Modeling of fluid flow through fractured porous media ...

Engineering Analysis with Boundary Elements 59 (2015) 166–171

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Modeling of fluid flow through fractured porous media by a single boundary integral equation M.N. Vu a,n, S.T. Nguyen a,c, M.H. Vu a,b a

R&D Center, Duy Tan University, Da Nang, Viet Nam CurisTec, 3 rue Claude Chappe, Parc d'affaire de Crécy, 69370 Saint-Didier-au-Mont-d'Or, France c Euro-Engineering, Pau, France b

art ic l e i nf o

a b s t r a c t

Article history: Received 31 October 2014 Received in revised form 7 June 2015 Accepted 10 June 2015 Available online 1 July 2015

The objective of this work is to provide theoretical materials for modelling two-dimensional fluid flow through an anisotropic porous medium containing intersecting curved fractures. These theoretical developments are suitable for numerical simulations using boundary element method and thus present a great advantage in mesh generation term comparing to finite volume discretization approaches when dealing with high fracture density and infinite configuration. The flow is modelled by Darcy’s law in matrix and Poiseuille’s law in fractures. The mass conservation equations, at a point on the fracture and an intersection point between fractures in the presence of a source or a sink, are derived explicitly. A single boundary integral equation is developed to describe the fluid flow through both porous media and fractures, i.e. the whole domain, which includes particularly the mass balance condition at intersection between fractures. Numerical simulations are performed to show the efficiency of this proposed theoretical formulation for high crack density. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Fractured porous media Fluid flow Mass exchange BIE Intersection

1. Introduction Modelling of fluid flow within porous geological formations containing a high fracture density is a subject of interest, as well as one of the most challenging problems for many application fields such as petroleum engineering, groundwater hydrology, geothermal energy, etc. To deal with real complex problems and to obtain local flow information, the numerical methods impose naturally. The domain discretization methods, such as finite element method (FEM) [1,2], finite volume method (FVM) [3,4], raise a major difficulty to generate an appropriate mesh for a domain containing numerous randomly distributed fractures. The methods, based on boundary integral equations (BEM), are computationally efficient and accurate for modelling the fluid flow in porous media thank to its advantage of reduction of problem dimension [5,6]. However, the classical BEM exhibits mathematical degeneracy for domain containing discontinuities. Alternative techniques are proposed for overcoming this difficulty such as Accelerated Perturbation BEM [7], Multi-domain Dual Reciprocity Method [8,9], Green element method [10,11], and Multi-region BEM [25–27]. However, these advanced techniques require a complicated numerical implementation in comparison with the standard BEM and also become inefficient for high fracture density.

n

Corresponding author. Tel.: þ 84 511 36 56 109. E-mail address: [email protected] (M.N. Vu).

http://dx.doi.org/10.1016/j.enganabound.2015.06.003 0955-7997/& 2015 Elsevier Ltd. All rights reserved.

Numerical developments, based on the symmetric Galerkin boundary element method, are presented by Rungamornrat and Wheeler [12] and Rungamornrat [13] for a full consideration of fluid flow through heterogeneous and anisotropic porous media containing non-conductive surface of discontinuity. The nonhomogeneous media consist of several subdomains with different properties. In those works, weak singular weak-form equations are established for pressure and its derivatives, i.e. flux flow. These interesting works are thus ready for application fields of fluid flow within porous media containing fractures or faults that act as barriers to flow. As a matter of fact, a fracture is generally much more conductive than the surrounding matrix, i.e. fluid flow presents both in the porous matrix and the fractures system. A system of boundary integral equations for flow in fractured porous media was first introduced by Rasmussen et al. [14], in which the matrix and fractures are treated as separate systems having a common interface made up of the fracture boundaries that are contained in the matrix. Therefore, this system is constituted by four equations: two boundary integral equations for the matrix and the fractures, and two conditions for pressure and velocity at fracture–matrix interfaces. Numerical computation requires a fine mesh to calculate accurately the difference between the unknowns at the collocation points on the two sides of the fractures. This difficulty is partially overcome in Lough et al. [15] by modelling the fractures as planar sources within the matrix. Nevertheless, this new model also consists in coupling two integral equations between matrix and fractures with the added unknowns of

