A Deterministic Method for Evaluating Safety Factor of Deep Excavation Stability against Groundwater Inrush Equivalently Considering Soil Uncertainty

Abstract

In current design specifications for deep excavation, the determination of safety factors against groundwater inrush only considers the self-weight effect of soil mass at the bottom. However, the shear strength and its spatial variability in a cement-solidified bottom layer also plays an important role in safety factor estimation. Based on that, in this study, the strength reduction method was introduced into reliability analysis of deep excavation engineering, and the influence of shear strength and its spatial variability of cement-solidified soil on the stability of deep excavation is studied. Furthermore, a deterministic analysis method that can equivalently consider soil uncertainty is proposed and verified.

1. Introduction

Groundwater inrush of deep excavation in soft soil with a confined aquifer may not only lead to severe safety consequences for construction personnel and equipment but also brings about unrecoverable damage or detrimental effects to nearby environmental facilities [1,2]. Therefore, the prevention of groundwater inrush remains a crucial problem for the design of deep excavation in soft soil with a confined aquifer.

In particular, in order to control the unfavorable environmental influence of dewatering in soft soil with a confined aquifer, the bottom soil mass (i.e., the passive zone) of deep excavation is commonly treated by continuous cement deep-mixing columns or jet grouting columns to form an enclosed waterproof layer together with the vertical cut-off wall [3,4,5,6,7]. Under this circumstance, the cement-solidified bottom layer (i.e., cement–soil slab) will show a much larger shear strength than the in situ soil, and thus bring a very positive effect on deep excavation stability against groundwater inrush. As a consequence, the pressure balance method seems to be excessively conservative, leading to significant increase in total cost. This suggests that, in the case of cement–soil slabs, it is very necessary to account for the shear strength of cement–soil in analyzing the stability of deep excavation against groundwater inrush.

However, it should be acknowledged that the engineering properties of cement–soil generally show very strong heterogeneity and spatial variability [8,9,10,11,12,13,14,15]. According to Navin [8] et al. and Naqshabandy [9] et al., the coefficient of variation (COV) for the shear strength of cement–soil can be up to 0.67–0.79, which is much larger than the COV value of in situ soil strength. Kasama [16] et al. reported that the mean values of unconfined compressive strength obtained from core samples cured in the field of cement-treated ground range from 100 to 7500 kPa with coefficients of variation between 0.14 and0.99. Some existing studies have demonstrated well the substantial influence of the strong spatial variability of cement–soil properties on the mechanical behavior of deep excavation. For instance, Liu [12] verified that the spatial variability of the cement-solidified bottom layer has a significant effect on the horizontal deformation control of the passive zone improvement area of excavation. Honjo [17] collected statistics on the strength variability of soil–cement deep-mixing piles at ports, such as Hiroshima and Yokohama in Japan, and the results showed that the coefficient of variation of soil–cement strength was up to 0.592. Tsutomu Namikawa [18] found that the current quality assurance method, without considering statistical uncertainty, may lead to conservative estimates of the constructed column strength. Furthermore, the probabilistic incorporation of stochastic uncertainty (random variability) was also applied to the seismic response analysis of structures and the typical spatiotemporal evolution of corroded pipelines and so on, considering vertical heterogeneous soil structure or properties [19,20,21,22,23]. Therefore, it is reasonable to believe that the strong spatial variability of cement–soil strength will also significantly affect the stability of deep excavation against groundwater inrush.

To date, how to evaluate the influence of spatial variability of cement–soil shear strength on the stability of deep excavation against groundwater inrush still remains an unclear issue, to the authors knowledge. In current design specification [24], the safety factor of deep excavation stability against groundwater inrush is generally calculated by the pressure balance method, which considers only the self-weight effect of soil mass at the bottom of deep excavation but does not account for the resistance provided by soil shear strength. However, the soil shear strength can play an important role in promoting the stability against groundwater inrush; simply omitting it can lead to inaccurate estimates [25,26,27].

The novelty of this paper is to gain an insight into the aforementioned issue and to develop a deterministic method for the estimation of safety factor of stability against groundwater inrush equivalently, considering the spatial variability in deep excavation with the cement-stabilized bottom layer. To be specific, the main content can be divided into four parts. First, the numerical model is established and the analysis cases are decided. Second, the deterministic analysis is conducted. Third, the probabilistic analysis is conducted and the results are analyzed. Finally, a deterministic method equivalently considering soil uncertainty is proposed and verified.

