Factorial Experiments of Soil Conditioning for Earth Pressure Balance Shield Tunnelling in Water-Rich Gravel Sand and Conditioning Effects’ Prediction Based on Particle Swarm Optimization–Relevance Vector Machine Algorithm

Abstract

The high permeability of gravel sand increases the risk of water spewing from the screw conveyor during earth pressure balance (EPB) shield tunnelling. The effectiveness of soil conditioning is a key factor affecting EPB shield tunnelling and construction safety. In this paper, using polymer, a foaming agent, and bentonite slurry as conditioning additives, the permeability coefficient tests of conditioned gravel sand are carried out under different injection conditions based on the factorial experiment design. The interactions between different concentrations of conditioning additives are analyzed. A prediction model for soil conditioning during shield tunneling based on particle swarm optimization (PSO) and relevance vector machine (RVM) algorithms is proposed to accurately and efficiently obtain the soil conditioning parameters in the water-rich gravel sand layer. The experimental results indicate that the improvement effect of the foaming agent on the permeability of the conditioned gravel sand gradually diminishes with the growing concentration of bentonite slurry. Under conditions of high polymer concentration, further increasing the concentration of bentonite slurry and foaming agent has a weak impact on the permeability coefficient when the concentration of bentonite slurry exceeds 10%. The significance of main effects, first-order interactions, and second-order interaction on the permeability of conditioned gravel sand are as follows: polymer concentration (A) > foaming agent concentration (B) > bentonite slurry concentration (C) > first-order interactions (A × B, A × C, B × C) > second-order interaction (A × B × C). The first-order interaction mainly manifests as a synergistic effect, while the second-order interaction primarily exhibits an antagonistic effect. Case studies show that the maximum relative error between predicted and experimental values is less than 3%. A field application of shield tunneling demonstrates the good performance of real-time optimization of soil conditioning parameters based on the PSO–RVM algorithm. This research provides a new method for evaluating the effectiveness of soil conditioning in the water-rich gravel sand layer.

1. Introduction

The earth pressure balance (EPB) shield machine, featuring construction safety, a high excavation rate, and minimal environmental disturbance, is extensively utilized in urban metro construction projects and the development of underground spaces worldwide [1,2,3]. During EPB tunneling, conditioning additives are injected into the soil chamber and ahead of the cutterhead to alter the properties of excavated soil, enhancing its flowability and impermeability, facilitating spoil removal, and maintaining excavation face stability [4]. Highly permeable strata such as gravel sand are frequently encountered in constructing urban subway tunnels [5]. When EPB shield machines excavate through the water-rich gravel sand layer, the high permeability of the gravel sand increases the risk of water spewing from the screw conveyor and the instability of the excavation face [6], posing significant challenges to construction safety [7,8,9]. Therefore, studying soil conditioning on water-rich gravel sand is important in improving construction efficiency and ensuring safety [10].

Many scholars have researched soil conditioning methods for highly permeable soils. Wang et al. [11] employed foaming agents to adjust the workability of gravel sand and investigated the influence of low foaming injection rates (FIRs) and water content on the slump of conditioned soil. Mori et al. [12] utilized foaming agents to condition gravel sand with different void ratios, identifying void ratio and effective stress as the main factors affecting the performance of conditioned soil. Sun et al. [13] developed a new type of slurry conditioning additive and found the new slurry had great advantages for improving the soil adhesion resistance coefficient and slump value. Souwaissi et al. [14] evaluated the conditioning of gravel sand using slump and rheological tests, deriving relationships between foam and conditioned soil properties at micro, meso, and macro scales. Huang et al. [15] conducted conditioning experiments on sand and cobble soil using foaming agents and bentonite slurry separately, observing an increased slump with the increasing FIR. Budach et al. [16] used foaming agents, polymers, polymer suspensions, and high-density slurries to condition gravel sand, analyzing the effects of different conditioning ratios on workability, compressibility, and water permeability. Wang et al. [17] examined the influence of different FIRs and water content of unconditioned sand on the permeability coefficient and slump of sand–foam mixtures, proposing reasonable theories regarding rheology mechanisms. Yang et al. [18] investigated the parameters of sand under the action of foaming agents, bentonite slurries, and polymers. They found that the conditioned sand exhibited dilatancy, enhanced cohesive strength, and a reduced internal friction angle.

The research results described above have contributed to a better understanding of the mechanical properties of conditioned sand, providing many new insights into soil conditioning methods for sand formations. Using various conditioning additives during EPB shield tunneling to condition highly permeable soils like gravel sand has become a trend [19]. However, the interaction effects among conditioning additives have received less research attention. The individual effects of one conditioning additive can vary with the changing injection conditions of another. If the differences exceed the range of random fluctuations, interaction effects exist among these factors. The factorial experiment is a multifactor cross-grouping experiment that can test the differences between each factor and examine the interaction effects between factors [20]. Therefore, the interactions between different concentrations of conditioning additives based on the factorial experiment need to be further studied.

Using experimental methods to evaluate and analyze the effectiveness of soil conditioning has drawbacks, such as long cycles and high costs [21], which cannot meet the real-time prediction demands in EPB tunneling. In recent years, researchers have gradually applied data-based methods to solve many complex engineering problems [22,23,24,25,26]. Relevance vector machine (RVM) is a machine learning method based on Bayesian theory, Markov properties, automatic relevance determination priors, and maximum likelihood theory. RVM has been preliminarily applied in various fields, such as reliability evaluation [27], agriculture [28], noise detection [29], and surface settlement prediction [30], demonstrating its great development potential. Currently, there are few reported applications of RVM in geotechnical engineering.

In light of this, this study conducts permeability tests on conditioned gravel sand using three commonly used conditioning additives, namely, polymer, foaming agent, and bentonite slurry, to analyze the main effects of each factor and the interaction effects between factors. Further, we apply the RVM algorithm to predict the effectiveness of soil conditioning in highly permeable gravel sand formation. Considering that the independent RVM model has shortcomings in parameter finding and computational efficiency, a particle swarm optimization (PSO) algorithm is used for parameter optimization. Finally, a prediction model for soil conditioning in shield tunnel construction based on the PSO–RVM algorithm is proposed to enhance accuracy and computational efficiency. By employing the PSO–RVM algorithm, real-time soil conditioning during shield tunneling can be achieved.

