Seminal papers by Brutsaert (1964) and Brutsaert and Luthin (1964) provided the first rigorous theoretical framework for examining the poroelastic behavior of unsaturated soils, including an important application linking acoustic wave propagation to soil hydraulic properties. Theoretical developments during the 50 years that followed have led Lo et al., (2005) to a comprehensive model of these phenomena, but the relationship of its elasticity parameters to standard poroelasticity parameters measured in hydrogeology has not been established. In the present study, we develop this relationship for three key parameters, the Gassman modulus, Skempton coefficient, and Biot-Willis coefficient by generalizing them to an unsaturated porous medium. We demonstrate the remarkable result that well-known and widely applied relationships among these parameters for a porous medium saturated by a single fluid are also valid under very general conditions for unsaturated soils. We show further that measurement of the Biot-Willis coefficient along with three of the six elasticity coefficients in the model of Lo et al. (2005) is sufficient to characterize poroelastic behavior. The elasticity coefficients in the model of Lo et al. (2005) are sensitive to the dependence of capillary pressure on water saturation and its viscous-drag coefficients are functions of relative permeability, implying that hysteresis in the water retention curve and hydraulic conductivity function should affect acoustic wave behavior in unsaturated soils. To quantify these as-yet unknown effects, we performed numerical simulations for Dune sand at two representative wave excitation frequencies. Our results show that the acoustic wave investigated by Brutsaert and Luthin (1964) propagates at essentially the same speed during imbibition and drainage, but is attenuated more during drainage than imbibition. Overall, effects on acoustic wave behavior caused by hysteresis become more significant as the excitation frequency increases.
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