u^h;nþ1_m ;xÞ ¼ ðp^h;nþ1_m ;r xÞ; ð52Þ

f₂ðp^h_f^;nÞp^h_f^;nþ1p^h_f^;n t ;u

!

þ ðr u^h_f^;nþ1;uÞ

ðSðp^hm^;nþ1;p^h_f^;nþ1Þ;uÞ ¼ ðQðp^hf^;nþ1Þ;uÞ; ð53Þ ðDfðp^h;n_f Þ¹u^h;nþ1_f ;xÞ ¼ ðp^h;nþ1_f ;r xÞ hp_w;xi_CD

f: ð54Þ Given p^h_m^;n andp^h;n_f , the numerical procedure to calculate pressure and velocity is presented here,

1. Update the thermodynamical variables explicitly.

2. Solve the equations(51)and(52)to obtainp^h_m^;nþ1and u^h;nþ1_m .

3. Solve the equations(53)and(54)to obtainp^h;nþ1_f and u^h_f^;nþ1.

5 Stability analysis

In this section, we carry out the stability analysis of the proposed MFE method, which ensures that the discrete solutions are bounded in a physically reasonable range.

The key issue encountered in the stability analysis is the nonlinearity of the matrix and fracture pressures. In order to resolve this issue, we need to define some auxiliary func- tions and analyze their boundedness. Now let us define the following functions,

F₁ðp^h_dÞ ¼ Z p^h_d

0

f₁ðp^h_dÞdp^h_d;G₁ðp^h_dÞ ¼ Z p^h_d

0

F₁ðp^h_dÞdp^h_d; ð55Þ

F₂ðp^h_dÞ ¼ Z p^h_d

0

f₂ðp^h_dÞdp^h_d;G₂ðp^h_dÞ ¼ Z p^h_d

0

F₂ðp^h_dÞdp^h_d: ð56Þ Therefore, one may

F₁ðp^h_mÞ ¼ /_r;mþC_1;mMw

ZRTp^h_me^aphm þM_wV_Lqs

V_std

C1;m

ae^aPLð1þPLÞEiðxÞj^aðP_aPL^LþphmÞ

þ P_Le^aphm P_Lþp^h_m1

þð1/r;mÞ p^h_m P_Lþp^h_m

; ð57Þ

G1ðp^h_mÞ ¼ /_r;mþC1;m

_Mw

a²ZRTh1 ðap^h_mþ1Þe^aphmi þMwVLqs

V_std

C_1;mae^aPLð1þP_LÞ ðP Lþp^h_mÞEiðxÞj^aðP_aPL^LþphmÞ 1

að1e^aphmÞ

P_Le^aPLþEiðxÞj^aðP_aPL^LþphmÞ

þð1/r;mC1;mÞp^h_mþ ð1/r;mÞPLln PL

PLþp^h_m

; ð58Þ

F₂ðp^h_fÞ ¼ /r;fþC_1;f M_w

ZRTp^h_fe^aphf; ð59Þ and

G₂ðp^h_fÞ ¼ /r;fþC_1;f M_w

a²ZRT 1 ðap^h_f þ1Þe^aphf

h i

: ð60Þ EiðxÞ is a special function called the exponent integral function. As stated above in the model formulation that the coefficient C_1;d; d¼m;f is positive in the case of increasing porosity and negative in the case of decreasing porosity.

Lemma 5.1 For the case of increasing matrix porosity, C_1;m>0, and sufficiently large p^h_m, there exists a positive constant,

c1¼ /_r;mþC1;m

C^l₄ M_w

ZRTþM_wV_Lqs

V_std

ð1/_r;mÞC^l₃; ð61Þ such that,

ðF1ðp^h_mÞ;p^h_mÞ c1jjp^h_mjj²: ð62Þ Proof. Note that the value of the exponent integral function EiðxÞis negative and (small for big values ofx), therefore the quantity, EiðxÞj^aðP_aPL^LþphmÞ has always a positive value which has lower and upper bounds, namely,

