Page 97 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

84 4 Algorithm for Simulating Fluid-Rock Interaction Problems

Eq. (4.23) states that the dissolution of one mole of K-feldspar needs to consume
+
+
2/3 moles of H and then produces 1/3 moles of muscovite, 2/3 moles of K and
two moles of quartz. Similarly, Eq. (4.24) states that the dissolution of one mole
+
of muscovite needs to consume one mole of H and two moles of quartz and then
produces 1.5 moles of pyrophyllite and one mole of K . This indicates that we
+
only need to consider six primary chemical reactive species, namely two aqueous
+
+
species (K and H ) in the pore-fluid and four solid minerals/species (K-feldspar,
muscovite, pyrophyllite and quartz) in the rock mass, in the computation.
As shown in Fig. 4.1, the computational domain to be used in the numerical anal-
ysis is a rectangle of 3000 m by 1000 m in size. This computational domain is dis-
cretized into 2700 4-node quadrilateral elements. Since the problem to be considered
here is essentially an initial value problem, the following initial conditions are used
in the computation. The initial porosity of the porous medium is 0.1. The initial val-
ues of the generalized concentrations of quartz and K-feldspar are 36.82 kmol/m 3
3
and 0.87 kmol/m respectively. It is assumed that all the aqueous reactive species
involved in the chemical reactions are in chemical equilibrium at the beginning of
the computation. Under this assumption, the initial values of the conventional con-
3
3
centrations of K + and H + are 0.1kmol/m and 1.6 × 10 −5 kmol/m . The hori-
zontal Darcy velocity of pore-fluid flow is 10 −8 m/s in the horizontal aquifer. The
2
dispersion coefficient of K + is 2 × 10 −6 m /s. Since the injected carbon dioxide
(CO 2 ) gas diffuses much faster than aqueous species, the dispersion coefficient of
the injected H , which is the equivalence of the injected carbon dioxide (CO 2 )gas,
+
is 100 times that of K + in the computation. The concentration of the injected H +
3
−3
is 6.4 × 10 kmol/m at the left vertical boundary of the computational domain.
For the purpose of simulating the chemical kinetics of heterogeneous reactions,
the following thermodynamic data are used. The chemical equilibrium constants
7
are 3.89 × 10 and 8318.0 for the K-feldspar and muscovite dissolution reactions,
whereas the chemical reaction rate constants for these two dissolution reactions are
2
2
5.03 × 10 −12 kmol/(m s) and 4.48 × 10 −12 kmol/(m s) respectively. α j ( j = 1, 2)
and q are assumed to be unity and 0.5 in the numerical analysis. In addition, the
8
integration time step is set to be 3×10 s, which is approximately equal to 10 years,
in the computation.
Figures 4.2, 4.3 and 4.4 show the generalized concentration distributions of
K-feldspar, muscovite and pyrophyllite in the fluid-rock interaction system at four
9
10
10
different time instants, namely t = 3 × 10 s, t = 1.5 × 10 s, t = 6 × 10 s and
11
t = 1.5×10 s respectively. It is observed the dissolution front (i.e. the region from
red to blue in Fig. 4.2) of K-feldspar propagates from the left side to the right side of
the computational domain. Since muscovite can be precipitated and dissolved at the
same time, there are two propagation fronts, the precipitation front and dissolution
front, in Fig. 4.3. The precipitation front of muscovite is the region from blue to
red in the right half of the computational domain, whereas the dissolution front is
the region from red to blue in the left half of the computational domain. Both the
precipitation and dissolution fronts propagate from the left side to the right side of
the computational domain. In the case of the generalized concentration distribution
of pyrophyllite (see Fig. 4.4), there only exists the precipitation front (i.e. the region