Page 166 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

156 7 Simulating Thermal and Chemical Effects of Intruded Magma Solidification Problems

reviews the development of this subject (Carslaw and Jaeger 1959, Crank 1984,
Alexiades and Solomon 1993, Zhao and Heinrich 2002). The problem associated
with numerical modelling of the thermal effect of solidification of intruded magma
is that the characteristic dimension of the whole geological system under considera-
tion is on the scale of tens and hundreds of kilometers, but the characteristic dimen-
sion of the intruded magma, such as a sill or dike (Johnson and Pollard 1973, Lister
and Kerr 1991, Rubin 1995, Weinberg 1996) is on the scale of meters and tens of
meters. As a result, the detailed solidification process of the intruded magma might
not be important, but the thermal effect caused by the heat release during the solidifi-
cation of the intruded magma is important, at least from the ore body formation and
mineralization point of view. Since heat release during solidification of the intruded
magma can be represented by a physically equivalent heat source, it is possible to
transform the original heat transfer problem with phase change due to solidification
of the intruded magma into a physically equivalent heat transfer problem without
phase change but with the equivalent heat source. The above physical understand-
ing means that from the numerical modelling point of view, we can remove the
moving boundary problem associated with the detailed solidification process of the
intruded magma, so that we can use a fixed finite element mesh to consider the ther-
mal effect of the intruded magma by solving the physically equivalent heat transfer
problem with an equivalent heat source. This is the basic idea behind the proposed
equivalent algorithm for simulating the thermal effect of the intruded magma solid-
ification in this study. The similar, even though not identical, approaches, such as
the immersed boundary method (Beyer and LeVeque 1992), the level-set method
(Osher and Sethian 1998, Sethian 1999), the segment projection method (Tornberg
and Engquist 2003a) and so forth, have been used to tackle other different prob-
lems with moving interfaces or fronts (Smooke et al. 1999, Mazouchi and Homsy
2000). In terms of solving partial differential equations with delta function source
terms numerically, Walden (1999), and Tornberg and Engquist (2003b) have inves-
tigated the convergence and accuracy of the numerical method. Recently, the fixed
grid approach has been successfully used to solve the phase change problem associ-
ated with moisture transport in high-temperature concrete materials (Schrefler 2004,
Schrefler et al. 2002, Gawin et al. 2003). In this chapter, it will be extended to the
solution of the phase change problem associated with intruded magma solidification
in geological systems. The proposed algorithm is only valid when the characteristic
length scale of the system is much larger than that of the intruded magma, which is
true for most geological systems (Johnson and Pollard 1973, Lister and Kerr 1991,
Rubin 1995, Weinberg 1996).
In what follows, we will present the proposed equivalent algorithm for simulating
the chemical effect of solidification of the intruded magma in geological systems.
Since the volatile fluids released during solidification of the intruded magma may
have many different chemical components/species, it is ideal to describe the gov-
erning equation of each chemical component separately. However, if the diffusivity
of each chemical component is assumed to be identical, then the concept of the
total concentration, which is the summation of the concentration of all the chemical
components, can be used to describe the governing equation of the released volatile