Calculating equivalent permeability: a review
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 = ∂ux/∂x + ∂uy/∂y + ∂uz/∂z 〈 〉 averaging operator E() mathematical expectation C() covariance σ2 variance D space dimension k,K isotropic or anisotropic local permeability h, H hydraulic head Keq equivalent permeability tensor Kef effective permeability tensor Kb block permeability tensor μa arithmetic mean μh harmonic mean μg
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. with μh = harmonic mean and μa = arithmetic mean.
Hashin and Shtrikman bounds
They are used for isotropic binary media: where f0 and f1 are the fractions of the permeability phases k0 and k1. k1 is higher than k0 and μa = 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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