A Semi-Analytical Solution of Two-Dimensional Unsteady Groundwater Flow Outside a Long-Strip Excavation Pit with a Cut-Off Wall

Abstract

In deep excavation engineering, the implementation of cut-off walls stands as a crucial measure to ensure structural support stability. However, the existing theories for dewatering design often overlook the variations in cut-off wall penetration depths, potentially compromising the efficacy of groundwater control strategies. Addressing this gap, this study conducts a comprehensive investigation into the dynamics of groundwater levels within confined aquifers during the dewatering process of strip excavation pits with cut-off walls. Central to this inquiry is the conceptualization of the entire excavation pit as a singular large-diameter well, with the open part beneath the cut-off wall in the confined aquifer serving as a constant-flux boundary. Employing an advanced analytical modeling approach, the study formulates a robust framework to describe the intricate interplay of unsteady groundwater flow phenomena. Leveraging the techniques of the Laplace and finite cosine transform methods, a semi-analytical solution is derived to elucidate groundwater drawdown patterns over time. The validation of the proposed solution against finite element method results underscores its fidelity and applicability. The parametric analysis reveals a dynamic evolution in drawdown characteristics within the confined aquifer, transitioning from initially cone-shaped distributions to more linear profiles that eventually stabilize with prolonged dewatering. This evolution is governed by the aquifer’s inherent anisotropy and the barrier effect exerted by the depth of cut-off wall penetration. The parametric research also underscores the critical role of lateral boundary distance in influencing groundwater drawdown patterns. The presented solutions can be used to identify optimal penetration depth ratios tailored to specific parameters, thereby offering insights for optimizing dewatering strategies for deep excavation groundwater control. Moreover, a case study was included in this study using the proposed analytical solution, and a comparison was made with field data to validate the practical applicability of the approach.

1. Introduction

For the past few years, foreshore land reclamation has been a solution to the problem of land scarcity in coastal regions due to the increasing population and economic boost [1]. Many deep excavations are constructed in the area where the stratums often have a high groundwater head [2,3,4]. The dewatering of deep excavation is essential for ensuring the safety of the excavation works, especially in well-permeable confined aquifers [5]. Cut-off walls are widely used to block groundwater seepage into the excavation, which can result in quicksand and internal piping erosion [6,7]. For deep excavation pits that require groundwater pumping in thick confined aquifers, a suspended cut-off wall is usually used to increase the seepage path [8,9]. As a result, hydraulic connection inside and outside the excavation pit is reduced, and the groundwater level inside the excavation pit can be lowered below the pit excavation bottom [10]. Therefore, it is crucial to predict the distribution of groundwater head induced by the dewatering of excavation before construction [11].

The existing research on dewatering in the excavation pits with cut-off walls are commonly based on the following methods: the analytical method [12,13,14,15], the numerical method [16,17,18,19] and the experimental method [20,21]. The commonly used numerical methods are the finite difference method [22,23], finite volume method [24] and finite element method [25], which can adapt to complex boundary conditions to calculate the head variations during dewatering. However, numerical analysis depends on the appropriate selection of parameters and does not directly reveal underlying mechanisms. Additionally, computational modeling requires significant time, which can be challenging for engineers needing on-site solutions. In order to provide the experimental basis for numerical results, Xu et al. conducted a series of laboratory investigations and studied the distribution of groundwater head during dewatering [21]. However, the experimental method has limitations, including issues with its applicability in general engineering scenarios and challenges in simulating complex hydrogeological conditions. While analytical solutions offer convenience, they are often based on several assumptions [26,27], which makes them unsuitable for the calculation of groundwater head during dewatering in an excavation pit with a cut-off wall, because of the existence of a mixed boundary condition [28,29]. With the help of a conformal transformation method, Banerjee and Muleshkov gave an implicit expression of the seepage field around the foundation pit in a homogeneous soil layer [30]. On this basis, Bereslavskii assume that one side of a cut-off wall is a free water surface and obtained an implicit expression of a seepage field [31]. When it comes to complex problems such as group well dewatering in an excavation pit, the simplification of the large well method is necessary [32]. With this method, Li et al. derived an analytical solution based on the confined–unconfined unsteady groundwater pumping flow model to assess groundwater level trends during excavation [33]. However, the anisotropic permeability of soil has not been considered in these analytical solutions. To solve this problem, Huang et al. derived a semi-analytical solution of two-dimensional steady seepage around the excavation pit in a confined aquifer with anisotropic permeability [34]. In summary, the hydraulic property of the aquifer, hydrogeological conditions, and the embedding depth of cut-off walls, among other engineering parameters, all influence the dewatering effectiveness of excavation pits. Considering these factors is crucial for developing groundwater control schemes that not only meet economic considerations but also fulfill environmental impact requirements. In addition, the problem of variation in groundwater level over time during dewatering has not been addressed. As a result, the existing theory may lead to the unreasonable design of excavation pit dewatering, which can bring hidden dangers to excavation construction.

