Research ArticleAn Extended Finite Element Model for Fluid Flow inFractured Porous Media

Fei Liu1 Li-qiang Zhao1 Ping-li Liu1 Zhi-feng Luo1 Nian-yin Li1 and Pei-shan Wang2

1State Key Laboratory of Oil amp Gas Reservoir Geology and Exploitation Southwest Petroleum UniversityChengdu Sichuan 610500 China2Exploration Utility Department Southwest Oil and Gas Field Company PetroChina Chengdu Sichuan 610000 China

Correspondence should be addressed to Fei Liu liufei10628aliyuncom

Received 31 December 2014 Accepted 5 January 2015

Academic Editor B Rush Kumar

Copyright copy 2015 Fei Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a numerical model for the fluid flow in fractured porous media with the extended finite element method Thegoverning equations account for the fluid flow in the porousmedium and the discrete natural fractures as well as the fluid exchangebetween the fracture and the porous medium surrounding the fracture The pore fluid pressure is continuous while its derivativesare discontinuous on both sides of these high conductivity fractures The pressure field is enriched by the absolute signed distanceand appropriate asymptotic functions to capture the discontinuities in derivatives The most important advantage of this methodis that the domain can be partitioned as nonmatching grid without considering the presence of fractures Arbitrarily multiplekinking branching and intersecting fractures can be treated with the new approach In particular for propagating fractures suchas hydraulic fracturing or network volume fracturing in fissured reservoirs this method can process the complex fluid leak-offbehavior without remeshing Numerical examples are presented to demonstrate the capability of the proposed method in saturatedfractured porous media

1 Introduction

Flow models estimating the flow in fractured porous mediamainly include the equivalent continuum model [1] dualcontinuum model [2] discrete fracture model [3] and dis-crete fracture networkmodel [4 5]Thesemodels fall into twocategories continuum models and discrete fracture modelsIn the equivalent continuummodel a representative elemen-tary volume (REV) is required the fractures are assumed todistribute regularly and the equivalent permeability is hardto determine [6] which make it unavailable while severallarge fractures existed such as hydraulic fractures complexfractures network and big faults Dual continuum modelcannot describe the flow in fracture because the fractures arenot treated explicitly in it [7 8] Discrete fracture networkmodel is more real and gains worldwide concern recentlybecause the fractures are treated explicitly and the effect ofone crack on the whole flow could be considered [9] YaoJun et al [10] presented a two-dimensional two-phase finiteelement flow model based on explicit treating of discrete

fracture network In the model matrix domain is partitionedusing triangle finite element method (FEM) grids while largefracture is partitioned using one-dimensional line elementsThe grids are required to match the explicit fractures andnodes are needed on both fracture intersections and cracktips to better solve such static fracture problems Moreovertedious remeshing is necessary for propagating fractures [11]The extended finite element method (XFEM) is an effectiveapproach to avoid remeshing and is now widely used to solvediscontinuity problem [12ndash14] Khoei et al [15] simulated thecoupled thermohydromechanical (THM) model for imper-meable discontinuities in saturated porous medium withXFEMMohammadnejad andKhoei [16] presented a coupledhydromechanical (HM) model with combined XFEM andcohesive crack Lamb [17] and Lamb et al [18] combinedXFEM with the dual-porosity and dual-permeability modelto describe the fluid flow deformation and fracture prop-agation in fractured porous medium By virtue of XFEMand lower-dimensional interface elements Watanabe et al[19] researched coupled HM problems in discrete fractures

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 604212 10 pageshttpdxdoiorg1011552015604212

2 Mathematical Problems in Engineering

porous media systems Based on the methodology of XFEMin this paper the pressure field is enriched by the absolutesigned distance and appropriate asymptotic functions todevelop an XFEM model for fluid flow in fractured porousmedium The results demonstrate that the XFEM is anefficient method for simulating fluid flow in fractured porousmedium with nonmatching grids especially when the fissureis propagating such as hydraulically driven fractures

2 Governing Equation

The strong form as well as the associated weak form ofgoverning equations for fluid flow inside bothmatrix domainand fracture domain is demonstrated in the section

21 Strong Form According to the law of mass conservationthe continuity equation for flow in the matrix domain iswritten as

(120572119898

minus 120601119898

119870119904

+ 120601119898

119870119908

) 120597119901119908

119898

120597119905 + nabla sdot (minusK119898

120583119908

nabla119901119908

119898) + 119876

119908119904119905119888119861119908

= 0(1)

Similarly the continuity equation for flow inside thefracture domain is given by

1119870119908

120597119901119908

119891

120597119905 + nabla sdot (minus119870119891

120583119908

nabla119901119908

119891) = 0 (2)

where 120572119898is the Biot constant 120593

119898is the porosity of matrix

rock 119870119904is the bulk modulus of solid phase Pa 119870

119908is the

bulkmodulus of pore fluid Pa119901119908

119898is the pore fluid pressure in

matrix Pa 119905 is time sK119898is the permeability tensor ofmatrix

m2 120583119908is the pore fluid viscosity Pasdots119876

119908119904119905119888is the sourcesink

term on the ground m3(m3sdots) 119861119908is the bulk coefficient 119901119908

119891

is the fluid pressure in fracture Pa and119870119891is the permeability

of fracture m2 calculated according to the following formula[20]

119870119891

= 1119891

119887212 (3)

where 119891 is a morphological parameter accounting for thedifference between real fracture and ideal parallel fractureranging from 104 to 165 119887 is the mechanical width offracture m

22 Weak Form In order to deduce the weak form ofgoverning equations a two-dimensional domain Ω boundedby the boundary Γ is considered as shown in Figure 1 Naturalfracture with high permeability insideΩ is regarded as a one-dimensional discontinuous line because the fracture width ismuch less than the length Γ+

119891and Γminus

119891are used to represent two

faces of the fracture respectively

qw

Γpw

Γqw

ΓΩ

+minus

minus

+

pw

Ωf

x

y 119847Γ119891

119847minusΓ119891

119847+Γ119891

Figure 1 The domain and boundary for fractured porous media

The initial and boundary conditions are as follows

119901119908 = 1199011199080 forallx isin Ω119901119908

119898(x) = 119901119908

119898x isin Γ

119901119908

(minusK

119898

120583119908

nabla119901119908

119898) sdot n

Γ= 119902

119908

120588119908

forallx isin Γ119902119908

119901119908

119898= 119901119908

119891forallx isin Γ

119891

(minusK119898

120583119908

nabla119901119908

119898) sdot n

Γ119891

= minus (minus119870119891

120583119908

nabla119901119908

119891) sdot n

Γ119891

= 119902119908

120588119908

(4)

where nΓis the unit normal vector of the boundary 120588

119908is

fluid density kgm3 and 119902119908is the flow rate on the boundary

kg(m2sdots)The weak forms of equations for flow in matrix and

fracture are derived by weighted residual method

intΩ

w119879

119901(120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

) 120597119901119908

119898

120597119905 119889Ω + intΩ

(nablaw119901)119879 K

119898

120583119908

nabla119901119908

119898119889Ω

+ intΩ

w119879

119901119876119908119904119905119888

119861119908

119889Ω + intΓ+

119891

w119879

119901

119902+119908

120588119908

119889Γ + intΓminus

119891

w119879

119901

119902minus119908

120588119908

119889Γ

+ intΓ119902119908

w119879

119901

119902119908

120588119908

119889Γ = 0

(5)

intΩ119891

w119879

119901

1119870119908

120597119901119908

119891

120597119905 119889Ω + intΩ119891

(nablaw119901)119879 119870

119891

120583119908

nabla119901119908

119891119889Ω minus int

Γ+

119891

w119879

119901

119902+119908

120588119908

119889Γ

minus intΓminus

119891

w119879

119901

119902minus119908

120588119908

119889Γ = 0(6)

in whichΩ119891is the fracture domainThe integrals over Ω

119891are

performed in the local Cartesian coordinate system (1199091015840 1199101015840)whose 1199091015840 axes and 1199101015840 axes are consonant with tangentialdirection and normal direction respectively

With regard to the flow inside the fracture both fluidpressure and its interpolating function are assumed to beuniform along fracture width [16] as shown in Figure 2

Mathematical Problems in Engineering 3

p+

p+

pminus

pminus

pf b

b

Ωminus

Ω+

A

A A998400

A998400

Figure 2 Fluid flow and pore fluid pressure around the fracture

minus1minus05

005

1

minus1minus05

005

10

02

04

06

08

1

r cos 120579r sin 120579

rco

s 1205792

(a) The first branch function

minus1

minus05

0

05

1

minus1minus05

005

1

minus1

minus05

0

05

1

r cos 120579

r sin 120579

minusr2

sin 1205792

(b) Its derivative for pore pressure

Figure 3 The first branch function and its derivative for porepressure

Thus the first integral of (6) can be rewritten as [16]

intΩ119891

w119879

119901

1119870119908

120597119901119908

119891

120597119905 119889Ω = intΓ119891

int1198872

minus1198872

w119879

119901

1119870119908

120597119901119908

119891

120597119905 1198891199101015840119889Γ

= intΓ119891

w119879

119901

119887119870119908

120597119901119908

119891

120597119905 119889Γ(7)

No

flow

No

flow

p = 0kPa

p = 286 kPa

L = 10m

W = 16m

Kf = 80DKm = 8mD

b = 1mm

Figure 4 2D fractured domain with applied pressure at top andbottom boundary

Because of the supposed uniform distribution of fluidpressure on the 1199101015840 axes the derivative does not vary with 1199101015840thus the second integral is expressed as [16]

intΩ119891

(nablaw119901)119879 119870

119891

120583119908

nabla119901119908

119891119889Ω = int

Γ119891

int1198872

minus1198872

(nablaw119901)119879 119870

119891

120583119908

nabla119901119908

1198911198891199101015840119889Γ

= intΓ119891

119887119870119891

120583119908

120597119908119879

119901

1205971199091015840

120597119901119908

119891

1205971199091015840119889Γ

(8)

Substituting the constituents of (6) and rearranging it theweak form of continuity equation for flow inside fracture isgained Adding it to the weak form for flow in matrix regionthe fluid exchange term can be eliminated as (9) shows Thusit is not necessary to express fluid exchange explicitly

intΩ

w119879

119901(120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

) 120597119901119908

119898

120597119905 119889Ω

+ intΩ

(nablaw119901)119879 (K

119898

120583119908

nabla119901119908

119898) 119889Ω + int

Γ119902119908

w119879

119901

119902119908

120588119908

119889Γ

+ intΓ119891

w119879

119901

119887119870119908

120597119901119908

119891

120597119905 119889Γ + intΓ119891

120597w119879

119901

1205971199091015840

119887119870119891119870119903119908

120583119908

120597119901119908

119891

1205971199091015840119889Γ = 0

(9)

3 Pressure Enrichment with XFEM

For permeable natural fracture the pressure field is continu-ous at the fracture surface while the derivative of pressure(ie flow rate) is discontinuous The enrichment of thediscontinuity in derivative is constructed as follows [21]

4 Mathematical Problems in Engineering

(a) DFM (b) FM-FEM (c) FM-Mfree

(d) XFEM

Figure 5 Domain discretization for different methods

For a node with a bisected support the absolute signeddistance function is applied to realize the enrichment so as tomeet the condition of continuous pressure and discontinuousderivative

119865 (x) = 1003816100381610038161003816119891 (x)1003816100381610038161003816 (10)

For a support which is slit by the discontinuity the branchfunction is applied to realize the enrichment

119861119897 (x) = [119903 cos 1205792 1199032 cos 120579

2 radic119903 cos 1205792] (11)

Its derivate with respect to 120579 is

120597119861119897 (x)120597120579 = [minus 119903

2 sin 1205792 minus1199032

2 sin 1205792 minusradic119903

2 sin 1205792] (12)

in which 120579 is the intersection angle between local coordinatesystem for fracture and global coordinate system

The first branch function and its derivative for porepressure are shown in Figure 3 On the surface of fracture(ie 120579 = 120587 and 120579 = minus120587) the interpolation function iscontinuous while its derivative is discontinuous The othertwo interpolation functions share the same characteristics

