1 Introduction

In urban areas, the use of TBMs to construct tunnels in soft soils has become the optimal solution for mitigating stresses caused by limited ground space, which offers several advantages, including increased efficiency, safety, and environmental friendliness [28, 74, 99]. Slurry TBM is a type of TBM specifically designed to tackle the challenges posed by tunneling in saturated non-cohesive soils and a complex hydrogeological environment [13, 100]. In this method, bentonite slurry within the TBM excavation chamber is pressurized to penetrate the unexcavated soil, with the aim of forming a low-permeability filter cake to stabilize the tunnel face (Fig. 1). This infiltration process has two effects. Firstly, it reduces soil permeability, facilitating the efficient sealing of the tunnel face. As a consequence of this effective sealing, the slurry pressure is successfully transferred to the soil skeleton, thereby ensuring the stability the tunnel face [53, 63, 101]. Secondly, it can also affect the groundwater flow by infiltrating the soil, which in turn increases pore water pressure. Excess pore water pressure can cause instability and deformation of the tunnel face, posing potential hazards such as tunnel collapse [8, 12, 88]. Hence, the pressure infiltration behavior of bentonite slurry in the vicinity of a slurry TBM is a crucial factor that determines the effectiveness of tunnel face support. Ensuring the success of a slurry TBM tunneling project hinges on a comprehensive understanding of bentonite slurry infiltration behavior in sand.

Fig. 1
figure 1

Schematic of tunnel face support principle in front of a slurry TBM

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Prior to its application in slurry TBM tunneling, bentonite slurry has already gained extensive use as a drilling fluid when constructing bored piles [49] and diaphragm walls [33, 50, 65]. Despite their different applications, they share a common objective, which is to form an effective seal on the excavation surface and provide temporary stabilization of the excavation. In addition, bentonite slurry must provide adequate yield stress to carry the drill cuttings out.

Bentonite slurry is known to exhibit shear-dependent viscosity [23], yield stress [38], and thixotropy [76, 77] even at very dilute concentrations of only a few percent solids. This leads to diverse and complex behavior when infiltrating the sand. Essentially, the infiltration of bentonite slurry into the sand can be conceptualized as a problem involving the flow of non-Newtonian fluids within porous media. In this context, four primary approaches exist for modeling this process: continuum models [15, 79], bundle of tubes models [27, 30], numerical methods [39, 92, 93], and pore-scale network modeling [5, 52]. However, it is important to note that achieving such modeling entails simplifying the situation, and typically involves making several assumptions about the properties of the fluid and the porous media. For example, steady-state and incompressible flow assumptions are often applied, and the porous media is assumed to be homogeneous. As a result, it is inevitable to encounter discrepancies between the modeling results and the reality.

Due to space limitations and the concealed nature of the slurry infiltration process in the soil, direct observations during a tunnel project are impractical. Therefore, as an indirect approach, field monitoring of pore water pressure induced by TBM tunneling has been successfully implemented in many tunnel projects, including prominent examples like the Green Heart Tunnel in the Netherlands [88], the London Underground Limited Central Line in England [84], and Line No.1 of the Hangzhou metro in China [17]. This monitoring technique enables the inference of slurry infiltration condition at the front face of the TBM. For instance, during the construction of the Green Heart Tunnel, the pore water pressure in the ground increased significantly during the TBM drilling process, indicating that the expected filter cake was unlikely to shape due to the cyclic rotation of the cutterhead. Instead, the generated pore water pressure gradually dissipated during the standstill, suggesting that the effective sealing of the tunnel face was achieved. Nevertheless, detailed information regarding the pressure infiltration behavior of bentonite slurry, like the time span of filter cake formation, infiltration distance, pressure transfer characteristics, and particle migration, is still not obtainable. These aspects are crucial for evaluating the effectiveness of the tunnel face support. For that purpose, Hu et al. [41] conducted experiments using a miniature slurry TBM with a diameter of 30 cm, which closely approximated real tunnel construction. These tests allowed for the analysis of the impact of slurry properties, ground permeability, and TBM advance rate on ground settlement. However, it should be noted that conducting such tests can often be cost-prohibitive and time-consuming, demanding an extended testing duration for individual samples. To date, laboratory-scale infiltration column tests have emerged as the predominant method for modeling the infiltration of bentonite slurry in saturated soils. In addition, with the advancements in computer hardware and software, numerical simulations, notably computational fluid dynamics (CFD) coupled with the discrete element method (DEM), have gained increasing prominence in studying slurry infiltration. Both approaches have undoubtedly yielded valuable insights, greatly enriching our comprehension of bentonite slurry infiltration in saturated sand. However, these investigations still face challenging issues, particularly in establishing a clear bridge between the obtained results and their direct applicability to tunnel practices. Hence, it is imperative to address the existing gaps in our current understanding, confront the challenges posed by diverse conditions, and decipher the implications for ensuring efficient and secure slurry TBM tunneling operations.

This paper attempts to provide a systematic comprehension of the pressure infiltration behavior of bentonite slurry in saturated sand by reviewing related studies, including laboratory tests and numerical simulations. The limitations of these two methods are discussed, and a simple new strategy is proposed to create comparable hydraulic conditions in the laboratory tests similar to those formed in the field. Furthermore, this paper summarizes the membrane behavior of filter cake from macro to micro scales and discusses the factors that affect the sealing quality of the filter cake.