2. Tunnel Overview and Engineering Geology

The tunnels from the Chaoyang Station to Qingfeng Station on Guangzhou Metro Line 13 in China were constructed using the EPB shield excavation method. The external diameter of the tunnels was 6.4 m, and the buried depth ranged from 5.3 to 16.7 m. The thickness of the segment was 0.3 m, and each circle consisted of six segments with a longitudinal length of 1.5 m. The cutterhead had a diameter of 6.68 m and an opening ratio of 32%. The groundwater level was between 2.6 and 7.8 m below the surface. Due to lateral replenishment from the nearby river, surface water and groundwater had a close hydraulic connection. As shown in Figure 1, the EPB shield excavated long distances in a water-rich sand layer. The sand of the engineering site was saturated with a permeability coefficient of 8.98 × 10⁻⁴ m/s and a saturated water content of 21.3%. The gravel sand was loose, typical of unstable soil, with a natural unit weight of 20.5 kN/m³, a cohesion of approximately 0, an internal friction angle of 33°, and a modulus of deformation of 22 MPa. The stability of the excavation face during shield tunneling was very poor. Further, the high permeability of the gravel sand increased the risk of water spewing from the screw conveyor during EPB shield tunneling.

The grain size distribution is illustrated in Figure 2. The uniformity coefficient (C_u) and curvature coefficient (C_c) of the sand are 10.83 and 0.86, respectively. According to the ASTM D2488-17e1 standard [31], the soil is categorized as poorly graded sandy (SP). The proportion of particles with a size of 2 to 20 mm in the soil sample was as high as 72.3%. Meanwhile, the fine particles smaller than 0.075 mm only comprised 3.6%, significantly lower than the ideal range of 20% to 30%. This resulted in the fine particles in the soil being easily carried away by the water flow from the interstices between the larger particles during shield tunneling, creating water ingress channels. This type of soil is extremely prone to water spewing from the screw conveyor and to excavation face instability when EPB shield tunneling.

3. Laboratory Tests on Soil Conditioning

3.1. Factorial Experimental Design

A full-factorial experiment is characterized by the ability to comprehensively display and reflect the impact of various factors on experimental indicators. At the same time, it can analyze the interaction effects of various factor combinations. The number of repeated experiments for each factor is increased, thereby improving the accuracy of each factor experiment.

A factorial experiment designs multiple factors at multiple levels to study their impact on a particular indicator. It can investigate the degree of influence of different levels within each factor and whether they interact. One can compute the main and interaction effect values between each factor. Further, the variance of the main effects and interaction effects can be calculated. After calculating the error effects, it is possible to compute the mean square error of the errors. Based on this, the variance estimates of the main effects of each factor and the interaction effects can be obtained. Statistical tests can be conducted based on these calculations.

$$T={\displaystyle \frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}}$$

(1)

where $\overline{x}$ is the sample mean, μ₀ is the population mean, s is the sample standard deviation, and n is the number of samples in the experiment.

In significance analysis, hypothesis testing from statistics is required. Based on the proposed null hypothesis H₀, each variance estimates maps out the p-value applied to the hypothesis test. When setting the significance level judgment value α (usually α = 0.05), the following definitions apply: if p < 0.05, reject H₀; if p ≥ 0.05, accept H₀. The absolute value of the test statistic T is compared with the distribution boundary value t, which satisfies the degrees of freedom, thus determining the significance order of each effect on the response value.

In experiments with multiple factors, the relationships between factors generally include no interaction, synergistic effect, or antagonistic effect. A synergistic effect relationship exists if different factors can increase the response value. Conversely, an antagonistic effect relationship exists if multiple factors weaken the response value. There is no interaction relationship if the factors do not affect each other. Only the main effect analysis is needed when there is no interaction between factors. When there is interaction between factors, both main and interaction effects must be analyzed simultaneously.

In current shield tunneling constructions in China, the most commonly used conditioning additives for sandy soil include foaming agents, sodium bentonite slurry, and polymer (carboxymethyl cellulose). Foam can reduce the soil’s permeability, enhance plasticity, and decrease wear on cutting tools. As shown in Figure 3, both bentonite slurry and polymer solutions are fluids with certain viscosities that can effectively fill the soil pores, significantly improving soil permeability.

Based on the factorial experiment design method, the coupling effects of polymer concentration, foaming agent concentration, and bentonite slurry concentration were fully considered. The three factors were represented by the three letters A, B, and C. Permeability tests were conducted to analyze the influence of these three factors on the soil conditioning effect. Given that the number of factorial experiments increases dramatically with the number of levels, limiting the number of levels for each factor is essential to manage the experimental complexity effectively. Referring to the commonly used concentration of additives in engineering [11,12,14,15,21,32,33,34,35], A was set at three levels: 25% (A₁), 50% (A₂), and 75% (A₃); B was set at five levels: 1% (B₁), 2% (B₂), 3% (B₃), 4% (B₄), and 5% (B₅); and C was set at five levels: 6% (C₁), 8% (C₂), 10% (C₃), 12% (C₄), and 14% (C₅). The mass injection ratios of A, B, and C to saturated gravel sand were 0.1%, 2%, and 10%, respectively, resulting in 75 experiments, as shown in

Table 1. Factorial experiment design.

	A1					A2					A3
	B1	B2	B3	B4	B5	B1	B2	B3	B4	B5	B1	B2	B3	B4	B5
C1	A1B1C1	A1B2C1	A1B3C1	A1B4C1	A1B5C1	A2B1C1	A2B2C1	A2B3C1	A2B4C1	A2B5C1	A3B1C1	A3B2C1	A3B3C1	A3B4C1	A3B5C1
C2	A1B1C2	A1B2C2	A1B3C2	A1B4C2	A1B5C2	A2B1C2	A2B2C2	A2B3C2	A2B4C2	A2B5C2	A3B1C2	A3B2C2	A3B3C2	A3B4C2	A3B5C2
C3	A1B1C3	A1B2C3	A1B3C3	A1B4C3	A1B5C3	A2B1C3	A2B2C3	A2B3C3	A2B4C3	A2B5C3	A3B1C3	A3B2C3	A3B3C3	A3B4C3	A3B5C3
C4	A1B1C4	A1B2C4	A1B3C4	A1B4C4	A1B5C4	A2B1C4	A2B2C4	A2B3C4	A2B4C4	A2B5C4	A3B1C4	A3B2C4	A3B3C4	A3B4C4	A3B5C4
C5	A1B1C5	A1B2C5	A1B3C5	A1B4C5	A1B5C5	A2B1C5	A2B2C5	A2B3C5	A2B4C5	A2B5C5	A3B1C5	A3B2C5	A3B3C5	A3B4C5	A3B5C5

. In the table, A₁B₁C₁ represents the polymer concentration of 25%, the foaming agent concentration of 1%, and the bentonite slurry concentration of 6%.

