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Control-Volume Model for Simulation ofWater Injection in Fractured Media:

Incorporating Matrix Heterogeneity andReservoir Wettability Effects

J.E.P. Monteagudo* and A. Firoozabadi, SPE, Reservoir Engineering Research Institute

SummaryThe control-volume discrete-fracture (CVDF) model is extendedto incorporate heterogeneity in rock and in rock-fluid properties. Anovel algorithm is proposed to model strong water-wetting withzero capillary pressure in the fractures. The extended method isused to simulate: (1) oil production in a layered faulted reservoir,(2) laboratory displacement tests in a stack of matrix blocks witha large contrast in fracture and matrix capillary pressure functions,and (3) water injection in 2D and 3D fractured media with mixed-wettability state. Our results show that the algorithm is suitable forthe simulation of water injection in heterogeneous porous mediaboth in water-wet and mixed-wettability states. The novel ap-proach with zero fracture capillary and nonzero matrix capillarypressure allows the proper prediction of sharp fronts in the fractures.

IntroductionThis work is focused on the numerical treatment of two mainphysical aspects of multiphase flow in fractured porous media:heterogeneity in rock-fluid properties and reservoir wettability.

In a previous work (Monteagudo and Firoozabadi 2004), aCVDF method was used to discretize the system of equationsgoverning water injection in fractured media with strong-water-wettability state and homogeneous matrix and rock-fluid proper-ties. The method was restricted to a finite contrast in matrix-fracture capillary pressure. In this work, we extend the CVDFmodel for simulation of water injection in fractured media com-prised of heterogeneous rocks and wettability conditions fromstrong-water-wetting to mixed-wetting conditions. We also presenta formulation for infinite contrast in capillary pressures of matrixand fractures (zero capillary pressure in the fracture and finitecapillary pressure in the matrix).

The control volume (CV) method, first proposed by Baliga andPatankar (1980), is a finite-volume formulation over dual cells(CVs) of a Delaunay mesh. It is locally conservative and suited forunstructured grids. It has been widely employed for the simulationof multiphase flow in porous media (Monteagudo and Firoozabadi2004; Verma 1996; Helmig 1997; Helmig and Huber 1998; Bas-tian et al. 2000; Geiger et al. 2003) and the convergence of themethod for two-phase immiscible flow in porous medium has al-ready been proved (Michel 2003).

Numerical treatment of heterogeneity in the framework of theCV method has been extensively studied in the past (Edwards2002; Edwards and Rogers 1998; Prevost 2000; Aavatsmark et al.1998a, b). Nevertheless, those works have focused on absolutepermeability heterogeneity and anisotropy in single-phase flow.The main concern in those works is the use of full tensor perme-ability and the accurate generation of streamlines (required by thestreamline numerical method). It is well known that the standardCV method produces inaccurate velocity fields around the inter-faces of heterogeneous media as the contrast in permeability is

increased (Durlofsky 1994). In the standard CV method, Delaunaytriangles are locally homogeneous and the polygonal CV cell maybe heterogeneous (see Fig. 1a). For accurate streamlines, severalauthors (Verma 1996; Edwards 2002; Edwards and Rogers 1998;Prevost 2000; Aavatsmark et al. 1998a) have proposed that thepolygonal CV cell must be locally homogeneous, implying het-erogeneous Delaunay triangles (see Fig. 1b). The latter configura-tion, however, generates additional problems in the simulation ofmultiphase flow in porous media. Basically, from mesh generationstandpoint, it may not be possible to generate an unstructured meshwhere the boundaries of the CV median-dual cell conform to het-erogeneous interfaces in the domain. Conforming mesh is impor-tant for the discrete-fracture approach. Therefore, it would be nec-essary to first generate a standard CV cell mesh, and later a ho-mogenization procedure would be required to obtain CV cells withconstant permeability. The homogenization or upscaling of per-meability is somehow possible, but the same is not true for rock-fluid properties; most challenging is capillary pressure with dif-ferent endpoints. Therefore, the approach with the homogeneousCV cell may be suitable for single-phase simulation where rock-fluid interactions are not part of the problem. However, rock-fluidinteractions have to be taken into account for simulation of mul-tiphase flow in fractured porous medium. Frequently, capillarypressure is disregarded in two-phase flow simulations; however,capillary pressure is of importance for simulation of multiphaseflow in fractured porous media (Monteagudo and Firoozabadi2004; Karimi-Fard and Firoozabadi 2003). Predictions of flowpattern and oil recovery may be severely affected if capillary pres-sure effect is neglected.

