Kinetics of sorption and volume change in three-component systems
Authors: Philip JR , Smiles DE 
Published: April 2005

The paper analyses one-dimensional absorption (swelling) and desorption (shrinkage) in media subject to volume change (such as soils of high colloid content) and containing both water and air. The void ratio, the hydraulic conductivity, and the moisture potential are taken to be arbitrary known functions of the volumetric moisture content �. Then the combination of Darcy's law, applied to water flow relative to the soil particles, and the continuity requirement yields the general flow equation for systems of this type. In terms of material coordinates, the equation takes a nonlinear heat-conduction form. For media exhibiting (a) zero and (b) normal volume change, the equation reduces, as it should, to two known equations: (a) the nonlinear diffusion equation in physical coordinates, and (b) the nonlinear diffusion equation in material coordinates. The specific problem is that of sorption consequent on a step-function change in moisture potential at the column surface. The column is taken to be effectively semi-infinite (and constrained at infinity). The relevant solution of the general flow equation is of the similarity formm(�,t) = �(�)t1/2 where m is the material coordinate, t is time, and 4(8) is found by the solution of a nonlinear, ordinary, integrodifferential equation. A rapid and accurate numerical method of evaluating �(�) from any given set of soil characteristics is provided by a minor variant of an established technique for solving the nonlinear diffusion equation subject to similar conditions. With �(�) known, it is an elementary matter to restate the solution in physical coordinates, and to deduce a great variety of properties of the sorption process : profiles of moisture content, moisture potential, and void ratio; cumulative total volume change and the instantaneous rate of total volume change; cumulative total sorption and the instantaneous sorption rate; profiles of soil-particle velocity; profiles of volume flux density of water (i) relative to the soil particles and (ii) due to mass flow; profiles of absolute volume flux density of water; and the displacement history of soil particles initially at various positions in the column. Two examples are worked out fully, one for absorption and one for the converse desorption process. Graphs are presented illustrating the various aspects of the solution mentioned above. It is found, in keeping with previous work on two-component systems exhibiting normal volume change, that swelling is propagated more rapidly than is shrinkage but that it is associated with a slower rate of exchange of water. As in the earlier study (Philip 1968), this seeming paradox arises from the influence of the mass flow of water on the phenomena. For the example of absorption, the total column swelling amounts to only 43% of normal swelling; on the other hand, the converse desorption process produces over 99% of normal shrinkage. The moisture profiles computed for absorption exhibit a shallow surface layer of large moisture gradient, reminiscent of the transition zone sometimes observed in infiltration and absorption experiments. The present analysis applies to the drying of a cracking soil, so long as the (properly defined) void ratio function exists.