Multicomponent fluid flow by discontinuous Galerkin and ...Galerkin (DG) methods. We use the cross-flow equilibrium concept to approximate the fractured matrix mass transfer. The discrete

Multicomponent fluid flow by discontinuous Galerkin and mixed

methods in unfractured and fractured media

H. Hoteit and A. Firoozabadi1

Reservoir Engineering Research Institute, Palo Alto, California, USA

Received 9 June 2005; revised 25 July 2005; accepted 9 August 2005; published 10 November 2005.

[1] A discrete fracture model for the flow of compressible, multicomponent fluids inhomogeneous, heterogeneous, and fractured media is presented in single phase. In thenumerical model we combine the mixed finite element (MFE) and the discontinuousGalerkin (DG) methods. We use the cross-flow equilibrium concept to approximate thefractured matrix mass transfer. The discrete fracture model is numerically superior tothe single-porosity model and overcomes limitations of the dual-porosity models includingthe use of a shape factor. The MFE method provides a direct and accurate approximationfor the velocity field, which is crucial for the convective terms in the flow equations. TheDG method associated with a slope limiter is used to approximate the species balanceequations. This method can capture the sharp moving fronts. The calculation of thefracture-fracture flux across three and higher intersecting fracture branches is a challenge.In this work, we provide an accurate approximation of these fluxes by using the MFEformulation. Numerical examples in unfractured and fractured media illustrate theefficiency and robustness of the proposed numerical model.

Citation: Hoteit, H., and A. Firoozabadi (2005), Multicomponent fluid flow by discontinuous Galerkin and mixed methods in

unfractured and fractured media, Water Resour. Res., 41, W11412, doi:10.1029/2005WR004339.

1. Introduction

[2] This paper considers the flow of miscible and com-pressible multicomponent fluids in unfractured and frac-tured media. The compositional modeling has broadapplications in various disciplines including petroleumengineering (hydrocarbon-gas injection and recycling ingas-condensate reservoirs), hydrology and geochemicalengineering (contamination of groundwater aquifers andradioactive waste management in the subsurface). Thenumerical simulation of the process should overcome manynumerical difficulties associated with the unstructured ge-ometry of the fractured field, sharp variations in the rockproperties, and high nonlinearity of the multicomponentsystem due to the compressibility and compositional effects.[3] The partial differential equations that describe the

flow processes are coupled and highly nonlinear. Tradi-tional algorithms that employ the classical finite difference(FD), finite volume (FV) or finite element (FE) methodsgenerally do not provide satisfactory results. These meth-ods may not represent the physics correctly because of twomain deficiencies: First, the flow processes are usuallyconvection dominated. These methods combined withfirst-order upstream techniques produce significant numer-ical dispersion [Coats, 1980] especially in the neighbor-hood of shocks in the solution. Second, it is known in theliterature [Durlofsky, 1994; Mosé et al., 1994] that for

discontinuous and anisotropic permeability these methodsmay not provide accurate velocity field, because only thepressure is computed as a primary variable and thevelocity field has to be approximated from Darcy’s lawby a postprocessing step.[4] The objective of this work is to develop a robust and

efficient numerical model that has the following features:(1) the mass is conserved locally at the element level,(2) sharp fronts are predicted without introducing spuriousoscillations or excessive numerical dispersion, (3) thevelocity field is correctly approximated in anisotropic andin highly heterogeneous media, and should have low meshdependence, and (4) unstructured grids are used for spatialdiscretization. To fulfill these requirements, we combinedthe mixed finite element (MFE) method and the discontin-uous Galerkin (DG) method. The MFE method is used todiscretize Darcy’s law. The main features for this choice areas follows: the pressure and the fluxes are approximatedsimultaneously with the same order of convergence, and themethod is locally conservative and it can easily accommo-date full permeability tensor. This method also producesminimal mesh orientation effect [Darlow et al., 1984]. Thefact that the MFE method is more reliable in flux calculationthan the FVand FE methods is well known [Durlofsky, 1994;Mosé et al., 1994]. The MFE formulation in our methoduses the lowest-order Raviart-Thomas space [Raviart andThomas, 1977; Thomas, 1977]. The original MFE formula-tion leads to a saddle point problem for elliptic or parabolicequations i.e., the linear system to solve, that has the cellpressure averages and the interelement fluxes as primaryunknowns, is indefinite. The remedy is to use the hybrid-ization technique [Brezzi and Fortin, 1991; Chavent andRoberts, 1991] where new degrees of freedom are appended

1Also at Chemical Engineering Department, Yale University, NewHaven, Connecticut, USA.

Copyright 2005 by the American Geophysical Union.0043-1397/05/2005WR004339$09.00

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WATER RESOURCES RESEARCH, VOL. 41, W11412, doi:10.1029/2005WR004339, 2005

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at the element edges. The additional unknowns representthe edge pressure averages (pressure traces). In our method,this approach will not help to make the linear system tobecome positive definite because of the Newton-Raphsonlinearization. However, it is still useful since it reduces thesize of the linear system which has the traces of the pressureas primary unknowns. The DG method is used to approx-imate the species balance equations. The state unknown(concentration) is approximated by using linear and bilinearshape functions on triangular and quadrilateral elements,respectively. The use of high-order approximation spacesproduces nonphysical oscillations near the shocks. Tostabilize the scheme, we use a multidimensional slopelimiter [Chavent and Jaffré, 1986; Hoteit et al., 2004b],which reconstructs the solution profile over the grid blocksby imposing some geometrical constraints. Recently, theDG method has received a lot of attention mainly because itcan be applied to structured and unstructured grids, itconserves mass locally at the element level and produceslow numerical dispersion compared to classical methods.Various versions of DG method have been developed.Cockburn and Shu [1989, 1998] developed the Runge-Kutta discontinuous Galerkin (RKDG) method. The RKDGmethod is an extension of the DG method to highertemporal and spatial approximation spaces. Recent develop-ments have extended this method to approximate elliptic,diffusion and convection-diffusion problems [Chen et al.,1995; Rivière et al., 1999]. Different problems have beensolved by using both the MFE and DG methods. Chavent etal. [1990] used these methods to approximate a two-phaseincompressible, immiscible fluid flow. Siegel et al. [1997]and Hoteit et al. [2004a] also combined these methods tosolve convection-diffusion equations in porous media. In asimilar fashion, our numerical model combines the MFEmethod to approximate the pressure unknown and the DGmethod to approximate the flow equations. In this work,however, the compressibility and compositional effectsare taken into account. The extension to compressible-compositional systems makes our method distinct fromthe work of others.[5] In this work, we also extend the MFE-DG method