2. Numerical Model and Analysis Cases

2.1. Numerical Model

The following analysis is based on an actual deep excavation project in soft soil. Figure 1 plots the geometric and geologic conditions of the deep excavation. The excavation process is divided into five steps, the thickness of each excavation step from top to bottom is 2 m, 5 m, 5 m, 5 m and 3 m, respectively, and the maximum excavation depth is 20 m.

The support system consists of three parts, inner bracing, passive zone improvement and enclosing diaphragm wall. The diaphragm wall belongs to the reinforced C30 concrete and inserts into a depth of 30 m below the ground surface. The concrete brace is also formed by reinforced C30 concrete with a cross section of 600 mm × 600 mm and is installed at a depth (central axis) of 1.2 m below the ground surface with a horizontal spacing of 6 m. The other three layers of braces are all made of Q235 steel pipes with 609 mm outer diameter and 16 mm thickness, and they are installed at depths (central axis) of 6.2 m, 11.2 m and 16.2 m, respectively, with a horizontal spacing of 3 m. The passive zone improvement is performed by cement deep-mixing method. The cement-solidified bottom layer is 3 m in thickness, and the cement–soil columns, 1.0 m in diameter, are arranged triangularly with 800 mm of center-to-center spacing.

In this study, a three-dimensional model is established to simulate the excavation process and analyze the safety factor of deep excavation stability against groundwater inrush. The computational domain is 136.2 m in length (x direction), 50 m in height (z direction) and 3 m in thickness (y direction). As demonstrated in Figure 2, the mesh gradually thins out along the direction away from the excavation, and the mesh of the passive zone improvement area (the cement slab) is locally refined. Regarding the boundary conditions, the left and right boundaries constrain displacement in the x direction, the front and rear boundaries (y direction) constrain displacement in the y direction, but the hydraulic conditions of both are fixed; for the bottom boundary, displacements in three directions are constrained but hydraulic condition is free [28,29,30,31].

Hardening soil model (HS model) is used to simulate the miscellaneous fill layer, clayey soil layers and sand layers; the elastic model is used to simulate the bed rock layer, diagram wall, and concrete/steel braces, while the Mohr–Coulomb model (MC model) is taken to characterize the mechanical behavior of cement–soil layer. The detailed mean parameters of the properties of soil layers and supporting structures are listed in

Table 1. Mean parameters for the properties of soil layers (HS model).

Soil Layer	Thickness (m)	γ (kN/m3)	cu (kPa)	φ (°)	E50 (kPa)	Eoed (kPa)	Eur (kPa)	p (cm/s)
Landfill	2	18.5	6.0	15	2.90 × 103	2.40 × 103	3.30 × 104	5.0 × 10−3
Clayey soil A	3	17.9	10.0	25.2	4.40 × 103	3.40 × 103	7.72 × 104	2.0 × 10−5
Clayey soil B	5	17.5	10.37	29	4.70 × 103	3.70 × 103	1.55 × 104	1.5 × 10−5
Clayey soil C	4	17.29	8.13	34	3.0 × 103	2.50 × 103	1.38 × 105	1.2 × 10−5
Clayey soil D	6	18.14	10.0	20.1	1.6 × 103	1.20 × 103	1.36 × 105	1.40 × 10−5
Sand soil A	4	18.8	0	34.5	1.36 × 104	1.06 × 104	1.27 × 105	7.34 × 10−2
Sand soil B	6	19.5	0	35.1	1.58 × 104	1.26 × 104	1.51 × 105	4.9 × 10−2
Clayed soil E	5	19.29	16.9	34	6.40 × 103	5.90 × 104	1.70 × 105	3.0 × 10−5

,

Table 2. Mean parameters for bedrock (elastic model).

Soil Layer	Thickness(m)	γ (kN/m3)	cu (kPa)	φ (°)	p (cm/s)
Bed rock	15	21.0	5.20 × 106	29.9	1.0 × 10−6

,

Table 3. Mean parameters for cement–soil (MC model).

Soil Layer	Thickness(m)	γ (kN/m3)	cu (kPa)	φ (°)	E (kPa)	p (cm/s)
Cement–soil	3	19	1.5 × 103	18	4.2 × 106	6.3 × 10−5

and

Table 4. Parameters for support structures.

