Numerical Simulation of Fluid-Solid Coupling in Fractured Porous Media with Discrete Fracture Model and Extended Finite Element Method

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1 Coputation 2015, 3, ; doi: /coputation Article OPEN ACCESS Coputation ISSN dpi.co/journal/coputation Nuerical Siulation o Fluid-Solid Coupling in Fractured Porous Media ith Discrete Fracture Model and Extended Finite Eleent Method Qingdong Zeng and Jun Yao * School o Petroleu Engineering, China University o Petroleu (East China), Qingdao , China; upc.zengqd@163.co * Author to ho correspondence should be addressed; b @s.upc.edu.cn or RCOGFR_UPC@126.co; el.: Acadeic Editors: Qinjun Kang and Li Chen Received: 25 August 2015 / Accepted: 27 October 2015 / Published: 30 October 2015 Abstract: Fluid-solid coupling is ubiquitous in the process o luid lo underground and has a signiicant inluence on the developent o oil and gas reservoirs. o investigate these phenoena, the coupled atheatical odel o solid deoration and luid lo in ractured porous edia is established. In this study, the discrete racture odel (DFM) is applied to capture luid lo in the ractured porous edia, hich represents ractures explicitly and avoids calculating shape actor or cross lo. In addition, the extended inite eleent ethod (XFEM) is applied to capture solid deoration due to the discontinuity caused by ractures. More iportantly, this odel captures the change o ractures aperture during the siulation, and then adjusts luid lo in the ractures. he inal linear equation set is derived and solved or a 2D plane strain proble. Results sho that the cobination o discrete racture odel and extended inite eleent ethod is suited or siulating coupled deoration and luid lo in ractured porous edia. Keyords: luid-solid coupling; porous elasticity; ractured porous edia; discrete racture odel; extended inite eleent ethod

2 Coputation 2015, Introduction he technology o hydraulic racturing has been idely used or reservoir stiulation, especially or unconventional reservoirs [1]. Coupled rock deoration and luid lo in ractured porous edia is iportant or reservoir siulation because rock deoration exerts an iportant inluence on reservoir production [2]. he general theory o 3D consolidation ith elasticity constitutive relationship and Darcy la has been established by Biot [3], and the eective stress orulation has been put orard by erzaghi [4]. A great nuber o researches about luid-solid coupling have been done based on these theories in petroleu engineering, ro conventional reservoirs to ractured reservoirs [5]. here exist several ethods to siulate luid lo in ractured porous edia [6]. Warren and Root introduced the dual continuu concept to characterize naturally ractured reservoirs [7]. he dual continuu approaches treat racture and atrix both as continua distributed ithin reservoir doain. he racture-atrix cross lo is based on the analytical solution o pseudosteady-state lo ithin the atrix syste ith a siple geoetry o atrix blocks. Moreover, shape actors are calculated or dierent geoetries o atrix blocks. he dual continuu approaches consist o dual porosity and dual pereability odels [8]. Scheatics or dual porosity concept and dual pereability concept are shon in Figures 1 and 2 respectively. he dierence beteen dual porosity odel and dual pereability odel is that dual pereability odel takes global atrix-atrix lo into account hile dual porosity odel does not account or it. (a) (b) Vugs Matrix Fractures Matrix Fractures Figure 1. Scheatic illustration o dual porosity odel o ractured reservoirs. (a) Actual reservoir; (b) Reservoir odel. Figure 2. Scheatic illustration o dual pereability odel o ractured reservoirs.

