Comput Geosci (2010) 14:527–538DOI 10.1007/s10596-009-9168-4

ORIGINAL PAPER

Simulation of multiphase flow in fractured reservoirsusing a fracture-only model with transfer functions

Evren Unsal · Stephan K. Matthäi · Martin J. Blunt

Received: 17 April 2008 / Accepted: 5 October 2009 / Published online: 13 November 2009© Springer Science + Business Media B.V. 2009

Abstract We present a fracture-only reservoir simula-tor for multiphase flow: the fracture geometry is mod-eled explicitly, while fluid movement between fractureand matrix is accommodated using empirical transferfunctions. This is a hybrid between discrete fracture dis-crete matrix modeling where both the fracture and ma-trix are gridded and dual-porosity or dual-permeabilitysimulation where both fracture and matrix continuaare upscaled. The advantage of this approach is thatthe complex fracture geometry that controls the mainflow paths is retained. The use of transfer functions,however, simplifies meshing and makes the simula-tion method considerably more efficient than discretefracture discrete matrix models. The transfer functionsaccommodate capillary- and gravity-mediated flow be-tween fracture and matrix and have been shown tobe accurate for simple fracture geometries, capturingboth the early- and late-time average behavior. Weverify our simulator by comparing its predictions withsimulation results where the fracture and matrix areexplicitly modeled. We then show the utility of the ap-proach by simulating multiphase flow in a geologically

E. Unsal · S. K. Matthäi · M. J. Blunt (B)Department of Earth Science and Engineering,Imperial College London, London, UKe-mail: m.blunt@imperial.ac.uk

Present Address:E. UnsalSchlumberger Cambridge Research,Cambridge, CB3 0EL, UK

Present Address:S. K. MatthäiDepartment of Mineral Resources and Petroleum Eng.,Montan University of Leoben, Leoben, Austria

realistic fracture network. Waterflooding runs revealthe fraction of the fracture–matrix interface area that isinfiltrated by water so that matrix imbibition can occur.The evolving fraction of the fracture–matrix interfacearea turns out to be an important characteristic ofany particular fracture system to be used as a scalingparameter for capillary driven fracture–matrix transfer.

Keywords Fractured reservoirs ·Countercurrent imbibition · Dual porosity ·Finite element/finite volume method

1 Introduction

In a fractured reservoir, fluids flow primarily throughthe interconnected fracture network, while the matrixstores the vast majority of the oil and water in thesystem [17]. The hydraulic properties of a reservoirare determined by the nature of the fracture network,while the matrix determines the ultimate recovery ofoil. One of the principal recovery mechanisms is im-bibition, controlled by capillary forces, where waterenters the matrix from the fractures, displacing oil [16].This process can either be co- or countercurrent. Incocurrent imbibition, both phases flow in the samedirection, while in countercurrent imbibition, they flowin opposite directions. In a fractured system where thefractures are the main flow path, it is possible to observeboth imbibition modes; the water flows cocurrently withoil through the fractures, while there is countercurrentimbibition between the fracture and the matrix, asshown schematically in Fig. 1. Countercurrent imbibi-tion is a much slower process than cocurrent imbibition

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Fig. 1 Countercurrent imbibition between the rock matrix andfracture. Due to capillary forces, as soon as injected water invadesa fracture, water (shown hatched) begins to imbibe the oil satu-rated matrix displacing oil into the fractures

[21], hence is possible for water to displace oil in thematrix cocurrently with flow parallel to the direction ofthe fracture.

The most physically realistic and computationallyaccurate way to model flow and transport in fracturedmedia is by using a discrete fracture/discrete matrix(DFDM) approach where both fracture and matrix aregridded explicitly [7, 8, 10, 13, 14, 19]. However, cap-turing displacement in a geologically realistic fracturenetwork requires a finely resolved grid and intricateindirect discretization approaches. Simulations on suchmeshes require very large amounts of time.