Support Structure	ρ (kg/m3)	E (kPa)	μ
Diagram wall	24	2.0 × 107	0.20
Concrete brace	24	2.0 × 107	0.20
Steel brace	78.5	1.3 × 108	0.25

.

The software called Flac3D was used for the finite element calculation in these cases. The detailed simulation process can be described as follows: Firstly, the complete excavation model is established, and the life and death element method is applied for the process of excavation and support structure erection in order to simulate the initial stress field, which considers the excavation unloading effects. Secondly, the safety factor of deep excavation stability against groundwater inrush is obtained based on strength reduction method. Since deep excavation stability against groundwater inrush depends primarily on the bottom layer, only the cement–soil in the passive zone is selected as the object for strength reduction analysis, and it is regarded as a cement-solidified soil slab for anti-surge safety factor analysis. The strength reduction method is used based on the slab model and the initial stress field is inherited from the excavation model. Therefore, the cement–soil layer in the passive zone will be extracted from the complete model with the stress field unchanged before the second step is conducted. Moreover, to be on the conservative side, the cement–soil is regarded as an impermeable material during the analysis of deep excavation stability against groundwater inrush via the strength reduction method.

In the analysis of the strength reduction method, after obtaining the relation curve between the reduction coefficient and the maximum upheaval, the safety factor against groundwater inrush is judged by the intersection point between the asymptote of upheaval curve and the horizontal coordinate axis, and the corresponding maximum upheaval will be defined as critical displacement u_f. When an uncertainty calculation result is greater than u_f, the deep excavation will be considered instable and damaged.

2.2. Analysis Cases

Both deterministic numerical modelling and probabilistic numerical modelling are performed in this study. For the deterministic numerical modelling, all the mean parameters presented in Table 1 are directly taken as the input parameters of the analysis model; whereas for the probabilistic numerical modelling, the cement–soil slab is considered as a random field. According to the results from Zhang [15], the average value of COV of cement-solidified soil could be seen as 0.416. Based on that, in this study, the reference COV is 0.4.

According to informed research [6,7], the variability of friction angle φ, unit weight γ or Poisson’s ratio μ of cement-solidified soil is generally insignificant in practice; in contrast, c_u and E have non-negligible spatial variability, and there is a strong correlation between the two parameters, which can be assumed as E = 200~400c_u. Therefore, in the probabilistic numerical modelling of this study, φ, γ and μ of cement–soil are considered as deterministic parameters, whereas c_u and E are considered as correlated random variables following the same distribution pattern. In summary, the characterization of the spatial variability of cement–soil parameters is based on c_u random field, and its generation is based on the K–L decomposition method and characterization method proposed by Liu [12] et al. The horizontal fluctuation scale of c_u random field is 1.0 m, while the vertical fluctuation scale is 3.0 m. The autocorrelation function of c_u is a single exponential function, and its distribution form is lognormal distribution.

3. Analysis of Modelling Results

3.1. Deterministic Analysis Results

This section mainly concentrates on the influence of cement-solidified soil shear strength on safety factor of deep excavation stability against inrush under the same COV condition. The analyses were carried out under the conditions of cement-solidified soil cohesion c_u = 0.75 MPa, 1.0 MPa, 1.25 MPa and 1.5 MPa, respectively. The curves of the maximum bottom heave u_max against the strength reduction coefficient F_s are plotted in Figure 3.

For the condition of c_u = 0.75 MPa, the critical displacement u_f is 60 cm, and the corresponding critical strength reduction factor (i.e., safety factor) F_n is 1.37; for the condition of c_u = 1.0 MPa, u_f is 58 cm and F_n is 1.72; for the condition c_u = 1.25 MPa, u_f is 64 cm and F_n is 2.03; for the condition c_u = 1.5 MPa, u_f is 64 cm and F_n is 2.43.