With these considerations, we have opted for the use of thestandard CV method with a novel approach for rock-fluid hetero-geneity based on the capillary pressure continuity concept. Thenumerical inaccuracies of the velocity field at the interface ofheterogeneous media may not be as relevant as the ones generatedby homogenization of rock-fluid properties or the neglect of cap-illary effects in multiphase flow in fractured media.

The second focus of this work is proper accounting for reser-voir wettability. In some reservoirs, wettability covers a widerange from strong water-wetting to strong oil-wetting (Rao et al.1992; Morrow 1990; Robin 2001). Even in a single reservoir,wettability can change from strong water-wetting at the bottom tooil-wetting at the top of the oil column.

The paper is outlined as follows: We first present the governingequations for two-phase immiscible and incompressible flow forfractured media. Next, we briefly describe the CV discretization ofthe governing equations with the numerical treatment for hetero-geneity in rock-fluid properties and different wettability states.Then, we provide numerical simulation of field-scale and lab-scaleexamples and numerical tests for mixed-wettability conditions in2- and 3D fractured media. Finally, concluding remarks are made.

Governing EquationsTwo-Phase Incompressible Flow in Porous Media. The two-phase incompressible, immiscible-flow displacement in porousmedia can be described by two equations (Monteagudo andFiroozabadi 2004; Aziz and Settari 1979):

−� � ��w + �n��w − � � �n��c − �qw + qn� = 0 . . . . . . . . . . . (1)

* Now with ConocoPhillips.

Copyright © 2007 Society of Petroleum Engineers

Original SPE manuscript received for review 16 July 2005. Revised manuscript received 10November 2006. Paper (SPE 98108) peer approved 17 November 2006.

355September 2007 SPE Journal

��Sw

�t− � � �w��w − qw = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

where � is the porosity, �i is the mobility of phase i defined by�i=kkri /�i, and k is the absolute permeability; kri, �i, and qi are therelative permeability, viscosity and source/sink term of phase i,respectively, �w is the water flow potential, Sw is the water satu-ration, and �c is the capillary flow potential which is defined(Karimi-Fard and Firoozabadi 2003) by

�c = �n − �w = Pc + ��n − �w�gz. . . . . . . . . . . . . . . . . . . . . . . . (3)

In Eq. 3, Pc is the capillary pressure, �i is the density of phase i, gis the acceleration of gravity, and z is the vertical coordinate,positive in the upward direction.

Eq. 1 is referred to as the flow-potential equation, which iselliptic in nature, while Eq. 2 is referred to as the saturation equa-tion, which can be seen as a diffusion-convection equation (Peace-man 1977). The main variables are the wetting-phase flow poten-tial, �w, and saturation, Sw. In most of our numerical simulations,the boundary conditions are assumed to be impervious:vi≡−�i��i�0 for both phases i�{n, w}, where vi is the velocityvector of phase i. We have also incorporated the Dirichlet bound-ary condition to include an aquifer in a field-scale problem thatwill be simulated in this work. In that case, �w��D and Sw�1 atthe bottom of the reservoir.

Formulation for Fractured Media. Details of the formulation forfractured media are provided in Monteagudo and Firoozabadi(2004). The most remarkable feature of the formulation is that nocomputation of matrix-fracture transfer term is required. This ispossible because of the integration in the CV cell and the use of thecrossflow equilibrium concept, which leads to capillary continuity(Firoozabadi and Hauge 1990; van Duijn et al. 1994). Using thecrossflow equilibrium, we write the equality of flow potentials atthe interface between the matrix and fractures:

�i

m = �i

f for i = �n, w�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

From Eq. 4, we readily obtain

�c

m�Sw

m� = �c

f�Sw

f �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

Eq. 5 implies that water saturation may be discontinuous across thematrix-fracture interface with different matrix and fracture capil-lary pressure functions. A clear physical relationship is establishedbetween Sw

m and Swf :

Sw

f = �Pc

f�−1Pc

m�Sw

m�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

Therefore, the system of Eqs. 1 and 2 can be integrated with theCV method in terms of matrix variables only.

