Page 83 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

3.4 Examples of the Proposed Consistent Point-Searching Interpolation Algorithm 69

numerical results, can we find something of more theoretical importance? In the fol-
lowing we will answer the above questions through some theoretical analysis. For
the hydrothermal system considered, the maximum change in the density of pore-
fluid due to temperature is about 4%, while the maximum change in the density of
pore-fluid due to the concentration of any reactant chemical species is only about
0.1%. This means that the buoyancy produced by temperature is much greater than
that produced by the concentration of the reactant chemical species. As a result,
the convective pore-fluid flow is predominantly driven by the temperature gradient,
rather than the concentration gradients of chemical species, even though the contri-
bution of chemical species to the buoyancy is taken into account. More specifically,
for the hydrothermal system considered, the velocity components of pore-fluid in the
system are strongly dependent on the temperature gradient, rather than the concen-
tration gradients of chemical species. This means that the concentrations of chemi-
cal species have minimal influence on the convective pore-fluid flow. Based on this
recognition, we can further explore why both the pattern and the magnitude of the
normalized concentration of reactant chemical species are very weakly dependent
on the rates of the chemical reaction. For the purpose of facilitating the theoretical
analysis, we need to rewrite the transport equation (see Eq. (3.5)) for one of the
reactant chemical species (i.e. species 1) as follows:

2 2
∂C 1 ∂C 1 ∂ C 1 ∂ C 1 −E a
ρ f 0 u + v = ρ f 0 D ex + D ey − φk A C 1 C 2 exp .
∂x ∂y ∂x 2 ∂y 2 RT
(3.90)
Since the reactant chemical species constitutes only a small fraction of the whole
matrix in a porous medium, as is commonly assumed in geochemistry (Phillips
1991), we can view the normalized concentration of the reactant chemical species
as the first order small quantity in the above equation, at least from the mathematical
point of view. Thus, the reaction term in this equation is at least the second order
small quantity. This implies that the distribution of the reactant chemical species
is controlled by the pore-fluid velocity and the dispersivity of the porous medium,
rather than by the chemical reaction term unless the rate of chemical reaction is very
fast. This is the reason why both the pattern and the magnitude of the normalized
concentration of the reactant chemical species are very weakly dependent on the
rates of the chemical reaction in the related numerical results (Figs. 3.14 and 3.15).
Next, we will examine, analytically, why the distribution pattern of the normal-
ized 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). In this case, we
need to rewrite the transport equation for chemical species 3 as follows:

2 2
∂C 3 ∂C 3 ∂ C 3 ∂ C 3 −E a
ρ f 0 u + v = ρ f 0 D ex + D ey + φk A C 1 C 2 exp .
∂x ∂y ∂x 2 ∂y 2 RT
(3.91)