Hydraulic Conductivity - Groundwater Flow and Contaminant Transport - Lecture Handout, Exercises for Groundwater Flow and Contaminant Transport

Groundwater Flow and Contaminant Transport

Description: A complete set of lecture sires for course Groundwater Flow and Contaminant Transport is available at docsity. This lecture includes: Hydraulic Conductivity, Permeability as Tensors, Intuitive Thinking, Formal Definition, Scalar, Dot, or Inner Product, Vector, or Cross Product, Transformation of the Components, Cartesian Tensor, Hydraulic Conductivity, Permeability Tensors
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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
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