CV DiscretizationWe use the CV methodology for the spatial discretization of boththe flow potential and saturation equations. In this method, theintegration is performed over the CV cells, which are constructedas the median dual of a Delaunay triangulation in 2D or tetrahe-drization in 3D. Fig. 1a shows the 2D CV cell formed by joiningthe barycenters of Delaunay triangles with their edge midpoints forunfractured media. In Fig. 2, we show the CV cell for fracturedmedia with homogeneous matrix (Fig. 2a) and with heterogeneousmatrix (Fig. 2b) media.

Eqs. 1 and 2 have three types of terms:1. Time derivative, ��Sw /�t.2. Source/sink, qi for i�{n, w}.3. Divergent term �·F, where the flux vector F�F(Sw) may be

either (�w+�n)��w and �n��c in Eq. 1, or �w��w in Eq. 2.All mobilities are computed at the boundaries of the CV cell

with the proper upwinding, based on the flow direction at theboundary. Next, we provide the discretization of all these terms for2D fractured media with homogeneous and heterogeneous matrix.Extension to 3D media is straightforward.

Discretization for Fractured Media With Homogeneous Ma-trix. In the following, we first add the matrix and fracture termsand then approximate the integration in the finite-volume.

Time Derivative.

�A

�

�t��Sw�dA ≈ �Am�m

�Swm

�t+ Af� f

�S wf

�t �, . . . . . . . . . . . . . . (7)

where superscripts m and f denote matrix and fracture variables.Eq. 7 represents the change in the volume of the wetting phase inthe volume element per unit time. It consists of two terms, onefrom the matrix contribution and the other from the fracture con-tribution. If there is no fracture in the volume elements, then thecontribution is from the matrix only. Note that in 2D, the productAm�m (Am is the area of the matrix in the element; in 3D it shouldbe replaced by the volume of the matrix in the element) and Af� f

(Af is the area of the fracture in the element) provide the volume ofthe matrix and fracture in the volume element, respectively. Wecan express Eq. 7 in terms of Sw

m by using Eq. 6 and the chain rule:

�A

�

�t�SwdA ≈ �Am�m + Af� f

dS wf

dS wm� �S w

m

�t. . . . . . . . . . . . . . . . (8)

Source/Sink Term.

�A

qwdA≈Amqwm + Afq w

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

Fig. 1—Control volume cell for heterogeneous media. (a) Standard CV method; (b) locally homogeneous CV.

356 September 2007 SPE Journal

Divergence Term.

�A

� · FdA = �A

F · n dA ≈ �j=1

nb

�F · ne�j . . . . . . . . . . . . . . . . (10)

The approximation in Eq. 10 allows us to relate the rate of changein the volume of the wetting phase in the volume element in termsof the normal fluxes across various edges (boundary elements) j ofmeasure ej (in 3D we need to use the edge surface area). Thenumber of boundary elements is nb. The summation over all theboundary elements provides the net flux of the wetting phase overthe control volume. We remark that for the CV cell containing afracture, the summation in Eq. 10 contains also the 1D flow alongthe fracture direction.

A detailed derivation of the system of equations for the IMPESformulation can be found in Monteagudo and Firoozabadi (2004).

Discretization for Fractured Media With Heterogeneous Ma-trix. Fig. 2b shows a CV cell with a fracture and two different rockmatrices in its interior. Similar to the matrix-fracture interface, weinvoke capillary continuity at the interface between different rocks.Therefore, we can relate the saturation of all rocks at the interfaceof a heterogeneous medium.

Time Derivative. The integration of the time-derivative term isapproximated by extending the idea of Eqs. 7 and 8.

�A

�

�t��Sw�dA ≈ �

k=1

nm �

�tSw

k �kAk, . . . . . . . . . . . . . . . . . . . . . . . (11)

where nm is the number of different media inside the CV cell. Eq.11 may be expressed in terms of the saturation of the referencemedium Sw

+. The reference medium can be chosen on the basis ofparameter B in the capillary pressure model of the form

P ck = −Bk lnSw

k , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

where superscript k�1 . . . nm is an index for the different mediainside the control volume. Note that by using the capillary conti-nuity concept, a relationship can be established between the dif-ferent media saturations. For example, in the case of fracturedmedia: Sw

m /Swf �exp(−Bm/Bf ).

In this work, we select the reference medium with the highestvalue of Bk in Eq. 12.

