Page 96 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

4.3 Application Examples of the Term Splitting Algorithm 83

Fig. 4.1 Geometry and 3000 m
initial conditions for the
fluid-rock interaction
problem in a pore-fluid
saturated aquifer CO 99% Quartz + 1% K-feldspar 1000 m
2
φ 0 = , 1 . 0 u x0 = 10 8 − m/ s

ple considered in this section is an isothermal fluid-rock interaction problem in a
pore-fluid saturated horizontal aquifer within a sedimentary basin. The topographi-
cally induced pore-fluid flow is horizontally from the left to the right of the aquifer.
This means that the horizontal Darcy velocity is constant within the aquifer. The
rock of the aquifer is initially composed of K-feldspar (KAlSi 3 O 8 ) and quartz (SiO 2 ).
There is an injection of carbon dioxide gas (CO 2 ) at the left inlet of the aquifer. The
injected carbon dioxide flow in the aquifer may dissolve K-feldspar and precipitate
muscovite (KAl 3 Si 3 O 10 (OH) 2 ). If the injected carbon dioxide flow is strong enough,
the precipitated muscovite may be dissolved and pyrophyllite (Al 2 Si 4 O 10 (OH) 2 )
will be precipitated. Basically, the following three overall chemical reactions may
take place in the aquifer.

fast
+
−
CO 2 + H 2 O =⇒ HCO + H , (4.22)
3
k 1
+
+
3KAlSi 3 O 8 + 2H =⇒ 2K + KAl 3 Si 3 O 10 (OH) 2 + 6SiO 2 , (4.23)
k 2
+
2KAl 3 Si 3 O 10 (OH) 2 + 2H + 6SiO 2 =⇒ 2K + 3Al 2 Si 4 O 10 (OH) 2 . (4.24)
+
The first reaction (i.e. Eq. (4.22)) states that when the injected carbon dioxide
(CO 2 ) gas enters the fluid-rock system, it reacts very fast with water (H 2 O), so that
the chemical equilibrium can be reached quasi-instantaneously. This means that the
injection of carbon dioxide (CO 2 ) gas is equivalent to the direct injection of H into
+
the system. As will be demonstrated later, the use of this equivalence may result in
a reduction in the number of primary reactive chemical species and therefore, a
considerable reduction in the degrees of freedom of the whole system. As a direct
consequence, the number of reactive species transport equations can be reduced to
the minimum in computation. This can lead to a significant reduction in require-
ments of both the computer storage and CPU time in the numerical modelling of
fluid-rock interaction problems.
It needs to be pointed out that Eq. (4.23) describes a heterogeneous chem-
ical reaction, in which K-feldspar (KAlSi 3 O 8 ) is dissolved and muscovite
(KAl 3 Si 3 O 10 (OH) 2 ) is precipitated, while Eq. (4.24) describes a heterogeneous
chemical reaction, in which muscovite (KAl 3 Si 3 O 10 (OH) 2 ) is dissolved and pyro-
phyllite (Al 2 Si 4 O 10 (OH) 2 ) is precipitated in the fluid-rock system. Chemically,