Figure 4 illustrates the change trend of F_n against c_u. According to the curve in Figure 4, it can be proposed that there is a linear positive correlation between F_n and c_u. By linearly fitting the curve, one equation is obtained:

$$F_{\mathrm{n}}=1.396c_{\mathrm{u}}+0.317,$$

(1)

The correlation coefficient (R²) of the fitting result is 0.998. It means that there is a good linear one-to-one correspondence between F_n and c_u. Therefore, in the deterministic analysis (that is, the condition of COV = 0), it can be assumed that there is a linear function relationship between F_n and c_u, which can be written as:

$$F_{\mathrm{n}}=a{c}_{\mathrm{u}}+b,$$

(2)

where a and b are constants obtained from curve linear fitting.

However, in current specifications, the safety factor k of deep excavation stability against inrush is calculated based on the pressure balance method, and the definition of k is as follows:

$$k=\frac{D\gamma }{H{\gamma }_{\mathrm{w}}},$$

(3)

where k is the safety factor, D and γ are the thickness and density of the impermeable layer of soil, γ_w is water density and H is the height of water table above impermeable layer surface.

For the deep excavation project shown in Figure 1, H = 18, γ = 19, γ_w = 10, D = 3 and k is hence solved to be 0.32. This means that the deep excavation will collapse because of the bottom water pressure. However, according to the computation result, the deep excavation does not undergo groundwater inrush during the excavation process. This means that the cement-solidified soil layer (passive zone improvement area) can effectively enhance the deep excavation stability against inrush. Therefore, the current safety factor calculation method is not suitable for deep excavation with passive zone improvement.

3.2. Probabilistic Modelling RESULTS

Pan [32] et al. applied a random finite element method to investigate the failure mechanism and failure criterion for cement-treated soil slabs, considering the effect of spatial variability. Wang [33] explored the probabilities of exceedance of the maximum tensile stress in cement-stabilized soft clay. Shen et al. [34] investigated the effect of soil spatial variability on failure mechanisms and capacities of strip foundations, and the shapes of failure enveloped between deterministic and probabilistic cases are similar. Nevertheless, it is worth noting that the influence of spatial variability on the horizontal deformation control efficiency of cement–soil slabs can be evidently different than that on the stability of deep excavation against groundwater inrush. In the case of the former, the most concerned force direction is primarily parallel to the plane of the cement–soil slab, and thus the horizontal deformation control efficiency depends on the “averaged” engineering properties (i.e., the “average effect” is significant); whereas for the latter case, the most concerned force direction is perpendicular to the plane of cement–soil slab, and hence the stability against groundwater inrush depends mainly on the weakest shear strength (i.e., the “average effect” is insignificant).

3.2.1. Effect of the Mean Value of Cohesion

Figure 5 plots a set of histograms of probability distribution of safety factor k under the reference COV condition with different shear strength.

To characterize the collapse of deep excavation, failure probability pf is introduced and defined as follows:

$$p_{\mathrm{f}}=1-{\displaystyle {\int }_{x_{\mathrm{f}}}^{\infty }g(x;\widehat{\mu };\widehat{\sigma })},$$

(4)

where x_f is the quantile value; ${\displaystyle {\int }_{x_f}^{\infty }g(x;\widehat{\mu }};\widehat{\sigma })$ is the probability density of lognormal distribution, and it can be written as follows:

$${\displaystyle {\int }_x^{\infty }g(x;\widehat{\mu };\widehat{\sigma })}=\frac{1}{\sqrt{2\pi {\widehat{\sigma }}^2}}{e}^{-\frac{{(\ln x-\widehat{\mu })}^2}{2{\widehat{\sigma }}^2}},$$

(5)

where $\widehat{\sigma }$ is the logarithmic standard deviation, $\widehat{\mu }$ is the logarithmic mean and β is the reliability index.

Taking β as 2.5, the corresponding p_f is 0.5%. The computation results of quantile value x_f for the safety factor (that is, the safety factor obtained from the uncertainty analysis) k_f are summarized in

Table 5. c_u = 0.75 MPa reliability analysis on safety factor.

cu(MPa)	$\widehat{\mathsf{\mu }}$	$\widehat{\mathsf{\sigma }}$	kf	η
0.75	0.2524	0.0525	1.12	0.81
1.0	0.4805	0.0597	1.39	0.80
1.5	0.8188	0.0590	1.95	0.80

.

In order to further analyze the change of law of k_f versus COV of c_u, the calibration coefficient η is introduced and defined as follows:

$$\eta =\frac{k_{\mathrm{f}}}{F_{\mathrm{n}}},$$

(6)

where k_f is safety factor corresponding to a certain reliability index, and F_n is the safety factor obtained from deterministic analysis.

