A Closed-Form Solution for Water Inflow into Deeply Buried Arched Tunnels

Abstract

The analytical solutions for groundwater inflow into tunnels are usually developed under the condition of circular tunnels. However, real-world tunnels often have non-circular cross-sections, such as arched, lens-shaped, or egg-shaped profiles. Accurately assessing water inflow for these diverse tunnel shapes remains challenging. To address this gap, this study developed a closed-form analytical solution for water inflow into a deeply buried arched tunnel using the conformal mapping method. When the tunnel circumference degenerates to a circle, the analytical solution degenerates to the widely used Goodman’s equation. The solution also showed excellent agreement with numerical simulations carried out using COMSOL. Based on the analytical solution, the impact of various factors on water inflow Q was further discussed: (1) Q decreases as the boundary distance $D^{\ast }$ increases. And the boundary inclination angle ($\alpha -\pi /2$) significantly affects Q only when the boundary is close to the tunnel ($D^{\ast }<20$); (2) Q increases quickly with the upper arc radius $r_1^{\ast }$, while it shows minimal variation with the change in the lower arc radius $r_2^{\ast }$. The findings provide a theoretical foundation for characterizing water inflow into arched tunnels, thereby supporting improved tunnel planning and grouting system design.

1. Introduction

The inflow of groundwater into tunnels is a critical issue in the fields of hydrology and geological engineering. Subsurface tunnels constructed below the phreatic surface are particularly susceptible to water inflow during both the excavation phase and long-term operation. The excessive inflow of uncontrolled tunnel water could lead to potential destabilization of surrounding rocks [1], safety hazards for personnel and infrastructure [2,3], ecological and environmental disasters [4,5,6], and substantial economic expenses [7]. Consequently, accurately assessing the flow of water into these tunnels is essential for tunnel construction and long-term tunnel stability.

To address these challenges, extensive research has been conducted to predict the tunnel inflow rate and the surrounding flow field. Various methods have been employed in previous studies, including analytical methods, empirical regression methods, numerical simulation methods, and data mining methods. Analytical methods, although requiring certain simplifications, have been widely used due to their convenience and solid theoretical foundations. Notably, by employing the image method to generalize a tunnel as a “point sink”, Goodman et al. [8] derived the widely used Goodman’s equation for water inflow into deep-buried tunnels. Subsequently, several modifications and extensions have been made to Goodman’s equation to broaden its applicability for shallow tunnels and variable water heads [9,10,11]. Additionally, using conformal mapping methods, Park et al. [12] derived the exact analytical solution for the water inflow in circular tunnels within a semi-infinite aquifer. Subsequently, researchers have further applied the method of conformal mapping to tunnels with lining and grouting rings [13,14,15,16]. However, due to the complexity of tunnel water inrush processes [17,18], the existing analytical methods are mostly limited to tunnels with regular cross-sectional shapes, such as circular, elliptical, and square tunnels [15,19,20]. For tunnels in practical engineering with more complex cross-sectional shapes (e.g., arched, lens-shaped, or egg-shaped profiles), regression methods, numerical simulation methods, and data mining methods are often employed to predict the water inflow rates [21,22,23,24].

Although regression methods and data mining methods are commonly used for the quick estimation of tunnel water inflow, they are not capable of accurately depicting the flow field around the tunnel. Moreover, the applicability of regression and data mining methods still requires further investigation under practical conditions [25]. Numerical simulation methods, although capable of calculating the flow field around tunnels, may encounter challenges when dealing with complex tunnel cross-sections. The detailed characterization of tunnel shapes leads to increased computational costs, and difficulties can also arise in determining an appropriate model domain as well as boundary conditions [2,26]. The tunnel water inflow may be overestimated/underestimated if the distance between the tunnel and the external model boundary is insufficient. Furthermore, the flow field around the tunnel may be significantly affected by the shape of the external model boundary [27]. Butscher [28] suggests that the model domain for tunnel water inflow simulations should extend to a distance of at least 20 times the tunnel diameter. Within this large model domain, the computational cost associated with the detailed characterization of the tunnel cross-section can be particularly expensive.

