PNG 550
Reactive Transport in the Subsurface

9.3 Governing Equations

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A general governing equation for the mass conservation of a chemical species for a primary species i is the following:

\begin{equation}\frac{\partial\left(\phi C_{i}\right)}{\partial t}+\frac{\partial\left(\phi v C_{i}\right)}{\partial x}-\frac{\partial^{2}\left(\phi D_{h} C_{i}\right)}{\partial x^{2}}=r_{i, t o t}, \quad i=1,2, \ldots, N_{p}\end{equation}

where $\phi$ is porosity, Ci is the concentration (mol/m3), Dh is the hydrodynamic dispersion (m2/s), v is the flow velocity (m/s), ri,tot is the summation of rates of multiple reactions that the species i is involved (mol/ m3/s), and Np is the total number of primary species. Here we notice that this equation is very similar to the advection-dispersion equation that we discussed in previous lessons except for two major differences. One is that it has an additional reaction term ri,tot to take into account sources and sinks coming from mineral dissolution and precipitation. The other difference is that this is written for the species i, which is one of the Np primary species. Note that if the species is K+, the ri,tot in Equation (3) will be the summation of the K-feldspar reaction rate rK-feldspar and Kaolinite reaction rate rKaolinite. This is because K+ is involved in both of the kinetic reactions, the K-feldspar and Kaolinite reactions (either dissolution or precipitation). The mineral reaction rate laws follow the Transition State Theory (TST):

\begin{equation}r_{K-f e l d s p a r}=\left(k_{H_{2} O}+k_{H} a_{H^{+}}^{0.5}+k_{O H} a_{O H^{-}}^{0.54}\right) A\left(1-\frac{a_{A l^{3+}} a_{K^{+}} a_{S i O_{2}}^{3}}{a_{H^{+}}{ }^{4} K_{e q, K-F e l d s p a r}}\right)\end{equation}

Where kH2O, kH, and kOH are the reaction rate constants with values of 10-13.0 mol/m2/s, 10-9.80 mol/m2/s under acidic conditions, and is the reaction rate constant of 10-10.15 mol/m2/s under alkaline conditions as shown, respectively (Table 1). The exponents of the H+ and OH- activities indicate the extent of rate dependence on H+ and OH-. A is the reactive mineral surface area (m2/m3 porous media). For kaolinite we have

\begin{equation}r_{\text {Kaolinite }}=k A\left(1-\frac{a_{A l^{3+}}{ }^{2} a_{S i O_{2}}{ }^{2}}{a_{H^{+}}{ }^{6} K_{\text {eq,Kaolinite }}}\right)\end{equation}

Where kH2O is the reaction rate constant of 10-13.0 mol/m2/s as shown in Table 1.

In the chemical weathering context, K-feldspar dissolves and releases Al3+, K+, SiO2(aq), whereas Kaolinite precipitates and is the sink for Al3+ and SiO2(aq). The rate term in the parenthesis (1-IAP/Keq) is positive when minerals dissolve and is negative when minerals precipitate so we don’t need to change the sign of the rate term in the equation.

Species. The chemical species from the minerals are Al3+, K+, SiO2(aq). The species from the rainwater include H+, OH-, CO2(aq), HCO3-, CO32-, Na+, and Cl-. Here the major anions are the carbonate species so we can assume the aqueous complexes are KHCO3 (aq) and KCl (aq) and we have a total of 12 species. We can pick Al3+, K+, SiO2(aq), H+, HCO3-, Na+, and Cl- as primary species (7). This means we have secondary species including KHCO3(aq), KCl(aq), OH-, HCO3-, and CO32-. We need to solve for 12 (total number of all species) – 5 (total number of secondary species) = 7 primary species (governing equations).