Research ArticleEffect of Natural Fractures on Stress Evolution ofUnconventional Reservoirs Using a Finite ElementMethod and a Fracture Continuum Method

Qiang Wang , Yongquan Hu , and Jinzhou Zhao

State Key Laboratory of Oil-Gas Reservoir Geology & Exploitation, Southwest Petroleum University, Chengdu 610500, China

Correspondence should be addressed to Jinzhou Zhao; 201811000142@stu.swpu.edu.cn

Received 23 May 2019; Revised 16 November 2019; Accepted 25 November 2019; Published 27 December 2019

Guest Editor: Frédéric Nguyen

Copyright © 2019 Qiang Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Refracturing, temporary plugging, and infilling well design play an important role in the development of reservoirs. The predictionof stress distribution can provide the basic guiding theory for the design and implementation of these techniques. In this paper, afully-coupled three-dimensional production model based on the finite element method (FEM) and fracture continuum method(FCM) for naturally fractured reservoirs is presented to study the effects of fluid consumption on the reservoir stress.Furthermore, the effects of natural fractures on the stress re-distribution and stress re-orientation are also studied. The modelalso considers the influence of natural fractures on the permeability, and the effect of the effective stress on natural fractureopenings, pore-elastic deformation, and fluid consumption. An analytical solution model and Eclipse were used for thecomparison, which verifies the accuracy of the model results. Based on two cases of one cluster of fractures and three clusters ofnon-planar fractures, the research results revealed that natural fractures have a significant influence on the surroundingdrainage, stress distribution, and stress re-orientation during the development. Under the influence of natural fractures, theproduction of the fluid along the direction of natural fractures is significantly easier, and it is highly probably that theinsufficient consumption area is perpendicular to the direction of natural fractures. Compared with the conventional model, thestress distribution in the proposed model is deflected to a certain extent under the flow mode dominated by natural fractures,which is significantly prominent in the non-planar fracture model. Due to the effect of natural cracks, the absolute values of thestress, displacement, and stress difference in this model are relatively larger than those in the conventional model. Moreover, there-orientation angles of the maximum principal stress are significantly different. After considering the natural cracks, there wasan increase in the change in re-orientation and the re-orientation range. The research findings reported in this paper can beused to predict the initiation, extension, and steering process of temporary plugging fracturing fractures and refracturingfractures in fractured reservoirs.

1. Introduction

In oil and gas exploitation, the pore pressure in reservoirsgradually decreases in accordance with the development. Atthis point, the pore pressure gradient due to the heterogene-ity of the reservoir pore pressure distribution induces achange in the initial reservoir stress distribution [1]. In con-ventional reservoir research, the reservoir is generally consid-ered as a porous medium elastomer. When evaluating thecoupling law between fluid flow and geological stress in theporous elastic model, the fluid flow is generally consideredto occur in three main directions. However, the fluid flow is

due to the pressure gradient. For reservoirs such as shale,wherein natural fractures develop and dominate the reservoirfluid flow, the fluid flow is mainly along the direction of nat-ural fractures. In this flowmode, the conventional permeabil-ity representation cannot accurately describe the flow offluid. Therefore, the relationship between the natural fracturepermeability and permeability tensor [2], the relationshipbetween the permeability and fracture width [3], and therelationship between the effective stress and natural fracturewidth were established [4]. The permeability tensor can thenbe used to describe the flow of the fluid in other directions,and the variation in the width of the natural fracture can be

HindawiGeofluidsVolume 2019, Article ID 1481960, 16 pageshttps://doi.org/10.1155/2019/1481960

used to describe the permeability evolution during produc-tion. Hence, a three-dimensional fluid-solid coupling math-ematical model in porous media was developed as anequivalent simulator for the analysis and evaluation of thestress evolution and stress steering in fractured reservoirsduring depletion.

Several studies have been conducted on the fluid-solidcoupling in porous media; however, the effect of natural frac-tures on the reservoir stress was considered in a few of thesestudies. The study on the coupling between fluid and geome-chanics in porous media can be divided into two parts. Thefirst aspect is the coupling of fluid and geomechanics duringinjection [5–10]. The other aspect is the coupling of the fluidand geological stress in the depletion [11–13]. The change inthe reservoir stress due to the fluid loss during hydraulic frac-turing is the result of the coupling between the flow and geo-mechanics, which is a process that occurs within a short-timeperiod. Ghassemi et al. established a three-dimensional fluid-solid coupling model for hydraulic fracturing and then solvedthe model using the finite element and displacement discon-tinuity methods. The distribution of the pore pressure andstress in the vicinity of the hydrofractures was obtained,and the failure of the rocks in the vicinity of the hydrofrac-tures was analyzed using the Mohr-Coulomb failure criterion[14]. Salimzadeh et al. proposed a fully coupled three-dimensional (3D) finite element model for the hydraulic frac-turing of permeable rocks and then evaluated the applicablerange of classical analytic solutions in the limit case using thismodel [15]. Gao and Ghassemi established a pore elasticmodel in the process of full 3D heterogeneous hydraulicfracturing using the coupling method of fluid flow and geo-mechanics and then solved it using the finite elementmethod. The induced stress distribution and steering aroundhydraulic fractures in heterogeneous reservoirs were con-firmed as functions of time and space [16].

In addition to changes in the reservoir stress during fluidinjection, changes were observed when the reservoir wasdepleted. Oil and gas production is a significantly long-termreservoir consumption process [17–19]. Gupta et al. evaluatedthe influence of production consumption and completionmigration on the reservoir stress, and the findings of the studyindicated that the stress is more easily diverted under low-stress difference conditions.Moreover, the fracture character-istics, pressure exhaustion, and interwell interference have asignificant influence on the infill well design and refracturingdesign in the latter stages [20]. Using a state-of-the-artcoupled symmetric Galerkin boundary-element method(SGBEM) and the finite element method (FEM)model, Rous-sel and Sharma simulated the influence of prior productionfrom the offset wells on the propagation direction of fracturesinitiated in an infill horizontal well. A stress reorientationzone was observed near the offset well after several years ofproduction, which exhibited a significant influence on thefracture extension in the infill well [11]. Safari and Ghassemiproposed a numerical evaluation model for the changes instress due to the depletion of the infill wells and then analyzedthe bending of hydraulic fractures in the disturbed stress fieldand the connectivity of the hydraulic fractures between wells[21]. The failure effect and critical parameters that determine