Therefore, for tunnels with complex cross-sections, further research on analytical solutions is necessary to reduce the cost of water inflow evaluation and flow field description. Such insights are instrumental in tunnel excavation planning and the design of drainage and seepage control systems.

Based on the objectives described above, this study employed the conformal mapping method to obtain analytical solutions for the intrusion of water into deeply buried arched tunnels. The performance of the obtained analytical solutions was then verified by comparing them with the numerical results simulated by COMSOL Multiphysics (version 6.3, COMSOL Inc., Burlington, MA, USA). Finally, by varying the parameters of the tunnel, the impact of various factors on tunnel water inflow is discussed to obtain further insights on the tunnel water inflow process.

2. Materials and Methods

2.1. Problem Generalization

To obtain the analytical solution for the groundwater flow field around an arched tunnel, several assumptions should be made for simplification:

(1)

The tunnel is a straight long tunnel; thus, the analysis of the flow field can be reduced to a 2-D flow problem on the plane perpendicular to the tunnel’s axis.

(2)

The tunnel is located in a homogeneous and isotropic porous medium, which means that the permeability is a uniform scalar value.

(3)

Groundwater flows from a specified-head boundary to the tunnel in a steady and laminar manner; thus, the flow can be described using the Laplace equation.

(4)

The drainage system on the tunnel circumference functions well, and the distance from the tunnel to the specified-head boundary is sufficiently large. Thus, compared to the head loss in the aquifer, the head loss from the tunnel circumference to the internal drainage channel is negligible, suggesting that the water head on the tunnel circumference can be considered a constant value.

(5)

The water inflow is mainly supplied from a linear boundary (e.g., a fracture, a large surface/ground water body with a flat bank, or a groundwater table with sufficient recharge), and the head change along this boundary is negligible compared to the head loss in the aquifer, allowing the water head on the linear boundary to be treated as a constant value.

For a problem that satisfies the above generalizations, the shape of the tunnel cross-section on the z plane (physical plane) is shown in Figure 1a: the radius of the upper arc of the tunnel circumference is denoted as $r_1$, the radius of the lower circular arc is denoted as $r_2$ (for a tunnel with a flat bottom, $r_2\gg r_1$), the length of the common chord of the two arcs is denoted as $2L$, the distance from the tunnel to the linear boundary is denoted as D ($L\le r_1,r_2\ll D$), and the inclination angle of the linear boundary is denoted as $\alpha -\pi /2$.

2.2. Conformal Mapping

To solve the flow field around the arched tunnel shown in Figure 1a, the aquifer outside the tunnel is transformed to the upper half-plane on the mapping plane through conformal mapping. Specifically, the transformation involves the following two steps.

2.2.1. Möbius Transformation

Firstly, a Möbius transformation is applied to the physical plane (z plane) containing the tunnel circumference—for any point on the z plane, the mapping is defined as follows:

$$\zeta ={\displaystyle \frac{z}{z-2L}}={\displaystyle \frac{z^{\ast }}{z^{\ast }-2}}$$

(1)

where $z^{\ast }=z/L$ represents the dimensionless complex coordinate normalized with respect to the unit length L. This transformation maps the upper and lower circular arc of the tunnel circumference to two rays on the $\zeta$ plane (Figure 1b). Point ➀ (the origin) on the z plane remains unchanged after the mapping, while point ➂ $z_3^{\ast }=2+0i$ (i stands for the imaginary unit) becomes the point at infinity on the $\zeta$ plane. Additionally, an arbitrary point on linear boundary ➄ on the original plane (z plane) can be represented as

$$z_5^{\ast }\left(\theta \right)=1+{\displaystyle \frac{D^{\ast }}{\cos (\theta -\alpha )}}{e}^{i\theta }$$

