Johnson *et al.* (1987) have shown that, for a single-porosity
medium, the frequency dependence of the dynamic permeability and tortuosity
can be well-approximated by

k() = k_0 [1 - 4ik_0^2_^2_f/^2^2]^12 - ik_0__f/ and

() = _ + ik_0_f
[1 - 4ik_0^2_^2_f/^2^2]^12.
The symbol stands for the fluid viscosity (in units of
), while is the kinematic viscosity
(in units of *m ^{2}*/

For single porosity media, Johnson *et al.* (1987) show that
the lambda parameter approximately satisfies

^2 = 8k_0F = 8k_0_/. For the present purposes, we will assume that this relation holds independently for the storage porosity and the fracture porosity. Then we have

^(1) = [8 k^(11) ^(1)/^(1)]^12 and

^(2) = [8 k^(22) ^(2)/^(2)]^12 = [8 k^(22)]^12.

Finally, we see that for the double-porosity medium, the corrections due to frequency dependence can be viewed alternatively as a frequency dependent viscosity, since equations (komega) and (tauomega) follow by assuming that

() (1-i^2_f/16)^12 These corrections need to be made separately for the two types of pores. This interpretation of the frequency dependence as being associated specifically with the viscosity is the one advocated by Biot (1956b), and has some advantages in the present context as it makes it quite straightforward to determine what the corrections should be for the multiple porosity problem. Note that we used (GLapproximation) to simplify the factors inside the square root in (etaGw).

It is not yet clear how to generalize these expressions for the
permeability coupling terms *k ^{(12)}*, but our assumption following
(b23) that

TABLE 1. Stress-strain parameters in double-porosity modeling
as derived by Berryman and Wang (1995).

Parameter | Formula | Berea |

TABLE 2. Material Properties for Berea Sandstone and Water

^{a}From Coyner (1984)

Berea Parameter |

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

4/20/1999