Gassmann results for isotropic poroelastic media

To understand the significance of the results to follow, we briefly review a well-known result due to Gassmann (1951) [also see Berryman (1999b) for a tutorial]. Gassmann's equation relates the bulk modulus K^* of a saturated, undrained isotropic porous medium to the bulk modulus K_dr of the same medium in the drained case:

$$\begin{eqnarray} K^* = K_{dr}/(1-\alpha B), \end{eqnarray}$$ (6)

where the parameters $\alpha$ and B [respectively, the Biot-Willis parameter (Biot and Willis, 1957) and Skempton's pore-pressure buildup coefficient (Skempton, 1954)] depend on the porous medium and fluid compliances. For the shear moduli of drained $(\mu_{dr})$and saturated $(\mu^*)$ media, Gassmann's quasi-static theory gives

$$\begin{eqnarray} \mu^* = \mu_{dr}. \end{eqnarray}$$ (7)

We want to emphasize once more that (7) is a result of the theory, not an assumption. The derivation of (6) and (7) shows that both results are elementary (and coupled) consequences of the theory. Furthermore, the two equations (6) and (7) taken together show that, for isotropic microhomogeneous media, the entire fluid effect on the overall elastic behavior is all contained in the parameter $\lambda^* = K^* - {2\over3}\mu^*$,where $\lambda$ and $\mu$ are the well-known Lamé parameters. This result is crucial for understanding the significance of our later results to oil and gas exploration.