A molecular model is developed for the transport of a conservative solute at low concentrations in a homogeneous isotropic water-saturated porous medium. In this model, a solute molecule undergoes contact collisions with the solid grains in the medium at successive random times; in between these collisions, the velocity of the molecule is governed by the Langevin equation; the effect of the collisions with the solid grains is to scatter a molecule in a random direction. This molecular model is employed to derive rigorously a parabolic differential equation for the solute concentration at the macroscopic level. In the absence of solute convection, the coefficient of molecular diffusion in a porous medium is proved to be less than the coefficient of molecular diffusion in bulk solution, a finding which is in agreement with experimental observations. For non-zero convection, the assumption of isotropicity of the medium is employed to prove that the solute dispersion tensor is diagonal. For small magnitudes of the liquid velocity, the isotropicity assumption also implies that the coefficients of longitudinal and transverse dispersion are approximately parabolic functions of the liquid velocity. The expressions derived for the dispersion coefficients, when compared with their experimentally observed values, suggest that their dependence on the liquid velocity comes primarily through solute-liquid molecular collisions instead of through collisions of the solute molecules with the grains of the solid phase. The solute convective velocity is shown to be less than or equal to the liquid velocity. Precisely when the difference between the two velocities will be significant remains to be established.
ASJC Scopus subject areas
- Water Science and Technology