Page 75 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

3.4 Examples of the Proposed Consistent Point-Searching Interpolation Algorithm 61

Table 3.1 Parameters used for the first coupled problem involving reactive mass transport
Material type Parameter Value
−3
pore-fluid dynamic viscosity 10 N × s/m 2
reference density 1000 kg/m 3
0
−4
volumetric thermal expansion coefficient 2.07 × 10 (1/ C)
0
specific heat 4185 J/(kg× C)
0
thermal conductivity coefficient 0.6W/(m× C)
−6
2
chemical species diffusivity coefficient 3 × 10 m /s
3
−7
pre-exponential reaction rate constant 10 kg/(m ×s)
4
activate energy of reaction 5 × 10 J/mol
0
gas constant 8.315 J/(mol× K)
porous matrix initial porosity 0.1
initial permeability 10 −14 m 2
10
elastic modulus 2 × 10 Pa
Poisson’s ratio 0.25
−5
0
volumetric thermal expansion coefficient 2.07 × 10 (1/ C)
0
thermal conductivity coefficient 3.35 W/(m× C)
studied, the temperature gradient is the main driving force to initiate the convective
pore-fluid flow. There is a criterion available (Phillips 1991, Nield and Bejan 1992,
Zhao et al. 1997a) for judging whether or not the convective pore-fluid flow is pos-
sible in such a hydrothermal system as considered here. The criterion says that if the
Rayleigh number of the hydrothermal system, which has a flat bottom and is heated
uniformly from below, is greater than the critical Rayleigh number, which has a the-
2
oretical value of 4π , then the convective pore-fluid flow should take place in the
hydrothermal system considered. For the particular hydrothermal system consid-
ered in this application example, the Rayleigh number is 55.2, which is greater than
the critical Rayleigh number. Therefore, the convective pore-fluid flow should occur
in the hydrothermal system, from the theoretical point of view. This indicates that
the numerical solution obtained from this application example can be qualitatively
justified.
Figure 3.12 shows the normalized concentration distributions of reactant and
product chemical species, whereas Fig. 3.13 shows the final distribution of porosity
and permeability in the computational domain of this application example, where
both the material deformation due to thermal effects and the reactive mass transport
are taken into account in the numerical computation. Obviously, the distributions
of the three chemical species are different. This indicates that the convective pore-
fluid flow may significantly affect the chemical reactions (i.e. reaction flow) in the
deformable porous medium. In particular, the distribution of chemical species 3,
which is the product of the chemical reaction considered in this application exam-
ple, demonstrates that the produced chemical species in the chemical reaction may
reach its equilibrium concentration in some regions, but may not necessarily reach
its equilibrium concentration in other regions in the computational domain. This
finding might be important for the further understanding of reactive flow transport