3 Coputation 2015, An alternative to the dual continuu approaches is the discrete racture odel [9,10]. he ractures are represented explicitly ithin the doain and discretized along ith the atrix doain. Lab [11] presented racture apping approach (FM) to siulate luid lo in ractured porous edia. In the FM approach, an eleent intersected by a racture is treated as a superposition o to eleents, naely a atrix eleent and a racture eleent. he atrix eleent and racture eleent interact via a transer unction. he scheatic o racture apping approach is shon in Figure 3. he approach adopts the transer unction presented by Barenblatt [12] to account or cross lo beteen the overlapping atrix and racture eleents, hich as a dual continuu odel to this extent. Figure 3. Scheatic representation o racture apping approach (FM). Due to stress singularity o racture tip, it needs esh reineent around the racture in the standard inite eleent raeork, hich is coputational burdensoe. An alternative to standard inite eleent ethod is the extended inite eleent ethod (XFEM). he extended inite eleent ethod as introduced by Belytschko [13] to discontinuous probles, hich has been idely used in any ields due to lexibility in eshing [14 17]. he racture is represented by level set ethod. Ghaouri and Leis [18,19] presented a inite eleent continuu approach to describe the coupled deoration and luid lo in ractured porous edia, hich as based on double porosity odel. ran [20] presented high level boundary eleent ethod ith periodic boundary conditions and lux continuous inite volue eleent ethod to siulate coupled luid lo through discrete racture netork; Al-Khoury [21] used the partition o unity ethod to describe the racturing process and double porosity odel to describe the resulting luid lo; Vire et al. [22] presented coupling an adaptive esh inite eleent luid odel ith a cobined inite-discrete eleent solid odel to investigate luid-solid interactions. he ethod is lexible in ters o discretization schees used or each aterial. Recently, Vire et al. [23] presents an iersed-shell ethod or odeling luid-structure interactions. he ethod consists o iersing the solid structures in an extended luid doain, and exchanging the coupling orces through a thin shell surrounding the solid structures. Lab [13] presented FM approach and the extended inite eleent ethod or coupled deoration and luid lo in ractured porous edia. he dierence beteen this paper and Lab s is that here discrete racture odel is used to avoid calculating cross lo and the odel captures the change o ractures aperture during the siulation. he advantages o cobination o discrete racture odel and extended inite eleent ethod over other ethods are that discrete racture odel avoids the coputation o shape actor and is ore accurate than dual continuu odel or siulating luid lo ith large ractures, eanhile the extended inite eleent ethod avoids esh reineent around the racture and is ell suited or

4 Coputation 2015, discontinuity probles. Furtherore, the odel is capable o capturing change o ractures aperture during the siulation. In this paper, the cobination o discrete racture odel and extended inite eleent ethod is used to couple deoration and luid lo in ractured porous edia. he governing equations and initial and boundary conditions are presented in Section 2. he nuerical solution is presented in Section 3. In the section, the extended inite eleent ethod and the discrete racture odel are briely described to capture deoration and luid lo respectively, and spatial and teporal discretization are conducted. Finally, nuerical siulation and result analysis are perored in Section 4 and the conclusion are dran in Section Matheatical Model 2.1. Governing Equations or Rock Deoration Under the assuptions o sall-strain situation, isotheral equilibriu and negligible inertial orces, Biot s theory describes the linear oentu balance equation or a to-phase ediu, hich is coposed by rock and ater. sσ + ρ g = 0 (1) here σ is total stress tensor, g is gravity, ρ is the averaged density o the ultiphase syste, and s is the syetric gradient operator atrix. s ρ = (1 φ) ρ + φρ (2) x s = 0 y 0 y x s here φ is porosity, ρ is the density o solid, and ρ is the density o ater. he relationship beteen total stress and eective stress is given by e σ = σ α p (4) e here σ is the eective stress, p is the ater pore pressure in the porous atrix, α is the Biot s copressibility coeicient, and = [1 1 0] in to diensions. he constitutive stress-strain relationship o the solid phase is given by σ = here D is the linear elastic aterial atrix and ε is the strain o the syste. he geoetric equation beteen strain and displaceent is given by here u is the displaceent o the syste. (3) D ε (5) ε = s u (6)

