Calculation method for the formation time of dynamic filter cake in slurry shield tunneling

Abstract

In slurry shield tunneling, the stability of tunnel face is closely related to the filter cake. The cutting of the cutterhead has negative impact on the formation of filter cake. This study focuses on the formation time of dynamic filter cake considering the filtration effect and rotation of cutterhead. Filtration effect is the key factor for slurry infiltration. A multilayer slurry infiltration experiment system is designed to investigate the variation of filtrate rheological property in infiltration process. Slurry mass concentration C_L, soil permeability coefficient k, the particle diameter ratio between soil equivalent grain size and representative diameter of slurry particles d₁₀/D₈₅ are selected as independent design variables to fit the computational formula of filtration coefficient. Based on the relative relation between the mass of deposited particles in soil pores and infiltration time, a mathematical model for calculating the formation time of dynamic filter cake is proposed by combining the formation criteria and formation rate of external filter cake. The accuracy of the proposed model is verified through existing experiment data. Analysis results show that filtration coefficient is positively correlated with slurry mass concentration, while negatively correlated with the soil permeability coefficient and the particle diameter ratio between soil and slurry. As infiltration distance increases, the adsorption capacity of soil skeleton to slurry particles gradually decreases. The formation time of external filter cake is significantly lower than internal filter cake and the ratio is approximately 3.9. Under the dynamic cutting of the cutterhead, the formation time is positively associated with the rotation speed of cutter head, while negatively with the phase angle difference between adjacent cutter arm. The formation rate of external filter cake is greater than 98% when d₁₀/D₈₅≤ 6.1. Properly increasing the content or decreasing the diameter size of solid-phase particles in slurry can promote the formation of filter cake.

Graphical abstract

Keywords

slurry infiltration / filtration coefficient / dynamic filter cake / formation time / rotation speed / phase angle

1 Introduction

In the field of oil exploration, slurry wall protection technology is widely used to prevent the instability and collapse of hole in saturated strata [1,2]. Since 1967, slurry has been applied in shield tunneling due to its good rheological properties [3,4]. Slurry suspension is commonly used as a supporting medium of tunnel face, which consists of bentonites particles, clay particles and suspending agent slime like carboxymethylcellulose sodium (CMC) [5–7]. In slurry shield tunneling, solid particles can be pressed into the soil and clog the soil pores under the pressure difference between the slurry pressure in chamber and the hydrostatic pressure. Furthermore, filter cake can form on the surface of tunnel face and transfer the slurry pressure to strata in effective stress. In this condition, slurry pressure maintains the stability of tunnel face completely in uniform surface force. However, the cutter head of the shield machine rotates continuously in tunneling. The cutter continuously cut the existed filter cake, meanwhile, the clean soil is exposed. The slurry in pressure chamber infiltrate into the easily permeable clean soil zone and form fresh filter cake. Consequently, if the formation time exceeded the sweep time interval of adjacent cutter arms, complete filter cake cannot form on tunnel face, leading to slurry loss and pressure fluctuation in the chamber. Hence, the formation time of dynamic filter cake is crucial for the analysis of tunnel face stability.

The development of filter cake is primarily investigated through laboratory test. Weiss summarized the stabilization condition of slurry support [8]: 1) the slurry needs to reach a certain yield strength to maintain the force equilibrium of soil skeletons; 2) slurry pressure needs greater than the resultant force of underground water and soil. Min classified the filter cake into three types by the ration size of soil grain size to representative diameter of slurry particles d₁₅/D₈₅ [9]. The results show that the particle size distribution of the soil and the pressure gradient is closely related with the type of filter cake. 1) External filter cake (d₁₅/D₈₅ ≤ 5.26): In fine grained stratum such as silt, almost all particles in slurry directly accumulated on soil surface, forming a thin and dense particles layer, which is called external filter cake. A sharp boundary can be observed between the layer and the soil. 2) External + Internal filter cake (5.26 ≤ d₁₅/D₈₅ < 10.53): In coarser soil like fine sand, the slurry infiltration process can be divided into two stages. The first stage is characterized by rapid infiltration of viscous slurry into the soil pores. In the second stage, suspended particles gradually deposited in soil pores and the infiltration velocity also slows down. Finally, the suspended particles reached equilibrium state of shear stress, and the slurry begins to lose water and consolidate on soil surface [10], which is also consistent with the research results of Müller-Kirchenbauer [11] and Thienert and Pulsfort [12]. In this case, internal filter cake formed in deep soil pores, while micro-permeable external filter cake forms on the soil surface but no sharp boundary generates. 3) Internal filter cake (d₁₅/D₈₅ ≥ 10.53): In coarse sand, gravel and other coarse grain formation, only external filter cake exists.

