## 3 Materials and Methods
### Model of the Study
#### 3.2.5 Calcium Carbonate Production and Influence on Permeability  
The amount of calcium carbonate precipitation per unit of time at a given point can be calculated by multiplying the bacteria number by the mass of soil in the element and the urea hydrolysis rate. It is assumed that calcium carbonate is formed in the pores of the soil. As the volume of pores in a given soil is limited, no further precipitation will occur once this volume is filled. This upper limit value can be calculated by the porosity of the soil and density of the calcium carbonate as follows.  
Taking Soil 1 as an example, for each element in the domain, assuming the length in the y direction of the element is also 0.5 m and based on the data in Table 1, the volume of the element, the soil mass in the element and the void space in the element can be calculated as follows:  
\(V_{\text{element}}=0.5\times 0.5\times 0.5=0.125\text{ m}^{3}=1.25\times 10^{5}\text{ cm}^{3}\)  
\(m_{\text{element}}=125000\times 1.76=2.2\times 10^{5}\text{ g}\)  
\(V_{\text{void}}=125000\times 0.34=4.25\times 10^{4}\text{ cm}^{3}\)  
For each soil, the maximum mass of calcium carbonate in an element in the model is shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Soil 1** & **Soil 2** & **Soil 3** \\ \hline Initial permeability in x direction \(k_{\text{x}}\) (m/s) *1 & \(2.25\times 10^{-4}\) & \(7.5\times 10^{-6}\) & \(6\times 10^{-8}\) \\ Initial permeability in z direction \(k_{\text{z}}\) (m/s) *1 & \(1.5\times 10^{-4}\) & \(5\times 10^{-6}\) & \(4\times 10^{-8}\) \\ Average coefficient of volume change \(m_{\text{v}}\) (kPa\({}^{-1}\)) *1 & \(3\times 10^{-4}\) & \(4.5\times 10^{-4}\) & \(6\times 10^{-4}\) \\ Bulk density \(\rho_{\text{bulk}}\) (g/cm\({}^{3}\)) *1 & \(1.76\) & \(1.60\) & \(1.50\) \\ Initial porosity *1 & \(0.34\) & \(0.4\) & \(0.50\) \\ British Standard Classification *2 & SM & Silt & CL \\ Maximum mass of calcium carbonate per element (g) & 115,175 & 135,500 & \(169\),375 \\ Bacteria concentration at the top of soil (CFU/g) *3 & & \(8.64\times 10^{4}\) & \\ CaCO\({}_{3}\) precipitation rate (mM/CFU/hour) *3 & & \(1.03\times 10^{-6}\) & \\ Density of the calcium carbonate (g/cm\({}^{3}\)) *4 & & \(2.71\) & \\ \hline \hline \end{tabular}  
*1 Results in ref. [40]. *2 BSI Standard [41]. *3 Results in ref. [42]. *4 Results in ref. [43].  
\end{table}
Table 1: Material properties.  
Figure 4: Bacteria test results: (**a**) Bacteria distribution in the model domain; (**b**) _S. pasteurii_ DSM 33 colony forming units plotted against different OD600 values.  
As calcium carbonate is precipitated and fills the pore space, it causes a reduction in the porosity and permeability of the soil. This variation in the permeability also influences the rate of pore water pressure dissipation in both x and z directions, so this is included in the model. Figure 5 shows data from previous studies on the influence of the calcium carbonate content ratio (gravimetric content to the dry soil) on the permeability of the fine sand/silty sand soils. A fitting curve is introduced to describe a general trend of these results and this relationship is used for all soils in this study.  
Three soils are used in the model, with properties as shown in Table 1. The coefficient of volume change (\(m_{\text{v}}\)) changes with pressure; however, an average value is adopted. Soil 1 is a silty sand used for the laboratory tests of bio-cementation described in Section 3.2.5 Bacteria Distribution. To assess the performance of the pressure-responsive MICP system in realistic geotechnical scenarios, two other soils have also been considered: silt (Soil 2) and low-plasticity clay (Soil 3), with properties as shown in Table 1. Experimental results for the bacteria concentration at the top of sand samples are also included in Table 1.