Page 54 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

40 3 Algorithm for Simulating Coupled Problems in Hydrothermal Systems

Equation (3.6) clearly indicates that the density of the pore-fluid considered in
this study is a function of both temperature and chemical species concentrations.
This means that the double diffusion effect (Phillips 1991, Nield and Bejan 1992,
Alavyoon 1993, Gobin and Bennacer 1994, Nguyen et al. 1994, Goyeau et al. 1996,
Nithiarasu et al. 1996, Mamou et al. 1998, Zhao et al. 2000b, 2005b, 2006a) is taken
into account in the first sub-problem.
Generally, the chemical reaction term involved in a chemical species transport
equation is strongly dependent on the specific chemical reaction considered in the
analysis. For a non-equilibrium chemical reaction consisting of aqueous chemical
species only, the following type of equation can be considered as an illustrative
example.

k 1
A + B ⇒ F (3.9)
This equation states that species A and species B react chemically at a rate con-
stant of k 1 and the product of this reaction is species F. For this type of chemical
reaction, the reaction term involved in Equation (3.5) can be expressed as follows:

R 1 =−k 1 C 1 C 2 , (3.10)
R 2 =−k 1 C 1 C 2 , (3.11)

R 3 = k 1 C 1 C 2 , (3.12)
where C 1 and C 2 are the normalized concentrations (in a mass fraction relative to the
pore-fluid density) of species A (i.e. species 1) and B (i.e. species 2) respectively; C 3
is the normalized concentration (in a mass fraction relative to the pore-fluid density)
of species F (i.e. species 3).
Note that for most of the chemical reactions encountered in the field of geoscience,
the rate of a chemical reaction is dependent on the temperature at which the chemical
reaction takes place. From a geochemical point of view (Nield and Bejan 1992), the
temperaturedependentnatureofthereactionrateforthechemicalreactionconsidered
here can be taken into account using the Arrhenius law of the following form:

−E a
k 1 = k A exp (3.13)
RT
where E a is the activation energy; R is the gas constant; T is the temperature in Kelvin
and k A is the pre-exponential chemical reaction constant.
The second sub-problem is a static deformation problem under plane strain con-
ditions. If the hydrothermal system is initially in a mechanically equilibrium state,
then body forces can be neglected in the corresponding force equilibrium equations.
This means that we assume that the material deformation of the hydrothermal sys-
tem due to gravity has completed before the system is heated by some thermal event
from below. Under this assumption, the governing equations for static deformation
(which is labeled as the second sub-problem) in the porous medium under plane
strain conditions are: