A linear constitutive model for unsaturated poroelasticity by micromechanical analysis

Abstract

SummaryThis paper extends the Biot theory of poroelasticity from the saturated to unsaturated case. The Biot phenomenological model uses parameters that are easily observable, such as the deformation of porous frame, total stress, pore pressure, and fluid specific discharge. Such model is preferred for engineering applications. At this macroscopic level, the extension of Biot theory from saturated to unsaturated is straightforward. The constitutive constants, however, are combined properties of solid, pore space, and fluids. In the unsaturated case, the constants are functions of the degree of saturation. Their measurements and tabulation over a range of saturation is generally not feasible for practical applications. In this work, a Biot‐Willis type analysis is performed for the unsaturated case to provide a theory that the bulk material constants can be evaluated using laboratory measurable micromechanical constants under saturated condition, plus a capillary pressure curve (saturation versus suction pressure) typically available for unsaturated porous medium, without the need of measurement at each state of saturation. In particular, it is demonstrated that the surface energy contained in the meniscus interface manifests as a “capillary modulus,” given by the negative inverse slope of the capillary pressure curve. A rigorous analysis based on the thermodynamic variational energy approach is also conducted to lend theoretical support to the phenomenological approach. The presented model can bring a closure to the practical engineering modeling of the deformation of partially saturated porous medium that lacks the information of material constants over the range of saturation.