C^u_2;mEiðxÞj^aðP_aPL^LþphmÞC^l_2;m; ð63Þ where C^l_2;m;C^u_2;m>0. On the other hand, in shale reser- voir, it is well known that the initial pressure has the max- imum value,i.e.,maxðp^h_mÞ ¼P₀, while, the pressure of the well has the minimum value, i.e., minðp^h_mÞ ¼P_w. There- fore, we have,

C^u₃¼ 1

P_Lþp_w 1

P_Lþp^h_m 1

P_Lþp₀¼C^l₃; ð64Þ C^u₄ ¼e^apwe^aphm e^ap0 ¼C^l₄; ð65Þ and,

C^u₅¼ e^apw

PLþp_w e^aphm

PLþp^h_m e^ap0

PLþp₀¼C^l₅; ð66Þ where C^l₃;C^u₃;C^l₄;C^u₄;C^l₅;C^u₅0. Therefore, for the case of increasing matrix porosity, C_1;m>0, and sufficiently large p^h_m, and holding(63)–(66), the following coefficient is positive,i.e.,

ae^aPLð1þPLÞEiðxÞj^{a P}ð ^LþphmÞ

aPL þP_Le^aphm

P_Lþp^h_m1>; ð67Þ

ae^aPLð1þP_LÞC^l_2;mþP_LC^l₅1¼C_3;m >0: ð68Þ Therefore,

ðF₁ðp^h_mÞ;p^h_mÞ ð/_r;mþC_1;mÞC^l_4ZRT^Mw ðp^h_m;p^h_mÞ þM_wV_Lqs

V_std C_1;mC_3;mþ ð1/r;mÞC^l₃ðp^h_m;p^h_mÞ /_r;mþC_1;m

C^l₄ Mw

ZRTðp^h_m;p^h_mÞ þMwVLqs

Vstd

ð1/_r;mÞC^l₃ðp^h_m;p^h_mÞ

¼ /_r;mþC1;m

C^l₄ M_w

ZRTþM_wV_Lqs

V_std ð1/_r;mÞC^l₃

jjp^h_mjj²¼c1jjp^h_mjj²; ð69Þ and this completesLemma 5.1proof.

Lemma 5.2 For the case of decreasing matrix porosity, C_1;m <0, assuming,

ae^aPLð1þP_LÞEiðxÞj^{a P}ð ^LþphmÞ

aP_L þP_Le^aphm

P_Lþp^h_m1<0; ð70Þ there exists a positive constantc1, such that,

ðF1ðp^h_mÞ;p^h_mÞ c1jjp^h_mjj²: ð71Þ Proof. It is clear that, C_1;m</r;m, so, we always have, /_r;mC1;m>0. Holding the assumption (70), one can easily prove this lemma in a similar way toLemma 5.1.

Lemma 5.3 For the case of increasing matrix porosity, C_1;m >0, and for sufficiently large p^h_m, there exist two posi- tive constants,

c2¼_a2^MZRT^w /_r;mþC_1;m

1C^u_4;m

þM_wV_Lqs

V_std C1;m ae^aPLð1þPLÞPLC^u_2;mþPLe^aPLC^u_2;m

h þ 1/r;m

P_Lln P_LC^u₃i

; ð72Þ

and

c⁰2¼MwVLqs

Vstd

C_1;mae^aPLð1þPLÞC^u_2;mþ ð1/r;mC_1;mÞ

h i

; ð73Þ such that,

ðG1ðp^h_mÞ;1Þ c2jXj þc⁰₂jjp^h_mjj: ð74Þ Proof. Holding (63)–(66) with C^u_4;m¼^aphmþ1

e^aphm <1 for suffi- ciently largep^h_m, and integrate (58) over Xm for the case of increasing porosity, C_1;m>0, it is easy to prove (74), which completes the lemma proof.

