Advances in Water Resources

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1 Advances in Water Resources 31 (28) Contents lists available at ScienceDirect Advances in Water Resources journal homepage:.elsevier.com/locate/advatres An efficient numerical model for incompressible to-phase flo in fractured media Hussein Hoteit a,1, Abbas Firoozabadi a,b, * a Reservoir Engineering Research Institute, Palo Alto, CA, USA b Yale University, Ne Haven, CT, USA article info abstract Article history: Received 1 October 27 Received in revised form 1 February 28 Accepted 19 February 28 Available online 7 March 28 Keyords: To-phase flo Water injection Fractured media Heterogeneous media Capillary pressure Discrete fractured model Mixed finite element method Discontinuous Galerkin IMPES methods Various numerical methods have been used in the literature to simulate single and multiphase flo in fractured media. A promising approach is the use of the discrete-fracture model here the fracture entities in the permeable media are described explicitly in the computational grid. In this ork, e present a critical revie of the main conventional methods for multiphase flo in fractured media including the finite difference (FD), finite volume (FV), and finite element (FE) methods, that are coupled ith the discrete-fracture model. All the conventional methods have inherent limitations in accuracy and applications. The FD method, for example, is restricted to horizontal and vertical fractures. The accuracy of the vertex-centered FV method depends on the size of the matrix gridcells next to the fractures; for an acceptable accuracy the matrix gridcells next to the fractures should be small. The FE method cannot describe properly the saturation discontinuity at the matrix fracture interface. In this ork, e introduce a ne approach that is free from the limitations of the conventional methods. Our proposed approach is applicable in 2D and 3D unstructured griddings ith lo mesh orientation effect; it captures the saturation discontinuity from the contrast in capillary pressure beteen the rock matrix and fractures. The matrix fracture and fracture fracture fluxes are calculated based on poerful features of the mixed finite element (MFE) method hich provides, in addition to the gridcell pressures, the pressures at the gridcell interfaces and can readily model the pressure discontinuities at impermeable faults in a simple ay. To reduce the numerical dispersion, e use the discontinuous Galerkin (DG) method to approximate the saturation equation. We take advantage of a hybrid time scheme to alleviate the restrictions on the size of the time step in the fracture netork. Several numerical examples in 2D and 3D demonstrate the robustness of the proposed model. Results sho the significance of capillary pressure and orders of magnitude increase in computational speed compared to previous orks. Ó 28 Elsevier Ltd. All rights reserved. 1. Introduction * Corresponding author. Address: Reservoir Engineering Research Institute, Palo Alto, CA, USA. address: af@rerinst.org (A. Firoozabadi). 1 No at ConocoPhillips Co. Modeling of multiphase fluid flo in fractured media is of interest in many of the environmental and energy problems. Examples include tracing a NAPL migration, radioactive aste management in the subsurface, and the enhanced oil recovery in naturally fractured reservoirs. Geological characterization and multiphase flo simulation are challenging tasks in complex fractured media. Various conceptual models have been used to describe multiphase flo in fractured media. The main models include the dual-porosity and its variations, single-porosity, and the discrete-fractured approaches. The dual-porosity model has traditionally been used to simulate the flo in fractured hydrocarbon reservoirs [1 6]. In this model, the matrix fracture mass transfer is described by empirical functions that incorporate ad hoc shape factors hich are based on theoretical derivations in single phase. In to-phase ith capillary and gravity, there is no theory for the calculation of shape factors despite the fact that the accuracy of flo calculations depend on their values. The problem may be even more fundamental. The driving force in the expression of the transfer function in tophase flo may not have a sound basis. We should point out that advanced variations of the dual-porosity model such as the multiple interaction continua method [7] have better accuracy and features than the conventional model. A more accurate and physics-based approach is the singleporosity model that describes fractures explicitly in the medium. This model requires the fracture-netork characterization and the fracture fluid interaction functions. Modeling fractures ith the single-porosity model can be classified as a complex case of heterogeneous porous media, here the fracture gridcells are treated similarly to the matrix gridcells. The implementation of this model is, hoever, almost impossible in field-scale problems because of excessive fine gridding and severe restrictions on the time steps due to the contrast in geometrical scales beteen the /$ - see front matter Ó 28 Elsevier Ltd. All rights reserved. doi:1.116/j.advatres

2 892 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) a c Fig. 1. Different spatial discretizations of the fractures and the adjacent matrix gridcells in FD, FE, CVFE, and MFE. (a) FD; (b) FE; (c) CVFE; (d) MFE. b d fracture and the adjacent rock-matrix gridcells [8]. The matrix inversion due to ill-conditioning becomes a challenge. The discrete-fracture model [9,1] describes the fractures explicitly in the medium similarly to the single-porosity model. Hoever, unlike the single-porosity model, the fractures gridcells are geometrically simplified by using ðn 1Þ-dimensional gridcells in an n-dimensional domain. In other ords, in 3D space, the fractures are represented by the matrix gridcell interfaces, hich are 2D. This simplification removes the length-scale contrast caused by the explicit representation of the fracture aperture as in the single-porosity model. As a result, computational efficiency is improved considerably. Hoever, the discrete-fracture model can only handle a limited number of fractures (of the order of thousands) for reasons related to computational resources. One approach to overcome this limitation is to model explicitly only the major fractures and use an