for fractured media by using the discrete fracture model.Different numerical approaches have been proposed in theliterature to model the flow in fractured porous media.The dual-porosity model [Warren and Root, 1963;Kazemi, 1969; Thomas et al., 1983; Arbogast et al.,1990] is widely used in the flow simulators in fracturedmedia. This model represents the fracture network by anequivalent porous medium which generally has the sugarcube configuration. The dual-porosity model is numerical-ly very efficient since computations are only performed inthe fracture network. The interaction between the matrixand the fractures is described with empirical transferfunctions. Because appropriate transfer functions are notwell established especially with gravity, compressibility,and compositional effects, the model is inadequate todescribe the compositional effects in fractured media.Another limitation is the modeling of discrete fracturessince this model assumes the medium to be described by adense connected fractured network. One can also use thesingle-porosity model with explicit grid blocks to describethe fractures in the same way as the matrix is described.

In such a model, the geological parameters vary sharplybetween the matrix and the fractures. The high contrastand different length scales in the matrix and fracturesmake this approach unpractical. An alternative is thediscrete fracture model, which can be considered as asimplification of the single-porosity model. Assuming thatthe fracture aperture is small compared to the matrix scale,fractures are represented by (n � 1)-dimensional elementsin an n-dimensional domain [Noorishad and Mehran,1982; Baca et al., 1984; Granet et al., 1998]. Thissimplification makes the latter much more efficient thanthe single-porosity model.[6] Unlike immiscible fluid flow, which has been studied

extensively [Kim and Deo, 1999, 2000; Bastian et al., 2000;Karimi-Fard and Firoozabadi, 2003], very few studies arereported in the literature for the compositional modeling infractured media. Here we intend to demonstrate the possi-bility of modeling multicomponent compressible flow indiscrete fractured media. The reliability and efficiency ofthis approach are conditional to two essential approxima-tions; the matrix-fracture and the fracture-fracture fluxes.On the basis of the cross-flow equilibrium concept, thepressure in a fracture element is assumed to be equal tothe pressure in the surrounding matrix elements. Integratingthe flow equations in a control volume that includes thefracture element and the adjacent matrix elements alleviatesthe computation of the matrix-fracture fluxes. This simpli-fication makes the approach much more efficient than thesingle-porosity model and overcomes the limitations of thedual-porosity models.[7] Computing the flux across the intersection of more

than two fracture branches is a challenge. Most of the cell-based finite volume schemes that are adapted to a discretefracture model have the fracture pressure unknowns at thediscretized element centers. On structured grids, Slough etal. [1999a, 1999b, 1999c] compute the fluxes between avertical fracture and a horizontal fracture by assuming asteady state flow in the control volume where the fracturesare intersected. However, such technique cannot be appliedin unstructured grids with multiintersecting fractures. In ourapproach, this deficiency is solved definitively due to theMFE formulation.[8] This paper is organized as follows: First, the differ-

ential and algebraic equations describing the displacementof compressible, compositional flow of multicomponentfluids are presented. Second, we describe the numericalmodel in unfractured media. Different aspects of the nu-merical approach, the discretization of Darcy’s law by MFEmethod, the flow equations by the DG method, the linear-ization technique and the slope limiter are discussed indetail. We then present the extension of the MFE-DGmethod to fractured media, where we show how to calculatethe matrix-fracture and fracture-fracture fluxes. Finally, wepresent numerical examples in unfractured and fracturedmedia to demonstrate the efficiency and robustness of ournumerical approach.

2. Mathematical Model

[9] In a subsurface flow problem, we are interested in thedisplacement of miscible and compressible fluid of nccomponents. By neglecting the molecular diffusion, the

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W11412 HOTEIT AND FIROOZABADI: MULTICOMPONENT FLUID FLOW IN FRACTURED MEDIA W11412

mass conservation for the ith component through the porousmedium is expressed by the equations

f@ci@t

þr: ciJð Þ ¼ fi i ¼ 1; . . . ; nc in W� 0; tð Þ

ci ¼ zicXnci¼1

zi ¼ 1 ð1Þ

where f is the porosity, c is the overall molar density, ci andzi are, respectively, the molar density and the mole fractionof component i, fi is a sink/source, W is the 2-Dcomputational domain and t denotes the simulation time.[10] The volumetric velocity field J appearing in

equation (1) is given by the generalized Darcy’s law:

J ¼ � km

rp� rgð Þ in W� 0; tð Þ ð2Þ

where k is the absolute permeability tensor, m, p, r and g are,respectively, the viscosity, pressure, mass density, and thegravitational force field.[11] Equations (1) and (2) are subject to initial and

boundary conditions (BC) of Dirichlet and Neumann type:

zi x; 0ð Þ ¼ z0i xð Þ; i ¼ 1; . . . ; nc in Wp x; 0ð Þ ¼ p0 xð Þ in Wp x; tð Þ ¼ pD x; tð Þ; i ¼ 1; . . . ; nc on GD � 0; tð Þczi x; tð ÞJ:n ¼ qNi x; tð Þ; i ¼ 1; . . . ; nc on GN � 0; tð Þ

ð3Þ

where qiN is the injection rate of component i across the

boundary GN, p0 and pD are the initial pressure and theimposed pressure on the Dirichlet BC, n denotes the unit-outward normal to the domain boundary GD. The flowequations (1)–(3) are coupled with an equation of state(EOS) that describes the phase molar density as a functionof the composition, temperature and pressure. In this work,the Peng-Robenson EOS is used:

r ¼ cM

c ¼ pZRT

Z3 � 1� Bð ÞZ2 þ A� 3B2 � 2B� �

Z � AB� B2 � B3� �

¼ 0 ð4Þ

where M is the molecular weight, T the temperature, R thegas constant, A and B are the PR EOS parameters [see, e.g.,Firoozabadi, 1999].

