Page 13 - Percolation Models for Transport in Porous Media With

2 INTRODUCTION

measurements of parameters for identical systems with further comparison of the
results.
Detailed reports, as well as reviews of works accomplished in this field, are
presented in the reviews [25-29, 115-118) and monographs [30-36) of Russian and
foreign authors. However the approached used is limited by the framework of
standard percolation theory where heterogeneity bears a threshold nature. In
other words, within the framework of these approaches, a medium consists of only
two types of elements - conducting ones with identical conductivities and non-
conducting ones.
This narrows the possibilities for the percolation approach to simulation of fluid
flow dramatically, since actual porous media contain a large variety of conducting
pore channels. The nature of transfer processes in such media is substantially
dependent on the structure of heterogeneity in them (i.e., on pore-space structure
of the medium and properties of its surface) and on the way it is filled with fluids.
In the majority of actual porous media, a commensurate contribution to effec-
tive conductivity can be made by groups of conducting elements with substantially
different intrinsic conductivities. Rocks, with their exclusive variety of pore space
structures, represent a typical example of such media. Thus a need for more ade-
quate percolation models arises, so that the latter be able to describe transfer in
heterogeneous media when the distribution of conducting elements with respect
to values of intrinsic conductivities is known.
The problem of effective conductivity for the media with the stochastic het-
erogeneity can be effectively solved for the case of a small variation of permeabil-
ity [104]. In this case perturbation of flow caused by the heterogeneity is small,
and the linearized theiry is applicable. It allows not only to describe an aver-
aged flow (effective conductivity), but also estimate the covariance. Perturbation
method provides exact formulas [104, 105). The self-consistant approach [104, 106)
provides the analytical solution for the highly heterogeneous systems, but under
the assumption of self-similarity with the variation of the scale.
It is worth mentioning that simple formulas for the arithmetic average and
the geometric average often give reasonable results (for the layer cake reservoir,
horizontal and vertical permeabilities, respectively, see [107, 108)). The com-
bined arithmetic/harmonic and harmonic/arithmetic averages give even better es-
timates [108, 109]. Renormalization method provides further improvement to the
accuracy of the estimate of the effective permeability for the media with stochastic
heterogeneity [1-4).
Nevertheless analytical models do not give sufficient results in the number of
important practical cases, and so numerical models are applied [108, 110).
Calculation of the effective permeability for the two-phase flow in the media
with stochastic heterogeneity is more complex and in the general case can be done
only numerically [111, 112).