the fracture bending were systematically evaluated [21, 22].The results revealed that the depletion perturbed the stresstensor of the stratum around the fractured horizontal well.The perturbation stress field is related to stress/formationanisotropy, fluid fluidity, pore pressure, bottom hole workingpressure, and Biot’s constant. Sangnimnuan et al. evaluatedthe influence of the hydrofracture morphology on the stressevolution in the production process using the embedded dis-crete fracture method (EDFM) and the finite volume method(FVM) [1]. The study revealed that the fracture morphologyhas a significant influence on the stress distribution, sur-rounding drainage, and stress reorientation. Moreover, witha decrease in the stress difference, the steering is easier toachieve. The propagation path and radius of curvature of thehydrofractures can be predicted using the fracture propaga-tion model combined with the varying stress field. In thispaper, reservoir simulation was integrated with rock geome-chanics to predict the well poststimulation productivities[23]. Several numerical and field studies have been presentedto study the impact of the reservoir depletion on the pore pres-sure and the stress variation in the infill well zones [24]. Rezaeiet al. presented refracturing analysis using a 2D poroelasticplain strain displacement discontinuity model built on thework of Zhang, Ghassemi and Zhang, and Chun [25–28].Kumar and Ghassemi presented a geomechanical perspectiveto better understand the problem of “frac-hits” in horizontalwell refracturing and to design solutions for it using geome-chanics analysis and modeling [13]. The numerical analysiswas based on a fully coupled 3D model “GeoFrac3D” withthe capabilities to simulate multistage fracturing of multiplehorizontal wells.

However, for reservoirs with natural fractures, the evolu-tion of the reservoir-induced stress is influenced by the pres-sure consumption, hydrofracture morphology, and naturalfractures, among others. Moreover, the influence of naturalfractures is significant. Most researchers know that naturalfractures have a significant impact on production during oiland gas production. However, when natural fractures andfluid-solid coupling are considered at the same time, thetreatment of natural fractures becomes a difficult task. Insummary, no research was found on the effect of the naturalfracture heterogeneity on the induced stress in reservoir pro-duction. This paper summarizes and presents the results ofthe analysis conducted on the effect of natural fractures onthe stress and stress diversion under the condition of planefractures and nonplane fractures. This study was a combina-tion and expansion of the fluid flow/geomechanics couplingtheory and natural fracture heterogeneity in the pore elasticmodel. Compared with previous studies, our model is closerto the actual geological situation of the reservoir, so the resultsaremore reliable and accurate. The research findings reportedin this paper can be used to predict the initiation, extension,and steering process of temporary plugging fracturing frac-tures and refracturing fractures in fractured reservoirs.

2. Mathematic Model

2.1. Natural Fracture Continuum Method. In naturally frac-tured reservoirs such as shale and tight sandstone, the

2 Geofluids

permeability of the reservoir matrix is significantly small andit is difficult to extract the fluid based only on the matrix[23, 29]. In engineering, multicluster fracturing is generallyemployed in horizontal wells to activate reservoirs withnatural fractures. The natural fractures then become themain channel of the oil and gas production and thendominate the flow mode of the reservoir fluid. Given thatthe permeability of natural fractures is significantly greaterthan that of the matrix, the contribution of the matrixpermeability to fluid flow is neglected in this model andonly the influence of natural fractures is considered [29].The distribution angle of natural fractures developed in shalereservoirs is typically consistent, i.e., the inclination angle andapproximation angle are generally distributed within a certainrange of the main direction [30]. Therefore, it was assumedthat the approximation angle and inclination angle of all-natural fractures in the model are consistent. In this model,the fracture continuum method proposed by McKenna andReeves is used to describe the characteristics of natural frac-tures. The permeability of the element grid is characterizedby the natural fracture inclination angle, approximationangle, opening angle, and fracture spacing [31]. The advan-tage of this method is that the natural fracture grid cannotbe processed separately, and the results were obtained underthe condition that the element is sufficiently small to meetthe precision requirements [3, 32]. Based on previous research[3], the natural fracture permeabilitywas dispersed to the con-tinuous unit. The permeability tensor of natural fracturesdefines the permeability for each grid cell in the modeldomain. For one fracture set, the permeability tensor can beexpressed as follows:

k = knf

n2ð Þ2 + n3ð Þ2 −n1n2 −n3n1−n1n2 n2ð Þ2 + n3ð Þ2 −n2n3−n3n1 −n2n3 n2ð Þ2 + n3ð Þ2

2664

3775,

ð1Þ

where k is the permeability tensor; n1, n2, and n3 are the unitsnormal to the fracture plane in the x, y, and z directions,respectively; and knf is the permeability of the natural fracture.Moreover, n1, n2, and n3 can be expressed as follows:

n1 = cos ςπ

180� �

sin ξπ

180� �

,

n2 = cos ςπ

180� �

cos ξπ

180� �

,

n3 = − sin ςπ

180� �

,

ð2Þ

where ς = 90° − dip and ξ = strike − 90°. Using Equations (1)and (2), the permeability tensor can be used to represent theheterogeneity caused by natural fractures. This heterogeneityis determined by the opening, spacing, inclination, andapproach angle of natural fractures. In the model, the fluid-solid coupling of the depletion is considered and the stressdistribution of the reservoir changes with respect to time, thusleading to changes in the natural fracture permeability. The

natural fracture permeability is mainly determined by thefracture opening as follows [3]:

knf =δ3

12d , ð3Þ

where δ is the reduction of the fracture aperture (fractureclosure) and d is the fracture spacing.

To accurately describe the natural fracture permeabilityas a function of time, a coupling model of the natural fractureopening and effective stress was introduced. Bandis et al.evaluated the crack opening and effective normal stress andthen proposed a semi-logarithmic relationship [33]:

log σn = ω + χδ, ð4Þ

where σn is the normal stress on the crack surface and ω andχ represent the constants in the semilog fracture closure/-stress relationship.

The effective normal stress that acts on the crack surfacecan be expressed as follows:

σn′ =12 σH′ + σh′� �

+ 12 σH′ − σh′� �

cos 2θ, ð5Þ

where θ is the angle between the normal to the fracture andthe σH direction and σH′ and σh′, respectively, represent themaximum effective principal stress and the minimum effec-tive principal stress, MPa.

By combining Equations (4) and (5), the following wasobtained [4]:

δ = 1χlog n + 1

2n + n − 12n cos 2θ

� �+ log σH′

χ−ω

χ, ð6Þ

where n is the effective stress ratio, which can be expressed asn = σH′ /σh′.

2.2. Mechanism of Fluid-Solid Coupling Mechanics

2.2.1. Principle of Effective Stress. Rock is a porous mediumcomposed of a solid skeleton and pores. In addition, thereare fluids (oil, gas, and water) stored in the pores of the skel-eton. The fluid in the rock can bear or transfer pressure,which is defined as the pore pressure. Moreover, the stresstransmitted through the contact surface between the rockparticles is the effective stress. The Biot effective stress canbe expressed as follows [34, 35]:

σij′ = σij + αδijpf , ð7Þ

where σij′ is the effective stress, σij is the total stress, δij is theKronecker symbol, and α represents the Biot constant, whichcan be defined as follows [36]:

α = 1 − KvKs

, ð8Þ

3Geofluids

where Kv and Ks represent the volume compression modulusof the rock and the compression modulus of the solid par-ticles, respectively. Moreover, α is typically close to 1; thus,α = 1 was used.

