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
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
Vector, or cross product
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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