Compositional Modeling ofDiscrete-Fractured Media Without Transfer

Functions by the Discontinuous Galerkinand Mixed Methods

H. Hoteit, Reservoir Engineering Research Inst. (RERI), and A. Firoozabadi, RERI and Yale U.

SummaryIn a recent work, we introduced a numerical approach that com-bines the mixed-finite-element (MFE) and the discontinuousGalerkin (DG) methods for compositional modeling in homoge-neous and heterogeneous porous media. In this work, we extendour numerical approach to 2D fractured media. We use the dis-crete-fracture model (crossflow equilibrium) to approximate thetwo-phase flow with mass transfer in fractured media. The dis-crete-fracture model is numerically superior to the single-porositymodel and overcomes limitations of the dual-porosity model in-cluding the use of a shape factor. The MFE method is used to solvethe pressure equation where the concept of total velocity is in-voked. The DG method associated with a slope limiter is used toapproximate the species-balance equations. The cell-based finite-volume schemes that are adapted to a discrete-fracture model havedeficiency in computing the fracture/fracture fluxes across threeand higher intersecting-fracture branches. In our work, the prob-lem is solved definitively because of the MFE formulation. Severalnumerical examples in fractured media are presented to demon-strate the superiority of our approach to the classical finite-difference method.

IntroductionCompositional modeling in fractured media has broad applicationsin CO2, nitrogen, and hydrocarbon-gas injection, and recycling ingas condensate reservoirs. In addition to species transfer, the com-pressibility effects should be also considered for such applications.Heterogeneities and fractures add complexity to the fluid-flowmodeling. Several conceptually different models have been pro-posed in the literature for the simulation of flow and transport infractured porous media.

The single-porosity approach uses an explicit computationalrepresentation for fractures (Ghorayeb and Firoozabadi 2000; Riv-ière et al. 2000). It allows the geological parameters to varysharply between the matrix and the fractures. However, the highcontrast and different length scales in the matrix and fracturesmake the approach unpractical because of the ill conditionality ofthe matrix appearing in the numerical computations (Ghorayeb andFiroozabadi 2000). The small control volumes in the fracture gridsalso add a severe restriction on the timestep size because of theCourant-Freidricks-Levy (CFL) condition if an explicit temporalscheme is used.

The dual-porosity/dual-permeability model (Warren and Root1963; Kazemi 1969; Thomas et al. 1983; Bourgeat 1984; Arbogastet al. 1990) is the widely used approach. Under the assumption thatthe medium is composed of many well-connected fractures, thismodel describes the fracture network by an equivalent porous me-dium that generally has the configuration of a sugar cube. Despitethe numerical efficiency of this approach, it has many limitations.The accuracy depends on the description of the transfer functions

between the matrix and the fractures. These functions may not beproperly defined with gravity, compressibility, and compositionaleffects. Another limitation in this approach is the assumption thatthe medium is composed of densely connected fractures, and itmay lose accuracy when treating few discrete or connected fractures.

The discrete-fracture model can be considered as an alternativeto the single- and dual-porosity models. The discrete-fracturemodel treats fractures explicitly in the medium. However, unlikethe single-porosity model, fractures are represented by (n-1)-dimensional elements in an n-dimensional domain (Noorishad andMehran 1982; Baca et al. 1984; Granet et al. 1998). This simpli-fication makes the latter much more efficient than the single-porosity model. Discrete-fracture models have been successfullyemployed for immiscible oil-water displacement; in 2D space byKim and Deo (1999, 2000) and Karimi-Fard and Firoozabadi(2003) using the finite-element method; and in 2D and 3D byBastian et al. (2000) and Monteagudo and Firoozabadi (2004)using the control-volume finite-element method.

Unlike the case of black-oil systems, very few studies are foundin the literature for compositional modeling in fractured reservoirs.In this work, we use the discrete-fracture model to approximate thetwo-phase multicomponent fluid flow in 2D fractured media.Based on the crossflow equilibrium concept, the pressure in afracture element is assumed to be equal to the pressure in thesurrounding matrix elements. This assumption, which is similar tothe one in vertical equilibrium, has validity as long as matrixelements are not too large (Tan and Firoozabadi 1995). Integratingthe flow equations in a control volume that includes the fractureelement and the adjacent matrix elements alleviates the computa-tion of the matrix/fracture fluxes with the crossflow equilibriumassumption. This simplification results in computational effi-ciency, and the reduction in CPU time can be two to three ordersof magnitude compared to the single-porosity models. It also over-comes limitations of the dual-porosity models.

Most of the cell-based finite-volume schemes that are adaptedto a discrete-fracture model have the fracture pressure unknowns atthe discretized-element centers. Using two-point approximationcould be sufficient to evaluate the flux across the intersection oftwo fracture branches. However, computing the flux across theintersection of three or more branches is a challenge. The currenttechniques that describe the flux across multi-intersecting fracturesmay have some deficiency. In our approach, this deficiency issolved definitively because of the MFE formulation that provides,in addition to the cell pressure average, the pressure at the inter-faces of the fracture element. We can write the global mass balanceat the intersection assuming zero accumulation. The fluxes acrossthe branches are then computed readily.

The governing flow equations in the 2D fractured and unfrac-tured media are solved by using a new numerical approach thatcombines the MFE and DG methods integrated with the discrete-fracture model. In this work, we neglect capillary pressure anddiffusion processes (in our forthcoming publication, both pro-cesses will be included). The DG method is used to solve thespecies-balance equations. It can be applied to structured and un-structured grids, and it conserves mass locally at the element level.The state unknown (concentration) is approximated by using linearand bilinear shape functions on triangular and rectangular ele-

Copyright © 2006 Society of Petroleum Engineers

This paper (SPE 90277) was first presented at the 2004 SPE Annual Technical Conferenceand Exhibition, Houston, 26–29 September, and revised for publication. Original manuscriptreceived for review 26 October 2004. Revised manuscript received 26 January 2006. Paperpeer approved 5 February 2006.

341September 2006 SPE Journal

ments, respectively. In this work, an implicit-pressure-explicit-concentration scheme is used, where the backward-Euler and for-ward-Euler schemes are used to discretize the time operator in thepressure equation and in the species-balance equations, respec-tively. Runge-Kutta methods can be used with the DG method(Cockburn and Shu 1989; 1998). However, high-order Runge-Kutta methods are usually associated with high-order spatial dis-cretization (Cockburn and Shu 2001). Using a multistep temporalscheme for our problem, which requires phase-split calculations,increases the computational cost. Moreover, if a high-order tem-poral scheme is used for the pressure equation to preserve consis-tency with the high-order temporal scheme for the species-balanceequations, the computational cost can increase substantially. Theuse of high-order spatial approximations such as the DG methodmakes the scheme unstable (Chavent and Cockburn 1989). A 2Dslope limiter (Chavent and Jaffré 1986; Hoteit et al. 2004), whichreconstructs the local solutions, is used for stabilization. Otherhigh-order upwind schemes such as TVD, essentially nonoscilla-tory (ENO), and weighted-ENO schemes have been used for thesimulation of compositional modeling in 1D (Mallison et al. 2005)and black-oil modeling in 2D (Rubin and Blunt 1991). In multi-dimensional streamline-based models, these methods are also usedto approximate the 1D solution along the streamlines (Peddibhotla etal. 1997).

In this work, the MFE method based on the lowest-order Ra-viart-Thomas space (Raviart and Thomas 1977) is chosen to solvethe pressure equation, where the total velocity concept is imple-mented (Ewing and Heinemann 1983; Durlofsky and Chien 1993;Chen and Ewing 1997a,b). To avoid simultaneous calculation ofthe pressure and flux unknowns, the hybridized MFE method(Brezzi and Fortin 1991; Chavent and Roberts 1991) is used. Withthis approach, the primary unknown is the pressure average at thegrid interfaces (pressure traces). In Hoteit and Firoozabadi (2006,2005), we demonstrate the superiority of the combined MFE-DGmethods to the classical finite-difference (FD) method associatedwith a single-point upstream weighting scheme.

In the following, we first review the governing equations oftwo-phase multicomponent fluid flow in porous media. We thenpresent the extension of the MFE-DG method to fractured media,where we show how to calculate the matrix/fracture and fracture/fracture fluxes. In the results section, we present numerical ex-amples for binary and ternary mixtures to demonstrate the effi-ciency and performance of the proposed algorithm. We concludethe work with a summary of the main results.

Mathematical ModelThe two-phase (gas and oil) flow of a multicomponent fluid isgoverned by the component balance equations, Darcy’s law, andthe thermodynamic equilibrium between the phases. In this work,we neglect the capillary and physical diffusion. In future work,these effects will be taken into account.