M.N. Vu et al. / Engineering Analysis with Boundary Elements 59 (2015) 166–171

source strength on the planar fractures. This numerical procedure is used by Teimoori et al. [16,17] to simulate the naturally fractured reservoir. The integral equation has been also used to study the heat extraction by circulating water in a fracture embedded geothermal reservoir [18]. This approach allows eliminating the discretization of reservoir and takes into account multi-dimensional effects compared to theoretical solutions. The complicated numerical implementation of this method does not show the advantage of BEM over the FEM and FVM. This could be a reason why BEM community is less active than FEM and FVM one when dealing with fluid flow through porous media in the presence of high crack density. The present work aims to develope a single boundary integral equation (BIE) in order to describe fluid flow through twodimensional fractured porous media that allows dealing with high crack density. As a matter of fact, the fractures distributed within the porous media are obviously three-dimensional configuration. However, three-dimensional problems could be simplified to twodimensional ones in several applications such as Excavation Damaged Zone around a tunnel or a borehole [19], fault zone [20], etc. In the two-dimensional consideration, we make an assumption that fracture has zero thickness and an infinitely transversal permeability, i.e. there is no pressure jump across the fracture. The discontinuity of fluid flow across the fracture is related to fluid flow within the fracture by the mass balance equation at a point on the fracture. Mass exchanges between fractures and porous matrix, at intersection between fractures, in the presence of source and sink points, are explicitly formulated based on the recent works [21–24]. Considering the fractures as the internal boundary, the BIE is written for fluid flow within the porous matrix. This equation links to fluid flow by means of fracture by the boundary condition on the internal boundary, i.e. the pressure and flow at the fracture–matrix interfaces. The mass exchange between matrix and fracture, as well as the fluid flow constitutive law within the fracture result in a single BIE that describes fluid flow within whole fractured porous media. This BIE presents the pressure field as a function of the pressure and the flow on boundary of domain and the pressure on the fracture system. It is worth recalling that the condition at the intersection between fractures, in numerical simulation by whatever methods, has been ignored in the literature by the lack of an explicit mass conservation at this point. The development procedure of BIE allows integrating this mass conservation at intersection points between fractures. This cancel the singularity at these points. A single BIE, developed to describe fluid flow within both the fractures and embedding porous matrix, presents a great advantage in numerical implementation in comparison to a system of four equations [14–17]. A quick numerical resolution based on collocation method is performed in order to validate and show the efficiency of the proposed BIE.

2. Governing equations Considering a homogeneous medium Ω embedding a set of n interconnected fractures Γ ¼ [ Γ i (i¼1,n) (Fig. 1). In mathematical model, the fracture, supposed to have zero-thickness, is modelled by a smooth function zi(s) of the curvilinear abscise s. This function represents the positions of fracture i within the domain. The porous matrix corresponds to Ω  Г. There are m sources or sinks located at points of coordinates xk (k¼ 1,m) within Ω with corresponding intensities qk. These sources or sinks can be allocated within the porous matrix (mp points), on fractures (mf points) or at fractures intersection points (ms points), thus m¼mp þ mf þ ms. S denotes the set of intersection points between fractures, fracture endpoints and source or sink points on fractures.

167

Fig. 1. Fluid flow within fractured porous media.

Fluid flow is governed by Darcy’s law (1) in the matrix and Poiseuille’s law (2) in the fractures: 8 x A Ω  Γ;

8 s A Γ;

vðxÞ ¼ 

kðxÞ

μ

∇pðxÞ

qðsÞ ¼  cðsÞ∂s p

ð1Þ

ð2Þ

where v(x), p(x) are the fluid velocity and pressure fields within the porous matrix, respectively; k(x) the matrix’s intrinsic permeability tensor; q the fluid flow in the fracture and c the fracture conductivity. The fracture conductivity is determined commonly by cubical law as c ¼e3/(12fμ), in which e is the fracture aperture, μ the dynamic viscosity of fluid and f the roughness factor of fracture surfaces [28]. From geometrical point of view, the fracture has zero thickness to be modelled as a one dimensional curve within two dimensional spaces. However, in the physical model, there exists an aperture of facture, i.e. the fracture conductivity according to Poiseuille’s law (2). The continuity equation of fluid in the porous matrix reads: 8 x A Ω  Γ;