Previous experimental studies have shown that filtration effect is the main reason for filter cake formation [13,14]. During the infiltration process of slurry in sandy soil, some suspended slurry particles are adsorbed by soil skeleton, the seepage pore size and slurry mass concentration also decreased. The migration and deposition process of particles in porous media is influenced by various factors like inertia force, physical interception, particles adhesion, charge repulsion, Brownian diffusion and hydrodynamic force [13–19]. Filtration coefficient is the critical physical quantity which always used to quantify the adsorption capacity of soil skeleton to slurry particles in macroscopic analysis. On the basic of classics filtration theory, some scholars have verified the adsorption capacity is closely related to the interception, sedimentation and diffusion and proposed some computing methods [20–23]. However, the existing methods contain too many indispensable parameters and are difficult to apply in practical engineering. Furthermore, the variation of slurry rheological property and soil pore characteristic in infiltration process is neglected.

Obviously, cutting has a negative effect on the formation of filter cake. The formation mechanism and characteristics of filter cake considering the interaction with cutter and soil mainly through model experiments [24,25]. On the basis analysis of cutter head arrangement and cutters track, Mao [24] carried out a dynamic slurry infiltration test. The results show that lower rotation speed of cutter head and higher slurry pressure is contribute to the formation of dense filter cake. Effective conversion rate of slurry pressure lowered approximately 18% for every 0.5 r/min increase of rotation speed. For dynamic infiltration tests, the formation time of filter cake and volume of filtration water is positively related to the number of cutter arms. To date, there is no theoretical model to solve the formation time of dynamic slurry filter cake.

Based on the above understanding, this study aims to develop a calculation method for the formation time of dynamic external and external + internal filter cake. The specific objectives of the study include: 1) A multilayer infiltration column experiment system is developed to investigate the content change of slurry particles in slurry infiltration process. Based on numerous experimental results, slurry mass concentration, permeability coefficient, the diameter ratio between soil equivalent grain size and representative diameter of slurry particles are selected as independent variables to fit the multivariable calculation formula of filtration coefficient. 2) Based on the relative relation between the mass of deposited particles in soil pores and infiltration time, a model for calculating the formation time of filter cake is proposed in combination with formation criterion of external filter cake. The accuracy of the proposed model is verified through the previous static and dynamic slurry infiltration experiment studies. The analysis results reveal that the formation of filter cake is negatively correlated with the phase angle difference between adjacent cutter under dynamic cutting. Section 5 performs a parametric sensitivity analysis to discuss the effect of slurry mass concentration, soil porosity, the particle diameter ratio between soil and slurry and rotation speed of cutter head on the formation time of filter cake.

2 Filtration coefficient measurement experiment

2.1 Experimental setup

As shown in Fig.1(a), a multilayer infiltration column experiment system is developed to investigate the infiltration behavior of slurry in soil with different particle size. The system consists of a multilayer infiltration column and several auxiliary devices, including pressure control program, air compressor, slurry storage bucket, water collector, electric furnace, electric balance and data processing program. The multilayer infiltration column is composed of organic glass infiltration column elements, slurry inlet and outlet valve. Each infiltration column element filled with soil individually, and connected through bolts. Rubber gaskets are installed between each element to ensure the airtightness of the multilayer infiltration column. The element is composed of a double-column and two partitions. The inner diameter of upper and lower column are 10 and 3 cm, respectively, the height is 3 cm. As illustrated in Fig.1(b), a tube with 0.5 cm inner diameter 7 cm long at the side wall center of the lower column is connected with hand valve to collect slurry filtrate. Circular holes with 1 cm diameter are opened at the center of the partitions. Pressurized air is supplied to the slurry storage bucket by the pressure device, and then slurry is injected into the multilayer infiltration column.

