next up previous print clean
Next: NUMERICAL EXAMPLES OF THE Up: ANALYTICAL EXAMPLES OF THE Previous: 3D duct

Discussion of analytical examples

All three of these examples, and indeed any other example as well, show that $F(\xi) \to 1$ as $\omega \to 0$. This result is universal. The other limit for $\omega \to \infty$ gives somewhat different results for the three cases considered, but they can all be approximated by using the form
   \begin{eqnarray}
F(\xi) \simeq \left(1 - iP\xi^2\right)^{1/2},
 \end{eqnarray} (40)
where P is a number that depends on the duct model. Assuming for the moment that R = a1 = a: For spherical particles, P = 1. For the 2D duct, P=1/9. For the 3D duct, P=1/16.

Since one of the implicit goals of the dynamic permeability analysis is to determine a universal form for the function $F(\xi)$ and thereby determine a universal form for the dynamic permeability, it is important to consider how these problems differ from each other. The radius R is a measure of the particle size, but this size is not easy to relate to the radius of the cylindrical pore in the third case, or to the duct height in the second case. It would make more sense to relate these quantities in some more general way since the typical rock sample will not have any of these geometries. Perhaps the obvious choice is to use $\kappa_0$ itself, since we need a pertinent measure of length squared, and that is exactly what the low frequency permeability is.

In fact, if we first consider the form of (37) and take the point of view that the factor $\kappa_0\alpha\omega/\eta\phi$in the denominator determines a ``natural'' characteristic frequency for the problem given by
   \begin{eqnarray}
\omega_0 = \eta\phi/\alpha\kappa_0,
 \end{eqnarray} (41)
then, we can choose to approximate the dynamic viscosity factor by
   \begin{eqnarray}
F(\xi) \simeq \left(1 - iP\xi^2\right)^{1/2}
 \end{eqnarray} (42)
where now $\xi \equiv (\omega/\omega_0)^{1/2}$ and P is a real numerical factor that is at least approximately problem independent. Experimental and computational results show (Sheng and Zhou, 1988; Johnson, 1989; Sheng et al., 1989) that many rocks and other porous systems can be successfully modeled this way using P's such that $0.4 \le P \le 0.5$. So the range of values for P is really quite small in many cases.

There are some exceptions to these rules but they are beyond our present scope, so the reader is encouraged to see the paper by Pride et al. (1993) for an extended discussion.


next up previous print clean
Next: NUMERICAL EXAMPLES OF THE Up: ANALYTICAL EXAMPLES OF THE Previous: 3D duct
Stanford Exploration Project
7/8/2003