BASICS OF ELASTIC WAVE PROPAGATION

For isotropic elastic materials there are two bulk elastic wave speeds (Ewing et al., 1957; Aki and Richards, 1980), compressional $v_p = \sqrt{(\lambda+2\mu)/\rho}$and shear $v_s = \sqrt{\mu/\rho}$.Here 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 $\rho = (1-\phi)\rho_s + \phi[S\rho_l + (1-S)\rho_g]$,where $\rho_s$, $\rho_l$, $\rho_g$ are the densities of the constituent solid, liquid and gas, respectively, and S is the liquid saturation, i.e. the fraction of liquid-filled void space in the range $0 \le S \le 1$ (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). 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. At low liquid saturations, the fluid bulk modulus is dominated by the gas, and therefore the effect of the liquid on $\lambda$ is negligible until full saturation is approached. This means that both seismic velocities v_p and v_s will decrease with increasing fluid saturation (Domenico, 1974; Wyllie et al., 1956; Wyllie et al., 1958) due to the ``density effect,'' i.e., the only quantity changing is the density which increases in the denominators of both v_p² and v_s². As full saturation is approached, the shear velocity continues its downward trend, while the compressional velocity suddenly (over a very narrow range of change of saturation) shoots up to its full saturation value. An example (Murphy, 1982; 1984) of this behavior is shown in Figure 1a. This is the expected (ideal Gassmann) behavior of porous rocks at low frequencies (sonic and below).