To facilitate slurry injection, the top column element A is always kept empty and the hand valve is also closed. The single-layer 170 mesh filter screen is laid at the bottom of upper column and 40 mesh for each lower column, which can ensure the sieving rate of bentonite slurry is 100% while the fine sand is 0. Each lower column is filled with glass beads (8 mm in diameter) to reduce the infiltration velocity and facilitate extraction. The measured result results indicate that the concentration difference of slurry after passing through the lower column is less than 10⁻⁴ kg/m³. The influence of glass beads on the measurement of filtrate concentration is neglected. 5 mm distance is maintained between the soil surface and the bottom partition of the previous element to ensure the slurry can infiltrate uniformly on the complete cross section of the upper column.

2.2 Physical properties of materials

2.2.1 Slurry material

The slurry sample mainly consists of Na-bentonite, water and CMC. The bentonite is produced by Halliburton in Wyoming, which the ignition loss at 1000 °C is 5.1%, the dry density ρ_S = 2620 kg/m³ and loose paving porosity φ_S = 0.583. The test water is tap water stirred to no bubbles. CMC is dissolved in water aforehand and mixed with the expanded bentonite solution before formal experiments. The viscosity of the bentonite slurry is time-varying, and the expansion time in the experiment is 24 h. In this study, five kinds of slurry are prepared with different mass concentration and the CMC content is 0.5‰. The basic rheological parameters of the slurry samples are shown in Tab.1. Dynamic viscosity u_f and shear strength τ_f are measured by 12-speed viscometer in normal atmospheric temperature condition. As shown in Fig.2, the size distribution of the slurry is measured by laser particle size analyzer. The representative diameter D₈₅ = 19.85 μm.

2.2.2 Soil material

Fig.3 shows the particle size distribution of six sandy soil samples in this study. Samples S1 and S2 are obtained from standard sand through sifting. Samples S3, S4, S5, and S6 are river sand. All soil samples are cleaned and dried at 110 °C in drying box for 24 h to eliminate the influence of silt particles on the mass concentration measurement of filtrate. Tab.2 lists the basic physical properties of soil samples. The particles size of soil is measured after cleaned and dried. The permeability coefficient at standard water temperature (20 °C) is determined by constant head test [26]. Based on the research of Min et al. [27], the type of filter cake can be distinguished by the ratio between soil diameter d₁₅ and slurry representative diameter D₈₅. Only infiltration zone exists when d₁₅/D₈₅ > 10.53. In this study, the d₁₅ of fine particle soil sample S1 is 0.55 mm, d₁₅/D₈₅ = 27.71, so no filter cake retarded the downflow of slurry in multilayer infiltration column, which is convenient for filtrate extraction.

2.3 Experiment procedure

In this study, the change of filtrate mass concentration during infiltration process is measured. The detailed steps are as follows.

1) Assemble the experimental devices. Install hand valves on each infiltration column element and switch to off state. Connect the bottom valve of the device with the water collection tank which used to saturate soil samples. 2) Preparation and saturation of soil samples. To ensure the slurry can penetrate through all samples, the particle size of the soil samples filled in the multilayer infiltration column decreases sequentially from top to bottom. The filling sequence of the soil samples is S1 to S6 in infiltration column elements B to G. The installation sequence of the infiltration column elements is G-F-E-D-C-B-A shown in Fig.1(a). Then, fill the tank with water and open the outlet valve to slowly saturate the soil samples from bottom to top and vent the air in pores. Close the outlet valve of the device after the saturation is complete. 3) Bentonite slurry preparation. Mix the bentonite powder and water in proportion and stir for 30 min. Then mix the CMC solution with the expanded slurry and continue to stir for 10 min. Stir again for 5 min before formal experiment to ensure the slurry particles distributed uniformly in slurry. 4) Slurry infiltration experiment. Start the pressure program and set to 50 kPa. Open the hand valve at the bottom of the multilayer column. Initially, the slurry slowly fills element A and then slowly seeps down. After the slurry flows through element B and fills the lower column, immediately open the manual valve and collect 5 mL filtrate sample with measuring cylinder. Then, take the filtrate samples from top to bottom in sequence. Stop pressurizing immediately when the slurry flows from the bottom out valve, which marked the end of single experiment. 5) Measurement of filtrate concentration. First, introduce the filtrate from the measuring cylinder into the beaker. Then, clean the measuring cylinder for 2–3 times to dissolve the slurry particles adhere to side wall and introduce the slurry suspension liquid into the beaker. Next, heat each filtrate samples at 60 °C until the water is completely evaporated. Measure the mass of remaining slurry particles immediately to reduce the absorption of moisture in air, and then calculate the mass concentration of filtrate samples.

