Page 79 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

3.4 Examples of the Proposed Consistent Point-Searching Interpolation Algorithm 65

involved in ore body formation and mineralization in the upper crust of the Earth.
Since the feedback effect of the medium deformation on the porosity and perme-
ability has been considered in the numerical simulation, the distributions of the final
porosity and permeability are totally different from that of their initial values, which
are uniformly distributed in the computational domain.
Again, we can justify the numerical solution for the final porosity and perme-
ability in the hydrothermal system. From the physics point of view, we know that
the higher the relative temperature in a porous medium, the larger the deforma-
tion of the porous medium due to the thermal effect. On the other hand, since the
solid aggregates in the porous medium are relatively stiff, the larger the deformation
of the porous medium, the greater the porosity of the porous medium, indicating
that large deformation of a porous medium results in greater permeability of the
porous medium. This implies that for a porous medium, a region of relatively high
temperature favours the formation of flow channels because of an increase in the
porosity of the region due to the thermal effect. This kind of phenomenon can be
clearly observed from the related numerical solutions for the temperature distribu-
tion (Fig. 3.11) and the final porosity/permeability distributions (Fig. 3.13) in the
hydrothermal system. This further justifies the numerical solutions obtained from
this application example, at least from a qualitative point of view.
The next application example is to investigate how the chemical reaction rate
affects the distribution of product chemical species (i.e. produced new minerals),
if all other parameters are kept unchanged in the hydrothermal system considered
above. Since the pre-exponential reaction rate constant, to a large extent, repre-
sents how fast the chemical reaction proceeds, three different values of the pre-
3
3
−5
−7
exponential reaction rate constant, namely 10 kg/(m ×s), 10 kg/(m ×s) and
3
10 −10 kg/(m ×s), have been used in the corresponding computations.
Figures 3.14, 3.15 and 3.16 show the effects of chemical reaction rates on the nor-
malized concentration distributions of reactant and product chemical species in the
hydrothermal system. It is obvious that the chemical reaction rate has little influence
on either the distribution pattern or the magnitude of the reactant chemical species
concentration. Even though the chemical reaction rate may affect the magnitude
of the product chemical species concentration, it does not affect the overall distri-
bution pattern of the product chemical species concentration. This finding implies
that if the reactant minerals constitute only a small fraction of the whole matrix
in a porous medium, which is a commonly accepted assumption in geochemistry
(Phillips 1991), the distribution pattern of the normalized concentration of the new
mineral produced by chemical reactions is strongly dependent on the characteristics
of the convective pore-fluid flow, even though the magnitude of the normalized con-
centration of the product mineral may also strongly depend on the rates of chemical
reactions.
As mentioned previously, any numerical solutions have to be justified before they
can be safely accepted and used. For this purpose, we must answer the following
questions: Are the numerical results reported in Figs. 3.14, 3.15 and 3.16 correct?
Do they represent the physical and chemical characteristics of the hydrother-
mal system rather than something else (i.e. numerical rubbish)? From these