upscaling technique to mimic the effect of the minor fractures [11]. Various numerical methods based on the discrete-fracture model have been used to simulate single and multiphase flo in fractured media. Methods that are developed for single-phase flo may not be trivially extended to to-phase flo. In to-phase flo, high permeable fractures may not be the main flo conducts due to the contrast in capillary pressure beteen the fracture and the rock matrix. Capillary pressure heterogeneity may also cause fluid trapping (discontinuous saturation) and, in some cases, pressure discontinuously [12 1]. The fluid streamlines may, therefore, change significantly ith time. The discrete-fracture model has been used in the finite difference (FD), finite volume (FV), and the Galerkin finite element (FE). These methods may have some inherent limitations, as discussed belo. Slough et al. [16,17] used the FD method to simulate the migration of a NAPL in 2D and 3D fractured media in structured grids. They used distinct degrees of freedom for the saturation and pressure in the fracture and matrix control-volumes. In Cartesian gridding, the intersection of horizontal and vertical fractures corresponds to a control-volume that has the dimensions of the fracture thicknesses (see Fig. 1a). Such a control-volume may reduce significantly the size of the time step in the numerical model. To eliminate the degrees of freedom at the fracture junctions, Slough et al. used a technique based on steady-state flo equations. They suggested to use to-point upstream eighting to approximate the mobilities. The extension of their technique to a fracture gridcell that has several feeding fractures as not discussed. When the control-volume at the fractures junction in Fig. 1a is eliminated, one fracture gridcell may have up to three upstream fractures. Lee et al. [11] used a hierarchical approach based on fracture lengths such that short fractures are treated implicitly by increasing the effective matrix permeability hile long fractures are described explicitly. Similarly to the model by Slough et al., the fractures ere aligned ith the matrix gridcells; that is, either vertical or horizontal. They used a technique for ell productivity indices for single-phase flo (capillary pressure as neglected) to calculate the transmissibilities beteen the fracture and the matrix gridcells. The Galerkin FE method has been used to model single-phase [9,1,18 2] and to-phase flo [21 23] in fractured media. In 2D space, the fractures are represented by 1D entities and the degrees of freedom are located at the mesh nodes (see Fig. 1b). The fractures and the adjacent matrix gridcells share the same degrees of freedom, therefore, there is an implicit assumption of the continuity of pressure and saturation at the matrix fracture interface. As a result, the FE suffers from a eakness in to-phase flo because it cannot account properly for saturation discontinuity from capillary pressure contrast at the matrix fracture interface. To FV methods ith different spatial discretizations have been used for fractured media: (1) the cell-centered FV, and (2) the vertex-centered FV, hich is also knon as the control-volume finite element (CVFE) method. In the cell-entered FV, similarly to the FD (Fig. 1a), the fracture gridcells are located at the boundaries of the matrix gridcells. This approach can describe heterogeneities ithin the mesh, hoever, there is numerical complexity in defining the matrix fracture and the fracture fracture fluxes. Karimi- Fard et al. [24] introduced a simplified cell-centered FV method for to-phase flo in fractured media. The authors, hoever, mentioned that their method could create numerical errors in nonorthogonal grids because the fluxes are calculated by the to-point flux approximation method. They used an analogy beteen the conductance through resistors and flo in porous media to eliminate the control-volumes at fracture intersections, here capillary pressure and gravity effects are neglected. Grant et al. [2] also used the cell-centered FV method to model to-phase flo in 2D fractured media. To calculate the fracture fracture flux, they introduced special nodes at the fracture intersections here they calculated the saturation at an intermediate time step. Many authors have used the CVFE method and some have combined it ith the Galerkin FE method to solve the to-phase flo equations in fractured media [26 31]. In the CVFE method, the fracture entities are embedded ithin the matrix control-volume (see Fig. 1c). In this approach, the calculation of the matrix fracture flux is avoided and there is no difficulty in computing the fracture fracture flux. Hoever, consistent modeling of the saturation discontinuity from the capillary pressure contrast at the fracture matrix interface has not been fully examined. Monteagudo and Firoozabadi [32] used the capillary pressure continuity equation at the matrix fracture interface and the capillary pressure saturation functions to calculate the saturations in the fracture and the matrix blocks ithin the same control-volume. In other ords, the saturation ithin a control-volume is distributed beteen the matrix and fracture blocks so that the capillary-pressure continuity is preserved at the interface. This approach may become inefficient hen the fracture capillary pressure is zero. Reichenberger et al. [33] used discontinuous approximation functions for the saturation in the fractures to account for the saturation discontinuity at the matrix fracture interface. Similarly to the method of Monteagudo and Firoozabadi [32], this approach suffers from zero capillary pressure in the fracture.