A common alternative is to consider the reservoiras containing two interacting media: the fractures thatcarry the flow connected to a relatively stagnant matrix.The geometry of fracture and matrix is not representedin any detail; instead, they are replaced by averagedproperties in a regularized (grid-block) representationof the field. This is the dual-porosity approach that isuniversally used for field-scale flow modeling of frac-tured reservoirs [1, 6, 8, 9, 24]. The fluid exchangebetween the two media is accommodated using a trans-fer function. This method works well if the fracturesare interconnected and the permeability contrast be-tween the matrix and the fractures is very high. Thismethod is computationally much less expensive. How-ever, it makes three main approximations. The first isthat there is no flow in the matrix. This can be over-come using a dual-permeability model [4, 5, 23] thatallows flow between matrix blocks. However, in manycases, it is appropriate to consider that the majorityof the flow is confined to the fractures. The secondapproximation is to replace the complex interactionbetween fracture and matrix mediated by capillary andgravitational forces by an empirical transfer function

that acts as a source/sink in the governing transportequations (see the next section). While the originalformulation for the transfer function was based on fluidexpansion and assumed a pseudosteady-state betweenfracture and matrix [9], recent work has led to thedevelopment of transfer functions that do accuratelycapture the dynamics of the displacement, albeit forsimple fracture geometries [12]. The third—and mostserious—approximation is to use averaged propertiesfor the fracture network. It is not possible to capturethe full complexity of the connectivity, topology, andpermeability of a network by a simple grid-block aver-age, even if tensor permeabilities are used.

We propose a hybrid model called a fracture-onlymodel that captures the advantages of both dual-porosity and DFDM models. It can also be consideredas a dual-porosity discrete fracture model. The matrixis treated using a dual-porosity approach, where allthe matrix properties are averaged and the transferbetween fracture and matrix is captured by a transferfunction. Since most of the reservoir volume is ma-trix, we save significant computational time by makingthis simplification. Fractures, on the other hand, aremodeled using a discrete fracture approach, where allthe fracture heterogeneities are considered. We will as-sume, however, that all the fractures are interconnectedand there is no viscous flow in the matrix.

In the following section, we introduce the CSMP++reservoir simulator which we used for our numericalsimulations. A detailed description of the formula-tion is provided. Next, we present the implementationof transfer functions into CSMP++, and finally, wepresent results in one, two, and three dimensions. Thecode is verified against analytical solutions (in onedimension) and explicit fracture–matrix simulations intwo dimensions.

2 Simulation method

2.1 CSMP++

The fracture-only model is implemented using theCSMP++ reservoir simulator [14]. It uses a combina-tion of finite element (FEM) and finite volume methods(FVM) using operator splitting [3, 18]. The geometricflexibility of the finite element method is retained, butthe finite volumes are better suited for the solutionof the hyperbolic part of the equation. Therefore, thisfinite element–finite volume method (FEFVM) givesbetter results. The FEFVM used in this article involvesan implicit pressure explicit saturation discretization oftime to simulate two-phase fracture flow.

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The new transfer calculations are performed aftereach implicit pressure explicit saturation solution cycleand represent a simple saturation update. We haveaugmented the standard CSMP fracture-only code toinclude matrix/fracture transfer. In the next sections,the conceptual model and governing equations used inthe numerical work are presented.

2.2 Conceptual model

We assume that all the flow occurs in an interconnectedfracture network. Only the fractures are gridded anda lower 2D representation is used; consequently, thenumber of grid blocks and computation time is re-duced by one to two orders of magnitude over DFDMmethods while the complex fracture geometry is stillretained. The fracture geometry is modeled explicitlyusing the FEFVM mesh, and each fracture element isassociated with a virtual matrix element via a trans-fer function (Fig. 2). The number and orientation ofthe elements vary depending on the dimensions ofthe system and the meshing method used. A virtualmatrix network is also formed; each matrix connectsto a fracture element via the transfer function. In thegoverning transport equations—see the next section—we need to assign an effective porosity to fracture andmatrix. Essentially, we need to know the ratio of thevolume of the fracture element to the pore volume ofthe matrix block. While it may appear that we haveto divide up the matrix and assign every region inspace to a specific fracture element, in this work, wechose a simpler approach. We compute the total porevolume of the matrix and the total pore volume of thefracture: this is the ratio φm/φf, where φf is the fractureporosity and φm is the matrix porosity. This fixed valueis then assigned to each fracture–matrix element. Ina heterogeneous medium, we could assign differentvolume ratios to each element dependent on the localgeometry.

In the fracture network, the pressure and the velocitydistributions are calculated by CSMP++. In the matrixnetwork, the viscous flow velocity is assumed to be zero,meaning there is no flow between the matrix elements.

