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Biot (1956a) defines the kinetic energy *T* of the porous medium
by

| |
(21) |

where and 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:
| |
(22) |

and
| |
(23) |

Furthermore, as a matter of definition for the ``structure factor'' , we also have
| |
(24) |

And Biot's discussion makes it clear, furthermore, that 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
| |
(25) |

where *r* is a measure of the geometry of the solid. In particular,
it is well-known [see Lamb (1936)] that the factor 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
| |
(26) |

or, if we take a conservative value for *r* and set it to
, then we have
| |
(27) |

The formula (27) was derived by Berryman (1980, 1981), and it
is frequently used with success when fitting real data.

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** Up:** Berryman: Dynamic permeability
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Stanford Exploration Project

7/8/2003