2.2.2. Stress Balance Equation. The theory of saturated-unsaturated seepage evaluates the movement of two or morefluids through pores. This study addressed the flow of asingle-phase fluid in a porous medium. Based on the princi-ple of virtual work, it can be known that at a certain moment,the virtual work of the rock mass of a unit is equal to thevirtual work generated by the force that acts on the unit(physical force and surface force); thus,

ðVδεTdσdV −

ðVδuTdf dV −

ðSδuTdFds = 0, ð9Þ

where F is the surface force, f is the body force, and δεand δu are the virtual strain and virtual displacement,respectively.

The constitutive relationship expressed in an incrementalform is

dσ′ =Dep dε − dεlð Þ, ð10Þ

where Dep is the elastoplastic matrix and dεl is the particlecompression due to the pore fluid pressure, which onlydeforms in the positive direction without shear direction.The specific expression is as follows:

dεl = −mdpf3Ks

, ð11Þ

where m = ½1, 1, 1, 0, 0, 0�T .Formula (7) can be expressed as follows:

σ′ = σ + αmpf : ð12Þ

For α = 1, the combination of Formula (12) and the sumFormula (9) can be expressed as follows:

ðVδεTDep dε +m

dpf3Ks

� �dV −

ðVδεTmdpfdV

−ðVδuTdf dV −

ðSδuTdf dV = 0:

ð13Þ

Given that the seepage continuity equation contains atime-related term, to couple the stress with the seepage, thetime derivative of the virtual work Equation (13) is required.The specific expression is as follows:

ðVδεTDep

∂ε∂t

+m1

3Ks

∂pf∂t

� �dV −

ðVδεTm

∂pf∂t

dV

−ðVδεTm

∂pf∂t

dV −ðSδuT

∂F∂t

dS = 0:ð14Þ

2.2.3. Continuity Equation. By combining Darcy’s law withthe conservation of fluid mass, a matrix flow model can beestablished to characterize the flow of porous media. Theamount of fluid that flows into this volume in time dt shouldbe equal to the increase in its internal water storage. Throughderivation, the continuity equation of seepage flow can beobtained as follows:

ϕcf +α − ϕ

Ks

� �∂pf∂t

+ α∂ε∂t

−∇kμf

∇pf + ρfgð Þ� �

= 0, ð15Þ

where g is the acceleration vector of gravity, cf is the com-pressibility of the fluid, and ϕ is the matrix porosity.

In the process of oil and gas development, hydraulic frac-turing is generally used to generate hydrofractures to increasethe productive drainage area. As shown in the Figure 1, ahydrofracture was located at the y-z section. Hence, thecontinuity equation can be expressed as follows:

ϕcf +α − ϕ

Ks

� �∂pf∂t

+ α∂ε∂t

−∇kμf

∇pf + ρfgð Þ� �

+ qΓ = 0,

ð16Þ

where qΓ represents the interchange term between thefracture and matrix.

3. Model Calculation and Validation

3.1. Finite Element Dispersion. The shape function can bedefined as follows:

u =Nuu,ε = Bu,pf =Np�pf ,

8>><>>: ð17Þ

where B and Nu represent the shape function of the dis-placement, Np is the shape function of pore pressure, �uis the nodal displacement, and pw is the pore pressure ofthe element node.

y

x

200 m

400

m

z

50 m

Hydraulic fracture

Figure 1: Single-phase solution validation model.

4 Geofluids

By substituting Equation (17) into Equation (15), thesolid phase finite element formula can be obtained aftersimplification, as follows:

Kd�udt

+ Cd�pwdt

= dfdt

, ð18Þ

where

K =ðVBTDepBdV ,

C =ðVBT Depm

3KsNpdV −

ðVBTmNpdV ,

df =ðVNT

u df dV +ðSNT

u dtdS:

ð19Þ

In the analysis of the seepage field, there are two types ofboundary conditions. The first is the flow boundary condi-tion, and the second is the pore pressure boundary condition.The flow boundary conditions can be expressed as follows:

qΓ =kμf

∂pf∂nc

, ð20Þ

where nc is the normal unit of the flow boundary. Thisboundary condition can be used as the outer boundary con-dition. When there are production fractures in the reservoirafter fracturing, the flow boundary condition can also be con-sidered as the internal boundary condition or fractureboundary condition.

The pore pressure boundary conditions can be expressedas follows:

pf = pfb, ð21Þ

where pfb is the pore pressure value at the known boundary.The Galerkin method can be used, which can be

expressed as follows:

ðVaT �AdV +

ðSbT�BdS = 0, ð22Þ

where a and b are arbitrary functions, �A is the governingequation, and �B is the continuity equation through theboundary.

By substituting Equation (15) as �A, Equation (20) as �B,and the formal function expression (17) into Equation (22)and by setting a = −b, after the simplification, the followingcan be obtained:

Ed�udt

+ F�pw +Gd�pwdt

= f̂ , ð23Þ

where

E =ðVNT

p αBdV ,

F =ðV

∇Np� T k

μf∇NpdV ,

G =ðVNT

p ϕcf +α − ϕ

Ks

� �NpdV ,

f̂ =ðsNT

p qΓdS −ðV

∇Np� T k

μfρfgdV :

ð24Þ

Table 1: Primary parameters of calculation.

Parameter Value Unit

Young’s modulus 35.5 GPa

Poisson’s ratio 0.2 —

Biot coefficient 1 —

Initial porosity 0.25 —

Fluid compressibility 8 × 10−4 MPa-1

Fluid viscosity 1.1 MPa s

The principal stress in x-direction 50 MPa

The principal stress in y-direction 54 MPa

The principal stress in z-direction 57 MPa

The initial pore pressure 30 MPa

The initial opening of natural fracture 0.2 mm

Average natural fracture spacing 20 m

Natural fracture dip 80 °

Natural fracture strike 40 °

Well radius 0.0762 m

0

50

100

150

200

250

0

1

2

3

4

5

6

0 100 200 300

Cum

ulat

ive o

il pr

oduc

tion

(t)

Dai

ly o

il pr

oduc

tion

(t)

Time (d)

Our modelEclipse

Figure 2: Flow rate comparison between the proposed model andEclipse for one year of production.