A preliminary experiment is conducted with high concentration slurry sample SL5 first to ascertain whether the mass concentration of filtrate after infiltrated from each soil samples is reasonable. The results indicate that the slurry cannot infiltrate uniformly on the complete cross section in soil sample S6. The concentration difference is less than 0.01 kg/m³. The measurement result is easily affected by measuring error. Hence, only S1–S5 soil samples are adopted in formal experiment and six infiltration column elements are needed.

2.4 Measurement method of filtration coefficient

The deposition of slurry particles in soil pores is commonly expressed by filtration coefficient λ, which is equal to the mass ratio of deposited particles m_d to inflowing particles m_i per unit length of porous medium [14]. In this study, the deposition mass of slurry particles is converted by the concentration difference of filtrate. λ can be determined by the following equation:

(1)\lambda =\frac{m_{\mathrm{d}}}{l{m}_{\mathrm{i}}},

where l is the length of porous medium, which is equal to the filling height of soil sample in infiltration column element.

(2)m_{\mathrm{d}}=\left(C_{\mathrm{L}\mathrm{i}}-C_{\mathrm{L}\mathrm{o}}\right)L,

(3)m_{\mathrm{i}}=v_{\mathrm{p}}tA{C}_{\mathrm{L}\mathrm{o}},

where C_Li and C_Lo represent the slurry concentration of inflow and outflow, respectively, which correspond to the filtrate concentration of the adjacent column elements, L is the volume of inflow slurry, v_p is the pore velocity which can simplified as L/φ₀At, t is the infiltration time, and A is the sectional area of upper column.

Inserting Eqs. (2) and (3) into Eq. (1), λ can be expressed as:

(4)\lambda =\frac{\left(C_{\mathrm{L}\mathrm{i}}-C_{\mathrm{L}\mathrm{o}}\right){\phi }_0}{C_{\mathrm{L}\mathrm{o}}l}.

2.5 Regression analysis of filtration coefficient

Five infiltration experiments were conducted which contain 25 sets of data. C_Li, C_Lo and measured λ are shown in Tab.3. The results investigate that the filtration coefficient is positively correlated with slurry mass concentration while negatively correlated with soil particle size.

Multiple regression equation of filtration coefficient λ is fitted by nonlinear analysis module in SPSS. In this study, slurry mass concentration C_L, soil permeability coefficient k₀, the particle diameter ratio between soil equivalent grain size and representative diameter of slurry particles d₁₀/D₈₅ are selected as independent design variables. C_L is the average of C_Li and C_Lo. The function relationship between each variable and response value is preliminarily determined based on the experimental data. As shown in Fig.4 and Fig.5, the scattered three-dimensional points are the measured filtration coefficient λ. The projection on the plane investigates that C_L is linearly associated with λ, and exponent relation exists between k₀, d₁₀/D₈₅ and λ.

The regression equation of filtration coefficient λ is as follows:

(5)\lambda =0.042C_{\mathrm{L}}+13.75{\mathrm{e}}^{-7.37k_0}+5.50{\mathrm{e}}^{\frac{19.51}{d_{10}{D}_{85}}}-6.11.

In addition, the regression analysis results indicate that the determination coefficient R² = 0.964. The average relative error is 16.96%. The significance probability P = 0.002 (much less than 0.05), which demonstrates the good relationship. Fig.4 and Fig.5 shows the variation of measured and fitted filtration coefficient with independent variables, which proved the consistency of experimental data and fitting result. It should be noted that the influence of gravity in the measurement is ignored, may overestimate the filtration coefficient.

3 Solution of filter cake formation time

The proposed model is developed in accordance with mass conservation of solid particles and volume conservation of liquid phase in slurry infiltration process. and is reasonable based on the following assumptions. 1) The infiltration of slurry in soil pores is laminar flow, and inertial forces is negligible [17]. 2) The migration of slurry is assumed to be one-dimensional owing to the infiltration distance of slurry particles is much smaller than the size of the tunnel face [28]. 3) The drilling velocity is slower than slurry infiltration, so the filter cake can form to the tunnel face and balance the earth pressure and hydrostatic pressure [29].