(2)

where $D^{\ast }=D/L\gg 1$ represents the dimensionless boundary distance, $\theta$ represents the argument of the point on the linear boundary, and $\theta \in (\alpha -\pi /2,\alpha +\pi /2)$. As shown in Appendix A, on the $\zeta$ plane, the linear boundary is mapped to a circle

$${\zeta }_5\left({\theta }^{\prime }\right)=1+\eta e^{-i\alpha }+\eta e^{i{\theta }^{\prime }}$$

(3)

where ${\theta }^{\prime }$ is a single-valued function of $\theta$, ${\theta }^{\prime }\in [0,2\pi )$. Since the head change along the linear boundary can be neglected, the mapped circle ${\zeta }_5\left({\theta }^{\prime }\right)$ on the $\zeta$ plane can be seen as a constant head well with its center located at

$${\zeta }_{5c}=1+\eta e^{-i\alpha }$$

(4)

and its radius is

$$\eta ={\displaystyle \frac{1}{D^{\ast }-\cos \alpha }}\to 0$$

(5)

On the other hand, by substituting the midpoint ➁ of the upper arc and the midpoint ➃ of the lower arc into Equation (1), the arguments of the two rays on the $\zeta$ plane (Figure 1b) can be determined as

$$\begin{array}{ccc}& & {\alpha }_1=\arg \left({\zeta }_2\right)=-\arctan {\displaystyle \frac{1}{\sqrt{r_1^{\ast 2}-1}}}\hfill \\ & & {\alpha }_2=\arg \left({\zeta }_4\right)=-\arctan {\displaystyle \frac{1}{\sqrt{r_2^{\ast 2}-1}}}+\pi \hfill \end{array}$$

(6)

where $\arg (·)$ represents the argument of a complex number, and $r_1$ and $r_2$ represent the dimensionless radius of the upper and lower arc with respect to the unit length L, $1\le r_1^{\ast },r_2^{\ast }\ll D^{\ast }$. From Equation (6), it can be found that ${\alpha }_1\in [-\pi /2,0)$ and ${\alpha }_2\in [\pi /2,\pi )$.

2.2.2. Rotation and Scaling

Secondly, the two rays on the $\zeta$ plane are further transformed onto the real axis on the $\xi$ plane:

$$\xi ={\left(\zeta e^{-i{\alpha }_1}\right)}^m$$

(7)

where

$$m={\displaystyle \frac{\pi }{{\alpha }_2-{\alpha }_1}}\in \left({\displaystyle \frac{2}{3}},2\right)$$

(8)

Here, m can be seen as a shape factor of the tunnel circumference: as the value of m becomes closer to 1, the tunnel circumference approaches the shape of a circle. Equation (7) maps the two rays on the $\zeta$ plane to the positive and negative real axes on the $\xi$ plane (Figure 1c). As a result, the flow field outside the tunnel is mapped to the upper half-plane.

The transformation defined by Equation (7) maps the well center ${\zeta }_{5c}$ to ${\xi }_{5c}$ on the $\xi$ plane:

$${\xi }_{5c}=e^{-im{\alpha }_1}{\zeta }_{5c}^m$$

(9)

and in treating the well as a point source, the well radius on the $\xi$ plane can be calculated by

$$r_w=\eta \left|{\displaystyle \frac{\mathrm{d}\xi }{\mathrm{d}\zeta }}\right|=m\eta {\left|{\zeta }_{5c}\right|}^{m-1}=m\eta {\left|1+\eta e^{-i\alpha }\right|}^{m-1}$$

(10)

where $\left|·\right|$ denotes the modulus of a complex number.

2.2.3. The Closed-Form Solution of Flow Field and Water Inflow

After conformal mapping, the drainage of the arched tunnel on the z plane can be treated as water flow from the equivalent injection well ${\xi }_{5c}$ to a constant head linear boundary located at the real axis of the $\xi$ plane (Figure 1c). The closed-form solution for the flow field on the $\xi$ plane is given by Strack [29] (Figure 1d):

$$\mathsf{\Omega }\left(\xi \right)=-{\displaystyle \frac{Q}{2\pi }}\ln {\displaystyle \frac{\xi -{\xi }_{5c}}{\xi -\overline{{\xi }_{5c}}}}+{\mathsf{\Phi }}_a$$