For the nodes cut by two intersecting discontinuities theenrichment is realized through the product of two absolutesigned distance functions related to the two discontinuities

119869119897 (x) = 10038161003816100381610038161198911 (x)1003816100381610038161003816 10038161003816100381610038161198912 (x)1003816100381610038161003816 (13)

According to the fundamental XFEM the local enrich-ment function and extra degrees of freedom are introduced

Mathematical Problems in Engineering 5

(a) DFM (b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) FM-Mfree

0 48 95 143 191 238 286

Pressure (kPa)

(d) XFEM

Figure 6 Pressure field obtained from different methods (independent fracture)

for interpolation of fluid pressure field Then the pressureafter enrichment can be expressed as

119901ℎ (x) = sum119894isin119878

119873119894(x) 119901

119886119894+

119873119888sum

119895=1

sum119894isin119878119867119895

119901119887119894119895

119873119894(x) [10038161003816100381610038161003816119891119895 (x)10038161003816100381610038161003816 minus 10038161003816100381610038161003816119891119895 (x

119894)10038161003816100381610038161003816]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

enrichment of bisected support for fracture 119895

+119873119905sum

119895=1

sum119894isin119878119862119895

119873119894(x) 119877119895 (x)

3

sum119897=1

119901119897

119888119894119895[119861119897

119895(x) minus 119861119897

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of slit support for fracture tip 119895

+119873119909sum

119895=1

sum119894isin119878119869119895

119901119889119894119895

119873119894(x) [119869

119895(x) minus 119869

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of intersecting support for cross point 119895

(14)

where 119873119894is the node interpolation function 119877 is the ramp

function 119878 is the set of all nodes 119878119867119895

is the set of nodesenriched by absolute signed distance function for fracture119895 119878

119862119895is the set of nodes enriched by appropriate asymptotic

function for fracture tip 119895 119878119869119895is the set of nodes enriched

by intersecting function for cross point 119895 119901119886is the freedom

degree of regular nodes (119886-freedomdegree)119901119887is the freedom

6 Mathematical Problems in Engineering

degree of extra nodes in bisected support (119887-freedomdegree)119901119888is the freedom degree of extra nodes in slit support (119888-

freedom degree) and 119901119889is the freedom degree of extra nodes

in intersecting support (119889-freedom degree)

4 Discretization Equation of XFEM

Regarding trial function w119901as the variation of pore pressure

interpolation function the XFEM discretization equation of(9) in space is then gained considering the irrelevance of thevalues of 119901

119886 119901

119887 119901

119888 and 119901

119889

([[[[[[

H119908

119898119886119886H119908

119898119886119887H119908

119898119886119888H119908

119898119886119889

H119908

119898119887119886H119908

119898119887119887H119908

119898119887119888H119908

119898119887119889

H119908

119898119888119886H119908

119898119888119887H119908

119898119888119888H119908

119898119888119889

H119908

119898119889119886H119908

119898119889119887H119908

119898119889119888H119908

119898119889119889

]]]]]]

+[[[[[[

H119908

119891119886119886H119908

119891119886119887H119908

119891119886119888H119908

119891119886119889

H119908

119891119887119886H119908

119891119887119887H119908

119891119887119888H119908

119891119887119889

H119908

119891119888119886H119908

119891119888119887H119908

119891119888119888H119908

119891119888119889

H119908

119891119889119886H119908

119891119889119887H119908

119891119889119888H119908

119891119889119889

]]]]]]

)

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

+ ([[[[[[

W119908

119898119886119886W119908

119898119886119887W119908

119898119886119888W119908

119898119886119889

W119908

119898119887119886W119908

119898119887119887W119908

119898119887119888W119908

119898119887119889

W119908

119898119888119886W119908

119898119888119887W119908

119898119888119888W119908

119898119888119889

W119908

119898119889119886W119908

119898119889119887W119908

119898119889119888W119908

119898119889119889

]]]]]]

+[[[[[[

W119908

119891119886119886W119908

119891119886119887W119908

119891119886119888W119908

119891119886119889

W119908

119891119887119886W119908

119891119887119887W119908

119891119887119888W119908

119891119887119889

W119908

119891119888119886W119908

119891119888119887W119908

119891119888119888W119908

119891119888119889

W119908

119891119889119886W119908

119891119889119887W119908

119891119889119888W119908

119891119889119889

]]]]]]

) 119889119889119905

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

=

119902119908119886

119902119908119886

119902119908119886

119902119908119886

(15)

where flow matrix H compressibility matrix W flow ratevector q at the equivalent node the unit tangent vector I119905

119891of

local coordinate system and coordinate transform matrix Rare respectively expressed as follows

H119908

119898120572120573= int

Ω

(nablaN119901

120572)119879 K

119898

120583119908

nablaN119901

120573119889Ω 120572 120573 = 119886 119887 119888 119889

H120587

119891120572120573= int

Γ119891

(nablaN119901

120572)119879R119879I119905

119891

119887119870119891

120583119908

(I119905119891)119879RnablaN119901

120573119889Γ

120572 120573 = 119886 119887 119888 119889

0

50

100

150

200

250

300

0 4 8 12 16

Pres

sure

(kPa

)

FM-FEM DFMXFEM FM-Mfree

Height in Y direction (m)

Figure 7 Pressure profile for a vertical section taken from the centerof the domain

W119908

119898120572120573= int

Ω

(N119901

120572)119879 (120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

)N119901

120573119889Ω

120572 120573 = 119886 119887 119888 119889W119908

119891120572120573= int

Γ119891

(N119901

120572)119879 119887

119870119908

N119901

120573119889Γ 120572 120573 = 119886 119887 119888 119889

119902119908120572

= intΓ119902119908

(N119901

120572)119879 119902

119908

120588119908

119889Γ 120572 = 119886 119887 119888 119889

I119905119891

= [1 0]119879

R = [ cos 120579 sin 120579minus sin 120579 cos 120579]

(16)

Equation (15) can be rewritten as

AX + B119889X119889119905 = C (17)

in which A is the sum of matrix H119898

of flow in matrixregion and matrix H

119891of flow in fracture B is the sum of

matrixW119898of rock compressibility andmatrixW

119891of fracture

compressibility X is the vector of pore pressure field and Cis the vector of flow rate at the equivalent node

In the time domain (17) can be discretized by the use ofNewmark method as follows [22]

(B + Δ119905120599A)X119899+1

= Δ119905C|119899+120599

+ [B minus (1 minus 120599) Δ119905A]X119899 (18)

in which parameter 120599 satisfies 0 le 120599 le 1 Only when 120599 ge 12the solution is unconditionally stable when 120599 = 12 it is thecentral difference in the time domain and when 120599 = 1 it isfully implicit

Mathematical Problems in Engineering 7

0 48 95 143 191 238 286

Pressure (kPa)

(a) DFM

0 48 95 143 191 238 286

Pressure (kPa)

(b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) XFEM

Figure 8 Pressure field obtained from different methods (intersecting fractures)

5 Numerical Examples

51 Steady State Flow For a two-dimensional fractureddomain with an independent fracture shown in Figure 4Lamb [17] compared the calculated results of discrete fracturemodel (DFM) fracturemapping withmesh free (FM-Mfree)and fracturemapping with finite element method (FM-FEM)and verified the correctness of fracturemappingmethodThedomain discretization pressure field and pressure profile fora vertical section taken from the center of the domain for eachmethod are respectively demonstrated in Figures 5 6 and 7

It is observed in Figure 5 that the mesh edges need to beset on the fracture and nodes need to be set on the fracturetip for DFM model and the mesh node needs to be set onthe fracture for FM-Mfree model However the FM-FEMand XFEM models can simulate the fluid flow in fracturedporous medium with nonmatching grid The permeabilityof elements with fracture is processed equivalently for FM-FEM model while the permeability of fracture is calculatedonly with cubic law or others for XFEM model The redsquare represents the enriched node with a bisected supporthollow red circle represents the branch function enrichednode with a silt support and solid red circle representsthe improved branch-function-enriched node for XFEM asshown in Figure 5(d) [23]

As can be seen from Figures 6 and 7 the calculatedpressure distribution and pressure profile are basically thesame for 4 methods with different cell number and nodenumber From this the XFEM approach proposed in thispaper is verified

Because of the introduction of enrichment functions withdiscontinuous derivate and extra degrees of freedom fornode it is not necessary to consider the location of fracture

x

y

O

(minus75 161)

(minus13 11)

(72 164)

50

m

40m

(12 4)

(minus33 minus53)(116 minus91)

(157 minus124)

(minus125 minus134)

(minus96 minus194)

(77 minus165)

(14 minus21)

(minus67 minus16)

Figure 9 Domain and boundary conditions for single phase fluidflow

in the process of domain discretization which makes itmore convenient when treating complex network fracturesIn particular when fracture propagation is concerned suchas that during hydraulic fracturing process there is no needto remesh and transform nodal pressure filed with obviousmerit

Adding one more fracture in the domain shown inFigure 4 it becomes intersecting fracture with the rest ofparameters and boundary conditions unchanged Lamb [17]

8 Mathematical Problems in Engineering

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 60 s

(a) 119905 = 60 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)Y

(m)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 120 s

(b) 119905 = 120 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 600 s

(c) 119905 = 600 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 1200 s

(d) 119905 = 1200 s

Figure 10 Pore fluid pressure distribution at different moments

computed the pressure distributions in the existence of inter-secting fractures by use of DFM and FM-FEM respectivelyComparing them in Figure 8 with the result using XFEM itis recognized that the result of XFEM has a good consistencywith that of DFM and FM-FEM which confirms that thefractured-porous-media flow model proposed in this papercan describe not only the flow process in both fractureand matrix but also process independent and intersectingfractures Thus it builds a solid foundation for coupledHM simulation and analysis of fracture propagation duringvolume stimulation under the concept of XFEM

52 Transient State Flow Consider a 40 meters long and 50meterswide horizontal reservoir with several 15-millimeters-wide fractures as shown in Figure 9 The permeability andporosity ofmatrix rock is 100times 10minus15m2 and 018 respectivelyThe initial pore fluid pressure is 20MPa Both the lowerboundary and the upper boundary are impermeable Theleft boundary has a constant pressure of 25MPa while theright boundary has a constant pressure of 20MPa The bulkmodulus of water phase and solid rock grains is 20GPaand 1346GPa respectively The viscosity of water phase is5mPasdots

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

porous media systems Based on the methodology of XFEMin this paper the pressure field is enriched by the absolutesigned distance and appropriate asymptotic functions todevelop an XFEM model for fluid flow in fractured porousmedium The results demonstrate that the XFEM is anefficient method for simulating fluid flow in fractured porousmedium with nonmatching grids especially when the fissureis propagating such as hydraulically driven fractures

2 Governing Equation

The strong form as well as the associated weak form ofgoverning equations for fluid flow inside bothmatrix domainand fracture domain is demonstrated in the section

21 Strong Form According to the law of mass conservationthe continuity equation for flow in the matrix domain iswritten as

(120572119898

minus 120601119898

119870119904

+ 120601119898

119870119908

) 120597119901119908

119898

120597119905 + nabla sdot (minusK119898

120583119908

nabla119901119908

119898) + 119876

119908119904119905119888119861119908

= 0(1)

Similarly the continuity equation for flow inside thefracture domain is given by

1119870119908

120597119901119908

119891

120597119905 + nabla sdot (minus119870119891

120583119908

nabla119901119908

119891) = 0 (2)

where 120572119898is the Biot constant 120593

119898is the porosity of matrix

rock 119870119904is the bulk modulus of solid phase Pa 119870

119908is the

bulkmodulus of pore fluid Pa119901119908

119898is the pore fluid pressure in

matrix Pa 119905 is time sK119898is the permeability tensor ofmatrix

m2 120583119908is the pore fluid viscosity Pasdots119876

119908119904119905119888is the sourcesink

term on the ground m3(m3sdots) 119861119908is the bulk coefficient 119901119908

119891

is the fluid pressure in fracture Pa and119870119891is the permeability

of fracture m2 calculated according to the following formula[20]