2.3 Governing equations

For a fluid phase a in a fracture network (the fracturesystem uses no-flow boundary conditions on the facesof the fractures), the mass balance equation for incom-pressible flow is given by the following equation [3],

φf∂Saf

∂t= −∇ · ua + qa, a ∈ {w, n} (1)

where S is saturation, t is time, u is fluid velocity, andq are fluid sources or sink terms including those origi-nating due to flow from the rock matrix. The subscripta will refer to either the wetting (water) phase w orthe nonwetting (oil) phase n. Subscript f refers to thefracture network. The saturations S satisfy,

Sw + Sn = 1. (2)

The fluid velocity of phase a is given empirically byDarcy’s law

ua = −λak (∇ pa − ρag) , (3)

where λa is the mobility and is equal to the ratio of therelative permeability kra and to the viscosity μa of phasea (λa = kra/μa). Tensor k represents the permeabilityof the porous medium, pa is the fluid pressure, and ρa

is the fluid phase density, while g is the gravitationalacceleration. The fluid pressure for the wetting andnonwetting phases are related through the capillarypressure Pc,

Pn = Pw + Pc. (4)

Fig. 2 Fracture-only virtualmatrix discretization of themethod. The matrix isrepresented by a series ofvirtual blocks

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Capillary effects in the fractures are neglected such thatPw = Pn. Then, Darcy’s law can be expressed for bothphases as

ut = −λtk∇ P + kg (λwρw + λnρn) (5)

where ut is the total velocity and λt is the total mobility.If we assume that the fluids and rock are incompress-ible, the divergence of the flow field is equal to the totalfluid source or sink qt = qw + qn

∇ · ut = qt. (6)

Equation 6 can be used to derive an elliptic equation forpressure. Once the total velocity is known, Eq. 1 can besolved for saturation. An operator-splitting techniqueis used to solve for pressure with the FEM and forsaturation with the FVM.

For this work, a transfer term T is added to Eq. 1:

φf∂Saf

∂t= −∇ · ua + qa − Ta. (7)

Since flow is treated as incompressible Tw + Tn = 0,transfer has no direct effect on the solution for pressure.The formulation of transfer function will be given inthe next section. We also present the implementationof relevant transfer functions into the CSMP++ code.

3 Fracture/matrix transfer functions

In this section, we will briefly review the dual-porosityapproach, including alternative ways of formulating thematrix/fracture transfer function.

3.1 Dual-porosity models

In dual-porosity models, the fractured porous mediumis divided into two main regions: (1) fractures, wherethe vast majority of flows occur due to their highpermeability, and (2) matrix, which has much lowerpermeability but stores the vast majority of the fluid.It is assumed that there is no viscous flow in the matrix.Fluid movement between the two regions is modeledusing a transfer function. The transfer term is includedin the relevant mass conservation equation for the frac-tures, Eq. 7, while for the matrix:

φm∂Sam

∂t= Ta, (8)

where the subscript m represents the matrix.

3.2 Transfer function

In this numerical study, we use two types of transferfunction. The first one is a relatively simple lineartransfer function from Di Donato and Blunt [2]:

T = φmβ(S∗ − Swm

), (9)

where β is a rate constant, S* is the maximum watersaturation reached in the displacement, and Swm is thewater saturation in the matrix. We have dropped thesubscript w on T in Eq. 9 for convenience: Tn = −Tw ≡−T. This function represents the late-time behavior ofcapillary-controlled displacements accurately, but doesnot model the early-time behavior properly [20]. Animproved formulation is given below; however, we willuse the linear form since it allows us to compare withanalytical solutions in one dimension (see Section 3.4).

The generic behavior of recovery for constant frac-ture conditions is well known: initially, it scales at√

t before the advancing front reaches any boundariesand, approximately, as (1 − e−βt) at late time withsome geometry and fluid-dependent rate β [22, 25].Lu et al. [12] extended the linear transfer functionfurther to capture both early and late time behaviors.They included a Vermuelen-type correction factor Be

in the transfer function:

T = Beβφm(S∗ − Swm

), (10)

where

β = 3λwmλom

λt

[σwo J∗

L2

√Km

φm+ Km |�ρwo| g

φm L

]

, (11)

Be =∣∣S∗

wm − Swim∣∣ + |Swm − Swim|

2 max ((εSwm) , |Swm − Swim|) , (12)

where λwm, λom, and λt are water, oil, and total mo-bilities in the matrix, Km is the matrix permeability,J* is the dimensionless capillary pressure, σ wo is theoil/water interfacial tension, L is the effective distancethrough which oil moves to reach the fracture (it isrelated to the size of the matrix block), Swim is the initialwater saturation in the matrix, and ε is a dimensionlessconvergence factor. We will refer to Eq. 10 as thenonlinear transfer function.