(11)

where $\mathsf{\Omega }\left(\xi \right)$ represents the complex potential of the flow field on the $\xi$ plane, $\overline{{\xi }_5}$ denotes the conjugate point of ${\xi }_{5c}$ (Figure 1d), and ${\mathsf{\Phi }}_a$ is the discharge potential along the tunnel circumference, defined as [29]:

$${\mathsf{\Phi }}_a=K{H}_a+C$$

(12)

where K is the hydraulic conductivity of the aquifer, $H_a$ is the head at the tunnel circumference, and C is an arbitrary constant because only the differences or gradients in ${\mathsf{\Phi }}_a$ and $\mathsf{\Omega }\left(\xi \right)$ matter. For the sake of simplicity, C was set to be $-K{H}_a$ in this study, and thus, the value of ${\mathsf{\Phi }}_a$ equals 0; Q represents the water inflow into the tunnel on the z plane, and can be seen as the injection rate of well ${\xi }_5$ on the $\xi$ plane. Thus, on the $\xi$ plane, Q can be calculated using the formula of the well injection rate near a linear boundary given by [29]:

$$Q={\displaystyle \frac{2\pi K(H_0-H_a)}{{\displaystyle ln\frac{2a}{r_w}}}}$$

(13)

where $H_0$ is the head of the equivalent injection well on the $\xi$ plane, i.e., the head at the original linear boundary; $r_w$ is the wellbore radius of the equivalent injection well on the $\xi$ plane, of which the value is given by Equation (10); a is the distance between the equivalent injection well and the real axis on the $\xi$ plane, i.e.,

$$\begin{array}{c}\hfill a=\mathrm{Im}\left\{{\xi }_{5c}\right\}=\mathrm{Im}\left\{e^{-im{\alpha }_1}{\zeta }_{5c}^m\right\}=\mathrm{Im}\left\{e^{-im{\alpha }_1}{\left(1+\eta e^{-i\alpha }\right)}^m\right\}\end{array}$$

(14)

where $\mathrm{Im}\{·\}$ represents the imaginary part of a complex number.

Under the condition of $r_1=r_2=L$ (i.e., the tunnel circumference degenerates to a circle with $m=1$ and ${\alpha }_1=-\pi /2$ as mentioned in Section 2.2.2), it can be calculated from Equation (14) that the value of a should be $1+\eta cos\alpha$, and from Equation (10), the value of $r_w$ is calculated to be $\eta$. Thus, when $r_1=r_2=L$, we have $a/r_w=1/\eta +cos\alpha =D^{\ast }=D/L$, which means that Equation (13) degenerates to Goodman’s equation, which has been widely used in practice for quasi-circular tunnels [8]:

$$Q^{\prime }={\displaystyle \frac{2\pi K(H_0-H_a)}{{\displaystyle ln\frac{2D}{r_{eq}}}}}$$

(15)

where $r_{eq}$ is the equivalent radius of tunnel circumference on the z plane, and $r_{eq}=L$ when $r_1=r_2=L$. The degeneration of Equations (13)–(15) partially verifies the correctness of Equations (10)–(14).

Finally, by substituting Equations (1), (7) and (9) into Equation (11), the flow field on the physical plane (z plane) can be obtained (Figure 1f):

$$\mathsf{\Omega }\left(z\right)=-{\displaystyle \frac{Q}{2\pi }}\ln {\displaystyle \frac{{\displaystyle {\left({\displaystyle \frac{z^{\ast }}{z^{\ast }-2}}\right)}^m-{\left(1+\eta e^{-i\alpha }\right)}^m}}{{\displaystyle {\left({\displaystyle \frac{z^{\ast }}{z^{\ast }-2}}\right)}^m-e^{i2m{\alpha }_1}{\left(1+\eta e^{i\alpha }\right)}^m}}}+{\mathsf{\Phi }}_a$$

(16)