In order to consider the influence of boundary conditions on groundwater seepage over time, this paper develops a mathematical model to describe the drawdown distribution in a homogeneous but anisotropic confined aquifer, considering the barrier effect of the cut-off wall. Assuming that the entire excavation pit is a large well and the part below the bottom of the cut-off wall is a constant-flux boundary under constant pumping in the excavation pit, a semi-analytical solution is developed by using the Laplace transform, finite cosine transform, and numerical inverse Laplace transform methods. The solution is then compared against the finite element results to demonstrate its effectiveness. Based on the presented solution, additional parametric studies are presented to investigate the barrier effect of the cut-off wall during dewatering in an excavation pit, which could provide a scientific basis for the dewatering design of excavation pits. Lastly, a case study is presented to demonstrate the applicability of this analytical solution.

2. Mathematical Model

2.1. Model Description and Basic Assumptions

For the narrow and long excavation pit, its length is much larger than its width. The seepage field can be approximated as two-dimensional plane flow. Figure 1 shows the cross-section of the model for the strip excavation pit. The width of the strip pit is 2a, the thickness of the confined aquifer is M and the penetrating depth of the cut-off wall is d. Additionally, the total pumping rate is set as Q. The distance from the center of the excavation pit to the lateral boundary is R. To establish the coordinate system for the problem, we take the center of the excavation pit bottom as the origin o, with the z-axis pointing downwards to represent the depth below the base.

In order to simplify the seepage model and obtain the solution, the main assumptions are listed as follows:

(1)

The recharge of the overlying unconfined aquifer is not considered;

(2)

The head of groundwater is constant at a distance R away from the strip excavation pit;

(3)

The cut-off wall is a no-flow boundary;

(4)

The entire excavation pit is regarded as a large well.

2.2. Governing Equation and Solving Conditions

Based on the model’s symmetry, the right half of the model is used as the focus of the study. The governing equation for the drawdown within the confined aquifer is formulated based on Darcy’s law and mass conservation as well as the aforementioned assumptions.

$$k_{\mathrm{h}}{\displaystyle \frac{{\partial }^{2}s}{\partial x^2}}+k_{\mathrm{v}}{\displaystyle \frac{{\partial }^{2}s}{\partial z^2}}={\mu }_{\mathrm{e}}{\displaystyle \frac{\partial s}{\partial t}}, \left(0\le x\le R, 0\le z\le M\right)$$

(1)

where $s(x,z,t)$ [L] is the drawdown of groundwater; $k_{\mathrm{h}}$ [LT⁻¹] and $k_{\mathrm{v}}$ [LT⁻¹] are the hydraulic conductivity in the x and z direction; and ${\mu }_{\mathrm{e}}$ [L⁻¹] is the specific storage.

The initial conditions are, respectively, described as

$${\left.s\right|}_{t=0}=0 (0\le x\le R, 0\le z\le M)$$

(2)

The lateral boundary conditions are expressed as

$${\left.{\displaystyle \frac{\partial s}{\partial x}}\right|}_{x=a}=\left\{\begin{array}{l}0 0<z<d\\ -{\displaystyle \frac{Q}{2k_{\mathrm{h}}(M-d)}} d\le z<M\end{array}\right.$$

(3)

$${\left.s\right|}_{x=R}=0 (t>0,0\le z\le M)$$

(4)

Without considering the recharge of the adjacent aquifers, the vertical boundary conditions can be written as

$${\left.{\displaystyle \frac{\partial s}{\partial z}}\right|}_{z=0}={\left.{\displaystyle \frac{\partial s}{\partial z}}\right|}_{z=M}=0 (t>0,0\le x\le R)$$

(5)

where $Q$ [L²T⁻¹] is the discharge per unit width; $a$ [L] is half the width of the excavation pit; $M$ [L] is the thickness of the confined aquifer; $R$ [L] is position of the constant-head boundary; and z [L] stands for the depth below the excavation pit.