The material balance for component i in a two-phase nc-component mixture is given by

��czi

�t+ �.Ui = 0, i = 1, . . . , nc , . . . . . . . . . . . . . . . . . . . . . . (1)

where Ui = �coxio�o + cgxig�g�.

The compositions zi , xio , and xig are constrained by

�i=1

nc

zi = �i=1

nc

xio = �i=1

nc

xig = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

In the above equations, � denotes the porosity, c is the overallmolar density, the subscripts g and o refer to gas and oil phases, iis the component index, zi is the overall mole fraction of compo-nent i, xi� is the mole fraction of component i in phase � (��o,g),and c� is the molar density of phase �. Other variables are Ui, themolar flux of component i, and �� is the phase velocity.

The velocity for each phase is given by the Darcy law,

�� = −kk

r�

��

��p − ��g�, � = o, g, . . . . . . . . . . . . . . . . . . . . . . (3)

where k is the absolute permeability of the porous medium; kr� ,�� , and �� are the relative permeability, viscosity, and mass den-sity of phase �, respectively; p denotes the pressure; and g is thegravitational vector.

Based on the concept of volume balance, one can write thepressure equation (Ács et al. 1985; Watts 1986):

���p

�t+ �

i=1

nc

vi�.Ui = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where � is the total fluid compressibility and vi is the total partialmolar volume for component i.

The local thermodynamic equilibrium implies the equality ofthe fugacities of each component in the two phases; that is,

foi�T, p, xjo, j=1, ..., nc−1� = fgi�T, p, xjg, j=1, ..., nc−1�, i = 1, . . . , nc .. . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

Numerical Approach. The description of the combined MFE andDG methods for unfractured media is provided in Hoteit andFiroozabadi (2006). In this work, we extend the approach to dis-crete-fractured media and focus on the calculation of fracture/matrix and fracture/fracture flux in two-phase flow.

Fracture-Matrix Fluxes. The fracture-matrix flux calculationis avoided to overcome two serious limitations of the single-porosity and dual-porosity models. In a single-porosity model, thefracture and matrix elements are gridded in the same way. Such adiscretization places a severe limitation on the timestep size. Inmost of the explicitly temporal discretization schemes, the size ofthe timestep is restricted to the CFL condition. These schemes arestable if CFL where, generally, 1. The CFL condition isproportional to the total flux across the cell interfaces and in-versely proportional to the cell area (Putti et al. 1990) (in 2Dspace). As a result, the timestep size in the fracture can be ordersof magnitude smaller than the timestep in the matrix because of thelength-scale difference. The dual-porosity model does not havethis limitation because the flow is calculated in the fracture net-work fed from the matrix. The mass exchange between the matrixand fracture is described by a predetermined transfer function.Defining this function with gravity, compressibility, and compo-sitional effects is a complicated task.

To overcome these limitations, we assume that the state un-knowns (pressure and composition) in the fracture and in the ad-jacent matrix cells are the same (see Fig. 1). This assumptionremoves the need for calculating the matrix/fracture flux; only thematrix/matrix and the fracture/fracture fluxes are required becausethe governing equations are integrated over the control volume thatincludes the fracture element and the two adjacent matrix cells. Asshown in Fig. 1, the dual-control volume has a quadrilateral shapefor both triangular and rectangular grids. Consequently, our meshmay have triangular, rectangular, and slice finite elements. Thismixing in the mesh elements can be readily treated in the MFE andDG methods (Hoteit and Firoozabadi 2005).

Matrix/Matrix and Fracture/Fracture Flux. In this section,we describe how to calculate the volumetric and molar fluxesacross the interface of a control volume that contains a fracture.

Fig. 1—Typical control volumes containing a fracture.

342 September 2006 SPE Journal

We also present a procedure to calculate these fluxes at the inter-section of several fracture branches.

Similar to Hoteit and Firoozabadi (2006), we introduce �, thetotal velocity (Ewing and Heinemann 1983; Durlofsky and Chien1993; Chen and Ewing 1997a,b) which is defined as the sum of gasand oil velocities ���o+�g. The total velocity � can be written incompact form,

� = −K��p − �g�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

where K��kkr� /�� is the mobility of phase

� �� = o, g�, K = �Ko + Kg�

and

� = �Ko�o + Kg�g���Ko + Kg�.

The phase velocity �� can be expressed in terms of total ve-locity � independent of �p, or

�� = ���� − G��, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

where ���K� /K, and

G� = �Ko��o − �g�g, � = g

Kg��g − �o�g, � = o.

Approximation of the Total Velocity. The total velocity is ap-proximated by using the lowest order Raviart-Thomas space (RT0).The total velocity �K in the matrix control volume K is expressedin terms of the normal flux qK,E across the edges E of the controlvolume, that is,

�K = �E∈�K

qK,E

wK,E

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

Similarly, the total velocity �f in the 1D fracture slice f is writtenas a function of the flux qf,e along the fracture ends e; that is,

�f = �e∈�f

qf,e

wf,e

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

In the previous two equations, wK,E and wf,e are the RT0 basisfunction in the surface and line elements, respectively.

Multiplying Eq. 8 and Eq. 9 by wK,E and wf,e, respectively, andintegrating by parts, the total fluxes qK,E and qf,e are expressed asa function of the cell pressure average pK and slice pressure av-erage pf and the pressure traces tpK,E and tpf,e, respectively. Thenone obtains (see the Appendix)

qK,E = �K,EpK − �E�∈�K

� K�E,E�

tpK,E� − �K,E, E ∈ �K, . . . . . . (10)

and

qf,e = �f,epf − �e�∈�f

� f�e,e�

tpf,e� − �f,e

, e ∈ �f. . . . . . . . . . . . . . (11)

In Eqs. 10 and 11, the coefficients �K,E , K, �K,E , �f,e , f , and �f,e

depend on the geometrical shape of the element and the localmobility coefficient (see the Appendix).

The pressure and flux degrees of freedom in the matrix andfracture elements are shown in Fig. 2. We assume that the control-

volume pressure average and the fracture slice pressure averageare equal; that is, pk=pf . For rectangular elements, we also assumethat the pressure traces at the intersection between the fractureslice and the control-volume edges are the same; that is, tpf,e=tpK,E

if e∈E (see Fig. 2). We then impose the flux continuity across thecontrol-volume interfaces, (qK,E+qK�,E�0; E=K∩K�). At the in-tersection point � of Nf (Nf�2) connected fractures fi , we assumethat there is no volumetric accumulation; that is,

�i=1

Nf

qfi,�= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

The flux unknown can be eliminated from Eqs. 10 and 11. As aresult, the following linear system with main unknowns, the pres-sure average P and the pressure trace TP , is obtained:

RTP − MTP = I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

In Eq. 13, M is a sparse square matrix of dimension NE; R is a sparseNK×NE rectangular matrix and I is a vector of dimension NE; NK

is the number of control volumes; and NE is the number of controlvolume interfaces plus the number of fracture element intersections.

Approximation of the Pressure Equation. We replace thephase velocity �� in Eq. 4 with Eq. 7. Then, the pressure equationtakes the following form:

���p

�t+ �

i=1

nc

vi�.�mi� − si� = 0, . . . . . . . . . . . . . . . . . . . . . . . . (14)

where mi = ��=o,g

c�xi��� and si = ��=o,g

c�xi���G� .

Eq. 14 is then integrated over all control volumes of the grid.Let us consider a control volume K that includes a fracture elementf. The integration of Eq. 14 over the control volume is subdividedinto the integral over the matrix control volume and the integralover the fracture element. The integral over the fracture can besimplified when multiplied by the fracture aperture �. The use ofthe divergence theorem yields:

��K�K |K | + | f |�f � f��pK

�t= �

i=1

nc

vi,K��K

�mi,K JK.nK − si,k.nK�

+ � �i=1

nc

vi,f ��f

�mi,f Jf.nf − si,f .nf�,

. . . . . . . . . . . . . . . . . . . . . . . . . . (15)

where nK and nf are the outward unit normal to the cell boundary�K and the fracture ends �f, respectively; |K | and | f | are the areaof the matrix and fracture elements K and f, respectively.

Using Eqs. 10 and 11 in Eq. 15, the flux unknown can beeliminated to obtain an expression in terms of the pressures. Byapplying the backward-Euler scheme to discretize the time opera-tor and considering all the coefficients explicitly in time, we obtaina second linear system in terms of the pressure average P and thepressure trace TP:

DP − RTP = G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

In the above equation, D is a square diagonal matrix of dimensionNK, R is a sparse NK×NE rectangular matrix, and G is a vector ofdimension NK .