2.3. Numerical Simulations for Comparison

To test the new analytical solution of this study (i.e., Equations (13) and (16)), a series of synthetic numerical simulations were performed. The mathematical model was numerically solved using the partial differential equation solver in COMSOL Multiphysics (version 6.3, COMSOL Inc., Burlington, MA, USA), which is a Galerkin finite-element software package. The simulation domain was designed as a semicircle (Figure 2a), with the chord located on the linear boundary (which is set to be a specified-head boundary in the simulations) and the radius being R ($R^{\ast }=R/L>>1$). The dimensionless radius $R^{\ast }$ determines the size of the simulation domain, as well as the distance of the far-field boundary (which is set to be a no-flow boundary in the simulations); thus, the value of $R^{\ast }$ should be carefully chosen. In this study, the values of $R^{\ast }$ were tested in the range of $(D^{\ast },1000D^{\ast }]$, and the change in Q with the increase in $R^{\ast }$ is shown in Figure 2b. It can be found in Figure 2b that when $R^{\ast }$ is larger than $10D^{\ast }$, the value of Q becomes steady and merely changes with $R^{\ast }$. To further improve simulation accuracy and avoid boundary effects, the domain radius was set to be $1000D^{\ast }$ in the following simulations. The model domain was then discretized using triangular elements, and the spatial discretization near the tunnel was depicted non-uniformly to minimize the numerical cutoff errors. There were 3,762,690 to 3,802,597 triangular elements in total (the number varies depending on the tunnel shape). For the sake of simplicity, the values of K were set to be 0.05 m/d, and the values of ($H_0-H_a$) were set to be 20 m for all mathematical and numerical models.

3. Results and Discussion

3.1. Solution Verification

The analytical solutions of the flow field around the tunnel using Equation (16) and the numerical solutions using COMSOL are compared in Figure 3 for nine different cases, with the parameters summarized in

Table 1. Comparison of the analytical and numerical solutions for tunnel water inflow.

No.	Parameters				Water Inflow Q and Relative Errors ${\mathit{E}}_{\mathit{r}}$
	${\mathit{r}}_{\mathbf{1}}^{\ast }$	${\mathit{r}}_{\mathbf{2}}^{\ast }$	${\mathit{D}}^{\ast }$	$\mathit{\alpha }$	${\mathit{Q}}_{\mathit{sim}}$, ${\mathbf{m}}^{\mathbf{2}}·$${\mathbf{d}}^{-\mathbf{1}}$	${\mathit{Q}}^{\prime }$, ${\mathbf{m}}^{\mathbf{2}}·$${\mathbf{d}}^{-\mathbf{1}}$	${\mathit{E}}_{\mathit{r}}^{\prime }$, ‰	$\mathit{Q}$, ${\mathbf{m}}^{\mathbf{2}}·$${\mathbf{d}}^{-\mathbf{1}}$	${\mathit{E}}_{\mathit{r}}$, ‰
1	1.2	3	30	$\pi /3$	1.5937	1.5975	2.39	1.5936	0.08
2	1	3	30	$\pi /3$	1.4567	1.4691	8.53	1.4564	0.20
3	1.5	3	30	$\pi /3$	1.7106	1.7119	0.73	1.7103	0.16
4	1.2	1.2	30	$\pi /3$	1.6143	1.6141	0.12	1.6141	0.12
5	1.2	100	30	$\pi /3$	1.5870	1.5959	5.62	1.5868	0.12
6	1.2	3	10	$\pi /3$	2.2491	2.2546	2.45	2.2454	1.65
7	1.2	3	100	$\pi /3$	1.2171	1.2193	1.78	1.2170	0.05
8	1.2	3	30	0	1.5847	1.5882	2.22	1.5845	0.13
9	1.2	3	30	$\pi /2$	1.5951	1.5990	2.43	1.5950	0.07

. It can be seen in Figure 3 that the solutions from Equation (16) show excellent agreement with the solutions of COMSOL.