3 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) There is another concern related to the matrix gridding that may affect the accuracy of the CVFE method in hich one invokes a cross-flo equilibrium beteen the fracture and the adjacent matrix blocks. Here, the pressure in the fracture and that in the matrix are assumed equal ithin the same control-volume. In the CVFE approaches that ignore the capillary pressure heterogeneity, the same saturation is assumed ithin the control-volume. These assumptions are justifiable if the matrix gridcells next to the fractures are small enough. For an acceptable accuracy for an enhanced oil recovery problem (gas injection), Hoteit and Firoozabadi [34] found that the matrix-gridcells size next to the fractures should be of the order of ten centimeters in a fractured reservoir of length-scale in kilometers. Reichenberger et al. [33] also used very fine griddings in their numerical experiments. The mixed finite element (MFE) method [3,36] has received much attention for its poerful features in approximating the velocity field in highly heterogeneous and anisotropic media and has been idely used to model single-phase flo in fractured media. In Refs. [37,38], the method is used to simulate steady-state single-phase flo in fracture-netorks and the mass transfer beteen the fractures and the rock matrix as neglected. Alboin et al. [39] employed the MFE to solve the transport equation in fractured media. Martin et al. [4] extended the ork of Alboin et al. to model the cases hen the fracture is less permeable than the matrix. In this ork, e extend the MFE method to to-phase flo in fractured media ith gravity and capillarity and account for matrix fracture interactions. In our algorithm, the use of the hybridized MFE method [41,36] overcomes the limitations of the numerical models discussed earlier. In our ork, the degrees of freedom for the saturation and pressure in a fracture gridcell are not the same as those in the neighboring matrix gridcells (see Fig. 1d). As a result, one can describe readily the discontinuity in saturation from capillary pressure heterogeneity. The fractional flo (global pressure) formulation [42] in the MFE does not allo proper modeling of saturation discontinuity. We use instead a consistent formulation from Ref. [43] in hich phase pressure serves as the primary unknon. Most of the conventional approaches in the literature use topoint upstream eighting to approximate the mobilities. For some applications, this technique is knon to provide poor accuracy [44,4]. In this ork, e use the discontinuous Galerkin (DG) method [42,46,47] stabilized by a multidimensional slope limiter introduced by Chavent and Jaffré [42] to reduce the numerical dispersion in the saturation. The to-phase immiscible flo equations are solved by an implicit-pressure explicit-saturation (IMPES) approach. The IMPES method may perform more efficiently than the fully-implicit method [48]. Its stability is, hoever, restricted to a CFL condition, hich is inversely proportional to the gridcell pore-volumes. Therefore, the fracture gridcells may impose a severe CFL condition. To allo for high computational efficiency, e implement an implicit scheme in the fracture netork and an explicit scheme in the matrix. With above literature revie on to-phase flo in fractured media, e are no set to first present the essence of the proposed model and to discuss the governing equations for to-phase immiscible flo in porous media. We then rerite the flo equations in an equivalent form hich is appropriate for the MFE formulation. The MFE formulations in the matrix and fracture domains are then applied to approximate the volumetric flux and the etting-phase potential, here the capillary pressure is taken into account. This is folloed by a description of the DG formulation and the proposed time scheme. Several examples in 2D and 3D are presented to sho the applicability of our method. In Ref. [43], e presented the formulation and the numerical method for to-phase flo in unfractured media. We focused on modeling capillary pressure continuity/discontinuity and verified the accuracy of our method ith analytical solutions in heterogeneous media. Some details presented in that ork are not repeated in this paper. 2. Proposed method In the conventional methods, the flux through gridcell interfaces is calculated in a post-processing step from the pressure field. In the MFE method, the flux and the pressure unknons are approximated simultaneously. The original MFE method ith the loest order Raviart Thomas space [3,49] leads to a saddle point problem, here the linear system to solve is indefinite. The hybridized MFE method [36,41] overcomes this problem by adding ne degrees of freedom (Lagrange multipliers) that represent the pressure averages (traces of the pressure) on the gridcell interfaces. The original MFE and the hybridized MFE are algebraically equivalent [41,]. The difference is only in the selection of the primary variables. In this ork, e use the hybridized MFE, and for the sake of brevity e denote it also by MFE. Its primary variables are the traces of pressure; the secondary variables are the gridcell average pressures and the averages fluxes across the gridcell interfaces (see Fig. 2). In our ork, e assume capillary pressure continuity (that is, pressure continuity) at the matrix fracture interface, except hen the pressure of the non-etting phase entering a different medium saturated ith the etting-phase is less than the threshold (entry) pressure [13 1]. The non-etting phase becomes immobile and, therefore, there may be discontinuity in the non-etting phase pressure. Modeling this phenomenon in heterogeneous media by the MFE method is discussed in Ref. [43]. For the sake of clarity, the discontinuity in capillary pressure is not discussed further in this ork; e assume that the etting phase is alays displacing the non-etting phase (imbibition) and so the pressures are alays continuous. In a previous ork [1,2], e used the MFE method to simulate single and to-phase compositional flo in 2D fractured media but ith neglected capillary pressure. We also invoked crossflo equilibrium beteen the fracture and the adjacent matrix gridcells in a similar fashion as the CVFE method. In other ords, the fracture pressure and the adjacent matrix gridcells pressures are assumed equal (see Fig. 2). As discussed in Section 1, this simplification is justifiable if the matrix gridcells next to the fracture are small. A central idea in our ork is a ne formulation that does not invoke the cross-flo equilibrium across the fractures and the neighboring matrix blocks. We assume a constant potential along the idth of the fracture gridcells (see Fig. 3) and therefore cross-flo equilibrium is assumed across the fractures only. As sketched in Fig. 3, the degrees of freedom for the potential and sat- Face-average potential Cell-average potential Simplification Fig. 2. Modeling of flo in fractured media using the cross-flo equilibrium in the fractures and the adjacent matrix gridcells.