Combining the linear systems in Eqs. 13 and 16, one ob-tains the final linear system, where the pressure traces are theprimary unknowns:

�M − RTD−1R�TP = RTD−1G − I. . . . . . . . . . . . . . . . . . . . . . . . . (17)

Note that D is a diagonal matrix; it can be readily inverted. Aftercalculating TP, the cell pressure average and the flux unknownscan be computed locally through Eqs. 15, 10, and 11.

Solution of the Species-Balance Equation. The DG method isused to discretize the species balance in Eq. 1. The method consistsof a discontinuous, piecewise, linear (on triangles) or bilinear (onFig. 2—Pressure degrees of freedom in the control volume.

343September 2006 SPE Journal

rectangles) approximation of the unknowns czi , i=1, . . . , nc. De-tails on the implementation of the DG method for the species-balance equations in unfractured media are provided in Hoteit andFiroozabadi (2005, 2006). In this work, for the sake of brevity, weonly describe the implementation of our method to the overallspecies balance equation; that is,

��c

�t+ �.��

�=o,g

c���� = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

Let K be a matrix block that contains a fracture slice f. As withEq. 15, the integration of Eq. 18 over the matrix block and fractureslice is written as

��K

�K

�cK

�t+ �

f

�f

�cf

�t �+ �

�=o,g��

�K

cK,�in⁄outJ�.n + �

�f

cf,�in⁄outJ�.n� = 0, . . . . . . . (19)

where the superscript “in/out” denotes the upstream values, whichare used to calculate the upstream weighted numerical fluxes [seeHoteit and Firoozabadi (2006)].

Details on the implementation of the method and the slopelimiter can be found in Hoteit and Firoozabadi (2006). In appli-cation to fractured media for this work, we assume that the con-centration unknowns in the fracture block and in the matrix blockare the same. The main difficulty in Eq. 19 is the calculation offracture/fracture flux (the last term in Eq. 19) at the intersectionpoint of several fractures.

The current techniques in the literature for the calculation of thefluxes at the intersection node of several fractures may have de-ficiencies. For two intersecting fractures, fluxes along each branchcan be easily evaluated by introducing an unknown approximatingthe pressure at the intersection node. Then, by writing the balanceat that node, the pressure unknown can be eliminated and thefluxes can be evaluated properly (Karimi-Fard et al. 2004). Here,no accumulation of material is assumed at the intersection node.This simple technique is not useful for three and higher intersect-ing fracture branches. Karimi-Fard et al. (2004) used an analogybetween flow in fractured porous media and conductance througha network of resistors. They used the so-called star-delta rule tocalculate the fluxes. Granet et al. (2001) introduced an additionalnode at the intersection in order to calculate the saturation at anintermediate time. They assumed that the transport between anintersection node and a joint node is twice as fast as the transportbetween two joint nodes.

In this work, we have discovered that the MFE method cannaturally solve the problem without any special treatment. Becausethe MFE formulation provides the pressure traces, the volumetric

fluxes can be computed locally (see Eq. 11). The remaining dif-ficulty is to define the material fluxes (mobility) at that node.

Let us consider an intersection point � with Nf connections,where fi; i�1, . . . , Nf are the labels of the connected fracturebranches (see Fig. 3). Denote the mobilities and the total volumet-ric fluxes by mi; i�1, . . . , Nf and qi; i�1, . . . , Nf , respectively,at � for the fractures fi; i�1, . . . , Nf. We also need to classify theinflux and efflux with respect to I. There exists an integer �,1�Nf−1 such that

qi � 0 for 1 i �

qi 0 for � � i Nf .. . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

Note that the effluxes are assumed to have positive sign. Indeed, themass balance at � assures the existence of �. One can write the totalvolumetric balance and the total material balance for each component:

�i=1

�

qi = −�i=�+1

Nf

qi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

�i=1

�

miqi = −�i=�+1

Nf

m�qi = −m� �i=�+1

Nf

qi , . . . . . . . . . . . . . . . . . . . . (22)

where m� refers to the mobility at �; it is the upstream mobility for alloutgoing fluxes. From Eqs. 21 and 22, m� can be readily calculated:

m� =�i=1

�

miqi

�i=1

�

qi

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)

This expression for the mobility at the intersection � may be con-sidered as a generalization of the classical single-point upstreamtechnique at the intersection of two fractures. If Nf�2 (��1), for

Fig. 3—Intersection node of five intersecting fractures (Nf=5,�=4).

Fig. 4—Fracture configuration for Example 1 (4 connected frac-tures), dimensions in meter.

Fig. 5—Fracture configuration for Example 2 (sugar-cube con-figuration), dimensions in meter.