Due to the high polymer concentration and the small injection mass, the polymer was first added to the bentonite slurry and thoroughly mixed during the experiment. Subsequently, the bentonite slurry was mixed with the gravel sand to be conditioned. The conducted experiments are illustrated in Figure 4, and the experimental procedure was as follows:

(1)

Bentonite slurry performance test: We prepared bentonite slurry with different concentrations, and then tested the rheological properties of the slurry using a Marsh funnel and a rotational viscometer. We studied the variation law of the rheological properties of the bentonite slurry with time and concentration.

(2)

Foam performance test: The foam expansion ratio (FER) and half-life time (H-T) are important parameters reflecting the quality of the foaming agent and the foam’s performance. During the experiment, the gas–liquid flow ratio was maintained at 9. The foaming pressure was 3 bar. The H-T and FER were tested under standard atmospheric pressure conditions.

(3)

Constant head permeability test: This test studied the variation in permeability of saturated gravel sand before and after conditioning following the GB/T 50123-2019 [36] specification.

3.2. Analysis of Test Results

Performance test results of bentonite slurry with different concentrations are shown in Figure 5. The symbol c_b in the figure represents the concentration of bentonite slurry. From Figure 5a, it can be observed that the funnel viscosity of the slurry increased with the increase in c_b and hydration time. When c_b ≤ 10%, the funnel viscosity gradually increased with c_b and hydration time. At bentonite slurry concentrations of 6%, 8%, and 10%, the increases in 24 h funnel viscosity were 19.1%, 27.2%, and 37.5%, respectively, and the increases in 48 h funnel viscosity were 66.7%, 147.6%, and 117.5%, respectively. When c_b > 10%, the viscosity increased significantly with c_b. When c_b = 14.0%, the 12 h funnel viscosity of the bentonite slurry reached 115 s, and the 24 h funnel viscosity of the bentonite slurry reached 150 s, exceeding the pumping capacity of the tunneling machine (usually, the funnel viscosity is within 100 s).

The apparent viscosity can qualitatively analyze the flow behavior of non-Newtonian fluids. Increasing the slurry’s apparent viscosity can enhance the conditioned soil’s cohesion and reduce its internal friction angle. Plastic viscosity represents the ease with which different shear rates disrupt a fluid. Figure 5b shows that the apparent viscosity was below 15 mPa·s, and the plastic viscosity was below 10 mPa·s when c_b ≤ 10%, indicating a relatively thin slurry with segregation after mixing. This segregation gradually disappeared with the increase in c_b. At the same time, the suspension, encapsulation, and sand-carrying capacity of the slurry improved. The apparent viscosity and plastic viscosity were almost linearly positively correlated with the bentonite slurry concentration. For every 2% increase in bentonite slurry concentration, the apparent viscosity increased by 4.34 mPa·s, and the plastic viscosity increased by 3.91 mPa·s on average. When c_b = 14%, the apparent viscosity approached 25 mPa·s, and the plastic viscosity approached 20 mPa·s, indicating an overall viscosity that was too high and unstable.

The yield point reflects the fluid’s ability to carry and encapsulate sand particles. When the ratio of yield point to plastic viscosity is too low, sand particles settle in the soil tank. Figure 5c shows that the yield point was almost linearly positively correlated with the bentonite slurry concentration. For every 2% increase in bentonite slurry concentration, the average yield point increased by 1.78 Pa. When c_b was within the range of 6% to 14%, the ratio of yield point to plastic viscosity of the slurry ranged from 0.53 to 0.82.

Gel strength reflects the settling speed of larger solid particles in the fluid and the ability to suspend smaller particles. Figure 5d shows that the slurry’s initial and 10 min gel strength showed the same trend with concentration and was basically linearly correlated with concentration. When c_b ≤ 14%, for every 2% increase in bentonite slurry concentration, the slurry’s initial and 10 min gel strength increased by 0.67 and 0.66 Pa on average, respectively. When c_b > 14%, for every 2% increase in bentonite slurry concentration, the slurry’s initial and 10 min gel strength increased by 1.68 and 1.95 Pa on average, respectively.

Figure 6 shows the performance test results of foaming agents with different concentrations. The symbol c_f represents the concentration of the foaming agent. It can be observed from the figure that the FER increased first and then stabilized with the increase in c_f. When c_f was less than 4%, the rise in FER was significant, ranging from 10 to 20, meeting the construction requirements. When c_f exceeded 5%, the FER stabilized at around 26.

The foam H-T increased with the increase in c_f. When c_f was greater than 4%, the H-T difference was insignificant. The FER and H-T tended to stabilize at a certain value with the increased c_f. The main reason is as follows. The surfactants in the foaming agent can reduce the surface tension of the liquid, resulting in the bubble volume changing with concentration. However, there is a critical micelle concentration (CMC) for surfactants. When the surfactant concentration reaches the CMC, a mixed adsorption layer forms on the foam surface, causing the surface tension to reach a constant value. At this point, changing c_f has little effect on the FER and H-T of the foam.

Figure 7 shows the variation curve of the permeability coefficient of the conditioned soil under the joint action of the foaming agent, polymer, and bentonite slurry. The symbol c_p in the figure represents the concentration of the polymer. With the increase in c_p, the permeability coefficient of the conditioned soil gradually decreased. With the rise in c_b, the improvement effect of the foam on the permeability of the conditioned soil gradually weakened. When c_p = 25%, and c_f was the same, the permeability coefficient of the conditioned soil decreased linearly with the increase in c_b. When c_p = 50% or 75%, and c_f was the same, the permeability coefficient of the conditioned soil decreased exponentially with the increase in c_b. Despite the relatively large polymer concentration compared to foaming agents and bentonite slurry, its injection mass was only 1/1000 of the soil to be conditioned, making the polymer content in the conditioned soil the smallest. Even small changes in polymer content significantly impacted the permeability coefficient of the conditioned soil. A further analysis of the improvement effect of polymer is provided below.

When c_b and c_f were the same, with the increase in c_p, the permeability coefficient of the conditioned soil decreased significantly—observing the permeability coefficient of the conditioned soil when c_b = 8% and c_f = 3%, the permeability coefficients under the conditions of c_p = 25%, 50%, and 75% were 4.085 × 10⁻⁶, 2.633 × 10⁻⁶, and 3.04 × 10⁻⁷ m/s, respectively, with corresponding reduction rates of 35.5% and 92.6%. Under conditions of relatively high c_p (75%) and c_b exceeding 10%, further increasing c_b and c_f had a weak effect on reducing the permeability coefficient. Although the method of “bentonite slurry + foaming agent” can reduce the permeability of the soil, its weakening effect is not as significant as that of polymer. Therefore, when conditioning the soil in permeable strata, adding a small amount of polymer agent can effectively reduce the permeability coefficient of the conditioned soil. It can achieve the requirements of fluidity and prevent water spewing.

3.3. Normalized Effect Analysis

Based on the test results, the main effects of polymer concentration (A), foaming agent concentration (B), and bentonite slurry concentration (C) on the permeability coefficient test of conditioned soil, as well as the first-order and second-order interaction effects, could be obtained. The calculation results are shown in

Table 2. Analysis of variance for the soil conditioning test.

Data Source	Degree of Freedom	Adjusted Sum of Squares	Adjusted Mean Squares	T-Value	F-Value	p-Value
A	2	60,784	30,391.8	−178.69	37,947.47	0.0011
B	4	37,638	9409.5	−139.66	11,748.75	0.0013
C	4	26,362	6590.5	−110.60	8229.01	0.0041
A × B	8	18,060	2257.5	102.17	2818.77	0.0062
A × C	8	6833	854.2	61.44	1066.52	0.0068
B × C	16	2766	172.9	31.16	215.83	0.0071
A × B × C	32	3070	95.9	−7.45	119.80	0.0077

.

For the shield tunneling construction of sand strata in Guangzhou Metro, the requirement for conditioned soil’s permeability coefficient (k) was less than 3 × 10⁻⁶ m/s. Based on the principle of hypothesis testing, the null hypothesis H₀: k ≤ 3 × 10⁻⁶ m/s was established for this factorial experiment. Based on H₀, the standardized effect normal distribution line suitable for the experiment was obtained. The main, first-order, and second-order interaction effects were distributed on both sides of the normal distribution line according to their significance, resulting in a normal probability plot, as shown in Figure 8a. To intuitively and effectively identify the main influencing factors of the response values, a Pareto chart of standardized effects was plotted, as shown in Figure 8b. In the chart, the abscissa represents the test statistic, and the ordinate represents the main and interaction effects. The test statistic for each effect was calculated using Equation (1). The significance threshold value was 2.1, as obtained from the t-distribution table. When the test statistic T-value was less than 2.1, it was defined as insignificantly affecting. When it was greater than 2.1, it significantly affected. The effects are listed in the chart according to the T-values.

From Figure 8, the deviation of each effect point from the normal distribution line was significant, indicating significant effects for all factors without any non-significant effect points. The significance of main effects, first- and second-order interactions on the permeability of conditioned soil was as follows: polymer concentration (A) > foaming agent concentration (B) > bentonite slurry concentration (C) > first-order interactions (A × B, A × C, B × C) > second-order interaction (A × B × C). The polymer concentration (A) had the greatest impact on the experimental results, with a test statistic of 178, 1.28 times that of the B test statistic, and 1.62 times that of the C test statistic. The foaming agent concentration (B) had a test statistic of approximately 139, 1.26 times that of the C test statistic. The bentonite slurry concentration (C) had a test statistic of 110. In terms of interactions, the test statistic for the first-order interaction A × B was approximately 102, 3 times that of B × C. The test statistic for the first-order interaction A × C was approximately 61, 2 times that of B × C.

3.4. Main Effect Analysis

The empirical cumulative distribution functions of each main effect are shown in Figure 9. It can be seen from the figure that the empirical cumulative distribution functions of c_f and c_b are very close to the fitting line. It indicates that these data follow a normal distribution pattern. The fitting lines of both c_f and c_b shift to the left. These two fitting lines have steep slopes. The standard deviations of the two conditioning additives concentrations were 1.417 and 2.835, respectively. The standard deviation of the polymer concentration was 20.46. Based on the relationship between the step-change magnitude and the standard deviation, the main effect relationship could be deduced as follows: A > B > C.

3.5. Interaction Analysis

Figure 10 shows the interaction plot of the three factors. The interaction between factors A, B, and C was primarily characterized by a synergistic effect, with antagonistic effects playing a secondary role. The soil conditioning mechanisms of polymer, foam, and bentonite slurry mainly determined the synergism effect. The high-molecular-weight polymers have a long-chain structure composed of many monomers. They attach to the surfaces of soil particles, and their hydrophilic groups interact with water molecules, improving the soil’s water-sealing performance. The injection of foam increases a large amount of sealing gas in the sand pores, which blocks the seepage channel of sand pore water and reduces the permeability of sand. The injection of bentonite slurry increases the content of fine particles in sandy soil, which narrows the permeability channel and reduces the permeability coefficient of sandy soil. The conditioning mechanisms of foam and bentonite slurry are the plugging effect of small forms and fine particles on sand pores. However, when the concentration of polymer and bentonite slurry increased to a certain extent, the cohesiveness of the soil was greatly enhanced, resulting in many foam bursts. The burst foams failed to condition the soil. At this stage, the interaction state was antagonistic.

3.6. Equivalence Relationship Prediction

The relationships between permeability coefficient and polymer concentration (A), foaming agent concentration (B), and bentonite slurry concentration (C) are shown in Figure 11. Figure 11a indicates that when c_p ranged from 25% to 31% and c_f ranged from 1% to 1.8%, the range of permeability coefficient for the conditioned soil was from 70 to 90 cm/d. When c_p ranged from 31% to 43% and c_f ranged from 1.8% to 2.5%, the range of permeability coefficient was from 50 to 70 cm/d. When c_p ranged from 43% to 57% and c_f ranged from 2.5% to 3.3%, the range of permeability coefficient was from 30 to 50 cm/d. When c_p ranged from 57% to 73% and c_f ranged from 3.3% to 5%, the range of permeability coefficient was from 10 to 30 cm/d. When c_p ranged from 73% to 75% the foaming agent’s influence was insignificant, and the conditioned soil’s permeability coefficient was smaller than 10 cm/d.

