EQUATIONS OF MOTION

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 = mass$\times$acceleration and 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 $\tau_{ij,j}$ and the gradients of the two fluid pressures p⁽¹⁾_,i and p⁽²⁾_,i for the matrix and fracture fluids, respectively. (In this notation, i,j index the three Cartesian coordinates x₁,x₂,x₃ and a comma preceding a subscript indicates a derivative with respect to the specified coordinate direction.) Then, the only work we need to do to establish the equations of motion for dynamical double-porosity systems concerns the inertial terms arising from the kinetic energy of the system, and drag terms arising from dissipation due to relative motion of the constituents.

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 ${\bf u}$ is the displacement of the solid, ${\bf U}^{(k)}$ is the displacement of the kth fluid which occupies volume fraction $\phi^{(k)}$,and the various coefficients $\rho_{11}$, $\rho_{12}$, etc., are inertial 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 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 [Miksis, 1988], should also be considered, but will not be discussed here.) We expect the coefficient b₂₃ will generally be small and probably be 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.   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 b₁₂ and b₁₃ are related to Darcy's constants for two single phase flow and b₂₃ is the small coupling coefficient. Explicit relations between the b's and the appropriate permeabilities (see Eqs.(53) and (54) of Berryman and Wang [1995]) are not difficult to establish (see the next section). The harder part of the analysis concerns the constitutive equations required for the right hand side of (solid). After the section on inertia and drag, the remainder of the paper will necessarily be devoted to addressing some of these issues concerning stress-strain relations.

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 identify the inertial and drag coefficients with physically measureable quantities.