344 September 2006 SPE Journal

example, Eq. 23 gives m��m1, which is the classical single-pointupstream solution.

Numerical ResultsIn this section, we present numerical examples for binary andternary mixtures for fractured media in 2D with linear relative

permeabilities. The configuration of the fractures is shown inFigs. 4 through 6. All the results are for the structured grids.Results for the unstructured grids will be presented in future work.The examples are selected to show the robustness and efficiency ofthe numerical procedure introduced in this work. In Hoteit andFiroozabadi (2006), we compared the MFE-DG method for homo-geneous and heterogeneous media to a commercial software thatuses the single-point upstream weighting FD method. The resultsdemonstrated that the MFE-DG method with higher-order approxi-mation is orders of magnitude more efficient than the single-pointupstream weighting FD method for a comparable accuracy in thesolution. However, in the examples that follow, we compare re-

Fig. 6—Fracture configuration for Example 3 (complex fracturedmedia), dimensions in meter.

Fig. 7—Methane composition in the gas phase for various grid refinements by the MFE-DG and MFE-FD methods: Example 1, PVinjection=0.41, dimensions in meter.

345September 2006 SPE Journal

sults from the MFE-DG method to those from the MFE-FDmethod. In the MFE-FD method, the shape functions for the DGmethod are assumed to be constant. In this case, the DG method isreduced to the first-order upstream FD method. In all our com-parisons, we use the single-point upstream weighting scheme withthe first-order FD method.

Example 1. In this example, we consider a 2D horizontal domainwith four connected fractures. The fracture configuration is shownin Fig. 4. The domain area is 100×100 m2. Methane is injected atone corner to displace propane, which saturates the domain, to theopposite producing corner (see Table 1). At the production well,the pressure is kept constant and equal to the initial pressure. InFigs. 7 and 8, the composition profile of C1 calculated by theMFE-FD and MFE-DG codes is shown at 41 and 82% PV injec-tion. Different griddings (40×40, 60×60, and 100×100) are used toexamine the solution convergence. The results by the MFE-FDmethod have a pronounced numerical dispersion compared to theMFE-DG solution. The MFE-DG method for a 40×40 griddingintroduces less numerical dispersion than the MFE-FD on a100×100 grid (see Figs. 7 and 8). There is significant merit inusing the higher-order approximation in the DG method.

Example 2. We consider the displacement of a binary liquid mix-ture of C2/C3 by a gas mixture of C1/C3. The gas (90% of C1 and10% of C3) is injected at one corner of the 2D fractured domain.

We use a sugar-cube configuration for the fractures (see Fig. 5).The relevant data for fluid, fracture, and rock properties are givenin Table 2. The composition profiles of methane and propane inthe oil and gas phases at three different positions (at Points A, B,and C, see Fig. 5) in the domain are depicted vs. the injected PVin Figs. 9 and 10. The reference solution is obtained by the MFE-DG method on a 100×100 grid. The MFE-FD solution showssignificant numerical dispersion compared to the MFE-DG solution.

Example 3. This example is chosen to show the effect of fractureson the compositional flow performance by injecting gas in thefractured and unfractured media. The fluid system in Example 1 isused in this example. The matrix and fracture permeabilities are 10

Fig. 8—Methane composition in the gas phase for various grid refinements by the MFE-DG and MFE-FD methods: Example 1, PVinjection=0.82, dimensions in meter.

346 September 2006 SPE Journal

md and 106 md, respectively. The fracture aperture is 1 mm. Therelevant data for fluid, rock, and fracture properties are presentedin Table 1. The fracture configuration can be seen in Fig. 6. Thecompositional contours of methane in the fractured and unfrac-tured domains are depicted in Fig. 11 at 0.13, 0.31, and 1.2 PV. InFig. 12, we show the methane composition at the production wellvs. the injected PV.

Summary and ConclusionsIn Hoteit and Firoozabadi (2006), we demonstrated the superi-ority of the MFE-DG method over the single-point upstreamweighting FD method in compositional modeling. In this work, wehave extended the MFE-DG method to the compositional model-

ing in fractured media. The following are introduced in the exten-sion:1. Fluid flux between the matrix and the fracture is based on the

assumption that the pressure and the concentration unknowns ina fracture element are equal to those in the adjacent cells thathave the fracture element in common. This technique allowseliminating the calculation of the flux between the fracture andthe matrix.