3.1 Model for slurry infiltration

The deposition of solid particles in soil pores leading to the decline of soil porosity. Meanwhile, the original pore water is pushed forward and replaced by slurry, resulting in the generation of excess pore water pressure in front of the tunnel face [30]. The two main reasons which leading to the dissipation of excess pore pressure in the infiltration process includes two parts: 1) the infiltration resistance between soil pores and slurry; 2) the retardation effect of broken external and internal filter cake [31]. Yang et al. [32] proposed a mathematical model to describe the generative mechanism of excess pore pressure in the internal filter cake zone and forward soil, which revealed the effect of filtration effect between soil and slurry particles on pressure dissipation.

On the basis of volume conservation, the difference between the inflow and outflow of slurry slime in soil microelement can be expressed as [32]:

(6)\begin{array}{rl}\mathrm{\Delta }V=& v{C}_{\mathrm{w}}A\mathrm{d}t-\left(v+\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\mathrm{d}x\right)\left(C_{\mathrm{w}}+\frac{\mathrm{\partial }{C}_{\mathrm{w}}}{\mathrm{\partial }x}\mathrm{d}x\right)A\mathrm{d}t\\ =& -\frac{\mathrm{\partial }v{C}_{\mathrm{w}}}{\mathrm{\partial }x}A\mathrm{d}x\mathrm{d}t,\end{array}

where v is the seepage velocity of slurry, C_w is volume fraction of the slime in slurry. A is the cross-sectional area of computation element, which is perpendicular to infiltration direction, x is infiltration distance and t represents pressuring time.

In the proposed model, the migration of particles in porous medium is nonlinear. For simplicity, the seepage velocity v of slurry is assumed as Darcy flow [33]:

(7)v=-\frac{k}{{\gamma }_{\mathrm{F}}}\frac{{\mu }_{\mathrm{w}}}{{\mu }_{\mathrm{f}}}\left(\frac{\mathrm{\partial }p}{\mathrm{\partial }x}-\frac{2{\tau }_{\mathrm{F}}}{r_{\mathrm{p}}}\right),

where γ_F represents the unit weight of the slurry, μ_w is the viscosity of water at room temperature, τ_F is the shear yield strength of the slurry, r_p is the equivalent pore radius of infiltration zone, p is the excess pore water pressure in infiltration process, k and μ_f are equivalent permeability coefficient and dynamic viscosity in infiltration process, respectively.

k is the equivalent permeability coefficient with deposited slurry particles in soil. On the basis of K–C formula [34], k is expressed as:

(8)k=k_0\frac{u_{\mathrm{w}}}{u_{\mathrm{f}}}{\left(\frac{1-\phi }{1-{\phi }_0}\right)}^{-4/3}{\left(\frac{\phi }{{\phi }_0}\right)}^3.

The evaluation formula of slurry viscosity and mass concentration of slurry particles can be expressed by Einstein Eq [35]:

(9){\mu }_{\mathrm{f}}={\mu }_{\mathrm{w}}\left(1+\alpha \frac{C_{\mathrm{L}}}{{\rho }_{\mathrm{S}}}\right),

where μ_f represents the dynamic viscosity of slurry suspension, μ_w is the initial viscosity of water, α is the viscosity coefficient, ρ_S is the dry density of bentonite particles.

The paper aims directly at uniform stratum. The change of soil computation element caused by excess pore water pressure is:

(10)\mathrm{\Delta }S=\frac{S_{\mathrm{S}}}{{\gamma }_{\mathrm{W}}}\frac{\mathrm{\partial }p}{\mathrm{\partial }t}A\mathrm{d}x\mathrm{d}t-\frac{{\phi }_{0}A}{{\rho }_{\mathrm{s}}}\frac{\mathrm{\partial }{C}_{\mathrm{S}}}{\mathrm{\partial }t}\mathrm{d}x\mathrm{d}t-\frac{A}{{\rho }_{\mathrm{s}}}\frac{\mathrm{\partial }\phi C_{\mathrm{L}}}{\mathrm{\partial }t}\mathrm{d}x\mathrm{d}t,

where γ_W is the unit weight of water, φ₀ is the initial porosity of soil while φ is the porosity considering the deposition of slurry particles, ρ_s is the dry destiny of slurry particles, C_L is the mass concentration of slurry, C_S is the mass of slurry particles deposited in soil pores, S_s represents the specific storage coefficient of stratum. In unconfined aquifers, elastic water storage can be ignored, S_s is approximately equal to the specific yield μ [36].