Figure 11b indicates that when c_p ranged from 25% to 48% and c_b ranged from 6% to 9.2%, the range of permeability coefficient for the conditioned soil was from 50 to 70 cm/d. When c_p ranged from 48% to 65% and c_b ranged from 9.2% to 12.9%, the range of permeability coefficient was from 30 to 50 cm/d. When c_p ranged from 65% to 75% and c_b ranged from 12.9% to 14%, the permeability coefficient of the conditioned soil ranged from 10 to 30 cm/d.

The main effects and interaction effects between the factors were determined based on the test results, and a regression equation was fitted based on the least squares method.

$$\begin{array}{l}L=203.46-2.2319A-24.9450B-8.2200C+0.2625A\times B\\ \hspace{1em}\hspace{1em}+0.07891A\times C+0.6255B\times C-0.004046A\times B\times C\end{array}$$

(2)

where L is the response value of the permeability coefficient.

The concentration of polymer (A), foaming agent (B), and bentonite slurry (C) can only be positive numbers. Equation (2) reveals that the first-order interactions (A × B, A × C, B × C) mainly exhibit a synergistic effect, while the second-order interaction (A × B × C) primarily demonstrates an antagonistic effect. The regression coefficients for A, B, C, and A × B × C were negative, with values of −2.2319, −24.9450, −8.2200, and −0.004046, respectively. The absolute value of the regression coefficient for B was the largest, indicating that B significantly impacted the permeability coefficient. When c_f increased, the permeability coefficient decreased significantly, reducing the synergistic effect and increasing the antagonistic effect. The regression coefficients for A × B, A × C, and B × C were positive, indicating that the first-order interactions positively contributed to the permeability coefficient. When the first-order interactions were strong, the synergistic effect increased, and the antagonistic effect decreased.

4. Soil Conditioning Prediction Based on PSO–RVM

Evaluating and analyzing soil conditioning effects on excavated soil have disadvantages, such as long cycles and high costs using experimental methods, which cannot meet the real-time prediction needs in tunneling. In recent years, faster and more economical methods for evaluating soil conditioning effects have attracted attention from researchers. RVM is a machine learning method based on Bayesian theory, Markov property, automatic relevance determination prior, and maximum likelihood theory. It has been applied to solve many complex engineering problems. This study applied the RVM algorithm to predict the effect of soil conditioning in water-rich gravel sand strata and utilized the PSO algorithm for parameter optimization, proposing a shield tunneling soil conditioning prediction model based on the PSO–RVM algorithm.

4.1. PSO–RVM Algorithm

RVM is a sparse probabilistic learning model based on SVM based on Bayesian theory, defining the prior probability of hyperparameter $\mathsf{\alpha }$ on the weight $\mathsf{\omega }$. In the dataset, the training samples are denoted as $\{x_n, t_n|n=1,\dots , 2, N\}$, where the input and output values of the training dataset are $x_n$ and $t_n$. The distributions of $t_n$ are independent of each other. $N$ is the number of samples. The functional relationship between the output and input values can be derived as follows:

$${\xi }_{\mathrm{n}}{t}_n=y(x_n; \mathsf{\omega })={\displaystyle \sum _{n=1}^N{\omega }_n}k(\mathit{x},x_n)+{\xi }_n$$

(3)

where $k(\mathit{x},x_n)$ is a kernel function, $\mathit{x}=\{x_1, x_2, \dots , x_N\}$; ${\omega }_n$ is the weight of each input quantity; and ${\xi }_n$ is data noise and follows a Gaussian distribution, ${\xi }_n\sim N(0, {\sigma }^2)$, where ${\sigma }^2$ is variance. The equation satisfying the Gaussian distribution is as follows.

$$p(\left.t_n\right|x_n)=N(\left.t_n\right|y(x_n), {\sigma }^2)$$

(4)

where $t_n$ and $y(x_n)$ are relative with ${\sigma }^2$.

The likelihood function of the training sample set can be expressed as

$$p(\left.\mathit{t}\right|\mathsf{\omega }, {\sigma }^2)={(2\pi {\sigma }^2)}^{-N/2}\exp \left\{-{\displaystyle \frac{1}{2{\sigma }^2}}\parallel \mathit{t}-\mathit{\Phi }\mathsf{\omega }{\parallel }^2\right\}$$

(5)

where $\mathit{t}={(t_1, \dots , t_N)}^T$ is the target; $\mathsf{\omega }={\left[{\omega }_0,{\omega }_1,\dots ,{\omega }_N\right]}^T$ represents the weight; and $\mathit{\Phi }={\left[\mathit{\varphi }(x_1), \mathit{\varphi }(x_2), \dots , \mathit{\varphi }(x_N)\right]}^T$ is an $N\times (N+1)$ matrix where the expression of each column in the matrix is $\mathit{\varphi }(x_n)=[1, k(x_n, x_1), k(x_n, x_2), \dots k(x_n, x_N)]$. Introducing hyperparameter $\mathsf{\alpha }={({\alpha }_0, {\alpha }_1,\dots , {\alpha }_N)}^T$ for solving $\mathsf{\omega }$ and ${\sigma }^2$ in Equation (5) and assuming ${\omega }_n$ follows a Gaussian distribution with mean 0 and variance ${\alpha }_n^{-1}$, then

$$p(\left.\mathsf{\omega }\right|\mathsf{\alpha })={\displaystyle \prod _{n=0}^{N}N}\left(\left.{\omega }_n\right|0, {\alpha }_n^{-1}\right)$$

(6)

The equation for the input values $x^{*}$ and output values $t^{*}$ of the prediction dataset is given by:

$$P(t^{*}\mid \mathit{t})={\displaystyle \int P(t^{*}\mid \mathsf{\omega },\mathsf{\alpha },{\sigma }^2)P(\mathsf{\omega },\mathsf{\alpha },{\sigma }^2\mid \mathit{t})\mathrm{d}\mathsf{\omega }\mathrm{d}\mathsf{\alpha }\mathrm{d}{\sigma }^2}$$

(7)

According to the properties of Bayes and Markov, and by simplifying Equation (7), the following can be obtained:

$$\mathit{P}(\mathsf{\omega }\mid \mathit{t},\mathsf{\alpha },{\sigma }^2)={(2\pi )}^{-{\displaystyle \frac{N+1}{2}}}{\left|\sum \right|}^{-{\displaystyle \frac{1}{2}}}\exp \left\{-{\displaystyle \frac{1}{2}}\times {(\mathsf{\omega }-\mathit{\mu })}^T{\sum }^{-1}(\mathsf{\omega }-\mathit{\mu })\right\}$$

(8)

where $\sum = {({\sigma }^2{\mathit{\Phi }}^T\mathit{\Phi }+\mathit{A})}^{-1}$ represents the variance; $\mathit{\mu }={\sigma }^{-2}\sum {\mathit{\Phi }}^T\mathit{t}$ represents the average value; and $\mathit{A}=\mathrm{diag}(a_0, a_1,\dots , a_N)$ is a diagonal matrix.

