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Hydraulic Conductivity and Permeability as Tensors )

1. Intuitive Thinking

– Scalar: a quantity that has only a magnitude and no direction associated with it, e.g., hydraulic

head, temperature, contaminant concentration

– Vector: a quantity that has both a magnitude and a direction, e.g., hydraulic gradient and flow

velocity

– Tensor: a quantity whose intrinsic properties are invariant under coordinate transformations, e.g.,

K, k, thermal conductivity, diffusion and dispersion coefficients

– Intrinsic properties: (i) length and (ii) orientation relative to some absolute coordinate system

– Discussion restricted to Cartesian tensors

2. Formal Definition

– Scalar—a zero-order tensor. The magnitude of a scalar h(>1, >2, >3, t) is not altered by the change

of translation/rotation of coordinate system

– Vector—a first-order tensor. Similarly, the length of a vector does not change with the

transformation of coordinate system

Let (x1, x2, x3) and (x1', x2', x3') be orthonormal basis sets, i.e., . A vector X can

be expressed in terms of either basis, as:

X = >1x1 + >2x2 + >3x3

X = >1'x1' + >2'x2' + >3'x3'

where (>1, >2, >3) and (>1', >2', >3') are referred to as components/coordinates of the vector X with

respect to the two bases

Vector Algebra

œ V1=a1i+b1j+c1k, V2=a2i+b2j+c2k as two vectors

Scalar, dot, or inner product

V1@V2 2: the angle between V1, V2

The following rules apply

V1@V2 V2@V1

2

Vector, or cross product

V1×V21

1: unit vectorzV1, V2 plane (right-handed screw)

The following rules apply

V1×V2 !V2×V1

Transformation of the components of X when the reference basis is changed can be written as

where 2ij is the angle between the axis and the >j axis, leading to

(i = 1, 2, 3)

where the direction cosines cos2ij are replaced by aij

– It can be proved that

– Generally, an nth order Cartesian tensor with components in the coordinate system >1,

>2, >3 has components in the coordinate system , , , defined as

(i=1,2,3; j=1,2,3;...; m=1,2,3)

where ain, ajo,..., ams are the direction cosines.

– This statement serves as tensor’s definition

–An nth order tensor has thus 3n components. Yet NOT any 3n scalars qualify to define a tensor

– A scalar, a vector, and a 2nd order tensor have 1, 3, and 9 components, respectively

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– Dyadic multiplication: the dyadic/cross product of an nth and mth order tensor is an (n+m)th

order tensor

– Single contraction: the dot product of an nth and mth order tensor is an (n+m!2)th order tensor

3. Hydraulic Conductivity/Permeability Tensors

–2

nd order, 9 components, defined as

(i=1,2,3; j=1,2,3)

Written in matric form (k may replace K)

or or

4. Darcy’s Law for Anisotropic Material

– When the coordinate axes are not oriented parallel to the principal axes of an anisotropic porous

medium, Darcy’s law is written as

– Here the K tensor is assumed to be symmetric