5Geofluids

By combining Equation (18) with Equation (23), thegeomechanic-seepage coupling equation can be expressedas follows:

K C

E G

" #ddt

�u

�pw

( )+

0 00 F

" #�u

�pw

( )=

dfdt

f̂

8><>:

9>=>;: ð25Þ

3.2. Model Validation

3.2.1. Single-Phase Flow Solver. To verify the oil-water flowmodel, a single fracture seepage model was established (asshown in Figure 1) and the same model was established byEclipse software for a simulation comparison. The perme-ability, porosity, and reservoir thickness of the model were5md, 0.1, and 50m, respectively. The boundary of the modelwas closed; thus, the reserves in the model were constant. Theother main parameters were the same as those shown inTable 1. In the proposed model, the flow model was estab-lished without considering natural fractures. Fluid-solid cou-pling is not considered in Eclipse software. The daily outputand cumulative oil production after one year of productionwere obtained by adopting the method of constant pressureproduction, as shown in Figure 2. The results of the modelwere approximately consistent with those of Eclipse. Due tothe different factors considered by the models, the changeprocess was slightly different. Therefore, the model in thispaper demonstrated a high accuracy.

4. Case Studies

4.1. Basic Parameters. As discussed in this section, a three-dimensional reservoir physical model was established to eval-uate the influence of anisotropy due to natural fractures onthe reservoir stress during the oil and gas development. Tofully demonstrate the significant influence of natural frac-tures on the generation of induced stress, a model was estab-lished that does not consider the influence of kxz , kyz , and kxy.However, the influence of the permeability in the main direc-tion was considered for comparison. As shown in Figure 3,the length, width, and height of the model were 400m,400m, and 100m, respectively, and the hydraulic fracture

was located at the cross section of the y-z-axis. The domainwas discretized to 50 four-node cells in the x- and y-direc-tions and 20 four-node cells in the z-direction. For the treat-ment of the reservoir boundary, the nodes on the left andright surfaces of the model were free to move along thex-axis and not the y-axis. All the nodes at the top andbottom outer surfaces were free to move along the y-axisand not the x-axis. Only the vertical displacement of all thenodes on the outer surface of the top and bottom could beachieved. Moreover, the six boundaries were nonflow bound-aries, i.e., the pore pressure in the reservoir was continuouslyreduced in the process of production. In all the calculationsbelow, the downhole flow velocity was calculated using thePeaceman equation [37]. The other parameters in the calcu-lation are shown in Table 1. Figures 4(a) and 4(b) present thelocations of the single fractures and three clusters of nonpla-nar fractures, respectively.

4.2. Single Fracture Study. Based on the model and physicalparameters established in Section 4.1, in this section, thefocus is on the influence of natural fractures on the stress dis-tribution and steering in the production process of a singlefracture in the y-z section. To fully reflect the effect of naturalfractures on the stress, a conventional model was establishedwithout considering the effect of natural fractures, for

y(m)z-x section

y-z section

(a) (b)

x(m)

z

0400

400

100200

200

x-y section

Hydraulic fracture

Hydraulic fracture

50

Figure 3: A schematic diagram of the 3D physical model.

Path B

Path Ay

x

(a) (b)

Figure 4: A schematic diagram of the hydrofracture layout ((a)presents the planar single fracture, and (b) presents the nonplanarmultiple fractures).

6 Geofluids

comparison. Compared with the boundary conditions in themodel, the calculated parameters were consistent with theproposed model.

During the production, the pore pressure decreases con-tinuously due to the continuous fluid extraction and the pres-sure is the lowest near the fracture. Figure 5 compares thedistribution of the reservoir pore pressure after 5 years ofproduction with and without considering the influence ofnatural fractures. As can be seen from the figure, due to theinfluence of natural fractures, there was a change in the fluidflow pattern. Compared with the conventional model, thepore pressure dissipation area was greater and it was easierto reduce the pore pressure along the direction of naturalfractures for the formation of a clockwise deflecting pressuredrop funnel. Therefore, under the natural fracture distribu-tion condition, the fluid in the upper right and lower left cor-ners could be easily extracted, whereas the fluid in the upperleft and lower right corners could not be easily extracted. Thiscan serve as an essential guide to the design of infill wells.

According to the principle of effective stress, the reservoirstress changes under the influence of the constant dissipationof the pore pressure. For the conventional model, the distri-bution of the stress σxx, σyy , and σzz in the reservoir after 5years of production is shown in Figures 6(a), 6(b), and 6(c).Due to fluid consumption, the reservoir stress distribution

exhibited a high complexity. Moreover, there was a decreasein σxx near the crack as a result of the pressure dissipation,whereas σxx increased on the left and right parts of thedomain to support the pressure depletion in the x-direc-tion. In addition, there was a decrease in σyy near the crackas a result of the pressure dissipation, whereas σyy increasedat the top and bottom parts of the domain to support thepressure depletion in the y-direction. Moreover, σzz mainlyexhibited a decrease in stress in the central region of thedomain. Considering the influence of natural fractures onthe stress evolution, the results are shown in Figures 7(a),7(b), and 7(c). Compared with the conventional model, thestress distribution of the proposed model is similar to thatof the conventional model. However, considering the influ-ence of tensors, the pressure dissipation of the proposedmodel was more rapid and the changes in stress weregreater. The most significant difference is that the distribu-tions of σxx, σyy, and σzz exhibited a degree of deflection.The stress distribution deflection was found to be significantlydifferent from the stress distribution near the fractures in theconventionalmodel, and this difference has a significant influ-ence on the fracture initiation direction and fracture extensionof refracturing. As can be seen in Figures 8(a) and 8(b), thedistribution of the shear stress on the x-y section exhibited aclockwise deflection when compared with the conventional

0 400200100 3000

400

200

100

300

y

x

x-y section

15.00

17.50

20.00

22.50

25.00

27.50

30.00Por (MPa)

(a)

0 400200100 30015.00

17.50

20.00

22.50

25.00

27.50

30.00Por (MPa)y

x

x-y section

0

400

200

100

300

(b)

Figure 5: Panels (a) and (b), respectively, represent the pore pressure distribution without the influence of the natural fractures and the porepressure under the influence of the natural fractures.

0 400200100 3000

400

200

300

100

y

x

x-y section

45.00

46.83

48.67

50.50

52.33

54.17

56.00MPa

(a)

0 400200100 3000

400

200

300

100

y

x

x-y section MPa

43.00

45.17

47.33

49.50

51.57

53.83

56.00

(b)

0 400200100 3000

400

200

300

100

x-y section

49.00

51.17

53.33

55.50

57.67

59.83

62.00MPay

x

(c)

Figure 6: (a), (b), and (c) present the principal stress in the x-direction, y-direction, and z-direction without considering the influence of thepermeability tensor of natural fractures.

7Geofluids

model, due to the influence of natural fractures. From acomparison between Figures 9(c) and 9(d) and Figures 9(a)and 9(b), the distribution of the displacement was found tobe deflected to a certain extent under the influence of naturalfractures. Figures 10(a) and 10(b) present the displacementprofiles on Paths A and B, respectively. In all the figures, thesolid lines represent the proposed model and the dotted linesrepresent the conventional model. The displacement trend ofthe two models was similar. The main difference was thatthere was a more significant change in the absolute valueof the displacement after the influence of natural fractureswas considered.