3.3 Implementation of the transfer functionsinto CSMP++

The transfer term, T, in Eq. 7 is implemented usingthe appropriate transfer function, Eq. 9 or 10. At eachtime step, the oil and water saturations at each nodein the fractures are calculated using Eqs. 1–6, ignor-ing transfer. Then, the saturations in the matrix and

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Fig. 3 Flowchart summarizing how the fracture/matrix satura-tions are calculated

fracture finite volumes are updated using the transferfunction, ignoring advection. If there is no transfer,the matrix saturation stays unchanged. To determinewhether there should be a transfer or not, the followingcondition is tested:

Sof〈Sofi, (13)

where Sof is the current oil saturation in each finite vol-ume and Sofi is the initial fracture oil saturation. If thewater has reached a certain finite volume via advection,then this decreases its oil saturation from the initialvalue. If this condition is satisfied, then transfer occurs.The saturations in the fracture nodes are updated usingthe following equations for oil and water, respectively.

Snof = Sn−1

of + T�tφf

, (14)

Snwf = Sn−1

wf − T�tφf

, (15)

where the superscript n refers to the updated satura-tion at the current time level and n− 1 refers to thesaturation updated to accommodate advection only. InCSMP++, there is a Courant–Friedrichs–Lewy (CFL)criterion which limits the time steps for the explicitsimulations [15]. However, in the fracture-only model,the time steps are also limited by the transfer terms,which are sources/sinks. Especially at early times, whenthe transfer rates are high, the time step is chosen to besufficiently small that during the update, the saturations

never drop lower than 0 or become greater than 1. Oncethe transfer terms are smaller, longer time steps aretaken. Finally, the matrix saturations at the correspond-ing node are updated for oil and water, respectively.

Snom = Sn−1

om − T�tφm

, (16)

Snwm = Sn−1

wm + T�tφm

. (17)

This procedure is repeated for fracture and then matrixfinite volume at each time step. A flowchart whichsummarizes this process is given in Fig. 3.

3.4 Analytical solution

To verify our numerical scheme with linear transferfunction, we use the time-dependent analytical solutionfrom Di Donato and Blunt [2]. They constructed 1Danalytical solutions for the fracture water saturationthat is valid at early and late times for the front locationusing the linear transfer function. To develop a trans-port equation in 1D at early times, they defined the timeof flight τ (s) as the time taken for a particle to move adistance x in the fracture.

τ (t) = φf

utx (t) . (18)

Assuming Swf(τ ,t = 0) = 0, Swf(τ = 0,t) = 1 and linearrelative permeabilities such that uw = Swut. the leadingedge of water front moves at unit speed, meaning thatimbibition has not had sufficient time to reduce fracturesaturation to zero anywhere. Hence, water first entersthe fracture at time τ = t, and at that moment, watertransfer from fracture to matrix starts with an exponen-tially decaying function:

Swf (τ, t) = (1 − γ τ e−β(t−τ)

)H (t − τ) , (19)

where

γ = βφm

φf(1 − Somr − Swmi) . (20)

and Somr is the residual oil saturation in the matrix,H(t− τ ) is the Heaviside step function: H(t − τ) = 0 ift > τ and H(t − τ) = 1 if t ≤ τ .

At late times, when the amount of imbibition intothe matrix exceeds the pore volume of the fracture,the advancing front slows down, and its advance isgiven by:

τ (t) = 1

(γ + β)2

[β (γ + β) t + γ

(1 − e−( γ+β)t))] . (21)

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Table 1 Parameters andvalues for the 1D CSMP++solution

Kf is the fracturepermeability

Parameter CSMP++parameters

φf 0.01φm 1qi 10−5 m s−1

μw 0.001 Pa sμo 0.001 Pa sρw 1,000 kg m−3

ρo 1,000 kg m−3

krwf Swf

krof 1 − Swf

β 5.15 × 10−7

Kf 10−13 m2

Swmi 0.03Somr 0.03S* 0.7

4 Results

The transfer term is evaluated by comparing simulationresults to those obtained using equivalent 1D analyticalsolutions and 2D discrete fracture simulations in simplegeometries.

4.1 1D simulations

In this section, we compare the 1D numerical solutionwith the corresponding analytical solutions for the lin-ear transfer function. We used a 1D grid with 200 nodeswith an internode distance of 0.005 m (see Table 1 forparameters used in CSMP++; Table 2 for the analyt-ical model). The first finite volume contains the fluidsource and the last one the sink. Water is injected at aconstant rate (Fig. 4) while transfer rate constant β iskept constant. The fracture water saturation is plottedagainst distance at different times. A time step of 10 swas used.

