A Method for Calculating Permeability Tensors using Perturbed Boundary Conditions

For reservoir simulation, it is usually necessary to represent fine-scale permeability heterogeneities by larger scale effective permeabilities. The effective permeability of a heterogeneous medium is a tensor and depends on the boundary conditions which dictate the direction of flow through the medium. We have reviewed current methods for determining effective permeability tensors, and find that existing methods either apply one type of boundary conditions, or give approximate results for a range of boundary conditions. This paper presents a new method for calculating the effective permeability tensors for single phase flow. The method is based on a pressure perturbation scheme which uses two flow cases. The first case uses boundary conditions which reflect the actual flow conditions for the medium. The second case uses a perturbation of the pressures calculated from the first case. This perturbation is applied first in the horizontal and then in the vertical directions. By using perturbed pressures, the flow is not distorted by unrepresentative boundary conditions. Each term of the effective permeability tensor is proportional to the ratio of the increment in flow to the increment in pressure gradient. This method has been tested using a variety of 2D permeability fields, both stochastic and deterministic, and gives good agreement with analytical results. We have applied the method to study the effects of no-flow boundaries in deterministic fields, representing certain types of elementary bedform. We have also investigated the effect of coarse-block size on the effective permeability in correlated random fields.