Abstract

The interaction between groundwater flow and soil skeleton plays an important role in the excavation of the foundation pit. This interaction phenomenon affects the water head, the soil settlement, the deformation, and responses of the excavation wall. However, the current simulation of excavation problems usually employs uncoupled analysis between flow and deformation. This paper presents a numerical coupling simulation of groundwater flow and deformation in the excavation process of the foundation pit. The numerical model is verified by published work. Subsequently, the groundwater flow and total discharge entering the excavation were investigated for four types of soils, namely clay, silt loam, sandy loam, and loamy sand. In addition, the dissipation of excess pore pressure was analyzed with different ratios of excavation rate (ER) to soil permeability k. Problems with ER/k >10⁴ can be analyzed using an undrained assumption, while a drained analysis is appropriate for those with ER/k < 10⁰. A fully coupled approach is suggested for the remaining values of ER/k. Finally, the differences in the responses of the retaining wall due to those assumptions were evaluated.

1 Introduction

Underground excavation has become more and more common in dense urban areas with many buildings containing basements (U.S. Census Bureau 2024). This construction progress requires a proper retaining wall system to prevent the surrounding soil from collapsing into the pit. Frequently, the water exists underground at the construction site, and inevitably interacts with the soil and wall, and complicates the excavation process. Many water-related incidents in foundation excavation have been reported around the world (Chow and Ou 1999; Jiang and Tan 2021; Zhang et al. 2021; Liu and Tan 2023) among many others, which prioritizes the analysis of groundwater movement and its effect on the excavation of the foundation pit.

The two most widely used assumptions for the soil in the interaction with groundwater are drained and undrained behaviours (Lees 2016). The former means that the water is free to flow through the soil, typically used when the loading rate is relatively small compared with soil’s permeability, also known as hydraulic conductivity. In this scenario, the applied load is directly transferred to the effective stress of the soil skeleton without generating any excess pore pressure (EPP). This assumption is applicable to sandy soils with high values of permeability. Meanwhile, the latter means that there is no water flow in the soil; the effective stress of the soil skeleton remains unchanged because the load is transferred to increase the total pore pressure (TPP). In the undrained process, the loading rate is fast relative to the soil’s permeability, and thus, there is not enough time for the EPP to dissipate. This undrained behaviour is typical for clay soils with very low values of permeability. These two ideal assumptions help to simplify the water–soil skeleton interaction problem, and reduce the computational workload, but might not accurately predict the generation and dissipation of the EPP (Lees 2016).

On the other hand, the fully coupled flow-deformation investigation of water–soil interaction has received much attention recently. This time-dependent analysis can account for the variation of TPP with time during the excavation process; therefore, estimating the movements of soil and retaining structures that fit the field observational data. Many published manuscripts have pointed out the differences between results obtained from a coupling analysis and those from drained/undrained investigation. Yong et al. (1989) analyzed strutted excavations in clay using a fully coupled consolidation algorithm and reported that the estimate of wall deformation is significantly lower in undrained analysis, compared to that of coupling simulation and field measurement. Meanwhile, Mahdavi et al. (2016) pointed out that a drained analysis of road embankments on ground improved with controlled modulus columns provides 32% and 62% lower estimates of base settlement and geosynthetic tension, respectively. Similarly, much work has been performed to analyze the coupling between the nonlinear behaviour of the soil skeleton and the changes in TPP (Hsi and Small 1992; de Lyra Nogueira et al. 2009; Uribe-Henao et al. 2023).

This study provides valuable insight into the analysis of the excavation of the foundation pit, in which a novel numerical coupling simulation of groundwater flow and soil deformation was used. The groundwater flow and total discharge entering the excavation pit were investigated for different types of soil during the excavation process. In addition, the excavation rate relative to the soil permeability was analyzed and serves as a selection criterion of the assumption (drained, partial drained, undrained) used in the analysis. The influences of this ratio on the dissipation of excess pore water pressure and on responses of retaining structures were evaluated parametrically to provide reference to practitioners and researchers on the excavation performance.

