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General Criteria for the Validity of the Buckingham-Darcy Flow Law¹

ABSTRACT

A detailed, first-principles study is undertaken on the exact equation of linear momentum balance for water in an unsaturated soil. First, it is shown that an approximate momentum balance equation, presented originally by Raats and Klute, can be used to demonstrate unequivocably that the flow of water through a rigid, homogeneous, isotropic, unsaturated soil will obey the Buckingham-Darcy law within 10^–12 to 10^–5 sec after a gradient in the total potential of soil water has been applied. Next, an exact equation of motion for the Fourier component of the water mass flux density vector is derived using standard methods in nonequilibrium statistical mechanics. This exact equation is employed to deduce the general physical criteria required for the Raats-Klute equation to be an accurate expression. The results of the investigation lead to the conclusion that the Raats-Klute equation will be accurate if: (i) the water mass density and mass flux density vector constitute a complete set of strongly coupled, macroscopic dynamical variables; and (ii) the time scale over which these two variables change significantly is much longer than that over which any other dynamical quantities vary. If these two conditions are met, then, according to statistical mechanics, the flow of water through unsaturated soil will be described accurately over all macroscopic time intervals by the Buckingham-Darcy law.

¹ Contribution from the Dep. of Soil and Environmental Sciences, University of California, Riverside, CA 92521.

² Professor of Soil Science.

Received for publication September 13, 1979. Accepted for publication May 7, 1980.