Time-dependent transport in heterogeneous formations of bimodal structures: 2. Results

Abstract

The theoretical results of part 1 [Dagan and Fiori, 2003] for modeling time-dependent, advective transport of a conservative solute in porous formations of bimodal structure are applied to illustrate the behavior of a few trajectory statistical moments as function of time, of the permeability contrast κ, and of the inclusions volume density n. The computations are carried out for circular (2D) and spherical (3D) inclusions to represent isotropic media. Advective transport is solved by studying the distortion of a thin plume, linear (2D) or planar (3D), normal to the mean velocity U and moving through a single inclusion. The deformation of the plume is determined from the residual trajectories of solute particles that are derived numerically by a quadrature. The longitudinal macrodispersivity is defined by α L(t; n, κ) = (2U)-1dX11/dt, where X 11 is the trajectories second moment in the mean flow direction. The general behavior of the time-dependent longitudinal dispersivity α L(t; n, κ) and, in particular, its constant, large time limit are examined. The tendency of αL to the "Fickian" limit with time depends strongly on the conductivity contrast; in particular, for low permeable inclusions (κ ≪ 1) it may be extremely slow. It is shown that the first-order approximation in the conductivity contrast κ is of limited validity (0.3 < κ < 2). The transverse moment X 22 tends asymptotically to a constant value. The analysis of the trajectory high order moments shows that the probability density function (pdf) of the solute trajectories tends to normality at large time. Similar to the "Fickian" limit the normal distribution may be reached at very large time in presence of low conductivity inclusions, with the pdf characterized by significant tailing for the trailing part of the pdf.