4 894 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) Face-average potential Cell-average potential uration in the matrix and fracture gridcells are distinct. This ne approach alleviates the size constraint on the matrix gridcells next to the fractures and allos accurate calculation of the saturations in the fracture and in the adjacent matrix gridcells ithout a need for small neighboring matrix blocks. Our proposal has similarities to the conventional single-porosity model, hoever there is a significant advantage as it allos accurate and efficient calculation of the matrix fracture and fracture fracture flux, as ill be discussed later. 3. Governing equations The governing equations in porous media for the flo of to incompressible and immiscible phases, a etting phase (referred to by the subscript ), and a non-etting phase (referred to by the subscript n), are given by the saturation equation and the generalized Darcy la of each phase. Saturation constraint and the capillary pressure relation complete the formulation: / os a ot þrðv aþ¼f a ; a ¼ n; ; ð1þ v a ¼ k ra Kðrp l a þ q a grzþ; a a ¼ n; ; ð2þ S n þ S ¼ 1; ð3þ p c ðs Þ¼p n p : ð4þ In the above equations, K is the absolute permeability tensor, S a, p a, q a, k ra, F a, and l a are the saturation, pressure, density, relative permeability, sink/source term, and viscosity of phase a, respectively, g is the gravity acceleration, z is the depth, and p c is the capillary pressure. As discussed in Ref. [43], the MFE method as initially proposed for single-phase flo in porous media. A number of authors have used the fractional flo formulation (also knon as the global pressure formulation) to extend the MFE for to-phase flo. The MFE method ith the fractional flo formulation, hoever, becomes inconsistent for the simulation of flo in heterogeneous media ith different capillary functions. A consistent formulation, that overcomes the fractional flo deficiency, is provided in Ref. [43]. A brief revie of the formulation follos. We define the flo potential, U a, of phase a as follos: U a ¼ p a þ q a gz; a ¼ n; : ðþ The capillary potential, U c, becomes: U c ¼ U n U ¼ p c þðq n q Þgz: Then, Eqs. (1) and (2) are reritten in the folloing equivalent forms: rðv a þ v c Þ¼F t ; ð7þ / os ot þrðf v a Þ¼F ; Simplification Fig. 3. Modeling of flo in fractured media using our proposal ithout the crossflo equilibrium. ð6þ ð8þ here the velocity variables v a and v c, hose sum is the total velocity, are given by: v a ¼ k t KrU ; v c ¼ k n KrU c : ð9þ ð1þ In the above equations, k a ¼ k ra =l a is the mobility of phase a, k t ¼ k n þ k is the total mobility, f ¼ k =k t is the etting-phase fractional function, and F t ¼ F n þ F is the total sink/source term. The system of Eqs. (7) and (8) are subject to Neumann (flo rate control) and Dirichlet (pressure control) boundary conditions. The system of Eqs. (7) (1) and the initial and boundary conditions are our orking equations ith the MFE and DG methods. 4. Approximation of the volumetric flux The pressure and the saturation equations are decoupled and solved sequentially in an IMPES-like approach. Hoever, e suggest the use of a different temporal scheme for the saturation equation in the fractures, as ill be discussed later. The MFE method is used to approximate implicitly the pressure and the velocity in the matrix and the fractures. Since the matrix and fracture gridcells have different geometrical dimensions (matrix gridcells are n- dimensional and fracture gridcells are ðn 1Þ-dimensional), the governing equations are approximated separately. In the matrix domain (referred to by superscript m), the equations describing the volumetric balance and velocity are given by: v m a ¼ km t K m ru m ; ð11þ rðv m a þ vm c Þ¼Fm t : ð12þ In the fractures, the governing equations are integrated along the fracture idth. The integrals are readily computed since the variations in potential and saturation are averaged across the fracture idth. The fracture idth is taken into consideration in the fracture fracture flux and the accumulation term in the fracture gridcells. The equations in the fractures (referred to by the superscript f) inðn 1Þ-dimensional are expressed by: v f a ¼ kf t Kf ru f ; rðv f a þ vf c Þ¼Q f þ F f t : ð13þ ð14þ The transfer function Q f in Eq. (14) accounts for the volumetric transfer across the matrix fracture boundaries. This transfer function ill be evaluated in the folloing section Discretization in the matrix The governing equations of the volumetric flo in the matrix expressed in Eqs. (11) and (12) are discretized separately by the hybridized MFE method ith the loest-order Raviart Thomas space. The MFE discretization of the velocity expression in Eq. (11) in matrix gridcell K yields the explicit expression for the flux q m a;k;e across a gridcell-face E in terms of the of the cell-average potential U m ;K and all the face-average potentials Km ;K;E in all the faces of K, that is: q m a;k;e ¼ am K;E Um ;K X b m K;E;E Km ;K;E ; E 2 ok; ð1þ E 2oK here the terms a m K;E and bm K;E are constants, independent of the potential and flux variables. Details are provided in Ref. [43]. The flux in Eq. (1) is ritten locally for all faces ithin each matrix gridcell. To link the gridcells in the mesh together, the folloing continuity conditions for the flux and the potential are imposed at each interface E of to neighboring gridcells K and K (that is, E ¼ K \ K ):

5 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) If E is neither a fracture nor a barrier, the continuity of flux and potential is imposed. ( q m a;k;e þ qm ¼ ; a;k ;E K m ;K;E ¼ Km ;K ;E ¼ Km ;E : ð16þ If E is a fracture, the total flux across both sides of the matrix fracture interface defines the transfer function Q f a;e at E, hich acts as a sink/source term. There is, hoever, continuity of the potential across the matrix fracture interface E (see Fig. 3). ( q m a;k;e þ qm ¼ Q f a;k ;E a;e ; K m ;K;E ¼ Km ;K;E ¼ Uf ;E : ð17þ If E is a barrier, the flux across E is zero and face potentials K m ;K;E and K m ;K ;E may not be equal. ( q m a;k;e ¼ qm ¼ ; a;k ;E K m ;K;E 6¼ Km ;K ;E : ð18þ a b matrix fracture matrix matrix fracture matrix We like to make the comment that Eq. (18) takes care of a barrier in simple ay. In addition to the above conditions, the flux and potential at the domain boundaries are described by the Neumann or Dirichlet boundary conditions (see Ref. [43]). For the sake of simplicity, e assume that the domain boundaries are impermeable. From Eqs. (1) (18), one can