Page 208 - gas transport in porous media

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Chapter 11: Scaling Issues in Porous and Fractured Media
where ω is the power coefficient. With this approach the selection of an appropriate
scaling rule is predicated on finding the proper power coefficient. Where flow is
oriented parallel or normal to an infinitely stratified system ω is simply set to 1
or −1 to yield an arithmetic or harmonic average, respectively. Alternatively, the
geometric mean is defined as ω = 0, and thus Equation (11.4) must undergo a limited
expansion as currently defined. In general, empirical methods are used to determine
ω. Journel et al. (1986) and Deutsch (1989) provide algorithms for estimating ω for
bimodal sand-shale sequences, while Desbarats (1992b) provides direction in the case
of log-normally distributed data.
Stochastic methods (e.g., Bakr et al., 1978; Gelhar, 1993) have been developed that
allow direct calculation of effective properties based on the heterogeneous character-
istics of the aquifer/reservoir. Application of these methods requires an unbounded
domain, and uniform flow (i.e., the extent of the domain and the characteristic scale
of the flow nonuniformity are much larger than the correlation length scale of the
medium). The permeability distribution must also be a weakly stationary and ergodic
random variable with a relatively small variance (generally Var[lnk] < 1, how-
ever under certain circumstances this limit can be exceeded). In this approach, the
aquifer/reservoir is viewed as an ensemble of homogenous, isotropic blocks whose
spatial distribution is fully characterized by its first two moments. Using a small per-
turbation, first-order approximation of the governing stochastic differential equation,
an expression for the effective permeability has been obtained. Gutjahr et al. (1978),
extending the earlier work of Matheron (1967), found the effective permeability of a
heterogeneous, isotropic medium to be:
2
k eff = k g [1 − σ /2] (11.5)
y
in one dimension,

k eff = k g (11.6)
in two dimensions, and
2
σ
y
k eff = k g 1 + (11.7)
6
2
in three dimensions, where k g and σ are the geometric mean and variance of the
y
natural log permeability distribution, respectively.An interesting result of this work is
the dependence of k eff on the dimensionality of the flow domain. Gelhar and Axness
(1983) extended this work to a 3-D statistically anisotropic medium. In this case, the
effective permeability is expressed as:
2
k ii = k g [1 + σ (0.5 − g ii )] (11.8)
y
where g ii is a geometric factor accounting for the degree of anisotropy and orientation
of flow relative to the principal permeability axes and the subscript ii designates the
tensor components. For the case of infinite stratification and flow parallel or normal
to the bedding, Equation (11.8) reduces to that of an arithmetic and harmonic mean,