## 3 Model Concept
### Supplementary Equations  
The permeability decreases due to calcite precipitation and urease adsorption calculated based on the reduction of porosity using a power law (e.g., [40]) with an exponent of three as used by both [29] and [21] for modeling MICP in the same type of porous medium as in this study:  
\[\frac{K}{K_{0}}=\left(\frac{\phi}{\phi_{0}}\right)^{3}. \tag{24}\]  
Here, \(K_{0}\) is the initial permeability and \(\phi\) and \(\phi_{0}\) are the current and the initial porosity, respectively. The porosity \(\phi\) decreases as the volume fractions of adsorbed urease and calcite increase:  
\[\phi=\phi_{0}-\phi_{\rm c}-\phi_{\rm au}. \tag{25}\]  
The capillary-pressure-saturation and relative-permeability-saturation relations of [PERSON] and [PERSON] [41; 42] are used to calculate the capillary pressure, using an entry pressure \(p_{\rm d}=10^{4}\) Pa and a pore-size distribution parameter \(\lambda=2\) as previously used for similar porous media [43]. The relative permeabilities of the wetting and the non-wetting phase are also calculated using the relations given by [PERSON] and [PERSON] [41; 42]. The impact of the calcite precipitation on capillary pressure \(p_{\rm c}\) is accounted for by using Leverett scaling [44] to adapt the capillary pressure of the initial porous medium \(p_{\rm c,0}\), assuming that both contact angle and surface tension do not change significantly:  
\[p_{\rm c}=p_{\rm c}\rho\sqrt{\frac{K_{0}\phi}{K\phi_{0}}}. \tag{26}\]  
The density and the viscosity of the CO\({}_{2}\) phase are calculated using the relation given by [45] and [46], respectively. In these calculations, the effects of the small amounts of water in the CO\({}_{2}\)-phase are neglected. The density and the viscosity of the aqueous phase are calculated according to [47] as a function of salinity. Sodium, chloride, and calcium are considered to contribute to the salinity and to thus affect the aqueous phase properties. The dissolution of CO\({}_{2}\) in the aqueous phase and the dissolution of water in CO\({}_{2}\) is calculated according to [48], with the equilibrium conditions in both phases being dependent on the pressure, temperature, and salinity of the aqueous phase. For further details on the phase composition calculations, see [49]. The temperature- and solution-composition-dependent speciation of NH\({}_{3}\) and H\({}_{2}\)CO\({}_{3}\) produced by ureolysis (see Equations (2)-(4)) is important for modeling EICP as the precipitation rate is dependent on the activity of CO\({}_{3}\)\({}^{2-}\), which is influenced by the solution's chemistry. The speciation of inorganic carbon is calculated using the apparent dissociation constants depending on ionic strength and temperature given by [50]. The temperature dependence of the speciation of NH\({}_{4}\)\({}^{+}\) is calculated according to [51], and the effects of salinity are included using the ionic strength-dependency provided by [52], as no relation was found accounting for both salinity and high temperature. The activity of H\({}^{+}\) is calculated using the charge balance of the resulting geochemical system as well as the law of mass action for the dissociation of water. The charge balance in general requires the following:  
\[\sum_{\kappa=1}^{\text{charged components}}z^{\kappa}m^{\kappa}=0, \tag{27}\]  
where \(z^{\kappa}\) is the charge of component \(\kappa\) and \(m^{\kappa}\) is in charge of its molality. The resulting charge balance for the specific geochemical system can be written as follows:  
\[0=2m^{\text{Ca}^{2+}}+m^{\text{Na}^{+}}+m^{\text{NH}^{4+}}+m^{\text{H}^{+}}-2m ^{\text{CO}_{3}{}^{2-}}-m^{\text{HCO}_{3}{}^{-}}-m^{\text{CI}^{-}}-m^{\text{OH} ^{-}}. \tag{28}\]  
In this equation, the molalities \(m^{\kappa}\) of NH\({}_{4}\)\({}^{+}\), CO\({}_{3}\)\({}^{2-}\), HCO\({}_{3}\)\({}^{-}\), and OH\({}^{-}\) are expressed as a function of \(m^{\text{H}^{+}}\) using the laws of mass action for the dissociation reactions.