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dependent flow/transport processes (e.g., hydrodynamic dispersion is superceded by
macrodispersion at larger length scales). Tidwell
Below we briefly consider the role of scaling in the modeling of fluid flow and
mass transport in porous and fractured media. We begin by reviewing some basic
approaches to modeling material property scaling. We then present examples from
the laboratory and field demonstrating scaling effects. The focus of this section is on
the scaling of permeability and dispersivity.

11.1 SCALING THEORIES
Simple averaging rules (arithmetic, geometric, and harmonic) represent the point of
entry for classical scaling theory. These averaging rules are only valid for a narrow
range of aquifer/reservoir conditions. Specifically, an infinitely stratified formation
comprised of individual layers with constant permeability that is subject to steady, lin-
ear flow. Where flow is oriented parallel to the stratification the effective permeability
is given by the arithmetic mean k a ,
n

k i d i
k a = (11.1)
d
i=1
where n is the sample set size, d is the total thickness of the system, and k i and, d i
are the permeability and thickness of each discrete bed, respectively. When flow is
oriented normal to stratification the harmonic mean k h , given by
d
k h = : (11.2)
n
6
d i k i
i=1
represents the appropriate averaging rule. These two effective permeabilities are
derived by solving the governing flow equation (Darcy’s Law) for each layer in
parallel and in series, respectively. Where the formation lacks spatial correlation,
Warren and Price (1961) found the geometric mean to provide a good estimate of the
effective permeability. The geometric mean k g is given by

n

k g = exp 1/n ln(k i ) (11.3)
i=1
It follows that, k a > k g > k h .
Unfortunately, most natural systems lie somewhere in-between these extremes as
do the appropriate scaling rules. The power law average provides a convenient means
of handling this wide range of behavior. The power law average k p is given by:
1/ω
n
ω
k p = 1/n k (11.4)
i
i=1