3. Solving Procedures

For convenience, the dimensionless variables for the governing equation and boundary conditions are introduced as $m=a/M$, $n_1=d/M$, $k=k_{\mathrm{h}}/k_{\mathrm{v}}$, $x_{\mathrm{D}}=x/a$, $z_{\mathrm{D}}=z/M$, $R_{\mathrm{D}}=R/a$, $t_{\mathrm{D}}=k_{\mathrm{h}}t/{\mu }_{\mathrm{e}}{a}^2$, $s_{\mathrm{D}}(t_{\mathrm{D}},x_{\mathrm{D}},z_{\mathrm{D}})={\displaystyle \frac{k_{\mathrm{h}}}{Q}}s(t,x,z)$. Incorporating the dimensionless variables into Equations (1)–(5), the governing equation and solving conditions can be transformed into the following:

$${\displaystyle \frac{{\partial }^2{s}_{\mathrm{D}}}{\partial x_{\mathrm{D}}^2}}+{\displaystyle \frac{m^2}{k}}{\displaystyle \frac{{\partial }^2{s}_{\mathrm{D}}}{\partial z_{\mathrm{D}}^2}}={\displaystyle \frac{\partial s_{\mathrm{D}}}{\partial t_{\mathrm{D}}}} (0\le x_{\mathrm{D}}\le R_{\mathrm{D}}, 0\le z_{\mathrm{D}}\le 1)$$

(6)

$${\left.{\displaystyle \frac{\partial s_{\mathrm{D}}}{\partial x_{\mathrm{D}}}}\right|}_{x_{\mathrm{D}}=1}=\left\{\begin{array}{l}0 0<z_{\mathrm{D}}<n_1\\ -{\displaystyle \frac{m}{2(1-n_1)}} n_1\le z_{\mathrm{D}}<1\end{array}\right.$$

(7)

$${\left.s_{\mathrm{D}}\right|}_{x_{\mathrm{D}}=R_{\mathrm{D}}}=0 (t_{\mathrm{D}}>0, 0\le z_{\mathrm{D}}\le 1)$$

(8)

$${\left.{\displaystyle \frac{\partial s_{\mathrm{D}}}{\partial z_{\mathrm{D}}}}\right|}_{z_{\mathrm{D}}=0}={\left.{\displaystyle \frac{\partial s_{\mathrm{D}}}{\partial z_{\mathrm{D}}}}\right|}_{z_{\mathrm{D}}=1}=0 (t_{\mathrm{D}}>0, 0\le x_{\mathrm{D}}\le R_{\mathrm{D}})$$

(9)

and

$${\left.s_{\mathrm{D}}\right|}_{t_{\mathrm{D}}=0}=0 (0\le x_{\mathrm{D}}\le R_{\mathrm{D}}, 0\le z_{\mathrm{D}}\le 1)$$

(10)

where subscript D represents a dimensionless symbol.

Applying the Laplace transform to Equations (6)–(9) with the initial conditions of Equation (10) gives

$${\displaystyle \frac{{\partial }^2{\overline{s}}_{\mathrm{D}}}{\partial {x_{\mathrm{D}}}^2}}+{\displaystyle \frac{m^2}{k}}{\displaystyle \frac{{\partial }^2{\overline{s}}_{\mathrm{D}}}{\partial {z_{\mathrm{D}}}^2}}-p{\overline{s}}_{\mathrm{D}}=0$$

(11)

$${\left.{\displaystyle \frac{\partial {\overline{s}}_{\mathrm{D}}}{\partial x_{\mathrm{D}}}}\right|}_{x_{\mathrm{D}}=1}=\left\{\begin{array}{l}0 0<z_{\mathrm{D}}<n_1\\ -{\displaystyle \frac{m}{2(1-n_1)p}} n_1\le z_{\mathrm{D}}<1\end{array}\right.$$

(12)

$${\left.{\overline{s}}_{\mathrm{D}}\right|}_{x_{\mathrm{D}}=R_{\mathrm{D}}}=0$$

(13)

and

$${\left.{\displaystyle \frac{\partial {\overline{s}}_{\mathrm{D}}}{\partial z_{\mathrm{D}}}}\right|}_{z_{\mathrm{D}}=0}={\left.{\displaystyle \frac{\partial {\overline{s}}_{\mathrm{D}}}{\partial z_{\mathrm{D}}}}\right|}_{z_{\mathrm{D}}=1}=0$$

(14)

where ${\overline{s}}_{\mathrm{D}}(p,x_{\mathrm{D}},z_{\mathrm{D}})={\displaystyle {\int }_0^{\infty }{s}_{\mathrm{D}}(t_{\mathrm{D}},x_{\mathrm{D}},z_{\mathrm{D}}){\mathrm{e}}^{-p{t}_{\mathrm{D}}}\mathrm{d}{t}_{\mathrm{D}}}$, and $p$ is the Laplace transform parameter.

