Terzaghi's model

Terzaghi simplifies Biot's model by decoupling the stress tensor into hydrostatic stress and stresses of the solid skeleton. Terzaghi's approximation, which are popular in the field of geomechanics, imply

$$\begin{eqnarray} \beta &=& \phi \\ K_f &=& K_s (1-\phi) + K_{f} \phi \\ M &=& K_{f} / \phi \\ p &=& - K_{f} \nabla {\bf U} \\ \end{eqnarray}$$ (36) (37) (38) (39)

If we further assume that the tortuosity a=1 is one and that we can ignore dissipation due to viscosity, we find again two non-dissipative P-waves with velocities

$$\begin{eqnarray} v_{P1} &=& \left(\frac{\lambda_0 + 2 \mu}{\rho_2 (1-\phi)}\right)^{1/2} \\ v_{P2} &=& \left(\frac{K_{f}}{\rho_f}\right)^{1/2}. \end{eqnarray}$$ (40) (41)

The first P-wave only involves the rock matrix, the second only involves the fluid.

If we continue to ignore loss due to viscosity but allow for tortuosity $a \ne 1$ and a fluid bulk modulo much smaller than the solid bulk modulo ($K_{f} \ll K_{m}$)than the wave velocities are

$$\begin{eqnarray} v_{P1} &=& \left(\frac{\lambda_0 + 2 \mu} {\rho_s (1-\phi) + \... ...\frac{\mu}{\rho_s (1-\phi) + \phi \rho_f (1-a^{-1})}\right)^{1/2} \end{eqnarray}$$ (42) (43) (44)

But Tzeraghi's assumption of decoupled stresses is considered unrealistic for usually highly consolidated hydrocarbon reservoir rocks.