2. The MFE method is used to approximate the pressure and thetotal flow velocity along the fractures. The degrees of freedomof this approximation are the pressure average over the fractureelement, the pressure traces, and the volumetric fluxes acrossthe element extremities. The MFE method allows the represen-tation of the fracture entity by the cell edges.

Fig. 9—Methane composition in the gas and liquid phases at different points vs. the injected PV by the MFE-DG and MFE-FDmethods: Example 2 (grid 30×30).

347September 2006 SPE Journal

We also presented a new method to calculate the fracture/fracture flux in two-phase flow. Generally, the cell-based finite-element methods face difficulty to approximate flow in multi-intersecting fractures. As mentioned previously, the MFE approxi-mation in the fractures provides, in addition to the pressureelement averages, the pressure traces and the fluxes across theelements. Consequently, the MFE approximation does not need aspecial treatment even with gravity. One simply writes the materialbalance equations at the intersection node. Our method allowsnatural evaluation of the upstream function values.

Nomenclaturec � overall molar density, mole/m3

c� � molar density of the � phase, mole/m3

e � node of fracture elements

E � mesh edgef � 1D fracture element

fgi � fugacity of component i in gas phasefoi � fugacity of component i in oil phaseg � acceleration of gravity, m/s2

kr� � relative permeabilityK � mesh cell

K� � effective mobility, m.s/kgnc � number of componentsNe � number of edges of a cellNE � number of edgesNf � number of intersecting fracture elements

NK � number of cells in a meshp � pressure, Pa

Fig. 10—Methane composition in the gas and liquid phases at different points vs. the injected PV by the MFE-DG and MFE-FDmethods: Example 2 (grid 60×60).

348 September 2006 SPE Journal

qf,e � flux across edge e in fqK,E � flux across edge E in K

t � time, sT � temperature, Ktp � pressure trace, PaUi � molar flux of component i

wf,e � RT0 basis function in the line elementwK,E � RT0 basis function in the surface element

xi� � mole fraction of component i in phase �zi � overall mole fraction of component i� � total velocity, m/s

�� � velocity of phase �, m/s� � viscosity, kg/m/s�i � total partial molar volume of i� � total fluid compressibility

�� � mass density of phase �, kg/m3

� � porosity, %

AcknowledgmentsThis work was supported by the member companies of theReservoir Engineering Research Inst. (RERI). This support isgreatly appreciated.

Fig. 11—Methane composition in the gas phase in homogeneous and fractured domains for various PV injection by the MFE-DGmethod: Example 3, dimensions in meter.

349September 2006 SPE Journal

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Fig. 12—Methane composition at the production well vs. the PVinjection for homogeneous and fractured domains by the MFE-DG method: Example 3.

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AppendixIn this appendix, we provide details of the approximation of thepressure and velocity field by the MFE method. The MFE method,which is based on the lowest-order Raviart-Thomas space, ap-proximates separately the total velocity equation (Eq. 6) and thepressure equation (Eq. 14).

Approximation of the Total Velocity Equation. From the Ravi-art-Thomas approximation space, the vectors � and g over eachelement K can be expressed in terms of the lowest-order Raviart-Thomas basis function wK,E and the fluxes across the elementboundaries; that is,

� = �E∈�K

qK,EwK,E and g = �E∈�K

qK,Eg wK,E, . . . . . . . . . . . . (A-1)

where qK,E = �E

�.nK,E and qK,Eg = �

E

g.nK,E = g |E | cos�Ang�g,nK,E��.

By inverting the mobility tensor K, Eq. 6 becomes

K−1� = −��p − �g�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

Multiplying Eq. A-2 by the basis function wK,E and integrating byparts yields

�K

wK,EK−1J = −�K

wK,E��p − �g�

= −�K

wK,E�p + �K

�wK,Eg

= �K

p�.wK,E − ��K

pwK,E.nK,E

+ �K

�wK,Eg E ∈ �K. . . . . . . . . . . . . . . . . . . . . (A-3)

The Raviart-Thomas basis function wK,E satisfies the follow-ing properties:

�.wE =1

|K | , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4)

wE.nE� = �1� |E | if E = E�

0 if E � E�.. . . . . . . . . . . . . . . . . . . . . . . (A-5)

Using Eqs. A-4 and A-5, the integral terms in the right side ofEq. A-3 are simplified to

�K

wK,EK−1� =1

|K | �K

p −1

|E | �E

p + �K

�wK,Eg E ∈ �K.

