Page 19 - Physical Principles of Sedimentary Basin Analysis

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Preliminaries

1.1 Notation
Most problems in this book are solved in 1D along the vertical axis. It is natural to let the
surface be at z = 0 and to have the z-axis pointing downwards, with positive z-coordinates
for the subsurface. An advantage with this choice is that the acceleration of gravity is
positive. A potential problem with the z-axis pointing downwards is that Fourier’s law
gives negative heat flow – heat that flows in the opposite direction to the positive z-axis.
A simple solution to this problem is to drop the minus-sign in Fourier’s law when the heat
flow is computed in practical problems. There is a similar problem with Darcy’s law, with
the same simple solution. But Fourier’s law and Darcy’s law retain their minus signs when
equations are derived. The full xyz-axis system is right-handed as shown in Figure 1.1b.
Vectors are written with lower case bold letters, as for instance, v, n or as
T
n = (n 1 ,... n 2 ), where T denotes the transpose. Matrices are written with upper case
bold letters, for instance like A and R. The matrix elements are A ij or R ij , where the
indices may be x, y and z for the respective spatial directions. Another example of a
matrix is
⎛ ⎞
k xx k xy k xz
K = ⎝ k yx k yy k yz ⎠ . (1.1)
k zx k zy k zz
Scalar products can be written in several different ways depending on what is most
convenient. Here are some examples:
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T
x · y = x y = x 1 y 1 + x 2 y 2 + x 3 y 3 = x i y i . (1.2)
i=1
The second example shows the scalar product as a matrix product, where the vectors are
written as row and column matrices. It is often convenient to write summations using what
is called Einstein’s summation convention, which says that summation is understood for
every pair of equal indices. Here is an example: the scalar product
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x · y = x i y i (1.3)
i=1

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