Applying the finite cosine transform to Equations (11)–(13) with the vertical boundary conditions of Equation (14) yields

$${\displaystyle \frac{{\partial }^2{\tilde{\overline{s}}}_{\mathrm{D}}}{\partial x_{\mathrm{D}}^2}}-\left[{\displaystyle \frac{{\left(n\mathsf{\pi }m\right)}^2}{k}}+p\right]{\tilde{\overline{s}}}_{\mathrm{D}}=0$$

(15)

$${\left.{\displaystyle \frac{\partial {\tilde{\overline{s}}}_{\mathrm{D}}}{\partial x_{\mathrm{D}}}}\right|}_{x_{\mathrm{D}}=1}={\displaystyle {\int }_{n_1}^1-\frac{m}{2(1-n_1)p}} \Theta ({\nu }_n,z_{\mathrm{D}})\mathrm{d}{z}_{\mathrm{D}}$$

(16)

and

$${\left.{\tilde{\overline{s}}}_{\mathrm{D}}\right|}_{x_{\mathrm{D}}=R_{\mathrm{D}}}=0$$

(17)

where ${\tilde{\overline{s}}}_{\mathrm{D}}(p,x_{\mathrm{D}},n)={\displaystyle {\int }_0^1{\overline{s}}_{\mathrm{D}}(p,x_{\mathrm{D}},z_{\mathrm{D}})\Theta ({\nu }_n,z_{\mathrm{D}})\mathrm{d}{z}_{\mathrm{D}}}$; $n$ is the finite cosine transform parameter; $\Theta ({\nu }_{n,}{z}_{\mathrm{D}})=\cos {\nu }_n{z}_{\mathrm{D}}$; and ${\nu }_n$ is the root of the equation $\sin {\nu }_n{z}_{\mathrm{D}}=0$, which can be expressed as ${\nu }_n=n\pi$.

According to the boundary conditions of Equations (16) and (17), the governing Equation (15) can be solved as

$${\tilde{\overline{s}}}_{\mathrm{D}}(p,x_{\mathrm{D}},n)={\displaystyle {\int }_{n_1}^1\frac{m}{2(1-n_1)p}\Theta ({\nu }_n,z_{\mathrm{D}})\mathrm{d}{z}_{\mathrm{D}}}{\displaystyle \frac{\sinh [{\lambda }_n(p)(R_{\mathrm{D}}-x_{\mathrm{D}})]}{{\lambda }_n(p)\cosh [{\lambda }_n(p)(R_{\mathrm{D}}-1)]}}$$

(18)

where ${\lambda }_n(p)=\sqrt{{\displaystyle \frac{{\left(mn\mathsf{\pi }\right)}^2}{k}}+p}$.

Applying the inverse finite cosine transform to Equation (18), the general solution in the Laplace domain can be written as

$$\begin{array}{ll}{\overline{s}}_{\mathrm{D}}(p,x_{\mathrm{D}},z_{\mathrm{D}})& ={\tilde{\overline{s}}}_{\mathrm{D}}(p,x_{\mathrm{D}},0)+2\cdot {\displaystyle \sum _{n=1}^{\infty }{\tilde{\overline{s}}}_{\mathrm{D}}(p,x_{\mathrm{D}},n)\cos \left(n\mathsf{\pi }{z}_{\mathrm{D}}\right)}\\ & ={\displaystyle \frac{m}{2p}}[{\displaystyle \frac{\sinh [{\lambda }_0(p)(R_{\mathrm{D}}-x_{\mathrm{D}})]}{{\lambda }_0(p)\cosh [({\lambda }_0(p)(R_{\mathrm{D}}-1)]}}-{\displaystyle \frac{2}{\pi (1-n_1)}}{\displaystyle \sum _{n=1}^{\infty }\frac{\sinh [{\lambda }_n(p)(R_{\mathrm{D}}-x_{\mathrm{D}})]}{{\lambda }_n(p)\cosh [({\lambda }_n(p)(R_{\mathrm{D}}-1)]}{R}_n}]\end{array}$$

(19)

where ${\lambda }_0(p)=\sqrt{p}$; $R_n={\displaystyle \frac{1}{n}}\sin (n\mathsf{\pi }{n}_1)\cos (n\mathsf{\pi }{z}_{\mathrm{D}})$. After the general solution in the Laplace domain is obtained, the numerical inverse Laplace transform should be used to obtain the time domain solution. In this study, the inverse transform is performed based on the work of Stehfest due to its ease of implementation and accuracy [35].

