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The statistical mechanical theory of water transport through unsaturated soil: 2. Derivation of the Buckingham‐Darcy Flux Law
Author(s) -
Sposito Garrison
Publication year - 1978
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr014i003p00479
Subject(s) - hydraulic conductivity , buckingham , darcy's law , geotechnical engineering , mechanics , water flow , mathematics , statistical physics , physics , soil science , geology , soil water , porous medium , quantum mechanics , porosity
The Buckingham‐Darey flux law, an empirical equation that relates the volumetric flux density of water in an unsaturated soil to the gradient of the total potential, is derived from first principles by using the methods of statistical mechanics. The derivation given, a direct application of well‐known techniques in nonequilibrium statistical mechanics, proceeds through a detailed molecular description of two laboratory experiments for measuring the hydraulic conductivity tensor of a homogeneous unsaturated soil. In the first experiment the steady flow of water is induced by a gradient in the matric potential, while in the second, flow is induced by a gradient in the gravitational potential. In both cases the appropriate form of the Buckingham‐Darcy law is derived on the basis of a linear response approximation, and an expression for the hydraulic conductivity is given in terms of a time integral of the correlation function for the velocities of the water molecules in the soil. The problem of calculating the hydraulic conductivity of a soil thereby is reduced to quadrature and to the task of developing a molecular model of the velocity correlations among the water molecules. A recent successful model of this type is discussed briefly.