Page 168 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

158 7 Simulating Thermal and Chemical Effects of Intruded Magma Solidification Problems

(x I , y I ) ∈ Γ RM , (7.4)
T R = T M

∂T R ∂T R ∂T M ∂T M
λ R n x + n y − λ M n x + n y
∂x ∂y ∂x ∂y
(7.5)

∂x I ∂y I
= ρ M [L + c pM (T IM − T m )] n x + n y (x I , y I ) ∈ Γ RM ,
∂t ∂t
where x I and y I are the x and y coordinate components of the interface position;
n x and n y are the x and y components of the unit normal to the interface between
the rock and intruded magma; T IM is the temperature of the intruded magma; T m
is the solidification temperature of the intruded magma; L is the latent heat of
fusion of the intruded magma; Γ RM is the interface between the rock and intruded
magma.
Except for the boundary conditions on the interface between the rock and
intruded magma, the boundary conditions on the other boundaries of the rock
domain and intruded magma domain can be either of the Dirichlet, Neumann or
mixed type (Carslaw and Jaeger 1959, Crank 1984, Alexiades and Solomon 1993,
Zhao and Heinrich 2002). Since the boundary conditions on the other boundaries of
the rock domain and intruded magma domain are trivial, it is not necessary to repeat
them here.
In order to develop an efficient and effective numerical algorithm for dealing with
solidification of intruded magma in sills and dikes, it is necessary to introduce some
related concepts. As shown in Fig. 7.1, supposing the initial position of the interface
between the magma and rock is at position 1, the interface moves to position 2
due to magma solidification during a time period, Δt M . As a result, the magma
solidification thickness is expressed by ΔL M . If the length of the interface front is
ΔL R , then the magma solidification area is the product of ΔL M and ΔL R during
the time period Δt M . Since heat and volatile fluids are only released during magma
solidification, we may consider the released heat and volatile fluids either as surface
heat and mass sources in a two-dimensional problem or as volumetric heat and mass
sources in a three-dimensional one. This means that it is physically possible to use

Position 1 Position 2
Magma
Rock Δ L
R
Interface movement
direction

Δ L
M

Fig. 7.1 Basic concepts related to the magma solidification problem