Approximation equation of apparent permeability in a two-dimensional, heterogeneous medium

This paper proposes a new approximation equation combined the arithmetical mean with the harmonic one of permeability distribution, in order to estimate an apparent permeability of two-dimensional, heterogeneous medium. The equation estimates the apparent permeability better than the geometric mean usually used in literature. In this theoretical study, apparent permeabilities of a heterogeneous medium with radial/lateral fluid flow are calculated by the Monte Carlo method and the finite element method. Also, the Bernoulli trials, the normal distribution or the logarithmic normal distribution is assumed to be a probability density function of permeability distribution. In the results, the apparent permeabilities are summarized in the features as follows: (1) the apparent permeability is not always equal to the arithmetical mean of the distribution function. (2) In a case that the skewness of the distribution function is equal to zero, the apparent permeability depends on the standard deviation, but not on the distribution function. (3) When the skewness is not equal to zero, the apparent permeability depends on the standard deviation, but not on the distribution function. (3) When the skewness is not equal to zero, the apparent permeability depends not only on the standard deviation, but also on the skewness. These facts are similarly appeared in the radial and lateral flow systems.