From Eq. 6 and the chain rule we derive:

�A

�

�t��Sw �dA ≈��

k=1

nm dSwk

dS w+ �kAk� �

�tS w

+ . . . . . . . . . . . . . . . . (13)

Source/Sink Term. This term can be written as

�A

qwdA ≈ �k=1

nm

Akq wk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

Divergence Term. Integration of the divergence term is pre-sented in Eq. 10. Each triangle contained in the CV cell is locallyhomogeneous, and the saturation of different media can be relatedthrough Eq. 6.

Formulation Using Normalized Saturations. Normalized satu-ration is defined as

Sw =Sw − Swi

1 − Swi − Sor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)

where Swi is the irreducible-water saturation and Sor is the residual-oil saturation.

By using the chain rule, Eq. 2 can be expressed as:

��Sw

�t− � � �w�Sw��w − qw = 0, . . . . . . . . . . . . . . . . . . . . . (16)

where �1−Swi−Sor

Numerical Treatment for Large ContrastBetween Matrix And FractureCapillary PressuresIn our previous work (Monteagudo and Firoozabadi 2004), thecontrast between matrix and fracture capillary pressure was ex-pressed by the ratio Bm /Bf using the capillary pressure model inEq. 12. All the simulations in that work were carried out for a finiteratio, that is, Bf >0. When Bm�Bf�0, the model of Monteagudoand Firoozabadi (2004) is applicable because Sw

m�Swf and dS w

f /dS w

m�1. There is a numerical problem when Bm>0 and Bf�0 orwhen Bm/Bf is large; with strong water wetting and relatively thickfractures, Bm/Bf would be large. Below we present a solution tothis problem.

Fig. 3 illustrates the imbibition process for the fracture and thematrix media. Fig. 3a shows the fracture and matrix capillarypressure curves, for Bf�0.6 atm and Bm�1.0 atm, respectively(Bm /Bf�1.67). Fig. 3b depicts the fracture and matrix capillarypressure for Bf�0.1 atm and Bm�1.0 atm, respectively (Bm /Bf�10). Imbibition of a CV cell containing a matrix-fracture in-terface with these capillary pressure curves will follow the paththrough the points A, B, and C as indicated by the arrows, providedthe initial water saturation is zero. Because capillary continuityexpressed by Eq. 6 must be satisfied at the interface during theimbibition, S w

f �0 and dSwf /dSw

m�0 along the path from A to B forcapillary pressures depicted in Figs 3a and 3b. On the other hand,

Fig. 2—Control volume cell containing a fracture. (a) Homogeneous matrix; (b) heterogeneous matrix.


along the path from B to C the derivative dS wf /dS w

m will bemuch larger when Bf�0.1 atm (Fig. 3b) than when Bf�0.6 atm(Fig. 3a). From this analysis, one can observe that in the event ofBf�0 then Bm /Bf→� and there is apparently no way to relate S w

m

and Swf along the path B to C with the formulation outlined pre-

viously because dS wf /dSw

m��. To avoid this numerical problem,the left term in Eq. 8 may be expressed in an alternative way:

�Am�m + Af� fdS w

f

dSwm� �Sw

m

�t≡ �Am�m

dSwm

dS wf

+ Af� f� �S wf

�t. . . (17)

Then we have two equivalent expressions for the left term in Eq.8 that can be distinctly used along different sections of the imbi-bition path shown in Fig 3b. If chosen correctly, the numericalproblem described here is avoided. Eq. 17 also provides a physicalinterpretation of the imbibition process inside the CV cell; for thepath A-B, dSw

f /dSwm�0, which implies imbibition in the matrix

alone, and for the path B-C dSwm /dSw

f �0, mainly only the fracturesaturation changes. We have implemented this criterion in ourmodel to switch between the two equivalent expressions in Eq. 17.That is,

If Pc>�, then

�A

�

�t��Sw�dA ≈ �Am�m�

�Swm

�t. . . . . . . . . . . . . . . . . . . . . . . . . . (18)

Otherwise,

�A

�

�t��Sw�dA ≈ �Af� f�

�S wf

�t, . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

where � is a tolerance.

Numerical ExamplesIn this section, we present several numerical examples. First, wesimulate oil production from data extracted from a large fracturedreservoir. The second example covers data from a laboratory testof water injection in a stack of matrix blocks at different injectionrates. Next, we present the results of numerical simulation forwater injection in 2D and 3D for mixed-wettability state. The 2Dand 3D meshes were generated with the packages triangle (Shew-chuk 1996) and tetgen (Si 2002), respectively. All simulationswere run with a Pentium PC 2.66 GHz.

