Page 169 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

7.1 An Equivalent Source Algorithm for Simulating Thermal and Chemical Effects 159

a fixed finite element mesh to consider the released heat and volatile fluids during
the magma solidification process. The key issue associated with the finite element
analysis using a fixed mesh is that, for a given magma solidification thickness ΔL M ,
which can be easily simulated by the fixed mesh, we need to determine the time
period Δt M so that the interface between the magma and rock just moves a distance
being equal to ΔL M in the interface movement direction. Since the magma is fully
solidified within the region of the solidification thickness ΔL M , the delta function
used in Eq. (7.3) should have a value of unity. Except in this solidification region,
the delta function should have a value of zero, meaning that there is no magma
solidification taking place in other regions. This is the reason why the delta function
only has values of unity and zero in the present numerical algorithm. Generally, for
the same magma solidification thickness ΔL M , the solidification time period can
vary significantly as the magma solidification proceeds. This requires us to develop
a numerical algorithm, in which the finite element mesh is fixed but the integration
time-step being equivalent to the solidification time period must be variable.
Based on the above considerations, Eqs. (7.1), (7.2), (7.3), (7.4) and (7.5) can be
represented by the following equations:

2 2
∂T R ∂ T R ∂ T R
(ρ R c pR ) = λ R + +δ(x−x I , y−y I ) f (x, y, t) (x, y) ∈ V R −V M ,
∂t ∂x 2 ∂y 2
(7.6)
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
(7.7)
where δ is the delta function with values equal to unity and zero; f (x, y, t)isthe
physically equivalent heat source due to the solidification of the intruded magma.

∂x I ∂y I
) *
ρ M L + c pM (T IM − T m ) n x + n y
∂t ∂t
f (x, y, t) = , (7.8)
ΔL Mk
where ΔL Mk is the solidification thickness of the intruded magma during a time
period Δt Mk ; k is the time-step index of integration in the finite element analysis.
The mass source of the released volatile fluids in Eq. (7.7) during the intruded
magma solidification can be determined using the solubility of H 2 O in the
NaAlSi 3 O 8 melt (Burnham 1979, Barns 1997). In this regard, the mole fraction of
H 2 O in the NaAlSi 3 O 8 melt is expressed as follows:

1
m m
w
w
X = % (X ≤ 0.5), (7.9)
mf
k w