Prediction by the method of moments of transport in a heterogeneous formation

Abstract

Natural geologic formations are highly heterogeneous. It is impossible and often unnecessary to describe in deterministic terms the spatial variability of their properties. However, the hydrogeologic parameters may be represented in probabilistic terms. Prediction of solute transport may then be defined as the derivation of the probabilistic properties of concentration. This work deals with the first two integral (or spatial) moments of solute concentration in a heterogeneous formation of infinite extent. The first moment is the vector of the mean position of the centroid of the plume and, in a generalized sense, represents advection. The second moment is the matrix of dyadics of the mean squared displacement about the average position of the centroid of the plume and, again in a generalized sense, represents dispersion. Assuming that the mixing at the laboratory scale is Fickian, with random but time-invariant velocities and lab-scale dispersion matrices, the differential equations satisfied by the first two moments are derived. An analytical first-order (or small perturbation) solution is obtained for stationary velocity, and compared with a numerical solution.