Diffusive instability of pore pressure transients in deformable unsaturated soils

Rapidly changing pore pressures are a major cause of soil instability, especially in the context of large-scale infrastructural collapses and geo-hazards. In this paper, the governing equations that control pore pressure transients in deformable unsaturated soils are studied from an analytical standpoint, with the purpose of connecting the mathematical properties of the field equations to the mechanics of saturation-induced soil instabilities. New criteria are proposed to identify the loss of parabolicity of the differential problem that governs the mass balance of the pore fluids and the temporal dynamics of their pressure evolution. It is shown that the fulfillment of such criteria yields an ill-posed mathematical problem that can be related to singularities of the pore pressure rate, as well as to violations of Lyapunov-like stability principles. It is further demonstrated that these expressions bear resemblance to the conditions of loss of controllability in unsaturated soils, thereby linking the occurrence of diffusive instabilities to the lack of uniqueness and/or existence of the underlying constitutive behavior. On the one hand, these findings point out that suction-dependent soil nonlinearity may promote the ill-posedness of the field equations, thus causing possible lack of robustness of the algorithms used for their numerical solution. On the other hand, they establish a connection between the loss of strength of unsaturated soils subjected to wetting and the temporal patterns of suction evolution, thus offering new tools to interpret catastrophic failures in natural settings and engineered geo-systems.