Page 85 - Physical Principles of Sedimentary Basin Analysis

3.17 An important Lagrange coordinate 67

(Notice that the summation convention tells us that there is a summation over all pairs of
the same index, in this case s.) We therefore have

∂b i ∂v i ∂x s ∂v i ∂x s ∂v i ∂x s ∂v i
= , , = b s . (3.150)
∂t ∂x s ∂a 1 ∂x s ∂a 2 ∂x s ∂a 3 ∂x s
Properties (1) and (2) from the definition of a determinant give that
∂ J ∂v 1 ∂v 2 ∂v 3
= d(b s , b 2 , b 3 ) + d(b 1 , b s , b 3 ) + d(b 1 , b 2 , b s ) = (∇· v) J (3.151)
∂t ∂x s ∂x s ∂x s
which is what we wanted to show.
Exercise 3.21 Show that the acceleration in Eulerian coordinates is Dv/dt.

3.17 An important Lagrange coordinate

We will in later chapters deal with subsiding sedimentary basins where the sediments
compact and loose porosity during deposition and burial. Either an Euler or a Lagrange
coordinate system can be used to describe the compaction of a sedimentary basin. The
Euler coordinate system has the advantage that it can in principle be any coordinate sys-
tem, but the disadvantage is that the material (time) derivative makes the equations more
difficult to solve. A Lagrange description avoids the material derivative, but it might be
difficult to find a convenient Lagrangian coordinate system. Fortunately, it turns out that
the net (porosity-free) amount of sediments, when it is measured as the height above the
base of the basin, is a convenient choice for a vertical Lagrangian coordinate. We will use
ζ as notation for this Lagrangian coordinate, and the ζ-position of a sedimentary “grain”
is constant, given that the movement of the sediments is restricted to the vertical direction.
The relationship between the ζ and the real z-coordinate is given by

dζ = (1 − φ)dz (3.152)

which clearly shows that dζ is the net volume of solid in the interval dz. The net volume of
solid is a constant given that the density of the solid is constant, an assumption that is nor-
mally justified. The ζ-coordinate is measured from the base of the basin (or a sedimentary
layer) while the z-coordinate is measured from the basin or water surface with the z-axis
pointing downwards. The z-coordinate for a given ζ-coordinate is therefore

ζ ∗
dζ
z = (3.153)
ζ 1 − φ(ζ)
where the porosity φ is a function of ζ, and the ζ-coordinate of the basin surface is
∗
denoted ζ . Figure 3.17 gives an example of what a 2D basin looks like in a xz-coordinate
system and in the corresponding xζ-coordinate system. The Lagrangian ζ-coordinates
in the next chapters are used in several applications that involve sediment compaction
(decreasing porosity with depth).