5 Coputation 2015, Governing Equations or Fluid Flo Fluid lo in ractured porous edia is typically siulated using dual-porosity odels, but dual-porosity odels are not ell suited or the odeling o a sall nuber o large-scale ractures, hich ay doinate the lo. Discrete racture odel, in hich the ractures are represented individually, has been broadly applied to siulate lo in ractured porous edia. In this paper, discrete racture odel is used or lo siulation. he equation o otion or luid lo in porous edia is given by Darcy la as ollos k v = ( p ρ g ) μ (7) here v is velocity, k is pereability, μ is viscosity o ater. he continuity equation takes into consideration the grain and luid volue variation resulting ro pressure change (the irst ter) and total volue strain resulting ro solid deoration (the second ter), hich is given by α φ φ p ε k ( + ) + α + [ ( p + ρ g )] = 0 (8) K K t t μ here s K s is the bulk odulus o the grain aterial, 2.3. Initial and Boundary Conditions K is the bulk odulus o ater. he initial conditions speciy the displaceent and ater pressure ields at tie t = 0. u= u 0, p = p0 in and on Γ (9) here is the doain o interest and Γ is the boundary. Boundary conditions o solid deoration include displaceent condition and orce condition, hich can be given as ollos u= u on Γu (10) l σ = t on Γ t here l is related to the unit outard noral vector n = { nx, ny} by nx 0 l = 0 ny (11) ny n x Boundary conditions o luid lo include ater pressure condition and lux condition, hich can be given as ollos here p = p on Γp k μ ( p ρ g) q on + = Γ q is the iposed volue lux noral to the boundary. q (12)

6 Coputation 2015, Nuerical Solution 3.1. Application o the Extended Finite Eleent Method In the inite eleent raeork, discontinuity odeling needs esh reineent around the crack, hich is coputationally burdensoe. he extended inite eleent ethod as introduced by Belytschko and Black, hich had been idely used or discontinuous proble. he advantage o extended inite eleent ethod is avoiding esh reineent around ractures (especially racture tips) ithout reducing accuracy. he ethod is adopted to solve solid deoration in this study. he extended inite eleent ethod exploits the partition o unity property o inite eleent. he displaceent ield is decoposed into to parts: the continuous displaceent ield and the discontinuous part. he displaceent approxiation can be ritten as ollos 4 k NI I NI H φ H φ I I NI γ γ I I I N cr tip I N I N k = 1 h u ( x) = ( x) u + ( x)( ( ( x)) ( ( x ))) a + ( x) ( Φ ( x) Φ ( x )) b (13) here x is the position vector; u I is the nodal displaceents; N I is the shape unction or non-enriched and enriched nodes, respectively; a I and Heaviside step unction; φ( x ) is signed distance unction; ( ) k b I are degrees or enriched nodes; H ( ξ ) is Φ x is enriched unctions or tip eleents; N is the set o all nodes in discretized odel; N cr is the set o nodes o all eleents tip containing cracks but not crack tips; N is the set o nodes o all eleents containing the crack tip. Heaviside step unction is deined as 1, x > 0 H( x) = 0, x= 0 1, x < 0 γ (14) Fro Equation (13), the opening beteen the to suraces o the racture can be given by u + = n ( u u ) = 2 n N a + 2 n r N b 1 I I I I cr I N tip I N (15) here n is the unit outard noral vector Application o Discrete Fracture Model In discrete racture odel, ractures are sipliied into 1D line eleent or 2D proble, and 2D surace area eleent or 3D proble, as shon in Figure 4. he hole ractured porous edia is decoposed into to parts: atrix syste and racture syste, hich can be given as ollos = + a (16) here represents atrix, represents racture, and a is the aperture o racture. Assuing that representative eleent volues o both atrix and racture syste exist, lo equations are applicable to the hole research region. hen or the discrete racture odel, the integral or o the lo equation can be given as ollos.