2 Bentonite slurry as the drilling fluid

As a commonly used type of drilling fluid, bentonite slurry plays a crucial role in stabilizing the soil around excavation sites. By forming a low-permeability barrier, it prevents water and other fluids from infiltrating the excavation and maintains the stability of the surrounding soil. In addition, it is designed to carry drill cuttings to the surface, ensuring a smooth and efficient drilling process.

2.1 Basic properties

Bentonite slurry is prepared by mixing bentonite clay powder and water. Most bentonites fall into two categories: sodium bentonite and calcium bentonite. The main distinction lies in the types of external cation that becomes absorbed onto the surface of clay particle during mineral formation. Owing to its superior swelling capacity, sodium bentonite sees more widespread usage compared to calcium bentonite [2, 3, 37, 61].

Montmorillonite clays, the main component of sodium bentonite, have the highest swelling capacity, and are responsible for viscosity build-up and filter cake formation. When dispersed in water under condition of high shearing mixing, these clays can disintegrate into tiny plate-like particles, with negative charges on their surfaces and positive charge along their edges (Fig. 2a). The platelets’ permanent negative surface charge can create sufficiently large and long-range repulsive forces to counter the attractive van der Waal forces (which can cause the particles to adhere upon contact), yielding a stable colloidal dispersion [18, 45]. When the slurry is left undisturbed, the bentonite platelets flocculate and an interlocking structure could be formed due to electrical bonding forces, causing the slurry to gel (Fig. 2b). If agitated again, these electrical bonds can be disrupted, returning the slurry to a fluid state with the particle resembling that shown in Fig. 2a. This phenomenon is known as thixotropy.

Fig. 2
figure 2

Schematics of two arrangements of bentonite platelets in water: a sheared with parallel orientation; b flocculation (adapted and modified from the [34])

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2.2 Rheological behavior

Bentonite slurries exhibit strong colloidal properties and behave as non-Newtonian fluids, causing an increase in liquid viscosity [44]. Typically, bentonite slurries used for drilling fluids are prepared at concentrations ranging from 3 to 7% [11]. The rheological behavior of bentonite slurries can vary due to differences in the properties of bentonites from different sources. Therefore, there has been considerable research on rheology of bentonite slurries, which is influenced by factors such as clay composition [4, 6, 60], temperature or pH [70, 81, 83], absorbed additives [1, 40], and electrolyte types [43, 66].

To characterize the rheological behavior of bentonite slurry, it is important to obtain its flow curve. The flow curve of a bentonite slurry describes the correlation between the shear stress (τ) and shear rate (\(\dot{\gamma }\)) of the slurry, and can be acquired by measuring the slurry sample using a viscometer. Since the flow curve is a fundamental parameter for predicting the flow behavior of a bentonite slurry, it is widely used in drilling fluid engineering and tunneling engineering. Based on the flow curve, the rheological behavior of a bentonite slurry can be further characterized using various rheological models, but three models are commonly used: the Bingham plastic model, the Power law model, and the Herschel–Bulkley model [42, 47, 85]. Their mathematic expressions are shown as follows:

  • The Bingham plastic model: \({\tau =\tau }_{0}+K\dot{\gamma }\)  

  • The Power law model: \(\tau =K{\dot{\gamma }}^{n}\)

  • The Herschel–Bulkley model: \({\tau =\tau }_{0}+K{\dot{\gamma }}^{n}\)

where τ0 denotes the yield stress; K represents the flow consistency coefficient; n is the flow behavior index. Besides, many other models, such as the Casson model [46], Robertson-Stiff model [78], and Sisko model [67], have also been verified to achieve satisfactory accuracy. However, due to the resulting equations’ complexity, which often involve four or even five parameters, these models have not been widely adopted.

Table 1 presents the rheological measurement test programs used in previous studies. It is evident that the laboratory test conditions were not consistent across the research. For instance, the mixing speed and hydration time while preparing the slurry sample, and the shearing history when measuring the flow curve varied significantly. These differences can significantly impact the test results, either directly or indirectly, leading to incomparable outcomes between studies. To obtain meaningful laboratory measurements, it is crucial to fully reconstitute the slurry samples. Additionally, when comparing measurements of different slurries, it is essential to ensure that the test conditions are stable as closely as possible.

Table 1 Rheological measurement test programs from some previous studies
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2.3 Workability improvement

The use of pure bentonite slurry (water–bentonite slurry) may not always be suitable in all circumstances when operating a slurry TBM. This is particularly true in very coarse sand and gravel, where the slurry would quickly penetrate the ground at the TBM’s face without generating the necessary support pressure. A good example of this is the Hermetschloo Sewage Gallery tunnel construction in Zurich, where various surface failures occurred due to the high permeability of the ground [35]. According to Fritz [35], the critical range of permeability values for using pure bentonite slurries is expected to be between 10–4 to 5 × 10–3 m/s. Accordingly, various additives have been considered to enhance the adaptability of bentonite slurry. In particular, Fritz [35] identified the best combination of modified slurry by testing potential additives, including polymer, sand, vermiculite, mica flakes, and sawdust, for use in the Zimmerberg Base Tunnel. Meanwhile, Yang et al. [95] discovered that the workability of calcium bentonite slurry could be significantly improved through the addition of sodium hexametaphosphate, as evidenced by microscopic analysis. Similarly, Bohnhoff and Shackelford [9] verified the effectiveness of polyacrylate-modified bentonite in providing increased resistance to salt degradation of membrane behavior. Moreover, there have been numerous attempts to improve the physical and chemical compatibility of bentonites through polymer modification, such as carboxymethyl cellulose (CMC) modified bentonite [22, 51, 86] and Xanthan gum [10, 14, 69]. Figure 3 illustrates how the rheological behavior of bentonite slurry can be improved by incorporating varying amounts of CMC.

