On the distribution of multicomponent mixtures over generalized exposure time in subsurface how and reactive transport: Batch and column applicationsinvolving residence-time distributions and non-Markovian reaction kinetics

Generalized differential equations that track the evolution of material den sities over space, time, and exposure time during reactive transport are sp ecified for simple cases involving linear reversible reactions between two states. Solutions are obtained for demonstration problems involving batch a nd column conditions. The exposure-time coordinate serves as a measure of r esidence time of materials that are "convected" (aged) along this dimension depending on the phase in which the material resides, and this exposure-ti me convection is used to determine the way in which material residence-time (to a particular phase) distributions evolve during reactive transport. Th e model is simplified to the form of a generalized batch reactor, and the s olution is developed by recognizing that this model is identical to the one -dimensional purely convective reactive transport model involving the same boundary conditions and reactions. This places the derived differential equ ation as the governing equation for the classical Giddings and Eyring [1955 ] solution for residence time distributions in the two-state Markov chain r epresentation of the batch. In the more general case where reaction rate va ries with memory of phase association, the present formulation may be viewe d as an extension of composite Markov process modeling to generally non-Mar kovian reactions. The model is specified for reactive transport in a porous medium in a one-dimensional column and applied to bacterial transport data from a published study where residence time to surfaces controlled the rea ction. The formulation and numerical solution are described, and the simula tions illustrate the evolution of material density over space, time, and ex posure time representing residence time sorbed.