ELASTIC AND POROELASTIC WAVE PROPAGATION

For isotropic elastic materials there are two bulk elastic wave speeds (Aki and Richards, 1980), compressional $v_p = \sqrt{(\lambda+2\mu)/\rho}$and shear $v_s = \sqrt{\mu/\rho}$.Here $\rho$ is the overall density, and the Lamé parameters $\lambda$ and $\mu$are the constants that appear in Hooke's law relating stress to strain in an isotropic material. The constant $\mu$ gives the dependence of shear stress on shear strain in the same direction. The constant $\lambda$ gives the dependence of compressional or tensional stress on extensional or dilatational strains in orthogonal directions. For a porous system with porosity $\phi$ (void volume fraction) in the range $0 < \phi < 1$, the overall density of the rock or sediment is just the volume weighted density given by

= (1-)_s + [S_l + (1-S)_g],   where $\rho_s$, $\rho_l$, $\rho_g$ are the densities of the constituent solid, liquid and gas, respectively. S is the liquid saturation, i.e., the fraction of liquid-filled void space in the range $0 \le S \le 1$ [see Domenico (1974)]. When liquid and gas are distributed uniformly in all pores and cracks, Gassmann's equations say that, for quasistatic isotropic elasticity and low frequency wave propagation, the shear modulus $\mu$ will be mechanically independent of the properties of any fluids present in the pores, while the overall bulk modulus K ($\equiv \lambda + {2\over3}\mu$)of the rock or sediment including the fluid depends in a known way on porosity and elastic properties of the fluid and dry rock or sediment (Gassmann, 1951; Berryman, 1999). Thus, in the Gassmann model, the Lamé parameter $\lambda$ is elastically dependent on fluid properties, while $\mu$ is not. The density $\rho$ also depends on saturation, as shown in equation (rho). At low liquid saturations, the bulk modulus of the fluid mixture is dominated by the gas, and therefore the effect of the liquid on $\lambda$ is negligible until the porous medium approaches full saturation. This means that both velocities v_p and v_s will decrease with increasing fluid saturation (Domenico, 1974) due to the ``density effect,'' wherein the only quantity changing is the density, which increases in the denominators of both v_p² and v_s². As the medium approaches full saturation, the shear velocity continues its downward trend, while the compressional velocity suddenly (over a very narrow range of saturation values) shoots up to its full saturation value. A well-known example of this behavior was provided by Murphy (1984). Figure 1 shows how plots of these data for sandstones will appear in several choices of display, with Figure 1(a) being one of the more common choices. This is the expected (ideal Gassmann-Domenico) behavior of partially saturated porous media. The Gassmann-Domenico relations hold for frequencies low enough (sonic and below) that the solid frame and fluid will move in phase, in response to applied stress or displacement. The fluid pressure must be (at least approximately) uniform throughout the porous medium, from which assumption follows the homogeneous saturation requirement.