Page 84 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

70 3 Algorithm for Simulating Coupled Problems in Hydrothermal Systems

Since the magnitude of the normalized concentration of the reactant chemical
species is very weakly dependent on the rates of the chemical reaction, the product
of C 1 and C 2 is almost independent of the chemical reaction rate in Eq. (3.91). As
mentioned before, the product of C 1 and C 2 is at least the second order small quan-
tity in the theoretical analysis. As a result, the normalized concentration of chemical
species 3 is at least the second (or higher) order small quantity. This means that the
magnitude of the normalized concentration of chemical species 3 is determined by
the reaction term in Eq. (3.91). More specifically, the magnitude of the normalized
concentration of chemical species 3 is strongly dependent on the rate of the chemi-
cal reaction because the product of φ, C 1 and C 2 are eventually independent of the
chemical reaction rate in Eq. (3.91). If C 3A is a solution corresponding to the reac-
tion rate constant k AA for Eq. (3.91), we can, mathematically, deduce the solution
for another reaction rate constant k AB below.
Since C 3A is a solution for Equation (3.91), we have the following equation:

2 2
∂C 3A ∂C 3A ∂ C 3A ∂ C 3A −E a
ρ f 0 u + v = ρ f 0 D ex + D ey + φk AA C 1 C 2 exp .
∂x ∂y ∂x 2 ∂y 2 RT
(3.92)
Multiplying Eq. (3.92) by k AB /k AA yields the following equation:

2 2
k AB ∂C 3A ∂C 3A k AB ∂ C 3A ∂ C 3A −E a
ρ f 0 u + v = ρ f 0 D ex + D ey +φk AB C 1 C 2 exp .
∂x ∂y ∂x 2 ∂y 2 RT
k AA k AA
(3.93)
Therefore, the solution corresponding to the reaction rate constant k AB can be
straightforwardly expressed as
k AB
C 3B = C 3A , (3.94)
k AA

where C 3B is the solution corresponding to the reaction rate constant k AB for
Eq. (3.91).
Equation (3.94) states that for any particular point in the hydrothermal system
considered, the normalized concentration of the product chemical species varies
linearly with the reaction rate constant. This is the reason why the distribution pat-
tern of concentration of the product chemical species is almost independent of the
rates of the chemical reaction, but the magnitude of the product chemical species
is strongly dependent on the rates of the chemical reaction (Fig. 3.16). If we fur-
ther observe the numerical solutions in Fig. 3.16, then we find that the normalized
concentration of the product chemical species, indeed, varies linearly with the reac-
tion rate constant. Therefore, the numerical analysis carried out for this application
example is further validated by the related theoretical analysis.
It needs to be pointed out that the proposed consistent point-searching interpola-
tion and the related solution methodology have been successfully applied to more