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Seepage flow through anisotropic permeable soil in three
dimensions
– Tensor of coefficients of permeability, principal value and
direction of the tensor –
Tsutomu Tanaka
Arnold Verruijt
Abstract
Soil generally has an anisotropic permeability with respect to seepage flow. This anisotropic
permeability has marked effects on the characteristics of seepage flow through soil. The mechanism of
seepage flow through anisotropic porous media should be clarified to accurately estimate the safety against
seepage failure when designing hydraulic structures. In this paper, the mechanism of water flow through a
pipe and channel (or crack) in 3D soil, the coefficients of permeability as a symmetric tensor, permeability
tensor at a point, principal value and direction of permeability tensor, invariants of permeability tensor,
and relationship between structural permeabilities and mathematically equivalent permeability tensor are
discussed. The following topics are also considered supplementarily:
(1) Methods of three multiplications of vectors, which are useful to formulize equations, solve examples,
etc.
(2) Questions and answers with respect to the anisotropic permeability tensor, which are based on
developing the seepage-flow theory.
(3) Rotational-coordinates transformation matrix, which is useful to understand the concept of the
coordinate-conversion system.
Key words: Three-dimensional soil with anisotropic permeability, seepage flow through soil, tensor of
coefficient of permeability (or permeability tensor), principal value and direction of permeability tensor,
invariant of permeability tensor, structural permeabilities and mathematically equivalent permeability
tensor
©2023 Research Center for Urban Safety and Security, Kobe University, All rights reserved.
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