## 1 Introduction  
In the past two decades, researchers have extensively studied biological techniques as a replacement for traditional methods [1]. As a biological technique, MICP technology offers advantages over traditional methods like cement, lime, and sintered bricks by reducing resource consumption [2]. It is also an efficient and eco-friendly approach for removing contaminants, including heavy metals, from polluted environments [3]. This technology has been used effectively in enhancing the properties of soil and concrete [4, 5, 6] and numerous researchers in the past have concentrated on enhancing the parameters of soils, such as erosion, strength, cohesion, shear strength, crack healing, seepage, hydraulic conductivity, and permeability [4, 7, 8, 9, 10, 11, 12, 13], making it a viable option for various geotechnical engineering applications. While MICP has many advantages in soil improvement, implementing it at the field scale is still challenging because of the inhomogeneous distribution of calcite precipitation [14]. A variety of studies, both experimental and theoretical, have been conducted to gain insight into this effect. For example, [PERSON] et al. [15] examined the homogeneity and mechanical properties of sand using a one-phase temperature-controlled MICP method, demonstrating that specimens treated with this approach exhibited increased CaCO\({}_{3}\) content with a nearly homogeneous distribution along the specimen's height, leading to a significant enhancement in strength and the development of noticeable dilatancy, attributed to the effective bonding network created by the homogeneous CaCO\({}_{3}\) precipitation. They also noted that a non-uniform MICP-treated specimen would form with the majority of CaCO\({}_{3}\) precipitation near the grouting inlet and proposed improving homogeneity by inhibiting bacterial activity and uniformly distributing the bacteria and cementation solution before the MICP reaction. [PERSON] et al. [16] investigated microfluidic chips to examine the pore-scale behavior of MICP. The results showed that calcium carbonate first forms on the bacterial surfaces before gradually spreading more evenly across the microchip. This approach provides valuable insights into distribution patterns and can aid in refining conditions to achieve more uniform calcite precipitation. [PERSON] and [PERSON] [17] focused on microfluidic micromodel design, calcite functionalization, and the investigation of biogeochemical interactions in porous media. In another study, [PERSON] et al. [14] developed an immersion method for MICP treatment, where soil samples are submerged in cementation media, allowing the media to diffuse freely into the soil, resulting in a uniform distribution of precipitated CaCO\({}_{3}\) and suggesting its potential for more consistent soil improvement. [PERSON] et al. [18] utilized X-ray \(\upmu\)-CT imaging to assess the efficacy of MICP in repairing cement defects, enabling the visualization of changes over time and across different spatial regions. Their findings revealed that increasing the injection flow rate and providing more frequent nutrient solution stimulation to the bacterial community resulted in a larger reduction in void fraction and a more uniform MICP seal. To gain insight into a quantitative and effective analysis for the MICP process, numerical models have also been investigated, in addition to experimental studies [19; 20]. For instance, [PERSON] et al. [21] developed a transport model to predict the MICP treatment of sand in one-dimension. The results showed a strong correlation between the simulated outcomes and experimental data, effectively representing the magnitudes and trends of various MICP treatment scenarios in half-meter, one-dimensional flow columns.