Page 173 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

7.3 Verification and Application of the Equivalent Source Algorithm 163

Due to the geometrical symmetry of the ideal experiment, analytical solutions
for the temperature distribution during the solidification of the intruded magma can
be expressed as follows.

√
L M
T R = T m 0 ≤ x ≤ − 2β αt, t ≤ t MMax , (7.21)
2
√
2x + L M − 4β αt

(T m − T R0 )erfc √
4 αt
T R = T R0 +
1 + erf(β) (7.22)

L M √
− 2β αt ≤ x ≤ L R , t ≤ t MMax ,
2
) *
ρ M L M c pM (T m − T R0 ) + L x 2
T R = T R0 + √ e − 4αt ( t > t MMax ), (7.23)
2ρ R c pR απt
where erfc(x) = 1 − erf(x) is the complementary error function of variable x;
2
t MMax = L 2 " (16αβ ) is the maximum time instant to complete the solidification
M
of the intruded magma.
In summary, the proposed equivalent algorithm for simulating the chemical effect
of the intruded magma solidification includes the following five main steps: (1) For
the given values of the initial temperature of the rock, T R0 , and the solidification
temperature of the intruded magma, T m , Eq. (7.17) is solved to determine the value
of β; (2) Substituting the value of β into Eq. (7.20) yields the value of the phys-
ically equivalent heat source due to the solidification of the dike-like and sill-like
intruded magma; (3) Eqs. (7.12) and (7.15) are used to calculate the mass source of
the released volatile fluids during the intruded magma solidification; (4) The inte-
gration time step is determined using Eqs. (7.15) and (7.16); (5) The conventional
finite element method is used to solve Eqs. (7.6) and (7.7) for the temperature and
concentration distribution in the whole domain during and after solidification of the
intruded magma. Note that both the mass source and the physically equivalent heat
source are only applied to some elements in the finite element analysis during the
solidification process of the intruded magma. The automatic transition from solid-
ification to post-solidification of the intruded magma is another advantage in using
the proposed equivalent source algorithm for simulating the chemical effect of the
intruded magma solidification in the conventional finite element analysis.

7.3 Verification and Application of the Equivalent
Source Algorithm

Since the key part of the proposed algorithm for simulating the chemical effect of the
intruded magma during solidification is to transform the original moving interface
(i.e. the solidification interface between the rock and intruded magma) problem into