7 Coputation 2015, Fd= Fd + afd (17) In the Equations (16) and (17), the apertures o ractures are calculated ro Equation (15). In the ipleentation o algorith, ractures are discretized into sall segents. Moreover, the aperture o each segent is set equal to average opening o its endpoints, hich is easy to calculate ro Equation (15). Hoever, the initial aperture o racture is given in the irst tie step and updated ro the second tie step. (1D) (2D) 3.3. Discretization in Space Figure 4. Scheatic o a discrete racture odel. According to strong or o solid deoration equation, the eak or can be expressed ith eighted residual ethod ε( u): D: ε( v)d α ε ( v) p d= ρg vd+ t vd Γ, v U Γ 0 (18) t 0 here U0 = { v v C ( / Γ cr ), v Γ u =0} is the trial unction space, and Γ cr is the racture in the doain. he eighted residual ethod is applied to the continuity equation or ater lo and to its natural boundary condition, hich yields k u α φ φ p ( p ρ g) + α s + ( + ) d+ q dγ= 0 μ t K K t (19) Γq s here is the eight unction, and is the region o integration expressed by Equation (16), hich consists o atrix and racture syste. Equation (19) turns to k u α φ φ p ( p ρ g) + α s + ( + ) d + μ t Ks K t k u α φ φ p a [ ( p ρ g) + α s + ( + ) ]d + q dγ= 0 μ t K K t Γ q s here k is the pereability o atrix, and k is the pereability o racture, hich can be given by cubic la as ollos (20)

8 Coputation 2015, a k = (21) 12 he expression or displaceent is given by Equation (13), and the expression or ater pressure is given as ollos here p is the vector o the nodal values o ater pressure, and p = Np p (22) N p is the shape unction or ater pressure. By integrating Equations (18) and (20), the eak or o the hole syste is discretized into the olloing set o equations dx AX + B = C (23) dt here Keuu Keua Keub L u Keau Keaa Keab La A = Kebu Keba Kebb Lb H + ah B = Lu La Lb S + as = [ ] (24) (25) X u a b p (26) C = [ F Fa Fb ] (27) Eleents o the aoreentioned listed atrices are given in Appendix Discretization in ie Using the ully iplicit tie discretization schee, the approxiation is given as ollos d X (28) dt dt n+ 1 n = X X hen, the inal discrete equation can be ritten as ollos (d t A + BX ) n+ 1 = BXn + dt C (29) he schee is ully iplicit and iposes no requireents o the tie step size hich is usually chosen or both stability and the eliination oscillatory eects in the solution. In the above equation, the unknon vector X includes standard degrees o reedo o all nodes, enriched degrees o reedo o enriched nodes in to directions and ater pressures o all nodes. he atrices are constructed according to Appendix A and the linear equation set is solved or X n+ 1 (value

9 Coputation 2015, or n + 1 tie step) providing that X n (value or n tie step) is given. Moreover, the initial value or X is given ith the initial conditions by setting nodal displaceents and enriched degrees to zero and ater pressures to p 0. he racture apertures are calculated according to Equation (15) using the already solved displaceents, then the atrix or lo in the racture is updated. he loop continues until end tie is reached. 4. Nuerical Exaple he DFM-XFEM odel is applied to a 2D plain strain proble shon in Figure 5. he doain is ully saturated and alloed to reely drain at the top, that is, excess pore ater pressure is equal to zero. he doain has an area o he racture is 8 long, is inclined at 45 and is centered in the iddle o the doain. he right, let and botto boundaries are assued to be undrained. he lateral boundaries o the doain are constrained to vertical translation only, and the botto boundary is constrained to be ixed. Static load is applied at the top boundary and aintained throughout the duration o the siulation. he input paraeters are given in able 1. he discrete racture odel has been idely used or siulating lo in ractured porous edia, and the extended inite eleent ethod has been broadly applied to the discontinuity proble. he cobination DFM-XFEM odel gives ull play to their advantages or siulating lo solid coupling in ractured porous edia. he solid odule shares the sae esh coniguration ith the luid lo odule. he nuerical ethod is ipleented using Matlab. Lab presented racture apping approach and the extended inite eleent ethod to couple deoration and luid lo in ractured porous edia, denoted by FM-XFEM odel. Moreover, the odel proposed in this paper is denoted by DFM-XFEM odel. Because the racture geoetry reains constant throughout the siulation or the FM-XFEM odel, that is, the racture does not close or open during the siulation, the assuption is applied to the DFM-XFEM odel or odel veriication by coparison ith FM-XFEM odel. he assuption is done by ixing racture aperture in the luid lo odule during the siulation. he coparison results o the to odels are shon in Figures 6 and 7. Figure 5. o-diensional ractured doain ith assigned boundary conditions.

