The theory of absorption in aggregated media

Abstract

The question of the applicability of the 'diffusion analysis' to transient flows in unsaturated aggregated media has been, till now, an open one. It has been uncertain whether significant deviations from the predictions of the diffusion analysis occur as a consequence of local disequilibrium. The present paper develops the theory of absorption in aggregated media and so provides a means of studying this question. An aggregated medium is regarded as one made up of macroporosity, through which flow on the Darcy scale occurs, and microporosity, which is free to exchange water with the macroporosity. Absorption in such a medium is analysed with the aid of two extreme models of macropore flow, one a wet-front model and the other a linearized model (the mathematical details of which are developed elsewhere). The wet-front analysis leads to an integral equation that is solved by means of the Faltung theorem of the Laplace transformation. It is found that there is little difference between the solutions for the two models. The wet-front model is chosen for further study since it appears to be more realistic and is more versatile. Various models of absorption into the microporosity are studied, and are found to lead to essentially similar results. All models (and general physical considerations) indicate that the aggregated medium behaves initially as a classical medium with sorptivity equal to that of the macroporosity S; and that the apparent sorptivity increases with time, ultimately approaching a limiting value (> S). The cumulative uptake into the microporosity appears to be proportional to time initially, but is finally proportional to (time)1/2. The characteristic time of absorption into the microporosity tends to be small for most media of interest. Ln such cases, deviations from the diffusion analysis due to aggregation occur only during a short initial period. These considerations hold also for other transient phenomena such as infiltration and capillary rise in various geometries. It is concluded that the diffusion analysis may usually be applied to aggregated media. The present work provides means of analysing the exceptional cases where this is not so, such as media containing large aggregates of low sorptivity.