eliminate the flux variables and construct a system consisting of the unknons; the fracture cell-potentials, U f, matrix cell-potentials, Um, and the matrix face-potentials, K m. The system of equations corresponding to the matrix gridcells connected to fractures and those not connected to fractures are ritten separately as follos: R T;m;m R T;m;f! U m Mm;m M m;f M f;m M f;f! K m U f ¼ Q f : ð19þ c matrix fracture matrix Fig. 4. Modeling of the flo in fractures in our proposed approach. The system given in the above equations is split into to parts to explicitly sho the connectivity at the matrix fracture interfaces. Note that M m;f ¼ M f;m in Eq. (19). The matrix fracture volumetric transfer function Q f ill be eliminated later hen e couple the flo beteen the matrix and the fractures. The definitions of the matrices in Eq. (19) are provided in Appendix A. The next step is the integration of Eq. (12) over K and to obtain flux unknons in terms of the potentials given in Eq. (1) [43], that is, a m K Um ;K X E2oK a m K;E Km ;K;E ¼ Fm K ; ð2þ here a m K ¼ P E2oK am K;E, and Fm K ¼ P E2oK qm c;k;e þ RRR KðF m t Þ is a function of capillary pressure that ill be discussed in section Calculation of the capillary flux. In matrix form, Eq. (2) and the potential continuity equations given in Eqs. (16) (18), yield: D m U m ðrm;m R m;f K m Þ ¼ F m : ð21þ U f The diagonal matrix D m in Eq. (21) is defined in Appendix A, and R m;m and R m;f appear in Eq. (19) Discretization in the fracture In n D space ðn ¼ 2; 3Þ, the fracture netork is ðn 1Þ D. The MFE formulation in the fracture netork is basically similar to the conventional formulation in unfractured ðn 1Þ D domain. There are, hoever, to differences: (1) the transfer function hich acts as a sink/source in the fracture (see Fig. 4), (2) the fracture fracture connectivity that may have multiple fracture-gridcells intersecting at the same interface as in Fig.. Similarly to the discretization in Fig.. Mesh ith five intersecting fractures at I; the fractures k 1, k 2, k 3 are in the upstream, and k 4, k are in the donstream. the matrix previously discussed, one discretizes the volumetric flo given in Eqs. (13) and (14) in the fractures. With the MFE discretization of Eq. (13) in a fracture gridcell k (k 2ðn 1Þ Dspace), the flux q f a;k;e across an interface e (e 2ðn 2Þ D space) of the fracture gridcell k becomes: q f a;k;e ¼ af k;e Uf ;k X b f k;e;e Kf ;k;e; e 2 ok: ð22þ e 2ok The constants a f and k;e bf k;e are similar to those defined in Eq. (1). Let n e be the number of intersecting fractures at the interface e. The fracture fracture interface e has a negligible volume, therefore, no accumulation is assumed and the potential continuity is imposed at e, that is,

6 896 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) P >< ne q f a;k i ;e ¼ ; By imposing the continuity of capillary pressure and flux at the matrix matrix and fracture fracture interfaces, the flux unknons i¼1 ð23þ >: K f ;k i ;e ¼ Kf ;e ; i ¼ 1;...; n e: q m c;k;e and qf c;k;e are eliminated, and the folloing system is obtained:! Using Eqs. (22) and (23), the flux unknon is eliminated and a bm m K m c system ith the unknons consisting of the fracture-gridcells M b f K f ¼ b!! V m br T;m U m c : ð3þ c bv f br T;f U f c potentials, U f, and the fracture-interfaces potentials, Kf, is obtained: The matrices in the above equation are defined in Appendix A. er T;f U f M ~ f K f ¼ : ð24þ The capillary potentials U m c and U f c in the right-hand side of Eq. (3) are calculated by using the capillary pressure functions and The volumetric balance given in Eq. (14) is integrated over each fracture gridcell. In matrix form, a system similar to Eq. (21) is obtained: the etting-phase saturations from the previous time step. The capillary pressure at the fracture gridcells are also calculated from the corresponding saturations. The system in Eq. (3) is diagonal per block and the matrices M b m, M b f are symmetric and positive definite, ed f U f R ef K f ¼ Ff þ Q f : ð2þ therefore, solvers such as preconditioned-conjugate-gradient In the above equation, Q f is the matrix fracture transfer previously (PCG) can be used efficiently. defined. The matrices D, e R ef, and F f in Eq. (2), and R et;f and M e f in Eq. (24) are defined in Appendix A.. Approximation of the etting-phase saturation 4.3. Combining the discretization in the matrix and fractures The MFE formulations of the volumetric flo in the matrix and fractures are coupled through the matrix fracture transfer function and the continuity of the potential given in Eq. (17). By substracting Eq. (2) from the second equation in Eq. (19), the transfer function Q f cancels out. The resulting equation from the first equation in Eq. (19) and Eqs. (21) and (24) becomes: 1 D m R m;m R m;f R T;m;m M m;m M m;f R T;m;f M m;f ðm f;f þ D e f Þ R e f CB A@ e R T;f e M f U m K m U f K f 1 C A ¼ F m F f 1 C A : ð26þ The matrix D m is diagonal and invertible, therefore, a simple Gaussian eliminate can be performed to eliminate the unknon U m. The final linear system reads as: A m;m A m;f K m V m A m;f A f;f R ef CB A U f C A ¼ V f A: ð27þ R et;f M e f K f The sought unknons are, therefore, the traces of the ettingphase potential in the matrix, K m, the fracture gridcell potentials, U f, and the traces of the potential in the fracture netork, Kf. The system in Eq. (27) has a size equal to the number of mesh interfaces plus the number of fracture fracture interfaces; it is sparse, symmetric and positive definite. There is no particular difficulty in constructing the system in Eq. (27). A simple example for a 2D fractured domain is given in Appendix B to sho the structure of the linear system. After solving Eq. (27), the matrix potentials and the flux in the matrix and fracture netork are calculated locally from the first equation in Eq. (26), and Eqs. (1) and (22) Calculation of the capillary flux As e did for the volumetric flo equation, the saturation equation describing the mass conservation is ritten separately in the matrix and fractures. In the matrix, the etting-phase saturation is defined by S m and the governing equation is given by: / m os m ot þrðf m vm a Þ¼Fm : ð31þ The discontinuous Galerkin (DG) method is used to approximate Eq. (31) ith an explicit, second-order Runge Kutta time scheme. A multidimensional slope limiter [42] is applied in a post-processing step to remove potential