5.1. Validation of the Numerical Model
In the PFC-CFD coupling calculation, the permeability coefficient of the slurry penetration region will be constantly changing with the slurry particles’ transport and clogging. The permeability coefficient of the penetration zone will vary when the slurry–soil contact mechanism is different. There are many factors that affect the permeability coefficient; the more important ones are the particle size and grading of the soil, the soil porosity and the soil mineral composition. The influence of mineral composition in clay soils is more significant and is not considered in the sand soils in this paper. Since the large pores formed by coarse particles can be filled by fine particles, the soil pores’ size is generally controlled by fine particles. The slurry permeation pattern can be studied by analyzing the pattern of variation in the ground permeability coefficient. To calculate the permeability coefficient, Zizka et al. [
37] proposed an expression between the permeability coefficient and the drainage volume within a very small range of time variation for laboratory tests.
where
Q(
t) is the time-dependent drainage (m
3/s).
Acyl is the drainage cross-sectional area (m
2). △
L is the macroscopic permeation distance (m). △
h is the time-dependent pressure difference between the two ends of the permeation distance (m), and
ξ is the soil porosity which corresponds to the fluid element porosity in the numerical model.
In order to improve the efficiency of the calculations, the dimensions of the numerical models (
Table 5) in this paper are different from the laboratory test models (
Figure 2). However, the difference in the model size does not affect ground porosity and the permeability coefficient trend during the slurry infiltration, but only shortens the infiltration time [
34]. Using Darcy’s law, the total duration of the slurry penetration is proportional to the square of the penetration path and inversely proportional to the slurry pressure difference. Because of the large gap between the physical time of the slurry penetration in the numerical calculations and the real slurry penetration time in laboratory tests, the time scale needed to be transformed for a comparative analysis. The test model in
Figure 2 was selected as the standard to transform the numerical time, as shown in
Table 6.
To analyze the permeability coefficient variation during the slurry infiltration, the excess pore water pressure drop on both sides of the infiltration zone and the amount of discharge water in the numerical case ④ (
Table 5) were monitored and the permeability coefficient was calculated according to Equation (15). At the same time, the porosity and the average particle size of the infiltration zone in case ④ were monitored, and the permeability coefficient of the sand column was calculated according to Equation (12). The calculation results are shown in
Figure 6. Zizka et al. [
37] monitored the discharge water variation and pore water pressure over time in experiments and calculated the permeability coefficient based on Equation (15). The soil particle size in the third set of tests was in the range between 0.25 and 0.5 mm, and the slurry support pressure was equal to 50 kPa, which was basically consistent with the parameters of numerical case ④. From
Figure 6, the results obtained using numerical test data based on Equation (15) were slightly smaller than the permeability coefficients in the numerical simulations. It is probably because the porosity used in Equation (15) is a constant, while the coupling process in the numerical calculations updates the porosity with time. However, in general, the results of the two methods were basically the same, so the numerical model in this paper can reflect the slurry infiltration with accuracy. It should be noted that Zizka et al. [
37] used industrial bentonite slurry in their laboratory test, which has a better particle grading, so the slurry penetration process is relatively slow. However, the overall trend of the permeability coefficient is almost the same as in the numerical calculation.
5.2. Type of Slurry Infiltration
The existing laboratory tests [
27,
37] showed that the penetration pattern of slurry particles in the soil can be divided into three types, type Ⅰ: intact filter cake; type Ⅱ: filter cake + infiltration zone; and type Ⅲ, only infiltration zone. Additionally, the equivalent pore size between the soil particles and the
d85 of the slurry particles were used to determine the contact type between the slurry and the soil. The results of the above 12 sets of numerical calculations considering different soils and slurry combinations were analyzed, and the contact types are shown in
Table 7. The numerical test results showed that when
d85/
DP ≥ 1.1, the slurry particles could penetrate the soil pores. An intact filter cake was then formed at the excavation face, as shown in
Figure 7b. When 0.6 <
d85/
DP < 1.1, some slurry particles could enter through the soil pores, and some particles were accumulated at the excavation surface, resulting in a mixed contact form of filter cake + infiltration zone, as shown in
Figure 7c. When
d85/
DP ≤ 0.6, almost all the slurry particles flowed through the soil pores and therefore did not stay at the excavation face, as shown in
Figure 7d.