Fig. 3—Imbibition paths in fracture (left column) and in matrix (right column) at the interface. (a) Bf =0.6 atm, and Bm=1.0 atm; (b)Bf =0.1 atm, and Bm=1.0 atm.


Field-Scale Example With Horizontal Production Well. A hori-zontal view of the reservoir simulated is shown in Fig. 4a. Aregion with an areal dimension of 1200×500m (shown as a rect-angle with dashed lines) was selected for simulation. The regioncontains two vertical wells, 13V and 6V, and one horizontal well,6H. Only the horizontal well is producing. The region containsseveral faults (the dark solid line). In Fig. 4a, only the three mainfaults (the grey solid lines) that cross the region are shown. Thehorizontal well is contained in one of the main faults. Fig. 4bshows the fault plane XZ joining wells 13V and 6V with details ofthe location of the horizontal well. We use a simple approach torepresent wells by a sink term and assign a triangular flow distri-bution along the horizontal well (Joshi 1991) (see Fig. 5). Thereservoir has a thickness of 424 m with an aquifer at the bottom.

Fig. 4—Cross-sectional study. (a) Horizontal view; (b) XZ cross section between Wells 13V and 6V showing horizontal Well 6H.

Fig. 5—Triangular flow distribution in a horizontal well.


The aquifer is treated as the Dirichlet boundary, where water flowpotential and saturation are specified.

Fig. 6 shows the conforming tetrahedrization in the fracturedregion. The horizontal well 6H is shown as a dashed thick line. Wehave incorporated three main and seven minor fractures as quad-rilaterals (see Table 1 for vertex coordinates). Fracture thicknesswas set to 0.1 cm. The mesh contains 4,000 nodes. Preliminarysensitivity study showed that with this refinement, there is verylittle change in oil-recovery predictions. Fluid and rock propertiesare shown in Tables 2 and 3, respectively. The reservoir is lay-ered. Fig. 7 shows the permeability distribution. Rock-fluid inter-actions are shown in Fig. 8. Note that the relative permeability andcapillary pressure curves may not show a consistent wettability state.

The initial distribution of fluids is determined using the verticalequilibrium criterion (Pooladi-Darvish and Firoozabadi 2001):

Pc�z� = Pco − g��n − �w�z , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

where Pco is the capillary pressure at the bottom of the reservoir(z�0) and z is the vertical coordinate (positive in the upwarddirection). Once Pc is known, Sw is computed from Eq. 12.

The simulation was carried out to 140 years of production,keeping a constant total flow rate of 2400 B/D. The total CPU timefor this simulation was 12 hours. Water saturation contours attime�70 years are shown in Fig. 9. The corresponding simulatedoil recovery curve is shown in Fig. 10. The initial reservoir pres-sure was 2,500 psia at z�0. The predicted average pressure acrossthe horizontal well after 17 years of production was approximately1.500 psia. The reported data for the same time period is 1,550psia. Our preliminary results are on the order of the reported data

Lab-Scale Example for a Stack of Matrix Blocks. We simulatethe displacement of n-decane by water from a stack of chalk matrixblocks. Description of the equipment and experimental data areprovided by Pooladi-Darvish and Firoozabadi (2001). Previousattempts to model the waterflooding have not been successfulwhen one focuses on the water/oil front in the fractures (Pooladi-Darvish and Firoozabadi 2001; Terez and Firoozabadi 1999).

In the setup, 12 matrix blocks were assembled; fractures are thespace between blocks and between blocks and the container walls.The 3D conforming mesh for the porous media with 1,800 nodesis presented in Fig. 11 (note that there are three matrix blocks perrow). Tables 4 and 5 present fluid and rock properties, respec-tively. Rock-fluid interactions parameters are shown in Table 6.The capillary pressure model used is defined by Eq. 12, and therelative permeabilities are modeled with power laws:

krw = krw0 S w

nw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

krn = krn0 �1 − Sw�nn, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)

where Sw is the normalized saturation defined by Eq. 15. All therock-fluid interaction parameters are reported in Pooladi-Darvishand Firoozabadi (2001). Fig. 12 depicts the capillary pressure andrelative permeability. Note that capillary pressure in the fracture isconsidered zero, as was evidenced by observation of a flat oil/water interface in the fractures. The relative permeability for thematrix shows a strong-water-wetting state.