oscillations in the saturation. A detailed description is provided in Ref. [43]. The saturation equation in the fracture netork is ritten as: / f os f ot þrðf f vf a Þ¼ Q b f þ Ff : ð32þ The sink/source term, Q b f, in the above equation is equal to the volumetric transfer function, Q f in Eq. (14), multiplied by the etting-phase fractional flo function f. From the assumption of incompressibility of the fluids, Q b f represents the mass transfer function through the matrix fracture interfaces. Because the flo in the fracture netork can be much faster than that in the matrix, a first-order implicit time scheme is used to approximate the time operator in Eq. (32). This approach removes the CFL restriction on the time step in the fracture netork and allos different time steps in the matrix and the fractures. Unlike the approximations in the matrix, a first-order spatial scheme is used for the saturation equation in the fractures. As a result, one may expect more numerical dispersion in fracture saturation, hoever, this may not have a significant influence as a second-order method is used in the matrix. We do not use an implicit DG method in the fracture for its expensive computational overhead. The first-order temporal and spatial discretizations of Eq. (32) in a fracture gridcell k at a time step n reads as: The degenerate flux from the capillary pressure and gravity, q c, in the right-hand side of Eqs. (1) and (22) is ritten in terms of the cell capillary potential U c and the face capillary potential K c in the matrix and fractures. In a matrix gridcell, K, the flux across an interface E is given by Hoteit and Firoozabadi [43]: q m c;k;e ¼ ^am K;E Um c;k X ^b m K;E;E Km c;k;e ; E 2 ok: ð28þ E 2oK Similarly to Eq. (28), in a fracture gridcell, k, the flux across an interface e is given by: q f c;k;e ¼ ^af k;e Uf c;k X ^b f k;e;e Kf c;k;e; e 2 ok: ð29þ e 2ok / f k S f;n Sf;n 1 Dt þ X e2ok f f;n ;e qf;n ¼ b a;k;e Q fþ ;n 1 þ Q b f ;n fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} þff;n ;k : ð33þ The second term on the left-hand side of Eq. (33) represents the net fracture fracture flux. The volumetric flux q f;n is calculated in a;k;e a preprocessing step from the MFE approximations, previously discussed. The matrix fracture mass transfer function on the righthand side of Eq. (33) is split into to parts that are approximated differently according to the flo direction beteen the fracture and the to adjacent matrix gridcells. The notation Q b fþ ;n 1 refers to a source term and Q b f ;n refers to a sink term for the fracture gridcell. We distinguish three possible cases as follos:

7 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) Y(m). Y(m) X(m) X(m).1 Fig. 6. Water saturation contours at PVI ¼ :: ith and ithout the cross-flo equilibrium, j ¼ :4; Example 1. (a) With cross-flo equilibrium; (b) ithout cross-flo equilibrium. If the to adjacent matrix gridcells are in the upstream of the fracture, as appears in Fig. 4a, then Q b f ;n ¼ and the total source term for the fracture, Q b fþ ;n 1, is approximated explicitly in time. If the to adjacent matrix gridcells are in the donstream of the fracture, as appears in Fig. 4b, then Q b fþ ;n 1 ¼ and the total sink term for the fracture, Q b f ;n, is approximated implicitly in time. If one matrix gridcell is in the upstream and the other in the donstream (see Fig. 4c), then the source term is approximated explicitly and the sink term implicitly. fracture gridcells, k i ; i ¼ 1;...; N I, as sketched in Fig.. The purpose is to evaluate the mobility at I hich is given by fractional flo function f ;I (see Eq. (33), for simplicity e drop the notation f). In each fracture cell k i, the fractional flo function and the volumetric flux The approach presented above is locally conservative in the matrix and fractures. The system of equations given in Eq. (33) is linearized by the Neton Raphson (NR) method. If the NR method fails to converge for a certain time step, one can cut the time step and does multiple iterations in the fractures ithout reducing the global time step in the matrix. The time step is our method is constrained by a CFL condition in the matrix here the explicit time method is used and by the nonlinearity (Neton iterations) oing to the flo interaction at the matrix fracture interfaces. A simple dynamic time-step algorithm is adopted, here the time step is controlled by the change in saturation in the gridblocks ithin the previous time step and the number of Neton iterations required for convergence. The mobilities at the fracture fracture interface are approximated by the conventional single-point upstream-eighting technique. An important point to be addressed next is ho to define the upstream-eighting in case of multiple intersecting fractures, here one fracture gridcell may have multiple upstream fractures..1. Upstream-eighting at multiple intersection fractures In the past, e developed a technique for multiple intersecting fractures in 2D space [2,3]; the extension to fractures in 3D domain is straightforard. Let I be the interface (line) connecting N I Table 1 Relevant data for Examples 1 and 2 Domain dimensions: 1 m 1 m Matrix properties: / m ¼ :2, K m ¼ 1md Fracture properties: / f ¼ 1:, K f ¼ 1 6 md, ¼ 1mm Fluid properties: l ¼ l n ¼ 1 cp, q ¼ q n ¼ 1 kg=m 3 Relative permeabilities: linear (Eq. (39)) Capillary pressure: neglected Residual saturations: S r ¼ :, S rn ¼ : Injection rate:. PV/day Mesh number: 1 triangles Fig. 7. Matrix elements ith different aspect ratios; Example 1. (a) j ¼ 2h ¼ 1:; (b) b j ¼ 2h ¼ :1. b

8 898 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) at I are denoted by f ;ki and q a;;ki. According to the signs of the fluxes, influx or efflux at I, e define an integer ð < <N I Þ, such that: q a;;ki > for < i ðinfluxþ; q a;;ki 6 for <i < N I ðeffluxþ: ð34þ Fig. 8. Oil recovery and ater production versus PVI for different mesh aspect ratios ith the cross-flo equilibrium; Example 1. (a) Oil recovery; (b) ater production. Fig. 9. Oil recovery and ater production versus PVI for different mesh aspect ratios ith and ithout the cross-flo equilibrium; Example 1. (a) oil recovery; (b) ater production barrier 6 barrier 6 barrier 4 fracture 4 fracture Fig. 1. Gridding of the domain ith one fracture and one barrier; Case I: the fracture does not cross the barrier; Case II: the fracture crosses the barrier; Example 2. (a) Case I; (b) Case II.

