Page 171 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

7.2 Implementation of the Equivalent Source Algorithm 161

y L R

x
0
L M
L R L R

Fig. 7.2 A representative model for the magma solidification problem

be determined analytically through the following ideal experiment. As shown in
Fig. 7.2, the sample in the ideal experiment is a rectangular domain filled with the
intruding magma in the middle and the intruded rocks on the either side. All the
external boundaries of the sample are insulated and the temperatures of the rock and
intruded magma are T R and T m , respectively. The characteristic length of the rock
"
is L R , while the characteristic length of the intruded magma is L M 2. It is assumed
that the characteristic length of the rock is much larger than that of the intruded
magma, which is true for sills and dikes in geology (Johnson and Pollard, 1973,
Lister and Kerr 1991, Rubin 1995, Weinberg 1996). Under this assumption, the
solidification problem in the ideal experiment can be treated as a one-dimensional
Stefan solidification problem, for which the analytical solution is already available
(Carslaw and Jaeger 1959).
If the characteristic length of the intruded magma is divided into K equal parts,
which are modelled by K finite element meshes of equal length in the magma solid-
ification direction, then the length of the finite element mesh in this direction is
"
Δx = L M (2K). In order to use the fixed finite element mesh, it is useful to keep
Δx constant. For this purpose, we define that the solidification thickness of the
intruded magma during a time period Δt Mk is equal to the length of the finite ele-
ment mesh in the magma solidification direction and, therefore, ΔL Mk = Δx =
"
L M (2K). This implies that for this ideal experiment, the scalar product of (n x , n y )
" "
and (∂x I ∂t, ∂y I ∂t) can be expressed as follows:
Δx
∂x I ∂y I ∂x I ΔL Mk
n x + n y = n x ≈ = , (7.14)
∂t ∂t ∂t Δt Mk Δt Mk
where Δt Mk is the time period when the solidification boundary moves from one
side of the kth finite element in the magma solidification direction at time t = t Mk−1
to another side of the kth finite element in the magma solidification direction at time
t = t Mk .

(k = 1, 2, 3,. . . , K). (7.15)
Δt Mk = t Mk − t Mk−1