3. Numerical Approximation inUnfractured Media

[12] The numerical approach used to solve the coupledsystem of equations (1)–(4) is based on combining the DGand MFE methods. This system is linearized by using theNewton-Raphson (NR) method. The primary unknown, thepressure, is solved implicitly and the concentrationunknowns are computed explicitly in time. The constructionof the numerical procedure can be organized in the follow-ing steps: (1) discretization of Darcy’s law by the MFEmethod, (2) discretization of the flow equations by the DGmethod, (3) combining the MFE and DG discretizations,and (4) data reconstruction by using a slope limiter. Beforedetailing these steps, we consider a spatial triangulation ofthe computational domain W consisting of triangles or

quadrilaterals. Unlike some other numerical methods, norestrictions are imposed on the element geometrical shape.We also use the notation K, discretized block or cell, E,edges of the cell K, Ne, number of edges for each cell (Ne =3 or 4), NK, number of cells in the mesh, and NE, number ofedges in the mesh not belonging to GD.

3.1. Discretization of Darcy’s Law

[13] The essential idea of the MFE method is to approx-imate simultaneously the pressure and its gradient. Thevelocity field is approximated in the so-called Raviart-Thomas space of lowest order (RT0) [see, e.g., Chaventand Roberts, 1991; Hoteit et al., 2002]. The main idea is toexpress the velocity over each grid cell with respect to thefluxes across the cell edges. A detailed description of theMFE formulation is presented in appendix A.[14] On the basis of Raviart-Thomas approximation

space, the vectors J and g in equation (2) can be expressedas

J ¼XE2@K

qK;EwK;E and g ¼XE2@K

qgK;EwK;E ð5Þ

where wK,E is a RT0 basis function, qK,E is the total fluxacross an edge E and qK,E

g is the flux due to the gravitationalforce. These vectors are determined by their normal fluxesacross the cell edges. Let K = k/m be the effective mobilitytensor. By inverting K, Darcy’s velocity equation becomes

K�1J ¼ � rp� rgð Þ ð6Þ

Multiplying equation (6) by the test function wK,E andintegrating by parts, then the total flux qK,E is expressedthrough each edge E as a function of the cell pressureaverage pK and the edge pressure averages tpK,E for eachcell K, i.e.,

qK;E ¼ aK;EpK �XE02@K

bK;E;E0 tpK;E0 � gK;E E 2 @K ð7Þ

where the coefficients aK,E, bK,E,E0 and gK,E depend on thegeometrical and rock properties of block K.[15] The continuity of the fluxes across the interelement

boundaries provides

qK;E ¼�qK0 ;E if E ¼ K \ K 0

qNE if E 2 GN

8<: ð8Þ

Equations (7) and (8) lead to the following algebraic linearsystem with main unknowns; the pressure cell averages in Pand the pressure edge averages in TP

RTP �MTP � V ¼ 0 ð9Þ

where

R ¼ RK;E� �

NK ;NE; RK;E ¼ aK;E E 2 @K

M ¼ ME;E0� �

NE ;NE; ME;E0 ¼

XE;E0�@K

bK;E;E0 E =2 GD

W11412 HOTEIT AND FIROOZABADI: MULTICOMPONENT FLUID FLOW IN FRACTURED MEDIA

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V is a vector of size NE that describes the boundaryconditions.[16] Equation (9) is valid at any time, so we can write

RTDP �MDTP ¼ G ð10Þ

where G = �RTP + MTP + V.

3.2. Discretization of the Flow Equations

[17] The DG method is used to discretize the flowequation (1). This method consists of a discontinuous,piecewise, linear (on triangles) or bilinear (on quadrilater-als) approximation of the concentration unknowns ci, i =1, . . ., nc. Over each cell K, the unknown ci,K is approxi-mated in a discontinuous finite element space, such that,

ci;K ¼XNe‘¼1

c‘i;KjK;‘ ð11Þ

where jK,j are some finite element shape functions.Multiplying equation (1) by the test functions andintegrating by parts yield

ZK

f@ci;K@t

jK;j �ZK

ci;KJ:rjK;j þZ@K

~ci;KjK;jJ:n

¼ZK

f ijK;j j ¼ 1; . . . ;Ne ð12Þ

where ~ci,K denotes the edge concentration in the upstreamblock. Depending on the influx or efflux with respect to thecell K, we have

~ci;K ¼ci;K if J:n � 0 effluxð Þ

ci;K0 if J:n < 0; influx from block K 0ð Þ

8<:

Let ci,K; i = 1, . . ., nc, denote the cell average of ci over K.In order to have explicitly the average values ci,K; i =1, . . ., nc from the flow equations, it is convenient to sumup equation (12) for j = 1, . . ., Ne. With some simplemanipulations, one gets

fK Kj jdci;K

dtþ

XE2@K

qK;E~cEi;K ¼ Kj jf i;K ð13Þ

where f i,K is the cell average of fi over K.[18] The line integrals in equation (12) are simplified by

using the equation

ZE

~ci;KJ:n ¼ qK;E~cEi;K ð14Þ

where ~ci,KE is the edge average concentration in the upstream

block. Replace the flux expression from equation (7) inequation (13) to get

dci;K

dtþ ~ai;KpK þ

XE2@K

~bi;K;EtpK;E þ ~gi;K ¼ 0 i ¼ 1; . . . ; nc ð15Þ

where the coefficients ~ai,K, ~bi,K,E and ~gi,K; i = 1, . . ., nc arefunctions of ~ci,K

E , aK, bK,E,E0, and gK. The discretization ofthe time operator in equation (15) by the forward Eulerscheme and the linearization by the NR method give

Dci;K þ Dt~ai;KDpK þ DtXE2@K

~bi;K;EDtpK;E ¼ �gi;K i ¼ 1; . . . ; nc

ð16Þ

where gi,K i = 1, . . ., nc denotes the residual functions. Thecoefficients aK, bK,E,E0 in equation (16) are evaluated usingedge average concentrations ~ci,K

E from old time level.

3.3. Combining the MFE and DG Discretizations

[19] The PR EOS is applied by using the cell averagevalues of the pressure, molar density, mass density, compo-sition and the temperature. Define the residual error func-tion gK as follows:

hK ci;K ; i ¼ 1; . . . ; nc; pK� �

¼ cK �pK

ZRTover each cell K ð17Þ

The Newton-Raphson linearization of equation (17) yields

dpKDpK þXnci¼1

dci;KDci;K ¼ �hK ð18Þ

where dpK =@hK@pK

; dci;K =@hK@ci;K

; i = 1, . . ., nc

The derivatives can be calculated by using the PR EOS [see,e.g., Firoozabadi, 1999]. By replacing equation (16) inequation (18) and writing in a matrix notation, we get thesystem

~DDP þ ~RDTP ¼ F ð19Þ

where

~D ¼ ~DK;K� �

NK ;NK; ~DK;K ¼ Dt

Xnci¼1

dci;K ~ai;K � dpK

~R ¼ ~RK;E� �

NK ;NE; ~RK;E ¼ Dt

Xnci¼1

dci;K~bi;K;E

F ¼ FK½ �NK ; FK ¼ �Xnci¼1

dci;Kgi;K þ hK

The system of Equations 10 and 19 can be written togetheras

~D ~RRT �M

�DPDTP

�¼ F

G

�ð20Þ

Because ~D is a diagonal matrix, the Schur complementmatrix is readily computed. Thus the numerical procedureleads to the following final system whose primaryunknowns are the traces of the pressure on the grid edges.

M þ RT ~D�1~R� �

DTP ¼ RT ~D�1F � G ð21Þ

The cell pressure unknowns are then updated byusing equation (19). The nodal values for the concentrations

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ci,K; i = 1, . . ., nc in each cell K are computed locally fromequation (12).

3.4. Slope Limiting

[20] It is known that with a high-order approximation, theDG method becomes unstable. The use of an appropriateslope limiter is essential in order to avoid overshoots andundershoots in the solution. One such slope limiter wasintroduced by Chavent and Jaffré [1986]. This slope limitercan be interpreted as a generalization of the Van LeerMUSCL 1-D slope limiter [Van Leer, 1974]. The essentialidea of this technique is to impose some local constraints ina geometric manner so that the reconstructed solutionsatisfies an appropriate maximum principle. These con-straints state that the value of the state function at a nodeI, for example, should lie between the minimum and themaximum of the cell averages of all elements containing Ias a vertex. This slope limiter was later modified by Hoteitet al. [2004b]. It is found that limiting the state edge averageis more appropriate than limiting the nodal values asChavent and Jaffre proposed. These modifications improvedthe capability of the slope limiter in eliminating spuriousoscillation and reducing the numerical dispersion as well.More details about the slope limiter used in our method aregiven in appendix B.

4. Algorithm

[21] The principal steps of the MFE-DG algorithm can beillustrated as follows.1. Initialize geometry and physical parameters of theproblem.

1.1. For initial pressure, temperature and composi-tion, use the PR EOS to calculate the initial molar andmass densities.

1.2. For given pressure, temperature and composi-tion, calculate the viscosities.2. Repeat until the predetermined simulation time isreached.3. Iterate on the Newton-Raphson linearization steps.

3.1. Calculate DTP by solving the linear systemgiven in equation (21).

3.2. Calculate DP locally from equation (19).3.3. Calculate the fluxes qK,E across the edges from

equation (7).3.4. Calculate the nodal values of the molar densities

from equation (12).3.5. Reconstruct ci, i = 1, . . ., nc by applying the

slope limiter procedure.3.6. Check convergence (kDPk < TOLP and kDcik <

TOLi; i = 1, . . ., nc) if not repeat step 3.4. For given pressure, temperature and composition updatethe mass density and the viscosities. Note that, in all thenumerical experiments, only one Newton-Raphson itera-tion step was sufficient to attain convergence.

5. Extension to Fractured Media

[22] In this section, we extend the coupled MFE-DGmethod to fractured media. The key solution is to correctlyapproximate the matrix-fracture and fracture-fracturetransfers.

5.1. Matrix-Fracture Transfer

[23] Generally, fractures have very small aperture (width)compared to the matrix size. A small aperture places asevere constraint on the time step. The main idea of thediscrete fracture model is to use the cross-flow equilibriumconcept to account for flow across the fractures [Noorishadand Mehran, 1982; Baca et al., 1984; Granet et al., 1998].Thus the integrals of the governing equations over thefracture can be simplified by dropping one of the spatialvariables through multiplying by the fracture width. Con-sequently, fractures can be represented by the edges of thecontrol volumes in the grid.[24] Classical finite element methods can be divided into

two categories: the first group includes the methods that usenodal or vertex-based representation for the unknowns, likeGalerkin FE and vertex-centered control volume methodsand the second group includes the cell-based methods, likecell-centered FV, FD, DG and MFE (see Figure 1). Accord-ing to the spatial approximation of the unknowns, eachmethod can be adapted to represent the linear representationof the fractures. Unlike the methods of the first category[Bastian et al., 2000; Karimi-Fard and Firoozabadi, 2003;Monteagudo and Firoozabadi, 2004], all methods in the

Figure 1. Classification of numerical methods accordingto spatial approximations.

Figure 2. Typical control volumes containing a fracturewith the cross-flow equilibrium in the computational controlvolume.