2 Numerical coupled analysis of an open-cut excavation

In this section, a fully coupled flow-deformation analysis of an open-cut excavation is conducted using the finite element method (FEM) implemented in commercial software PLAXIS 2D V2023.2. This software can solve the mixed equations of nodal displacements and pore pressure simultaneously with an implicit scheme of time integration. The transient flow in porous media is governed by Darcy’s law (Darcy 1856):

(1)

Where:

q	=	specific discharge,
ksat	=	permeability in saturated states,
krel	=	ratio of the permeability in a given saturation to the permeability in saturated states,
ρw	=	water density,
∇pw	=	gradient of the water pore pressure, and
g	=	gravitational acceleration.

The coupling characteristics of the flow and soil skeleton deformation are based on Biot’s formula (Biot 1941), and are expressed via a system of equations (Galavi 2010):

(2)

Where:

K	=	stiffness matrix,
fu	=	load vector,
H	=	permeability matrix,
S	=	compressibility matrix,
Q, C	=	coupling matrices,
	=	nodal displacements vector,
pw	=	pore water pressures vector,
G	=	vector considering the effect of gravity on vertical flow, and
qp	=	flux on the boundaries.

In addition, the most commonly used model, the Van Genuchten function (Van Genuchten 1980), is used to represent the relationship between the saturation and the pressure head:

(3)

Where:

S(ψ)	=	saturation
ψ	=	pressure head
Sres	=	residual saturation
Ssat	=	saturation at saturated conditions, Ssat = 1 by default
ga, gn	=	two fitting parameters

More details about the groundwater flow and coupled analysis can be found in Galavi (2010).

This manuscript analyzes the ERTC7 excavation benchmark problem, which was organized by the International Society for Soil Mechanics and Geotechnical Engineering (Schweiger 2006). In this benchmark, a deep excavation supported by a single layer of struts is performed together with a drawdown of groundwater inside the excavation pit. The geometry of the benchmark problem is shown in Figure 1.

Figure 1 FEM mesh of the excavation problem.

The excavation pit is 12.0 m wide and 6.0 m deep. The original water table is located at 4.0 m depth from the ground surface. In the final stage of the excavation process, the water table inside the pit is lowered to 6.0 m depth. The permanent and variable loads are 10 and 50 kPa, respectively. An 11.0 m length and 0.8 m thick retaining wall was constructed to prevent the surrounding soil from collapsing into the pit. It is supported against lateral movements by a single layer of struts installed at the elevation -1.5 m. Bedrock is assumed at 20.0 m below the soil surface.

The elastic-perfectly plastic constitutive model with Mohr-Coulomb failure criteria is used to describe the behaviour of soil domain. The unsaturated and saturated unit weight of the soil are 19 and 20 kN/m³, respectively. Regarding the soil stiffness parameters, the Young’s modulus is E = 30 MPa, while Poisson’s ratio is taken as v = 0.3. The effective internal friction angle φ' = 27.5°, and the effective cohesion is c' = 10 kPa. The hydraulic parameters for the soil material are taken from the database of United States Department of Agriculture (USDA) for sandy clay, with S_res = 0.2632, S_sat = 1, g_a = 2.7 m^-1, and g_n = 1.230.

The retaining wall material is isotropic linear elastic with Young’s modulus, Poisson’s ratio, and self-weight being 3 × 10⁴ MPa, 0.18, and 24 kN/m³, respectively. A strength reduction factor of 0.67 is used for the interface elements around the wall. The axial stiffness of the strut is EA = 1500 MPa with an out-of-plane spacing of 1.0 m.

The excavation stages are modeled as follows:

1. Initial phase: generate geostatic initial stress condition with at-rest lateral earth pressure coefficient K₀ = 0.5;
2. Phase 1: activate diaphragm wall and surcharge loads;
3. Phase 2: excavate up to -2.0 m;
4. Phase 3: install the strut and excavate up to -4.0 m; and
5. Phase 4: dewater and excavate up to -6.0 m.

