Page 210 - gas transport in porous media

Chapter 11: Scaling Issues in Porous and Fractured Media
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where λ is the correlation length scale and γ = q 1 / K g J 1 (where q 1 and J 1 are
the flux and average gradient in the longitudinal direction, respectively). In strati-
fied media, the transverse mixing process is highly anisotropic; that is, the horizontal
transverse macrodispersivity in the plane of bedding is much larger than vertical trans-
verse macrodispersivity associated with a direction perpendicular to that of bedding.
Assuming finite correlation length scales, Dagan (1988) shows that the longitudinal
macrodispersivity grows with travel time to an asymptotic value that is independent
of the anisotropy ratio. These travel distances are on the order of tens of horizontal
log-conductivity correlation length scales to reach asymptotic mixing conditions. One
limitation of these and similar theories is the assumption of mild heterogeneity (i.e.,
2
σ < 1). For more heterogeneous materials, Neuman and Zhang (1990) employed
y
quasi-linear theory to describe the evolution of the dispersion process.
Fractal concepts have also been used to explain the scaling of basic transport pro-
cesses like diffusion (Spoval et al., 1985) and macrodispersivity (Wheatcraft and
Tyler, 1988). For porous media exhibiting evolving scales of heterogeneity, classical
Fickian dispersive and diffusive concepts fail; that is the macrodispersivity continues
to grow with scale rather than reaching an asymptotic limit. Wheatcraft and Tyler
(1988) developed a Lagrangian model for dispersion in a set of fractal streamtubes.
They found the dispersivity is proportional to the straight-line travel distance raised
to a power of 2D − 1 where D is the fractal dimension. This evolving scale depen-
dent behavior is seen when one plots dispersivity versus experimental scale (e.g.,
Gelhar et al., 1992). Assuming a hierarchy of fractal media and quasi-linear the-
ory, Neuman (1990) offers a theoretical basis for interpreting this long recognized
behavior.

11.2 RESULTS FROM LABORATORY AND FIELD TESTS
DOCUMENTING SCALING EFFECTS
Field and laboratory studies have been performed to gain a better understanding of
scaling processes. In general, these studies involve the measurement of some property
over a range of different scales. However, the manner in which the scaling studies are
conducted can vary significantly in terms of the measurement techniques employed,
the range in sample support (i.e., sample volume) investigated, and the geologic
media interrogated. Laboratory tests have the advantage that greater control over the
measurement process can be maintained, while field studies allow a larger range of
scales to be investigated.
One obvious approach to investigating scaling processes is to dissect a heteroge-
neous medium and then reconstruct its effective properties from its component parts.
Laboratory-based studies conducted by Henriette et al. (1989) reconstructed the effec-
tive permeability of two 15 × 15 × 50 cm blocks of rock, one a sandstone and the
other a limestone. The reconstruction was based first on permeability measurements
made on 15 × 15 × 15 cm intermediate-scale blocks cut from the former and then
300 core samples cut from them. Scaling was manifest as trends in the statistical
properties of the measured permeability values. Specifically, the sample variance of