next up previous print clean
Next: Another Interpretation of as Up: Berryman: Dynamic permeability Previous: EQUATIONS OF POROELASTICITY

INDUCED MASS EFFECT

Biot (1956a) defines the kinetic energy T of the porous medium by
   \begin{eqnarray}
2T = \rho_{11}\dot{\bf u}\cdot\dot{\bf u} +
2\rho_{12}\dot{\bf u}\cdot\dot{\bf u}_f +
\rho_{22}\dot{\bf u}_f\cdot\dot{\bf u}_f,
 \end{eqnarray} (21)
where $\dot{\bf u}$ and $\dot{\bf u}_f$ are the solid and fluid velocities at a point in the medium. In the standard way, the overdots indicate a time derivative. Then he shows that the inertial coefficients satisfy two sum rules:
   \begin{eqnarray}
\rho_{11} + \rho_{12} = (1-\phi)\rho_s
 \end{eqnarray} (22)
and
   \begin{eqnarray}
\rho_{22} + \rho_{12} = \phi\rho_f.
 \end{eqnarray} (23)
Furthermore, as a matter of definition for the ``structure factor'' $\alpha$, we also have
   \begin{eqnarray}
\rho_{12} = - (\alpha-1)\phi\rho_f.
 \end{eqnarray} (24)
And Biot's discussion makes it clear, furthermore, that $-\rho_{12}$ should be thought of as the added or induced mass of the solid when it oscillates in the presence of the fluid. So we have
   \begin{eqnarray}
\rho_{11} = (1-\phi)\rho_s - \rho_{12} \equiv (1-\phi)(\rho_s +
r\rho_f),
 \end{eqnarray} (25)
where r is a measure of the geometry of the solid. In particular, it is well-known [see Lamb (1936)] that the factor $r = {{1}\over{2}}$for well-separated spheres and that it can vary from zero to unity, depending on the shape for simple objects such as spheroids. Combining these results, we find in general that
   \begin{eqnarray}
\alpha = 1 + r(\phi^{-1}-1),
 \end{eqnarray} (26)
or, if we take a conservative value for r and set it to $r = {{1}\over{2}}$, then we have
   \begin{eqnarray}
\alpha = {{1}\over{2}}\left(1 + \phi^{-1}\right).
 \end{eqnarray} (27)
The formula (27) was derived by Berryman (1980, 1981), and it is frequently used with success when fitting real data.


next up previous print clean
Next: Another Interpretation of as Up: Berryman: Dynamic permeability Previous: EQUATIONS OF POROELASTICITY
Stanford Exploration Project
7/8/2003