The soil domain is discretized using 15-node triangular elements, as shown in Figure 1. For better numerical accuracy, the mesh is denser near the retaining wall and near the ground surface where the loads are applied. Due to the symmetry of geometry and applied loads, only half of the problem is calculated. The right boundary of the model is 30.0 m away from the retaining wall, which is five times larger than the excavation depth and, thus, sufficient to prevent any fictitious boundary effect. Regarding deformation, in all excavation stages, the left and right boundaries are restrained from lateral displacement, the bottom boundary is fully fixed, and the top boundary is free. Regarding the boundary conditions for the groundwater flow, the bottom one is assumed to be impermeable. Since there is no flux passing through the symmetrical surface, the left boundary is also considered impervious. Inside the excavation pit, after dewatering, the water level is at the base of the pit, which is 6.0 m below the ground surface. Meanwhile, outside of the pit, the water level drops down near the retaining wall because of the dewatering, but levels off at -4.0 m at the right boundary, which is an open boundary for the flow. In addition, the retaining wall is modeled as an impermeable boundary which does not allow water to pass through.

3 Results and discussion

3.1 Verification

The results obtained by Sterpi (2008) are used to verify the proposed numerical model. In this section, the dewatering and excavation in Phase 4 are simulated as drained condition, which is the assumption used in Sterpi’s work.

The groundwater head and steady state pore pressure are shown in Figures 2 and 3, respectively. The magnitudes and the distribution patterns of two parameters agree well with Sterpi’s results (Figure 3 from Sterpi 2008). Some observed differences are because Sterpi solved the problem of confined flow, while the current study deals with unconfined flow. Notice that the patterns shown in Figures 2 and 3 are very typical contours in excavation problems (Lees 2016).

Figure 2 Distribution of groundwater head.

Figure 3 Distribution of steady state pore pressures.

The deformed mesh of soil domain and lateral displacement of the diaphragm wall are shown in Figures 4 and 5. The results of the current study match well with those from Sterpi’s paper, proving the high fidelity of the presented numerical model. As shown in Figure 5, in all phases of the excavation process, the lateral displacements at the toe of the retaining wall obtained from this study are less than a 9% difference from those of Sterpi’s work, which is a good agreement.

Figure 4 Settlements at ground surface and excavation base at Phase 4.

Figure 5 Lateral displacements of the retaining wall in four phases.

3.2 Effect of soil permeability

A parametric study was conducted to investigate the effect of soil permeability on the hydraulic responses of the excavation problem. The groundwater flow and groundwater head were analyzed in different types of soil, namely clay, silt loam, sandy loam, and loamy sand. The permeability and parameters of Van Genuchten model were chosen as recommended by Plaxis for the corresponding soil types, as summarized in Table 1. The termination time of the couple analyses was sufficiently long so that the soil responses can be seen as drained.

Table 1 Parameters for the Van Genuchten model.

Soil type	Sres	Ssat	ga (m-1)	gn	k (m/day)
Clay	0.1789	1.000	0.800	1.090	10-2
Silt loam	0.1489	1.000	2.000	1.410	10-1
Sandy loam	0.1585	1.000	7.500	1.890	100
Loamy sand	0.1390	1.000	12.40	2.280	101

Although the magnitude of the flow is different, the pattern of the flow field is very similar in four cases of soil. A typical pattern is shown in Figure 6 for the case of loamy sand, with the maximum flow being 7.07 m/day. It is clear that the large flows gather around the wall toe. This is because the groundwater movement tends to follow the shortest path, which is around the impervious retaining wall. The other soil cases produce similar figures and are not presented here.

Figure 6 Groundwater flow in case of loamy sand.

Two cross-sections, denoted as A-A and B-B in Figure 6, were selected for the analysis. The values of total discharge Q and the directions of the groundwater flow entering the excavation at sections A-A and B-B are shown in Figure 7. The magnitude and direction of the arrows represent the magnitude and direction of the groundwater flow, respectively. Clearly, the directional patterns of the flow vectors are very similar in different soils. The flow vectors are distorted near the wall toe (section B-B), while being uniformly distributed at the base of the excavation (section A-A). However, the permeability of the soil plays a critical role in limiting the discharge entering the excavation pit. As illustrated in Figure 8, the total discharge Q shows a strongly correlated increase with the increasing soil permeability, which is as expected. In reality, the soil permeability depends largely on the porosity, and thus the grain size of the soil. The shape of the grain also plays a decisive role in soil permeability.

Figure 7 Groundwater flow and total discharge of four types of soil.