4. Solution Verification

To verify the semi-analytical solutions, a numerical model is established with the finite element method. In the numerical simulation, the computation domain is the confined aquifer, whose top and bottom boundaries are considered as non-flow boundaries. In the horizontal direction, the lateral boundary away from the excavation is the constant-head boundary. The boundary on the left side corresponds to the position of the cut-off wall of the excavation. Throughout its entire depth, the upper portion represents the location of the wall, serving as an impermeable boundary, while below the wall constitutes a constant flux boundary.

Table 1. Input data for verification.

Input Parameters	Value	Dimensionless Parameters	Value
Strip excavation pit width (2a)	100 m	Dimensionless thickness of confined aquifer (m)	2.5
Confined aquifer thickness (M)	20 m
Discharge per unit width (Q)	2 m2/day
Boundary distance (R)	500 m	Dimensionless boundary distance (RD)	10
Depth of the cut-off wall in the confined aquifer (d)	4 m
Specific storage (μe)	3 × 10−5 m−1	Penetrating depth ratio of the cut-off wall (n1)	0.2
Horizontal hydraulic conductivities (kh)	2.5 m/day
Vertical hydraulic conductivities (kv)	1.5 m/day	Anisotropic (k)	1.67

shows the input data of the verification tests in this study. The boundary conditions and mesh of the hydrogeological numerical model can be seen in Figure 2. In numerical simulations, a quadrilateral grid is employed with grid refinement near the cut-off wall to capture potential significant variations in flow direction and to obtain accurate results.

Figure 3 illustrates the comparison between the results from the presented solution and the numerical simulation at different times according to the above parameters. The symbol box indicates the results from the numerical solution while the solid line expresses the results from the semi-analytical solution. It can be observed that the results obtained from the semi-analytical solution have an excellent agreement with the numerical results. Therefore, the semi-analytical solution for the model of the strip excavation pit with the cut-off wall is reasonable.

5. Characteristics of Groundwater Seepage

As can be seen from Figure 3, the distribution of groundwater head along the x_D-axis gradually becomes a straight line and tends to be stable with time. In the early stage, the closer to the cut-off wall, the greater the rate of change in groundwater head, and there is almost no change in the groundwater head near the boundary of the constant water head.

With the progression of excavation dewatering, the drawdown near the excavation site undergoes temporal evolution, while the drawdown along the horizontal direction continuously increases. Spatially, the excavation is conceptualized as a large well. Computation results indicate a positive correlation between proximity to the excavation and the intensity of dewatering effects, leading to more significant drawdowns within the aquifer near the pit.

The implementation of a cut-off wall is an important measure for controlling groundwater inflow into the excavation. The cut-off wall intercepts the aquifer, reducing the cross-sectional area available for flow, thereby effectively obstructing the groundwater seepage path from the surrounding aquifer. This effect of the cut-off wall is known as the barrier effect [36]. Figure 4 and Figure 5 show the horizontal distribution and vertical distribution of the dimensionless drawdown s_D, respectively. It is evident that the dimensionless drawdown increases with the increase in depth and decreases with the increase in distance. But it is worth noting that the drawdown difference in the vertical direction gradually reduces as the distance from the strip excavation pit rises. There is almost no drawdown difference in the vertical direction when the distance is as large as half the width of the pit. On account of the barrier effect of the cut-off wall, the groundwater flow in the confined aquifer changes from a simple one-dimensional flow to a two-dimensional flow, which is most obvious near the cut-off wall. As the distance increases, the barrier effect gradually weakens and the flow of groundwater again becomes a one-dimensional flow.