Theoretically, ∆V = ∆S. Based on the mass conservation of slurry particles in infiltration process [32], the mass difference of inflow and outflow is equal to the mass sum of deposited and suspended in soil pores. The coupled governing equations is summarized as below:

(11)\{\begin{array}{rl}& \lambda v{\phi }_0{C}_{\mathrm{L}}+\frac{\mathrm{\partial }\phi C_{\mathrm{L}}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }v{C}_{\mathrm{L}}}{\mathrm{\partial }x}=0,\\ & \frac{\mu }{{\gamma }_{\mathrm{W}}}\frac{\mathrm{\partial }p}{\mathrm{\partial }t}-\frac{{\phi }_0}{{\rho }_{\mathrm{s}}}\frac{\mathrm{\partial }{C}_{\mathrm{S}}}{\mathrm{\partial }t}-\frac{1}{{\rho }_{\mathrm{s}}}\frac{\mathrm{\partial }\phi C_{\mathrm{L}}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }v{C}_{\mathrm{W}}}{\mathrm{\partial }x}=0.\end{array}

Equation (11) is solved by COMSOL Multiphysics, the boundary conditions and initial value conditions are defined as:

(12)\{\begin{array}{rl}& p(x,0)=0,\\ & C_{\mathrm{L}}(x,0)=0,\\ & p(0,t)=p_0,\\ & C_{\mathrm{L}}(0,t)=C_{\mathrm{L}0},\end{array}

where p₀ is the initial excess pore water pressure at tunnel face, and C_L0 represents the initial mass concentration of slurry.

3.2 Formation time of dynamic filter cake

Deposition of solid particles in soil pores is the main cause of filter cake formation [37]. The mass concentration or viscosity also decreases with slurry infiltration. Theoretically, the formation time of filter cake is negatively correlated with filtration coefficient. The relationship between the mass of deposited particles and infiltration time can be expressed as:

(13)\frac{\mathrm{\partial }{C}_{\mathrm{S}}}{\mathrm{\partial }t}=\lambda v{C}_{\mathrm{L}}.

For external filter cake, solid particles in slurry directly accumulated on the superficial soil pores and form a dense impervious film. The excess pore water pressure dissipates completely over a very short distance. Related researches show that the porosity of deposited slurry particles on soil surface reached loose paving porosity φ_S means external filter cake is formed [31]. The critical value of particles deposited mass is:

(14)C_{\mathrm{S}}^{\mathrm{c}\mathrm{r}}=\frac{{\rho }_{\mathrm{S}}\left(1-{\phi }_{\mathrm{S}}\right)}{\beta },

where β is the expansion coefficient of slurry particles. The particles of bentonite are assumed as sphere arranged in cube, β = 6/π [13].

The formation time of external filter cake t_E can be expressed as:

(15)t_{\mathrm{E}}=\frac{C_{\mathrm{S}}^{\mathrm{c}\mathrm{r}}}{\lambda v{C}_{\mathrm{L}}}=\frac{{\rho }_{\mathrm{S}}\left(1-{\phi }_{\mathrm{n}}\right)}{\beta {\lambda }_0{v}_0{C}_{\mathrm{L}0}},

where λ₀ is the initial filtration coefficient of soil, the thickness of external filter cake e_e is ignored.

The seepage velocity of slurry at soil surface v₀ is given by Bezuijen and Xu [38]:

(16)v_0=\frac{k_0{p}_0}{{\phi }_0{\gamma }_{\mathrm{W}}R},

where p₀ is the difference value of slurry pressure in chamber and hydrostatic pressure, γ_W is the unit weight of water, R is the grouting radius.

For internal + external filter cake, the fine particles in slurry first penetrate into the deep pores of soil and form the internal filter cake. With the decrease of porosity of deep soil pores, solid particles gradually deposit on the soil surface and form external filter cake [39]. In actual tunneling process, the cutter head is continuous rotating, so the external filter cake on soil surface is in the cycle of cutting and regenerating [33]. Consequently, the excess pore water pressure drop steeply at broken external cake and descends slowly within the internal filter cake zone [32,40]. On the basic of Jin and Yuan [31], the loss of slurry pressure in e_e zone Δp = ξp₀, where ξ is the formation ratio of external filter:

(17)\xi =1-\frac{t_{\mathrm{E}}\omega }{\theta },(0⩽\xi ⩽1)

where ω is the rotation speed of cutter head, θ is the phase angle difference between adjacent cutter.