10 Coputation 2015, able 1. Input paraeters or the odels. Paraeter Deinition Magnitude Units E Young s odulus 40 MPa v Poisson s ratio μ Fluid viscosity Pa. s φ Matrix porosity φ Fracture porosity k Matrix pereability Darcy k Fracture pereability Darcy K Bulk odulus o luid MPa K s Bulk odulus o solid MPa t Static load Pa y x x (a) (b) Figure 6. Displaceent distribution () ater 100 days. (a) Fracture apping extended inite eleent ethod (FM-XFEM) odel; (b) Discrete racture odel (DFM)-XFEM odel. (a) (b) Figure 7. Excess pore pressure distribution (Pa) ater 100 days. (a) FM-XFEM odel; (b) DFM-XFEM odel. he displaceent ields along y direction o the odel are shon in Figure 6. he results o to dierent odels are very close, and they are qualitatively identical. he displaceent ield is

11 Coputation 2015, discontinuous due to the pre-existing racture, and the extended inite eleent ethod captures discontinuity ithout esh reineent around the racture, hich is a great iproveent over the standard inite eleent ethod. he excess pore ater pressure distributions are shon in Figure 7. he results o to dierent odels are very close, and they are also qualitatively identical. he discrete racture odel represents racture explicitly, and it does not need to calculate shape actor, hich is not easy to deterine in the dual porosity odel. Because the top boundary is a zero pressure boundary, luid can reely lo out and the pressure is becoing loer and loer. he displaceent ields along y direction at point A (located at the top shon in Figure 5 and excess pore pressure at point B (located at the botto shon in Figure 5 or the FM-XFEM odel and DFM-XFEM odel at varying esh resolutions during 100 days are shon in Figures 8a and 9a respectively. he plots are zooed in or 10 days to clariy the discrepancies beteen the various curves and shon in Figure 8b and Figure 9b. hese to igures indicate that the results o DFM-XFEM odel sho great coincidence ith the result o FM-XFEM, hich validates the correctness o DFM-XFEM odel. Moreover, as the esh resolutions becoe higher, the results o DFM-XFEM odel sho little dierence, hich validates the convergence o the proposed algorith. x x Displacent/ FM-XFEM 128*128 Mesh DFM-XFEM 1492 Eleents DFM-XFEM 2336 Eleents DFM-XFEM 4238 Eleents Displacent/ FM-XFEM 128*128 Mesh DFM-XFEM 1492 Eleents DFM-XFEM 2336 Eleents DFM-XFEM 4238 Eleents ie/day (a) ie/day (b) Figure 8. Displaceent at point A. (a) 100 days; (b) 10 days Pressure/Pa FM-XFEM 128*128 Mesh DFM-XFEM 4238 Eleents DFM-XFEM 2336 Eleents DFM-XFEM 1492 Eleents Pressure/Pa FM-XFEM 128*128 Mesh DFM-XFEM 4238 Eleents DFM-XFEM 2336 Eleents DFM-XFEM 1492 Eleents ie/day (a) ie/day (b) Figure 9. Excess pore pressure at point B. (a) 100 days; (b) 10 days.

