Dynamical equations

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 $\xi= f,s$ refer to fluid or solid mineral, respectively, and the other symbols are density $\rho$, displacement ${\bf u}$, stress tensor ${\bf \tau}$, and body force $\bf f$, 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 K_m and G_m are, respectively, the bulk and shear moduli of the constituent mineral. The identity tensor is symbolized by ${\bf I}$.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 $\kappa_f$ and $\mu_f$ 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

- 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).