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Transformation of the tensor form of Darcy's law in inhomogeneous and anisotropic soils
Author(s) -
Case C. M.,
Cochran G. F.
Publication year - 1972
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/wr008i003p00728
Subject(s) - ellipse , anisotropy , tensor (intrinsic definition) , cartesian coordinate system , permeability (electromagnetism) , mathematical analysis , connection (principal bundle) , diagonal , transformation (genetics) , coordinate system , mathematics , hydraulic head , geometry , darcy's law , ellipsoid , classical mechanics , physics , geology , geotechnical engineering , porous medium , geodesy , biochemistry , chemistry , quantum mechanics , membrane , biology , porosity , gene , genetics
The transformation of the tensor form of Darcy's law from an arbitrary cartesian coordinate system to one in which the permeability tensor is diagonal is discussed. Matrix equations are developed that allow this transformation to be conveniently performed. This transformation leads in a natural way to a discussion of the ellipses of permeability, which have been used in attempts to find geometrically the direction of the gradient of head given the velocity, and vice versa, in anisotropic media. Equations are derived that allow the magnitude, as well as the direction, of the gradient of head or velocity, as the case may be, to be found from the geometrical constructions; thus the utility of the equations is extended. Finally, the connection between the permeability ellipse and the ellipse of direction, which does not seem to have been previously noted in the literature, is demonstrated, and the implications of the connection are briefly discussed.