Fig. 3
figure 3

Flow curves of bentonite slurries with different amounts of CMC (data extracted from [86])

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3 Methods for modeling slurry infiltration in sand

During excavation with a slurry TBM, bentonite slurry is injected into the excavation face to create a protective layer around the tunnel. However, slurry infiltration in front of the machine cannot be observed directly since it occurs underground and out of sight. Engineers must rely on indirect methods, such as measuring the pressure and flow rate of the injected slurry and monitoring changes in pore water pressure and soil stability behind the tunnel face to monitor slurry infiltration. To deepen our comprehension of the infiltration behavior of bentonite slurry during excavation, laboratory experiments and numerical simulations have been utilized to model the process.

3.1 Laboratory experiments

In order to directly model the interaction between the slurry and soil during infiltration, laboratory-scale infiltration column tests have become the primary method. An example of a conventional infiltration column test set-up is shown in Fig. 4. The infiltration column is a key component of the infiltration test, as it contains the test soil (in saturated state) and the prepared slurry and provides the space for the slurry to infiltrate into the soil. The slurry is pressurized by specifying an external air pressure or a hydraulic pressure. During the test, the water that is displaced by the infiltrating slurry in the soil is collected from the bottom. Additionally, the pore pressures of the soil at specific locations along the side wall are recorded.

Fig. 4
figure 4

Example of a conventional infiltration test set-up (a) and the schematic of the infiltration column (b) (adapted and modified after [97])

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The conventional infiltration column has been used to study various issues related to bentonite slurry infiltration in saturated soil. Min et al. [64] conducted an array of infiltration column tests involving nine different slurries and five varied soil types. Their observations revealed three distinct types of slurry infiltration: the presence of a filter cake, a filter cake coupled with an infiltration zone, and an infiltration zone in the absence of a filter cake. Similar results could also be found in some other studies [20, 80, 97]. These findings are important because they demonstrate the variability in slurry infiltration behavior depending on the specific slurry and soil properties, which has implications for the design and operation of slurry TBM excavation. Talmon et al. [82] identified two key processes during slurry infiltration: the initial mud spurt characterized by viscous invasion, and the subsequent development of a filter cake signifying the consolidation of the slurry). It is noteworthy that filter cake formation initiates when the slurry invasion velocity drops below the Peclet criterion (Peclet number Pe < 10), which corresponds to undrained behavior of the slurry. However, as reported by Xu and Bezuijen [90], the estimated Pe slightly surpassses the anticipated threshold (Pe < 10) at the initiation of the filter cake formation. Additionally, other important topics have been explored, such as the effects of slurry characteristics and salt on the infiltration behavior of bentonite slurry [21, 55, 58, 62]. These investigations have provided insights into how changes in slurry properties and environmental conditions can impact the effectiveness of the protective layer around the tunnel face.

Table 2 displays some basic infiltration test conditions with the conventional infiltration column from several representative studies. By analyzing the applied pressure and length of the soil sample in Table 2, the hydraulic gradient i over the soil sample in the study can be calculated. The calculation results are presented in Fig. 5. It can be seen that the majority of the hydraulic gradients in the studies mentioned were maintained at a high level (i > 10). However, according to Bezuijen, et al. [7], the hydraulic gradient at the center of a tunnel face with a diameter of 10 m is approximately 1, which is significantly lower than the values presented in Fig. 5. These high hydraulic gradients could result in a rapid slurry infiltration velocity in the soil, which is not representative of the actual field conditions. Additionally, the obtained test results regarding the time span may not be directly comparable when considering the dynamic removal of the fresh filter cake by the rotating cutterhead during the excavation process.

Table 2 Basic infiltration test conditions using the conventional infiltration column from several representative studies
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Fig. 5
figure 5

Hydraulic gradient over the soil sample from the studies in Table 2

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To mitigate the issue of large hydraulic gradients in conventional infiltration set-ups, Xu and Bezuijen [91] introduced an innovative modification to the infiltration column setup, which includes a small pipe positioned at the bottom of a large cylinder (Fig. 6b). This configuration causes a diameter contraction in the infiltration column, which leads to an increase in flow resistance and a decrease in flow velocity. Theoretically, this should result in a larger equivalent length (Le) and a reduced hydraulic gradient. Assuming the sand within the large cylinder and (D1 in diameter and L1 in height) the small pipe (D2 in diameter and L2 in height) have the same porosity, and there is a steady water flow from the large cylinder to the small pipe (Fig. 6c). The piezometric head at the sand surface is h0. ∆h1 and ∆h3 stand for the head loss in the large cylinder and the small pipe, respectively. ∆h2 represents the head loss due to the contraction of the flow section from D1 to D2. In this case, the flow rate Q should be the same everywhere. According to Darcy’s law, Q in the large cylinder and the small tube can be expressed as

$$Q=\frac{{k}_{\mathrm{w}}\pi \Delta {h}_{1}{{D}_{1}}^{2}}{{4L}_{1}}$$
(1)
$$Q=\frac{{{k}_{\mathrm{w}}\pi \Delta {h}_{3}{D}_{2}}^{2}}{{4L}_{2}}$$
(2)

where kw denotes the hydraulic conductivity of the sand for water. Assuming it is a half-sphere flow from the large cylinder to the small pipe, the head loss in a small area as indicated in Fig. 6c can be calculated by

$$Q=2\pi {k}_{\mathrm{w}}{r}^{2}\frac{\mathrm{d}h}{\mathrm{d}r}$$
(3)