∇:vðxÞ þ

mp X k¼1

qk δðx  x k Þ ¼ 0

ð3Þ

where δ represents the Dirac distribution. The mass conservation for flow within the fracture, excluding the fractures intersection points and source or sink points, is written as [3,4,21,24] 8 s A Γ  S;

½½vðzÞ UnðsÞ þ ∂s q ¼ 0

ð4Þ

where z is the point on the factures at abscise s, n(s) the normal unit vector to the fracture oriented from Γ  to Γ þ and ½½vðzÞ ¼ v þ ðzÞ  v  ðzÞ is the velocity jump across the fractures. For the mass balance condition at the intersection point having no source or sink, Pouya and Vu [21] showed that the sum of P outgoing fluid flow vanishes, i.e. i q0i ¼ 0, where q0i is the outflow in the fracture branch i from the intersection point. Their demonstration method is used to derive the mass conservation expressions within the fractures or at fracture intersection points including sources or sinks. Considering a small domain D surrounding a fracture intersection point z where there is the presence of a source with intensity qs(z). There are I  1 fractures come together at this point. Let us now replace the source by a fictitious fracture ΓI meeting with other fractures at z. The flow within ΓI is constant and equal to qs ¼ qI (Fig. 2a), i.e. there is no exchange between the fictitious fracture and the porous matrix.

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ΓI

Γ i+1 n

v

n ΓI

q DI

v

q Di +1

q 0I

vx v+ n

q 0i +1 D i,i+1 n

D

i 0

q

n

q Ds n

∂D i,i+1

D ql vx v+ n

qDi Γi

∂D

q Dl

qs n q

r

q Dr Γi

∂D

Fig. 2. Mass exchange at fracture intersection points (a), on the fracture (b) including sources (modified from [21]).

Fig. 3. Fractures are divided in fracture segments by intersection points.

The mass balance for the domain D reads as Z X v:n ds þ qD i ¼0 ∂D

ð5Þ

i

where qD i is the outflow at the domain boundary ∂D from the fracture i (i.e. at the intersection point between ∂D and Γi). Moreover, the domain D can be split into I subdomains Di,i þ 1 (with the convention I þ1 ¼1 and ΓI þ 1 ¼ Γ1) (Fig. 2a). The mass balance for each subdomain Di,i þ 1 becomes Z Z Z v U n ds  v þ Un ds þ v  U n ds ¼ 0 ð6Þ Γ Di

∂Di;i þ 1

Γ Diþ 1

where Γ i is the portion of Γi within the domain D and n the outward unit normal on the boundary ∂Di,i þ 1. Summing Eq. (6) for i run from 1 to I, we obtain Z XZ v U n ds ½½v U n ds ¼ 0 ð7Þ D

∂D

i

Γ Di

Applying Eq. (4) to the segment Γ i yields X XZ 0 ½½v Un ds ¼  ðqD i  qi Þ D

i

Γ Di

ð8Þ

i

Combination of Eqs. (5), (7) and (8) results in I X i¼1

q0i ¼

I1 X

q0i þ qs ¼ 0

impermeable rock [29,30]. The sum of outgoing flow in the fracture from the intersection point is equal to the source intensity. The fractures, meeting at the intersection, could have different conductivities. This expression helps avoid the numerical singularity at the fractures intersection points in modelling of fluid flow by using BEM as will be shown in the next section. Indeed, the Poiseuille’s law (2) and the mass exchange (4) at a point on the fracture are valid for transient fluid flow. Therefore, the derivation procedure of mass exchange at intersection between fractures from Eq. (5) to (8), as well as the final solution (9) are also true for transient fluid flow. This derivation method could also be applied for a point on the fracture where there is a source or sink of the intensity qs. The mass conservation is expressed as follows: ql þ qr þ qs ¼ 0 l

ð10Þ

r

where q , q are the outflow in the left and right branches of the fracture from the source point, respectively (Fig. 2b). Dirichlet and Neumann conditions are prescribed respectively on ∂Ω1, ∂Ω2 of the domain boundary: 8 x A ∂Ω1 ;

pðxÞ ¼ p0 ðxÞ

ð11Þ

8 x A ∂Ω2 ;

 k∇pðxÞ U n ¼ v0 ðxÞ

ð12Þ

ð9Þ

i¼1

Eq. (9) shows that the mass conservation at the fracture intersection point is independent of the fracture–matrix mass exchange, which is the same for a fracture network in an

