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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**,

**, thermal conductivity, diffusion and dispersion coefficients**

*k*– 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 (*x*1, *x*2, *x*3) and (*x*1', *x*2', *x*3') be orthonormal basis sets, i.e., . A vector **X** can

be expressed in terms of either basis, as:

**X** = >1*x*1 + >2*x*2 + >3*x*3
**X** = >1'*x*1' + >2'*x*2' + >3'*x*3'

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 **V**1=a1**i**+b1**j**+c1**k**, **V**2=a2**i**+b2**j**+c2**k** as two vectors

*Scalar, dot, or inner product
*

**V**[email protected]**V**2 2: the angle between **V**1, **V**2

The following rules apply

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*Vector, or cross product
*

**V**1×**V**2 **1
**

**1**: unit vectorz**V**1, **V**2 plane (right-handed screw)

The following rules apply

**V**1×**V**2 !**V**2×**V**1

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

where 2** ij** is the angle between the axis and the >

**axis, leading to**

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

where the direction cosines **cos**2** ij** are replaced by

*aij*– It can be proved that

– Generally, an *n*th 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 *n*th 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 *n*th and *m*th order tensor is an (*n*+*m*)th
order tensor

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

**3. Hydraulic Conductivity/Permeability Tensors
**

– 2nd 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