Then, integrating from D2/2 to D1/2 it comes to

$$Q=\frac{\pi {k}_{\mathrm{w}}{\Delta h}_{2}}{1/{D}_{2}-1/{D}_{1}}$$
(4)

The total head loss equals the piezometric head at the sand surface

$${h}_{0}=\Delta {h}_{1}+\Delta {h}_{2}+\Delta {h}_{3}$$
(5)

Combining the equations above to get

$${h}_{0}=\Delta {h}_{1}\left(1+\frac{{L}_{2}{{D}_{1}}^{2}}{{{L}_{1}{D}_{2}}^{2}}+\frac{{{D}_{1}}^{2}}{4{L}_{1}{D}_{2}}-\frac{{D}_{1}}{4{L}_{1}}\right)$$
(6)

To maintain consistent flow conditions, an equivalent cylinder height Le with the same diameter and flow rate as the large cylinder is introduced as

$$\frac{\Delta {h}_{1}}{{L}_{1}}=\frac{{h}_{0}}{{L}_{e}}$$
(7)

Finally, the equivalent cylinder height Le can be expressed as

$${L}_{e}={L}_{1}+{L}_{2}{\left(\frac{{D}_{1}}{{D}_{2}}\right)}^{2}+\frac{{{D}_{1}}^{2}}{{4D}_{2}}-\frac{{D}_{1}}{4}$$
(8)
Fig. 6
figure 6

Comparison between the conventional and modified infiltration column: a conventional infiltration column; b modified infiltration column; c hydraulic condition in modified column (adapted and modified from Xu and Bezuijen [91])

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Given the dimensions shown in Fig. 6b, the equivalent length Le measures 2.83 m. This is a significant increase when compared to the scenario without the small pipe, where L measures 0.17 m (Fig. 6a). In the case of a pressure of 50 kPa (~ 5 m) employed to the sand, the hydraulic gradient i across the sand is approximately 1.77. This value closely resembles the hydraulic gradient at the center of a tunnel face with a 10-m diameter [7]. Building upon the modified infiltration column setup, Xu and Bezuijen [89] conducted a comprehensive investigation. They examined the dynamics of mud spurt and filter cake formation of water-bentonite slurry both before and after removing the external filter cake. Futhermore, they explored the conditions under which filter cake forms during water-bentonite–sand slurry infiltration, accounting for the scenario where bentonite slurry was mixed with excavated soil in the front chamber of the TBM. The influence of sand particle size within the sand column and sand content within the bentonite slurry was also studied [72, 92, 93]. Moreover, the writers examined the infiltration behavior of seawater-based bentonite slurry in the medium and fine sand, all within the context of the modified infiltration column, maintaining a hydraulic gradient of 1.38 [73].

In fact, there is another way to provide a hydraulic gradient comparable to real tunnels. Herein, a simple new strategy is introduced based on the conventional infiltration column. As shown in Fig. 7, a finer sand layer with relatively lower hydraulic conductivity is set below the test sand in the column. Similarly, assuming a steady water flow from the top to the bottom, and the piezometric head at the surface of test sand is h0. ∆h1 and ∆h2 stand for the head loss in the test sand and finer sand, respectively. In this case, the flow rate Q in the column can be expressed as

$$Q=\frac{{k}_{1}\pi \Delta {h}_{1}{{D}_{1}}^{2}}{{4L}_{1}}$$
(9)
$$Q=\frac{{k}_{2}\pi \Delta {h}_{2}{{D}_{1}}^{2}}{4{L}_{2}}$$
(10)

where k1 and k2 are the hydraulic conductivities of test sand and finer sand for water, respectively. The total head loss equals the piezometric head at the surface of test sand

$${h}_{0}=\Delta {h}_{1}+\Delta {h}_{2}$$
(11)

By setting the finer sand layer below the test sand, the equivalent cylinder height Le could be estimated by

$${L}_{e}=\left(1+\frac{{k}_{1}{L}_{2}}{{k}_{2}{L}_{1}}\right){L}_{1}$$
(12)

The equation indicates that Le can be controlled by varying L1 and L2, as well as the ratio between k1 and k2. Therefore, it is possible to adjust the hydraulic gradient i over the soil sample without requiring any modifications to the column. It is worth noting that Eqs. (8) and (12) are only applicable for the cases where the slurry infiltrates within the larger cylinder and the test sand, respectively. In the new strategy, the repeatability of the infiltration test may become a challenging problem.

Fig. 7
figure 7

Hydraulic condition in the sand column with a finer sand layer set below

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3.2 Numerical simulations

Recent advances in computer hardware and software have enabled the simulation of slurry infiltration in sand at a particle-scale level. One significant advancement in computational modeling has been the development of CFD coupled with DEM simulations. In the past, CFD and DEM were two separate techniques used to model fluid and solid particles, respectively. However, with the development of CFD-DEM, these two techniques have been combined into one simulation, enabling more accurate and realistic modeling of fluid-particle interactions.

In CFD-DEM simulations, the fluid flow is considered as a continuous medium, whereas the particles are treated as individual entities that interact with each other and with the fluid. The fluid is modeled using the Navier–Stokes equations, while the particles are modeled using Newton’s laws of motion and contact mechanics. Figure 8 shows the main procedures of a coupled CFD-DEM simulations.

