Diffusion of strongly sorbed solutes in soil: a dual-porosity model allowing for slow access to sorption sites and time-dependent sorption reactions

We use homogenization techniques to derive a dual (or double) porosity model of solute diffusion and reaction in soil, allowing for slow access to sorption sites within micro-aggregates and time-dependent sorption reactions. We give a means for determining the conditions in which micro-scale concentration gradients affect macro-scale gradients and fluxes. We present equations for a unit volume of soil represented as a series of uniformly-spaced, porous spherical particles, containing and surrounded by solution through which solutes diffuse. The methods we use can, in principle, be applied to more complex geometries. We compare the model's predictions with those of the equivalent single porosity model for commonly used boundary conditions. We show that failure to allow for slow access to reaction sites can lead to seriously erroneous results. Slow access has the effect of decreasing the sorption of solute into soil from a source or desorption from soil to a sink. As a result of slow access, the diffusion coefficients of strongly-sorbed solutes measured at the macro-scale will be time-dependent and will depend on the method of measurement. We also show that slow access is more often likely to limit macro-scale diffusion than rates of slow chemical reactions per se. In principle, the unimportance of slow reactions except at periods longer than several weeks of diffusion simplifies modelling because, if slow access is correctly allowed for, sorption can be described with equilibrium relations with an understanding of speciation and rapid sorption-desorption reactions.