The maximum likelihood function is organized and simplified to

$$P(\mathit{t}\mid \mathsf{\alpha },{\sigma }^2)={(2\pi )}^{-{\displaystyle \frac{N}{2}}}{\left|\sum \right|}^{-{\displaystyle \frac{1}{2}}}\exp \left\{{\displaystyle \frac{1}{2}}\times {(\mathsf{\omega }-\mathit{\mu })}^T{\sum }^{-1}(\mathsf{\omega }-\mathit{\mu })\right\}$$

(9)

By taking the partial derivatives of $\mathsf{\alpha }$ and ${\sigma }^2$ in Equation (10) and setting them equal to zero, the equation becomes

$$a_n^{\mathrm{new} }={\displaystyle \frac{{\gamma }_n}{{\mu }_n^2}}$$

(10)

$${({\sigma }^2)}^{\mathrm{new}}={\displaystyle \frac{\Vert \mathit{t}-\mathit{\Phi }\mathit{\mu }{\Vert }^2}{N-{\displaystyle \sum _{n=0}^N{\gamma }_n}}}$$

(11)

where ${\gamma }_n=1-{\mathsf{\alpha }}_n{\sum }_{nn}$, and the element in the nth row and nth column of the matrix $\sum$ is denoted by ${\sum }_{nn}$. After updating iterations based on Equations (10) and (11), $\mathsf{\alpha }$ and ${\sigma }^2$ are obtained. At the same time, the weighted posterior mean $\mathit{\mu }$ and the covariance matrix Σ are iterated until the convergence condition is met or the maximum number of iterations is met. During the iteration process, new optimal solutions ${\mathsf{\alpha }}_{\mathrm{MP}}$ and ${\sigma }_{\mathrm{MP}}^2$ are be obtained, and most of the weights approach 0.

Assuming the predictive sample is $x^{*}$, and the predicted value is $t^{*}$, then the distribution of $t^{*}$ is given by:

$$P(t^{*}\mid \mathit{t}, {\mathsf{\alpha }}_{\mathrm{MP}}, {\sigma }_{\mathrm{MP}}^2)={\displaystyle \int P(t^{*}\mid \mathsf{\omega }, {\sigma }_{\mathrm{MP}}^2)P(\mathsf{\omega }\mid \mathit{t}, {\mathsf{\alpha }}_{\mathrm{MP}}, {\sigma }_{\mathrm{MP}}^2)\mathrm{d}\mathsf{\omega }}$$

(12)

Since both functions integrated in Equation (12) follow Gaussian distributions, the predicted value $t^{*}$ also follows a Gaussian distribution.

$$P(t^{*}\mid \mathit{t}, {\mathsf{\alpha }}_{\mathrm{MP}}, {\sigma }_{\mathrm{MP}}^2)=N(t^{*}\mid y^{*}, {\sigma }_{*}^2)$$

(13)

where the expected value is $y^{*}={\mathit{\mu }}^T\mathit{\varphi }(x^{*})$, and the variance is ${\sigma }_{*}^2={\sigma }_{\mathrm{MP}}^2+\mathit{\varphi }{(x^{*})}^T\sum \mathit{\varphi }(x^{*})$.

The kernel function is the core of the RVM model. It significantly reduces the computational complexity of the RVM model’s dataset. When using the RVM model for prediction, manually tuning the hyperparameters and kernel function parameters often leads to low prediction accuracy and high time costs. The PSO algorithm can assist RVM models in finding optimal parameters, overcoming the difficulty of manually adjusting parameters, and automating the entire analysis model process.

PSO is an evolutionary computing technique based on swarm intelligence. It involves initializing a group of random particles within the feasible solution space, each representing a potential optimal solution to the problem. These particles iteratively update their positions and velocities to find the optimal solution by continuously exploring the solution space. During the iterative process, particles update their positions and velocities by comparing and tracking individual best positions (P_id) and global best positions (P_gd). By combining PSO with RVM, the PSO–RVM model is formed, as depicted in Figure 12. As shown in the figure, PSO optimizes RVM by updating particles in the particle swarm algorithm, automatically reducing the fitness and mean squared error between predicted and actual results until the kernel width meets the precision requirements of the RVM computation.

4.2. Case Study

Various factors influence the conditioning effect of excavated soil in water-rich sand layers. The type of input variables significantly affects the predictive results of the model. Considering the difficulty of data acquisition, the concentration of polymer, foaming agent, and bentonite slurry, permeability coefficient (before conditioning), and resistivity were selected as input variables. The slump and permeability coefficient (after conditioning) were chosen as output variables. Resistivity is one of the intrinsic properties of water-rich sand stratum, an important parameter for distinguishing sand and other soil types. The resistivity reflects the basic physical and mechanical properties and structural characteristics of the water-rich sand stratum in Guangzhou. Therefore, the resistivity of sand was also included as an input variable in the case analysis. Based on the principles of PSO for RVM prediction models, a PSO–RVM model for predicting the conditioning effect of sand soil was established, as shown in Figure 13. In the figure, $k_1$ to $k_5$ represent the first to fifth kernel functions, and ${\omega }_1$ to ${\omega }_5$ represent the weights of the first to fifth kernel functions, respectively.

The implementation steps were as follows.

Step 1: Collect experimental data, then analyze and organize the collected sample data, as shown in

Table 3. Datasets.