Figures 11(a), 11(b), and 11(c) present the σxx , σyy, andσzx stress distributions of the proposed model and conven-tional model after 5 years of production on the z-x section,respectively. As can be seen from the figure, the stress distri-bution of the conventional model exhibited a symmetricaldistribution. Compared with the conventional model, theoverall trend of the stress distribution in the proposed modelwas in good agreement. However, the stress distribution wassymmetrically distributed under the influence of natural frac-tures, and there was a relative increase in the variation rangeof the stress value.

Significant attention has been directed toward the reser-voir stress evolution and surrounding drainage in the processof fluid recovery, given their significance for the latter refrac-

turing and completion of infill wells. The change in thedirection of the horizontal principal stress direction has a sig-nificant influence on the design of refracturing. Due to theheterogeneity of the permeability and natural fractures, theheterogeneity consumption of the pore pressure may resultin stress reorientation and changes in the direction of thehorizontal maximum principal stress. As can be seen fromFigures 12(a) and 12(b), a region with a stress difference lessthan 0 was observed near the fracture after one year of pro-duction. In addition, the stress in this region was reversed.It can also be found that the time when the stress differencechanges the most is not the 20 years with the longest produc-tion life but the 10th year. This is because with an increase inthe production life, most of the fluid in the easily flowingregion was produced, whereas the fluid in the previously rel-atively noneasily flowing region was gradually produced.Therefore, there was a decrease in the difference betweenthe pore pressures in the regions, which led to a decrease inthe stress differences and an increase in the influence range.Figures 12(c) and 12(d) present the shear stress profiles onPaths A and B. Compared with the conventional model, theshear stress absolute value of the proposed model is influ-enced more significantly by natural cracks.

Refracturing is one of the most effective methods for lat-ter reservoir development. The distribution of the reservoirstress after the production and the direction of the horizontal

–1.50

–1.00

–0.50

0.00

0.50

1.00

1.50MPay

x0 400200100 300

x-y section

0

400

200

300

100

(a)

–1.80

–1.20

–0.60

0.00

0.60

1.20

1.80MPay

x0 400200100 300

x-y section

0

400

200

300

100

(b)

Figure 8: (a) and (b) present the induced shear stress in the x-y section with and without considering natural fractures, respectively.

44.00

46.25

48.50

50.75

53.00

55.25

57.50MPay

x

0 400200100 300

x-y section

0

400

200

300

100

(a)

MPay

x0 400200100 300

x-y section

42.00

44.58

47.17

49.75

52.33

59.91

57.50

0

400

200

300

100

(b)

MPay

x

0 400200100 300

x-y section

47.00

49.67

52.33

55.00

57.67

60.33

63.00

0

400

200

300

100

(c)

Figure 7: (a), (b), and (c) present the principal stress in the x-direction, y-direction, and z-direction considering the influence of thepermeability tensor of natural fractures.

8 Geofluids

maximum principal stress are two critical factors that deter-mine the fracture pressure, fracture initiation direction, andfracture extension trajectory of refracturing. The stress evolu-tion in the production process was previously analyzed. Forthe solution of the change in reorientation of the maximumhorizontal principal stress (σH max) on the x-y section, theaction of the vertical principal stress (σzz) was neglected;thus, the original direction of the maximum horizontal prin-cipal stress is shown in Figure 13(a). After 5 and 10 years ofproduction, there was a significant change in the direction

of σH max. In the conventional model, the change area ofthe principal stress direction was in the shape of a verticalellipse, whereas the change area of the σH max direction inthe proposed model was in the shape of an inclined ellipse.The degree of inclination is related to the location of naturalfractures. The area with the largest reorientation change ofσH max was near the hydrofracture. Compared with the con-ventional model, the change in reorientation of σH max inthe proposed model was influenced more significantly bynatural fractures (θ1′ > θ1, θ2′ > θ2). In the conventional

–0.15

–0.1

–0.05

0

0.05

0.1

0.15

0 100 200 300 400

Disp

lace

men

t (m

)

x (m)

1 year 10 years20 years 1 year10 years 20 years

(a)

–0.2

–0.15

–0.1

–0.05

0

0.05

0.1

0.15

0.2

0 100 200 300 400

Disp

lace

men

t (m

)

y (m)

1 year 10 years20 years 1 year10 years 20 years

(b)

Figure 10: (a) and (b) present the displacement profiles on Paths A and B.

–12.00

–8.00

–4.00

0.00

4.00

8.00

12.00y

x0 400200100 300

0

400

200

300

100

x-y section m⨯10–3

(a)

–13.50

–9.00

–4.50

0.00

4.50

9.00

13.50y

x0 400200100 300

0

400

200

300

100

x-y section m⨯10–3

(b)

–13.00

–8.67

–4.33

0.00

4.33

8.67

13.00y

x

m⨯10–3

0 400200100 3000

400

200

300

100

x-y section

(c)

–14.60

–9.74

–4.87

0.00

4.87

9.74

14.60m⨯10–3y

x0 400200100 300

0

400

200

300

100

x-y section

(d)

Figure 9: (a) and (b) present ux and uy on the x-y section without considering natural fractures, respectively, and (c) and (d) present ux and uyon the x-y section considering natural fractures, respectively.

9Geofluids

model, the change in reorientation of σH max was 90°

(Figure 14(a)); however, the change in reorientation ofσH max in the proposed model was 110° under the influenceof natural cracks (Figure 14(b)). Similar to the change inthe stress difference, the change in the reorientation wasthe largest at the 10th year, not the 20th year. Moreover,the range of the reorientation areas is proportional toproduction time.

4.3. Nonplanar Multifracture Study. In the actual segmentedmulticluster fracturing process, there is stress interferencebetween the fractures when multiple fractures are simulta-neously extended, which leads to the case wherein the middlefractures are restrained and the fractures on both sides areextended in a nonplanar manner. Hence, the displacementdiscontinuity method was used to simulate the extended tra-jectories of three clusters of hydrofractures with spacings of

–6–4–2

02468

10

0 100 200 300 400σ yy–

σ xx/

MPa

x (m)

1 year 10 years20 years 1 year10 years 20 years

(a)

σ yy–

σ xx/

MPa

–6

–4

–2

0

2

4

6

8

0 100 200 300 400

y (m)

1 year 10 years20 years 1 year10 years 20 years

(b)

–0.4

–0.2

0

0.2

0.4

0 100 200 300 400σ xy (M

Pa)

x (m)

1 year 10 years20 years 1 year10 years 20 years

(c)

σ xy (M

Pa)

–0.4

–0.2

0

0.2

0.4

0.6

0 100 200 300 400

y (m)

1 year 10 years20 years 1 year10 years 20 years

(d)

Figure 12: (a) and (b) are present the stress difference profiles on Paths A and B of different production years, and (c) and (d) present theshear stress profiles on Paths A and B of different production years.

