Page 135 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

124 6 Fluid Mixing, Heat Transfer and Non-Equilibrium Redox Chemical Reactions

λ ex = φλ fx + (1 − φ)λ sx , λ ey = φλ fy + (1 − φ)λ sy , (6.7)

D ex = φD fx , D ey = φD fy , (6.8)

where u and v are the horizontal and vertical velocity components of the pore-fluid in
the x and y directions respectively; P is the pore-fluid pressure; T is the temperature
of the porous medium; C i is the concentration of chemical species i; N is the total
number of the active chemical species considered in the pore-fluid; K x and K y are
the permeabilities of the porous medium in the x and y directions respectively; μ is
the dynamic viscosity of the pore-fluid; ρ f and ρ s are the densities of the pore-fluid
and solid matrix; g is the acceleration due to gravity; ρ f 0 and T 0 are the reference
density and reference temperature used in the analysis; λ fx and λ sx are the thermal
conductivities of the pore-fluid and solid matrix in the x direction; λ fy and λ sy are the
thermal conductivities of the pore-fluid and solid matrix in the y direction; c pf and
c ps are the specific heat of the pore-fluid and solid matrix respectively; D fx and D fx
are the diffusivities of the chemical species in the x and y directions respectively; φ is
the porosity of the porous medium; β T is the thermal volume expansion coefficient
of the pore-fluid; R i is the source/sink term for the reactive transport equation of
chemical species i.
It is noted that if the aqueous mineral concentrations associated with ore body
formation and mineralization are relatively small, their contributions to the density
of the pore-fluid are negligible so that the mass transport process can be decoupled
from the pore-fluid flow and heat transfer processes. This means that the whole cou-
pled problem between fluids mixing, heat transfer and redox chemical reactions in
fluid-saturated porous rocks can be divided into two new problems. The first is a
coupled problem between the pore-fluid flow and the heat transfer process, while
the second is a coupled problem between the mass transport process and the redox
chemical reaction process. Since the first coupled problem, which is described by
Eqs. (6.1), (6.2), (6.3) and (6.4), can be solved using the existing finite element
method (Lewis and Schrefler 1998, Zienkiewicz 1977), the main purpose of this
study is to develop a new decoupling procedure to effectively and efficiently solve
the second coupled problem, which is described by Eq. (6.5) and the related chemi-
cal reaction equations.
If the reaction term in Eq. (6.5) can be determined and is linearly dependent
on the chemical species concentration, then the coupled problem defined between
fluids mixing, heat transfer and redox chemical reactions in fluid-saturated porous
rocks above is solvable using the numerical methods available (Zhao et al. 1998a,
2003a). This requires that the chemical reaction be of the first order. Since many
chemical reactions of different orders are associated with ore body formation and
mineralization in fluid-saturated porous rocks, both the second order and the high
order chemical reactions are very common in nature. Without loss of generality, the
second order redox chemical reaction is considered in order to develop a concept
resulting in a new decoupling procedure for removing the coupling between reac-
tive transport equations of redox chemical reactions. In principle, the new concept