3. Boundary integral formulation Consider a porous domain Ω containing n fractures that can intersect together. A fracture numbered i, intersected by other

M.N. Vu et al. / Engineering Analysis with Boundary Elements 59 (2015) 166–171

fractures at pi intersection points, is divided in pi þ1 segments in a way that a segment is limited by two consecutive intersection points except two extremity segments are bounded by a fracture tip and the closest intersection point. (Fig. 3). Supposing that the porous medium Ω presents n0 segments of fractures that are limited by ns fracture tips and intersection points, called by singular points S. A fracture segment Γ of normal n is constituted by two faces Γ þ and Γ  with two normal vectors: n þ ¼ n  ¼n, (Fig. 3). Across this facture segment, the pressure is continuous: p þ ¼p  while the velocity is discontinuous v þ av  . The normal velocities v þ n þ and v  n  prescribed on the positive and negative fracture lips are related to flow within the fracture by Eq. (4). Writing BIE for fluid flow via porous matrix Ω  Г, results in the following field pressure solution:

λðξ Þpðξ Þ ¼ 8ξ AΩГ

þ þ

n' Z X þ k ¼ 1 Γk

n' Z X

k¼1 Γ

 k

R

∂Ωi pðxÞ∇Gðx;

ξ Þ U m ds 

p þ ðxÞ∇Gðx; ξ Þ U m þ ds  p  ðxÞ∇Gðx; ξ Þ U m  ds 

R

n' Z X

∂Ωi Gðx;

þ i ¼ 1 Γk

n' Z X

 i ¼ 1 Γk

ξ Þ∇pðxÞ U m ds

Gðx; ξ Þ∇p þ ðxÞ U m þ ds Gðx; ξ Þ∇p  ðxÞ U m  ds

ð13Þ pffiffiffiffiffiffiffiffi where ξ is the field point, m ¼ k= j kj n with n the normal of the domain boundary ∂Ω. G(x,ξ) is the fundamental solution of Laplace’s equation for anisotropic material: 

Gðx; ξ Þ ¼

1 pffiffiffiffiffiffiffiffiLn‖ 2π j kj

qffiffiffiffiffiffiffiffiffi 1 k ðx  ξ Þ‖;

∇Gðx; ξ Þ U m ¼

  1 k x ξ Um 1 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi   2π j kj ‖ k  1 U x  ξ ‖2

ð14Þ where λ(ξ) is a coefficient that depends on the position of ξ relative to the subdomain Ω: λ(ξ)¼1 if ξ  Ω  ∂Ω; λ(ξ) ¼0 if ξ g Ω and 0 o λ(ξ)o 1 if ξ  ∂Ω. For the latter case when the field point on the boundary ∂Ω, the coefficient λ depends on the local boundary geometry at this point and it takes the value of 12 for smooth boundary. In general, this coefficient can be determined by the following equation: 8 ξ A ∂ Ω;

λðξ Þ ¼ lim

Z

ε-0 Sε

Gðx; ξ Þdx

ð15Þ

where Sε is an infinitesimal circular surface of centre ξ and radius ε enclosed in the solid Ω. Since two faces Γ þ and Γ  of fracture Γ are coincidental with the normal m þ ¼ m  ¼m and there is no pressure jump across it, Eq. (13) becomes

λðξ Þpðξ Þ ¼ 

Z ∂Ω

n' Z X k¼1

Γk

pðxÞ∇Gðx; ξ Þ Um ds 

Z ∂Ω

Gðx; ξ Þ∇pðxÞ:mds

  Gðx; ξ Þ ∇p þ ðxÞ  ∇p  ðxÞ U m ds

ð16Þ

This equation could be also obtained by using the idea of multiregion BEM, which has been successfully used for the linear fracture mechanic [6]. In this framework, the domain is divided into sub-domains by the fractures and additional fictitious interfaces. Fluid flow BIE are written at a field point for all subdomains. Assembling these all BIE and associating to consideration of fluid flow within fracture, as well as the mass exchange between fractures and porous matrix allow resulting in Eq. (16). Moreover, the gradient pressure jump appearing in Eq. (16) could be expressed through the fluid flow within the fractures by between matrix and fracture (4) as: the þ mass exchange   1 þ ∇p ðxÞ  ∇p  ðxÞ U m ¼ k v ðxÞ  v  ðxÞ Ukn ¼ ∂s q. Introducing

