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 the 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. It follows immediately from (6) for any isotropic poroelastic medium. Furthermore, the two equations (6) and (7) taken together show that, for isotropic microhomogeneous media, the fluid effect 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.