119870119891

= 1119891

119887212 (3)

where 119891 is a morphological parameter accounting for thedifference between real fracture and ideal parallel fractureranging from 104 to 165 119887 is the mechanical width offracture m

22 Weak Form In order to deduce the weak form ofgoverning equations a two-dimensional domain Ω boundedby the boundary Γ is considered as shown in Figure 1 Naturalfracture with high permeability insideΩ is regarded as a one-dimensional discontinuous line because the fracture width ismuch less than the length Γ+

119891and Γminus

119891are used to represent two

faces of the fracture respectively

qw

Γpw

Γqw

ΓΩ

+minus

minus

+

pw

Ωf

x

y 119847Γ119891

119847minusΓ119891

119847+Γ119891

Figure 1 The domain and boundary for fractured porous media

The initial and boundary conditions are as follows

119901119908 = 1199011199080 forallx isin Ω119901119908

119898(x) = 119901119908

119898x isin Γ

119901119908

(minusK

119898

120583119908

nabla119901119908

119898) sdot n

Γ= 119902

119908

120588119908

forallx isin Γ119902119908

119901119908

119898= 119901119908

119891forallx isin Γ

119891

(minusK119898

120583119908

nabla119901119908

119898) sdot n

Γ119891

= minus (minus119870119891

120583119908

nabla119901119908

119891) sdot n

Γ119891

= 119902119908

120588119908

(4)

where nΓis the unit normal vector of the boundary 120588

119908is

fluid density kgm3 and 119902119908is the flow rate on the boundary

kg(m2sdots)The weak forms of equations for flow in matrix and

fracture are derived by weighted residual method

intΩ

w119879

119901(120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

) 120597119901119908

119898

120597119905 119889Ω + intΩ

(nablaw119901)119879 K

119898

120583119908

nabla119901119908

119898119889Ω

+ intΩ

w119879

119901119876119908119904119905119888

119861119908

119889Ω + intΓ+

119891

w119879

119901

119902+119908

120588119908

119889Γ + intΓminus

119891

w119879

119901

119902minus119908

120588119908

119889Γ

+ intΓ119902119908

w119879

119901

119902119908

120588119908

119889Γ = 0

(5)

intΩ119891

w119879

119901

1119870119908

120597119901119908

119891

120597119905 119889Ω + intΩ119891

(nablaw119901)119879 119870

119891

120583119908

nabla119901119908

119891119889Ω minus int

Γ+

119891

w119879

119901

119902+119908

120588119908

119889Γ

minus intΓminus

119891

w119879

119901

119902minus119908

120588119908

119889Γ = 0(6)

in whichΩ119891is the fracture domainThe integrals over Ω

119891are

performed in the local Cartesian coordinate system (1199091015840 1199101015840)whose 1199091015840 axes and 1199101015840 axes are consonant with tangentialdirection and normal direction respectively

With regard to the flow inside the fracture both fluidpressure and its interpolating function are assumed to beuniform along fracture width [16] as shown in Figure 2

Mathematical Problems in Engineering 3

p+

p+

pminus

pminus

pf b

b

Ωminus

Ω+

A

A A998400

A998400

Figure 2 Fluid flow and pore fluid pressure around the fracture

minus1minus05

005

1

minus1minus05

005

10

02

04

06

08

1

r cos 120579r sin 120579

rco

s 1205792

(a) The first branch function

minus1

minus05

0

05

1

minus1minus05

005

1

minus1

minus05

0

05

1

r cos 120579

r sin 120579

minusr2

sin 1205792

(b) Its derivative for pore pressure

Figure 3 The first branch function and its derivative for porepressure

Thus the first integral of (6) can be rewritten as [16]

intΩ119891

w119879

119901

1119870119908

120597119901119908

119891

120597119905 119889Ω = intΓ119891

int1198872

minus1198872

w119879

119901

1119870119908

120597119901119908

119891

120597119905 1198891199101015840119889Γ

= intΓ119891

w119879

119901

119887119870119908

120597119901119908

119891

120597119905 119889Γ(7)

No

flow

No

flow

p = 0kPa

p = 286 kPa

L = 10m

W = 16m

Kf = 80DKm = 8mD

b = 1mm

Figure 4 2D fractured domain with applied pressure at top andbottom boundary

Because of the supposed uniform distribution of fluidpressure on the 1199101015840 axes the derivative does not vary with 1199101015840thus the second integral is expressed as [16]

intΩ119891

(nablaw119901)119879 119870

119891

120583119908

nabla119901119908

119891119889Ω = int

Γ119891

int1198872

minus1198872

(nablaw119901)119879 119870

119891

120583119908

nabla119901119908

1198911198891199101015840119889Γ

= intΓ119891

119887119870119891

120583119908

120597119908119879

119901

1205971199091015840

120597119901119908

119891

1205971199091015840119889Γ

(8)

Substituting the constituents of (6) and rearranging it theweak form of continuity equation for flow inside fracture isgained Adding it to the weak form for flow in matrix regionthe fluid exchange term can be eliminated as (9) shows Thusit is not necessary to express fluid exchange explicitly

intΩ

w119879

119901(120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

) 120597119901119908

119898

120597119905 119889Ω

+ intΩ

(nablaw119901)119879 (K

119898

120583119908

nabla119901119908

119898) 119889Ω + int

Γ119902119908

w119879

119901

119902119908

120588119908

119889Γ

+ intΓ119891

w119879

119901

119887119870119908

120597119901119908

119891

120597119905 119889Γ + intΓ119891

120597w119879

119901

1205971199091015840

119887119870119891119870119903119908

120583119908

120597119901119908

119891

1205971199091015840119889Γ = 0

(9)

3 Pressure Enrichment with XFEM

For permeable natural fracture the pressure field is continu-ous at the fracture surface while the derivative of pressure(ie flow rate) is discontinuous The enrichment of thediscontinuity in derivative is constructed as follows [21]

4 Mathematical Problems in Engineering

(a) DFM (b) FM-FEM (c) FM-Mfree

(d) XFEM

Figure 5 Domain discretization for different methods

For a node with a bisected support the absolute signeddistance function is applied to realize the enrichment so as tomeet the condition of continuous pressure and discontinuousderivative

119865 (x) = 1003816100381610038161003816119891 (x)1003816100381610038161003816 (10)

For a support which is slit by the discontinuity the branchfunction is applied to realize the enrichment

119861119897 (x) = [119903 cos 1205792 1199032 cos 120579

2 radic119903 cos 1205792] (11)

Its derivate with respect to 120579 is

120597119861119897 (x)120597120579 = [minus 119903

2 sin 1205792 minus1199032

2 sin 1205792 minusradic119903

2 sin 1205792] (12)

in which 120579 is the intersection angle between local coordinatesystem for fracture and global coordinate system

The first branch function and its derivative for porepressure are shown in Figure 3 On the surface of fracture(ie 120579 = 120587 and 120579 = minus120587) the interpolation function iscontinuous while its derivative is discontinuous The othertwo interpolation functions share the same characteristics

For the nodes cut by two intersecting discontinuities theenrichment is realized through the product of two absolutesigned distance functions related to the two discontinuities

119869119897 (x) = 10038161003816100381610038161198911 (x)1003816100381610038161003816 10038161003816100381610038161198912 (x)1003816100381610038161003816 (13)

According to the fundamental XFEM the local enrich-ment function and extra degrees of freedom are introduced

Mathematical Problems in Engineering 5

(a) DFM (b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) FM-Mfree

0 48 95 143 191 238 286

Pressure (kPa)

(d) XFEM

Figure 6 Pressure field obtained from different methods (independent fracture)

for interpolation of fluid pressure field Then the pressureafter enrichment can be expressed as

119901ℎ (x) = sum119894isin119878

119873119894(x) 119901

119886119894+

119873119888sum

119895=1

sum119894isin119878119867119895

119901119887119894119895

119873119894(x) [10038161003816100381610038161003816119891119895 (x)10038161003816100381610038161003816 minus 10038161003816100381610038161003816119891119895 (x

119894)10038161003816100381610038161003816]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

enrichment of bisected support for fracture 119895

+119873119905sum

119895=1

sum119894isin119878119862119895

119873119894(x) 119877119895 (x)

3

sum119897=1

119901119897

119888119894119895[119861119897

119895(x) minus 119861119897

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of slit support for fracture tip 119895

+119873119909sum

119895=1

sum119894isin119878119869119895

119901119889119894119895

119873119894(x) [119869

119895(x) minus 119869

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of intersecting support for cross point 119895

(14)

where 119873119894is the node interpolation function 119877 is the ramp

function 119878 is the set of all nodes 119878119867119895

is the set of nodesenriched by absolute signed distance function for fracture119895 119878

119862119895is the set of nodes enriched by appropriate asymptotic

function for fracture tip 119895 119878119869119895is the set of nodes enriched

by intersecting function for cross point 119895 119901119886is the freedom

degree of regular nodes (119886-freedomdegree)119901119887is the freedom

6 Mathematical Problems in Engineering

degree of extra nodes in bisected support (119887-freedomdegree)119901119888is the freedom degree of extra nodes in slit support (119888-

freedom degree) and 119901119889is the freedom degree of extra nodes

in intersecting support (119889-freedom degree)

4 Discretization Equation of XFEM

Regarding trial function w119901as the variation of pore pressure

interpolation function the XFEM discretization equation of(9) in space is then gained considering the irrelevance of thevalues of 119901

119886 119901

119887 119901

119888 and 119901

119889

([[[[[[

H119908

119898119886119886H119908

119898119886119887H119908

119898119886119888H119908

119898119886119889

H119908

119898119887119886H119908

119898119887119887H119908

119898119887119888H119908

119898119887119889

H119908

119898119888119886H119908

119898119888119887H119908

119898119888119888H119908

119898119888119889

H119908

119898119889119886H119908

119898119889119887H119908

119898119889119888H119908

119898119889119889

]]]]]]

+[[[[[[

H119908

119891119886119886H119908

119891119886119887H119908

119891119886119888H119908

119891119886119889

H119908

119891119887119886H119908

119891119887119887H119908

119891119887119888H119908

119891119887119889

H119908

119891119888119886H119908

119891119888119887H119908

119891119888119888H119908

119891119888119889

H119908

119891119889119886H119908

119891119889119887H119908

119891119889119888H119908

119891119889119889

]]]]]]

)

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

+ ([[[[[[

W119908

119898119886119886W119908

119898119886119887W119908

119898119886119888W119908

119898119886119889

W119908

119898119887119886W119908

119898119887119887W119908

119898119887119888W119908

119898119887119889

W119908

119898119888119886W119908

119898119888119887W119908

119898119888119888W119908

119898119888119889

W119908

119898119889119886W119908

119898119889119887W119908

119898119889119888W119908

119898119889119889

]]]]]]

+[[[[[[

W119908

119891119886119886W119908

119891119886119887W119908

119891119886119888W119908

119891119886119889

W119908

119891119887119886W119908

119891119887119887W119908

119891119887119888W119908

119891119887119889

W119908

119891119888119886W119908

119891119888119887W119908

119891119888119888W119908

119891119888119889

W119908

119891119889119886W119908

119891119889119887W119908

119891119889119888W119908

119891119889119889

]]]]]]

) 119889119889119905

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

=

119902119908119886

119902119908119886

119902119908119886

119902119908119886

(15)

where flow matrix H compressibility matrix W flow ratevector q at the equivalent node the unit tangent vector I119905

119891of

local coordinate system and coordinate transform matrix Rare respectively expressed as follows

H119908

119898120572120573= int

Ω

(nablaN119901

120572)119879 K

119898

120583119908

nablaN119901

120573119889Ω 120572 120573 = 119886 119887 119888 119889

H120587

119891120572120573= int

Γ119891

(nablaN119901

120572)119879R119879I119905

119891

119887119870119891

120583119908

(I119905119891)119879RnablaN119901

120573119889Γ

120572 120573 = 119886 119887 119888 119889

0

50

100

150

200

250

300

0 4 8 12 16

Pres

sure

(kPa

)

