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# 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 and shear .Here the Lamé parameters and are the constants that appear in Hooke's law relating stress to strain in an isotropic material. The constant gives the dependence of shear stress on shear strain in the same direction. The constant gives the dependence of compressional or tensional stress on extensional or dilatational strains in orthogonal directions. For a porous system with porosity (void volume fraction) in the range , the overall density of the rock or sediment is just the volume weighted density given by ,where , , 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 (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 will be mechanically independent of the properties of any fluids present in the pores, while the overall bulk modulus 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 is elastically dependent on fluid properties, while is not. The density 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 is negligible until full saturation is approached. This means that both seismic velocities vp and vs 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 vp2 and vs2. 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).

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10/25/1999