A SOLUTE FLUX APPROACH TO TRANSPORT IN HETEROGENEOUS FORMATIONS .1. THE GENERAL FRAMEWORK

It is common to represent solute transport in heterogeneous formations in terms of the resident concentration C(x, t), regarded as a random space function. The present study investigates the alternative representation by q, the solute mass flux at a point of a control plane normal to the mean flow. This representation is appropriate for many field applications in which the variable of interest is the mass of solute discharged through a control surface. A general framework to compute the statistical moments of q and of the associated total solute discharge Q and mass M is established. With x the direction of the mean flow, a solute particle is crossing the control plane at y = eta, z = zeta and at the travel (arrival) time-tau. The associated expected solute flux value is proportional to the joint probability density function (pdf) g1 of eta, zeta, and tau, whereas the variance of q is shown to depend on the joint pdf g2 of the same variables for two particles. In turn, the statistical moments of eta, zeta, and tau-depend on those of the velocity components through a system of stochastic ordinary differential equations. For a steady velocity field and neglecting the effect of pore-scale dispersion, a major simplification of the problem results in the independence of the random variables-eta, zeta, and tau. As a consequence, the pdf of eta and zeta can be derived independently of tau. A few approximate approaches to derive the statistical moments of eta, zeta, and tau are outlined. These methods will be explored in paper 2 in order to effectively derive the variances of the total solute discharge and mass, while paper 3 will deal with the nonlinear effect of the velocity variance upon the moments of eta, zeta, and tau.