The numerical results showed that both the soil pore size and slurry grade had a significant influence on the slurry permeation behavior. As the ratio
d85/
D0 decreased, more and more slurry particles entered inside the sand stratum, and fewer particles gathered at the surface, making it less likely to form an intact filter cake. Jia et al. [
38] showed that when the slurry density is high enough, even if the ratio
d85/
D0 is small at the beginning, a filter cake + infiltration zone accumulation pattern can be formed. This is because when the slurry density is high, more slurry particles can fill the soil pores. It will reduce the average pore size of the formation, resulting in an increase in
d85/
D0. However, it is worth noting that this process will take a long time. Sherard et al. [
43] proposed the ratio of
D15/
d85 to assess the type of slurry/soil contact. When
D15/
d85 ≤ 5.26, a type 1 filter cake is formed at the excavation surface. When 5.26 <
D15/
d85 < 10.53, a type 2 contact pattern is formed. When
D15/
d85 ≥10.53, no slurry particles are trapped at the excavation surface and, therefore, only an infiltration zone is present (type 3). For well-graded soils (
Cu ≥ 5 and 1<
Cc < 3), the above criteria are usually conservative, as the minimum pore size is smaller. However, when the soil is not particularly well-graded (
Cu < 5 or
Cc < 1 or
Cc ≥3), there is an obvious error for this method. This is because when the soil is poorly graded, the pores in the soil particles are not sufficiently filled by the micro grains. Since the soil particles used in this study are more uniformly graded, the above criteria proposed by Sherard et al. [
43] are no longer applicable.
5.3. Slurry Support Mechanism
To analyze the formation of the three types of slurry penetration mentioned above from a mechanical point of view, the microscopic distribution patterns of the slurry particles in the soil pores were extracted from the longitudinal profile as shown in
Figure 8. For the first type of slurry infiltration, most of the slurry particles are accumulated at the surface of the soil particles, forming a dense and intact filter cake. Only a small fraction of the slurry particles fill the shallow soil pores, while the number of slurry particles entering the soil is minimal and are unable to travel through large distances. They can eventually stagnate in the soil pores forming a loose filling structure. For the second infiltration type, the thickness of the filter cake is significantly reduced, although an intact filter cake can also be formed. Most of the slurry particles are transported to the deeper parts of the soil column. However, the stagnant particles in the shallow part of the soil can form a dense filling network. The sand particles are then tightly packed in the filling network. For the third type of infiltration, slurry particles are almost always transported into the soil pores. Dense filling structures can be formed; the distribution of the filling structures is generally longitudinal.
From the analysis above, during the process of slurry penetration, three main contact mechanisms will occur depending on the ratio of the slurry particle size to the soil pore size. The first type occurs when the slurry particle size is larger than the mean soil pore size (
d85 ≥ 1.1
DP) and the slurry particles are stagnantly deposited at the soil surface. As the deposition progresses, a filter cake is then formed at the soil surface. The second type occurs when the slurry particle size is close to the mean soil pore size (0.6
DP <
d85 < 1.1
DP); the slurry particles are gradually filled with the pores of the soil particles driven by the hydraulic gradient. With the penetration development, loose filling gradually transforms into dense filling and forms a horizontally distributed dense filling network. Finally, a dense filter cake will form on the sand column surface. The third type occurs when the size of the slurry particles is smaller than the mean pore channel (
d85 ≤ 0.6
DP), the slurry particles migrate through the soil pores driven by the hydraulic gradient. The slurry particles will stop moving and will stagnate in the soil pores until the hydraulic gradient is not enough to drive the slurry particles, or they encounter smaller pore channels. The retention of the slurry particles further alters the soil pore structure and affects the slurry penetration. In an engineering project, if bentonite is used as a base material for slurry, the clay particles usually form flocculated clusters [
44,
45]. As the particle size of the clusters is larger than the soil pore channel’s diameter, they will be stagnantly deposited at the excavation surface. As the deposition progresses, a filter cake will be formed.
