Calculating equivalent permeability: a review

doi:10.1016/S0309-1708(96)00050-4

Advances in Water Resources

Volume 20, Issues 5–6, October–December 1997, Pages 253-278

https://doi.org/10.1016/S0309-1708(96)00050-4 Get rights and content

Abstract

The purpose of this article is to review the various methods used to calculate the equivalent permeability of a heterogeneous porous medium. It shows how equivalence is defined by using a criterion of flow or of the energy dissipated by viscous forces and explains the two different concepts of effective permeability and block permeability. The intention of this review is to enable the reader to use the various published techniques and to indicate in what circumstances they can be most suitably applied. © 1997 Elsevier Science Ltd. All rights reserved

Section snippets

NOMENCLATURE

k,f	lightface notations: scalar
*x,K*	boldface notations: tensors
Δ	nabla operator (∂/∂x, ∂/∂y, ∂/∂z)
grad	gradient operator: gradh = Δh = (∂h/∂x,∂h/∂y, ∂h/∂z)
div	divergence operator: div u = Δ · u = ∂u_x/∂x + ∂u_y/∂y + ∂u_z/∂z
〈〉	averaging operator
E()	mathematical expectation
C()	covariance
σ²	variance
D	space dimension
k,K	isotropic or anisotropic local permeability
h, H	hydraulic head
K_eq	equivalent permeability tensor
K_ef	effective permeability tensor
K_b	block permeability tensor
μ_a	arithmetic mean
μ_h	harmonic mean
μ_g

INTRODUCTION

The use of numerical models for studying subsurface flow has become common practice in hydrology and petroleum engineering over the last 30 years. However, one of the major questions that still pose a problem is: what parameters to introduce into the models? Although contemporary computers are growing ever more powerful and capable of describing the relevant flows with increasing precision, it is impossible to measure all the parameters at all points.

Flow models can be divided into two groups:

Wiener bounds

This inequality is also called the fundamental inequality because it is always valid. It has been demonstrated by a great number of authors, e.g. Wiener,[112] Cardwell and Parsons,[16] Matheron[79] and Dagan,[19] among others. $μ_{h} ≤K_{ef} ≤μ_{a}$ with μ_h = harmonic mean and μ_a = arithmetic mean.

Hashin and Shtrikman bounds

They are used for isotropic binary media: $μ_{a} − f_{1} f_{0} (k_{1} −k_{0})^{2} (D−f_{0})k_{0} +f_{0} k_{1} ≤ K_{ef} ≤μ_{a} − f_{1} f_{0} (k_{1} −k_{0})^{2} (D−f_{1})k_{1} +f_{1} k_{0}$ where f₀ and f₁ are the fractions of the permeability phases k₀ and k₁. k₁ is higher than k₀ and μ_a = f₀k₀ + f₁k

Sampling

The first technique is simply not to change scales. A block is given the permeability measured at its center. This very basic technique is commonly used in the petroleum industry and consists in passing from a measurement at the 10 cm scale to a block on the meter scale.

Averaging means

The general idea is to take a value between two theoretical bounds.

DETERMINISTIC METHOD

In the deterministic method, the permeability field K(x, y, z) and the boundary conditions are assumed to be known. For a sufficiently simple permeability field (e.g. a stratified medium), exact analytical solutions can be found. For more general cases, there are theories (percolation, effective medium, streamline, renormalization) that can be used to make approximated calculations with varying precision. An approach which is, in principle, more general consists in solving numerically the

STOCHASTIC METHOD

In order to deal with the uncertainty arising from a partial knowledge of the reservoir properties, the stochastic method considers the studied variables as random functions in space. The definition of the effective permeability is then based on the notion of mathematical expectation (Eq. (3)).

The determination of the probability distribution function of the equivalent permeability is expressed in terms of a stochastic differential equation, i.e. as a differential equation linking several

Effective or block permeability?

The first possible choice is to describe the heterogeneous medium by a single value (the effective permeability) or by a set of values (block permeability). Durlofsky[30] compares three two-dimensional methods for media with correlated log-normal permeability distributions. These methods are: (1) a technique known as global where the heterogeneous medium is replaced by a uniform effective permeability; (2) the technique of sampling; (3) a local technique which consists in calculating the block

CONCLUSION

The purpose of this review of the literature is to give an account of the methods that are currently used to calculate the equivalent permeability for uniform, single-phase, steady-state flow. We have tried to present as complete an inventory as possible of the different techniques, divided into three main categories: deterministic, stochastic and heuristic. These groups have proved to be complementary rather than antagonistic.

It is clear that the stochastic methods are the only ones capable of

Acknowledgements

This work was supported by the Geoscience 2, reservoir Engineering Project funded by the Commission of the European Communities (1993–1995).

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Calculating equivalent permeability: a review

Abstract

Section snippets

NOMENCLATURE

INTRODUCTION

Wiener bounds

Hashin and Shtrikman bounds

Sampling

Averaging means

DETERMINISTIC METHOD

STOCHASTIC METHOD

Effective or block permeability?

CONCLUSION

Acknowledgements

Int. J. Engng Sci

Chem. Engng Sci.

Adv. Water Resour.

Adv. Water Res.

J. Phys. A: Math. Gen.

Effective permeability of log-normal isotropic random media

J. Phys. A: Math. gen.

The local change of scale method for modelling flow in natural porous media (1): Numerical tools.

Adv. Water Res.

The universal structure of the groundwater flow equations

Water Resour. Res.

Assigning effective values to simulation grid-block parameters for heterogeneous reservoirs

SPE Reservoir Engineering

Percolation theory and its application to groundwater hydrology

Water Resour. Res.

Analysis of an upscaling method based on conservation of dissipation

Trans. Porous Media.

Eléments de comparaison entre la méthode d'homogénéisation et la méthode de prise de moyenne avec fermeture

C. R. Acad. Sci. Paris

Effective model of two-phase flow in a porous medium made of different rock types

Applicable Analysis

Average permeabilities of heterogeneous oil sands

Trans. Am. Inst. Mining. Met. Pet. Eng.

Models of groundwater flow in statistically homogeneous porous formations

Water Resour. Res.

High-order correction of effective permeability of heterogeneous isotropic formations of log-normal conductivity distribution

Trans. Porous Media

La percolation: un concept unificateur

La Recherche

Quelques méthodes d'approche de la variabilité spatiale des reservoirs souterrains

Hydrogéologie

Quelques réflexions sur l'utilisation des modèles en hydrologie

Revue des Sciences de l'Eau

Numerical estimation of effective permeability in sand-shale formations

Water Resour. Res.

Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media

Math. Geol.

Calculating effective absolute permeability in sandstone/shale sequences

SPE Form. Eval.

Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media

Water Resour. Res.

Representation of grid block permeability in course scale models of randomly heterogeneous porous media

Water Resour. Res.

Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities

Water Resour. Res.

Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach1. Method

Water Resour. Res.

Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach2. Results

Water Resour. Res.

Estimations du tenseur de perméabilitié

C. R. Acad. Sci. Paris

Statistics of block conductivity through a simple bounded stochastic medium

Water Resour. Res.

Three-dimensional stochastic analysis of macrodispersion in aquifers

Water Resour. Res.

Stochastic analysis of spatial variability in suvsurface flows 2: Evaluation and application

Water Resour. Res.