As shown in Fig.6, the length of internal filter cake e_i is the x-intercept of C_S, which can be obtained from Eq. (9). The pressure boundary at the interface of internal and external filter cake p_e is set as (1−ξ)p₀.

The formation time of external + internal filter cake t_IE is expressed as:

(18)t_{\text{IE}}=\frac{{\rho }_{\text{S}}\left(1-{\phi }_{\text{n}}\right)}{\beta {\lambda }_0{v}_0{C}_{\text{L0}}}+\frac{1}{e_{\text{i}}}{\int }_0^{e_{\text{i}}}\frac{C_{\text{S}}}{\lambda v{C}_{\text{L}}}\text{d}x.

4 Model verification

Previous experimental research has shown that both the dissipation of excess pore water pressure and formation time of filter cake are affected by the rheological property slurry, the pore structure of soil and slurry pressure. Additionally, the influence of configuration and rotation speed of cutter head is also nonnegligible [25,41]. Several sets of data from static and dynamic slurry infiltration experiments are selected to verify the accuracy of the proposed model. The time corresponding to a significant decrease in water filtration is typically taken as the formation time of filter cake, which means the duration required from the beginning of infiltration until the filter cake can balance the forward earth pressure and hydrostatic pressure [10,42]. All the required input parameters of Eq. (11) are listed in Tab.4, where the specific yield of soil is referred to the empirical value of silty-fine sand and medium-coarse sand. The shear yield strength of slurry in cases 4–9 is measured after concocting as described in Ref. [25]. All of the filter cake types in cases 1–9 is external + internal filter cake. Cases 1–3 are static slurry infiltration test, the pressuring time is 500 s. As shown in Fig.7, both the test results and prediction of Eq. (18) investigate that the formation time t_IE is positively correlated with slurry pressure. However, the prediction result is lower, perhaps because the proposed model cannot consider the increase in the thickness of the external filter cake with slurry injection. Cases 4–9 are dynamic infiltration tests and pressuring time is set as 200 s. The results show that the configuration of cutter head is nearly has no effect on t_IE in static state. Under dynamic cutting conditions, t_IE and the phase angle difference between adjacent cutter arm are of negative correlation. In addition, t_IE increases with the increase of rotation speed, which means the external + internal filter cake gradually transmute to internal type. It is possible that the disturbance of cutter head increased the porosity of the soil within infiltration zone and slow down the deposition of slurry particles.

5 Parametric study

The variations of slurry rheological property and the pore structure of the soil in the infiltration process are also investigated. Fig.8 shows the change of slurry dynamic viscosity and soil porosity in infiltration process. The initial parameters of slurry and soil are referred to SL3 and S3. Pressuring time is set as 500 s and p₀ = 50 kPa. The results indicate that the dynamic viscosity decreases rapidly in short distance and followed by a smooth drop until the rheological properties of slurry is close to water. The slurry particles in slurry are almost completely adsorbed by the soil skeleton when μ_f ≈ 1.0 mPa·s. Similarly, the soil porosity increases rapidly and then slowly returns to the initial value φ₀, which also demonstrates that most slurry particles cannot penetrate into deep soil pores. Additionally, the effect of particle diameter ratio between soil equivalent grain size d₁₀ and representative diameter of slurry particles D₈₅ on filtration coefficient is discussed in Fig.9. The data indicates that filtration coefficient is not constant along the infiltration direction, while gradually decreases to a stable value. As infiltration distance increases, the content of slurry particles gradually decreases to 0, and the adsorption capacity of the soil skeleton also returns to original value. In addition, the steep drop of filtration coefficient at soil surface (x = 0) is more apparent to low particle diameter ratio, which can explain the fluid loss of slurry in coarse particle size strata is higher.

Fig.10 shows the effect of slurry mass concentration C_L0, soil porosity φ₀ on the formation time of external and internal filter cake. The initial input parameters are referred to [41]. Tunnel radius R is set to 3 m, slurry pressure p₀ = 200 kPa and pressuring time is 500 s. Research shows that the formation time of external filter cake t_E is significantly lower than internal filter cake t_I. The calculated results show that the ratio of t_I to t_E is approximately to 3.9. Both t_E and t_I are negatively correlated with C_L0. Besides, the formation time of external + internal filter cake t_IE also gradually increases with the increase of φ₀, which shows that appropriately increasing the content of solid particles in slurry is conducive to the rapid formation of filter cake.

