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_mu_s I + G_m(u_s + u_s^T - 23u_s I), where Km and Gm 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 + _fu_f)I + _f(u_f + u_f^T - 23u_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 Kf through
- p_f = K_fu_f.
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_sI - K_m I + (1-)G_m(B<>u_s + B<>u_s^T -23B<>u_sI) - G_m D and
B<>_f = K_f^*B<>u_fI + K_f^*I + _ft(B<>u_f + B<>u_f^T - 23 B<>u_f I) + _f tD, where
= 1V_I nu dS,
D = 1V_E (nu + un - 23nuI) dS, and
K_f^* = K_f(1 + _fK_ft).