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Variation of the permeability tensor ellipsoid in homogeneous anisotropic soils
Author(s) -
Liakopoulos Aristides C.
Publication year - 1965
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/wr001i001p00135
Subject(s) - permeability (electromagnetism) , ellipsoid , hydraulic head , isotropy , permeameter , mathematics , anisotropy , mathematical analysis , geometry , principal value , tensor (intrinsic definition) , principal axis theorem , hydraulic conductivity , geology , geotechnical engineering , soil water , soil science , physics , chemistry , geodesy , membrane , biochemistry , quantum mechanics
The flow of water through a homogeneous anisotropic soil is governed by Darcy's equation in which the velocity is proportional to the imposed hydraulic gradient. The constant of proportionality (coefficient of permeability) is a symmetric tensor of second rank, reducing to a scalar only in the case of isotropic soils. It is shown that for determining the velocity vector from a known hydraulic gradient the permeability tensor ellipsoid should be constructed with semiaxes equal to the inverse of the square root of the permeability values. For determining the hydraulic gradient from a known velocity, the permeability tensor ellipsoid should have semiaxes equal to the square root of the principal permeability values. The fact that two different tensor ellipsoids can be constructed for a given anisotropic soil may produce confusion and give rise to a significant error. Special attention should therefore be given in evaluating the appropriate permeability value to be used in Darey's equation. It is recommended that the ellipsoid used always be the one with semiaxes equal to the inverse square root of the principal permeability values. Permeability tests were conducted on anisotropic sandstone. Cylindrical cores were taken at various directions with respect to a fixed coordinate system, and permeability values were determined in a constant‐head permeameter. The values obtained when plotted on polar coordinate paper result in a two‐dimensional permeability ellipse. The necessary graphical constructions for determining the direction of the velocity or the direction of the hydraulic gradient from the corresponding tensor ellipses are also given.