Sample Number	Input Variables					Output Variables
	cp (%)	cf (%)	cb (%)	Permeability Coefficient (Before Conditioning) (×10−4 m/s)	Resistivity of Sand (Ω·m)	Slump Value (mm)	Permeability Coefficient (After Conditioning) (×10−7 m/s)
1	25	5	14	8.98	54.24	121	4.08
2	50	4	10	8.90	54.10	143	9.88
3	75	2	14	9.12	56.44	55	0.41
4	50	4	6	9.05	53.63	168	21.88
5	75	3	10	9.09	54.68	51	0.21
6	25	2	10	9.25	56.67	192	58.82
7	50	3	6	8.70	53.13	187	45.11
8	75	5	10	8.95	55.89	57	0.49
9	25	1	8	8.89	54.94	205	73.53
10	50	3	12	8.98	55.37	124	4.73
11	75	4	10	9.13	51.58	50	0.14
12	25	5	6	9.11	55.23	180	28.59
13	50	3	8	9.14	56.49	176	26.33
14	25	3	8	8.96	51.42	186	40.85
15	50	2	10	9.24	53.44	162	16.24
16	25	1	12	9.13	53.44	195	62.09
17	50	2	8	8.87	56.40	181	31.76
18	25	2	14	9.27	53.18	182	32.68
19	75	3	8	9.22	55.54	117	3.04
20	25	5	10	8.80	55.99	148	11.29
21	50	1	8	8.59	52.25	184	35.29
22	75	2	6	8.83	51.83	159	14.12
23	50	4	14	8.79	53.74	113	2.82
24	75	1	12	8.94	53.50	118	3.27
25	50	5	10	9.04	54.93	100	1.41
26	75	3	14	9.09	54.15	33	0.02
27	25	3	14	8.73	54.54	140	8.17
28	25	2	8	9.15	54.80	193	61.27
29	50	1	6	9.23	55.28	196	64.94
30	75	5	14	8.62	54.98	46	0.08
31	25	3	12	8.71	53.06	159	14.12
32	75	3	12	8.69	55.85	41	0.07
33	50	3	14	8.95	56.57	124	4.24
34	25	4	6	9.09	55.14	183	33.18
35	50	4	8	8.99	56.66	160	14.26
36	75	1	8	9.13	56.15	140	8.17
37	25	5	8	8.91	54.14	169	22.88
38	75	1	6	8.96	52.54	161	15.53
39	25	1	10	8.62	52.35	200	70.26
40	75	1	14	8.70	55.02	110	2.45
41	75	5	6	8.93	55.39	To be predicted	To be predicted
42	25	4	14	9.27	55.81
43	25	5	12	8.83	56.13
44	50	1	10	9.18	52.78
45	25	4	12	8.72	54.51
46	25	3	10	9.26	56.66
47	50	1	12	9.23	55.51
48	25	2	12	8.73	51.44
49	25	1	14	9.03	56.44
50	50	2	6	9.17	55.12

. The numerical values of variables in the dataset are not on the same scale, which may affect the model’s accuracy. The data need to be standardized to eliminate the influence of the magnitude of each data point on the prediction effect. The equations for standardization are as follows:

$$x_{i,n}^{*}={\displaystyle \frac{x_{i,n}-x_{i,\min }}{x_{i,\max }-x_{i,\min }}}$$

(14)

$$y_{j,n}^{*}={\displaystyle \frac{y_{j,n}-y_{j,\min }}{y_{j,\max }-y_{j,\min }}}$$

(15)

wherewith the following variables:

$x_{i,n}$ represents the nth data value of the ith influencing factor in the input layer, where i takes values of 1, 2, 3, 4, 5; it represents the concentration of polymer, concentration of foaming agent, concentration of bentonite slurry, permeability coefficient (before conditioning), and resistivity;

$x_{i,\max }$ and $x_{i,\min }$ represent the maximum and minimum values of all data for different influencing factors;

$x_{i,n}^{*}$ represents the normalized value of the nth data value of the ith influencing factor;

$y_{j,n}$ represents the nth data value of the jth output target in the output layer, with j taking values of 1, 2, representing the slump value and permeability coefficient (after conditioning);

$y_{j,\max }$ and $y_{j,\min }$ represent the maximum and minimum values of different output targets;

and $y_{j,n}^{*}$ represents the normalized value of the nth data value of the jth output target after normalization.

Step 2: Based on the standardized data, select the first 40 sets of data as learning samples for the mapping training of the PSO–RVM model to identify the nonlinear mapping relationship between input values and output values. The remaining 10 sets of data are used as prediction samples to verify the model’s predictive performance.

Step 3: Initiate the PSO program to generate particles, which can find the optimal kernel width through optimization. This kernel width is then passed to the RVM program to compute the prediction samples.

Step 4: Establish a prediction model that meets the requirements based on the optimal parameters. Use this model to predict the prediction samples. Compare the predicted values of the 10 prediction samples with the experimental values. Analyze the relative error and average relative error of predicted and experimental values to validate the accuracy and reliability of the model.

Table 4. Descriptive statistics of the sample data.

	Variables	Minimum	Maximum	Standard Deviation	Dispersion Coefficient	Coefficient of Skewness	Coefficient of Kurtosis
Input layer	cp (%)	25	75	20.530	0.416	0.046	−1.516
	cf (%)	1	5	1.382	0.481	0.111	−1.164
	cb (%)	6	14	2.810	0.282	0.153	−1.253
	Permeability coefficient (before conditioning) (m/s)	8.59	9.27	0.190	0.021	−0.302	−0.933
	Resistivity of sand (Ω·m)	51.42	56.67	1.479	0.027	−0.327	−0.833
Output layer	Slump value (mm)	33	205	50.752	0.362	−0.771	−0.606
	Permeability coefficient (after conditioning) (m/s)	0.02	73.53	22.260	1.049	0.996	−0.233

shows the statistical parameters of the sample data. The polymer concentration, foaming agent concentration, and bentonite slurry concentration in the input variables showed a positively skewed distribution. In contrast, the permeability coefficient (before conditioning) and resistivity showed a negatively skewed distribution. In the output variables, the slump value exhibited a negatively skewed distribution, and the permeability coefficient (after conditioning) exhibited a positively skewed distribution. The low dispersion coefficients of all variables indicated a low degree of data dispersion. The coefficient of kurtosis of the sample data was lower than that of a normal distribution, indicating a relatively uniform data distribution and good representativeness of the sample data.

The predicted results were obtained after inputting the data of 10 prediction samples into the model. The comparisons between the predicted and experimental values are shown in Figure 14. As seen from the figure, the prediction results obtained by the PSO–RVM model better agreed with the experimental values. For the PSO–RVM model, the maximum relative error, minimum relative error, and average relative error between the predicted and experimental values of the slump were 2.99%, 0.57%, and 0.53%, while the corresponding errors of the permeability coefficient were −2.79%, 0.67%, and 0.39%. For the BP model, the maximum relative error, minimum relative error, and average relative error between the predicted and experimental values of the slump were 4.88%, −4.94%, and 0.92%, while the corresponding errors of the permeability coefficient were 8.40%, −8.12%, and 2.75%. This demonstrated the high accuracy of the PSO–RVM model proposed in this study for predicting the effectiveness of soil conditioning.

