Stochastic-perturbation analysis of a one-dimensional dispersion-reaction equation: Effects of spatially-varying reaction rates

We carry out a stochastic-perturbation analysis of a one-dimensional convection-dispersion-reaction equation for reversible first-order reactions. The Damkohler number, Da, is distributed randomly from a distribution that has an exponentially decaying correlation function, controlled by a correlation length, xi. Zeroth- and first-order approximations of the dispersion coefficient, D, are computed from moments of the residence-time distribution obtained by solving a one-dimensional network model, in which each unit of the network represents a Darcy-level transport unit, and the solution of the transfer function in zeroth- and first-order approximations of the transport equation. In the zeroth-order approximation, the dispersion coefficient is calculated using the convection-dispersion-reaction equation with constant parameters, that is, perturbation corrections to the local equation are ignored. This zeroth-order dispersion coefficient is a linear function of the variance of the Damkohler number, [(Delta Da)(2)]. A similar result was reported in a two-dimensional network simulation. The zeroth-order approximation does not give accurate predictions of mixing or spreading of a plume when Damkohler numbers, Da much less than 1, and its variance, [(Delta Da)(2)] > 0.25[Da(2)]. On the other hand, the first-order theory leads to a dispersion coefficient that is independent of the reaction parameters and to equations that do accurately predict mixing and spreading for Damkohler numbers and variances in the range root[(Delta Da(2))]/[Da] less than or equal to 0.3.