Using the previously described data (without any adjustment),we simulated the experiments. There is good agreement betweenthe simulation results and measured recovery data for all three

Fig. 6—Conforming tetrahedrization mesh used in the simulation of field-scale problem. (a) Fracture mesh; (b) matrix mesh.


injection rates. Results are shown in Fig. 13. Our simulation resultsalso show the correct shape of the front in the fractures (seeFig. 14). The front based on our simulation is in better agreementwith measured data than in the previous works (Pooladi-Darvishand Firoozabadi 2001; Terez and Firoozabadi 1999). One couldobtain a better match by adjusting the matrix capillary pressurecurve slightly. The CPU time required for a displacement of 2 PVat each flow rate was 27 hours. Flow through fractures for Pc<�requires a small timestep to avoid instabilities. This results in theincreased CPU time.

Mixed-Wettability Examples. We present 2- and 3D numericalsimulations to evaluate the performance of the CVDF method tosimulate waterflooding in porous media with mixed wettabilitystate. Because of the way the problem has been formulated, themixed-wet state has not been implemented for two-phase immis-cible flow in fractured media (Bogdanov et al. 2003).

The 2D example consists of a square of 25×25 m containing 6fractures. The 2D mesh with 2,700 nodes is shown in Fig. 15, withthe fractures marked as thick lines. Water is injected at a flow rateof 3.96×10−7 m3/s (2.7×10−4 PV/day) at the lower left corner, andthe producing well is at the opposite corner. The initial watersaturation is zero in the whole domain.

The 3D example consists of a cube of 20×20×20 m with onefracture plane inside. A plot of the mesh for the fracture plane andthe matrix is shown in Fig. 16. Water is injected at a flow rate of3.7×10−4 m3/s (0.02 PV/day) at coordinates [0 m, 0 m, 0 m], andthe producing well is placed at coordinates [20 m, 20 m, 20m].

Tables 7 and 8 present fluid and rock properties used for theexamples. Rock-fluid interactions are shown in Fig. 17. Capillarypressure has two branches, positive and negative, indicating mixedwettability state. These rock-fluid properties are used for both 2Dand 3D simulations. Because the capillary pressure model in Eq.12 is only suitable for a water-wet media, we compute dSw

f /dSwm

numerically for the mixed-wetting state. Water saturation is ini-tialized using the positive Pc–Sw relationship shown in Fig. 17.

The CPU time for the 2D geometry was 24 hours for a 1-PVdisplacement. This represents a speed-up of three times when com-pared with a previous code (Karimi-Fard and Firoozabadi 2003)due to our formulation. For the 3D geometry, the CPU time was 20minutes. Water saturation contours at 25 and 50% PV displace-ment for the 2D configuration are shown in Fig. 18a and b, re-spectively. For the 3D configuration, we show in Fig. 19a and bthe water saturation contours at 20% PV displacement for fractureand matrix, respectively. It is observed that the water front in thefracture moves fast in this problem when compared to the contoursobtained at the faults in the field-scale reservoir configuration.These results reflect what we found in our previous work (Mon-teagudo and Firoozabadi 2004); the contrast in capillary pressureaffects the performance of waterflooding in fractured media. Be-cause of large nonlinearity of capillary pressure in mixed-wet me-dia, CPU time is much higher than in water-wet media.

Conclusions1. We have extended the control-volume discrete-fracture method

(Monteagudo and Firoozabadi 2004) to account for importantphysical aspects in reservoir simulation, such as heterogeneity inmatrix rock and rock-fluid properties, different wettability con-ditions, and large contrast in the matrix-fracture capillary pressures.

2. We propose the use of capillary continuity concept in order tosolve the problem of having a heterogeneous CV cell with rocks

that may have very different rock-fluid properties with differentendpoints. The idea is tested in the field-scale simulation of alayered reservoir.

3. A novel numerical procedure has been proposed when there islarge contrast in capillary pressure at the matrix-fracture inter-face (Pc

f ≈0). The procedure was tested in the simulation of thewaterflooding in a stack of matrix blocks. The method not onlyprovides proper prediction of oil recovery at different waterinjection flow rates, but it also predicts a sharp water front inthe fractures.

4. We also present the results of the numerical simulation for waterinjection in a mixed-wet medium using our CVDF model. In thepast, the discrete fracture approach had not been used for waterinjection in fractured media with a mixed-wet state that has ahigher nonlinearity than a water-wet state. Results show that ourformulation performs well in the mixed-wet state.

