## 3 Model Concept
The sources and sinks due to reactions \(q^{\kappa}\) and \(q^{\lambda}\) are specific to the following components:  
* water: \(q^{\mathrm{w}}=0\)
* sodium: \(q^{\mathrm{Na}}=0\)
* chloride: \(q^{\mathrm{Cl}}=0\)
* urea: \(q^{\mathrm{u}}=-r_{\mathrm{u}}\)
* ammonium and ammonia: \(q^{\mathrm{N_{tot}}}=2r_{\mathrm{u}}\)
* dissolved inorganic carbon: \(q^{\mathrm{Ca}}=r_{\mathrm{diss}}-r_{\mathrm{prec}}+r_{\mathrm{u}}\)
* calcium: \(q^{\mathrm{Ca}}=r_{\mathrm{diss}}-r_{\mathrm{prec}}\)
* calcite: \(q^{\mathrm{c}}=r_{\mathrm{prec}}-r_{\mathrm{diss}}\)
* suspended urease: \(q^{\mathrm{su}}=-r_{\mathrm{i}}^{\mathrm{su}}-r_{\mathrm{a}}+r_{\mathrm{d}}\)
* adsorbed urease: \(q^{\mathrm{au}}=-r_{\mathrm{i}}^{\mathrm{au}}+r_{\mathrm{a}}-r_{\mathrm{d}}\)  
The reaction rates used to calculate the source terms are as follows:  
* \(r_{\mathrm{u}}\): the rate of ureolysis
* \(r_{\mathrm{diss}}\) and \(r_{\mathrm{prec}}\): the rates of CaCO\({}_{3}\) dissolution and precipitation
* \(r_{\mathrm{i}}^{\mathrm{su}}\) and \(r_{\mathrm{i}}^{\mathrm{au}}\): the inactivation rates of suspended and adsorbed urease
* \(r_{\mathrm{a}}\) and \(r_{\mathrm{d}}\): the adsorption and desorption rates of urease  
We consider ureolysis by suspended urease \(r_{\mathrm{u}}^{\mathrm{su}}\) and ureolysis by adsorbed urease \(r_{\mathrm{u}}^{\mathrm{au}}\). Thermal ureolysis is due to high temperatures \(r_{\mathrm{u}}^{\mathrm{T}}\) (thermally induced calcium carbonate precipitation (TICP)). \[r_{\mathrm{u}}=r_{\mathrm{u}}^{\mathrm{su}}+r_{\mathrm{u}}^{\mathrm{au}}+r_{ \mathrm{u}}^{\mathrm{T}}. \tag{7}\]  
The reaction rates for urease-catalyzed ureolysis are modeled to be of first order with respect to urease and urea (substrate) concentration, as suggested by [22; 23]; the thermal ureolysis rate is assumed to be of first order with respect to only the urea concentration, inhibited by the presence of Ca\({}^{2+}\)[30]. The rates are as follows:  
\[r_{\mathrm{u}}^{\mathrm{su}}=\frac{k_{\mathrm{u}}^{\mathrm{e}}C_{\mathrm{w}}^ {\mathrm{u}}}{M^{\mathrm{u}}}C_{\mathrm{w}}^{\mathrm{su}}\phi S_{\mathrm{w}} \tag{8}\]  
\[r_{\mathrm{u}}^{\mathrm{au}}=\frac{k_{\mathrm{u}}^{\mathrm{e}}C_{\mathrm{w}}^ {\mathrm{u}}}{M^{\mathrm{u}}}\rho_{\mathrm{au}}\,\phi_{\mathrm{au}} \tag{9}\]  
\[r_{\mathrm{u}}^{\mathrm{T}}=\frac{k_{\mathrm{u}}^{\mathrm{T}}C_{\mathrm{w}}^ {\mathrm{u}}}{M^{\mathrm{u}}} \tag{10}\]  
where \(C_{\mathrm{w}}^{\mathrm{u}}\) is the mass concentration of urea in the aqueous phase in \(\
icefrac{{\mathrm{kg}}}{{\mathrm{m}^{3}}}\), \(M^{\mathrm{u}}\) is the molar mass of urea, \(\phi_{\mathrm{au}}\) and \(\rho_{\mathrm{au}}\) are the volume fraction and the density of the crude urease source, \(C_{\mathrm{w}}^{\mathrm{su}}\) is the mass concentration of suspended crude urease in the aqueous phase, \(S_{\mathrm{w}}\) is the water phase saturation, and \(k_{\mathrm{u}}^{\mathrm{e}}\) and \(k_{\mathrm{u}}^{\mathrm{T}}\) are the temperature-dependent rate coefficients for ureolysis by crude urease and elevated temperatures, respectively. Both rate coefficients are calculated using Arrhenius-type exponential relations:  
\[k_{\mathrm{u}}^{\mathrm{e}}=c_{\mathrm{u},0}^{\mathrm{e}}e^{\frac{c_{\mathrm{u}, T}^{\mathrm{e},T}}{T}}, \tag{11}\]\[k_{\rm u}^{\rm T}=c_{\rm u,0}^{\rm T}e^{\left(\frac{\sum_{\rm u,T}^{\rm T}}{T}-c_{ \rm u,Ca^{2}+}^{\rm T}m^{\rm Ca^{2}+}\right)}. \tag{12}\]  
where \(c_{\rm u,0}^{\rm e}\) and \(c_{\rm u,0}^{\rm T}\) are the pre-exponential factors for enzymatic and thermal ureolysis, \(T\) is the absolute temperature, \(c_{\rm u,T}^{\rm e}\) and \(c_{\rm u,T}^{\rm T}\) are the lumped exponents describing the temperature dependence of the rate coefficients for enzymatic and thermal ureolysis or the activation energy of ureolysis divided by the universal gas constant, \(c_{\rm u,Ca^{2}+}^{\rm T}\) is an exponent describing the influence of calcium on thermal ureolysis, and \(m^{\rm Ca^{2}+}\) is the molality of calcium. We use the apparent enzymatic ureolysis rate coefficients of [23] and the thermal ureolysis rate coefficients from [30].