To illustrate the use of the formulas derived for the coefficients of
the double-porosity system, we will now compute and plot the coefficients for
a realistic system. We will use data of Coyner (1984) for Navajo
sandstone, and modify it somewhat to produce a plot that will
highlight the results obtained from the equations. The first problem
we encounter in doing so is that, although we can make reasonable
direct estimates of the bulk and shear moduli of the constituents,
we also must have an estimate of the overall bulk modulus *K ^{*}* of
the composite double-porosity medium. And more than that, we need it
as a function of the volume fractions of the two constituents.
Our analysis has assumed that

TABLE 1. Input parameters for a Navajo sandstone model
of double-porosity system. Bulk moduli *K* have units of GPa.
Poisson's ratio and porosity are dimensionless.

The parameters used for Navajo sandstone are listed in Table 1.
Although Poisson's ratio does not appear explicitly in the equations
here, it is required in the CPA (or any but the most elementary)
effective medium calculation for the overall bulk modulus *K ^{*}*.
The results are shown in Figure 3.

Figure 1

Note that the off-diagonal coefficient *a _{23}*, which couples the fluid in
the storage porosity to the fluid in the transport porosity, is very
close to zero for all values of storage material volume fraction.
This behavior has been observed previously (Berryman and Wang 1995),
and is believed to be a strong indication that the double-porosity
approach is appropriate for the system studied.
If this coefficient is not small, then the fluids in the two types of
porosity are strongly coupled and therefore should not be treated as a
double-porosity system.

The behavior of the other coefficients is as one would expect: All the coefficients for the transport porosity tend to vanish as the volume fraction of this phase vanishes, and the medium again reduces to a single-porosity system in this limit.

6/8/2002