By observing the drawdown distribution at the boundary position, it can be found that the drawdown gradient ${\displaystyle \frac{\partial s_{\mathrm{D}}}{\partial x_{\mathrm{D}}}}$ in the horizontal direction and ${\displaystyle \frac{\partial s_{\mathrm{D}}}{\partial z_{\mathrm{D}}}}$ in the vertical direction is equal to zero at the top and bottom of the confined aquifer ($z_{\mathrm{D}}=0, z_{\mathrm{D}}=1$) and the position of the cut-off wall ($x_{\mathrm{D}}=1.0, 0<z_{\mathrm{D}}<n_1$). This is consistent with the no-flow boundary condition.

The flow line and piezometric line of the hydraulic head is shown in Figure 6, which visually shows the change from two-dimensional flow near the cut-off wall to one-dimensional flow far away from the cut-off wall. The flow at depth toward the opening is radial and has a large vertical component near the cut-off wall. As the density of the piezometric line directly reflects the velocity of the groundwater flow, it is obvious that the flow velocity decreases as the lateral distance increases in the early stage of pumping. In addition, the vertical component of the flow velocity is gradually reduced along the x-axis until the flow velocity becomes completely horizontal. The change in the magnitude of flow velocity reflects the effect of pumping, while the change in the direction of flow velocity reflects the barrier effect of the cut-off wall.

6. Parametric Study

6.1. The Effect of Hydraulic Conductivity Anisotropy

The permeability of an aquifer is a significant hydrogeological parameter that describes its hydraulic conductivity, which greatly influences the effectiveness of excavation pit dewatering: a higher permeability coefficient implies easier groundwater migration within the aquifer. Due to the natural sedimentation processes, aquifer permeability often exhibits anisotropic characteristics. In order to investigate the impact of this anisotropy on dewatering effectiveness, Figure 7 plots the horizontal distribution of the dimensionless drawdown at the top of the anisotropic confined aquifer with different anisotropic degrees (k = k_h/k_v). As can be seen, the anisotropy of the confined aquifer has a significant impact on the dimensionless drawdown near the cut-off wall, and this effect diminishes with increasing horizontal distance. With the increase in the anisotropic degree (k = k_h/k_v), the dimensionless drawdown gradually decreases and tends to become more uniform. It is noteworthy that, with the increase in the degree of anisotropy, which enhances the permeability in certain directions, the pressure transmission within the aquifer becomes more efficient. This, in turn, strengthens the hydraulic connection between the aquifer and the lateral boundaries. Consequently, under constant pumping rates, the variation in water level within the aquifer due to the dewatering of the excavation becomes less pronounced. In natural aquifers, sedimentation leads to particle alignment, resulting in aquifers often exhibiting anisotropy. However, conventional dewatering designs for excavations typically use permeability coefficients derived from pumping tests that do not differentiate between horizontal and vertical directions. Therefore, when dewatering confined aquifers, it is essential to consider the impact of this anisotropy. In specific scenarios, such as the one in this analysis, the dewatering effectiveness under designed pumping rates may deviate from expectations due to this anisotropic behavior.

6.2. The Effect of the Penetrating Depth of Cut-Off Wall

In instances where the depth or thickness of the aquifer is considerable, the containment of groundwater flow by the cut-off wall of a foundation pit often adopts a suspended configuration, thus failing to achieve complete containment. In such cases, determining the appropriate penetrating depth stands as one of the most important design parameters. Various factors can influence the efficacy of groundwater level reduction at varying depths of cut-off wall embedment, including the hydrogeological conditions of the aquifer and boundary recharge conditions. Nevertheless, given the determination of other parameters, delineating the relationship between the penetrating depths and the corresponding aquifer water level drawdown serves to ascertain the optimal embedment depth required. Figure 8 plots the distribution of dimensionless drawdown at the top of the confined aquifer at different penetrating depth ratios of the cut-off wall after dewatering the strip excavation. As the cut-off wall increases the seepage path, the deeper the cut-off wall is in the confined aquifer, the more obvious the barrier effect on the groundwater flow, and the smaller the drawdown outside the strip excavation pit. As shown in Figure 8, it demonstrates that the change in drawdown is more obvious with the increase in the penetrating depth when n₁ < 0.6, while the drawdown changes slightly when $n_1\in \left[0.6, 0.8\right]$. In other words, as the penetrating depth is large enough, increasing the penetrating depth could result in an uneconomic design for dewatering. Therefore, it is necessary to determine a reasonable penetrating depth ratio (n₁) to obtain a balance between the barrier effect and economy.

