The seismic equations of motion for a double-porosity medium have been derived recently by Tuncay and Corapcioglu (1996) using a volume averaging approach. (These authors also provide a thorough review of the prior literature on this topic.) We will present instead a quick derivation based on ideas similar to those of Biot's original papers (Biot, 1956; 1962), wherein a Lagrangian formulation is presented and the phenomenological equations derived.

Physically what we need is quite simple -- just equations
embodying the concepts of *force = massacceleration*,
together with dissipation due to viscous loss mechanisms. The forces
are determined by taking a derivative of an energy storage
functional. The appropriate energies are discussed at length later in
this paper, so for our purposes in this section it will suffice to
assume that the constitutive laws relating stress and
strain are known, and so the pertinent forces are the divergence of
the solid stress field and the gradients of
the two fluid pressures *p ^{(1)}*

Generalizing Biot's approach (Biot, 1956) to the
formulation of the kinetic energy terms, we find that, for a system
with two fluids, the kinetic energy *T* is determined by

2T = _11B<>uB<>u
&+& _22B<>U^(1)B<>U^(1)
+ _33B<>U^(2)B<>U^(2) &+& 2_12B<>uB<>U^(1)
+ 2_13B<>uB<>U^(2)
+ 2_23B<>U^(1)B<>U^(2),
where is the displacement of the solid,
is the displacement of the *k*th fluid which
occupies volume fraction ,and the various coefficients , , etc., are mass
coefficients that take into account the fact that
the relative flow of fluid through the pores is not uniform, and
that oscillations of solid mass in the presence of fluid
leads to induced mass effects. Clarifying the precise meaning of
these displacements is beyond our current scope, but other recent
publications help with these interpretations (Pride and Berryman, 1998).

Dissipation plays a crucial role in the motion of the fluids and so cannot be neglected in this context. The appropriate dissipation functional will take the form

2D = b_12(B<>u-B<>U^(1))(B<>u-B<>U^(1))
&+& b_13(B<>u-B<>U^(2))(B<>u-B<>U^(2))
&+& b_23(B<>U^(1)-B<>U^(2))
(B<>U^(1)-B<>U^(2)).
This formula assumes that all dissipation is caused by motion
of the fluids either relative to the solid, or relative to each
other. (Other potential sources of attenuation, especially for
partially saturated porous media (Stoll, 1985; Miksis, 1988), should also be
treated, but will not be considered here.) We expect the
fluid-fluid coupling
coefficient *b _{23}* will generally be small and probably negligible,
whenever the double-porosity model is appropriate for the system under study.

Lagrange's equations then show easily that

t(T u_i) + Du_i = _ij,j, for i=1,2,3, and that

t(T
U^(k)_i) + D
U^(k)_i = -p^(k)_i, for i=1,2,3; k = 1,2,
where the pressures are the macroscopic fluid pressures across
interfaces and are related to the internal pore pressures
*p*^{(k)} by factors of the porosity so that
and ,with *v ^{(2)}* being the total volume fraction of the fracture
porosity and and being the matrix and fracture
porosities, respectively. (Note that in this method of accounting
for the void space, .)
These equations now account properly for inertia and elastic
energy, strain, and stress, as well as for the specified types of
dissipation mechanisms, and are in complete agreement with those
developed by Tuncay and Corapcioglu (1996) using a different approach.
In (fluids), the parts of the equation
not involving the kinetic energy can be shown to be equivalent
to a two-fluid Darcy's law in this context, so

In summary, equations (solid) and (fluids) can be combined into

_11 & _12 & _13 _12 & _22 & _23 _13 & _23 & _33 ü_i Ü^(1)_i Ü^(2)_i + b_12+b_13 & -b_12 & -b_13 -b_12 & b_12+b_23 & -b_23 -b_13 & -b_23 & b_13+b_23 u_i U^(1)_i U^(2)_i = _ij,j - p_,i^(1) - p_,i^(2) , showing the coupling between the solid and both types of fluid components.

In the next section we show how to relate the inertial and drag coefficients to physically measureable quantities.

4/20/1999