Further, the correlation between the predicted and experimental values using the Pearson correlation coefficient was analyzed. The correlation coefficient between the slump values predicted by PSO–RVM and the experimental values was 0.9910, while that between the permeability coefficients predicted by PSO-RVM and the experimental values was 0.9992. The correlation coefficient between the slump values predicted by BP and the experimental values was 0.9780, while that between the permeability coefficients predicted by BP and the experimental values was 0.9829. This indicated that the prediction results based on the PSO–RVM algorithm had a higher correlation with the experimental results and better stability. The PSO–RVM algorithm therefore provides a new method for predicting the effectiveness of soil conditioning in water-rich sand layers.

5. Field Application

For the tunnels from Chaoyang Station to Qingfeng Station on Guangzhou Metro Line 14, a combination soil conditioning scheme of “polymer + foam agent + bentonite slurry” was adopted when shield tunneling in a water-rich gravel sand layer. We monitored the permeability coefficient and resistivity parameters of the original sand layer in real-time during shield tunneling. With the permeability coefficient of 3 × 10⁻⁶ m/s and slump of 170–200 mm as the target values, the PSO–RVM algorithm was used to back-analyze the polymer, foam agent, and bentonite slurry concentrations required to achieve the desired soil conditioning effect to obtain the optimal soil conditioning parameters in real time. Tunnel rings ranging from 740 to 820 were selected as the analysis objects to analyze the effects of this soil conditioning optimization method. The tunneling parameters of each ring, including thrust, torque, soil chamber pressure, and advance rate, were collected, as shown in Figure 15.

As shown in Figure 15a, there was a significant difference in total thrust between the unoptimized and optimized segments. When the conditioning additives were unoptimized, the total thrust values were unstable with large fluctuations, occasionally showing significant increases or decreases. The average thrust was 1.30 MN with a fluctuation range of 96 kN. After optimizing the conditioning additives, the total thrust of the shield machine became more stable, with an average thrust reduced to 0.98 MN and a fluctuation range of 37 kN. The optimization method could significantly reduce the total thrust of the shield machine, effectively improving the flow plasticity of the soil conditioning and demonstrating a remarkable improvement in soil conditioning effectiveness.

Figure 15b shows that the cutterhead torques were unstable, with large fluctuations when the conditioning additives were unoptimized. The average cutterhead torque was 1.58 MN·m with a fluctuation range of 95 kN·m. After optimizing the conditioning additives, the cutterhead torque and fluctuation range decreased, and the average torque remained at 1.29 MN·m with a reduction of approximately 0.29 MN·m compared to the unoptimized case. This was mainly due to the further reduction in the internal friction angle and shear strength of the conditioned soil with the optimized conditioning additives. It led to a decrease in the resistance encountered by the cutterhead during the excavation of the soil conditioning.

Figure 15c shows the soil chamber pressure stabilizing at around 96 kPa after optimizing the conditioning additives. The fluctuation ranges before and after optimization were 29 kPa and 6 kPa, respectively. In this state, the conditioned soil could better balance the water and soil pressure at the tunnel face, maintain the stability of the tunnel face, control ground deformation, and ensure the stable advancement of the shield tunneling.

From Figure 15d, the average torque of the screw conveyor decreased from 15.05 kN·m to 9.10 kN·m after optimizing the conditioning additives. The fluctuation range also decreased from 3.91 kN·m to 2.53 kN·m.

The above analysis indicates that compared to the unoptimized area, after optimizing the soil conditioning parameters, the construction parameters of the shield machine were relatively stable, with reduced variability. The real-time optimization of soil conditioning parameters based on the PSO–RVM algorithm achieved good results. The relatively stable construction parameters of the shield machine were more conducive to keeping the machine in good working condition, which would effectively improve the efficiency of tunnel boring.

Note that the cost of soil conditioning additives is an important concern in shield tunnel construction. The experienced cost of soil conditioning in the Guangzhou area was approximately 500–900 RMB/m. After implementing the project, the average cost of soil conditioning additives was approximately 860 RMB/m. Although the material cost increased after optimizing the parameters, it was still within an acceptable range. Considering the additional economic and social benefits brought by the reduction in shield machine losses, improvement in construction safety, and advancement of the project schedule, the real-time optimization of soil conditioning parameters based on the PSO–RVM algorithm has potential promotion value.

6. Conclusions

Conducting permeability coefficient tests on water-rich gravel sand under different concentrations of polymer, foaming agent, and bentonite slurry, the variation pattern of the permeability coefficient was studied. The normalized effect, main effect, and interactions were analyzed based on factorial experiments. The following main conclusions are drawn:

(1)

As the concentration of bentonite slurry increased, the foaming agent’s improvement effect on the permeability of the conditioned soil gradually weakened. Under conditions of high polymer concentration (75%) and a concentration of bentonite slurry exceeding 10%, further increasing the concentrations of bentonite slurry and foaming agent had a weak impact on the permeability coefficient.

(2)

The significance of main effects, first-order, and second-order interactions on the permeability of conditioned soil were as follows: concentration of polymer (A) > concentration of foaming agent (B) > concentration of bentonite slurry (C) > first-order interactions (A × B, A × C, B × C) > second-order interaction (A × B × C).

(3)

The interaction relationship was mainly characterized by synergistic effects with antagonistic effects as secondary. The first-order interactions A × B, A × C, B × C mainly manifested as synergistic effects, while the second-order interaction A × B × C mainly exhibited antagonistic effects.

(4)

The PSO algorithm was utilized for parameter optimization. A shield tunneling soil conditioning prediction model based on the PSO–RVM algorithm was proposed, with the PSO model finding the optimal parameters for the RVM model. The maximum relative error between the predicted values based on the PSO–RVM model and the experimental values was less than 3%.

The field application of the PSO–RVM algorithm showed that the tunneling parameters of the shield machine were relatively stable after optimizing the soil conditioning parameters, indicating that the proposed PSO–RVM algorithm achieved good results. Note that the PSO–RVM algorithm was used to back-analyze the polymer, foam agent, and bentonite slurry concentrations required to achieve the desired soil conditioning effect and obtain the optimal soil conditioning parameters in real time. The prediction results of the PSO–RVM model depend on the accuracy of the soil parameters in front of the shield machine. However, obtaining the parameters of the soil to be excavated in real time is a challenge in practical engineering.