Figure 9 displays the variation in the dimensionless drawdown with the penetrating depth of the cut-off wall at different distances. As shown in Figure 9, the influence of the penetrating depth on the dimensionless drawdown decreases as x_D increases. So, focusing on the curve for when x_D = 1.0, it can be seen that the rate of change of the dimensionless drawdown drops with the rise of the penetrating depth ratio n₁. When n₁ ≥ 0.7, the rate of change is less than 0.1069, which means a negligible slight growth in the barrier effect. Therefore, it is thought that there is an optimum penetrating depth ratio when determining the parameters. This conclusion is similar to the results obtained from the finite element method [37].

6.3. The Effect of Lateral Boundary Distance

In groundwater calculations affected by excavation dewatering, the distance of lateral boundary conditions represents the external recharge condition for the excavation’s dewatering area, significantly impacting its effectiveness. To investigate this influence, Figure 10 illustrates the effect of different lateral boundary distances on the depth of water level drawdown at z_D = 0.67 near the excavation pit when dewatering reaches stability. It can be observed from the figure that, when other parameters remain unchanged, with the increase in the lateral distance, the supply from the constant-head boundary becomes farther away from the excavation. This results in a greater drawdown of the water level inside the excavation, thus leading to improved dewatering effects. Conversely, as the lateral distance increases towards the excavation, the dewatering effect diminishes. Therefore, to meet specific drawdown requirements, an increase in the pumping rate or embedding depth of cut-off walls becomes increasingly necessary. In other words, when the lateral distance to the excavation is closer, the excavation is more easily replenished from external sources, necessitating a greater pumping rate to achieve the desired drawdown depth.

7. Case Study and Comparison with the Field Data

7.1. Project Overview

The excavation pit project in this case study, intended for the construction of a municipal tunnel, is located in the Yangtze River Delta region within the geological unit of the Yangtze River floodplain. The excavation pit has a typical elongated shape, with a total length of 1130 m and an average width of approximately 36 m, resulting in a length-to-width ratio of over 30 [38]. The excavation depth ranges from 7.8 m to 12.2 m. Due to its proximity to the Yangtze River, the site has abundant groundwater, which poses significant safety concerns for the excavation project. Reinforced concrete and steel struts are used for internal support, while deep cement mixing pile walls are employed as a waterproof curtain. The subsurface layers primarily consist of muddy clay, sandy silt, and silty fine sand, among which the fine sand is an aquifer. The remaining layers at the site have low permeability, all classified as weakly permeable strata or aquifuges.

Table 2. Physical and mechanical parameters of each soil layer.

Soil Layer	Soil Type	Thickness (m)	Unit Weight (kN/m3)	Effective Friction Angle (°)	Effective Cohesion (kPa)	Permeability Coefficient (cm/s)
I	Silty Clay	2.0	17.5	11.5	10.0	2.0 × 10−6
II	Sandy Silt	2.0	19.0	30.0	1.25	2.0 × 10−3
III	Silty Clay	0.5	18.0	11.0	10.0	1.0 × 10−6
IV	Sandy Silt	3.70	18.5	31.5	1.0	2.0 × 10−3
V	Silty Clay	2.3	17.3	10.0	5.0	2.0 × 10−6
VI	Fine Sand	29.5	19.0	31.5	1.0	2 × 10−2

provides the specific physical and mechanical parameters of the soil layers at the site.

According to the geological survey data, the muddy clay and sandy silt extend to a depth of about 10 m below the surface, followed by a thick silty fine sand layer reaching a depth of 40.0 m, down to the bedrock. The site is close to the Yangtze River, with abundant groundwater and a high groundwater table, approximately 0.4 m to 3.3 m below the surface. Pumping tests conducted during the survey phase indicate a significant hydraulic connection between the fine sand confined layer and the Yangtze River, with surface water capable of recharging the aquifer. In the groundwater control plan, deep cement mixing pile walls are used as a waterproof curtain for the excavation pit, combined with dewatering wells within the pit to reduce pressure in the confined sand layer. Since the length of the waterproof curtain is insufficient to penetrate the thick aquifer, it is classified as a suspended waterproof curtain.