169

this relationship to Eq. (16) leads to

λpðξ Þ ¼

Z ∂Ω

pðxÞ∇Gðx; ξ Þ U m ds 

Z ∂Ω

Gðx; ξ Þ∇pðxÞ U m ds 

n' Z X k¼1

Γk

Gðx; ξ Þ∂s q 1ds

ð17Þ Integrating by part the third integral on the right hand side of this equation yields 0 1 mk n Z n' Z X X X X ∂Gðx; ξ Þ j k ds Gðx; ξ Þ∂s q ds ¼ Gðz ; ξ Þ@ q0k A  qðsÞ k ∂s S j¼1 k ¼ 1 Γk k¼1 Γ ð18Þ where zk AS is an intersection point of mk fractures or an endpoint of a fracture (mk ¼ 1), qj0k the outgoing flow at point zk in the branch j. When the source or sink points are not considered, Pmk j j ¼ 1 q0k ¼ 0 according to Eq. (9), i.e. the first term on the right hand side of Eq. (18) vanishes. Besides, the derivation of fundamental solution regarding the curvilinear abscises s of fracture curve is given by   1 k xðsÞ  ξ U t ðsÞ ∂Gðx; ξ Þ 1  ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi  ð19Þ ∂s 2π j kj ‖ k  1 U xðsÞ  ξ ‖2 where t(s) is the unit vector tangential to the fracture. Substituting Eq. (18) into Eq. (17) gives the integral representation of solution for the whole fractured porous media

λpðξ Þ ¼

Z ∂Ω

pðxÞ∇Gðx; ξ Þ U m ds 

Z ∂Ω

Gðx; ξ Þ∇pðxÞ U m ds þ

n' Z X k¼1

Γk

qðsÞ

∂Gðx; ξ Þ ∂s

ds

ð20Þ Furthermore, the flow within the fracture can be expressed by Eq. (2), thus: λpðξ Þ ¼

Z ∂Ω

pðxÞ∇Gðx; ξ Þ U m ds 

Z ∂Ω

Gðx; ξ Þ∇pðxÞ U m ds

n' Z X k¼1

Γk

ck ðsÞ∂s p

∂Gðx; ξ Þ ∂s

ds

ð21Þ where ck is the hydraulic conductivity of fracture segment numbered k. This equation is the combination between the classical boundary integral equation for potential problem over a domain having no discontinuity and the contribution of conductive fractures. Considering the field point ξ on the boundary of domain ∂Ω or on the fractures Γ, we have a singular integral equation with the unknown variables of pressure and of pressure gradient on ∂Ω and the unknown pressure on Γ. As a conclusion, the use of BEM approach, only for porous matrix surrounding the fractures, then associating to the mass exchange between matrix and fractures at a point on the fracture (Eq. (4)) and at an intersection point between fractures (Eq. (9)), as well as the constitutive law of fluid flow through the fracture (Eq. (2)) leads to only one BIE (21) that describes the fluid flow through fractured porous media. Hence, this theoretical solution presents an advantage in comparison to a system of four equations obtained by Rasmussen et al. [14], Lough et al. [15] and Teimoori et al. [16,17].

4. Numerical applications For now, a numerical resolution, based on the collocation method, has been developed to deal with Eq. (21). General description of this method for potential problem in finite domain without fractures can be found in BEM books [5,6]. Once this equation is solved, the unknown obtained, on the boundary and on the fractures, are reinjected in Eq. (21) to compute the pressure field within the whole domain. A summary of the numerical algorithm is presented in the following to obtain the numerical solution of Eq. (21).

