Summary A simplified discrete-fracture model suitable for use with general-purpose reservoir simulators is presented. The model handles both 2- and 3D systems and includes fracture-fracture, matrix-fracture, and matrix-matrix connections. The formulation applies an unstructured control volume finite-difference technique with a two-point flux approximation. The implementation is generally compatible with any simulator that represents grid connections by a connectivity list. A specialized treatment based on a "star-delta" transformation is introduced to eliminate control volumes at fracture intersections. These control volumes would otherwise act to reduce numerical stability and timestep size. The performance of the method is demonstrated for several oil/water flow cases including a simple 2D system, a more complex 3D fracture network, a localized fractured region with strong capillary pressure effects, and a model of a strike-slip fault zone. The discrete-fracture model is shown to provide results in close agreement with those of a reference finite-difference simulator in cases in which direct comparisons are possible. Introduction Flow through fractured porous media is typically simulated using dual-porosity models. This approach, although very efficient, suffers from some important limitations. For example, dual-porosity models are not well suited for the modeling of a small number of large-scale fractures, which may dominate the flow. Another shortcoming is the difficulty in accurately evaluating the transfer function between the matrix and the fractures. For these reasons, discrete-fracture models, in which the fractures are represented individually, are beginning to find applications in reservoir simulation. These models can be used both as stand-alone tools as well as for the evaluation of transfer functions for dual-porosity models. Such models can also be used in combination with the dual-porosity approach. To accurately capture the complexity of a fractured porous medium, it is usually necessary to use an unstructured discretization scheme. There are, however, some effective procedures based on structured discretization approaches. For example, Lee et al.1 presented a hierarchical modeling of flow in fractured formations. In this approach, the small fractures were represented by their effective properties, and the large-scale fractures were modeled explicitly. In the case of unstructured discretizations, there are two main approaches: finite-element and finite-volume (or control volume finite-difference) methods. Baca et al.2 were among the early authors to propose a 2D finite-element model for single-phase flow with heat and solute transport. In a more recent paper, Juanes et al.3 presented a general finite-element formulation for 2- and 3D single-phase flow in fractured porous media. There has been some work on the extension of the finite-element method to handle multiphase flow. For example, Kim and Deo4 and Karimi-Fard and Firoozabadi5 presented extensions of the work of Baca et al.2 for two-phase flow. They modeled the fractures and the matrix in a 2D configuration with the effects of capillary pressure included. The two media (matrix and fractures) were coupled using a superposition approach. This entails discretizing the matrix and fractures separately and then adding their contributions to obtain the overall flow equations. The existing approaches based on finite-element procedures are successful in the case of single-phase flow and heat transfer, but in the case of multiphase flow in highly heterogeneous reservoirs, they do not ensure local mass conservation. Finite-element formulations based on mixed or discontinuous Galerkin methods (e.g., Riviere et al.6) can eliminate this difficulty, though these methods are generally more expensive than standard finite-volume procedures. Existing reservoir simulators are, in the great majority of cases, based on finite-difference or control volume finite-difference methods. Therefore, in order to maintain compatibility with existing general-purpose reservoir simulators, which is one of the intents of this work, it is important that the discrete-fracture model be based on control volume finite-difference techniques. The research on discrete-fracture modeling using finite-volume approaches is quite recent and is mainly limited to 2D problems. The work of Koudina et al.7 is an exception in that 3D fracture networks were considered, though the contribution of the matrix was ignored. A vertex-based finite-volume procedure was applied in this work for the solution of single-phase flow through the fracture network. A similar approach was used by Dershowitz et al.8 to calculate the dual-porosity parameters for a fractured porous medium. Cell-based approaches, in which control volumes can be readily aligned with the discontinuities of the permeability field, are probably more appropriate for reservoir simulation applications. Previous work on cell-based approaches is for 2D systems discretized on triangular meshes. For example, Caillabet et al.9,10 used a two-equation model for single-phase problems. A similar approach was used by Granet et al.11,12 for a single porosity model. They first applied the method to single-phase flow systems11 and then extended it to the two-phase flow case.12 In all of these approaches, the fracture intersections were treated through the introduction of a special node at each intersection. The purpose of this node is to allow for the redirection of flow. This treatment works well for single-phase flow, but problems can arise in the case of transport calculations (for multiphase flow). This is because of the very small size of the control volumes created at the fracture intersections, which influence the stability and allowable timestep of the numerical method. This issue was recognized and addressed by Granet et al.12 for the case of two-phase flow. They assumed that there is no accumulation term at the intersection, and in addition introduced a modified upwinding for the intersection control volumes. Because we are interested in compatibility with existing simulators, we prefer to avoid approaches such as this, which require special handling for control volumes associated with fracture intersections.

