## 1. Introduction

### 1.1. Acoustic Waves and Poroelasticity

[2] The theory of poroelasticity, a term coined by *Geertsma* [1966], aims to provide a fundamental mathematical framework to describe the time-dependent coupling between applied stress and pore fluid pressure that occurs when acoustic waves pass through a porous medium. It is based on foundations laid down more than 70 years ago in a famous series of papers by *Biot* [1941, 1956, 1962], who was motivated initially by the classic geomechanical problem of soil consolidation. *Biot* [1956, 1962] presented a theory of dynamic poroelasticity that provides a general continuum-scale description of wave propagation through an elastic porous medium containing a single fluid, with wave attenuation arising from fluid viscosity. In addition to the dilatational and shear waves like those observed in wave propagation through an isotropic elastic solid, *Biot* [1956, 1962] correctly predicted the existence of a second dilatational wave (“Biot slow wave”) caused by the coupled out-of-phase motions of the elastic porous framework and the viscous pore fluid.

[3] Extending the approach of *Biot* [1956], *Brutsaert* [1964] developed the first rigorous theoretical basis for modeling the effects of partial water saturation, such as occurs routinely in soils, on elastic wave behavior in unconsolidated porous media. As was the case in the pioneering study of *Brutsaert* [1964], most attempts to model acoustic wave propagation in porous media containing two immiscible fluids [*Garg and Nayfeh*, 1986; *Tuncay and Corapcioglu*, 1997; *Wei and Muraleetharan*, 2002] have neglected the possibility of coupling among the fluid phases and solid framework caused by differences between their accelerations (“inertial drag”), as opposed to coupling caused by differences between their velocities (“viscous drag”). Inertial drag was taken into account in the Biot-inspired model of elastic wave behavior in unsaturated porous media developed by *Berryman et al*. [1988], but their model was simplified significantly by imposing the condition that the capillary pressure between the two immiscible pore fluids remains constant during acoustic wave excitation.

[4] *Santos et al*. [1990] appear to have been the first to construct a model of wave propagation through elastic porous media containing two immiscible fluids, one wetting and one nonwetting (e.g., water and air), that accounts for both capillary pressure fluctuations and inertial drag. Since fluctuating capillary pressure allows the pore fluids to experience distinct time-dependent pressure changes, a dilatational mode in addition to the two predicted by the *Biot* [1962] model is expected. On the basis of numerical simulations, *Santos et al*. [2004] proposed that this third mode involves the nonwetting fluid moving out-of-phase with the wetting fluid while it moves in-phase with the solid framework. This proposal was elucidated by *Lo et al*. [2010], who demonstrated numerically that excitation of the third dilatational mode actually does not affect the solid framework significantly, but instead causes the two pore fluids to move out-of-phase, which is consistent with this mode arising from capillary pressure fluctuations.

[5] *Lo et al*. [2005] developed a comprehensive Eulerian model of acoustic wave behavior in an elastic porous medium permeated by two immiscible fluids as an exact counterpart of the Lagrangian approach taken by *Santos et al*. [1990]. The *Lo et al*. [2005] model includes all previous poroelasticity approaches as special cases. Invoking plane harmonic wave excitation and homogeneous boundary conditions, these authors also demonstrated the existence of three dilatational waves, that with the highest phase speed (P1) corresponding to the compressional wave that propagates through any elastic medium (“Biot fast wave”), followed by one of lower phase speed (P2) that corresponds to the Biot slow wave, and finally a slowest third wave (P3) associated with capillary pressure fluctuations and having a phase speed dependent on the slope of the relationship between capillary pressure and wetting fluid saturation. At seismic frequencies, the P2 and P3 waves are diffusive, with low phase speed and high attenuation; however, they are propagating waves when the excitation frequency lies in the ultrasound régime [*Santos et al*., 1990].

### 1.2. Connecting Theory to Experiment

[6] The poroelasticity model of *Lo et al*. [2005], although comprehensive, contains six linear elasticity coefficients that remain somewhat abstract, with their relation to standard poroelasticity parameters [*Wang*, 2000] not yet established. Although *Lo et al*. [2005] developed expressions for their elasticity coefficients in terms of poroelasticity and hydraulic parameters for the individual phases in a porous medium containing two fluids, these expressions were designed to provide mathematical generality more than physical insight. In the present paper, we remedy this situation by formulating the stress-strain relationships in the model of *Lo et al*. [2005] in terms of two directly measurable pressure variables, the confining pressure and the average fluid pore pressure. This reformulation permits three standard poroelasticity parameters [*Wang*, 2000], Gassman's modulus (“undrained bulk modulus”), Skempton's coefficient (“undrained pore pressure buildup coefficient”), and the Biot-Willis coefficient (“effective stress coefficient”), to be generalized to apply to a porous medium saturated by two fluids, with their connection to the elasticity coefficients in the linear stress-strain relations of *Lo et al*. [2005] specified precisely.

[7] Building on the pioneering efforts of *Brutsaert and Luthin* [1964] to relate the speed of the P1 wave to water content, a number of experimental studies have been undertaken to establish the relationship between acoustic wave behavior and soil hydraulic parameters [e.g., *Flammer et al*., 2001; *Lu and Sabatier*, 2009]. *Lo et al*. [2007b] examined the effect of soil texture on the phase speed and attenuation of the P1 wave under varying water content using numerical simulations based on the model of *Lo et al*. [2005] and experimental data on the water retention curve and hydraulic conductivity function reported by *Rawls et al*. [1992]. The phase speed exhibited a maximum at low-water content that grew more pronounced as the texture became more clayey, while the attenuation was relatively greater as the texture became increasingly sandy. The water content dependence of the phase speed was shown to be very sensitive to changes in the capillary pressure with water content, implying the importance of the matric potential, a point also noted nearly 50 years ago by *Brutsaert and Luthin* [1964].

[8] But it is well known that matric potential and water content have a path-dependent relationship which differs during imbibition (wetting) from that during drainage (drying) [*Warrick*, 2003; *Jury et al*., 2004]. This hysteresis effect has not received attention as to its impact on the behavior of acoustic waves in an unsaturated porous medium. Since the elasticity coefficients for a porous medium filled by two fluids are associated with changes in capillary pressure with respect to water saturation and the viscous-drag coefficients are functions of relative permeability [*Santos et al*., 1990; *Tuncay and Corapcioglu*, 1996; *Lo et al*., 2005], imbibition and drainage processes should influence the speed and attenuation of acoustic waves differently. To illustrate the effect of hysteresis in the water retention curve on these properties of the three dilatational waves, we performed numerical simulations at two representative seismic frequencies, 10 and 200 Hz, for unsaturated Dune sand using the boundary-curve approach of *Kool and Parker* [1987].