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

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 U-m, the macrodispersion tensor D-m, and an additional tensor Theta. The macroscopic velocities and macrodispersivities are well known from numerous previous studies, but Theta is introduced here for the first time. The tensor Theta is needed to describe the kinetics of dilution of a plume: Two cases with the same U-m and D-m, have different dilution characteristics depending on Theta. The characteristic times of dilution are given by tensor Theta D-m(-1). 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 of Kapoor ann 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.