FM-FEM DFMXFEM FM-Mfree

Height in Y direction (m)

Figure 7 Pressure profile for a vertical section taken from the centerof the domain

W119908

119898120572120573= int

Ω

(N119901

120572)119879 (120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

)N119901

120573119889Ω

120572 120573 = 119886 119887 119888 119889W119908

119891120572120573= int

Γ119891

(N119901

120572)119879 119887

119870119908

N119901

120573119889Γ 120572 120573 = 119886 119887 119888 119889

119902119908120572

= intΓ119902119908

(N119901

120572)119879 119902

119908

120588119908

119889Γ 120572 = 119886 119887 119888 119889

I119905119891

= [1 0]119879

R = [ cos 120579 sin 120579minus sin 120579 cos 120579]

(16)

Equation (15) can be rewritten as

AX + B119889X119889119905 = C (17)

in which A is the sum of matrix H119898

of flow in matrixregion and matrix H

119891of flow in fracture B is the sum of

matrixW119898of rock compressibility andmatrixW

119891of fracture

compressibility X is the vector of pore pressure field and Cis the vector of flow rate at the equivalent node

In the time domain (17) can be discretized by the use ofNewmark method as follows [22]

(B + Δ119905120599A)X119899+1

= Δ119905C|119899+120599

+ [B minus (1 minus 120599) Δ119905A]X119899 (18)

in which parameter 120599 satisfies 0 le 120599 le 1 Only when 120599 ge 12the solution is unconditionally stable when 120599 = 12 it is thecentral difference in the time domain and when 120599 = 1 it isfully implicit

Mathematical Problems in Engineering 7

0 48 95 143 191 238 286

Pressure (kPa)

(a) DFM

0 48 95 143 191 238 286

Pressure (kPa)

(b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) XFEM

Figure 8 Pressure field obtained from different methods (intersecting fractures)

5 Numerical Examples

51 Steady State Flow For a two-dimensional fractureddomain with an independent fracture shown in Figure 4Lamb [17] compared the calculated results of discrete fracturemodel (DFM) fracturemapping withmesh free (FM-Mfree)and fracturemapping with finite element method (FM-FEM)and verified the correctness of fracturemappingmethodThedomain discretization pressure field and pressure profile fora vertical section taken from the center of the domain for eachmethod are respectively demonstrated in Figures 5 6 and 7

It is observed in Figure 5 that the mesh edges need to beset on the fracture and nodes need to be set on the fracturetip for DFM model and the mesh node needs to be set onthe fracture for FM-Mfree model However the FM-FEMand XFEM models can simulate the fluid flow in fracturedporous medium with nonmatching grid The permeabilityof elements with fracture is processed equivalently for FM-FEM model while the permeability of fracture is calculatedonly with cubic law or others for XFEM model The redsquare represents the enriched node with a bisected supporthollow red circle represents the branch function enrichednode with a silt support and solid red circle representsthe improved branch-function-enriched node for XFEM asshown in Figure 5(d) [23]

As can be seen from Figures 6 and 7 the calculatedpressure distribution and pressure profile are basically thesame for 4 methods with different cell number and nodenumber From this the XFEM approach proposed in thispaper is verified

Because of the introduction of enrichment functions withdiscontinuous derivate and extra degrees of freedom fornode it is not necessary to consider the location of fracture

x

y

O

(minus75 161)

(minus13 11)

(72 164)

50

m

40m

(12 4)

(minus33 minus53)(116 minus91)

(157 minus124)

(minus125 minus134)

(minus96 minus194)

(77 minus165)

(14 minus21)

(minus67 minus16)

Figure 9 Domain and boundary conditions for single phase fluidflow

in the process of domain discretization which makes itmore convenient when treating complex network fracturesIn particular when fracture propagation is concerned suchas that during hydraulic fracturing process there is no needto remesh and transform nodal pressure filed with obviousmerit

Adding one more fracture in the domain shown inFigure 4 it becomes intersecting fracture with the rest ofparameters and boundary conditions unchanged Lamb [17]

8 Mathematical Problems in Engineering

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 60 s

(a) 119905 = 60 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)Y

(m)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 120 s

(b) 119905 = 120 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 600 s

(c) 119905 = 600 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 1200 s

(d) 119905 = 1200 s

Figure 10 Pore fluid pressure distribution at different moments

computed the pressure distributions in the existence of inter-secting fractures by use of DFM and FM-FEM respectivelyComparing them in Figure 8 with the result using XFEM itis recognized that the result of XFEM has a good consistencywith that of DFM and FM-FEM which confirms that thefractured-porous-media flow model proposed in this papercan describe not only the flow process in both fractureand matrix but also process independent and intersectingfractures Thus it builds a solid foundation for coupledHM simulation and analysis of fracture propagation duringvolume stimulation under the concept of XFEM

52 Transient State Flow Consider a 40 meters long and 50meterswide horizontal reservoir with several 15-millimeters-wide fractures as shown in Figure 9 The permeability andporosity ofmatrix rock is 100times 10minus15m2 and 018 respectivelyThe initial pore fluid pressure is 20MPa Both the lowerboundary and the upper boundary are impermeable Theleft boundary has a constant pressure of 25MPa while theright boundary has a constant pressure of 20MPa The bulkmodulus of water phase and solid rock grains is 20GPaand 1346GPa respectively The viscosity of water phase is5mPasdots

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

p+

p+

pminus

pminus

pf b

b

Ωminus

Ω+

A

A A998400

A998400

Figure 2 Fluid flow and pore fluid pressure around the fracture

minus1minus05

005

1

minus1minus05

005

10

02

04

06

08

1

r cos 120579r sin 120579

rco

s 1205792

(a) The first branch function

minus1

minus05

0

05

1

minus1minus05

005

1

minus1

minus05

0

05

1

r cos 120579

r sin 120579

minusr2

sin 1205792

(b) Its derivative for pore pressure

Figure 3 The first branch function and its derivative for porepressure

Thus the first integral of (6) can be rewritten as [16]

intΩ119891

w119879

119901

1119870119908

120597119901119908

119891

120597119905 119889Ω = intΓ119891

int1198872

minus1198872

w119879

119901

1119870119908

120597119901119908

119891

120597119905 1198891199101015840119889Γ

= intΓ119891

w119879

119901

119887119870119908

120597119901119908

119891

120597119905 119889Γ(7)

No

flow

No

flow

p = 0kPa

p = 286 kPa

L = 10m

W = 16m

Kf = 80DKm = 8mD

b = 1mm

Figure 4 2D fractured domain with applied pressure at top andbottom boundary

Because of the supposed uniform distribution of fluidpressure on the 1199101015840 axes the derivative does not vary with 1199101015840thus the second integral is expressed as [16]

intΩ119891

(nablaw119901)119879 119870

119891

120583119908

nabla119901119908

119891119889Ω = int

Γ119891

int1198872

minus1198872

(nablaw119901)119879 119870

119891

120583119908

nabla119901119908

1198911198891199101015840119889Γ

= intΓ119891

119887119870119891

120583119908

120597119908119879

119901

1205971199091015840

120597119901119908

119891

1205971199091015840119889Γ

(8)

Substituting the constituents of (6) and rearranging it theweak form of continuity equation for flow inside fracture isgained Adding it to the weak form for flow in matrix regionthe fluid exchange term can be eliminated as (9) shows Thusit is not necessary to express fluid exchange explicitly

intΩ

w119879

119901(120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

) 120597119901119908

119898

120597119905 119889Ω

+ intΩ

(nablaw119901)119879 (K

119898

120583119908

nabla119901119908

119898) 119889Ω + int

Γ119902119908

w119879

119901

119902119908

120588119908

119889Γ

+ intΓ119891

w119879

119901

119887119870119908

120597119901119908

119891

120597119905 119889Γ + intΓ119891

120597w119879

119901

1205971199091015840

119887119870119891119870119903119908

120583119908

120597119901119908

119891

1205971199091015840119889Γ = 0

(9)

3 Pressure Enrichment with XFEM

For permeable natural fracture the pressure field is continu-ous at the fracture surface while the derivative of pressure(ie flow rate) is discontinuous The enrichment of thediscontinuity in derivative is constructed as follows [21]

4 Mathematical Problems in Engineering

(a) DFM (b) FM-FEM (c) FM-Mfree

(d) XFEM

Figure 5 Domain discretization for different methods

For a node with a bisected support the absolute signeddistance function is applied to realize the enrichment so as tomeet the condition of continuous pressure and discontinuousderivative

119865 (x) = 1003816100381610038161003816119891 (x)1003816100381610038161003816 (10)

For a support which is slit by the discontinuity the branchfunction is applied to realize the enrichment

119861119897 (x) = [119903 cos 1205792 1199032 cos 120579

2 radic119903 cos 1205792] (11)

Its derivate with respect to 120579 is

120597119861119897 (x)120597120579 = [minus 119903

2 sin 1205792 minus1199032

2 sin 1205792 minusradic119903

2 sin 1205792] (12)

in which 120579 is the intersection angle between local coordinatesystem for fracture and global coordinate system

The first branch function and its derivative for porepressure are shown in Figure 3 On the surface of fracture(ie 120579 = 120587 and 120579 = minus120587) the interpolation function iscontinuous while its derivative is discontinuous The othertwo interpolation functions share the same characteristics

For the nodes cut by two intersecting discontinuities theenrichment is realized through the product of two absolutesigned distance functions related to the two discontinuities

119869119897 (x) = 10038161003816100381610038161198911 (x)1003816100381610038161003816 10038161003816100381610038161198912 (x)1003816100381610038161003816 (13)

According to the fundamental XFEM the local enrich-ment function and extra degrees of freedom are introduced

Mathematical Problems in Engineering 5

(a) DFM (b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) FM-Mfree

0 48 95 143 191 238 286

Pressure (kPa)

(d) XFEM

Figure 6 Pressure field obtained from different methods (independent fracture)

for interpolation of fluid pressure field Then the pressureafter enrichment can be expressed as

119901ℎ (x) = sum119894isin119878

119873119894(x) 119901

119886119894+

119873119888sum

119895=1

sum119894isin119878119867119895

119901119887119894119895

119873119894(x) [10038161003816100381610038161003816119891119895 (x)10038161003816100381610038161003816 minus 10038161003816100381610038161003816119891119895 (x

119894)10038161003816100381610038161003816]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

enrichment of bisected support for fracture 119895

+119873119905sum

119895=1

sum119894isin119878119862119895

119873119894(x) 119877119895 (x)

3

sum119897=1

119901119897

119888119894119895[119861119897

119895(x) minus 119861119897

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of slit support for fracture tip 119895

+119873119909sum

119895=1

sum119894isin119878119869119895

119901119889119894119895

119873119894(x) [119869

119895(x) minus 119869

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of intersecting support for cross point 119895

(14)

where 119873119894is the node interpolation function 119877 is the ramp

function 119878 is the set of all nodes 119878119867119895

is the set of nodesenriched by absolute signed distance function for fracture119895 119878

119862119895is the set of nodes enriched by appropriate asymptotic

function for fracture tip 119895 119878119869119895is the set of nodes enriched

by intersecting function for cross point 119895 119901119886is the freedom

degree of regular nodes (119886-freedomdegree)119901119887is the freedom

6 Mathematical Problems in Engineering

degree of extra nodes in bisected support (119887-freedomdegree)119901119888is the freedom degree of extra nodes in slit support (119888-

freedom degree) and 119901119889is the freedom degree of extra nodes

in intersecting support (119889-freedom degree)

4 Discretization Equation of XFEM

Regarding trial function w119901as the variation of pore pressure

interpolation function the XFEM discretization equation of(9) in space is then gained considering the irrelevance of thevalues of 119901

119886 119901

119887 119901

119888 and 119901

119889

([[[[[[

H119908

119898119886119886H119908

119898119886119887H119908

119898119886119888H119908

119898119886119889

H119908

119898119887119886H119908

119898119887119887H119908

119898119887119888H119908

119898119887119889

H119908

119898119888119886H119908

119898119888119887H119908

119898119888119888H119908

119898119888119889

H119908

119898119889119886H119908

119898119889119887H119908

119898119889119888H119908

119898119889119889

]]]]]]

