Page 172 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

162 7 Simulating Thermal and Chemical Effects of Intruded Magma Solidification Problems

The time, at which the intruded magma solidification boundary reaches both
sides of the kth finite element boundary in the magma solidification direction, can
be determined from the analytical solution (Carslaw and Jaeger 1959).

2
2
k (Δx) 2 (k − 1) (Δx) 2
t Mk = , t Mk−1 = (k = 1, 2, 3, ..., K),
4αβ 2 4αβ 2
(7.16)
"
where α = λ R (ρ R c pR ) is the thermal diffusivity of the rock; β can be determined
from the following transcendental equation:
√
e −β 2 L π
= , (7.17)
β[1 + erf(β)] c pR (T m − T R0 )
where T R0 is the initial temperature of the rock; erf(β) is the error function of vari-
able β.

2 β −r 2
erf(β) = √ e dr. (7.18)
π 0
Substituting Eqs. (7.14) into Eq. (7.13) yields the following equation:

ρ M L
f (x, y, t Mk ) = (k = 1, 2, 3,..., K). (7.19)
Δt Mk

Finally, substituting Eq. (7.16) into Eq. (7.19) yields the following equation:

4ρ M Lαβ 2
f (x, y, t Mk ) = 2 (k = 1, 2, 3,. . . , K), (7.20)
(2 k − 1)(Δx)
"
where Δx = ΔL Mk = L M (2K) and ΔL Mk is the constant solidification thickness
of the intruded magma during the variable time period Δt Mk .
It needs to be pointed out that since the magma solidification boundary passes
only through the kth finite element from the initial intruded interface between the
rock and magma during the time period Δt Mk , both the mass source of the released
volatile fluids and the physically equivalent heat source expressed in Eqs. (7.12) and
(7.20) only need to be applied in the kth finite element, which is numbered from the
initial intruded interface between the rock and magma, in the finite element analysis.
This means that if a fixed finite element mesh is employed, the variable time step
expressed by Eqs. (7.15) and (7.16) needs to be used during the magma solidification
period in the finite element analysis. From the numerical analysis point of view, a
change in the time step is much easier to implement than a change in the finite
element mesh in the finite element analysis. This is the main advantage in using the
proposed equivalent source algorithm to simulate the chemical effect of the intruded
magma solidification in geological systems.