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second category face difficulties and therefore need specialtreatments to handle the hybrid spatial approximations [see,e.g., Slough et al., 1999b; Karimi-Fard et al., 2004; Granetet al., 2001]. The cell-based methods require computing thefluxes across the cell edges. Consequently, with a fracture(see Figure 1), some fluxes have to be evaluated across thelateral sides of the fracture. One choice is to add a newdegree of freedom representing the edge average of the stateunknown (e.g., pressure, concentration or saturation). Sucha discretization places a sever limitation on the time stepsize. It is well known that the size of the time step that canbe used with time explicit schemes is restricted to theCourant-Freidricks-Levy (CFL) condition. The CFL condi-tion for incompressible fluid flow is expressed by [Putti etal., 1990] as

CFL ¼ Dt2 Kj j

Z@K

J:nj j ð22Þ

where the line integral in equation (22) represents the fluxesacross the boundary. This condition may be interpreted thatthe time step must be less than the necessary time for flow topass through one mesh block K. As a result, a fracture maycritically reduce the size of the time step especially if the

fracture is essentially perpendicular to the flow streamlines.In order to overcome this limitation, we assume that the stateunknowns (pressure and composition) in the fracture and inthe adjacent cells are the same (see Figure 2). Thisassumption removes the necessity of calculating thematrix-fracture flux; only the matrix-matrix and thefracture-fracture fluxes are required. As a result, there wouldbe a significant reduction in CPU time as we will see later.

5.2. Approximation of Darcy’s Law in the Fractures

[25] The same mixed hybrid finite element formulationcan be applied to discretize Darcy’s law in the fractureelement I. The flux qI,e across the extremity e of I can thenbe written as

qI ;e ¼ aI ;epI �Xe02@I

bI ;e;e0 tpI ;e0 � gI ;e e 2 @I ð23Þ

where pI and tpI,e are, respectively, the pressure average on Iand the pressure traces at the extremities of I. The pressureand flux degrees of freedom in the matrix and fractureelements are shown in Figure 3. In the grid blocks thatcontain fractures, we assume that the block pressure averageand the fracture slice pressure average are equal, that is,pK = pI. We impose the flux continuity across the blockinterfaces, (qK,E + qK0,E = 0; E = K \ K0). Similarly, at theintersection point o of NI (NI � 2) of connected fractures Ii(see Figure 4), we assume that there is no volumetricaccumulation, that is

XNIi¼1

qIi;o ¼ 0 ð24Þ

By using Equations (23) and (24) in the fracture elementsand Equations (7) and (8) in the matrix elements, the fluxunknown can then be eliminated and a linear system withmain unknowns; the pressure average and the pressure traceis obtained.

5.3. Approximation of Species Flow Equations

[26] The DG method can be readily extended to fracturedmedia. The only difficulty, here, is the calculation offracture-fracture flux at the intersection point of the frac-tures. Let K be a matrix block that contains a fracture slice Iand ci,K be the cell average of ci over K. The integration ofthe flow equation (1) over the matrix block and fractureslice is written as

ZK

fK@ci;K@t

þZI

fI@ci;I@t

0@

1A

þZK

r: ci;KJ� �

þZI

r: ci;IJ� �0@

1A ¼

ZK[I

fi;K j ¼ 1; . . . ;Ne

ð25Þ

Figure 3. Pressure degrees of freedom in the controlvolume.

Figure 4. Intersection node of five intersecting fractures(NI = 5, ‘ = 4).

Table 1. Critical Properties for C1 and C3

Properties C1 C3

Acentric factor 0.01 0.15Critical temperature, K 190 370Critical pressure, bars 46 42Molecular weight, g/mol 16 44

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Using the divergence theorem and assuming that theaverage concentrations in the matrix block and fractureslice are equal, equation (25) is simplified to

fK Kj j þ Ij jfIð Þdci;K

dtþ

XE�@K

qK;E~cK;E þ eXe�@I

qI ;e~cI ;e ¼ f i;K

ð26Þ

where e is the fracture aperture; ~cK,E and ~cI,e are theconcentrations from the upstream matrix block and fractureslice, respectively. The final linear system, which has thetraces of the pressure as primary unknowns, is constructedin a similar way to the algorithm described in unfracturedmedia. The only key step we need to discuss is thecalculation of the term in equation (26), which representsthe fracture-fracture fluxes.

5.4. Fracture-Fracture Transfer

[27] In discrete fracture models, the approximation ofthe fluxes at the intersection point of several fracture

branches is a challenge. For two intersecting fractures,fluxes along each branch can be easily evaluated byintroducing an unknown approximating the pressure atthe intersection node. Then, by writing the balance at thatnode, the pressure unknown can be eliminated and thefluxes can be evaluated. On structured grids, Slough et al.[1999a, 1999b, 1999c] computed the fluxes between avertical fracture and a horizontal fracture by assuming asteady state flow in the control volume where thefractures intersect. On unstructured grids, Karimi-Fardet al. [2004] used an analogy between flow in fracturedporous media and conductance through a network ofresistors. They used the so-called star delta rule tocalculate the fluxes. Granet et al. [2001] introduced anadditional node at the intersection in order to calculatethe saturation at an intermediate time. They assumed thatthe transport between an intersection node and a jointnode is twice faster than the transport between two jointnodes.[28] We have found that the MFE method can naturally

solve the problem on structured and unstructured gridswithout a special treatment. Since the MFE formulationprovides the pressure traces, the volumetric fluxes can becomputed locally (see equation (23)). The remaining aspectof the problem is to define the material flux or the mobilityat the intersection node. In other words, what upstreamweighting should be considered in case of multiple up-stream fracture branches.[29] Consider an intersection point o with NI connec-

tions, where Ii; i = 1, . . ., NI are the labels of theconnected fracture branches (see Figure 4). In each frac-ture Ii, we denote by mi and qi the mobility and the

Table 2. Rock and Fluid Properties

Property Value

Injection gas, mole fraction 1.0 C1, 0.0 C3Initial fluid, mole fraction 0.0 C1, 1.0 C3Pressure, bars 20Temperature, K 394Porosity, fraction 0.2Permeability, mdarcy 10

Figure 5. Methane composition profile (mole fraction) for various grid refinements by the MFE-DGand FD methods, PV injection = 0.7: Example 1. See color version of this figure in the HTML.


