Following Pride *et al.* (1992), we assume that the dynamical equations for constituents
of the solid/fluid mixture composing the porous medium can be linearized to

_ ^2 **u**_t^2 =
_ + **f**_,
where subscripts refer to fluid or solid mineral, respectively, and the
other symbols are density , displacement , stress tensor
, and body force , with *t* being the independent variable of time.
Assuming Hooke's law for the isotropic solid, we have

_s = K_m**u**_s **I** +
G_m(**u**_s + **u**_s^T -
23**u**_s **I**),
where *K*_{m} and *G*_{m} are, respectively, the bulk and shear moduli
of the constituent mineral.
The identity tensor is symbolized by .Similarly, the fluid is assumed to be a linearly viscous Newtonian fluid
obeying

_f = (-p_f + _f**u**_f)**I**
+ _f(**u**_f + **u**_f^T -
23**u**_f **I**),
where and are, respectively, the coefficients of bulk and
shear viscosity.
Dots over displacement indicate a single time derivative.
The increment of fluid pressure associated with conservative work
is related to the fluid dilatation by the bulk modulus *K*_{f} through

Performing the bulk averages on the microscopic stress/strain relations and using the averaging theorem gives the general constitutive relations for the solid and fluid stress tensors

(1-)B<>_s = (1-)K_mB<>u_s**I**
- K_m **I** + (1-)G_m(B<>u_s + B<>u_s^T
-23B<>u_s**I**)
- G_m **D**
and

B<>_f = K_f^*B<>u_f**I**
+ K_f^***I**
+ _ft(B<>u_f
+ B<>u_f^T - 23 B<>u_f
**I**) + _f t**D**,
where

**D** = 1V_E (**n****u** + **u****n**
- 23**n****u****I**) dS,
and

11/12/1997