To visually analyze the transport and accumulation of slurry particles during infiltration in the three typical cases, the changes in porosity during calculations are monitored as shown in
Figure 9. At the beginning of the calculations, the lower half of the model is filled with sand particles and the upper half is filled with slurry (
Figure 7a). With the infiltration time ongoing, case ④ showed an obvious decrease in porosity at the slurry–soil contact surface (
Figure 9a), while the porosity within the overlying sand column remained essentially unchanged. It indicates that a dense filter cake was formed at the excavation surface. As for case ⑨ (
Figure 9b), the decrease in porosity at the slurry–soil contact surface was not so obvious compared with case ④. Because the coarse-grid method was used to calculate the fluid motion, the formation of a thin filter cake over the contact surface did not significantly change the fluid element porosity. The minimum value of porosity occurred below the contact surface because the soil pores at this place were sufficiently filled. In case ⑫ (
Figure 9c), the porosity change inside the sand column during the slurry penetration was not so obvious. The porosity at the slurry–soil contact surface remained essentially unchanged, which means that there was no trapped accumulation of slurry particles.
5.4. Analysis of the Slurry Support Effect
The essence of the slurry support during slurry shield construction is the balancing effect on the soil and water pressure at the tunnel excavation face. Different slurry–soil interaction mechanisms will produce different support effects. The excess pore water pressure overlying the contact layer was monitored during the slurry infiltration as shown in
Figure 10. For the first infiltration mechanism (type 1: case ④), the excess pore water pressure in the overlying sand layer under the filter cake decreased significantly and changed at a fast rate. For the second infiltration mechanism (type 2: case ⑨), the excess pore water pressure also decreased but at a slower rate and to a significantly lesser extent than the first mechanism. This was mainly because it takes more time for the slurry particles to fill the soil pores and form a filter cake, which has a smaller thickness. For the third infiltration mechanism (type 3: case ⑫), there was no significant change in excess pore water pressure in the soil layer adjacent to the slurry–soil contact surface. This was because the slurry particles do not form an effective filling network in the soil pores that inhibit the slurry penetration development.
The distribution of excess pore water pressure can effectively reflect the supporting effect of the slurry infiltration.
Figure 11 shows the distribution of the excess pore water pressure inside the sand column from three different slurry infiltration types in cases ④, ⑨ and ⑫. As can be seen from
Figure 11, there was a small pressure gradient in the slurry itself, but it was negligible. For case ④ in
Figure 11a, as the slurry penetration time increased, the excess pore water pressure in the slurry–soil contact layer decreased significantly, about 26% of the entire slurry support pressure. Additionally, the pressure gradient in the slurry–soil contact layer was about 2.4 times the pressure gradient in the sand column. The filter cake formed at the slurry–soil contact surface can lead to a significant reduction in the excess pore water pressure. For case ⑨ in
Figure 11b, there was no significant change in the excess pore water pressure in the slurry–soil contact layer at the beginning of the slurry infiltration phase. A slight decrease in the excess pore water pressure was observed within the sand column. As the infiltration time progressed further, there was a decrease in the excess pore water pressure in the slurry–soil contact layer, about 19% of the entire slurry pressure. For case ⑫ in
Figure 11c, the excess pore water pressure inside the sand column remained basically unchanged during the whole slurry penetration process. The third type of slurry infiltration mechanism basically cannot achieve the purpose of slurry support.