7.2. Conceptualized Groundwater Calculation Model

In practical engineering, the stratigraphic conditions, hydrogeological conditions and boundary conditions are often complex and involve many factors. To facilitate the analytical solutions presented in this study, it is necessary to focus on the main aspects of the problem and simplify the original problem. Based on the overview of the excavation pit project and the site’s hydrogeological conditions, the upper layer consists of poorly permeable clay and silt, considered as an aquiclude, while the underlying fine sand layer is the main confined aquifer. The thickness of the confined aquifer in the calculation model is set to 30 m, and the depth of the waterproof curtain embedded in the confined aquifer is 20 m, which makes the curtain a suspended cut-off wall. Preliminary pumping tests indicate that the confined layer at the site primarily relies on surface water replenishment from the Yangtze River, so the lateral boundary is considered a constant-head boundary. Additionally, the aquifer is assumed to be isotropic, with a permeability coefficient of 17.3 m per day. The specific calculation parameters of the model are detailed in

Table 3. Calculation parameters of conceptualized dewatering project.

Calculation Parameters	Value
Length of the calculation model L	200 m
Thickness of the aquifer M	30 m
Depth of the waterproof curtain embedded lc	20 m
Half-width of the foundation pit B	18 m
Unit flow rate of dewatering wells Q	25 m2/day
Horizontal permeability coefficient Kx	17.3 m/day
Vertical permeability coefficient Kz	17.3 m/day

, and by utilizing symmetry, the right region of the conceptualized calculation model is depicted in Figure 11.

7.3. Results and Comparison

According to the semi-analytical solution of unsteady groundwater flow outside the pit with a suspended waterproof curtain established in this paper, an analysis was conducted on the conceptual model. Figure 12 presents the time–depth groundwater level decline of an observation well outside the excavation pit’s cut-off curtain. The pit is situated in an area of the Yangtze River floodplain, and preliminary pumping tests have shown hydraulic connectivity between the site’s confined layer and the Yangtze River. There are seasonal variations in the Yangtze River water level, which could significantly affect groundwater levels around the pit during the stages of dewatering. However, the established semi-analytical solutions, based on the assumption of constant-head lateral boundaries, cannot account for such complex time-variable boundary conditions. In order to mitigate the influence of this type of time-dependent boundary, observation well data were selected from the end furthest from the Yangtze River for comparative analysis.

It can be observed that, at this specific location, the groundwater level decreases with prolonged pumping time and tends towards a stable state. The figure also compares measured values with numerical results based on finite element simulations using quadrilateral elements, with refinement near the pumping wells and waterproof curtain locations. Overall, the presented semi-analytical and numerical simulation results are in good agreement, and they generally match the measured values from the observation well.

8. Conclusions

A mathematical model of unsteady groundwater flow in a limited confined aquifer was established in this investigation to obtain the drawdown distribution outside a strip excavation pit. The approach uses the Laplace transform and Fourier transform to derive an analytical solution to evaluate the two-dimensional drawdown. In addition, the time-domain result is obtained by the Laplace inversion transform which is based on the work of Stehfest. The semi-analytical solution was compared with numerical simulation via the finite element method. From the analytical analysis of the barrier effect of the cut-off wall on the groundwater seepage, these main conclusions can be drawn:

(1)

The existence of the cut-off wall has a barrier effect on the distribution of groundwater head, as the cut-off wall increases the seepage path. The groundwater flow in the confined aquifer changes from a simple one-dimensional flow to a two-dimensional flow around the cut-off wall, which indicates the change in the flow velocity and direction.

(2)

From the time-domain results, the distribution of drawdown in the horizontal direction changes from a cone-shaped distribution to a straight line over time in a limit confined aquifer, which shows the process of transition from unsteady flow to steady flow. The barrier effect reduces with the increase in pumping time.

(3)

The dimensionless drawdown decreases with the increase in anisotropic degree ($k=k_{\mathrm{h}}/k_{\mathrm{v}}$). The effect of anisotropy is obvious near the cut-off wall, and weakened with the horizontal distance. The effectiveness of excavation dewatering is significantly influenced by the lateral boundary distance, with closer boundaries requiring higher pumping rates to achieve desired drawdown depths, while increased distances result in improved dewatering effects.

(4)

The impacts on drawdown in a limited confined aquifer due to the barrier effect are correlated with the penetrating depth of the cut-off wall. When the cut-off wall is not buried deep enough to cut off the confined aquifer completely, it is thought that there is an optimum penetrating depth ratio to achieve a balance between the barrier effect and economy, when determining the parameters.

(5)

A case study was conducted on a practical municipal engineering scenario involving the dewatering of a long rectangular excavation. The practical scenario was conceptualized and analyzed using the presented analytical solution, as well as finite element numerical methods. Comparison with field data validates the engineering applicability of the approach proposed in this paper.