0 400200100 30044.00

46.25

48.50

50.75

53.00

55.25

57.50MPaz

x

z-x section

Our model

Conventional model

0

0100

100

(a)

0 400200100 300

MPa

42.00

44.58

47.17

49.75

52.33

54.91

57.00z

x

z-x section

Our model

Conventional model

0

0100

100

(b)

0 400200100 300

MPa

0 –1.80

–1.20

–0.60

0.00

0.60

1.20

1.80z

x

z-x section

0

Our model

Conventional model100

100

(c)

Figure 11: (a), (b), and (c) present the comparison of the principal stress along the x-direction, y-direction, and the induced shear stress onthe z-x section, respectively.

10 Geofluids

50m. The three hydraulic fractures are considered as produc-tion fractures. It was assumed that all three clusters of frac-tures pass through the reservoir in the z-direction. Theother calculation parameters are the same as those inTable 1. Under this condition, the reservoir stress distribu-tion after 1, 5, and 10 years of production was simulated.

Figure 15 presents the pore pressure distribution after 1,5, and 10 years of nonplanar fracture production usingthe conventional model. The pore pressure distribution inthe domain was axially symmetric. However, under theinfluence of natural fractures, there was an initial decreasein the pore pressure along the strike direction of the natural

0

20

40

60

80

100

120

0 100 200 300 400y (m)

5 years10 years20 years

Orie

ntat

ion

chan

ge o

f𝜎

Hm

ax (d

egre

e)

(a)

0

20

40

60

80

100

120

0 100 200 300 400

Orie

ntat

ion

chan

ge o

f𝜎

Hm

ax (d

egre

e)

y (m)

5 years10 years20 years

(b)

Figure 14: (a) and (b) present the steering angle of the maximum horizontal principal stress along Path B without and with considering theinfluence of natural fractures in different production years, respectively.

𝜃1 𝜃2

x

y

(a) (b) (c)

𝜃1 𝜃2

(d) (e)

Direction of horizontalmaximum principal stress

x-y section

5 year 10 year

5 year 10 year

Figure 13: (a) presents the direction of the maximum principal stress on the x-y section in the initial state. (b) and (c) present the direction ofthe maximum horizontal principal stress after 5 years and 10 years of production without considering natural fractures. (d) and (e) present thedirection of the maximum horizontal principal stress after 5 years and 10 years of production considering the influence of natural fractures.

11Geofluids

fractures, and the distribution exhibited central symmetryrather than axial symmetry (as shown in Figure 16). Theupper left part and the lower right part of the domain areunderexploited areas under such reservoir geological condi-tions and distributed fractures.

Due to the influence of the nonplanar fracture geometry,the stress distribution after production exhibited a high com-plexity. Without considering the effect of natural fractures,the stress distribution of the conventional model was axiallysymmetric. Compared with a single fracture, there was anincrease in the rates of the pore pressure dissipation and stressdecrease and there was an increase in the size of the influencerange, as shown in Figures 17(a), 17(b), and 17(c). The stress(σxx) on the left and right of the domain and the stress (σyy)on the top and bottom of the domain increased to supportthe pressure depletion of the middle area.When the influenceof natural fractures was considered, the stress distributionin the proposed model exhibited a significant degree ofdeflection, as shown in Figures 18(a), 18(b), and 18(c). Thedeflection of the stress distribution confirms that natural frac-tures have a significant influence on the reservoir fluid flowand production.

Compared with a single crack, the distribution of theprincipal stress was more complex. The change in reorienta-tion of σH max was θ1′ > θ1, θ2′ > θ2, as shown in Figure 19. Dueto the influence of the nonplanar crack geometry, therotation angle of the maximum principal stress direction inthe conventional model reached 135°. In the proposedmodel, the rotation angle of the principal stress direction

near the crack was concentrated at 140°. The reorientationregion affected by the natural fractures was larger than thoseof the conventional models. Therefore, under the action ofthe nonplanar fracture geometry and natural fractures, theevolution of the reservoir stress tends to be complex in theproduction process. From a comparison of Figures 20(a)and Figure 18(a), the profile of the change in reorientationon Path B was found to be significantly different due to theinfluence of the hydraulic fracture geometry. During the firstyear of production, the central region did not turn; however,it turned in accordance with an increase in the productiontime. From a comparison of Figures 20 and Figure 18,there was a similarity in that the greatest change in thereorientation of σH max occurred in the 10th year and thechange in reorientation was correlated with the reservoirdepletion time.

5. Conclusion

A fully coupled 3D pore elastic model based on the FEM wasdeveloped to analyze the stress evolution and stress reorien-tation in heterogeneous porous media in the reservoir deple-tion process. In the model, the continuous natural fracturemodel is used to characterize the effect of natural fractureson the reservoir fluid flow. The relationship between thenatural fracture width and effective stress was used to estab-lish a fluid cluster single fracture model and a three-clusternonplane fracture model for an example analysis. Therewas a good agreement between the analytical solutions and

x0 400200100 300

0

400

200

300

100

x-y sectiony

15.00

17.50

20.00

22.50

25.00

27.50

30.00MPa

(a)

x0 400200100 300

0

400

200

300

100

x-y sectiony

15.00

17.50

20.00

22.50

25.00

27.50

30.00MPa

(b)

15.00

17.50

20.00

22.50

25.00

27.50

30.00MPa

x0 400200100 300

0

400

200

300

100

x-y sectiony

(c)

Figure 15: (a), (b), and (c) present the pore pressure distribution after 1, 5, and 10 years of nonplanar fracture production without consideringthe influence of natural fractures.

12 Geofluids

43.00

45.50

48.00

50.50

53.00

55.50

58.00

x

0 400200100 3000

400

200

300

100

x-y sectiony MPa

(a)

42.00

44.67

47.33

50.00

52.67

55.33

58.00

x

0 400200100 3000

400

200

300

100

x-y sectionyMPa

(b)

49.00

51.50

54.00

56.50

59.00

61.50

64.00

x

0 400200100 3000

400

200

300

100

x-y sectiony MPa

(c)

Figure 17: (a), (b), and (c) present the σx , σy , and σz stress distributions after 10 years of nonplanar fracture production without consideringthe influence of natural fractures.

x

x-y section

0 400200100 3000

400

200

300

100

y

15.00

17.50

20.00

22.50

25.00

27.50

30.00MPa

(a)

x-y section

0 400200100 3000

400

200

300

100

y

x 15.00

17.50

20.00

22.50

25.00

27.50

30.00MPa

(b)

15.00

17.50

20.00

22.50

25.00

27.50

30.00MPax-y section

0 400200100 3000

400

200

300

100

y

x

(c)

Figure 16: (a), (b), and (c) present the pore pressure distribution after 1, 5, and 10 years of nonplanar fracture production considering theinfluence of natural fractures.