Table 2 Parameters and values for the 1D analytical solutions

Parameter Figures 5 and 6 Figure 7

φf 0.01 0.01φm 0.2 0.2qi 10−5 m s−1 1.8 × 10−5 m s−1

μw 0.0003 Pa s 0.0003 Pa sμo 0.0003 Pa s 0.0003 Pa sρw 1,000 kg m−3 1,000 kg m−3

ρo 1,000 kg m−3 1,000 kg m−3

krwf Swf Swf

krof 1 − Swf 1 − Swf

β 0 day−1 (Fig. 5); 1.86 × 10−6 day−1

5.15 × 10−7 day−1

Kf 10−13 m2 10−13 m2

Swmi 0.03 0.03Somr 0.03 0.03S* 0.7 0.7

Fig. 4 One dimensional fracture geometry

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atur

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Fig. 5 The fracture water saturation–distance profile when thereis no transfer. The analytical solution is piston-like advance

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Fig. 6 Comparison of numerical and analytical results at differ-ent times for 1D flow when there is transfer

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0

0.2

0.4

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0.8

0 20000 40000 60000 80000 100000

fron

t loc

atio

n [m

]

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analyticalnumerical

Fig. 7 Comparison between analytical and numerical 1D solu-tions for early and late times

Our first test case examines flow with no transferwhere the analytical solution is simply a step functionmoving at unit dimensionless speed. Figure 5 comparesthe analytical and numerical solutions; the smearing ofthe front in the CSMP++ implementation is due to thenumerical dispersion associated with the second-orderaccurate high-resolution method using the MINMODspatial limiter [11].

Figure 6 shows a comparison of numerical and an-alytical solution, Eq. 19, at early time when there istransfer. Here, the water saturation in the fracturedecreases with time and distance. The CSMP++ resultsare in good agreement with the analytical solution.

Figure 7 shows example results at late time long afterthe front passed through the right-hand boundary ofthe model. In CSMP++, the location of the front iswhere the water saturation is first nonzero. There is atransition from fast flow of water in the fracture at earlytimes to a slower flow associated with increasing waterimbibition into the matrix at later times—the leadingfront is slowed down due to complete removal of thewater into the matrix by imbibition.

4.2 2D model

For a 2D single fracture model, solutions for two-phaseflow are compared to a commercial finite differencereservoir simulator, ECLIPSE (Eclipse 100, Black OilModel). In the CSMP++ model, the fracture is repre-sented as a plane with dimensions of 10 m length and

Fig. 9 Regular grid of 2D fracture for comparison with commer-cial finite difference simulator and associated virtual matrix. Thefracture plane is at the top; the matrix block is at the bottom. Eachfracture finite volume surrounding each finite element node isconnected a matrix finite volume via a transfer function T. Forsimplicity, the connections are shown only for the first row ofelements

1 m width (Fig. 8). Each fracture element is assigned athickness, Wf, of 1 cm. The grid has 200 finite elementsin the X direction and five elements in the Y direction,1,000 elements in total. Initially, the fracture is fullyoil saturated, and water is injected from the left-sideboundary.

Each fracture element is associated with a virtualmatrix element via the transfer function as illustratedin Fig. 9. Each virtual matrix element has a cross-section equal to the fracture element cross-section itis connected to and also has a thickness, Wm, which isused in the definition of Eq. 11. The matrix thicknessWm is 0.42 m. The relative permeability of the fractureis assumed to be a linear function of saturation, and thecapillary pressure in the fracture is zero.

For the finite difference model, a total of 9,800 gridblocks were used in the simulation, with smaller blocksnear the inlet to capture accurately the initial advanceof water into the matrix (Fig. 10). The fracture has 200grid blocks, and the matrix has 48 grid blocks along thefracture (Table 3). All other faces in contact with thefracture were closed off (no flow boundaries). Again,the relative permeability of the fractures was treatedas a linear function of saturation, and the capillarypressure in the fracture was set to zero; the matrixcapillary pressure is given in Table 4. The fracture wasdefined explicitly as high permeability region with a

Fig. 8 Regular grid used torepresent a 2D fracture(dimensions 10 × 1 m)

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Fig. 10 Finite difference gridsystem for 2D simulations ofcountercurrent imbibition. Atotal of 9,800 grid blocks wereused

porosity of 1. Quadratic relative permeabilities weredefined in the matrix (Table 4).