12 Coputation 2015, he coputational cost o odels is shon in able 2. It can be shon that the coputational tie o DFM-XFEM is uch less than that o FM-XFEM ith approxiately equal nuber o eleents. he coparison shos the advantage o DFM-XFEM over FM-XFEM. able 2. Coputational cost o odels. Models Coputational ie/s FM-XFEM Mesh 31.7 FM-XFEM Mesh 88.1 DFM-XFEM 1492 Eleents 8.8 DFM-XFEM 2336 Eleents 14.3 DFM-XFEM 4238 Eleents 27.5 In the above exaple, the racture reains constant during the siulation. Hoever, the racture could close or open during the siulation, so it needs to capture change o the racture aperture in the luid lo. he DFM-XFEM odel is able to siulate the situation hen racture aperture changes as described in Section 3.2. he odel paraeters are the sae as the above exaple. he coparison results or a odel ith changed and ixed racture aperture are shon in Figures 10 and 11. Figure 10 illustrates the displaceent at point A or 100 days and indicates that the displaceent or odel ith ixed racture aperture is larger than that or odel ith changed racture aperture ater a short tie and the displaceents or these to odels sho little dierence ater 50 days. Figure 11 illustrates the excess pore pressure at point B or 100 days and indicates that excess pore pressure or a odel ith ixed racture aperture is larger than that or odel ith changed racture aperture ater a short tie and excess pore pressures or these to odels sho little dierence ater 50 days. hese results are seen because the racture aperture is becoing saller, as shon in Figure 12. Figure 12 shos that the racture aperture decreases quickly ater a short tie and slos don the decreasing rate ater a longer tie. As the racture aperture decreases, the racture conductivity decreases, hich results in pore pressure decreasing ore sloly. x Displaceent/ odel ith racture aperture changed odel ith racture aperture ixed ie/day Figure 10. Displaceent at point A or 100 days.

13 Coputation 2015, Pressure/Pa odel ith racture aperture changed odel ith racture aperture ixed ie/day Figure 11. Excess pore pressure at point B or 100 days. 3.5 x Fracture aperture/ t=10days t=25days t=50days t=75days t=100days Distance along racture/ Figure 12. Fracture aperture distribution at dierent tie. he ethod is applied to another case, hich consists o to intersecting ractures. he doain has an area o he results o FM-XFEM odel and DFM-XFEM are also copared, as shon in Figures 13 and 14. It can be shon ro Figure 13 that the displaceent ields along y direction are very close or to ethods. he y displaceents in the area above the ractures are uch larger than that belo the ractures. It can be shon ro Figure 14 that the excess pore pressure distributions o to ethods are identical. he pressure around the racture is saller than other area because o larger conductivities in the ractures. x10-3 x (a) (b) Figure 13. Displaceent distribution () ater 100 days. (a) FM-XFEM odel; (b) DFM-XFEM odel.

14 Coputation 2015, (a) (b) Figure 14. Excess pore pressure distribution (Pa) ater 100 days. (a) FM-XFEM odel; (b) DFM-XFEM odel. he ater pressures at point (located at the right botto) are shon in Figure 15a at varying esh resolutions during 100 days. he plot is zooed in or 10 days to clariy the discrepancies beteen the various curves as shon in Figure 15b. Also, the results o FM-XFEM odel are shon in the igures. It can be shon that as the esh resolutions becoe higher, the results converge to that o FM-XFEM. he dierences beteen the are very sall. With regard to coputational cost, the results are shon in able 3. It can be shon that the coputational cost o DFM-XFEM odel is uch less than that o FM-XFEM odel. he results prove that the proposed ethod is better than FM-XFEM odel Pressure/Pa FM-XFEM 70*70 Mesh DFM-XFEM 1500 Eleents DFM-XFEM 2684Eleents DFM-XFEM 5926 Eleents Pressure/Pa FM-XFEM 70*70 Mesh DFM-XFEM 1500 Eleents DFM-XFEM 2684Eleents 1000 DFM-XFEM 5926 Eleents ie/day ie/day (a) (b) Figure 15. Excess pore pressure at right botto point. (a) 100 days; (b) 10 days. able 3. Coputational cost o odels. Models Coputational ie/s FM-XFEM Mesh 44.2 FM-XFEM Mesh DFM-XFEM 1500 Eleents 11.4 DFM-XFEM 2684 Eleents 22.1 DFM-XFEM 5926 Eleents 50.1

