Modeling Fluid Flow Through Complex Reservoir Beds

Abstract

Summary Complex beds can significantly affect the flow of fluids in reservoirs. However, current simulators generally are not well-suited to model such flows, in part because they typically do not model permeability as a tensor quantity. This paper demonstrates the importance of treating permeability as a full tensor quantity in simulating flow through complex beds by considering two related problems. First, fluid flow through cross-stratified beds is studied. In this case, heterogeneities appear on the fine scale, and a scale-up procedure is required to model explicitly the effects of these beds. A general numerical procedure, which properly preserves the tensor nature of the procedure, which properly preserves the tensor nature of the effective permeability of such strata, is presented for this computation. The method is based on a finite-element solution of the fine-scale pressure equation with periodic boundary conditions imposed. Second, the modeling of flow through anisotropic, inclined reservoir beds is studied. It is shown that the commonly used techniques for modeling flow through such features, stratigraphic and horizontal layering, implicitly assume principal directions (or orientations) for the permeability tensor. With simple model problems, the differences between simulation results for different orientations of the permeability tensor and differences between results from horizontal and stratigraphic layering are quantified. These differences, which are substantial in the case of high reservoir dip and high anisotropy, emphasize the importance of a knowledge of the full permeability tensor in predicting flow through complex reservoir features. predicting flow through complex reservoir features. Introduction The incorporation of realistic reservoir information, such as fine-scale porosity and permeability data, has led to more predictive reservoir simulations. A next step in realistic reservoir simulation is to capture the effects of the complexities of the reservoir geometry, such as complicated beds, faults, and dipping strata. Many current simulators, however, are poorly suited for modeling complex strata. The reasons are that they typically use standard finite-difference methods, which lack the geometric flexibility needed to discretize complex strata, and that they treat permeability as a diagonal tensor (commonly called "vector permeability") rather than as a full tensor quantity. Permeability permeability") rather than as a full tensor quantity. Permeability must be treated in its full tensor form for geometrically complex strata (not as a diagonal tensor) because, for such strata, the principal directions of the permeability tensor generally are not principal directions of the permeability tensor generally are not aligned with the axes of the coordinate system in which the flow equations are solved. Finite-difference discretization of the reservoir flow equations with full tensor permeability fields requires a nine-point finite-difference stencil rather than the usual five-point stencil. The purpose of this paper is to elucidate and quantify the importance of properly modeling the permeability field as a full tensor quantity in the simulation of flow through irregular strata. Two related problems are addressed. First, the modeling of flow through cross-stratified beds is considered. Crossbedding typically is not modeled explicitly in reservoir simulation because it is on a scale much finer than that of the simulation gridblocks. Therefore, to include the effects of crossbedded strata in flow models, a procedure is required to scale up the fine-scale rock heterogeneities to scales more suitable for reservoir simulation (e.g., gridblock scale). Permeability on this larger scale is called effective permeability. A general numerical procedure, based on the application of finite-element methods and appropriately formulated periodic boundary conditions for the fine-scale problem, is presented to accomplish this scale-up. This method is quite general and yields effective permeabilities that provide realistic descriptions of large-scale flow through crossbedded strata.