Water was injected at a constant rate via an injectionwell into the first grid block of the fracture, and oilwas produced at a constant rate from a producingwell which was situated in the last fracture grid block(Fig. 10). The fluid, matrix, and fracture medium prop-erties are listed in Table 4. The matrix permeability wasset to 1 mD (10−15 m2) to ensure a huge disparity inpermeability between fracture and matrix. A time stepof 10 s was used.

4.2.1 Analysis of results

2D CSMP++ results are now compared with finitedifference simulations. In the CSMP++ model, non-linear transfer functions were used (Eqs. 10–12). Thefollowing equations were used to define fracture andmatrix porosities in the dual-porosity model:

φf = Wf

Wf + Wm(22)

φm = ϕmmWm

Wf + Wm(23)

where W is the thickness and φmm is the matrix porosityin the finite difference model.

In the first test, we considered just imbibition withno flow in the fractures. This was a test to determineif the nonlinear transfer function accurately captured

Table 3 Dimensions of the 2D finite difference model

Dimension Unit Fracture Matrix

X m 10 10Y m 0.01 0.42Z m 1 1

the transfer of water into the matrix. For this case, thefracture was represented as one big water saturatedblock in the finite difference simulations—the fractureacted as a water reservoir. There was no additionalwater injection. Water imbibed in the matrix only dueto spontaneous imbibition. In CSMP++, similar condi-tions were created. The fracture was kept fully watersaturated. Figure 11 shows a comparison between thefinite difference code and CSMP++ using a best-fit β

value of 9 × 10−7 s−1. Both the early- and late-timeimbibition behavior is accurately captured.

Now that the β value was determined, the simu-lations were run with water flowing in the fracture.Figures 12 and 13 compare the matrix and fracturesaturation profiles in CSMP++ and the finite differencecode at different times. The agreement was close forboth matrix and fracture.

4.3 3D model

Having validated the fracture-only model for simpleone- and 2D examples, we apply it to a geologically rep-resentative 3D system. An example of a 3D fracture-only geometry is shown in Fig. 14. It consists of about 30interconnected fractures; the fractures are representedas 2D surfaces; however, all assigned a thickness of 1 cmin the simulations. The thickness is the same for eachfinite volume. Only the fractures are gridded; there isno matrix. The fractures have an initial oil saturationof 1 (Sof). There is a water source in one of the frac-tures (green circle) at the top corner and a producerat the bottom corner. It can be seen how as the timeprogresses the water imbibes through the fractures.The injection rate is 1.2 × 10−7 m3 s−1. The matrixand fracture porosities are 0.2 and 1, respectively. Thetransfer rate constant, β, is set to 9 × 10−7 s−1. The timestep was 10 s.

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Table 4 Data used in 2Dsimulations

Parameter Unit Value

Fracture porosity, ϕf Fraction ≈0.023Matrix porosity, ϕm Fraction ≈0.2Fracture permeability, Kf m2 10−10

Matrix permeability, Km m2 10−15

Initial matrix water saturation, Swmi Fraction 0Water density, ρw kg m−3 1,000Oil density, ρo kg m−3 1,000Water viscosity, μw Pa s 1 × 10−3

Oil viscosity, μo Pa s 1 × 10−3

Fracture water relative permeability Kfw = Sw

Fracture oil relative permeability Kfo = (1 − Sw)

Fracture capillary pressure Pc = 0Matrix water relative permeability Kmw = S2

wMatrix oil relative permeability Kmo = (1 − Sw)2

Matrix capillary pressure Pc =√

φmKm

σom(Sn

wm − S∗n)

n = −0.17

Figure 15 shows the average matrix saturation acrossthe whole system. As water contacts more of the frac-tures, there is a greater surface area for transfer, andthe imbibition rate increases rapidly. After around1,000 days, however, the fraction of the fracture net-work contacted by water increases more slowly, as wa-ter is simply being cycled through the more conductiveportion of the network. The recovery rate decreases, asimbibition reaches its late-time limit in those regions ofthe matrix contacted by water-saturated fractures. In afield-scale model, we would need to find an empiricalformulation to capture this average behavior; the wholesystem is approximately the size of a single grid block ina field-scale simulation: note that the overall recoveryis dissimilar to that observed locally for a single matrixblock (Fig. 11).