Fig. 8
figure 8

The integrated process of CFD-DEM simulation (modified after Zhang et al. [98])

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For most of the associated studies, the coupled CFD-DEM model was employed to simulate the infiltration column test, an example was shown in Fig. 9. Furthermore, Table 3 summarizes the fundamental parameters and conditions used in CFD-DEM simulations of slurry infiltration in previous studies. From these studies, it can be seen that CFD-DEM simulations offer several advantages over laboratory experiments in modeling slurry infiltration. They provide a detailed understanding of the behavior of individual particles and the fluid at a microscopic scale, which is difficult to achieve experimentally. Moreover, CFD-DEM simulations can offer insight into the mechanisms and fundamental physics of slurry infiltration in sand, which is challenging to observe in experiments. They can be used to investigate a wide range of parameters and conditions, which can be difficult or impossible to control in experiments. Additionally, CFD-DEM simulations can be more cost-effective and efficient than experiments since they do not require extensive laboratory equipment and materials. However, the CFD-DEM method also faces several challenging issues. Due to limitations in computational capacity, the real size of bentonite particles (which typically have diameters less than 150 µm) is often amplified, and intrinsic microscopic forces between bentonite particles are typically ignored. While some studies have used the Johnson-Kendall-Roberts (JKR) model to estimate the cohesion forces between bentonite particles, it is important to note that the cohesion energy density parameter may still not precisely represent the real case. In addition, the heights of sand beds in these simulations are often limited to save calculation time, which may result in an incomplete infiltration process. The platelet shape and stiffness of bentonite particles have not been well reflected in simulations, despite being critical factors in the formation of filter cakes. Finally, the rheological characteristics of bentonite slurry, such as its non-Newtonian behavior, are typically not considered in these simulations.

Fig. 9
figure 9

Slurry infiltration modeling using CFD-DEM simulation: a geometry and boundary conditions of the column; b initial state of the infiltration; c filter cake formation (adapted and modified from Yin et al. [96])

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Table 3 Fundamental settings used in CFD-DEM simulations of slurry infiltration from previous studies
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It is important to keep in mind that laboratory tests and numerical simulations are subjected to a static condition and can only provide an approximation of the actual conditions encountered during tunneling. In fact, the filter cake or infiltration zone is periodically destroyed by the cutting tools under the dynamic excavation process, which can be mainly distinguished into two cases [102]. The tool's penetration depth during its passage is greater in Case A and less in Case B when compared to the distance of slurry infiltration (Fig. 10). Hence, it is advisable to emply these methods in ombination with indirect monitoring techniques to offer a holistic comprehension of soil and slurry behavior, along with the potential risks linked to tunneling. Ultimately, the combination of these methods can help engineers make informed decisions and ensure the safety and success of tunneling projects.

Fig. 10
figure 10

The cutting process of an active scraper at the tunnel face: a Case A; b Case B (modified from Zizka et al. [102])

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4 Pressure infiltration behavior of bentonite slurry

4.1 Slurry infiltration types

According to the FPS [34], when bentonite slurry infiltrates the soil, three distinct sealing mechanisms can be observed: surface filtration, deep filtration, and rheological blocking. Surface filtration takes place as hydrated bentonite particles create bridges at the entrance to soil pores, forming an external filter cake. This mechanism offers the advantage of rapid soil sealing with minimal penetration of bentonite into the soil. In contrast, deep filtration occurs as the slurry progressively penetrates the soil, gradually obstructing the pores and forming a filter cake within them. Finally, rheological blocking arises when slurry flows into the soil until it is constrained by its shear strength. Min et al. [64] categorized slurry infiltration into three distinct types based on the results of 45 infiltration tests conducted with polymer-modified bentonite slurries: Type I, characterized by a filter cake; Type II, characterized by a filter cake along with an infiltration zone; Type III, characterized by an infiltration zone without a filter cake. The average diameter of pore channel in the soil D0 was derived and calculated by

$${D}_{0}=\sqrt{\frac{{k}_{\mathrm{w}}}{31\times 1.5n}}$$
(13)

where n is the porosity of the soil. Furthermore, the criteria for identifying the three slurry infiltration types were established based on the ratio between D0 and the particle size for which 85% by weight of particles in the slurry (d85) (Fig. 11). By using pure bentonite slurry without any additives, the external filter cake along with the infiltration zone was also observed [71, 90].

Fig. 11
figure 11

Relationship between d85/D0 and types of slurry infiltration (adapted and modified after Min et al. [64])

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While the slurry infiltration types can vary depending on the properties of the soil and bentonite slurry, they can generally be categorized into two main types: one involving an external filter cake and an infiltration zone, and the other featuring only an infiltration zone. The term “filter cake” refers to the deposited layer of bentonite (bentonite cake) with extremely low permeability. The formation of an internal filter cake is unlikely, especially for pure bentonite slurry. This is because the effective stress in the soil is mainly borne by the soil skeleton, rather than the bentonite particles suspended in the pore fluids. As a result, it is difficult to achieve a dense state of bentonite particles within the soil. Since bentonite slurries have significant yield stress, the infiltration should, in theory, stop definitely even without forming a filter cake, and at a declining rate as the slurry penetrates further from the excavation.