9 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) Z Z Y Y X X 2 4 X(m) Pressure (bar) 2 4 X(m) Pressure (bar) Fig. 11. Pressure contours for Cases I and II; PVI ¼ :; Example 2. (a) Case I; (b) Case II Y(m) Y(m) X(m) X(m) Fig. 12. Water saturation contours for Cases I and II; PVI =.; Example 2. (a) Case I; (b) Case II Fig. 13. Triangulations of the fracture domain; Example 3. (a) Coarse: N m k ¼ 1718, nf j ¼ 14; (b) fine: Nm k ¼ 4667, Nf j ¼ 191. Table 2 Relevant data for Example 3 Domain dimensions: 2 m 2 m Matrix properties: / m ¼ :2, K m ¼ 1md Fracture properties: / f ¼ 1:, K f ¼ 8 1 md, ¼ :1mm Fluid properties: l ¼ 1 cp, q ¼ 1 kg=m 3, l n ¼ :4 cp, q n ¼ 66 kg=m 3 Injection rate:.27 PV/day Mesh number: 1718 (coarse) and 4667 (fine) triangles Note that, an integer that satisfies Eq. (34) does alays exist. By riting the total volumetric balance and the total mass balance at I, one gets, X i¼1 X i¼1 q a;;ki ¼ XN I i¼ þ1 f ;ki q a;;ki ¼ XN I q a;;ki ; i¼ þ1 X N I f ;I q a;;ki ¼ f ;I i¼ þ1 q a;;ki : ð3þ ð36þ

10 9 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) Table 3 Relative permeabilities and capillary pressure in the matrix: Example 3 S k k n p c ðbarþ E E E Table 4 Relative permeabilities and capillary pressure in the fractures: Example 3 S k k n p c ðbarþ E In Eq. (36), f ;I, hich refers to the mobility at I, is equal to the upstream mobility for the effluxes. From Eqs. (3) and (36), f ;I can be readily calculated: f ;ki ¼ P i¼1 f ;k i q a;;ki P i¼1 q a;;k i : ð37þ The expression for the mobility in Eq. (37) is a generalization of the conventional to-point upstream-eighting technique. 6. Numerical examples The combined MFE-DG method ith capillary effect as validated using analytical solutions for to-phase flo in unfractured media. We demonstrated that the algorithm enjoys accuracy and robustness especially in highly heterogeneous media ith unstructured griddings [43]. In fractured media, the MFE-DG associated ith the cross-flo equilibrium has been validated ith the single-porosity model [3]. In this ork, e provide four examples of various degrees of complexity. In the first example, e present results demonstrating that our method is not sensitive to the size of the matrix gridcells next to the fractures. The previous orks in the literature invoking the concept of cross-flo equilibrium beteen the fractures and the adjacent rock matrix depend on the matrix gridding. The second example shos the flexibility of our proposed model in numerical simulation of to-phase flo in porous media composed of a fracture and a barrier. The third example investigates the effect of the fracture capillary pressure on the fluid flo. The last example demonstrates the efficiency of our method in a large 3D problem ith layering, barriers, and fractures. We ould like to point that the gridding for all the problems is by no means optimal. The selection shos poerful features of the algorithm. All runs ere performed on an 1.8 GHz Intel-Centrino PC ith 1 GB of RAM Example 1: mesh sensitivity In this example, e examine the dependency of our model on the size of the matrix gridcells in the vicinity of the fractures. We consider a 2D horizontal porous medium of dimensions ð1 m 1 mþ ith one fracture along the diagonal, as shon in Fig. 6. The domain is initially saturated ith oil. Water is injected at the loer left corner to produce oil at the opposite corner. The fluid and medium properties are provided in Table 1. The domain is discretized into a mesh ith about 1 triangular matrix gridcells and 2 fracture gridcells (segments). We define an aspect ratio j for the matrix gridcells next to the fractures by j ¼ 2h=b, here h is the triangle height, and b is the base. Fig. 7a shos a mesh here the aspect ratio is one. In Fig. 7b, the aspect ratio is reduced to.1 by dragging the nodes of the triangles facing the fractures closer to its base (that is, the fracture). The aspect ratio of the mesh is, therefore, modified by changing the locations of some nodes ithout changing their number. Fig. 6 shos the ater saturation profiles ith and ithout the cross-flo equilibrium assumption. The former appears to be more dispersive close to the fracture. Oil recovery and ater breakthrough obtained ith the cross-flo equilibrium assumption for different aspect ratios are plotted in Fig. 8. Large matrix gridcells next to the fractures, that correspond to large aspect ratios, delay ater breakthrough and result in overestimation of the oil recovery. The results sho that convergence is obtained ith a small aspect ratio (about.1) indicating that the matrix gridcells should be relatively small. On the other hand, our ne model, shos almost identical results for aspect ratios of 1. and.1 (Fig. 9). Fig. 9 also shos that the results obtained ith cross-flo equilibrium ith fine matrix gridcells ðj ¼ :1Þ converges to the results by our method ith coarse matrix gridcells ðj ¼ 1Þ. We also note that both methods need comparable CPU time on the same mesh, hoever, our method ith j ¼ 1 is nearly one order of magnitude faster than the method ith j ¼ : Example 2: flexibility in modeling fractures and barriers In this example, e demonstrate the flexibility of the MFE-DG method in modeling fractures and barriers. We consider a 2D rectangular domain (1 m 1 m) initially saturated ith oil. The injector and producer locations are as in Example 1; the relevant data are provided in Table 1. One fracture and one barrier are embedded in the porous medium; to cases, I and II, are distinguished. In Case I, the fracture touches the barrier but does not cross it (see Fig. 1a). In Case II, the fracture intersects the barrier and crosses it (Fig. 1b). The fracture barrier intersection point in Case I has a dual role. The precise modeling by the Galerkin FE method ith consideration of the degrees of freedom at the mesh nodes, is not straightforard. In Case II, the injected fluid flos through the barrier via a single point located at the fracture barrier intersection. Modeling this case is a challenge for