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0 2e+006 4e+006 6e+006 8e+006

mat

rix w

ater

sat

urat

ion

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Finite differenceCSMP++

Fig. 11 Transfer rate comparison between finite difference simu-lator and CSMP++ model using transfer functions. Matrix watersaturation when there is no fracture flow, β is determined as 9 ×10−7 s−1

More realistic fracture geometries are also tested.Figure 16 shows a fracture system (6 × 6 × 3 km) fromthe Clair Field which is located in the North Sea, 75 kmwest of Shetland Islands. The reservoir is made up offractured sandstones; current interpretations suggest astotal volume of oil in place in excess of 410 millionmetric tonnes of oil. Water is injected from the sourcewhich is located in the center of the geometry. Theinitial fracture water saturation (Swf) is 0. Water movesthrough the fractures toward the sink, choosing a paththrough the better connected, higher permeability por-tions of the network.

It takes less time to run this simulation compared towhen the matrix is gridded; in these examples, it wasnot possible to run a simulation including the matrix,but similar runs have taken around 3 months of CPUtime [14] compared to 1 week for the Clair model.

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rix w

ater

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Fig. 12 Comparison of matrix water saturation profiles fromCSMP++ and finite difference simulations

536 Comput Geosci (2010) 14:527–538

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Fig. 13 Comparison of fracture water saturation profiles inCSMP++ and finite difference simulation

5 Conclusions and future work

We have presented a new extended DFN model offracture transport combined with capillary fluid ex-change with a virtual matrix. It is a combination ofdiscrete fracture and a dual-porosity model retainingthe realistic flow field of DFDM. Its domain of ap-plicability is naturally fractured reservoirs with a strong

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Fig. 15 Average water matrix saturation across the wholefracture–matrix system

influence of the fractures on the flow and matrix withlow permeability. The rock matrix is treated usingan effective medium representation where all matrixproperties are averaged and acts as the fluid reservoir.Exchange of fluid between fracture and the matrix iscaptured by a transfer function with a parameterizationthat can be customized for each fracture cell-virtualmatrix block couple. This ability is used to set transfer

Fig. 14 3D 30-fracture modelat different time steps, a0.1 days, b 20 days, c 200 days,and d 5,475 days. Greenindicates the movement ofinjected water along the oilsaturated fractures. The sizeof the system is 50 × 50 ×1 m. The fractures have aheight of 1 m and aperture of1 mm; β is 9 × 10−7 s−1

Comput Geosci (2010) 14:527–538 537

Fig. 16 An example of a 3D geometry of a fracture system fromClair Field. The system dimensions are 6 km × 6 km × 3 km

initiation times to the time of the water front arrivalin each fracture–matrix cell pair. We used both linearand nonlinear transfer functions in this approach whichcircumvents the need to grid the matrix via use oftransfer functions and thus represents an attractive,more efficient simulation technique as compared withDFDMs.

We tested the model for simple one- and 2D geome-tries. 1D results for fracture water saturation versusdistance at different times were successfully comparedagainst analytical solutions from Di Donato and Blunt[2], validating the simulator. 2D numerical model re-sults were compared with those from a finite differencereservoir simulator. The transfer function accuratelycaptures the dynamics of the matrix saturation changes.

Proof of concept, 3D results indicate that bulk re-covery from the rock matrix of a naturally fracturedreservoir is controlled by both the local transfer rateand the fraction of the fracture surface area contactedby injected water. Future work will study this behaviorfor a suite of geologically representative models todevelop field-scale dual-porosity models with an up-scaled transfer function that accommodates the effectsof fracture contact area in the transfer function.

Acknowledgements The authors would like to thank the DTI,EPSRC, BP, Petro-Canada, Total, ExxonMobil, and Chevron forfinancial support, BP for generously providing the Clair Fielddata, Virginie Mucha for her contributions in building the ClairModel, and Hamid Maghami Nick for his help during the simula-tions. This work was carried out as part of the project “Improvedsimulation of flow in fractured reservoirs”.

References

1. Barenblatt, G.I., Zheltov, I.P., Kochina, I.N.: Basic conceptsin the theory of seepage of homogeneous liquids in fissuredrocks. J. Appl. Math. Mech., Eng. Trans. 24, 1286–1303 (1960)

2. Di Donato, G., Blunt, M.J.: Streamline-based dual-porositysimulation of reactive transport and flow in fractured reser-voirs. Water Resour. Res. 40, W04203 (2004)

3. Geiger, S., Roberts, S., Matthai, S.K., Zoppou, C., Burri, A.:Combining finite element and finite volume methods for ef-ficient multiphase flow simulations in highly heterogeneousand structurally complex geologic media. Geofluids 4, 284–299 (2004)