13Geofluids

x

y

(a)

𝜃1 𝜃2

(b) (c)

𝜃1 𝜃2

(d) (e)

Direction of horizontalmaximum principal stress

x-y section

3 year 10 year

3 year 10 year

Figure 19: (a) presents the direction of the initial maximum horizontal principal stress on the x-y section. (b) and (c) present the direction ofthe maximum horizontal principal stress after 3 years of depletion without considering the natural cracks. (d) and (e) present the direction ofthe maximum horizontal principal stress after 10 years of production considering the natural cracks.

41.00

44.17

47.33

50.50

53.67

56.83

60.00

x

MPax-y sectiony

0

400

200

300

100

0 400200100 300

(a)

40.00

43.33

46.66

50.00

53.33

56.66

60.00

x

MPax-y sectiony

0

400

200

300

100

0 400200100 300

(b)

48.00

50.83

53.67

56.50

59.33

62.17

65.00

x

MPax-y sectiony

0

400

200

300

100

0 400200100 300

(c)

Figure 18: (a), (b), and (c) present the σx , σy , σz stress distribution after 10 years of nonplanar fracture production considering the influenceof natural fractures.

14 Geofluids

numerical results. By comparing the conventional models, itwas confirmed that natural cracks have a significant influenceon the stress evolution, surrounding drainage, and change inreorientation in the depletion process. Under the influence ofnatural fractures, the fluids in reservoirs along the directionof the natural fractures are easier to recover, and areas thatare perpendicular to the natural fracture surface are proneto underconsumption. In addition, under the flow modedominated by natural fractures, the stress distribution inthe proposed model is deflected to a certain extent, especiallyin the nonplanar fracture model. Due to the action of naturalcracks, the absolute values of the stress, displacement, andstress difference in the proposed model are relatively greaterand the change in the reorientation of the maximum princi-pal stress steering is significantly different. After consideringthe natural cracks, the steering angle and steering rangeincreased. Moreover, this directly results in significant differ-ences in the design of the late-stage fracturing temporaryplugging steering and refracturing technology for conven-tional reservoirs. The simulation results revealed that it isvery important to consider the effects of natural fractureson the stress redistribution, surrounding drainage, andreorientation in production. The findings of this study canserve as a theoretical guide for the design of refracturing,temporary plugging steering, and infilling wells and helppredict the direction of new fractures and the trajectory offracture extensions.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work presented in this paper was supported by the mainprogram of the China National Petroleum Corporation(2016E-01), National Science and Technology Major SpecialProject-Fu Ling Shale Gas Development DemonstrationProject (2016ZX05060), and the China National ScholarshipFund.

References

[1] A. Sangnimnuan, J. Li, and K. Wu, “Development of efficientlycoupled fluid-flow/geomechanics model to predict stress evo-lution in unconventional reservoirs with complex-fracturegeometry,” SPE Journal, vol. 23, no. 03, pp. 640–660, 2018.

[2] M. Chen, M. Bai, and J. C. Roegiers, “Permeability tensorsof anisotropic fracture networks,” Mathematical Geology,vol. 31, no. 4, pp. 335–373, 1999.

[3] X. Weng, V. Sesetty, and O. Kresse, “Investigation ofshear-induced permeability in unconventional reservoirs,”in 49th U.S. Rock Mechanics/Geomechanics Symposium, SanFrancisco, California, November 2015.

[4] R. H. Richard, “The influence of fracture stiffness and the insitu stress field on the closure of natural fractures,” PetroleumGeoscience, vol. 4, no. 1, pp. 57–65, 1998.

[5] T. K. Perkins and J. A. Gonzalez, “The effect of thermoelasticstresses on injection well fracturing,” Society of PetroleumEngineers Journal, vol. 25, no. 01, pp. 78–88, 1985.

[6] Y. Q. Hu, Q. Wang, J. Z. Zhao, Z. Guo, Y. Zhang, andC. Mao, “Development of efficiently coupled thermo-hydro-mechanical model to predict hydraulic fracture morphologyin heavy oil reservoirs,” Environmental Earth Sciences,vol. 77, no. 23, p. 778, 2018.

[7] L. Ren, R. Lin, J. Z. Zhao, V. Rasouli, J. Zhao, and H. Yang,“Stimulated reservoir volume estimation for shale gas frac-turing: mechanism and modeling approach,” Journal ofPetroleum Science and Engineering, vol. 166, pp. 290–304,2018.

0

20

40

60

80

100

120

140

160

0 100 200 300 400y (m)

1 year10 years20 years

Orie

ntat

ion

chan

ge o

f𝜎H

max

(deg

ree)

(a)

0

20

40

60

80

100

120

140

160

0 100 200 300 400

Orie

ntat

ion

chan

ge o

f𝜎H

max

(deg

ree)

y (m)

1 year10 years20 years

(b)

Figure 20: (a) and (b) present the change in reorientation of σH max along Path B without and with considering the influence of naturalfractures and under different depletion years, respectively.

15Geofluids

[8] Q.Wang, Y. Hu, J. Z. Zhao, L. Ren, C. Zhao, and J. Zhao, “Mul-tiscale apparent permeability model of shale nanopores basedon fractal theory,” Energies, vol. 12, no. 17, p. 3381, 2019.

[9] J. Z. Zhao, Q. Wang, Y. Q. Hu, L. Ren, and C. Zhao, “Numer-ical investigation of shut-in time on stress evolution and tightoil production,” Journal of Petroleum Science and Engineering,vol. 179, pp. 716–733, 2019.

[10] J. Z. Zhao, Q. Wang, Y. Q. Hu, C. Zhao, and J. Zhao, “Predic-tion of pore pressure-induced stress changes during hydraulicfracturing of heterogeneous reservoirs through coupled fluidflow/geomechanics,” Journal of Engineering Mechanics,vol. 145, no. 12, article 05019001, 2019.

[11] N. P. Roussel and M. M. Sharma, “Role of stress reorientationin the success of refracture treatments in tight gas sands,” SPEProduction & Operations, vol. 27, no. 04, pp. 346–355, 2012.

[12] R. Safari, R. Lewis, X. Ma, O. Mutlu, and A. Ghassemi, “Infillwell fracturing optimization in tightly spaced horizontalwells,” SPE Journal, vol. 22, no. 02, pp. 582–595, 2016.

[13] D. Kumar and A. Ghassemi, “Three-dimensional poroelasticmodeling of multiple hydraulic fracture propagation fromhorizontal wells,” International Journal of Rock Mechanicsand Mining Sciences, vol. 105, pp. 192–209, 2018.