NomenclatureA � area, L2, m2

B � parameter for Pc model Eq. 12, m/Lt2, Pa [atm]e � measure of a boundary segment of a CV cellF � generic flux crossing the boundaries of a CV cellk � absolute permeability, L2, m2

kri � relative permeability of phase i�{n, w}nb �number of boundary segments of a CV cellnm �number of different media inside a CV cellnn � exponent for the nonwetting-phase relative

permeability model Eq. 21nw � exponent for the wetting-phase relative permeability

model Eq. 22n � outward surface normal to the boundary of a CV cell

Pc � capillary pressure, m/Lt2, Pa [atm]qi � sink/source term of phase i, t−1, s−1

Si � saturation of phase i={n, w}Sor � residual oil saturation

Fig. 7—Permeability field of the layered reservoir.


Swi � irreducible water saturationSi � normalized saturation of phase i�{n, w} (Eq. 15)t � time, s

� parameter in Eq. 16A � boundary of CV cell A

� � tolerance�i � mobility of phase i, L3t/m, m2/Pa-s�t � total mobility, L3t/m, m2/Pa-s�i � viscosity of phase i, m/Lt, Pa-s�i � density of phase i, m/L3, kg/m3

� � porosity�c � capillary potential, m/Lt2, Pa�D � flow potential specified at Dirichlet boundary, m/Lt2,

Pa�i � flow potential of phase i, m/Lt2, Pa

Subscripts

n � nonwetting phasew � wetting phase

Fig. 9—Water saturation contours after 70 years of production. (a) Fracture contours; (b) matrix contours.

Fig. 10—Oil recovery for the field-scale simulation.

Fig. 8—Field-scale simulation (rock-fluid properties). (a) Capillary pressure; (b) relative permeability.


Superscripts

f � fracturem � matrix

AcknowledgmentsThis work was supported by the member companies of the Res-ervoir Engineering Research Institute (RERI).

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Fig. 11—Conforming tetrahedrization mesh for lab-scale simu-lation. (a) Fracture mesh; (b) matrix mesh.


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Fig. 12—Lab-scale waterflooding (rock-fluid properties). (a) Capillary pressure; (b) relative permeability.

Fig. 13—Oil recovery for different water injection rates in thestack of matrix blocks.

Fig. 14—Simulated and experimental fracture waterfront at different PV displacement in the stack of matrix blocks. (a) Front levelin the fracture; (b) water saturation in the fractures.


van Duijn, C., Molenaar, J., and de Neef, M. 1994. The effect of capillaryforces on immiscible two-phase flow in heterogeneous porous media.Tech. Report 94-103. Delft, The Netherlands: Delft University of Tech-nology.

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Jorge Palomino Monteagudo is a staff reservoir engineer in theReservoir Simulation Development Division of ConocoPhillips,Houston. He holds a BS degree from the Universidad Nacionalde Ingenieria in Lima, Peru, and MS and DSc degrees from theUniversidade Federal do Rio de Janeiro in Brazil, all in chemicalengineering. He is currently a Technical Editor for SPEREE. Ab-bas Firoozabadi is a senior scientist and director at RERI. He alsoteaches at Yale U. and at Imperial College, London. e-mail:[email protected]. His main research activities center on thermo-dynamics of hydrocarbon reservoirs and production and onmultiphase-multicomponent flow in fractured petroleum reser-voirs. Firoozabadi holds a BS degree from Abadan Inst. of Tech-nology, Abadan, Iran, and MS and PhD degrees from the Illi-nois Inst. of Technology, Chicago, all in gas engineering.Firoozabadi is the recipient of both the 2002 SPE Anthony LucasGold Medal and the 2004 SPE John Franklin Carll Award.

Fig. 16—Conforming tetrahedrization mesh for 3D mixed-wet fractured media. (a) Fracture; (b) matrix.

Fig. 15—Conforming triangulation mesh for 2D mixed-wet frac-tured media.

Fig. 17—Mixed-wet fractured system (rock-fluid properties). (a) Capillary pressure; (b) relative permeability.


Fig. 18—Water saturation contours for injection into a 2D mixed-wet fractured media with initial Sw=0. (a) 25% PV displacement;(b) 50% PV displacement.

Fig. 19—Water saturation contours at 20% PV displacement for injection into a 3D mixed-wet system where initial Sw field iscomputed with vertical equilibrium concept. (a) Fracture contours; (b) matrix contours.


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