Figure 8 Variation of total discharge with soil permeability.

3.3 Effect of excavation rate

A fully coupled flow-deformation algorithm is used to perform the parametric studies, in which the ratio of the excavation rate (ER) to soil hydraulic conductivity k is investigated. This ratio plays the role of an indicator to evaluate the appropriateness of drained, partially drained, and undrained assumptions. Large values of ER/k mean that the rate of excavation is fast relative to the rate water dissipates, and thus the soil tends to react in an undrained manner, although the problem was analyzed with a coupling algorithm. Meanwhile, small values of ER/k imply the excavation process is relatively slow compared with the rate of water dissipation, which is a drained condition. This section considers the permeability k = 1 x 10^-3 m/day, and the ratio ER/k ranges from 10⁰ to 10⁹.

EPPs obtained in the analyses are shown in Figure 9. Observably, the EPP is accumulated at the end of the excavation process in case of ER/k = 10⁶, which serves as an undrained boundary. When ER/k decreases, the problem becomes partially drained and EPP dissipates gradually. With ER/k = 1, the rate of excavation is comparable to the rate of water flow, and thus, the soil behaves like a drained material in which the EPP values are almost zero across the whole computational domain.

Figure 9 Excess pore pressures at the end of excavation process.

To quantify the rate of the EPP dissipation, two observational points, A and B, were chosen, as shown in Figure 1. Point A is located at the center of the excavation pit, at the elevation -10m, while point B is behind the retaining wall at 6.0 m depth. The EPP u_e is normalized with the total vertical stress relief Δσ_v at the end of excavation. The value of Δσ_v, in other words, is calculated from the self-weight of the excavated soil. Following the sign convention, the EPP is negative at point B (pressure), while it is positive at point A (suction). Figure 10 shows the EPP at point A and B, at the end of excavation process for different values of ER/k. u_e = 0 means that the excess pore water pressure dissipates completely at the end of the excavation (Phase 4), similarly to the excavation in drained material where excess pore water pressure is not built up. From Figure 10, it can be concluded that with ER/k < 10⁰, the excavation process can be analyzed approximately as a drained problem, while it can be simulated as an undrained problem in case ER/k > 10⁴ – 10⁶. In the middle region, the excavation problem should be analyzed using a fully coupled algorithm for better projection.

Figure 10 Excess pore pressure dissipation at point A and point B.

The responses of the retaining wall were also investigated. The shear force Q, the bending moment M, and the lateral displacement of the wall during the dewatering and excavation in Phase 4, the stage in which the dewatering occurs, are plotted in Figure 11. It can be observed that the assumption of drained, partially drained, and undrained causes substantial differences in the wall responses. The differences of maximum values can be up to 30% in the shear force, 20% in the bending moment, and 45% in the lateral displacement.

Figure 11 Shear force Q, bending moment M, and lateral displacement of the retaining wall in Phase 4.

4 Conclusion

This paper analyzes an excavation problem, using a numerical coupling hydro-mechanical approach to consider the interaction between groundwater flow and soil skeleton. The proposed numerical simulation shows a good agreement with the published data. Subsequently, a parametric study was conducted to investigate the effect of soil types on groundwater flow and total discharge. The ratio of excavation rate to soil permeability ER/k was also examined to indicate drained, partially drained, and undrained conditions. Some conclusions can be drawn:

1. The pattern of the flow field is consistent with different types of soils, with maximum values being located around the wall toe. The water flows are distorted near the wall toe, and are almost uniformly distributed at the excavation base.
2. The excavation problems with the value of ER/k < 10⁴ – 10⁶ can be analyzed with an undrained assumption, while those with the value ER/k < 10⁰ are more appropriate with a drained condition. A fully coupled approach is suggested for the remaining values of ER/k.
3. The assumption of drained, partially drained, and undrained conditions can cause significant differences in the responses of the retaining wall, which might be up to 30% in the shear force, 20% in the bending moment, and 45% in the lateral displacement.

The current study is limited to a homogeneous one-layer soil medium. Soil profiles might consist of many layers with a high contrast in permeability that can complicate the coupled flow-deformation problem. More work should be done to investigate such scenarios.

Acknowledgments

We acknowledge Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for supporting this study.

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