No abstract

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.

No abstract

The key ingredients to successful real-time reservoir management, also known as a Bclosed-loop^approach, include efficient optimization and model-updating (historymatching) algorithms, as well as techniques for efficient uncertainty propagation. This work discusses a simplified implementation of the closed-loop approach that combines efficient optimal control and model-updating algorithms for real-time production optimization. An adjoint model is applied to provide gradients of the objective function with respect to the well controls; these gradients are then used with standard optimization algorithms to determine optimum well settings. To enable efficient history matching, Bayesian inversion theory is used in combination with an optimal representation of the unknown parameter field in terms of a KarhunenYLoeve expansion. This representation allows for the direct application of adjoint techniques for the history match while assuring that the two-point geostatistics of the reservoir description are maintained. The benefits and efficiency of the overall closed-loop approach are demonstrated through real-time optimizations of net present value (NPV) for synthetic reservoirs under waterflood subject to production constraints and uncertain reservoir description. For two example cases, the closed-loop optimization methodology is shown to provide a substantial improvement in NPV over the base case, and the results are seen to be quite close to those obtained when the reservoir description is known a priori.

No abstract