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volumetric flux at o. We also classify the influx and effluxat o such that.

qi > 0 for 1 � i � ‘

qi � 0 for ‘ < i � NIð27Þ

where ‘, 1 � ‘ � NI � 1. Note that the effluxes are assumedto have positive sign. The material balance at o assures theexistence of ‘. By writing the total volumetric balance andthe total material balance for each component we get,

X‘i¼1

qi ¼ �XNfi¼‘þ1

qi ð28Þ

X‘i¼1

miqi ¼ �XNIi¼‘þ1

moqi ¼ �moXNIi¼‘þ1

qi ð29Þ

where mo refers to the mobility at o; it is the upstreammobility for the effluxes. From Equations 28 and 29, mo canbe readily calculated:

mo ¼

X‘i¼1

miqi

X‘i¼1

qi

ð30Þ

The above expression for the mobility at the intersection omay be considered as a generalization of the classical

upstream technique at the intersection of two fractures. IfNI = 2 (‘ = 1), for example, equation (30) gives mo = m1which is the classical upstream solution.

6. Numerical Results

[30] The validity of the model was first checked bycomparing the results to a commercial FD-based composi-tional simulator using the implicit saturation explicit pres-sure (IMPES) option. The simulator uses the conventional,five-point orthogonal Cartesian grid in 2-D. The linearsolver in the commercial simulator is the Orthomin precon-ditioned by ILU. The numerical experiments given belowwere performed in order to show the three main objectivesof this work: First, the DG method introduces low numer-ical dispersion compared to the FD-based methods. Evenwith very refined meshes the FD solution may not be asaccurate as the DG solution on a coarse mesh. Further, theFD method in such cases could require orders of magnitudemore CPU time compared to the DG method. We alsocompare the discrete fracture model with the single-porositymodel. We then present an example in fractured media byusing unstructured grids.[31] All numerical testes are performed by using a binary

mixture of methane (C1) and propane (C3). The propertiesof these species are given in Table 1. The initial pressure,temperature and matrix porosity are given in Table 2. At theproduction side, the pressure is kept at pD = p0. Differentpermeabilities and injection rates are used and viscositiesare computed by using the correction of Lohrentz et al.[1964]. The direct solver package UMFPACK of Davisand Duff [1999] is used to solve the linear system inequation (21). A simple adaptive time step algorithm isused. It controls the time step by limiting the maximal localvariations in pressure and concentrations. All test runs wereperformed on a 2 GHz Pentium 4 PC.

6.1. Example 1

[32] We consider a homogeneous 2-D horizontal domainof area 40 m � 40 m. Methane is injected at one corner todisplace propane to the opposite producing corner; thedomain is initially saturated with propane (see Table 2).

Table 3. CPU Time for Different Griddings: Example 1

Grid FD Method, s MFE-DGM Method, s

40 � 40 15 3880 � 80 269160 � 160 4658

Figure 6. Pressure distribution (KPa) and velocity field (shown by arrows) from the FD and MFE-DGmethods; 60 � 60 grid, PV injection = 0.7: Example 2. See color version of this figure in the HTML.

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At the production end, the pressure is kept constant. Wecompare our code to the FD commercial code. In Figure 5the composition profile of C1 by the FD and MFE-DGcodes are shown at nearly 71% PV displacement. The FDsolution has a pronounced numerical dispersion comparedto the MFE-DG solution with a 40 � 40 grid. Even withmore refined grids (80 � 80 and 160 � 160), the FD resultis still very dispersive; the MFE-DG method for a 40 � 40gridding introduces less numerical dispersion than the FDon a 160 � 160 grids (Figure 5). The MFE-DG algorithm is2–3 orders of magnitude faster than the FD method for thesame accuracy (see Table 3).

6.2. Example 2

[33] In this example we consider a 2-D vertical hetero-geneous domain of 40 m � 40 m where the gravity effect istaken into account. The medium has two zones of differentpermeabilities. One zone is nearly impermeable where itspermeability is 10�7 mdarcy (see Figure 6). Elsewhere thepermeability is 10 mdarcy. As shown in Figure 6, the resultsby the MFE-DG and FD methods for the velocity field are

very different. Because of the gravity effect (C1 is lighterthan C3), C1 does not propagate in the whole domain as inExample 1. The results for methane composition, which aredepicted in Figure 7, show that the FD method has asignificant numerical dispersion.

6.3. Example 3

[34] The aim of this example is to show the reliability ofthe MFE method in approximating the velocity field inheterogeneous media. We consider the same system as inExample 1 with the exception that the porous medium ishighly heterogeneous with 6 orders of magnitude for thecontrast in permeability. The permeability in each block ischosen at random (Figure 8). The composition distributionof C1 and the velocity filed are depicted in Figure 9.

6.4. Example 4

[35] In this example we compare the accuracy and theefficiency of the discrete fracture model and the single-

Figure 7. Methane composition (mole fraction) by the MFE-DG and FD methods, 2-D verticalgeometry with 60 � 60 grid, PV injection = 0.7: Example 2. See color version of this figure in theHTML.

Figure 8. Permeability distribution in the unstructuredgrid with 898 nodes: Example 3. See color version of thisfigure in the HTML.

Figure 9. Methane composition (mole fraction) andvelocity field distribution (shown by triangles): Example 3.See color version of this figure in the HTML.


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Figure 10. Methane composition (mole fraction) from the single-porosity and discrete fracture models,fracture thickness � = 10�2 m and � = 10�3 m and PV injection = 0.6: Example 4. See color version of thisfigure in the HTML.