The results of η are also summarized in Table 5, Figure 6 and Figure 7, respectively, depicting the curve of k_f and η against c_u.

By performing a linear function fitting on the trend line plotted in Figure 6, one can obtain:

$$k_{\mathrm{f}}=1.1086c_u+0.2857,$$

(7)

The correlation coefficient R² of the fitting result is 0.9999. That means k_f and c_u satisfy a good linear function relationship. However, the curve plotted in Figure 7 is nearly a horizontal line. This means η could be independent for c_u. Therefore, it can be assumed that, when COV of c_u is given, there is a linear relationship between k_f and c_u, i.e.,

$$k_{\mathrm{f}}=a{c}_u+b,$$

(8)

For η, the following equation exists:

$$\eta =d,$$

(9)

where a and b are constants obtained from curve fitting.

According to the probabilistic modelling results, the constant parameters can be set as follows: d = 0.80, a = 1.1086, b = 0.2857.

3.2.2. Effect of COV of Cohesion

This section analyzes the influence of COV of c_u on the safety factor against inrush. The computation results obtained from different conditions of COV = 0.2, 0.4, 0.6 and 0.8 are compared, and reliability analyses are carried out.

The frequency distribution histograms and the lognormal distribution fitting results of safety factor k are depicted in Figure 8. The statistical results of k and its coefficient of variation COVF are summarized in

Table 6. c_u = 0.75 MPa reliability analysis on safety factor.

COV	$\widehat{\mathsf{\mu }}$	$\widehat{\mathsf{\sigma }}$	kf	η	COVF
0.2	0.2984	0.0289	1.24	0.91	0.029
0.4	0.2524	0.0525	1.12	0.81	0.05
0.6	0.1801	0.0763	0.98	0.72	0.076
0.8	0.1109	0.0872	0.89	0.64	0.087

. The reference cohesion is 0.75 MPa.

The reliability index β is still taken as 2.5, and the failure probability p_f is 0.5%. The corresponding quantile safety factor k_f is listed in Table 6. Figure 9 plots the curve of COV of c_u against c_u.

As can be seen from Figure 9, η decreases linearly with COV of c_u. By performing linear fitting on the curve plotted in Figure 8, one can obtain:

$$\eta =-0.45\mathrm{COV}+0.995,$$

(10)

The correlation coefficient R² of the fitting result is 0.9975. That means when c_u is a definite value, a good linear function relationship can be found between F_n and c_u, and it can be presented as follows:

$$\eta =-m\mathrm{COV}+n,$$

(11)

Substitute Equation (11) into Equation (6), one can obtain the relationship between k_f and F_n (under the same cohesion condition):

$$k_{\mathrm{f}}=(-m\mathrm{COV}+n)F_{\mathrm{n}},$$

(12)

where m and n are constants obtained from linear fitting.

4. Discussion

4.1. Development of the Deterministic Method

Based on the above discussion, this study proposes an equivalent deterministic analysis method for the safety factor of deep excavation against inrush that can consider the spatial variability of shear strength of cement-solidified soil.

According to the conclusions in Section 3.1, the obtained deterministic safety factor F_n and the cement-solidified soil cohesion c_u satisfy a good linear function relationship. At the same time, there is also an ideal linear function relationship between the uncertainty safety factor k_f and the cohesion c_u when COV of c_u is a definite value. Therefore, it can be assumed that for different COV of c_u, there exists a general linear function relationship between the safety factor k and cement-solidified soil cohesion c_u, and it can be presented as follows:

$$k=a{c}_u+b,$$

(13)

where a and b are constants obtained from curve fitting. When COV of c_u is constant, for different c_u, a and b will remain the same.

Based on discussion in Section 3.2, when c_u is constant, there is a good one-to-one correspondence between η and COV of c_u. This correlation relationship is also a good linear function relationship:

$$\eta =-m\mathrm{COV}+n,$$

(14)

Therefore, under the same condition of c_u, the linear function relationship between the safety factor k_f obtained from uncertainty analysis and F_n from a deterministic analysis can be written as follows:

$$k_{\mathrm{f}}=(-m\mathrm{COV}+n)F_{\mathrm{n}},$$

(15)

where m and n are constant parameters.