4.2 Infiltration distance

The depth to which slurry can infiltrate into the soil is influenced by various factors, including soil type, slurry properties, and applied pressure. It is crucial to regulate the infiltration distance of the slurry during excavation stabilization to avoid it from penetrating too deeply into the soil and triggering instability. By means of the infiltration column test, the slurry infiltration distance l is estimated under the assumption that the volume of displaced water equals the volume of slurry invasion:

$$l=\frac{4Qt}{n{\pi D}^{2}}$$
(14)

Figure 12 shows the development of the estimated slurry infiltration distance in two infiltration tests. From Fig. 12, there appear to be two distinct stages of slurry infiltration: a rapid initial invasion followed by a very limited infiltration. These two stages were identified as the initial mud spurt and the subsequent filter cake formation by Talmon et al. [82]. Besides, when compared to the findings from the conventional column, the filter cake was formed at a relatively slowly pace under hydraulic gradients similar to those encountered in real tunneling scenarios.

Fig. 12
figure 12

Developments of the estimated slurry infiltration distance in two different set-ups (modified after Xu and Bezuijen [91])

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To predict the slurry infiltration distance, Krause [48] introduced a straightforward mathematical model aimed at determining the maximum theoretical infiltration distance during a mud spurt. This distance is achieved when the wall shear stresses within the pores of the soil equal the yield stress of the slurry. However, the results of many experiments have shown that the final infiltration distance is significantly smaller than the maximum theoretical value [82, 89]. This discrepancy can be to the formation of a filter cake, which halts the mud spurt before it can reach the maximum theoretical infiltration distance. Furthermore, based on the maximum infiltration distance, a time-dependent infiltration distance model was further introduced by Xu and Bezuijen [88]. While the model prediction exhibits good agreement with the experiments, it is still essential to experimentally determine the hydraulic conductivity of the soil for slurry to make accurate predictions.

4.3 Pressure transfer characteristics

The primary objective of using pressurized bentonite slurry used at the front face of the slurry TBM is to provide sufficient effective face support, a critical aspect reliant on the effective transfer of pressure around the tunnel face. According to the measured pore pressures during the tests, the writers identified the pressure transfer characteristics both with and without the filter cake [72]. In the experiments, a pure bentonite slurry with a specified concentration was used to examine the potential formation of a filter cake in both fine and coarse sand. The real-time measurements, locations of the pore pressure transducers used (P1–P4), and infiltration soil samples (after drying) are shown in Fig. 13.

Fig. 13
figure 13

Measured pore pressures during the infiltration column tests: a fine sand; b coarse sand (modified after Qin et al. [72])

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After the test began, the recorded pore pressures in the fine sand quickly decreased to a very low level. A significant pressure difference (~ 50 kPa) is observed between P1 and P2, indicating that the soil skeleton (effective stress) is bearing the applied slurry pressure (total pressure) rather than the pore pressure in the soil. This is due to the effective sealing on the sand bed surface, which indicates the formation of a low permeable filter cake. Furthermore, the pore pressures at P2–P4 decrease to their hydrostatic values, and there is insufficient driving force for further bentonite infiltration. The extracted filter cake above the fine sand curls up after drying because of the high concentration of bentonite. Without the filter cake, the pore pressures in the sand bed would be maintained at a high level but at a decreasing gradient as the infiltration distance increases.

4.4 Hydraulic conductivity of sand for the slurry

Based on the recorded water discharge and pore pressures in the infiltration test, the hydraulic conductivity of sand for the slurry ks can be estimated using Darcy’s law [71,72,73, 89,90,91,92,93]:

$${k}_{s}=\frac{4\Delta Q\Delta l}{\pi {{D}_{1}}^{2}\Delta {h}_{\mathrm{p}}}$$
(15)

where ∆Q represents the water discharge in one second, ∆l stands for the thickness of sand layer between two adjacent pore pressure transducers, and ∆hp indicates the piezometric head difference between two adjacent pore pressure transducers. Figure 14 illustrates the estimated values of ks plotted against the infiltration distance l for both fine and coarse sand, utilizing the test results from Qin et al. [72]. The ks values estimated through the pressure data from P1 and P2 are denoted as P1-P2, and so forth. It is important to note that in the tests, the ∆l is specified as 1.25 cm between P1 and P2, as the initial 1.25 cm represents solely the presence of slurry prior to the formation of the filter cake. In cases where filter cake formation prevails, the flow resistance within the cake substantially exceeds that within the sand bed. Consequently, the ks is regarded as the hydraulic conductivity of the filter cake above the sand.

Fig. 14
figure 14

Hydraulic conductivity against infiltration distance in a fine sand and b coarse sand (modified after Qin et al. [72])

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The experiment showed that as the slurry infiltrated the region between two adjacent pore pressure transducers, the presence of suspended bentonite particles and increased slurry viscosity significantly heightened the flow resistance while passing through the interconnected voids of the sand bed. As a result, the hydraulic conductivity of the local sand layer experienced a notable reduction. Based on the results of P1-P2 in Fig. 14a, the hydraulic conductivity of the filter cake was estimated to be on the order of 10–8 m/s. It is essential to emphasize that assuming a 1 mm thickness for the filter cake would likely result in an even lowers hydraulic conductivity, potentially reduced by a factor of 10. Furthermore, the writers investigated the relationship between the ks and flow velocity v (Darcy velocity) by applying various pressures in the absence of filter cake formation [71], as shown in Fig. 15. Although there is a great deviation in the estimated ks, there is a general linear relationship between ks and v. This suggests that reducing the flow velocity of the slurry can increase the flow resistance in the soil. This is due to the rheological properties of bentonite slurry, where a lower shear rate results in higher viscosity and increased viscous resistance. In practice, using a lower slurry pressure is beneficial for promoting the rheological resistance of bentonite slurry in the soil, which can help stabilize the excavation of TBM tunneling.

