A theoretical explanation of solute dispersion in saturated porous media at the Darcy Scale

Abstract

The transport of a nonreactive dilute solute in saturated porous media is explained at three distinct space‐time scales. These are the kinetic, microscopic, and Darcy scales. The transition from one scale to the next higher scale, i.e., from the kinetic to the microscopic to the Darcy, is a consequence of the central limit theorem of probability theory. At the microscopic scale, the solid and the liquid phases together form a heterogeneous continuum. The microscopic solute concentration is governed by a parabolic equation with spatially varying drift and diffusion coefficients. The so‐called dispersion phenomenon at the Darcy scale is shown to appear in the transition from the microscopic to the Darcy scale. In the computation of the dispersion coefficients, the Peclet number appears naturally as a dimensionless parameter. For large Peclet numbers the coefficients of dispersion at the Darcy scale are shown to be linear in the liquid convective velocity. A general expression is obtained which gives the order of magnitude of the dispersion coefficient for all Peclet numbers within the Darcy regime of the liquid convective velocities. This expression shows that for very small Peclet numbers, only the molecular diffusion provides the dominant contribution, whereas for intermediate values of Peclet numbers, both the liquid convection and the molecular diffusion contribute to the dispersion coefficients; in this range the dispersion coefficients are not linear in the liquid convective velocity. These findings are well supported by existing experimental observations. These theoretical insights into dispersion at the Darcy scale are also important in explaining the so‐called macrodispersion and the scale effect at field scales.