+[[[[[[

H119908

119891119886119886H119908

119891119886119887H119908

119891119886119888H119908

119891119886119889

H119908

119891119887119886H119908

119891119887119887H119908

119891119887119888H119908

119891119887119889

H119908

119891119888119886H119908

119891119888119887H119908

119891119888119888H119908

119891119888119889

H119908

119891119889119886H119908

119891119889119887H119908

119891119889119888H119908

119891119889119889

]]]]]]

)

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

+ ([[[[[[

W119908

119898119886119886W119908

119898119886119887W119908

119898119886119888W119908

119898119886119889

W119908

119898119887119886W119908

119898119887119887W119908

119898119887119888W119908

119898119887119889

W119908

119898119888119886W119908

119898119888119887W119908

119898119888119888W119908

119898119888119889

W119908

119898119889119886W119908

119898119889119887W119908

119898119889119888W119908

119898119889119889

]]]]]]

+[[[[[[

W119908

119891119886119886W119908

119891119886119887W119908

119891119886119888W119908

119891119886119889

W119908

119891119887119886W119908

119891119887119887W119908

119891119887119888W119908

119891119887119889

W119908

119891119888119886W119908

119891119888119887W119908

119891119888119888W119908

119891119888119889

W119908

119891119889119886W119908

119891119889119887W119908

119891119889119888W119908

119891119889119889

]]]]]]

) 119889119889119905

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

=

119902119908119886

119902119908119886

119902119908119886

119902119908119886

(15)

where flow matrix H compressibility matrix W flow ratevector q at the equivalent node the unit tangent vector I119905

119891of

local coordinate system and coordinate transform matrix Rare respectively expressed as follows

H119908

119898120572120573= int

Ω

(nablaN119901

120572)119879 K

119898

120583119908

nablaN119901

120573119889Ω 120572 120573 = 119886 119887 119888 119889

H120587

119891120572120573= int

Γ119891

(nablaN119901

120572)119879R119879I119905

119891

119887119870119891

120583119908

(I119905119891)119879RnablaN119901

120573119889Γ

120572 120573 = 119886 119887 119888 119889

0

50

100

150

200

250

300

0 4 8 12 16

Pres

sure

(kPa

)

FM-FEM DFMXFEM FM-Mfree

Height in Y direction (m)

Figure 7 Pressure profile for a vertical section taken from the centerof the domain

W119908

119898120572120573= int

Ω

(N119901

120572)119879 (120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

)N119901

120573119889Ω

120572 120573 = 119886 119887 119888 119889W119908

119891120572120573= int

Γ119891

(N119901

120572)119879 119887

119870119908

N119901

120573119889Γ 120572 120573 = 119886 119887 119888 119889

119902119908120572

= intΓ119902119908

(N119901

120572)119879 119902

119908

120588119908

119889Γ 120572 = 119886 119887 119888 119889

I119905119891

= [1 0]119879

R = [ cos 120579 sin 120579minus sin 120579 cos 120579]

(16)

Equation (15) can be rewritten as

AX + B119889X119889119905 = C (17)

in which A is the sum of matrix H119898

of flow in matrixregion and matrix H

119891of flow in fracture B is the sum of

matrixW119898of rock compressibility andmatrixW

119891of fracture

compressibility X is the vector of pore pressure field and Cis the vector of flow rate at the equivalent node

In the time domain (17) can be discretized by the use ofNewmark method as follows [22]

(B + Δ119905120599A)X119899+1

= Δ119905C|119899+120599

+ [B minus (1 minus 120599) Δ119905A]X119899 (18)

in which parameter 120599 satisfies 0 le 120599 le 1 Only when 120599 ge 12the solution is unconditionally stable when 120599 = 12 it is thecentral difference in the time domain and when 120599 = 1 it isfully implicit

Mathematical Problems in Engineering 7

0 48 95 143 191 238 286

Pressure (kPa)

(a) DFM

0 48 95 143 191 238 286

Pressure (kPa)

(b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) XFEM

Figure 8 Pressure field obtained from different methods (intersecting fractures)

5 Numerical Examples

51 Steady State Flow For a two-dimensional fractureddomain with an independent fracture shown in Figure 4Lamb [17] compared the calculated results of discrete fracturemodel (DFM) fracturemapping withmesh free (FM-Mfree)and fracturemapping with finite element method (FM-FEM)and verified the correctness of fracturemappingmethodThedomain discretization pressure field and pressure profile fora vertical section taken from the center of the domain for eachmethod are respectively demonstrated in Figures 5 6 and 7

It is observed in Figure 5 that the mesh edges need to beset on the fracture and nodes need to be set on the fracturetip for DFM model and the mesh node needs to be set onthe fracture for FM-Mfree model However the FM-FEMand XFEM models can simulate the fluid flow in fracturedporous medium with nonmatching grid The permeabilityof elements with fracture is processed equivalently for FM-FEM model while the permeability of fracture is calculatedonly with cubic law or others for XFEM model The redsquare represents the enriched node with a bisected supporthollow red circle represents the branch function enrichednode with a silt support and solid red circle representsthe improved branch-function-enriched node for XFEM asshown in Figure 5(d) [23]

As can be seen from Figures 6 and 7 the calculatedpressure distribution and pressure profile are basically thesame for 4 methods with different cell number and nodenumber From this the XFEM approach proposed in thispaper is verified

Because of the introduction of enrichment functions withdiscontinuous derivate and extra degrees of freedom fornode it is not necessary to consider the location of fracture

x

y

O

(minus75 161)

(minus13 11)

(72 164)

50

m

40m

(12 4)

(minus33 minus53)(116 minus91)

(157 minus124)

(minus125 minus134)

(minus96 minus194)

(77 minus165)

(14 minus21)

(minus67 minus16)

Figure 9 Domain and boundary conditions for single phase fluidflow

in the process of domain discretization which makes itmore convenient when treating complex network fracturesIn particular when fracture propagation is concerned suchas that during hydraulic fracturing process there is no needto remesh and transform nodal pressure filed with obviousmerit

Adding one more fracture in the domain shown inFigure 4 it becomes intersecting fracture with the rest ofparameters and boundary conditions unchanged Lamb [17]

8 Mathematical Problems in Engineering

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 60 s

(a) 119905 = 60 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)Y

(m)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 120 s

(b) 119905 = 120 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 600 s

(c) 119905 = 600 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 1200 s

(d) 119905 = 1200 s

Figure 10 Pore fluid pressure distribution at different moments

computed the pressure distributions in the existence of inter-secting fractures by use of DFM and FM-FEM respectivelyComparing them in Figure 8 with the result using XFEM itis recognized that the result of XFEM has a good consistencywith that of DFM and FM-FEM which confirms that thefractured-porous-media flow model proposed in this papercan describe not only the flow process in both fractureand matrix but also process independent and intersectingfractures Thus it builds a solid foundation for coupledHM simulation and analysis of fracture propagation duringvolume stimulation under the concept of XFEM

52 Transient State Flow Consider a 40 meters long and 50meterswide horizontal reservoir with several 15-millimeters-wide fractures as shown in Figure 9 The permeability andporosity ofmatrix rock is 100times 10minus15m2 and 018 respectivelyThe initial pore fluid pressure is 20MPa Both the lowerboundary and the upper boundary are impermeable Theleft boundary has a constant pressure of 25MPa while theright boundary has a constant pressure of 20MPa The bulkmodulus of water phase and solid rock grains is 20GPaand 1346GPa respectively The viscosity of water phase is5mPasdots

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

(a) DFM (b) FM-FEM (c) FM-Mfree

(d) XFEM

Figure 5 Domain discretization for different methods

For a node with a bisected support the absolute signeddistance function is applied to realize the enrichment so as tomeet the condition of continuous pressure and discontinuousderivative

119865 (x) = 1003816100381610038161003816119891 (x)1003816100381610038161003816 (10)

For a support which is slit by the discontinuity the branchfunction is applied to realize the enrichment

119861119897 (x) = [119903 cos 1205792 1199032 cos 120579

2 radic119903 cos 1205792] (11)

Its derivate with respect to 120579 is

120597119861119897 (x)120597120579 = [minus 119903

2 sin 1205792 minus1199032

2 sin 1205792 minusradic119903

2 sin 1205792] (12)

in which 120579 is the intersection angle between local coordinatesystem for fracture and global coordinate system

The first branch function and its derivative for porepressure are shown in Figure 3 On the surface of fracture(ie 120579 = 120587 and 120579 = minus120587) the interpolation function iscontinuous while its derivative is discontinuous The othertwo interpolation functions share the same characteristics

For the nodes cut by two intersecting discontinuities theenrichment is realized through the product of two absolutesigned distance functions related to the two discontinuities

119869119897 (x) = 10038161003816100381610038161198911 (x)1003816100381610038161003816 10038161003816100381610038161198912 (x)1003816100381610038161003816 (13)

According to the fundamental XFEM the local enrich-ment function and extra degrees of freedom are introduced

Mathematical Problems in Engineering 5

(a) DFM (b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) FM-Mfree

0 48 95 143 191 238 286

Pressure (kPa)

(d) XFEM

Figure 6 Pressure field obtained from different methods (independent fracture)

for interpolation of fluid pressure field Then the pressureafter enrichment can be expressed as

119901ℎ (x) = sum119894isin119878

119873119894(x) 119901

119886119894+

119873119888sum

119895=1

sum119894isin119878119867119895

119901119887119894119895

119873119894(x) [10038161003816100381610038161003816119891119895 (x)10038161003816100381610038161003816 minus 10038161003816100381610038161003816119891119895 (x

119894)10038161003816100381610038161003816]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

enrichment of bisected support for fracture 119895

+119873119905sum

119895=1

sum119894isin119878119862119895

119873119894(x) 119877119895 (x)

3

sum119897=1

119901119897

119888119894119895[119861119897

119895(x) minus 119861119897

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of slit support for fracture tip 119895

+119873119909sum

119895=1

sum119894isin119878119869119895

119901119889119894119895

119873119894(x) [119869

119895(x) minus 119869

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of intersecting support for cross point 119895

(14)

where 119873119894is the node interpolation function 119877 is the ramp

function 119878 is the set of all nodes 119878119867119895

is the set of nodesenriched by absolute signed distance function for fracture119895 119878

119862119895is the set of nodes enriched by appropriate asymptotic

function for fracture tip 119895 119878119869119895is the set of nodes enriched

by intersecting function for cross point 119895 119901119886is the freedom

degree of regular nodes (119886-freedomdegree)119901119887is the freedom

6 Mathematical Problems in Engineering

degree of extra nodes in bisected support (119887-freedomdegree)119901119888is the freedom degree of extra nodes in slit support (119888-

freedom degree) and 119901119889is the freedom degree of extra nodes

in intersecting support (119889-freedom degree)

4 Discretization Equation of XFEM

Regarding trial function w119901as the variation of pore pressure

interpolation function the XFEM discretization equation of(9) in space is then gained considering the irrelevance of thevalues of 119901

119886 119901

119887 119901

119888 and 119901

119889

([[[[[[

H119908

119898119886119886H119908

119898119886119887H119908

119898119886119888H119908

119898119886119889

H119908

119898119887119886H119908

119898119887119887H119908

119898119887119888H119908

119898119887119889

H119908

119898119888119886H119908

119898119888119887H119908

119898119888119888H119908

119898119888119889

H119908

119898119889119886H119908

119898119889119887H119908

119898119889119888H119908

119898119889119889

]]]]]]

+[[[[[[

H119908

119891119886119886H119908

119891119886119887H119908

119891119886119888H119908

119891119886119889

H119908

119891119887119886H119908

119891119887119887H119908

119891119887119888H119908

119891119887119889

H119908

119891119888119886H119908

119891119888119887H119908

119891119888119888H119908

119891119888119889

H119908

119891119889119886H119908

119891119889119887H119908

119891119889119888H119908

119891119889119889

]]]]]]

)