On the basis of Equations (13) and (15), an equivalent deterministic analysis method for the safety factor of deep excavation against inrush that can consider the spatial variability of the shear strength of cement-solidified soil is proposed. The basic idea is as follows:

i.

When the cement-solidified soil cohesion c_u is known as A₁, A₂, the deterministic safety factors are known as F_n1, F_n2. Substituting F_n1 and F_n2 into Equation (13) obtains the values of a and b in the deterministic analysis (that is, when the coefficient of variation COV of c_u is 0);

ii.

c_u is known as A₁, COV of c_u are known as $B_1^1$,$B_1^2$, the corresponding safety factor $k_{\mathrm{f}1}^1$,$k_{\mathrm{f}1}^2$ can be obtained from uncertainty analysis. Substituting $k_{\mathrm{f}1}^1$,$k_{\mathrm{f}1}^2$ and F_n1 into Equation (15) solves the values of m and n;

iii.

Knowing the parameters a, b, m, n and cohesion c_u as A₁, for the condition of c_u = A_j and COV=$B_j^i$, the equation to solve the deterministic safety factor F_nj is as follows:

$$F_{\mathrm{n}j}=a{A}_j+b,$$

(16)

Additionally, the equation to solve the uncertainty safety factor $k_{\mathrm{f} j}^i$ can be written as follows:

$$k_{\mathrm{f} j}^i=(-m{B}_j^i+n)F_{\mathrm{n}j}=(-m{B}_j^i+n)\cdot (a{A}_j+b),$$

(17)

Therefore, under the conditions of any cement-solidified soil cohesion and any coefficient of variation of cohesion, the corresponding uncertainty safety factor can be obtained. On the basis of the deterministic results, the spatial variability of the shear strength of cement-solidified soil can be considered in the reliability analysis.

4.2. Verification of the Developed Method

The uncertainty analysis was carried out under the condition of c_u = 1.0 MPa, COV = 0.2. The frequency distribution histogram of safety factor k and lognormal distribution fitting results are illustrated in Figure 10.

According to the lognormal distribution fitting results, the logarithmic mean of k is 0.529371, and the logarithmic standard deviation of k is 0.031022. Based on that, reliability analysis of the results is carried out. The reliability index β is still taken as 2.5, corresponding failure probability p_f is 0.5%, and the quantile safety factor k_f obtained from Equation (4) is 1.5656.

According to the equivalent deterministic analysis method described above, in this selected model the constant parameters are a = 1.4143, b = 0.3079, m = 0.45, n = 0.995. Substituting the four parameters into Equation (17) obtains Equation (18):

$$k_{\mathrm{f} }^{}=(-0.45\mathrm{COV}+0.995)\cdot (1.4143c_u+0.3079),$$

(18)

Substituting COV as 0.2 and c_u as 1.0 MPa into Equation (18), shows k_f is 1.5586.

The difference between an uncertainty analysis result and an equivalent deterministic analysis result is less than 0.01, and the error range is within 0.5%. Therefore, it can be concluded that the proposed equivalent deterministic analysis method is reasonable and has certain practical significance in the assessment of deep excavation stability against inrush. In summary, the method can calculate a safety factor of stability against groundwater inrush with the deterministic analysis and probabilistic analysis of two cases being carried out when checking the heaving stability of excavation. It is able to save the computing time and resources, and also simplify the design process.

5. Conclusions

In this study, the strength reduction method is introduced into reliability analysis of deep excavation stability against groundwater inrush, and the impacts of shear strength of cement–soil and its spatial variability on deep excavation stability against groundwater inrush are analyzed. The main conclusions drawn from the study are as follows:

(1)

For deep excavations with passive zone improvement, it is necessary to take the resistance provided by cement–soil shear strength into consideration.

(2)

The results show that under the same condition of the cohesion coefficient of variation, the ratio of the safety factor obtained from uncertainty analysis to that from deterministic analysis has nothing to do with the shear strength of the cement-solidified soil. However, with the same reliability index and same cohesion, the ratio of the uncertainty safety factor to deterministic safety factor has a good linear correspondence with the cohesion coefficient of variation.

(3)

A deterministic analysis method equivalently considering soil uncertainty is proposed and verified. The proposed method provides a simpler and more convenient approach to the assessment of deep excavation stability against groundwater.