Fig. 15
figure 15

Hydraulic conductivities against Darcy velocity (data extracted from Qin et al. [71])

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5 Membrane behavior of filter cake

The filter cake generated by the bentonite slurry during excavations can exhibit membrane-like behavior, which means that it behaves like a thin, impermeable membrane or layer. This layer acts as a barrier that prevents further infiltration of the slurry into the soil and helps to maintain pressure balance in the excavation face, preventing blowouts or collapses.

5.1 Physical properties of filter cake

After conducting an infiltration test, it is possible to extract a sample of the filter cake, and its basic properties, such as thickness lf, void ratio e, and hydraulic conductivity kf, can be analyzed and obtained. Since the pores of the filter cake are almost completely filled with water during its formation, it is assumed to have a saturation degree Sr of 100%. The void ratio e of the filter cake was calculated by Min et al. [62] using the following equation and found to be 2.36.

$$e=\frac{w{G}_{\mathrm{s}}}{{S}_{\mathrm{r}}}$$
(16)

where Gs represents the specific gravity of bentonite and w denotes the water content of filter cake sample. Moreover, using Darcy’s law, the hydraulic conductivity kf can be calculated by incorporating the volume of filtered water and the measured thickness. However, obtaining a clean and intact filter cake sample after the infiltration test can be quite challenging, particularly when dealing with cohesive filter cakes that are difficult to remove from the sand bed surface. This can lead to inaccuracies in measurements of thickness and water content. As a result, inaccuracies may arise when attempting to determine the hydraulic properties of the filter cake.

The modified fluid loss test, developed by Chung and Daniel [19], has been validated as a rapid and reliable method for estimating the hydraulic properties of a filter cake, particularly under lower pressures conditions [29, 57, 68, 87]. This method is derived from the standard fluid loss test commonly employed in laboratories to measure the ability of drilling fluids to prevent fluid loss in permeable soils. The test involves placing a sample of the drilling fluid in a filter press, applying pressure to force the fluid through a filter paper, and measuring the amount of fluid that passes through the filter over a specified time period. Using the modified fluid loss test can yield more convenient filter cake samples and provide information about their physical properties (Fig. 16). By incorporating the principles of cake filtration theory and Darcy’s law, it becomes feasible to estimate the hydraulic conductivity of the filter cake accurately. Figure 17 shows the calculation results from a series of modified fluid loss tests, and three bentonite slurries were tested under various pressure levels [68]. As shown in the results, there is a clear trend where increasing the applied pressure leads to a decrease in both the hydraulic conductivity kf and void ratio e of the filter cake, regardless of the types of bentonite slurry used. The hydraulic conductivity kf ranged from 2.16 × 10–11 to 2.58 × 10–10 m/s while the void ratio e ranged from 11.96 to 16.60. This is consistent with the test results by the writers [71].

Fig. 16
figure 16

Measuring the thickness of a filter cake sample after the fluid loss test

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Fig. 17
figure 17

Results from a series of modified fluid loss tests on three different bentonite slurries with a concentration of 6%: a hydraulic conductivity plotted against applied pressure; b hydraulic conductivity plotted against average void ratio (modified after Nguyen et al. [68])

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5.2 Micro-mechanism of filter cake formation

To obtain a deeper understanding of the micro-mechanisms involved in filter cake formation, a detail analysis of the filter cake’s structure and composition at a microscopic-particle level is imperative. Although the CFD-DEM simulations suggest that the filter cake is formed due to the initial bentonite particles trapped in the void between the sand particles and its subsequent sedimentation, this simplified representation of bentonite particles as small spheres falls short in fully explaining the particle aggregation and interaction during filter cake formation [32, 98]. As a result, the impact of the resulting structure on the hydraulic properties of the filter cake remains unclear. One effective approach to explore the micro-mechanism of filter cake formation is through microscopy and imaging techniques, with scanning electron microscopy (SEM) being particularly valuable. SEM can generate high-resolution images of the filter cake’s microstructure, allowing researchers to analyze the morphology and composition of the filter cake at the particle level. After Du et al. [25] used the freeze-drying method to prepare filter cake samples, they found that the filter cake exhibited a consistent honeycomb structure (Fig. 18a). Moverover, theire findings revealed that the polymer, specifically polyanionic cellulose, had the capacity to create a three-dimensional network structure amidst bentonite particles. This network structure effectively obstructed the the intergranular pore space, resulting in a constricted and tortuous flow path for liquid, consequently leading to a reduced hydraulic conductivity (Fig. 18b). Additionally, the filter cakes were also found to be composed of a number of stacked monolithic units and detrital mineral grains [59, 62]. To obtain a clear view of the particle shape and packing arrangement of the filter cake, the writers adopted a new method. They first oven-dried the wet filter cake sample and then placed two small pieces on two different bases, allowing for a thorough observation of the microstructure from two directions. As depicted in Fig. 19, the filter cake, having a thickness of 0.1 mm after drying process, takes on the appearance of a coverlet draped over the sand bed. These platelet layers are aligned perpendicular to the direction of flow, yielding a smooth surface along the path of infiltration [72, 73]. While the oven-drying process may impact the overall integrity of the filter cake sample, it effectively retains the relative positions and orientations of the bentonite particles.