The filter cake formation can be divided into three different stages by observing the transport of the slurry particles and analyzing the trends in the excess pore water pressure in the soil layer, as well as the soil porosity. In the first stage, soil particles were transported by the slurry particles. The surface soil particles were rearranged during this stage and caused the slurry particles to fill and accumulate in the soil particle pores, forming a permeable zone at the contact surface. This process resulted in a rapid water infiltration into the soil and an instantaneous increase in the excess water pressure in the stratum. In the second stage, the slurry particles continued to accumulate at the contact surface, and the thickness of the filter cake increased. The excess pore water pressure in the overlying layer of the filter cake continued to decrease as the filter cake gradually became denser and denser. In the third stage, the overall permeability coefficient of the denser filter cake became lower and the excess water pressure in the overlying soil body remained stable. It should be noted that the slurry particles are not well-graded in the numerical calculations, so the filter cake formed still has a certain degree of permeability. For the second type of slurry infiltration, both filter cake and slurry infiltration zones are present. Both the dense filling network formed by the slurry particles and the filter cake significantly reduced the excess pore water pressure in the soil body, which means that part of the reduction was successfully converted into an effective support pressure applied to the soil skeleton. For the third type of slurry infiltration, there was no complete dense filling network formed in the direction of the slurry penetration, but only longitudinally extended slurry veins. Therefore, it can hardly change the excess pore water pressure distribution in the soil and cannot effectively ensure the slurry support pressure.
5.5. Time Effect of the Slurry Infiltration
Observations of the three types of slurry–soil interaction reveal that the most obvious difference is the penetration distance of the slurry particles (
Figure 8). The transformation effect of the slurry support pressure is closely related to the penetration distance. Larger permeation distance indicates a lower support pressure conversion efficiency [
29] as well as a larger slurry particle infiltration loss, which also represents a lower stability coefficient of the tunnel face.
For the support mechanism with both filter cake and infiltration zones, the slurry support effect is mainly shared by the effective support pressure of the filter cake and the shear forces between the slurry particles and the soil pore channel walls. The support pressure generated by the filter cake can be calculated through the excess pore water pressure drop on both sides of it. However, the shear force between the slurry particles and the soil pore channel wall is a microscopic concept and cannot be calculated directly. It is only when the slurry particle transport reaches the maximum penetration distance that the shear force can be in balance with the penetration driving force. Kilchert and Karstedt [
43] studied the relationship between the stagnation gradient of the slurry penetration, the characteristic particle size of the soil particles, and the slurry yield strength based on experimental studies [
21,
46,
47]. The formula for the calculation of the maximum penetration distance was modified accordingly:
where
fso is the slurry stagnation gradient according to Kilchert and Karstedt [
16] (kPa/m).
a is the empirical factor from the experiments; the value can be 2 or 3.5.
d10 is the characteristic grain size of soil (10% passage in sieve analysis, mm).
τf,s is the static yield point of the slurry (Pa). △
S is the slurry excess pressure (kPa).
lmax is the maximum slurry penetration depth (m).
Zizka et al. [
37] carried out a series of experiments to study the patterns of the slurry infiltration and calculated the slurry infiltration distance by real-time drainage monitoring. A comparison of the slurry permeation distances in the experiments and the empirical formula calculations are shown in
Figure 12. The empirical formula calculates a larger permeation distance compared to the experimental monitoring. This is because the slurry will change the soil grading during the penetration process, which will reduce the characteristic grain size (
d10) of the soil body. Therefore, a larger slurry stagnation gradient is obtained, which will eventually lead to the reduction in the slurry permeation distance according to Equation (16).
The maximum penetration distance is determined by the final state of the slurry penetration and does not consider the time effect. Once the slurry penetrates over long distances in a short period and produces large slurry losses, it may lead to a destabilization of the tunnel face. Therefore, the development process of the slurry penetration distance has a significant influence on the slurry support effectiveness. The displacement of the slurry particles is monitored in the numerical calculations as shown in
Figure 13. It can be seen from the figure that for the first type of interaction mechanism, the slurry particles are not transported in the soil, but almost completely stagnate and accumulate at the soil surface. For the second type of slurry penetration mechanism, the slurry particles move faster at the beginning of the penetration, but with the gradual formation of the dense filling network, the penetration distance gradually stabilizes. For the third type of slurry penetration mechanism, the penetration distance of the slurry particles increases linearly during penetration, and when the sand column is not high enough, slurry particles tend to penetrate the entire sand column.