Apart from the flow field, the analytical solutions of water inflow from Equation (13) were also compared with the numerical solutions for all nine cases. Taking the numerical simulated water inflow $Q_{sim}$ as the reference value, the relative errors for the analytical solutions are defined as follows:

$$E_r={\displaystyle \frac{|Q-Q_{sim}|}{Q_{sim}}}$$

(17)

And to obtain a more comprehensive assessment for the performance of Equation (13), the water inflows into the tunnel calculated using the widely used Equation (15) are also presented in Table 1, as well as their relative errors (noted as $Q^{\prime }$ and $E_r^{\prime }$, respectively).

It can be seen in Table 1 that despite variations in parameters, $E_r$ remains consistently lower than $E_r^{\prime }$, indicating the superior performance of Equation (13). In comparing Cases 1, 4, and 5, it can also be found that as $r_2^{\ast }$ increases from 1.2 to 100, $E_r^{\prime }$ increases significantly (from 0.12‰ to 5.62‰), while $E_r$ remains stable around 0.10‰. The quick increase in $E_r^{\prime }$ with $r_2^{\ast }$ can be attributed to the changes in tunnel shape. Equation (15) calculates water inrush for equivalent circular tunnels, resulting in small errors when the tunnel shape approaches a circle (e.g., $r_2^{\ast }=r_1^{\ast }=1.2$ in Case 4, Figure 3d). However, the error becomes larger as the tunnel shape approaches a semi-circle (i.e., $r_2^{\ast }=100\gg r_1^{\ast }$ for Case 5 in Figure 3e). In contrast, Equation (13) captures the change in tunnel shape, resulting in consistently lower errors across different $r_2^{\ast }$ values.

3.2. Error Analysis

In comparing cases 1, 6, and 7 in Table 1, a rapid decline in $E_r$ is observed from 1.65‰ to 0.05‰ as $D^{\ast }$ increases from 10 to 100, contrasting with a more modest reduction in $E_r^{\prime }$ from 2.45‰ to 1.78‰. A closer examination of the relationship between $E_r$ ($E_r^{\prime }$) and $D^{\ast }$ is presented in Figure 4, with $D^{\ast }$ increasing from 10 to 100 and the other parameters remaining the same as the parameters in Case 1. Figure 4 shows a more pronounced decrease in $E_r$ with $D^{\ast }$ than $E_r^{\prime }$.

In fact, $E_r$ is mainly introduced through the approximation that well ${\zeta }_5$ on the $\zeta$ plane can be treated as a ’point source’ during the transformation of Equation (10), which means that well radius $\eta$ should be as close to 0 as possible. And if well radius $\eta$ is not small enough, then according to Equations (3), (4) and (7), the transformed well ${\xi }_5$ on the $\xi$ plane should be

$$\begin{array}{ccc}\hfill {\xi }_5\left({\theta }^{\prime }\right)& =& {\left[{\zeta }_5\left({\theta }^{\prime }\right)e^{-i{\alpha }_1}\right]}^m=e^{-im{\alpha }_1}{\left({\zeta }_{5c}+\eta e^{i{\theta }^{\prime }}\right)}^m\hfill \\ & =& e^{-im{\alpha }_1}{\zeta }_{5c}^m\left[1+m{\displaystyle \frac{\eta e^{i{\theta }^{\prime }}}{{\zeta }_{5c}}}+O\left({\eta }^2\right)\right]\hfill \\ & =& {\xi }_{5c}+e^{-im{\alpha }_1}m\eta {\zeta }_{5c}^{m-1}{e}^{i{\theta }^{\prime }}+O\left({\eta }^2\right)\hfill \end{array}$$

(18)

And ${\xi }_5\left({\theta }^{\prime }\right)$ can be seen as a quasi-circle with its center located at ${\xi }_{5c}$, with its radius being

$$\begin{array}{ccc}\hfill r_{w1}& =& \left|e^{-im{\alpha }_1}m\eta {\zeta }_{5c}^{m-1}\right|+O\left({\eta }^2\right)\hfill \\ & =& m\eta {\left|{\zeta }_{5c}\right|}^{m-1}+O\left({\eta }^2\right)\hfill \\ & =& r_w+O({\displaystyle \frac{1}{{(D^{\ast }-\cos \alpha )}^2}})\hfill \end{array}$$