Lemma 5.4 For the case of decreasing matrix porosity, C_1;m <0, with assuming that,

C_1;m ae^aPLð1þP_LÞPLC^u_2;mþP_Le^aPLC^u_2;m

þð1/r;mÞP_LlnðP_LC^u₃Þ 0; ð75Þ and

C_1;mae^aPLð1þP_LÞC^u_2;mþ ð1/r;mC_1;mÞ 0; ð76Þ one can prove,

ðG₁ðp^h_mÞ;1Þ c2jXj þc⁰2jjp^h_mjj: ð77Þ Proof. It is clear that, C_1;m</r;m, so,we always have, /_r;mC_1;m>0. Holding the assumptions (75) and (76), therefore wefind,c2>0; c⁰₂>0, and integrate(58)over Xm for the case of decreasing porosity, C1;m<0, which proves(74), and this completes the proof.

Lemma 5.5 For both cases of increasing or decreasing fracture porosity, namely, C1;f >0 or C1;f <0, and suffi- ciently large p^h_f, there exists a positive constant,

c3¼ /_r;f þC1;f

M_w

ZRTC^l₄; ð78Þ such that,

ðF2ðp^h_fÞ;p^h_fÞ c3jjp^h_fjj²: ð79Þ Proof. Again, we have,

C^u₄ ¼e^apw e^aphf e^ap0 ¼C^l₄; ð80Þ where C^l₄;C^u₄>0. Therefore, for the case of increasing fracture porosity, C_1;f >0, and sufficiently large p^h_f, Therefore,

ðF2ðp^h_fÞ;p^h_fÞ /_r;f þC_1;f M_w

ZRTC^l₄ðp^h_f;p^h_fÞ ¼c3jjp^h_fjj²: ð81Þ As stated above, for the case of the decreasing porosity, C_1;f <0, the coefficientð/_r;fþC_1;fÞremains positive, then the above inequality holds. This completes the lemma proof.

Lemma 5.6 For both cases of increasing or decreasing fracture porosity, namely, C_1;f >0 or C_1;f <0, and suffi- ciently large p^h_f, there exists a positive constant,

c4¼ Mw

a²ZRTð/_r;mþC_1;mÞð1C^u_4;mÞ; ð82Þ such that,

ðG2ðp^h_fÞ;1Þ c4jXj: ð83Þ Proof. Similar to the previous proofs.

Lemma 5.7Assuming that,

0<q_g qg q_g; 0<k_f k_f k_f;

0<l_g lg l_g; ð84Þ

then, Qðp^h_fÞis bounded.

Proof. Sincep_feis bounded byp_w andp₀, i.e.,p_w p_fe p₀, then, 0<p_fep_w p₀p_w ¼C_p, such thatC_pis positive number. Holding the conditions(84), one mayfind that,

0<k_fðp^h_fÞqgðp^h_fÞ½p_fep_w lg

k_fq_gC_p lg

¼C_Q >0: ð85Þ

Theorem 8 Holding the results in the above lemmas (5.1–5.6), namely,

F₁ p^h_m

;p^h_m

c1jjp^h_mjj²; G₁ p^h_m

;1

c2j j þX c⁰₂jjp^h_mjj;

ð86Þ

and

ðF₂ðp^h_fÞ;p^h_fÞ c3jjp^h_fjj²; ðG₂ðp^h_fÞ;1Þ c4jj1jj; ð87Þ with assuming that,

c1jjp^h_mjj²c2jXj þc⁰2jjp^h_mjj; ð88Þ and

c3jjp^h_fjj²c4jXj: ð89Þ Moreover, suppose that p_fe;p_w 2L²ð0;T;L²ðXÞÞ. Then,

jjp^h_mjj²_L1ð0;T;L²ðXÞÞþ jjD_mðp^h_mÞ¹⁼²u^h_mjj²_L2ð0;T;L²ðXÞÞ

þ jjp^h_fjj²_L1ð0;T;L²ðXÞÞþ jjDfðp^h_fÞ¹⁼²u^h_fjj²_L2ð0;T;L²ðXÞÞ

þ jjp^h_mp^h_fjj²_L2ð0;T;L²ðXÞÞ CðF1ðp₀Þ;p₀Þ þCðF2ðp₀Þ;p₀Þ þCðG1ðp₀Þ;1Þ þCðG2ðp₀Þ;1Þ þCðjjp_fejj²_L2ð0;T;L²ðXÞÞ