Anagnostou and Kovári [
29] proposed a time-dependent formula for the theoretical calculation of the slurry penetration distance based on the assumption of a one-dimensional infiltration at the tunnel face. The slurry penetration distance is determined by the soil permeability coefficient and the viscosity ratio of the slurry to water.
where
t is the timespan since the slurry penetration start (s).
n is the soil porosity.
μs is the slurry dynamic viscosity (Pa·s).
γw is the water unit weight (kN/m
3). △
S is the slurry excess pressure (kPa).
l is the penetration distance at the timespan
t (m).
lmax is the maximum penetration distance of the slurry (m).
kw is the permeability coefficient of soil for water (m/s).
μw is the dynamic viscosity of water (Pa·s).
fso is the slurry stagnation gradient (kPa/m).
The slurry penetration distance is compared as a dimensionless number with the empirical formula [
29] based on the slurry penetration development in numerical case ⑨. At the same time, the data of the slurry infiltration with similar combinations (soil particle size range: 0.5–1.0 mm, porosity: 0.41, slurry density: 1.03 g/cm
3, and slurry concentration of 5.5%) in the laboratory tests [
37] are shown in
Figure 14. The development of the penetration distance obtained by the three methods is almost identical. However, the empirical formula as well as the laboratory tests have a faster penetration rate, especially the empirical formula, which shows a significantly larger penetration rate at the beginning of the test. Although, the empirical formula uses constant calculation parameters such as the stagnation gradient and porosity, but once the maximum permeation distance can be determined, the empirical formula reflects the development of the slurry permeation distance. It is in good agreement with the numerical and test results. The exponential relationship of the empirical formula can well reflect the influence of the slurry penetration process on the permeation rate due to the stagnant accumulation of particles.
During shield tunneling, the filter cake formed at the tunnel face is a dynamic process, this is because the filter cake is always in a cycle of the following type: formed, destroyed and re-formed. Therefore, it is required that the filter cake is formed quickly, which means that the slurry particles fill the soil pores quickly with a small amount of slurry filtration loss. A dense filter cake will then be formed at the tunnel face, sealing the excavation surface to maintain the stability of the tunnel face. The filter cake formation time, as well as the amount of slurry filtration loss during the formation process and the excess pore water pressure in the ground after the formation are often used as important indicators for evaluating the filter cake quality and the effect of the slurry support. However, real-time observation of the filter cake quality and monitoring of the excess pore water pressures are not feasible during a real tunnel excavation. The slurry filtration loss and the water discharge during infiltration can be monitored. These indicators can be obtained by calculating the amount of slurry injection and slag discharge, and therefore can be used as key indicators to evaluate the filter cake quality. Water discharge for the three different slurry infiltration mechanisms were monitored during numerical calculations and are shown in
Figure 15.
For the first slurry infiltration type, the discharge water grew very slowly. It is because the slurry particle size distribution was not well graded; the filter cake formed was still permeable to a certain extent. The cumulative discharge water was around 4 × 10−3 m3/m2 for a slurry excess pressure of 50 kPa. For the second type of slurry infiltration, the water discharge increased more rapidly at the beginning of the infiltration, but after about 7 s, the discharge volume tended towards a stable value of 12 × 10−3 m3/m2. For the third infiltration type, the drainage trend at the beginning of the slurry infiltration was like the second type. However, with the infiltration development, the drainage volume quickly overtook the second infiltration type and did not reach a steady value. An analysis of the discharge water corresponding to the three types of slurry infiltration indicated that the formation of an intact filter cake could effectively inhibit the drainage increase, and the slurry support effect could be achieved in a short period. The purely infiltration mechanism could not effectively control the drainage development and could not achieve the slurry support effect, accompanied by the presence of a high excess pore water pressure in the stratum. For the second type of slurry penetration, as the slurry particles filled and stacked inside the soil pores, the pore structure of the soil changed and an intact filter cake could eventually be formed. However, the whole process was slower compared to the first type of slurry infiltration.