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

+ ([[[[[[

W119908

119898119886119886W119908

119898119886119887W119908

119898119886119888W119908

119898119886119889

W119908

119898119887119886W119908

119898119887119887W119908

119898119887119888W119908

119898119887119889

W119908

119898119888119886W119908

119898119888119887W119908

119898119888119888W119908

119898119888119889

W119908

119898119889119886W119908

119898119889119887W119908

119898119889119888W119908

119898119889119889

]]]]]]

+[[[[[[

W119908

119891119886119886W119908

119891119886119887W119908

119891119886119888W119908

119891119886119889

W119908

119891119887119886W119908

119891119887119887W119908

119891119887119888W119908

119891119887119889

W119908

119891119888119886W119908

119891119888119887W119908

119891119888119888W119908

119891119888119889

W119908

119891119889119886W119908

119891119889119887W119908

119891119889119888W119908

119891119889119889

]]]]]]

) 119889119889119905

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

=

119902119908119886

119902119908119886

119902119908119886

119902119908119886

(15)

where flow matrix H compressibility matrix W flow ratevector q at the equivalent node the unit tangent vector I119905

119891of

local coordinate system and coordinate transform matrix Rare respectively expressed as follows

H119908

119898120572120573= int

Ω

(nablaN119901

120572)119879 K

119898

120583119908

nablaN119901

120573119889Ω 120572 120573 = 119886 119887 119888 119889

H120587

119891120572120573= int

Γ119891

(nablaN119901

120572)119879R119879I119905

119891

119887119870119891

120583119908

(I119905119891)119879RnablaN119901

120573119889Γ

120572 120573 = 119886 119887 119888 119889

0

50

100

150

200

250

300

0 4 8 12 16

Pres

sure

(kPa

)

FM-FEM DFMXFEM FM-Mfree

Height in Y direction (m)

Figure 7 Pressure profile for a vertical section taken from the centerof the domain

W119908

119898120572120573= int

Ω

(N119901

120572)119879 (120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

)N119901

120573119889Ω

120572 120573 = 119886 119887 119888 119889W119908

119891120572120573= int

Γ119891

(N119901

120572)119879 119887

119870119908

N119901

120573119889Γ 120572 120573 = 119886 119887 119888 119889

119902119908120572

= intΓ119902119908

(N119901

120572)119879 119902

119908

120588119908

119889Γ 120572 = 119886 119887 119888 119889

I119905119891

= [1 0]119879

R = [ cos 120579 sin 120579minus sin 120579 cos 120579]

(16)

Equation (15) can be rewritten as

AX + B119889X119889119905 = C (17)

in which A is the sum of matrix H119898

of flow in matrixregion and matrix H

119891of flow in fracture B is the sum of

matrixW119898of rock compressibility andmatrixW

119891of fracture

compressibility X is the vector of pore pressure field and Cis the vector of flow rate at the equivalent node

In the time domain (17) can be discretized by the use ofNewmark method as follows [22]

(B + Δ119905120599A)X119899+1

= Δ119905C|119899+120599

+ [B minus (1 minus 120599) Δ119905A]X119899 (18)

in which parameter 120599 satisfies 0 le 120599 le 1 Only when 120599 ge 12the solution is unconditionally stable when 120599 = 12 it is thecentral difference in the time domain and when 120599 = 1 it isfully implicit

Mathematical Problems in Engineering 7

0 48 95 143 191 238 286

Pressure (kPa)

(a) DFM

0 48 95 143 191 238 286

Pressure (kPa)

(b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) XFEM

Figure 8 Pressure field obtained from different methods (intersecting fractures)

5 Numerical Examples

51 Steady State Flow For a two-dimensional fractureddomain with an independent fracture shown in Figure 4Lamb [17] compared the calculated results of discrete fracturemodel (DFM) fracturemapping withmesh free (FM-Mfree)and fracturemapping with finite element method (FM-FEM)and verified the correctness of fracturemappingmethodThedomain discretization pressure field and pressure profile fora vertical section taken from the center of the domain for eachmethod are respectively demonstrated in Figures 5 6 and 7

It is observed in Figure 5 that the mesh edges need to beset on the fracture and nodes need to be set on the fracturetip for DFM model and the mesh node needs to be set onthe fracture for FM-Mfree model However the FM-FEMand XFEM models can simulate the fluid flow in fracturedporous medium with nonmatching grid The permeabilityof elements with fracture is processed equivalently for FM-FEM model while the permeability of fracture is calculatedonly with cubic law or others for XFEM model The redsquare represents the enriched node with a bisected supporthollow red circle represents the branch function enrichednode with a silt support and solid red circle representsthe improved branch-function-enriched node for XFEM asshown in Figure 5(d) [23]

As can be seen from Figures 6 and 7 the calculatedpressure distribution and pressure profile are basically thesame for 4 methods with different cell number and nodenumber From this the XFEM approach proposed in thispaper is verified

Because of the introduction of enrichment functions withdiscontinuous derivate and extra degrees of freedom fornode it is not necessary to consider the location of fracture

x

y

O

(minus75 161)

(minus13 11)

(72 164)

50

m

40m

(12 4)

(minus33 minus53)(116 minus91)

(157 minus124)

(minus125 minus134)

(minus96 minus194)

(77 minus165)

(14 minus21)

(minus67 minus16)

Figure 9 Domain and boundary conditions for single phase fluidflow

in the process of domain discretization which makes itmore convenient when treating complex network fracturesIn particular when fracture propagation is concerned suchas that during hydraulic fracturing process there is no needto remesh and transform nodal pressure filed with obviousmerit

Adding one more fracture in the domain shown inFigure 4 it becomes intersecting fracture with the rest ofparameters and boundary conditions unchanged Lamb [17]

8 Mathematical Problems in Engineering

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 60 s

(a) 119905 = 60 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)Y

(m)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 120 s

(b) 119905 = 120 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 600 s

(c) 119905 = 600 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 1200 s

(d) 119905 = 1200 s

Figure 10 Pore fluid pressure distribution at different moments

computed the pressure distributions in the existence of inter-secting fractures by use of DFM and FM-FEM respectivelyComparing them in Figure 8 with the result using XFEM itis recognized that the result of XFEM has a good consistencywith that of DFM and FM-FEM which confirms that thefractured-porous-media flow model proposed in this papercan describe not only the flow process in both fractureand matrix but also process independent and intersectingfractures Thus it builds a solid foundation for coupledHM simulation and analysis of fracture propagation duringvolume stimulation under the concept of XFEM

52 Transient State Flow Consider a 40 meters long and 50meterswide horizontal reservoir with several 15-millimeters-wide fractures as shown in Figure 9 The permeability andporosity ofmatrix rock is 100times 10minus15m2 and 018 respectivelyThe initial pore fluid pressure is 20MPa Both the lowerboundary and the upper boundary are impermeable Theleft boundary has a constant pressure of 25MPa while theright boundary has a constant pressure of 20MPa The bulkmodulus of water phase and solid rock grains is 20GPaand 1346GPa respectively The viscosity of water phase is5mPasdots

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

(a) DFM (b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) FM-Mfree

0 48 95 143 191 238 286

Pressure (kPa)

(d) XFEM

Figure 6 Pressure field obtained from different methods (independent fracture)

for interpolation of fluid pressure field Then the pressureafter enrichment can be expressed as

119901ℎ (x) = sum119894isin119878

119873119894(x) 119901

119886119894+

119873119888sum

119895=1

sum119894isin119878119867119895

119901119887119894119895

119873119894(x) [10038161003816100381610038161003816119891119895 (x)10038161003816100381610038161003816 minus 10038161003816100381610038161003816119891119895 (x

119894)10038161003816100381610038161003816]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

enrichment of bisected support for fracture 119895

+119873119905sum

119895=1

sum119894isin119878119862119895

119873119894(x) 119877119895 (x)

3

sum119897=1

119901119897

119888119894119895[119861119897

119895(x) minus 119861119897

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of slit support for fracture tip 119895

+119873119909sum

119895=1

sum119894isin119878119869119895

119901119889119894119895

119873119894(x) [119869

119895(x) minus 119869

119895(x

119894)]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟enrichment of intersecting support for cross point 119895

(14)

where 119873119894is the node interpolation function 119877 is the ramp

function 119878 is the set of all nodes 119878119867119895

is the set of nodesenriched by absolute signed distance function for fracture119895 119878

119862119895is the set of nodes enriched by appropriate asymptotic

function for fracture tip 119895 119878119869119895is the set of nodes enriched

by intersecting function for cross point 119895 119901119886is the freedom

degree of regular nodes (119886-freedomdegree)119901119887is the freedom

6 Mathematical Problems in Engineering

degree of extra nodes in bisected support (119887-freedomdegree)119901119888is the freedom degree of extra nodes in slit support (119888-

freedom degree) and 119901119889is the freedom degree of extra nodes

in intersecting support (119889-freedom degree)

4 Discretization Equation of XFEM

Regarding trial function w119901as the variation of pore pressure

interpolation function the XFEM discretization equation of(9) in space is then gained considering the irrelevance of thevalues of 119901

119886 119901

119887 119901

119888 and 119901

119889

([[[[[[

H119908

119898119886119886H119908

119898119886119887H119908

119898119886119888H119908

119898119886119889

H119908

119898119887119886H119908

119898119887119887H119908

119898119887119888H119908

119898119887119889

H119908

119898119888119886H119908

119898119888119887H119908

119898119888119888H119908

119898119888119889

H119908

119898119889119886H119908

119898119889119887H119908

119898119889119888H119908

119898119889119889

]]]]]]

+[[[[[[

H119908

119891119886119886H119908

119891119886119887H119908

119891119886119888H119908

119891119886119889

H119908

119891119887119886H119908

119891119887119887H119908

119891119887119888H119908

119891119887119889

H119908

119891119888119886H119908

119891119888119887H119908

119891119888119888H119908

119891119888119889

H119908

119891119889119886H119908

119891119889119887H119908

119891119889119888H119908

119891119889119889

]]]]]]

)

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

+ ([[[[[[

W119908

119898119886119886W119908

119898119886119887W119908

119898119886119888W119908

119898119886119889

W119908

119898119887119886W119908

119898119887119887W119908

119898119887119888W119908

119898119887119889

W119908

119898119888119886W119908

119898119888119887W119908

119898119888119888W119908

119898119888119889

W119908

119898119889119886W119908

119898119889119887W119908

119898119889119888W119908

119898119889119889

]]]]]]

+[[[[[[

W119908

119891119886119886W119908

119891119886119887W119908

119891119886119888W119908

119891119886119889

W119908

119891119887119886W119908

119891119887119887W119908

119891119887119888W119908

119891119887119889

W119908

119891119888119886W119908

119891119888119887W119908

119891119888119888W119908

119891119888119889

W119908

119891119889119886W119908

119891119889119887W119908

119891119889119888W119908

119891119889119889

]]]]]]

) 119889119889119905

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

=

119902119908119886

119902119908119886

119902119908119886

119902119908119886

(15)

where flow matrix H compressibility matrix W flow ratevector q at the equivalent node the unit tangent vector I119905

119891of

local coordinate system and coordinate transform matrix Rare respectively expressed as follows

H119908

119898120572120573= int

Ω

(nablaN119901

120572)119879 K

119898

120583119908

nablaN119901

120573119889Ω 120572 120573 = 119886 119887 119888 119889

H120587

119891120572120573= int

Γ119891

(nablaN119901

120572)119879R119879I119905

119891

119887119870119891

120583119908

(I119905119891)119879RnablaN119901

120573119889Γ

120572 120573 = 119886 119887 119888 119889

0

50

100

150

200

250

300

0 4 8 12 16

Pres

sure

(kPa

)

FM-FEM DFMXFEM FM-Mfree

Height in Y direction (m)

Figure 7 Pressure profile for a vertical section taken from the centerof the domain

