A linear constitutive model for unsaturated poroelasticity by micromechanical analysis

Abstract

This 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.

MPeat—A fully coupled mechanical‐ecohydrological model of peatland development

“…The interpretation of Equation is that the compression of a fully saturated porous medium consists of the compression of pore water, solid skeleton and the amount of water expelled from it by the flow. The value of α is equal to one (Terzaghi, 1943), and M is equal to the inverse of the specific storage, that is, (Cheng, 2020; Green & Wang, 1990). In this formulation, the vertical head gradient is contained in the excess pore water pressure, which in turn influences the effective stress.…”

“…The parameters α w and M w depend on the degree of saturation of water (Cheng, 2020): where S w is the degree of saturation of water (−), γ w is the specific weight of water (N m −3 ), ϕ is the active porosity (−), λ is the first water retention empirical constant (−), μ is the second water retention empirical constant (m −1 ), ϵ is the strain (−), p w is the excess pore water pressure (Pa) and κ is the hydraulic conductivity (m s −1 ).…”

Characterization of unsaturated response of shale-like materials

“…The contributions of wetting and nonwetting fluids to the linear stress-strain relations in Biot poroelastic theory have been recently considered by Cheng (2019). Unsaturated porous media where wetting and nonwetting fluid exist simultaneously in voids do not adhere to the behavior of linear poroelastic materials since the interaction between the fluids and solid is highly related to the diffusion process of pore fluids.…”

Preconditioners for multiphase poromechanics with strong capillarity

“…Capillarity arises from surface tension between co‐existing fluid phases in the pore space. It plays a critical role in unsaturated soil mechanics, 19–26 and significantly influences oil recovery predictions 27,28 . The solid deformation is intricately coupled with fluid pressures via nonlinear constitutive laws 29–33 relating the capillary pressure to degree of saturation.…”