Fig. 18
figure 18

The microstructure of filter cake samples formed by a water–bentonite slurry; b polyanionic cellulose–amended bentonite slurry (modified after Du et al. [25])

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Fig. 19
figure 19

SEM images of the filter cake: a overview; b perpendicular direction of infiltration; c infiltration direction (modified after Qin et al. [72])

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5.3 Factors affecting the sealing quality of filter cake

A good filter cake is essential in excavation operations to maintain stability, prevent fluid loss into the soil, and minimize the risk of blowouts. However, the sealing quality, which refers to the hydraulic conductivity of filter cakes, may be seriously affected by the complex working conditions. Based on the findings of the fluid loss test conducted with various types of bentonite, it has been found that bentonite with a high specific surface area and swelling ability forms a denser and more impermeable filter cake [68]. Increasing the pressure on the slurry leads to a higher infiltration rate, which allows more bentonite particles to settle and form a more compact filter cake [19, 71]. When filter cakes encounter nonstandard permeant solutions, such as seawater, acidic or heavy metal liquids, the hydraulic conductivity tends to increase compared to water. This is mainly due to the flocculation of bentonite particles, which leads to a looser structure and wider flow paths for liquid to pass through [26, 31, 36, 57, 73]. In addition, during the excavation process, natural soils will inevitably mix with bentonite slurry. It is important to control the fine sand content in the slurry, as the presence of fine sand particles can hinder the formation of a dense and horizontally-oriented arrangement of bentonite platelets, resulting in a poorly formed filter cake [72].

6 Concluding remarks

This paper provides a comprehensive review of existing studies that are related to bentonite slurry infiltration at the front face of the TBM during tunneling in saturated sand. The rheological measurement of bentonite slurry as the drilling fluid has been summarized and discussed. Based on the review of laboratory experiments, specifically infiltration column tests, as well as numerical simulations utilizing CFD-DEM simulations to model slurry infiltration in sand, a comprehensive overview of the pressure infiltration behavior of bentonite slurry has been synthesized. This encompasses aspects such as slurry infiltration types, infiltration distance, pressure transfer, and hydraulic conductivity. Finally, the physical properties and micro-mechanisms of filter cake have been condensed, as well as the factors that can affect the sealing quality of the filter cake.

The review revealed that the rheological measurement results of bentonite slurry tend to vary among studies due to differences in testing conditions, such as the initial mixing speed and hydration time used to prepare the slurry sample, as well as the shearing history during the flow curve measurement. Because the shearing conditions in a rheometer fundamentally differ from those experienced by the slurry when it flows through soil pores, it is essential to establish a connection between the rheological parameters obtained from a rheometer and the actual flow behavior of bentonite slurry in porous media in future research.

The conventional infiltration column used in infiltration column tests would produce a high hydraulic gradient across the soil sample, leading to a rapid infiltration velocity of the slurry. Consequently, the findings derived from such tests may not accurately reflect the actual slurry infiltration behavior in TBM tunneling. The modified infiltration column by setting a small pipe can yield a reduced hydraulic gradient comparable to that formed in the field, based on which the slurry infiltration distance and time span of filter cake formation are more representative of the conditions in a real tunnel case. Additionally, a simple new strategy for reducing the hydraulic gradient in infiltration column tests is proposed based on the concept of the equivalent length of the soil sample. This involves adding a finer sand layer with a lower hydraulic conductivity below the test soil, which allows for more flexibility in adjusting the hydraulic gradient without requiring any modifications to the column.

The current CFD-DEM simulations on slurry infiltration face limitations in computational capacity, leading to simplifications in various aspects, such as the size and shape of bentonite particles, intrinsic microscopic forces between bentonite particles, length of the sand bed, and flow behavior of fluid phase. These simplifications, while enabling numerical simulation, may introduce discrepancies between the simulated and actual infiltration processes. Both laboratory tests and numerical simulations are conducted under static conditions and can only offer an approximation of the actual conditions encountered during tunneling. For more accuracy, it would be advisable to consider the dynamic excavation process in both laboratory experiments and numerical analyses in future studies.

Based on the observations from the infiltration column tests, two main slurry infiltration types have been identified: one with an external filter cake and an infiltration zone, and the other with only an infiltration zone. Incorporating the formation of a filter cake into existing theoretical models for predicting the slurry infiltration distance remains a challenging problem. The low permeable filter cake formed at the soil–slurry interface can function as a media to transfer the slurry pressure onto the soil skeleton. The viscoplastic behavior of bentonite slurry causes an increase in flow resistance at lower flow velocities, making it beneficial to use relative lower pressures to promote rheological resistance in the soil during TBM tunneling. This helps stabilize the excavation process and maintain the tunnel face.

The modified fluid loss test provides a quick and convenient approach for exploring physical properties of the filter cake, such as thickness, average void ratio, and hydraulic conductivity. SEM images of the filter cake reveal that it is composed of multiple layers of bentonite platelets, arranged perpendicular to the flow direction. This will result in a tortuous flow path for liquid, leading to a low hydraulic conductivity. It is crucial to pay attention to the sealing quality of the filter cake in situations where drilling encounters nonstandard permeant solutions, such as seawater, acidic or heavy metal liquids. In addition, during the tunneling process, it is important to manage the fine sand content in the slurry; otherwise, a poorly formed filter cake may be obtained.