þ jjp_wjj²_L²_ð0;T;L²_ðXÞÞÞ: ð90Þ Proof. Letu¼p^h_min(47)andx¼u^h_min(48), and sum the two equations, we get,

f1ðp^h_mÞop^h_m ot ;p^h_m

þ jjDmðp^h_mÞ¹⁼²u^h_mjj²þ ðSðp^hm;p^h_fÞ;p^h_mÞ ¼0:

ð91Þ It follows from the definitions ofF₁ andG₁ that,

f₁ðp^h_mÞop^h_m ot ;p^h_m

¼ o

otðF1ðp^h_mÞ;p^h_mÞ F₁ðp^h_mÞ;op^h_m ot

¼ o

otðF1ðp^h_mÞ;p^h_mÞ oG₁ðp^h_mÞ ot ;1

: ð92Þ

Integrating(92)from 0 tot (0<t T) yields, c1jjp^h_mjj²ðtÞ c2jXjðtÞ c⁰₂jjp^h_mjjðtÞ

þ Z t

0

jjD_mðp^h_mÞ¹⁼²u^h_mjj²þ Z t

0

ðSðp^h_m;p^h_fÞ;p^h_mÞ ðF1ðp₀Þ;p₀Þ ðG1ðp₀Þ;1Þ: ð93Þ It is similar to obtain,

c3jjp^h_fjj²ðtÞ c4jXjðtÞ þ Z t

0

jjDfðp^h_fÞ¹⁼²u^h_fjj² þ

Z _t

0

p_w;uw

h i_C^D

f

Z _t

0

ðSðp^hm;p^h_fÞ;p^h_fÞ ðF2ðp₀Þ;p₀Þ ðG2ðp₀Þ;1Þ

Z t 0

ðQðp^h_fÞ;p^h_fÞ: ð94Þ There exists a positive constant a0 such that,

S p^h_m;p^h_f

;p^h_m

S p^h_m;p^h_f

;p^h_f

a0jjp^h_mp^h_fjj²: ð95Þ

Thus, it is obtained from,(93)–(95)that,

c1jjp^h_mjj²ðtÞ c2jXjðtÞ c⁰₂jjp^h_mjjðtÞ þ

Z t 0

jjDmðp^h_mÞ¹⁼²u^h_mjj²þc3jjp^h_fjj²ðtÞ c4jXjðtÞ þ

Z _t

0

jjDfðp^h_fÞ¹⁼²u^h_fjj²þa0

Z _t

0

jjp^h_mp^h_fjj² ðF₁ðp₀Þ;p₀Þ þ ðF₂ðp₀Þ;p₀Þ ðG₁ðp₀Þ;1Þ ðG2ðp₀Þ;1Þ þC

Z t 0

ðjjp_fejj²þ jjp_wjj²þ jjp^h_fjj²Þ:

ð96Þ Finally, (90)is obtained by Gronwall’s lemma.

Remark 5.1.In Lemma 5.6, if we do not use the coeffi- cient, C^u_4;m¼^aphmþ1

e^aphm <1, and just use the coefficient, C^u_m¼e^apw, then the coefficient of the term, ap^h_m in C^u_4;m will be moved fromc2intoc⁰₂. Thus, the definitions of both c2andc⁰₂will be slightly changed without loss of the stabi- lity concept.

6 Conclusion

In this work, stability analysis of the MFE solution of the gas transport in a low-permeability fractured reservoir with stress-sensitivity effect has been considered. The dual- porosity dual permeability model with the slippage effect and the apparent permeability have been used to describe the flow in a low-permeability fractured porous media.

Stability analysis of the MFEM is presented theoretically and numerically. We proved a seven lemmas and a theorem on the stability of the MFEM. Stability conditions are stated and estimated. Variation in the porosity is correlated to the stress-sensitivity effect and depends on the values of the corresponding physical parameters. For example, the coefficientC_1;d is relatively small depending on the varia- tion of the porosity. It has a positive value in the case of porosity increasing while it has a negative value in the case of porosity decreasing.Lemmas 5.1and5.3discuss the posi- tive case, while Lemmas 5.2 and 5.4 discuss the positive case.Lemmas 5.5and5.6discuss the two cases.

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