The effective dry moduli of the different hydrate frameworks have to be related
to the effective moduli of the same rock-containing fluid. The
effective moduli of a saturated rock can be calculated at seismic
frequencies by means of Gassmann's formulas:

(9) |

(10) |

where *K*_{sat} and *G*_{sat} are the effective bulk and shear moduli of the
saturated rock; *K*_{dry} and *G*_{dry} the moduli of the dry rock;
*K*_{fluid} and *G*_{fluid} the moduli of the fluid; and *K*_{solid} the bulk
modulus of the mineral material making up the rock.
is the porosity of the saturated rock.

The effective moduli of the mineral *K*_{solid} can be calculated
using the Voigt-Reuss-Hill average, which is an arithmetic average of
the Voigt upper bound (*K*_{V}) and the Reuss lower bound (*K*_{R}):

(11) |

(12) |

(13) |

where *f* and *f*_{h} are the volume fractions of the grains and the hydrate,
respectively. *K* is the grain bulk modulus and *K*_{h} is the hydrate
bulk modulus.

In order to calculate the effective fluid bulk modulus *K*_{fluid} for
partially-saturated rocks, we used the following equation:

(14) |

where *f*_{i} is the volume fraction of the fluid, *N* is the number of fluids,
and *K*_{fluid<<188>>i} is the bulk modulus of the fluid.

After having derived the effective moduli of the saturated rock, we used the
following relationships to find the seismic velocities in saturated
rock:

(15) |

(16) |

where is the density of the saturated rock,

(17) |

is the density of the solid phase, and is the density of the fluid phase which can be obtained from:

(18) |

11/12/1997