. . . . . . . . . . . . . . . . . . . . . . . . (A-6)

Let pK and tpK,E denote the cell average pressure on K and the edgeaverage pressure on E, respectively (that is, the first and the secondterms on the right side of Eq. A-6). Replace Eq. A-1 in Eq. A-6 to get

�E�∈K

qK,E��K

wK,EKK−1wK,E� = pK − tpK,E�

+ �E�∈K

�KqK,E�g �

K

wK,EwK,E� E ∈ �K. . . . . . . . . (A-7)

Eq. A-7 can be written in matrix form

BKQK = pKe − TpK − �KBKQKg , . . . . . . . . . . . . . . . . . . . . . . . . (A-8)

where

BK = ��BK�E,E� �E,E�∈�K; �BK�E,E� = �K

wK,EKK−1wK,E�

BK = ��BK�E,E� �E,E�∈�K; �BK�E,E� = �K

wK,EwK,E�

QK = �qK,E�E∈�K; QKg = �qK,E

g �E∈�K;

TpK = �tpK,E�E∈�K; e = �1�E∈�K.

By inverting BK in Eq. A-8, the flux qK,E through each edge E isexpressed as a function of the cell pressure average pK and theedge pressure averages tpK,E , or

qK,E = �K,EpK − �E�∈�K

� K�E,E�

tpK,E� − �K,E , E ∈ �K, . . . . . (A-9)

where,

�K,E = �E�∈�K

�BK−1�E,E�,

� K�E,E� = �BK−1�E,E�,

and �K,E = −�K �E�∈�K

�BK−1BK�E,E�qK,E�

g .

In a similar way, the flux at the 1D fracture ends is written as

qf,e = �f,epf − �e�∈�f

� f�e,e�tpf,e� − �f,e, e ∈ �f. . . . . . . . . . . . (A-10)

Approximation of the Pressure Equation. The fluxes across theboundaries of a matrix element K and the ends of a fracture ele-ment f are defined, respectively, as

�E∈�K

qK,E = ��K

��K.nK�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-11)

�E∈�f

qf,E = ��f

��f .nf�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-12)

Replacing Eqs. A-11 and A-12 in Eq. 15 yields

�K,f

dpK

dt= �

i=1

nc

vi,K��E∈�K

mi,K,EqK,E − si,K,E�+ ��

i=1

nc

vi,f��e∈�f

mi,K,eqK,e − si,K,e�, . . . . . . . . . . . . . (A-13)

where �K,f = ��K�K|K | + | f |�f �f�,

si,K,E = �E

�si,K.nK,E�, and si,f,e = �E

�si,f.nf,e�.

Eq. A-13 can be written in terms of the pressure and the tracesof the pressure unknowns by placing Eqs. A-9 and A-10 intoEq. A-13; that is,

�K,f

�pK

�t= �KpK − �

E∈�K

K,EtpK,E − �e∈�f

f,etpf,e + �K, . . . . . (A-14)

where

�K = �E∈�K

vK,E�K,E; vK,E = �i=1

nc

�vi,Kmi,K,E�

K,E

= �E�∈�K

vK,E�� K�E,E�; f,e = � �e�∈�f

vf,e�� f�e,e�

�K = �E∈�K

vK,E�K,E + �i=1

nc

vi,Ksi,K,E + ��e∈�f

vf,e�f,e + � �i=1

nc

vi,fsi,f,e.

. . . . . . . . . . . . . . . . . . . . . . . .(A-15)

351September 2006 SPE Journal

Hussein Hoteit is a scientist at the RERI in Palo Alto, California.His research interests include numerical modeling of mul-tiphase miscible and immiscible flow in homogeneousand fractured media. He holds MS and PhD degrees in ap-plied mathematics from U. Rennes 1, France. Abbas Firooza-badi is a senior scientist and director at RERI. He also teachesat Yale U. and at Imperial College, London. e-mail:af@rerinst.org. His main research activities center on thermo-

dynamics of hydrocarbon reservoirs and production andon multiphase/multicomponent flow in fractured petroleumreservoirs. Firoozabadi holds a BS degree from Abadan Inst.of Technology, Abadan, Iran, and MS and PhD degreesfrom the Illinois Inst. of Technology, Chicago, all in gas engi-neering. Firoozabadi is the recipient of both the 2002 SPE An-thony Lucas Gold Medal and the 2004 SPE John Franklin CarllAward.

352 September 2006 SPE Journal