Figure 11. Methane composition (mole fraction) at different PV injections; one and three fractures withdiagonal orientation: Example 5. See color version of this figure in the HTML.

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Figure 12. Methane composition (mole fraction) at different PV injections; four connected fractures:Example 5. See color version of this figure in the HTML.

Figure 13. Methane composition (mole fraction) at different PV injections; sugar cube configuration:Example 5. See color version of this figure in the HTML.


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porosity model both by the combined MFE-DG method. Weconsider a rectangular horizontal domain of 40 m � 40 mcontaining one fracture (the fracture is shown by the thicksolid line in Figure 10). The domain is saturated withpropane (C3). Methane (C1) is injected at one corner todisplace C3 from the opposite corner. The permeability inthe matrix and in the fracture is 10 mdarcy and 104 mdarcy,respectively. For two different fracture thicknesses 10�2 mand 10�3 m, Figure 10 shows that methane composition byboth models is comparable. The results by the single-porosity model could have more numerical dispersionbecause of the use of small time steps. The 2-D represen-tation of the fracture with the single-porosity model imposesa sever restriction on the time step size. In the first casewhere the fracture thickness is 10�2 m, the CPU times forthe discrete fracture model and single-porosity model are,respectively, 45 and 840 s. The gain in CPU time becomesmore significant for thinner fractures. In the second case(fracture thickness is 10�3 m), the CPU times is 21 secondsfor the former and 1.2 � 104 seconds for the latter. Thinnerfracture relaxes the time step limitation in the discretefracture model, whereas, this has an opposite effect in thesingle-porosity model.

6.5. Example 5

[36] We present results by the MFE-DG method forfractured media. The computational domain is the unitsquare with dimension 1 m � 1 m. Methane (C1) is injectedat one corner to displace propane (C3) to the opposite cornerat a constant pressure of 20 bar. The thickness of thefractures is 10�2 m. The permeability in the matrix and inthe fractures is, respectively, 10 mdarcy and 104 mdarcy.The injection rate is 0.115 PV/day. Several configurationswith one, three, four and a sugar cube configuration for thefractures are considered. Figures 11–13 show the results forC1 composition for the different fracture configurations. All

the results presented in Figures 11–13 are based on theruns, which we consider smooth and efficient.

7. Concluding Remarks

[37] In this work, a robust and efficient numericalapproach has been developed for the solution of com-pressible flow of multicomponent fluids in homogeneousand in fractured porous media. This numerical approachcombines the MFE and DG methods. The hybridized MFEmethod is used to approximate Darcy’s law where theprimary unknowns are the pressure traces on the edges.The DG method is used to approximate the flow equa-tions. The numerical method leads to solve a linear systemwith the size equal to the number of edges in the mesh.The main features of this work can be summarized in threepoints.[38] 1. The MFE method provides a highly accurate

approximation of the velocity field. This method is lessmesh dependent than the FD approximation. Thus con-straints, such as the strict Delaunay conditions, are notrequired on the mesh generation which sometimes couldbe stiff to satisfy especially on complicated geometries likefractured media. With the MFE method, the flux throughmultiintersecting fractures is accurately approximated with-out any special treatment in structured and unstructuredgrids. It should be noted that this approach is independent ofthe methods chosen to approximate the flow equations orthe fluxes in the matrix. Therefore this approach is alsouseful for other techniques that are based on finite differ-ence or finite volume methods.[39] 2. This work shows the applicability of the DG

method in approximating compressible, multicomponentfluid flow in homogeneous and fractured media. The DGmethod has superiority to capture shocks or sharp gra-dients in the solution without creating spurious oscillationsor excessive numerical dispersion. Comparisons with aFD-based commercial software showed that the combinedMFE-DG method becomes 2–3 orders of magnitude moreefficient than the FD method for comparable accuracy inthe solution.[40] 3. The discrete fracture concept is a powerful con-

cept for application to connected or disconnected fractures.The geometrical simplification of the fractures significantlyreduces the CPU and memory requirements.

Appendix A

[41] In this appendix we provide details of the MFEapproximation, which is based on the Raviart-Thomasspace. The hybridized mixed finite element uses asunknowns the cell pressure averages, the edge pressureaverages and the interelement fluxes. We use the Raviart-Thomas space of lowest order. The main objective is toexpress the velocity over each cell with respect to the fluxesacross the edges. Consider the reference element K asdepicted in Figure A1. The element edges E and thecorresponding exterior normals n appear also in Figure A1.The Raviart-Thomas (RT0) basis functions on K are definedby

wE1 ¼u

v� 1

�; wE2 ¼

u� 1v

�; wE3 ¼

u

v

�

ðA1Þ

Figure A1. Reference element K with the unit normal andbasis functions.

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The patterns of these vector fields are shown in Figure A1.One can readily verify that these vector fields are linearlyindependent and that they satisfy the following properties:

r:wE ¼1

Kj j ðA2Þ

wE:nE0 ¼1= Ej j if E ¼ E0

0 if E 6¼ E0:

8<: ðA3Þ

The lowest-order Raviart-Thomas space RT0,K over K is thefunctional space generated by wEi; i = 1, . . ., 3. Thus anyvector cK 2 RT0,K can be written with respect to the basisfunctions equation (A1) as

cK ¼XE2@K

qK;EwE ðA4Þ

where @E = {E1, E2, E3}.From Equations A2 and A3, one can readily check that

ZE

cK :nE ¼ qK;E

[42] From the Raviart-Thomas approximation space, thevectors J and g can be expressed as

J ¼XE2@K

qK;EwK;E and g ¼XE2@K

qgK;EwK;E ðA5Þ

where qK,E =

ZE

J.nK,E and qK,Eg =

ZE

~g.nK,E = gjEjcos

(Ang(~g, nK,E)).By inverting the mobility tensor K, Darcy’s equationbecomes

K�1J ¼ � rp� rgð Þ ðA6Þ

Multiplying equation (A6) by the test function wK,E andintegrating by parts yield