W119908

119898120572120573= int

Ω

(N119901

120572)119879 (120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

)N119901

120573119889Ω

120572 120573 = 119886 119887 119888 119889W119908

119891120572120573= int

Γ119891

(N119901

120572)119879 119887

119870119908

N119901

120573119889Γ 120572 120573 = 119886 119887 119888 119889

119902119908120572

= intΓ119902119908

(N119901

120572)119879 119902

119908

120588119908

119889Γ 120572 = 119886 119887 119888 119889

I119905119891

= [1 0]119879

R = [ cos 120579 sin 120579minus sin 120579 cos 120579]

(16)

Equation (15) can be rewritten as

AX + B119889X119889119905 = C (17)

in which A is the sum of matrix H119898

of flow in matrixregion and matrix H

119891of flow in fracture B is the sum of

matrixW119898of rock compressibility andmatrixW

119891of fracture

compressibility X is the vector of pore pressure field and Cis the vector of flow rate at the equivalent node

In the time domain (17) can be discretized by the use ofNewmark method as follows [22]

(B + Δ119905120599A)X119899+1

= Δ119905C|119899+120599

+ [B minus (1 minus 120599) Δ119905A]X119899 (18)

in which parameter 120599 satisfies 0 le 120599 le 1 Only when 120599 ge 12the solution is unconditionally stable when 120599 = 12 it is thecentral difference in the time domain and when 120599 = 1 it isfully implicit

Mathematical Problems in Engineering 7

0 48 95 143 191 238 286

Pressure (kPa)

(a) DFM

0 48 95 143 191 238 286

Pressure (kPa)

(b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) XFEM

Figure 8 Pressure field obtained from different methods (intersecting fractures)

5 Numerical Examples

51 Steady State Flow For a two-dimensional fractureddomain with an independent fracture shown in Figure 4Lamb [17] compared the calculated results of discrete fracturemodel (DFM) fracturemapping withmesh free (FM-Mfree)and fracturemapping with finite element method (FM-FEM)and verified the correctness of fracturemappingmethodThedomain discretization pressure field and pressure profile fora vertical section taken from the center of the domain for eachmethod are respectively demonstrated in Figures 5 6 and 7

It is observed in Figure 5 that the mesh edges need to beset on the fracture and nodes need to be set on the fracturetip for DFM model and the mesh node needs to be set onthe fracture for FM-Mfree model However the FM-FEMand XFEM models can simulate the fluid flow in fracturedporous medium with nonmatching grid The permeabilityof elements with fracture is processed equivalently for FM-FEM model while the permeability of fracture is calculatedonly with cubic law or others for XFEM model The redsquare represents the enriched node with a bisected supporthollow red circle represents the branch function enrichednode with a silt support and solid red circle representsthe improved branch-function-enriched node for XFEM asshown in Figure 5(d) [23]

As can be seen from Figures 6 and 7 the calculatedpressure distribution and pressure profile are basically thesame for 4 methods with different cell number and nodenumber From this the XFEM approach proposed in thispaper is verified

Because of the introduction of enrichment functions withdiscontinuous derivate and extra degrees of freedom fornode it is not necessary to consider the location of fracture

x

y

O

(minus75 161)

(minus13 11)

(72 164)

50

m

40m

(12 4)

(minus33 minus53)(116 minus91)

(157 minus124)

(minus125 minus134)

(minus96 minus194)

(77 minus165)

(14 minus21)

(minus67 minus16)

Figure 9 Domain and boundary conditions for single phase fluidflow

in the process of domain discretization which makes itmore convenient when treating complex network fracturesIn particular when fracture propagation is concerned suchas that during hydraulic fracturing process there is no needto remesh and transform nodal pressure filed with obviousmerit

Adding one more fracture in the domain shown inFigure 4 it becomes intersecting fracture with the rest ofparameters and boundary conditions unchanged Lamb [17]

8 Mathematical Problems in Engineering

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 60 s

(a) 119905 = 60 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)Y

(m)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 120 s

(b) 119905 = 120 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 600 s

(c) 119905 = 600 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 1200 s

(d) 119905 = 1200 s

Figure 10 Pore fluid pressure distribution at different moments

computed the pressure distributions in the existence of inter-secting fractures by use of DFM and FM-FEM respectivelyComparing them in Figure 8 with the result using XFEM itis recognized that the result of XFEM has a good consistencywith that of DFM and FM-FEM which confirms that thefractured-porous-media flow model proposed in this papercan describe not only the flow process in both fractureand matrix but also process independent and intersectingfractures Thus it builds a solid foundation for coupledHM simulation and analysis of fracture propagation duringvolume stimulation under the concept of XFEM

52 Transient State Flow Consider a 40 meters long and 50meterswide horizontal reservoir with several 15-millimeters-wide fractures as shown in Figure 9 The permeability andporosity ofmatrix rock is 100times 10minus15m2 and 018 respectivelyThe initial pore fluid pressure is 20MPa Both the lowerboundary and the upper boundary are impermeable Theleft boundary has a constant pressure of 25MPa while theright boundary has a constant pressure of 20MPa The bulkmodulus of water phase and solid rock grains is 20GPaand 1346GPa respectively The viscosity of water phase is5mPasdots

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

degree of extra nodes in bisected support (119887-freedomdegree)119901119888is the freedom degree of extra nodes in slit support (119888-

freedom degree) and 119901119889is the freedom degree of extra nodes

in intersecting support (119889-freedom degree)

4 Discretization Equation of XFEM

Regarding trial function w119901as the variation of pore pressure

interpolation function the XFEM discretization equation of(9) in space is then gained considering the irrelevance of thevalues of 119901

119886 119901

119887 119901

119888 and 119901

119889

([[[[[[

H119908

119898119886119886H119908

119898119886119887H119908

119898119886119888H119908

119898119886119889

H119908

119898119887119886H119908

119898119887119887H119908

119898119887119888H119908

119898119887119889

H119908

119898119888119886H119908

119898119888119887H119908

119898119888119888H119908

119898119888119889

H119908

119898119889119886H119908

119898119889119887H119908

119898119889119888H119908

119898119889119889

]]]]]]

+[[[[[[

H119908

119891119886119886H119908

119891119886119887H119908

119891119886119888H119908

119891119886119889

H119908

119891119887119886H119908

119891119887119887H119908

119891119887119888H119908

119891119887119889

H119908

119891119888119886H119908

119891119888119887H119908

119891119888119888H119908

119891119888119889

H119908

119891119889119886H119908

119891119889119887H119908

119891119889119888H119908

119891119889119889

]]]]]]

)

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

+ ([[[[[[

W119908

119898119886119886W119908

119898119886119887W119908

119898119886119888W119908

119898119886119889

W119908

119898119887119886W119908

119898119887119887W119908

119898119887119888W119908

119898119887119889

W119908

119898119888119886W119908

119898119888119887W119908

119898119888119888W119908

119898119888119889

W119908

119898119889119886W119908

119898119889119887W119908

119898119889119888W119908

119898119889119889

]]]]]]

+[[[[[[

W119908

119891119886119886W119908

119891119886119887W119908

119891119886119888W119908

119891119886119889

W119908

119891119887119886W119908

119891119887119887W119908

119891119887119888W119908

119891119887119889

W119908

119891119888119886W119908

119891119888119887W119908

119891119888119888W119908

119891119888119889

W119908

119891119889119886W119908

119891119889119887W119908

119891119889119888W119908

119891119889119889

]]]]]]

) 119889119889119905

119901119908

119886

119901119908

119886

119901119908

119886

119901119908

119886

=

119902119908119886

119902119908119886

119902119908119886

119902119908119886

(15)

where flow matrix H compressibility matrix W flow ratevector q at the equivalent node the unit tangent vector I119905

119891of

local coordinate system and coordinate transform matrix Rare respectively expressed as follows

H119908

119898120572120573= int

Ω

(nablaN119901

120572)119879 K

119898

120583119908

nablaN119901

120573119889Ω 120572 120573 = 119886 119887 119888 119889

H120587

119891120572120573= int

Γ119891

(nablaN119901

120572)119879R119879I119905

119891

119887119870119891

120583119908

(I119905119891)119879RnablaN119901

120573119889Γ

120572 120573 = 119886 119887 119888 119889

0

50

100

150

200

250

300

0 4 8 12 16

Pres

sure

(kPa

)

FM-FEM DFMXFEM FM-Mfree

Height in Y direction (m)

Figure 7 Pressure profile for a vertical section taken from the centerof the domain

W119908

119898120572120573= int

Ω

(N119901

120572)119879 (120572

119898minus 120601

119898

119870119904

+ 120601119898

119870119908

)N119901

120573119889Ω

120572 120573 = 119886 119887 119888 119889W119908

119891120572120573= int

Γ119891

(N119901

120572)119879 119887

119870119908

N119901

120573119889Γ 120572 120573 = 119886 119887 119888 119889

119902119908120572

= intΓ119902119908

(N119901

120572)119879 119902

119908

120588119908

119889Γ 120572 = 119886 119887 119888 119889

I119905119891

= [1 0]119879

R = [ cos 120579 sin 120579minus sin 120579 cos 120579]

(16)

Equation (15) can be rewritten as

AX + B119889X119889119905 = C (17)

in which A is the sum of matrix H119898

of flow in matrixregion and matrix H

119891of flow in fracture B is the sum of

matrixW119898of rock compressibility andmatrixW

119891of fracture

compressibility X is the vector of pore pressure field and Cis the vector of flow rate at the equivalent node

In the time domain (17) can be discretized by the use ofNewmark method as follows [22]

(B + Δ119905120599A)X119899+1

= Δ119905C|119899+120599

+ [B minus (1 minus 120599) Δ119905A]X119899 (18)

in which parameter 120599 satisfies 0 le 120599 le 1 Only when 120599 ge 12the solution is unconditionally stable when 120599 = 12 it is thecentral difference in the time domain and when 120599 = 1 it isfully implicit

Mathematical Problems in Engineering 7

0 48 95 143 191 238 286

Pressure (kPa)

(a) DFM

0 48 95 143 191 238 286

Pressure (kPa)

(b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) XFEM

Figure 8 Pressure field obtained from different methods (intersecting fractures)

5 Numerical Examples

51 Steady State Flow For a two-dimensional fractureddomain with an independent fracture shown in Figure 4Lamb [17] compared the calculated results of discrete fracturemodel (DFM) fracturemapping withmesh free (FM-Mfree)and fracturemapping with finite element method (FM-FEM)and verified the correctness of fracturemappingmethodThedomain discretization pressure field and pressure profile fora vertical section taken from the center of the domain for eachmethod are respectively demonstrated in Figures 5 6 and 7

It is observed in Figure 5 that the mesh edges need to beset on the fracture and nodes need to be set on the fracturetip for DFM model and the mesh node needs to be set onthe fracture for FM-Mfree model However the FM-FEMand XFEM models can simulate the fluid flow in fracturedporous medium with nonmatching grid The permeabilityof elements with fracture is processed equivalently for FM-FEM model while the permeability of fracture is calculatedonly with cubic law or others for XFEM model The redsquare represents the enriched node with a bisected supporthollow red circle represents the branch function enrichednode with a silt support and solid red circle representsthe improved branch-function-enriched node for XFEM asshown in Figure 5(d) [23]

As can be seen from Figures 6 and 7 the calculatedpressure distribution and pressure profile are basically thesame for 4 methods with different cell number and nodenumber From this the XFEM approach proposed in thispaper is verified

Because of the introduction of enrichment functions withdiscontinuous derivate and extra degrees of freedom fornode it is not necessary to consider the location of fracture

x

y

O

(minus75 161)

(minus13 11)

(72 164)

50

m

40m

(12 4)

(minus33 minus53)(116 minus91)

(157 minus124)

(minus125 minus134)

(minus96 minus194)

(77 minus165)

(14 minus21)

(minus67 minus16)

Figure 9 Domain and boundary conditions for single phase fluidflow

in the process of domain discretization which makes itmore convenient when treating complex network fracturesIn particular when fracture propagation is concerned suchas that during hydraulic fracturing process there is no needto remesh and transform nodal pressure filed with obviousmerit

Adding one more fracture in the domain shown inFigure 4 it becomes intersecting fracture with the rest ofparameters and boundary conditions unchanged Lamb [17]

