Reactive transport and immiscible flow in geological media. I. General theory

Steady flow of incompressible fluids takes place in geological formations of spatially variable permeability. The permeability is regarded as a stationary random space function (RSF) of given statistical moments. The fluid carries reactive solutes and we consider, for illustration purposes, two types of reactions: nonlinear equilibrium sorption of a single species and mineral dissolution (linear kinetics). In addition, we analyse the nonlinear problem of horizontal flow of two immiscible fluids (the Buckley-Leverett flow). We consider injection at constant concentration in a semi-infinite domain at constant initial concentration and we neglect the effect of pore scale dispersion. The field-scale transport problem consists of characterizing an erratic plume, or displacement front, emanating from a given source area along distinct random flow paths. Reactive transport along three-dimensional flow paths is transformed to a one-dimensional Lagrangian-Eulerian domain ($\tau $, t), where $\tau $ is the fluid residence time and t is the real time. Due to nonlinearity, discontinuities (shock waves) along a flow path may develop. Close form solutions are obtained for the expected values of the spatial and temporal moments of a nonlinearly reacting solute plume, or of two immiscible fluids. These results generalize the previous results for linearly reacting solute (Cvetkovic & Dagan 1994). The general results are illustrated and discussed in part II.