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

Consider the advection and dispersion of a conservative nonsorbing solute i n a spatially variable but statistically homogeneous velocity field. A Lagr angian approach leads to expressions for the large-time mean and variance o f concentration. The expressions require only the macroscopic velocity vect or U-m, the macrodispersion tensor D-m, and an additional tensor Theta. The macroscopic velocities and macrodispersivities are well known from numerou s previous studies, but Theta is introduced here for the first time. The te nsor Theta is needed to describe the kinetics of dilution of a plume: Two c ases with the same U-m and D-m, have different dilution characteristics dep ending on Theta. The characteristic times of dilution are given by tensor T heta D-m(-1). It is demonstrated, for the first time through a Lagrangian a pproach, that the coefficient of variation of concentration at the center o f the plume becomes proportionate to 1/t, as was previously shown in the Eu lerian theory of Kapoor ann Gelhar [1994a, b]. Expressions for the geometri c mean of the dilution index and the reactor ratio are derived. Numerical s imulations support the validity of the approach.