15 Coputation 2015, Conclusions In this paper, the cobination o discrete racture odel and extended inite eleent ethod is proposed to siulate luid-solid coupling in ractured porous edia. he discrete racture odel is able to capture luid lo accurately ithout accessing cross lo beteen atrix and racture. he extended inite eleent ethod is capable o solving solid deoration ithout esh reineent around ractures tips. he odel captures change o racture aperture during the siulation. he results o nuerical exaple sho that the proposed ethod is ell suited or siulating luid-solid coupling in ractured porous edia. Acknoledgents his paper is supported by the National Natural Science Foundation o China (No ) and Progra or Changjiang Scholars and Innovative Research ea in University (IR1294). Author Contributions Jun Yao established the atheatical odel. Qingdong Zeng perored the nuerical siulations. Both authors perored the analysis o the siulation results. Both authors have read and approved the inal anuscript. Conlicts o Interest he authors declare no conlict o interest. Appendix he eleents o the aoreentioned listed atrices are given as ollos i j Keij = ( B ) DB d i, j = u, a, b (A1) i L = ( B ) N d = u, a, b (A2) i p i F = N I ρ g d + N t I d Γ (A3) Γt F = HN ρ gd+ HN t dγ a I Γt I (A4) F = Φ γ I ρ d + Φ γ I d Γ b N g N t (A5) Γt K H = ( N p) N pd μ (A6) K H = ( N p) N pd (A7) μ φ S = N p N pd (A8) K

16 Coputation 2015, Reerences φ S = N p N pd (A9) K K = N p ( ρ g n q )dγ Γq μ (A10) u B = s N I (A11) a B = sn I( H( φ( x)) H( φ( xk))) (A12) b B = sn I( Φγ( x) Φγ( xk)) (A13) 1. Cottrell, M.; Hosseinpour, H.; Dershoitz, W. Rapid discrete racture analysis o hydraulic racture developent in naturally ractured reservoirs. In Proceedings o the Unconventional Resources echnology Conerence, Denver, CO, USA, August Minko, S.E.; Stone, C.M.; Bryant, S.; Peszynska, M.; Wheeler, M.F. Coupled luid lo and geoechanical deoration odeling. J. Petrol. Sci. Eng. 2003, 38, Boit, M.A. General heory o hree-diensional Consolidation. J. Appl. Phys. 1941, 12, erzaghi, V.K. heoretical Soil Mechanics; John Wiley & Sons, Inc.: Ne York, NY, USA, 1943; pp Hou, G.; Wang, J.; Layton, A. Nuerical ethods or luid-structure interaction A revie. Coun. Coput. Phys. 2012, 12, Moinar, A.; Narr, W.; Hui, M.H.; Mallison, B..; Lee, S.H. Coparison o Discrete-Fracture and Dual-Pereability Models or Multiphase Flo in Naturally Fractured Reservoirs; SPE Reservoir Siulation Syposiu: he Woodlands, X, USA, Warren, J; Roor, P.J. he behavior o naturally ractured reservoirs. SPE J. 1963, 3, Nie, R.S.; Meng, Y.F.; Jia, Y.L.; Zhang, F.X.; Yang, X..; Niu, X.N. Dual porosity and dual pereability odeling o horizontal ell in naturally ractured reservoir. ransp. Porous Med. 2012, 92, Karii-Fard, M.; Durlosky, L.; Aziz, K. An eicient discrete-racture odel applicable or general-purpose reservoir siulators. SPE J. 2004, 9, Huang, Z.; Yao, J.; Wang, Y.; ao, K. Nuerical study on to-phase lo through ractured porous edia. Sci. China echnol. Sci. 2011, 54, Lab, A.R.; Goran, G.J.; Elsorth, D. A racture apping and extended inite eleent schee or coupled deoration and luid lo in ractured porous edia. Int. J. Nuer. Meth. Eng. 2013, 37, Barenblatt, G.; Zheltov, I.P.; Kochina, I. Basic concepts in the theory o seepage o hoogeneous liquids in issured rocks [strata]. J. Appl. Math. 1960, 24, Belytschko,.; Black,. Elastic crack groth in inite eleents ith inial reeshing. Int. J. Nuer. Eng. 1999, 45, Stolarska, M.; Chopp, D.L.; Moës N.; Belytschko,. Modelling crack groth by level sets in the extended inite eleent ethod. Int. J. Nuer. Eng. 2001, 51,

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