Page 209 - gas transport in porous media

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respectively. Indelman andAbramovich (1994) demonstrated g ii to also be a function
of the shape of the covariance function. Tidwell
Scaling has also been approached by purely numerical means. In this case (Ababou
et al., 1989; Bachu and Cuthiell, 1990; Desbarats, 1987; Kossack et al., 1990) the
effective permeability of a spatially heterogeneous permeability field is determined
by numerically solving the steady-state flow equation for a prescribed domain. Real-
izations of the heterogeneous permeability field are generated by statistical methods,
based on a variety of principles ranging from sedimentological process models to geo-
statistics (e.g., Koltermann and Gorelick, 1996). In formulating the flow model, care
must be taken to assign boundary conditions that reflect conditions the computational
element will experience within the larger flow domain (Lasseter et al., 1986; Gomez-
Hernandez and Journel, 1994). The statistical moments of the upscaled permeability
field can then be extracted though Monte Carlo simulation or from a single realization
provided that the ergodicity requirements are satisfied. Use of numerical methods in
applied scaling problems is limited due to the excessive, sometimes insurmountable
computational requirements.
In general, each of the aforementioned theories assume the porous media exhibits a
discrete hierarchy of scales (i.e., a finite correlation length scale exists). However, not
all porous media behave in such manner, but rather exhibit a continuous hierarchy of
scales or continuous evolving heterogeneity (i.e., infinite length scale). A convenient
means of modeling evolving heterogeneities is with fractals. In fact, a number of
researchers have found geologic materials to display fractal characteristics. Fractal
behavior was noted for the case of sandstone porosity (Katz and Thompson, 1985),
soil properties (Burrough, 1983), fracture networks (Barton and Larson, 1985), and
reservoir porosity (Hewett, 1986). Upon examining transmissivity and permeability
data measured over scales of 10 cm to 45 km, Neuman (1994) argued that the data
scale according to a power-law semivariogram (i.e., infinite length scale). Using
the stochastic framework of Gelhar and the apparent power-law scaling, Neuman
offers a model for the effective permeability. The model predicts that the effective
isotropic permeability will decrease with increasing sample support (i.e., sample
volume) in one-dimensional media, increase in three-dimensional media, and show
no systematic variation within two-dimensional media. The concept of multifractal
scaling of hydraulic conductivity distributions has also been developed to deal with
data sets where conductivity variations are more heterogeneous at smaller scales than
at larger scales (Liu and Molz, 1997).
Similar approaches have been used to predict effective transport parameters such
as the dispersion coefficient or macrodispersivity. The macrodispersivity tensor is a
measure of the influence hydraulic conductivity imposes on large-scale solute mixing.
Calculations based on stochastic theory and assuming a finite correlation length scale
predict that the longitudinal macrodispersivity, A ij increases directly with the variance
of log hydraulic conductivity in the isotropic case (Gelhar and Axness, 1983)
2
σ λ
y
A 11 = (11.9)
γ 2