## Materials and Methods
### Experimental Study on Isothermal Adsorption Characteristics of Solute  
On the basis of the study of the solute dynamic adsorption characteristics in the previous section, the time for dynamic adsorption to reach equilibrium was determined. In this section, the isothermal adsorption characteristics of \(S\). _pasteurii_ and Ca2+ are presented, and the corresponding parameters are obtained, laying the foundation for the further study of the microbial migration behavior in soil, migration theory, and model establishment. Isothermal adsorption refers to the relationship between the concentration of the solute adsorbed in the solid and liquid phases when the solute adsorption reaches equilibrium under isothermal conditions. When the sand can absorb a certain solute, \(\overline{C}\) is the adsorption concentration of the solute for the sand. The solute concentration in the solution is assumed in equilibrium with the solute concentration adsorbed by the porous medium. Sand contains a given substance with a solute concentration of \(C_{1}\) and the corresponding adsorption concentration of \(\overline{C}_{1}\) will appear in the solid phase. If the concentration of water changes to \(C_{2}\), then the system will reach a new equilibrium, and the adsorption concentration will reach a new concentration (\(\overline{C_{2}}\)). Multiple tests were repeated; a higher concentration was used than the previous one, and sufficient time was reserved for each replacement to achieve a new equilibrium between the adsorbed and dissolved phases. According to the results of this type of experiment, the curve obtained by plotting dissolved concentration \(C\) and adsorption concentration \(\overline{C}\) is called the isotherm diagram. For most solutes in groundwater, the isotherm can be expressed as follows [34]:  
\[\overline{C}=K_{f}C^{a}, \tag{5}\]  
The slope of the isotherm is expressed as follows:  
\[\frac{\partial\overline{C}}{\partial C}=K_{f}aC^{a-1} \tag{6}\]The constant (\(K_{J}\)) and index (\(\alpha\)) of each compound in each porous medium should be determined one by one. Equation (6) describes the Freundlich isotherm [23]. When the solute concentration is low, the adsorption curve is consistent with the Freundlich isotherm. When \(\alpha=1\), a linear relationship exists between \(\overline{C}\) and \(C\), as shown in Equation (7):  
\[\overline{C}=K_{d}C, \tag{7}\]  
where the slope (\(K_{d}\)) of the linear isotherm is called the partition coefficient.\(\overline{C}\) is in mg/kg; \(C\) is in mg/L; and \(K_{d}\) is in L/kg. The Freundlich isotherm assumes that the amount of solute that can be adsorbed by the solid substrate is infinite, so the adsorption concentration can increase indefinitely with the increase in the solute concentration. The Langmuir isotherm considers the maximum adsorption capacity and is defined as follows [34]:  
\[\overline{C}=\frac{K_{l}\overline{S}C}{1+K_{l}C}, \tag{8}\]  
The slope of the isotherm is equal to  
\[\frac{\partial\overline{C}}{\partial\overline{C}}=\frac{K_{l}\overline{S}}{ \left(1+K_{l}C\right)^{2}}, \tag{9}\]  
where \(K_{l}\) is the Langmuir constant, and \(\overline{S}\) is the maximum adsorption concentration, which indicates the maximum mass of solute adsorbed per unit mass of porous medium. In the case of a low solute concentration, the Langmuir isotherm is approximately linear, \(K_{d}=K_{l}\overline{S}\). (1) Isothermal adsorption test of \(S\). _pasteurii_  
In a 20 mL centrifuge tube, 5 g of Shanghai layer 3 fine sand and 10 mL of \(S\). _pasteurii_  
\(a\) solution with OD\({}_{600}=0.35\), OD\({}_{600}=0.63\), OD\({}_{600}=0.76\), OD\({}_{600}=0.95\), OD\({}_{600}=1.16\), and OD\({}_{600}=1.35\) were added. Combined with the actual working conditions of the site, the test environment temperature was set to 20 \({}^{\circ}\)C, and the oscillation was carried out at 200 rpm. According to the results of the dynamic adsorption test, the oscillation time was set to 100 min.