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Discretization of BIE (21) is performed by the geometry discretization and choosing the variable interpolation functions. The mesh is self-consistent at intersection points and is fine around fracture extremities. On the boundary, a linear variation of pressure and normal flux over each element are assumed. Whereas, two types of elements are distinguished on the fracture: current elements and extremity elements where fractures end in the matrix. A linear interpolation of pressure, i.e., constant flux, is used for current elements. However, an interpolation function corresponding to a variation as according to s1/2 (s is the distance from extremity point on fracture) is used for discharge q(s) on the extremity elements. On the boundary, the nodes are taken as the collocation point. For an element on fractures, a collocation point set is chosen to cancel the integral over this element, thus to eliminate the singularity. And then, the extremities of fractures are added as the supplementary collocation points. This choice makes the number of collocation points is always greater than the number of nodes. Nevertheless, the approximated solution may be computed by using the least square fitting method. In the following, two numerical applications using the theoretical description in the two previous sections are performed. The first one is devoted to validate the numerical procedure by comparing the fluid flow result obtained by boundary integral method against one of finite element simulation for a simple configuration. The second one is dedicated to show the efficiency of the proposed BIE for a high crack density configuration.

methods on the cut from the middle of left side to one of right side is presented in Fig. 5. A perfect agreement between these two solutions is observed.

4.2. High fracture density This section is devoted to apply the BEM numerical programme for computing fluid flow through a porous domain containing a high fracture density to show the efficiency of proposed method. Three fracture families are generated randomly by its orientation and its positions. Each fracture family, numbered by i (i¼ 1,2,3), is characterized by a length 2Li, a conductivity ci and a density ρi ¼ Ni/Ω, where Ni is the number of fracture within the area Ω. Fig. 6 displays the pressure field solutions for a fracture configuration, composed of 40 fractures of length 2L1 ¼0.4 m, 40 fractures of length 2L2 ¼0.2 m and 40 ones with 2L3 ¼0.1 m, that is distributed within a square B¼ 1 m for two boundary conditions p(x)¼x and p (x)¼y. These solutions also show the important pressure gradient in the vinicity of fracture extremities vis-a-vis in the porous matrix. As a matter of fact, it is not evident to obtain the numerical result for such configuration by using other numerical methods (FEM, FVM) owning to a complicated mesh generation.

5. Conclusion

Consider a simple configuration where a porous medium of permeability k contains a dilute presence of fractures (Fig. 4). Applying the BEM numerical programme for this case, with B¼ 1 m, k¼ 1 and c¼ 1000 and the boundary condition p(x)¼ x, yields the pore pressure field in Fig. 4 itself. The result shows that the pressure is almost constant along fractures since the fracture is much more conductive than the matrix. Moreover, the pressure contour is dense around the fracture extremities and the pressure gradient behaves asymptotically as r  0.5 in the vicinity of these singular points, where r is the distance to the fracture ends. Nevertheless, this phenomenon is not observed at the intersections between the fractures. As shown in Section 2, these points are not singular ones thank to the mass conservation expression (9). In order to validate the theoretical developments in this work, the same configuration is simulated by FEM in which the fracture is modelled by joint element of zero thickness. As expected, the finite element simulation gives the similar result compared to BEM approach. The comparison of pore pressure obtained by these two

Constitutive equations and mass balance equations, governing fluid flow through fractured porous media, are described. The fracture is mathematically represented by a curve of zerothickness with infinite transversal permeability, i.e. no pressure jump between its two fracture lips. The mass exchange between fractures and matrix, at a point on the fracture or at an intersection point between fractures in the presence of sources or sinks, are derived in an explicit form. The pressure field BIE, writing for porous matrix embedding the fractures, incorporates to these mass conservation expressions and Poisseuille’s flow within the fracture to obtain a single boundary integral solution that describes the fluid flow within the whole fractured porous media. The mass conservation condition at intersections between fractures is taken into account in this BIE. This proposed BIE presents an advantage when dealing with high fracture density in comparison with finite element or volume element methods and other BEM techniques. A simple numerical resolution of this BIE based on collocation method has been performed to validate and to show the advantage of the developed BIE. To reduce the resource of computation (time and memory), the fast multipole method

Fig. 4. Pressure field within a porous medium containing a dilute presence of fractures.

Fig. 5. Comparison of effective permeabilities obtained by BEM and FEM for a simple configuration.

4.1. Validation of numerical solution

M.N. Vu et al. / Engineering Analysis with Boundary Elements 59 (2015) 166–171

171

Fig. 6. Pressure field solution for the case: k ¼ 10  12 m/s, c ¼ 5  10  11 m2/s at: (a) A ¼ (1,0) and (b) A ¼ (0,1).

[31,32] could be used, in particular for modelling the unsteady fluid flow in three-dimensional finite fractured porous media. Acknowledgements The authors wish to thank the editor and three anonymous reviewers for their constructive comments on earlier versions of this article.

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