ZK

wK;EK�1J ¼ �

ZK

wK;E rp� rgð Þ

¼ �ZK

wK;ErpþZK

rwK;E~g

¼ZK

pr:wK;E �Z@K

pwK;E:nK;E

þZK

rwK;Eg E 2 @K ðA7Þ

The integral terms in the right-hand side of equation (A7)are simplified asZ

K

pr:wK;E ¼1

Kj j

ZK

p ;

Z@K

pwK;E:nK;E ¼XE2K

1

Ej j

ZE

p

Let pK =1

Kj j

ZK

p, tpK,E =1

Ej j

ZE

p be the cell and edge

averages, respectively and r = rK and K = KK be constantover each cell K. Equation (A7) becomesZ

K

wK;EK�1J ¼ 1

Kj j

ZK

p� 1Ej j

ZE

pþZK

rwK;Eg E 2 @K

ðA8Þ

Replace equation (A1) in equation (A8) to get

XE02K

qK;E0

ZK

wK;EK�1K wK;E0 ¼ pK � tpK;E0

þXE02K

rKqgK;E0

ZK

wK;EwK;E0 E 2 @K ðA9Þ

Equation (A9) can be written in matrix form

BKQK ¼ pKe� TpK � rK ~BKQgK ðA10Þ

where,

BK ¼ BKð ÞE;E0h i

E;E02@K; BKð ÞE;E0 ¼

ZK

wK;EK�1K wK;E0

~BK ¼ ~BK� �

E;E0

h iE;E02@K

; ~BK� �

E;E0¼

ZK

wK;EwK;E0

QK ¼ qK;E� �

E2@K ; QgK ¼ q

gK;E

h iE2@K

;

TpK ¼ tpK;E� �

E2@K ; e ¼ 1½ �E2@K :

By inverting BK, equation (A10) becomes

QK ¼ pKB�1K e� B�1K TpK þ rKB�1K ~BKQgK ðA11Þ

Thus equation (A11) expresses the fluxes qK,E through eachedge E as a function of the cell pressure average pK and theedge pressure averages tpK,E, i.e.,

qK;E ¼ aK;EpK �XE02@K

B�1K� �

E;E0tpK;E0 � bK;E E 2 @K

ðA12Þ

where, aK,E =XE02@K

(BK�1)E,E0, and bK,E = �rK

XE02@K

(BK�1~BK)E,E0qK,E0

g .

Appendix B

[43] In this appendix, we give details of the slope limiterusually needed to stabilize the DG scheme. We start by theone dimensional space.

B1. Slope Limiter in 1-D Geometry

[44] Let us denote the points and the discretized elementsof the 1-D domain by . . . < xi < xi+1 < . . . and by Ki =(xi, xi+1). The DG method seeks to approximate the state


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variable (say the overall composition c) in a discontinuousfinite element space. Over each element K, c is approxi-mated by

c x; tð Þ ¼ c x; tð ÞjK¼Xiþ1j¼i

cK;jjj

where, jj are the standard finite element shape functionsand cK,j are the nodal values of c on the boundaries of K.Note that this approximation allows a discontinuity of c atthe interfaces between elements. The DG method leadssometimes to nonphysical oscillation at the nodal values ofc; in this case the slope limiter is essential to stabilize themethod.[45] The basic idea of the slope limiter is to impose some

local constraints in a geometric manner so that the recon-structed solution satisfies an appropriate maximum principle.This procedure aims to eliminate local maxima or minimafrom the solution.[46] Let cK be the cell average over a 1-D element K:

cK ¼1

Kj j

ZK

c xð Þ ¼ 12

cK;i þ cK;iþ1� �

The reconstructed nodal values (cK,ir , cK,i+1

r ) of (cK,i, cK,i+1)should satisfy the following conditions: (1) conservation ofmass balance cK,i

r + cK,i+1r = cK,i + cK,i+1, (2) avoidance in

creating local extrema

min cK�1; cKf g � cK;i � max cK�1; cKf g

min cK ; cKþ1f g � cK;iþ1 � max cK ; cKþ1f g

and (3) minimum modification of (cK,i, cK,i+1), i.e.,

k cK;i; cK;iþ1� �T � crK;i; crK;iþ1

� �Tk2

is minimal. Therefore the slope limiter process is reduced toa minimization problem. In Figure B1 we see localmaximum and minimum at the boundaries of K (beforelimiting). The data is reconstructed (after limiting) so thatwe eliminate these extrema by keeping the same material in

the cell. Note that one may use a constant approximation ofc over each cell (i.e., cK = cK,i = cK,i+1). In this case, thescheme is reduced to the first-order upstream finitedifference method.

B2. Slope Limiter in Multidimensional Space

[47] The extension of the slope limiter to the multidimen-sional space is formulated in such a way that in each cell K,each state variable at a vertex i lies between the cellaverages of all neighboring elements containing i as a vertex(Figure B2). We define the notation

cmin;i ¼ min cK i is a vertex of Kjf g

cmax;i ¼ max cK i is a vertex of Kjf g

Then, the generalization of the slope limiter to multi-dimensional space seeks to solve the minimization problem:

minCrK

k CrK � CK k2;

subject to the linear constraints

Mass conservation

cK ¼1

Kj j

ZK

cdx

Avoiding local extrema

cmin;i � cK;i � cmax;i ; i ¼ 1; . . . ;Ne:

Note that this minimization problem can be solved readily.For details on this slope limiter, we refer to Hoteit et al.[2004b].

[48] Acknowledgment. This work was supported by the membercompanies of the Reservoir Engineering Research Institute (RERI) andthe U.S. DOE (grant DI-FG26-99BC15177).

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��A. Firoozabadi and H. Hoteit, Reservoir Engineering Research Institute,

385 Sherman Avenue, Suite 5, Palo Alto, CA 94306, USA. ([email protected];[email protected])


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Multicomponent fluid flow by discontinuous Galerkin and ...Galerkin (DG) methods. We use the cross-flow equilibrium concept to approximate the fractured matrix mass transfer. The discrete