Multidimensional flow of water in unsaturated soils

Abstract

In his studies of flow in porous media, John Philip favored the macroscopic (Darcian) scale, although he also regularly paid attention to processes at the underlying pore scale. At both of these scales, he recognized the connections between general concepts in continuum mechanics and their particular forms in soil physics. This paper specifically explores concepts from continuum mechanics that can be used to interpret multi-dimensional flow patterns at the Darcian scale. The standard model for flow of water in saturated and unsaturated soils is presented. Techniques for solving multi-dimensional flow problems for specific classes of unsaturated soils are indicated briefly. Some concepts of the kinematics of continuous media are introduced, with emphasis on the velocity vector field. Euler's general solution of the mass balance equation for a compressible continuum is adapted to soil water. Numerous special cases of this general solution are derived, thus unifying the scattered literature dealing with specific spatial and/or temporal simplifications. The implications of Darcy's law for the velocity and volumetric flux vector fields are explored. The rotation vector fields of the volumetric flux and velocity vector fields are analyzed by considering their decompositions in terms of the Frenet trihedron. The expressions for these rotation vectors are found to be surprisingly simple, involving explicitly the soil physical properties that are known to govern multi-dimensional flows. Finally, some implications of Darcy's law for the nature of flow patterns of soil water are related to properties of general lamellar and complex lamellar vector fields.