In Fig.11, the effect of the particle diameter ratio between soil equivalent grain size and representative diameter of slurry particles d₁₀/D₈₅ and rotation speed of cutter head ω on the formation ratio of external filter cake, formation time of filter cake is discussed. The initial input parameters come from Ref. [25], pressuring time is 200 s and R = 3 m. d₁₀/D₈₅ ranges from 1 to 50. The statistics indicate that the rotation speed of cutter head for slurry shield in medium-coarse sand is set to 0.5–1 r/min [42]. The analysis reveals that ξ gradually decreases as d₁₀/D₈₅ and ω increases. When 1 ≤ d₁₀/D₈₅ ≤ 6.1, ξ is greater than 98%. Filter cake can rapidly generated and t_IE ≤ 3.4 s. Besides, t_IE is positively associated with d₁₀/D₈₅ and ω. The difference of filter cake formation time also increases gradually under the same rotation speed of cutter head. It indicates that choosing fine-grained bentonite or silt is more conducive to the rapid formation of filter cake and improve the support efficiency of slurry. Only formation time is lower than the cutting time of adjacent arms can filter cake effectively support the tunnel face. Take spoked cutter head with five arms for instance, ω is set to 0.5, 0.75, and 1 r/min. Consequently, the cutting duration between adjacent cutter arms T_n is 24, 16, and 12 s, respectively. In the condition of ω = 1 r/min, filter cake can effectively support tunnel when 1 ≤ d₁₀/D₈₅ ≤ 12.7. When ω = 0.75 r/min, the limit value of d₁₀/D₈₅ increases to 19.3. In engineering practice, the rheological properties and particle size of solid-phase in slurry should be considered comprehensively in the settings of cutter head rotation speed.

6 Conclusions

This study carried out a series of multilayer infiltration tests with 5 different kinds of slurry with a slurry multilayer infiltration device to set up the multiple regression equation of filtration coefficient. The influence of slurry mass concentration, soil porosity, the particle diameter ratio between soil and slurry on filtration effect is discussed. Based on the relative relation relationship between the mass of deposited particles and infiltration time, a mathematical model for calculating the formation time of dynamic filter cake is proposed combining the formation criteria and cake formation rate of external filter cake. The main conclusions of this study are as follows.

1) Theoretical prediction results show the dynamic viscosity of slurry decreases rapidly in short distance and followed by a smooth drop for internal filter cake. Similarly, the soil porosity increases rapidly and then slowly increases to initial value, which indicate that most solid particles in slurry deposited on the surface of the soil, and only a few particles continue to migrate in deep pores under excess pore water pressure.

2) Filtration coefficient is positively correlated with slurry mass concentration while negatively correlated with the soil permeability coefficient and the particle diameter ratio between soil equivalent grain size and representative diameter of slurry particles. The adsorption capacity of soil to slurry particles decreases with the increasing of infiltration distance after slurry penetrated. As infiltration distance increases, the content of solid particles gradually decrease to 0, the adsorption capacity of soil skeleton also returns to original value. In addition, the steep drop of filtration coefficient at soil surface (x = 0) is more apparent to low particle diameter ratio, which can explain the fluid loss of slurry in coarse particle size strata is high. Decreasing the diameter of solid particles in slurry is more conducive to the formation of filter cake.

3) The formation time of external filter cake is significantly lower than internal filter cake. Properly increase the mass concentration of slurry in low porosity soil is conducive to the rapid formation of filter cake. Under the dynamic rotation and cutting, the formation time of filter cake is positively related to the rotation speed of cutter head while negatively related with the phase angle difference between adjacent cutter arm. Only formation time is lower than the cutting time of adjacent arms can filter cake effectively support the tunnel face. In practical engineering, choosing fine-grained bentonite or silt is more can contribute to the formation of filter cake and improve the support efficiency of slurry in coarse sand, gravel and other coarse grain stratum.

The proposed theoretical model still needs improvement. Microscopically, particle adhesion effect is also closely related to the formation of filter cake, which can describe the influence of size distribution of solid particles in slurry and protrusion height of the infiltration channel on the infiltration mechanism of slurry in soil. A more comprehensive model is the focus of our future research.

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