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Next: DYNAMIC PERMEABILITY AND TORTUOSITY Up: Berryman: Dynamic permeability Previous: INDUCED MASS EFFECT

Another Interpretation of $\alpha$ as Tortuosity

It has been shown by Berryman (1980, 1981) that the speed of the slow wave [i.e., the second compressional wave predicted by Biot (1956a)] can be written as
   \begin{eqnarray}
v_{-}^2 = {{K_f}\over{\alpha \rho_f}},
 \end{eqnarray} (28)
where Kf is the bulk modulus of the pore fluid. Or, first noting that the sound speed in the fluid is given by $v_f = \sqrt{K_f/\rho_f}$, we also have
   \begin{eqnarray}
v_{-} = {{v_f}\over{\sqrt{\alpha}}}.
 \end{eqnarray} (29)
Thus, since the slow wave is a compressional wave through the pore space which presumably travels at approximately the speed vf locally, the formula (29) can be interpreted as showing that the path length between two points in the pore space a distance L apart in space are a distance $\sqrt{\alpha}L$ apart when the travel path is restricted to the pore space. Thus, $\sqrt{\alpha}$ is a measure of the tortuosity of the pore space.

That $\alpha$ is a measure of the tortuosity has also been shown using a very different argument by Brown (1980). He shows that $\alpha$ is related to the electric formation factor for a porous system. In particular $\alpha = \phi {\cal F}$, where ${\cal F}$ is the electrical formation factor [which should not be confused with the dynamic viscosity factor $F(\xi)$ to be introduced later] defined by ${\cal F} = g_f/g \ge 1$, with gf being the electrical conductivity of the pore fluid and g being the effective overall conductivity of the system when the solid is nonconducting. The formation factor and the tortuosity are both equal to unity when $\phi = 1$. For some systems, it is known both from Archie's law and from theory too that ${\cal F} \simeq \phi^{-3/2}$ is often a reasonable estimate for pore systems. Thus, $\alpha \simeq \phi^{-1/2}$. Both this expression and (27) have the same repesentation as $\alpha \simeq
(3-\phi)/2$ when $\phi$ is large, showing that the two approaches using electrical tortuosity and the induced mass do in fact give similar results in this limit, even though it might seem hard to understand physically just exactly why this should be true.


next up previous print clean
Next: DYNAMIC PERMEABILITY AND TORTUOSITY Up: Berryman: Dynamic permeability Previous: INDUCED MASS EFFECT
Stanford Exploration Project
7/8/2003