(19)

Equation (19) indicates that when the boundary distance $D^{\ast }$ is not large enough, the approximation of well ${\zeta }_5$ being a ’point source’ may introduce a significant error into $r_w$. The introduced error concerns small higher-order quantities of $1/{(D^{\ast }-\cos \alpha )}^2$; thus, $Er$ should decrease quickly with $D^{\ast }$. On the other hand, the error in Goodman’s equation (i.e., Equation (15)) is mainly introduced through the approximation of the equivalent radius $r_{eq}$. As the equivalent radius $r_{eq}$ is determined only by the shape of the tunnel cross-section and remains a constant under different $D^{\ast }$ values, $E{r}^{\prime }$ should decrease slowly with the increase in $D^{\ast }$.

3.3. Parameter Sensitivity

Figure 5 illustrates the influence of parameters ($r_1^{\ast }$, $r_2^{\ast }$, $D^{\ast }$, and $\alpha$) on water inflow Q. As shown in Figure 5a, water inflow Q decreases with the increase in boundary distance $D^{\ast }$.

When the boundary is close to the tunnel circumference (i.e., $D^{\ast }<20$), the inclination angle of the boundary ($\alpha -\pi /2$) has a significant impact on Q, while as the boundary distance increases, Q becomes less sensitive to changes in $\alpha$. Additionally, Q reaches its maximum when $\alpha =\pi /2$, and the value of Q exhibits symmetry about the plane $\alpha =\pi /2$.

Figure 5b shows that Q increases quickly with the upper arc radius, $r_1^{\ast }$, but shows minimal change with the the increase in the lower arc radius, $r_2^{\ast }$. This difference can be explained by the relative positions of the arcs: the upper arc is closer to the boundary than the lower arc, implying that alterations in $r_1^{\ast }$ have a more pronounced effect on Q compared to those in $r_2^{\ast }$.

4. Conclusions

Steady-state analytical solutions for water inflow into a deeply buried arched tunnel were developed using the conformal mapping method together with the complex variable techniques. The developed analytical solution was verified by numerical models and has a superior performance under a wide range of parameters, with errors generally being <1‰. Based on the solutions, this study examined the impact of various factors on water inflow Q: (1) Q decreases with the boundary distance $D^{\ast }$, while the boundary inclination angle ($\alpha$) significantly affects Q only when the boundary is close to tunnel ($D^{\ast }<20$); (2) Q increases with the upper arc radius $r_1^{\ast }$, while changes little with the lower arc radius $r_2^{\ast }$.

It should be noted that the analytical solution as well as the numerical models presented in this study are based on 2D conditions. However, in real field conditions, the groundwater boundary and potential fractures are seldom strictly parallel to the tunnel axis, and the inherent heterogeneity of the geological medium, variable topography, and irregular boundary shapes play a significant role in influencing the inflow. As such, 3D numerical simulations, which can incorporate these complexities in greater detail, often provide a more accurate representation of the actual flow conditions.

Nevertheless, in the early stages of tunnel design and investigation, detailed geotechnical surveys and extensive experimental/numerical studies are often impractical due to time and resource constraints. Under these conditions, fast-evaluation formulas—such as the classical Goodman solution—are typically employed to estimate inflow and guide further investigation. Our approach, as outlined in Section 2, builds on these established methods while providing an improved sensitivity regarding the influence of tunnel shape parameters and boundary distances on water inflow. The proposed analytical solution offers a relatively accurate and efficient formula that helps designers roughly determine optimal tunnel shapes and locations, thereby serving as a valuable preliminary tool that lays the foundation for more detailed subsequent studies, including detailed field investigations and numerical modeling.

On the other hand, it should also be mentioned that the analytical solution obtained by this study is applicable only for deeply buried arched tunnels without grouting. Further investigation is required to extend the analytical solution to more complex scenarios, including shallow tunnels, tunnels with egg-shaped or lens-shapped cross-sections, and those equipped with either uniform or non-uniform grouting rings.