Page 167 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

7.1 An Equivalent Source Algorithm for Simulating Thermal and Chemical Effects 157

fluids. Thus, we can use one governing equation to describe variations of the total
concentration of the released volatile fluids in the system. In this way, computer
efforts can be reduced significantly in dealing with the intruded magma solidifi-
cation problems. Because the volumetric amount of the released volatile fluids is
relatively small to the intruded magma, their effects on the heat transfer process
can be neglected in the analysis of the intruded magma solidification problem. This
allows us to assume that thermal equilibrium between the released volatile fluids
and the intruded magma/rock has been achieved. Thus, the governing equations of
the original heat transfer and mass transport problem considering the phase change
during the intruded magma solidification (Carslaw and Jaeger 1959, Crank 1984,
Alexiades and Solomon 1993, Zhao and Heinrich 2002) can be described, for a
two-dimensional problem, as
2 2

∂T R ∂ T R ∂ T R
(ρ R c pR ) = λ R + (x, y) ∈ V R , (7.1)
∂t ∂x 2 ∂y 2
2 2
∂T M ∂ T M ∂ T M
(ρ M c pM ) = λ M + (x, y) ∈ V M , (7.2)
∂t ∂x 2 ∂y 2
2 2

∂C T ∂ C T ∂ C T
= D + + δ(x − x I , y − y I )Q(x, y, t) (x, y) ∈ V R + V M ,
∂t ∂x 2 ∂y 2
(7.3)
where T R and T M are the temperature of the rock and intruded magma; ρ R , c pR and
λ R are the density, specific heat and thermal conductivity of the rock; ρ M , c pM and
λ M are the density, specific heat and thermal conductivity of the intruded magma;
C T is the total concentration of the released volatile fluids; D is the diffusivity of the
released volatile fluids; Q is the mass source of the released volatile fluids during
the intruded magma solidification; x I and y I are the x and y coordinate components
of the interface position; δ is the delta function of values of unity and zero; V R and
V M are the spaces occupied by the rock and intruded magma.
It is noted that, although the released volatile fluids during the intruded magma
solidification are comprised of H 2 O, CO 2 , H 2 S, HCl, HF, SO 2 and other substances
(Burnham 1979, Barns 1997), H 2 O is the most abundant magmatic volatile and CO 2
is the second most abundant magmatic volatile in the intruded magma. For this rea-
son, the solubility of H 2 O in silicate melts has been investigated for many years.
These extensive studies (Burnham 1979, Barns 1997) have demonstrated that if the
constraints of the solution model for the NaAlSi 3 O 8 −H 2 O system are imposed, H 2 O
solubilities in the igneous-rock melts are essentially identical to those in NaAlSi 3 O 8
melts. Therefore, the solubility of H 2 O in the NaAlSi 3 O 8 melt can be used to approx-
imately determine the mass source of the released volatile fluids during the intruded
magma solidification.
Since the original magma solidification problem belongs to a moving inter-
face problem, the temperature and heat flux continuity conditions on the interface
between the rock and intruded magma are as follows: