Page 170 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

160 7 Simulating Thermal and Chemical Effects of Intruded Magma Solidification Problems
1 4
m m
X = 0.5 + ln (X > 0.5) (7.10)
w 2667 mf w
6.25 − k w
T
mf
m
where X is the mole fraction of H 2 O in the NaAlSi 3 O 8 melt; T is in Kelvin; k w is
w
the equilibrium constant for H 2 O in melts of feldspar composition.
mf −8 2 −4
ln k = 5 + (ln P)(4.481 × 10 T − 1.51 × 10 T − 1.137)
w
−8
2
−5
−2
2
+ (ln P) (1.831 × 10 T − 4.882 × 10 T + 4.656 × 10 )
(7.11)
−3
−4
3
4
−3
+ 7.8 × 10 (ln P) − 5.012 × 10 (ln P) + 4.754 × 10 T
2
−6
− 1.621 × 10 T ,
where P is the pressure of the intruded magma; P and T are in bars and Kelvin,
respectively.
Using the concept of molar mass, the mass source of the volatile fluids released
during solidification from the intruded magma can be expressed as
m
X W w m
w
X W (1 − X )W
Q(x, y, t) = m m m m , (7.12)
w w w albite
+ Δt Mk
ρ m ρ m
w albite
where Δt Mk is the time period required to complete the magma solidification within
a given solidification thickness ΔL Mk ; W m and ρ m are the molar mass and den-
w w
sity of the volatile fluids in the magma; W m and ρ m are the molecular mass
albite albite
and density of the albite (NaAlSi 3 O 8 ) melt. It is noted that using the definition in
Eq. (7.12), the mass source of the released volatile fluids has units of the density of
the albite (NaAlSi 3 O 8 ) melt divided by time.
In the case of the intruded magma temperature being equal to the solidification
magma temperature, Eq. (7.8) can be rewritten as

∂x I ∂y
I
ρ M L n x + n y
∂t ∂t
f (x, y, t) = . (7.13)
ΔL Mk
7.2 Implementation of the Equivalent Source Algorithm
in the Finite Element Analysis with Fixed Meshes
If the physically equivalent heat source term, f (x, y, t), is determined either ana-
lytically or experimentally, Eqs. (7.6) and (7.7) can be directly solved using the
conventional finite element method (Zienkiewicz 1977). For dike-like and sill-like
intruded magmas, the physically equivalent heat source due to the solidification can