8 Mathematical Problems in Engineering

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 60 s

(a) 119905 = 60 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)Y

(m)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 120 s

(b) 119905 = 120 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 600 s

(c) 119905 = 600 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 1200 s

(d) 119905 = 1200 s

Figure 10 Pore fluid pressure distribution at different moments

computed the pressure distributions in the existence of inter-secting fractures by use of DFM and FM-FEM respectivelyComparing them in Figure 8 with the result using XFEM itis recognized that the result of XFEM has a good consistencywith that of DFM and FM-FEM which confirms that thefractured-porous-media flow model proposed in this papercan describe not only the flow process in both fractureand matrix but also process independent and intersectingfractures Thus it builds a solid foundation for coupledHM simulation and analysis of fracture propagation duringvolume stimulation under the concept of XFEM

52 Transient State Flow Consider a 40 meters long and 50meterswide horizontal reservoir with several 15-millimeters-wide fractures as shown in Figure 9 The permeability andporosity ofmatrix rock is 100times 10minus15m2 and 018 respectivelyThe initial pore fluid pressure is 20MPa Both the lowerboundary and the upper boundary are impermeable Theleft boundary has a constant pressure of 25MPa while theright boundary has a constant pressure of 20MPa The bulkmodulus of water phase and solid rock grains is 20GPaand 1346GPa respectively The viscosity of water phase is5mPasdots

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 48 95 143 191 238 286

Pressure (kPa)

(a) DFM

0 48 95 143 191 238 286

Pressure (kPa)

(b) FM-FEM

0 48 95 143 191 238 286

Pressure (kPa)

(c) XFEM

Figure 8 Pressure field obtained from different methods (intersecting fractures)

5 Numerical Examples

51 Steady State Flow For a two-dimensional fractureddomain with an independent fracture shown in Figure 4Lamb [17] compared the calculated results of discrete fracturemodel (DFM) fracturemapping withmesh free (FM-Mfree)and fracturemapping with finite element method (FM-FEM)and verified the correctness of fracturemappingmethodThedomain discretization pressure field and pressure profile fora vertical section taken from the center of the domain for eachmethod are respectively demonstrated in Figures 5 6 and 7

It is observed in Figure 5 that the mesh edges need to beset on the fracture and nodes need to be set on the fracturetip for DFM model and the mesh node needs to be set onthe fracture for FM-Mfree model However the FM-FEMand XFEM models can simulate the fluid flow in fracturedporous medium with nonmatching grid The permeabilityof elements with fracture is processed equivalently for FM-FEM model while the permeability of fracture is calculatedonly with cubic law or others for XFEM model The redsquare represents the enriched node with a bisected supporthollow red circle represents the branch function enrichednode with a silt support and solid red circle representsthe improved branch-function-enriched node for XFEM asshown in Figure 5(d) [23]

As can be seen from Figures 6 and 7 the calculatedpressure distribution and pressure profile are basically thesame for 4 methods with different cell number and nodenumber From this the XFEM approach proposed in thispaper is verified

Because of the introduction of enrichment functions withdiscontinuous derivate and extra degrees of freedom fornode it is not necessary to consider the location of fracture

x

y

O

(minus75 161)

(minus13 11)

(72 164)

50

m

40m

(12 4)

(minus33 minus53)(116 minus91)

(157 minus124)

(minus125 minus134)

(minus96 minus194)

(77 minus165)

(14 minus21)

(minus67 minus16)

Figure 9 Domain and boundary conditions for single phase fluidflow

in the process of domain discretization which makes itmore convenient when treating complex network fracturesIn particular when fracture propagation is concerned suchas that during hydraulic fracturing process there is no needto remesh and transform nodal pressure filed with obviousmerit

Adding one more fracture in the domain shown inFigure 4 it becomes intersecting fracture with the rest ofparameters and boundary conditions unchanged Lamb [17]

8 Mathematical Problems in Engineering

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 60 s

(a) 119905 = 60 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)Y

(m)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 120 s

(b) 119905 = 120 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 600 s

(c) 119905 = 600 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 1200 s

(d) 119905 = 1200 s

Figure 10 Pore fluid pressure distribution at different moments

computed the pressure distributions in the existence of inter-secting fractures by use of DFM and FM-FEM respectivelyComparing them in Figure 8 with the result using XFEM itis recognized that the result of XFEM has a good consistencywith that of DFM and FM-FEM which confirms that thefractured-porous-media flow model proposed in this papercan describe not only the flow process in both fractureand matrix but also process independent and intersectingfractures Thus it builds a solid foundation for coupledHM simulation and analysis of fracture propagation duringvolume stimulation under the concept of XFEM

52 Transient State Flow Consider a 40 meters long and 50meterswide horizontal reservoir with several 15-millimeters-wide fractures as shown in Figure 9 The permeability andporosity ofmatrix rock is 100times 10minus15m2 and 018 respectivelyThe initial pore fluid pressure is 20MPa Both the lowerboundary and the upper boundary are impermeable Theleft boundary has a constant pressure of 25MPa while theright boundary has a constant pressure of 20MPa The bulkmodulus of water phase and solid rock grains is 20GPaand 1346GPa respectively The viscosity of water phase is5mPasdots

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 60 s

(a) 119905 = 60 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)Y

(m)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 120 s

(b) 119905 = 120 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)

20

205

21

215

22

225

23

235

24

245

25t = 600 s

(c) 119905 = 600 s

minus20 minus10 0 10 20minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

X (m)

Y(m

)

(MPa

)20

205

21

215

22

225

23

235

24

245

25t = 1200 s

(d) 119905 = 1200 s

Figure 10 Pore fluid pressure distribution at different moments

computed the pressure distributions in the existence of inter-secting fractures by use of DFM and FM-FEM respectivelyComparing them in Figure 8 with the result using XFEM itis recognized that the result of XFEM has a good consistencywith that of DFM and FM-FEM which confirms that thefractured-porous-media flow model proposed in this papercan describe not only the flow process in both fractureand matrix but also process independent and intersectingfractures Thus it builds a solid foundation for coupledHM simulation and analysis of fracture propagation duringvolume stimulation under the concept of XFEM

52 Transient State Flow Consider a 40 meters long and 50meterswide horizontal reservoir with several 15-millimeters-wide fractures as shown in Figure 9 The permeability andporosity ofmatrix rock is 100times 10minus15m2 and 018 respectivelyThe initial pore fluid pressure is 20MPa Both the lowerboundary and the upper boundary are impermeable Theleft boundary has a constant pressure of 25MPa while theright boundary has a constant pressure of 20MPa The bulkmodulus of water phase and solid rock grains is 20GPaand 1346GPa respectively The viscosity of water phase is5mPasdots

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Applying 39times 49 structured gird the distributions of porefluid pressure at different moments (119905 = 60 s 119905 = 120 s119905 = 600 s and 119905 = 1200 s) are demonstrated in Figure 10after calculating with XFEM Before pressure spreading tofracture (119905 lt 60 s) the isobaric lines appear to be parallelwhich is in the stage of linear displacement (Figure 10(a))During pressure encountering fracture (119905 gt 60 s) the isobaricline bents toward tip direction near the fracture tip and thefracture tip is presented as a pointsink source at the moment(Figures 10(b)sim10(d)) After pressure sweeping over thewholefractured region the pressure drop of whole fractured regionis very small and pressure loss occurs mainly in the matrix(Figures 10(c) and 10(d))

6 Conclusions and Discussions

(1) Both absolute signed distance function and appro-priate asymptotic function are continuous but dis-continuous in derivatives which enables them tobe enrichment functions during processing pressurefield across permeable fractures

(2) Based on the explicit expression of real distribution offractures locally enrichment with XFEM was used todevelop fluid flow in fractured porous medium Themodel is able to elaborately simulate the flow in reser-voir with arbitrary independent kinking branchingintersecting and complex network fractures

(3) Domain discretization is independent of fracturelocation with XFEM flow model Moreover remesh-ing is unnecessary when dealing with dynamic prop-agating fractures which enables it to be used forsimulating hydraulic fracturing with coupled HMmodel

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful for the research support of theNational Natural Science Foundation of China (NSFC no51404207)

References

[1] B Berkowitz J Bear and C Braester ldquoContinuum models forcontaminant transport in fractured porous formationsrdquo WaterResources Research vol 24 no 8 pp 1225ndash1236 1988

[2] G I Barenblatt I P Zheltov and I N Kochina ldquoBasic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks [strata]rdquo Journal of Applied Mathematics and Mechanicsvol 24 no 5 pp 1286ndash1303 1960

[3] M Karimi-Fard L J Durlofsky and K Aziz ldquoAn efficientdiscrete-fracture model applicable for general-purpose reser-voir simulatorsrdquo SPE Journal vol 9 no 2 pp 227ndash236 2004

[4] J Lee S-U Choi and W Cho ldquoA comparative study of dual-porosity model and discrete fracture network modelrdquo KSCEJournal of Civil Engineering vol 3 no 2 pp 171ndash180 1999

[5] S H LeeM F Lough and C L Jensen ldquoHierarchical modelingof flow in naturally fractured formations with multiple lengthscalesrdquo Water Resources Research vol 37 no 3 pp 443ndash4552001

[6] B G Chen E X Song and X H Cheng ldquoA numerical methodfor discrete fracture network model for flow and heat transferin two-dimensional fractured rocksrdquo Chinese Journal of RockMechanics and Engineering vol 33 no 1 pp 43ndash51 2014

[7] J E Warren and P J Root ldquoThe behavior of naturally fracturedreservoirsrdquo Old SPE Journal vol 3 no 3 pp 245ndash255 1963

[8] K Pruess and T N Narasimhan ldquoA practical method formodeling fluid and heat flow in fractured porousmediardquo Societyof Petroleum Engineers journal vol 25 no 1 pp 14ndash26 1985

[9] O Kolditz ldquoModelling flow and heat transfer in fractured rocksconceptual model of a 3-D deterministic fracture networkrdquoGeothermics vol 24 no 3 pp 451ndash470 1995

[10] J Yao Z S Wang Y Zhang and Z Q Huang ldquoNumericalsimulation method of discrete fracture network for naturallyfractured reservoirsrdquoActa Petrolei Sinica vol 31 no 2 pp 284ndash288 2010 (Chinese)

[11] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers amp Structures vol 81 no 3 pp 119ndash129 2003

[12] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[13] L X Li and T J Wang ldquoThe extended finite element methodand its applications a reviewrdquo Advances in Mechanics vol 35no 1 pp 5ndash20 2005

[14] T T Yu The Extended Finite Element Method Theory Appli-cation and Program Science Press Beijing China 2014 (Chi-nese)

[15] A R Khoei S Moallemi and E Haghighat ldquoThermo-hydro-mechanical modeling of impermeable discontinuity in sat-urated porous media with X-FEM techniquerdquo EngineeringFracture Mechanics vol 96 pp 701ndash723 2012

[16] T Mohammadnejad and A R Khoei ldquoAn extended finite ele-ment method for hydraulic fracture propagation in deformableporous media with the cohesive crack modelrdquo Finite Elementsin Analysis and Design vol 73 pp 77ndash95 2013

[17] A R Lamb Coupled Deformation Fluid Flow and FracturePropagation in Porous Media Imperial College London UK2011

[18] A R LambG J Gorman andD Elsworth ldquoA fracturemappingand extended finite element scheme for coupled deformationand fluid flow in fractured porous mediardquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 37no 17 pp 2916ndash2936 2013

[19] N Watanabe W Wang J Taron U J Gorke and O KolditzldquoLower-dimensional interface elements with local enrichmentapplication to coupled hydro-mechanical problems in discretelyfractured porous mediardquo International Journal for NumericalMethods in Engineering vol 90 no 8 pp 1010ndash1034 2012

[20] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[21] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[22] RW Lewis andBA SchreflerTheFinite ElementMothod in theStatic and Dynamic Deformation and Consolidation of PorousMedia John Wiley amp Sons 1998

[23] T T Yu and Z W Gong ldquoDetermination of enrichment typeof node in extended finite element methodrdquo Rock and SoilMechanics vol 34 no 11 pp 3284ndash3290 2013 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of