[14] A. Ghassemi, X. X. Zhou, and C. Rawal, “A three-dimensionalporoelastic analysis of rock failure around a hydraulicfracture,” Journal of Petroleum Science and Engineering,vol. 108, pp. 118–127, 2013.

[15] S. Salimzadeh, A. Paluszny, and R. W. Zimmerman, “Three-dimensional poroelastic effects during hydraulic fracturing inpermeable rocks,” International Journal of Solids and Struc-tures, vol. 108, pp. 153–163, 2017.

[16] Q. Gao and A. Ghassemi, “Pore pressure and stress distribu-tions around a hydraulic fracture in heterogeneous rock,” RockMechanics and Rock Engineering, vol. 50, no. 12, pp. 3157–3173, 2017.

[17] Y. Zeng, Z. Wang, Y. Zang et al., “Pore pressure disturbanceinduced by multistage hydraulic fracturing in shale gas:modelling and field application,” Geofluids, vol. 2019, ArticleID 1315451, 11 pages, 2019.

[18] H. Zhu, X. Tang, Q. Liu, K. Li, and J. D. Mclennan, “4D multi-physical stress modelling during shale gas production: a casestudy of Sichuan Basin shale gas reservoir, China,” Journal ofPetroleum Science and Engineering, vol. 167, pp. 929–943,2018.

[19] H. Zhu, X. Tang, Q. Liu et al., “Permeability stress-sensitivityin 4D flow-geomechanical coupling of Shouyang CBM reser-voir, Qinshui Basin, China,” Fuel, vol. 232, pp. 817–832, 2018.

[20] D. Kumar and A. Ghassemi, “A three-dimensional analysis ofsimultaneous and sequential fracturing of horizontal wells,”Journal of Petroleum Science and Engineering, vol. 146,pp. 1006–1025, 2016.

[21] R. Safari and A. Ghassemi, “Three-dimensional poroelasticmodeling of injection induced permeability enhancementand microseismicity,” International Journal of Rock Mechanicsand Mining Sciences, vol. 84, pp. 47–58, 2016.

[22] R. Safari and A. Ghassemi, “3D thermo-poroelastic analysis offracture network deformation and induced micro-seismicity inenhanced geothermal systems,” Geothermics, vol. 58, pp. 1–14,2015.

[23] M. Lin, S. Chen, E. Mbia, and Z. Chen, “Application ofreservoir flow simulation integrated with geomechanics in

unconventional tight play,” Rock Mechanics and Rock Engi-neering, vol. 51, no. 1, pp. 315–328, 2018.

[24] A. Morales, K. Zhang, K. Gakhar et al., “Advanced modeling ofinterwell fracturing interference: an Eagle Ford shale oil study-refracturing,” in SPE Hydraulic Fracturing Technology Confer-ence, Woodlands, TX, USA, February 2016.

[25] A. Rezaei, G. Bornia, M. Rafiee, M. Soliman, and S. Morse,“Analysis of refracturing in horizontal wells: insights fromthe poroelastic displacement discontinuity method,” Inter-national Journal for Numerical and Analytical Methods inGeomechanics, vol. 42, no. 11, pp. 1306–1327, 2018.

[26] Q. Zhang, A Boundary Element Method for Thermo-Poroelasticity with Applications in Rock Mechanics, Universityof North Dakota, Grand Forks, ND, USA, 2004.

[27] A. Ghassemi and Q. Zhang, “Porothermoelastic analysis of theresponse of a stationary crack using the displacement disconti-nuity method,” Journal of Engineering Mechanics, vol. 132,no. 1, pp. 26–33, 2006.

[28] K. H. Chun, Thermo-Poroelastic Fracture Propagation Model-ing with Displacement Discontinuity Boundary ElementMethod, Ph. D. Dissertation, Texas A & M University, CollegeStation, TX, USA, 2013.

[29] I. Walton and J. Mclennan, “The role of natural fractures inshale gas production,” in ISRM International Conference forEffective and Sustainable Hydraulic Fracturing, Brisbane,Australia, May 2013.

[30] N. R. Warpinski, “Hydraulic fracturing in tight, fissuredmedia,” Journal of Petroleum Technology, vol. 43, no. 02,pp. 146–209, 1991.

[31] S. A. McKenna and P. C. Reeves, Fractured ContinuumApproach to Stochastic Permeability Modeling, AmericanAssociation of Petroleum Geologists, Tulsa, Oklahoma, 2006.

[32] E. Kalinina, S. A. McKenna, K. Klise, K. T. Hadgu, and T. S.Lowry, “Incorporating complex three-dimensional fracturenetwork into geothermal reservoir simulations,” Transac-tions-Geothermal Resources Council, vol. 36, pp. 493–497,2012.

[33] S. C. Bandis, A. C. Lumsden, and N. R. Barton, “Fundamentalsof rock joint deformation,” International Journal of RockMechanics and Mining Sciences, vol. 20, no. 6, pp. 249–268,1983.

[34] M. A. Biot, “General theory of three-dimensional consolida-tion,” Journal of Applied Physics, vol. 12, no. 2, pp. 155–164,1941.

[35] M. A. Biot, “Theory of elasticity and consolidation for a porousanisotropic solid,” Journal of Applied Physics, vol. 26, no. 2,pp. 182–185, 1955.

[36] R. W. Zimmerman, “Coupling in poroelasticity and thermoe-lasticity,” International Journal of Rock Mechanics and MiningSciences, vol. 37, no. 1-2, pp. 79–87, 2000.

[37] D. W. Peaceman, “Interpretation of well-block pressures innumerical reservoir simulation (includes associated paper6988),” SPE Journal, vol. 18, no. 03, pp. 183–194, 1978.

16 Geofluids

Hindawiwww.hindawi.com Volume 2018

Journal of

ChemistryArchaeaHindawiwww.hindawi.com Volume 2018

Marine BiologyJournal of

Hindawiwww.hindawi.com Volume 2018

BiodiversityInternational Journal of

Hindawiwww.hindawi.com Volume 2018

EcologyInternational Journal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com

Applied &EnvironmentalSoil Science

Volume 2018

Forestry ResearchInternational Journal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

International Journal of

Geophysics

Environmental and Public Health

Journal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

International Journal of

Microbiology

Hindawiwww.hindawi.com Volume 2018

Public Health Advances in

AgricultureAdvances in

Hindawiwww.hindawi.com Volume 2018

Agronomy

Hindawiwww.hindawi.com Volume 2018

International Journal of

Hindawiwww.hindawi.com Volume 2018

MeteorologyAdvances in

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com

The Scientific World Journal

Volume 2018Hindawiwww.hindawi.com Volume 2018

ChemistryAdvances in

Scienti�caHindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Geological ResearchJournal of

Analytical ChemistryInternational Journal of

Hindawiwww.hindawi.com Volume 2018

Submit your manuscripts atwww.hindawi.com