Poroelastic theory of consolidation for a two-layer system with an upper unsaturated soil and a lower saturated soil under fully permeable boundary conditions

Soils are naturally present in the form of stratified layers, each being distinguishable from adjacent layers by a distinctive set of hydraulic and elasticity properties. Lo et al. (2020) have recently presented a theoretical model of poroelasticity that provides a detailed description of simultaneous solid framework deformation and immiscible fluid flow in a two-layer soil system that is made up of upper unsaturated and lower saturated zones caused by a vertical constant surface load under semi-permeable boundary drainage conditions. Due to the existence of different moisture contents (partial and full saturations) in the upper and lower layers, excess pore fluid pressure does not exhibit a symmetric distribution with respect to depth. Therefore, an exact, analytical solution for fully permeable boundary drainage conditions cannot be determined readily by applying scaling transformations and spatial translations from our recent result even though the coupled model equations possess an invariance feature. In this paper, a well-defined boundary-value problem based on the decoupled model equations of poroelasticity generalized for a two-fluid system is rigorously formulated for a double-layer (unsaturated-saturated) soil profile that allows water drainage at both top and bottom surfaces. Using Laplace time transformation, we derive the closed-form, exact-analytical solution accounting for the intricate interaction between deformable solid matrix and compressible interstitial fluids in such a two-layer system. The solution takes a more complicated form than that for the semi-permeable case. Boundary drainage conditions indeed affect the development of excess pore water and air pressures along with the time evolution of total settlement. We show that when the bottom boundary of the saturated layer overlain by the unsaturated layer with a lower hydraulic conductivity is impervious, the curve describing the relationship between the depth and excess water pressure at the interface has a negative slope, whereas the slope is positive when the boundary is pervious. The total settlement in the lower saturated layer would thus achieve an equilibrium state faster.