Large‐time behavior of concentration variance and dilution in heterogeneous formations

Abstract

Consider the advection and dispersion of a conservative nonsorbing solute in a spatially variable but statistically homogeneous velocity field. A Lagrangian approach leads to expressions for the large‐time mean and variance of concentration. The expressions require only the macroscopic velocity vector Um, the macrodispersion tensor Dm, and an additional tensor Θ. The macroscopic velocities and macrodispersivities are well known from numerous previous studies, but Θ is introduced here for the first time. The tensor Θ is needed to describe the kinetics of dilution of a plume: Two cases with the same Um andDm have different dilution characteristics depending on Θ. The characteristic times of dilution are given by tensor ΘD−1m. It is demonstrated, for the first time through a Lagrangian approach, that the coefficient of variation of concentration at the center of the plume becomes proportionate to 1/t, as was previously shown in the Eulerian theory ofKapoor and Gelhar [1994a, b]. Expressions for the geometric mean of the dilution index and the reactor ratio are derived. Numerical simulations support the validity of the approach.