A derivation of the macroscopic solute transport equation for homogeneous, saturated, porous media

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Abstract
The macroscopic transport equation for a conservative solute in a homogeneous, water‐saturated porous medium is derived on the basis of a rigorous cumulant expansion applied to the equation of mass balance. The essential physical concept underlying the derivation is that of a local volume‐averaged solute velocity which fluctuates on a time scale that is orders of magnitude smaller than its autocorrelation time scale, which, in turn, is much smaller than the time scale of interest in a typical solute transport experiment. This clear separation of time scales is illustrated with representative data on solute transport in homogeneous, water‐saturated soils and is employed to justify the truncation of an exact cumulant expansion of the divergence of the volume‐averaged solute mass flux density. With the cumulant expansion terminated at first order in the ratio of the solute velocity autocorrelation time to the macroscopic solute transport time interval, an expression for the macroscopic solute mass flux density is produced which is the same as Fick's Law extended to porous media.