Flow in Heterogeneous Porous Media

Abstract

Techniques for studying the performance characteristics of heterogeneous reservoirs have been developed. The effect of permeability variation on both the steady-state and the transient flow of a single fluid has been investigated. Limited comparisons with field and laboratory data have been made. The physical models studied consist of random three-dimensional arrays of homogeneous porous blocks. The permeabilities of the individual elements are assigned according to a specific distribution function; uniform anisotropy is introduced by varying the relative dimensions of the blocks. A particular model is perturbed simply by re-arranging its elements at random. The behavior of a physical model is determined by digitally solving its numerical analogue. Based on computational experiments, subject to the restrictions implied by the assumptions that were made, the following general conclusions have been drawn.The most probable behavior of a heterogeneous system approaches that of a homogeneous system with a permeability equal to the geometric mean of the individual permeabilities.The effects of flow geometry and anisotropy on the most probable value of the effective permeability of a heterogeneous medium are finite but not significant.The permeability determined from a pressure build-up curve for a heterogeneous reservoir gives a reasonable value for the effective permeability of the drainage area.A qualitative measure of the degree of heterogeneity and its spatial configuration are obtained from a comparative study of core analysis and pressure build-up data. It has been indicated that these conclusions are predicated on the assumption that the core analysis and the pressure build-up data represent true reservoir characteristics. Common sources of error in these data have been discussed. Introduction One of the most significant problems of reservoir engineering is that of determining the nature and the disposition of the heterogeneities that inevitably occur in petroliferous formations. It is with this problem that this paper is concerned.

Limitations to upscaling of groundwater flow models dominated by surface water interaction

“…The rule of geometric averaging is only satisfied for an isotropic lognormal medium and a checkerboard binary design [Warren and Price, 1961]. Moreover, it holds only for uniform (parallel) flow fields and is not satisfied for example for radial flow that is discretized by rectangles.…”

“…This method was first introduced by Warren and Price [1961], and after all, it simply consists of numerically simulating the experiment of Darcy upon a coarse grid block isolated from the total model domain (see Figure 4). The advantage of Darcian methods is that the shape of hydraulic transmissivities affects the final upscaled hydraulic transmissivity, that power averaging lacks (section 3.1).…”

“…Hydraulic permeability K for a checkerboard media versus the refinement of the mesh for a grid-centered and node-centered finite difference discretization. Both methods have their equilibrium at the exact hydraulic permeability, known as the geometric mean [Warren and Price, 1961].…”

Artificial Neural Network to Determine Dynamic Effect in Capillary Pressure Relationship for Two-Phase Flow in Porous Media with Micro-Heterogeneities

“…Warren and Price (1961) and Jensen (1991) suggested that the average permeability should be calculated at sub-block grid scale for uncorrelated permeability distribution. Motivated by these, the average permeability is computed using Eq.…”

How well can we predict permeability in sedimentary basins? Deriving and evaluating porosity–permeability equations for noncemented sand and clay mixtures

“…Warren & Price (1961) have shown that the effective permeability of randomly distributed components is equal to the geometric mean of the components, which for a random mixture of sand and clay yields:…”