the CVFE method due to consideration of the degrees of freedom at the center of the control-volumes. We model in a simple ay both cases ithout a special treatment. The implementation of simple expressions in Eqs. (17) and (18) in the MFE-DG method provides the tool to model to-phase flo ith capillarity and gravity in complex media containing fractures (Eq. (17)) and barriers (Eq. (18)). Fig. 11 depicts the pressure contours ith discontinuity at the barrier (Fig. 11a, Case I) and pressure continuity for the fracture crossing the barrier (Fig. 11b, Case II). Results for ater saturations at PVI ¼ : for Cases I and II are plotted in Fig. 12a and b, respectively Example 3: capillary pressure effect In the past, Karimi-Fard and Firoozabadi [23] have used the domain in Fig. 13 to study the effect of capillary pressure on ater injection in fractured media. In this example, e use our method to provide a comprehensive study in terms of model efficiency and capillary pressure effect. The domain ith the fracture net-

11 91 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) f Fig. 14. Water saturation contours and ater velocity field for cases I, II, and III at PVI ¼ :2 and PVI ¼ :; Example 3. (a) PVI ¼ :2: pm c ¼, pc ¼ (Case I); (a ) f m f m f m f PVI ¼ :: pm c ¼, pc ¼ (Case I); (b) PVI ¼ :2: pc 6¼, pc 6¼ (Case II); (b ) PVI ¼ :: pc 6¼, pc 6¼ (Case II); (c) PVI ¼ :2: pc 6¼, pc ¼ (Case III); (c ) PVI ¼ :2: f pm c 6¼, pc ¼ (Case III). ork is sketched in Fig. 13a and b corresponding to a coarse and a fine grid. The relative permeability and capillary pressure curves in the matrix and fractures are given in Tables 3 and 4, respectively. Other relevant data are provided in Table 2. We investigate the effect of capillary pressure on flo by considering three cases. In Case I, the capillary pressures in the matrix ðpm c Þ and in the fracture f ðpfc Þ are neglected (that is, pm c ¼ pc ¼ ). In Case II, both capillary pressures are nonzero, as provided in Tables 3 and 4. In Case III, only the fracture capillary pressure is zero and that of the matrix is given in Table 3. In Fig. 14, e sho the ater saturation distributions and ater velocity profiles for the three cases at PVI ¼ :2 and PVI ¼ : on the fine grid. In Case I. (Fig. 14a and a ), ater flos very quickly through the fractures and leads to early breakthrough. In Case II (Fig. 14b and b ), hoever, the effect of capillary-pressure contrast at the matrix fracture interface delays breakthrough. Note that due to capillary cross-flo, the displacement process becomes much more efficient in Case II than in Case I. As the contrast of capillary pressure increases in Case III (Fig. 14c and c ), the fractures permeability has little influence on ater mobility due to very large capillary effect hen ater saturation in the matrix is much loer in the matrix than in the fracture. Although, the fracture absolute permeability is orders of magnitudes greater than that in the matrix, ater in the fractures and the surrounding matrix flos ith comparable speed. The contrast in capillary pressure at the fracture matrix interface creates a significant cross-flo via the ater velocity vectors laterally crossing the fractures (see Fig. 14c and c ). The oil recovery and ater

12 92 H. Hoteit, A. Firoozabadi / Advances in Water Resources 31 (28) Fig. 1. Oil recovery and ater production versus PVI for the three cases I, II, and III; Example 3. (a) Oil recovery; (b) ater production. Table Relevant data for Example 4 Domain dimensions: 1 m 3 m 2 m Matrix properties: / m ¼ :2, K m ¼f2 md;1md;1 mdg a Fracture properties: / f ¼ 1:, K f ¼ 1 md, ¼ 1mm Fluid properties: as in Example 3 Relative permeabilities (layers): Eq. (39) ith m ¼f2; 3; 2g; a m n ¼f3; 3; 2g a S rn ¼ ; S r ¼ :2 (all layers) Relative permeabilities linear (fractures): (Eq. (39)); S rn ¼ S r ¼ Capillary pressure (Case I): neglected Capillary pressure (Case II): B f ¼ :1, B m ¼f1; ; 2g a bar Injection rate: 1 4 PV/day Mesh number: 8 prisms a In layers 1 (top), 2 and 3, respectively. Table 6 CPU time in minutes for Examples 3 and 4 Example 3 (coarse) Example 3 (fine) Example 4 Zero capillarity Nonzero capillarity breakthrough versus PVI for the three cases are plotted in Fig. 1.It is obvious that capillary pressure contrast improves the recovery and delays the breakthrough. The capability to explicitly describe the matrix fracture cross-flo due to capillary pressure contrast is a significant feature of our method. The results from the fine and coarse grids are almost the same. The CPU times on both grids for Cases I and II are shon in, Table 6. The CPU time in Case III is about 2% more than in Case II Example 4: complex 3D media ith layers, faults, and fractures In this example, e consider a 3D layered tilted domain ð1 m 3 m 2 mþ, as appears in Fig. 16a. The medium has to barriers and several intersection fractures (Fig. 16b). The layer thicknesses from top to bottom are 3 m, 4 m, and 3 m, respectively. The injector is located at the bottom corner of the medium (at ð; ; Þ) and the producer at the opposite top corner (at (1 m, 3 m, 2 m)). The domain is discretized into an unstructured mesh of 8 matrix gridcells (prisms) (Fig. 16a). The number of fracture gridcells (quadrilaterals) is nearly 17. In Ref. [43], e shoed the applicability of the MFE-DG method on different discretizations in 3D. The rock properties, and relative permeability functions used in the layers and the fractures are provided in Table. The relative permeabilities and the capillary pressure are given by: k r ¼ S m e ; k rn ¼ð1 S e Þ mn ; ð38þ p c ¼ B lnðs e Þ; ð39þ here m and m n are defined in Table, and S e is the normalized saturation defined in terms of S r and S rn, hich are the residual saturations for the etting and non-etting phases, respectively: S e ¼ S S r : ð4þ 1 S r S rn Z Y X 1 1 Z (m) Layer 1 Layer 2 Layer 3 Z (m) Fracture Barrier Fig. 16. Discretization of the layered domain and the embedded fractures and barriers; Example 4. (a) 3D layered domain; (b) 2D fractures and barriers.