Page 175 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

7.3 Verification and Application of the Equivalent Source Algorithm 165

modeled using 240 four-node quadrilateral finite elements, while the whole com-
putational domain is modeled with 640 four-node quadrilateral finite elements. The
length and width of the whole computational domain are 42 m and 20 m, respec-
tively. The width of the intruded dike-like magma is assumed to be 2 m in this
benchmark problem. For the purpose of comparing the numerical solutions with
the corresponding analytical ones, the thermal properties of the intruded magma are
assumed to be the same as those of the surrounding rocks. The following parameters
are used in the finite element analysis: the densities of both the magma and surround-
3
◦
ing rocks are 2900 kg/m ; specific heat is 1200 J/(kg× C); thermal conductivity is
5
1.74 W/(m× C); the latent heat of fusion of the intruded magma is 3.2 × 10 J/kg.
◦
Since the temperature difference between the intruded magma and the surround-
ing rocks is an important indicator of this benchmark problem, it is assumed to be
◦
1000 C in the numerical computation.
Using the above thermal properties and Eq. (7.17), the value of β is determined
to be 0.73 approximately. In order to examine the effect of time-steps of the intruded
magma solidification on the overall accuracy of the numerical solution, two compu-
tational models, namely a one-step solidification model and a three-step solidifica-
tion model, are considered in the transient finite element analysis. For the one-step
solidification model, the intruded magma is solidified just in one time step, which
is 10.86 days from Eq. (7.16). The corresponding physically-equivalent heat source
3
determined from Eq. (7.20) is 989.06 W/m for this one-step solidification model.
However, for the three-step solidification model, the intruded magma is solidified in
three variable time steps so that the intruded magma solidified equal distance from
the beginning interface between the magma and surrounding/solidified rocks. Using
" " "
Eq. (7.16), the three variable time steps are 10.86 9, 10.86 3 and (5 × 10.86) 9
3
days and the corresponding physically-equivalent heat sources are 8901.54 W/m ,
3
3
2967.18 W/m and 1780.31 W/m respectively. After completion of the intruded
magma solidification, the constant time step of 10.86 days is used throughout the
rest of the transient finite element analysis.
Figures 7.4 and 7.5 show the comparisons of the analytical solutions with the
corresponding numerical solutions from the one-step solidification model and the
three-step solidification model respectively. It is obvious that the numerical solu-
tions of the temperature difference between the intruded magma and the surround-
ing rocks, which are obtained from both the one-step solidification model and the
three-step solidification model, have very good agreement with the correspond-
ing analytical ones. For example, at the time instant of 21.73 days (i.e. t = 21.73
days), the numerical solutions of the maximum temperature difference between
the intruded magma and the surrounding rocks are 732.1 C and 733.5 Cfor the
◦
◦
one-step solidification model and the three-step solidification model respectively,
while the corresponding analytical solution of the maximum temperature difference
between the intruded magma and the surrounding rocks is 737.8 C. The relative
◦
numerical error between the numerical and analytical solutions for the maximum
temperature difference between the intruded magma and the surrounding rocks is
0.78% and 0.58% for the one-step solidification model and the three-step solidi-
fication model respectively. This demonstrates that the numerical model based on