4. Gerke, H.H., Van Genuchten, M.T.: A dual-porosity modelfor simulating the preferential movement of water andsolutes in structured porous media. Water Resour. J. 29, 305–319 (1993)

5. Gerke, H.H., Van Genuchten, M.T.: Evaluation of a first-order water transfer term for variably saturated dual porositymodels. Water Resour. Res. 29, 1225–1238 (1993)

6. Gilman, J.R., Kazemi, H.: Improvement in simulation of nat-urally fractured reservoirs. SPE J. 23, 695–707 (1983)

7. Hotetot, H., Firoozabadi, A.: An efficient model for incom-pressible two-phase flow in fractured media. Adv. Water Re-sour. 31, 891–905 (2008)

8. Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficientdiscrete-fracture model applicable for general purpose reser-voir simulators. SPE J. 9, 227–236 (2004)

9. Kazemi, H., Merrill, L.S., Porterfield, K.L., Zeman, P.R.: Nu-merical simulation of water–oil flow in naturally fracturedreservoirs. SPE J. 16, 318–326 (1976)

10. Kim, J.-G., Deo, M.D.: Finite-element, discrete-fracturemodel for multiphase flow in porous media. Alche J. 46(6),1120–1130 (2000)

11. LeVeque, R. J.: Finite Volume Methods for HyperbolicProblems. Cambridge Texts in Applied Mathematics, p. 558.Cambridge University Press, Cambridge

12. Lu, H., Di Donato, G., Blunt, M.J.: General transfer functionsfor multiphase flow in fractured reservoirs. SPE J. 13(3), 289–297 (2008)

13. Matthäi, S.K., Mezentsev, A., and Belayneh, M.: Finite-element node-centered finite-volume experiments withfractured rock represented by unstructured hybrid ele-ment meshes. SPE Reserv. Evalu. Eng. 10(6), 740–756(2007)

14. Matthäi, S.K., Geiger, S., Roberts, S.G., Paluszny, A.,Belayneh, M., Burri, A., Mezentsez, A., Lu, H., Coumou, D.,Driesner, T., Heinrich, C.A.: Numerical Simulation of Multi-phase Fluid Flow in Structurally Complex Reservoirs. SpecialPublications, 292, pp. 405–429. Geological Society, London(2007)

15. Matthäi, S.K., Nick, H.M., Pain, C., Neuweiler, I.: Simulationof solute transport through fractured rock: a higher-order ac-curate finite-element finite-volume method permitting largetime steps. Trans. Porous Media (2009). doi:10.1007/s11242-009-9440-z

16. Morrow, M.R., Mason, G.: Recovery of oil by spontaneousimbibition. Curr. Opin. Colloid Interface Sci. 6, 321–337(2001)

17. Nelson, R.A.: Geologic Analysis of Naturally FracturedReservoirs, p. 332. Gulf Professional, Boston (1987/2001)

18. Paluszny, A., Matthai, S.K., Hohmeyer, M.: Hybrid finiteelement-finite volume discretization of complex geologicstructures and a new simulation workflow demonstrated onfractured rocks. Geofluids 7(2), 186–208 (2007)

538 Comput Geosci (2010) 14:527–538

19. Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: Amixed-dimensional finite volume method for multiphase flowin fractured porous media. Adv. Water Res. 29, 1030–1036(2006)

20. Tavassoli, Z., Zimmerman, R.W., Blunt, M.J.: Analytic analy-sis for oil recovery during counter-current imbibition instrongly water-wet systems. Trans. Porous Media 58, 173–189(2005)

21. Unsal, E., Mason, G., Morrow, N.R., Ruth, D.W.: Co-currentand counter-current imbibition in independent tubes of non-axisymmetric geometry. J. Colloid Interface Sci. 306, 105–117(2007)

22. Vermuelen, T.: Theory of irreversible and constant-patternsolid diffusion. Ind. Chem. Eng. 45, 1664–1670 (1953)

23. Vogel, T., Gerke, H.H., Zhang, R., Van Genuchten, M.T.:Modeling flow and transport in a two-dimensional dual-permeability system with spatially variable hydraulic proper-ties. J. Hydrol. 238, 78–89 (2000)

24. Warren, J.E., Root, P.J.: The behavior of naturally fracturedreservoirs. SPE J. 3, 245–255 (1963)

25. Zimmerman, R.W., Chen, G., Hadgu, T., Bodvarsson, G.S.:A numerical dual-porosity model with semianalytical treat-ment of fracture/matrix flow. Water Resour. Res. 29(7),2127–2137 (1993)