Dynamic water potential waves in unsaturated soils

Quantitative descriptions of the temporal and spatial variations of volumetric water content and total water potential in unsaturated soils have for more than nine decades focused on diffusive mechanisms as represented by the Richards equation (Richards, 1931). However, recent laboratory studies (Lo et al., 2017) have revealed short-time transient behavior of water in non-deforming unsaturated soils that is distinctly oscillatory, not diffusive. In this paper, we establish a theoretical framework to generalize the Richards equation in a systematic way to account for non-diffusive mechanisms giving rise to oscillatory behavior in the water content and total water potential. A partial differential equation is developed, with the total water potential as the dependent variable, based on the coupling of mass and linear momentum balance within the continuum theory of mixtures. This new differential equation becomes equivalent to the Richards equation when non-diffusive mechanisms can be neglected, but when this condition is not met, the governing equation describes damped propagating wave behavior of the total water potential. This dynamic wave differs from a poroelastic wave in respect to solid framework motions and wave speed, as well as time and length scales. For a rigid, homogeneous unsaturated soil, the governing equation reduces to the telegraph equation, which can be solved analytically. An inherent feature of our approach is the appearance of natural time and length scales characteristic of total water potential waves. Consideration of these natural scales for three representative soils of differing texture helps to prescribe more precisely the domain of applicability of the Richards equation.