[1] A procedure for developing coarse-scale continuum models from detailed fracture descriptions is developed and applied. The coarse models are in the form of a generalized dual-porosity representation, in which matrix rock and fractures exchange fluid locally while large-scale flow occurs through the fracture network. The methodology developed here introduces local subgrids to resolve dynamics within the matrix and provides appropriate coarse-scale parameters describing fracture-fracture, matrix-fracture, and matrix-matrix flow. The geometry of the local subgrids, as well as the required parameters for the coarse-scale model, are determined from local flow solutions using the underlying discrete fracture model. The method is applied to two-and three-dimensional singlephase and two-phase flow problems, and the accuracy of the coarse models is assessed relative to fully resolved discrete fracture simulations. For the cases considered, it is shown that the technique is capable of generating highly accurate coarse models with many fewer unknowns than the detailed characterizations, and speedups of about a factor of 100 are achieved.

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Summary Numerical simulation of water injection in discrete fractured media with capillary pressure is a challenge. Dual-porosity models, in view of their strength and simplicity, can be used mainly for sugar-cube representation of fractured media. In such a representation, the transfer function between the fracture and the matrix block can be calculated readily for water-wet media. For a mixed-wet system, the evaluation of the transfer function becomes complicated because of the effect of gravity. In this work, we use a discrete-fracture model in which the fractures are discretized as ID entities to account for fracture thickness by an integral form of the flow equations. This simple step greatly improves the numerical solution. Then, the discrete-fracture model is implemented using a Galerkin finite-element method. The robustness and the accuracy of the approach are shown through several examples. First, we consider a single fracture in a rock matrix and compare the results of the discrete-fracture model with a single-porosity model. Then, we use the discrete-fracture model in more complex configurations. Numerical simulations are carried out in water-wet media as well as in mixed-wet media to study the effect of matrix and fracture capillary pressures.

Numerical simulation of water injection in discrete fractured media with capillary pressure is a challenge. Dual-porosity models in view of their strength and simplicity can be mainly used for sugar-cube representation of fractured media. In such a representation, the transfer function between the fracture and the matrix block can be readily calculated for water-wet media. For a mixed-wet system, the evaluation of the transfer function becomes complicated due to the effect of gravity. In this work, we use a discrete-fracture model in which the fractures are discretized as one dimensional entities to account for fracture thickness by an integral form of the flow equations. This simple step greatly improves the numerical solution. Then the discrete-fracture model is implemented using a Galerkin finite element method. The robustness and the accuracy of the approach are shown through several examples. First we consider a single fracture in a rock matrix and compare the results of the discrete-fracture model with a single-porosity model. Then, we use the discrete-fracture model in more complex configurations. Numerical simulations are carried out in water-wet media as well as in mixed-wet media to study the effect of matrix and fracture capillary pressures. Introduction Numerical simulation of oil recovery from fractured petroleum reservoirs remains a challenge. The heterogeneity of the porous media and the connectivity of the fractures have a significant effect on two-phase flow with capillary pressure and gravity effects. Dual-porosity models1,2,3 have been used to simulate two-phase flow with connected fractures; the sugar-cube model configuration has been studied in such a model. This approach, although very efficient, suffers from some important limitations. One limitation is that the method cannot be applied to discon-nected fractured media and cannot represent the heterogeneity of such a system. Another shortcoming is the complexity in the evaluation of the transfer function between the matrix and the fractures. In fact, in mixed-wet fractured media, a dual-porosity model may lose accuracy due to the effect of gravity. The single-porosity model provides the accuracy, but it is not practical due to very large number of grids. A large number of grids is required because of two different length scales (matrix size and fracture thickness). A geometrical simplification of the single-porosity model can make it applicable to larger configurations. The simplified model is called the discrete-fracture model. In this model, the fractures are discretized as one dimensional entities. The heterogeneity is accounted accurately and there is no need for the transfer function; it can also be applied to both water-wet and mixed-wet media. The discrete-fracture model was first introduced for single-phase flow. Noorishad and Mehran4 and Baca, Arnett and Langford5 were among the early authors to use one dimensional entities to represent fractures. These authors used finite element formulation to simulate 2D single-phase flow through fractured porous media. Noorishad and Mehran4 solved the transient transport equation in fractured porous media using an upstream finite element method to avoid oscillation for convective-dominated flow. Baca et al.5 considered a 2D single-phase flow with heat and solute transport. The two media (matrix and fractures) are coupled using the superposition principle.