Multipoint flux approximation (MPFA) methods were introduced to solve control-volume formulations on general grids. Although these methods are general in the sense that they may be applied to any grid, their convergence and monotonicity properties vary. We introduce a new MPFA method for quadrilateral grids termed the L-method. This method seeks to minimize the number of entries in the flux stencils, while honoring uniform flow fields. The methodology is valid for general media. For homogeneous media and uniform grids in two dimensions, this method has four-point flux stencils and seven-point cell stencils, whereas the MPFA O-methods have six-point flux stencils and nine-point cell stencils. The reduced stencil of the L-method appears as a consequence of adapting the method to the closest neighboring cells, or equivalently, to the dominating principal direction of anisotropy. We have tested the convergence and monotonicity properties for this method and compared it with the O-methods. For moderate grids, the convergence rates are the same, but for rough grids with large aspect ratios, the convergence of the O-methods is lost, while the Lmethod converges with a reduced convergence rate. Also, the L-method has a larger monotonicity range than the O-methods. For homogeneous media and uniform parallelogram grids, the matrix of coefficients is an M-matrix whenever the method is monotone. For strongly nonmonotone cases, the oscillations observed for the O-methods are almost removed for the L-method. Instead, extrema on no-flow boundaries are observed. These undesired solutions, which only occur for parameters not common in applications, should be avoided by requiring that the previously derived monotonicity conditions are satisfied. For local grid refinements, test runs indicate that the L-method yields almost optimal solutions, and that the solution is considerably better than the solutions obtained by the O-methods. The efficiency of the linear solver is in many cases better for the L-method than for the O-methods. This is due to lower condition number and a reduced number of entries in the matrix of coefficients.
Two discrete-fracture models (DFMs) based on different, independent numerical techniques have been developed for studying the behavior of naturally fractured reservoirs. One model is based on unstructured gridding with local refinement near fractures, while in the second model fractures are embedded in a structured matrix grid. Both models capture the complexity of a typical fractured reservoir better than conventional dual-permeability models, leading to a more accurate representation of fractured reservoirs. The accuracy of the DFM approaches is confirmed by their match with a structured, grid-aligned, explicit-fracture model in tests involving capillary imbibition during water flooding and gravity drainage in oil-gas systems. The DFMs are insensitive to grid orientation. Simulations also show consistency and agreement of results of the DFM methods in synthetic models with complex fracture patterns. Our simulations indicate that conventional dual-permeability approaches are appropriate when the fracture system is very sparse relative to the grid spacing. In these situations a DFM can be used as the basis for defining dual-permeability model parameters. However, conventional dual-permeability approaches are inadequate in the presence of high localized anisotropy and preferential channeling. When used with general purpose reservoir simulators, both DFMs show computational performance that is comparable to that of dual-permeability models.
Streamline-based methods can be used as effective post-processing tools for assessing flow patterns and well allocation factors in reservoir simulation. This type of diagnostic information can be useful for a number of applications, including visualization, model ranking, upscaling validation, and optimization of well placement or injection allocation. In this paper, we investigate finite-volume methods as an alternative to streamlines for obtaining flow diagnostic information. Given a computed flux field, we solve the stationary transport equations for tracer and time of flight by use of either single-point upstream (SPU) weighting or a truly multidimensional upstream (MDU) weighting scheme. We use tracer solutions to partition the reservoir into volumes associated with injector/producer pairs and to calculate fluxes (well allocation factors) associated with each volume. The heterogeneity of the reservoir is assessed with time of flight to construct flowcapacity/storage-capacity (F-vs.-U) diagrams that can be used to estimate sweep efficiency. We compare the results of our approach with streamline-based calculations for several numerical examples, and we demonstrate that finite-volume methods are a viable alternative. The primary advantages of finite-volume methods are the applicability to unstructured grids and the ease of implementation for general-purpose simulation formulations. The main disadvantage is numerical diffusion, but we show that a MDU weighting scheme is able to reduce these errors.
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