Kinetics of sorption and volume change in clay-colloid pastes

Abstract

The present paper analyses one-dimensional absorption (swelling) and desorption (shrinkage or consolidation) in two-component pastes of clay-colloid and electrolyte solution. The work deals specifically with the case where solution exchange with the paste column is possible only through its base. The extension to systems with 'top drainage' and 'double drainage' is trivial. The analysis combines Darcy's law applied to the flow of solution relative to the colloid particles with the continuity requirement. The hydraulic conductivity K and the moisture potential (i.e. minus the 'swelling pressure') � are taken to be arbitrary known functions of the volumetric solution content �. The fundamental flow equation which follows is basically a nonlinear partial integrodifferential equation. It can be restated as a Fokker-Planck equation with �-dependent 'diffusivity' and the 'drift velocity' spatially constant but unknown and time dependent. The dependent variable is conveniently taken as �, although it can be made � if this is desired. The specific problem is that of sorption consequent on a step-function change in applied load (i.e. of the value of � (or �) at the base of the column). The column is taken to be effectively semi-infinite, so that the solution applies only until swelling or shrinkage at the upper surface of the column first becomes significant. Under these conditions, the solution is of the similarity form x(�, t) = �(�)t1/2 where x is the space coordinate, t is time, and � is a function of � which is found by the solution of a nonlinear, ordinary, integrodifferential equation. With �(�) known, it is an elementary matter to deduce a great variety of properties of the sorption process: the instantaneous distribution of colloid content; the instantaneous rate of total volume change; the cumulative total volume change; the instantaneous local velocity and volume flux density of colloid; the instantaneous local mean velocity and volume flux density of solution relative to colloid; the instantaneous local absolute mean velocity and volume flux density of solution; the instantaneous local volume flux density of solution due to mass flow; and the displacement history of colloidal particles initially at various positions in the column.