# An Analytical Model of Seepage Field for Symmetrical Underwater Tunnels

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

_{0}represents the radius of the tunnels. The distance between the two tunnel centers is 2b. The tunnel depth from the center to the ground surface is h. The water table is above the ground surface. The water depth from the water table to the ground surface is h

_{w}. The permeability of the aquifer is expressed as k.

- (1)
- Symmetrical underwater tunnels are circular and are located in a fully saturated, homogeneous, isotropic, and semi-infinite aquifer.
- (2)
- A state of steady flow is assumed.
- (3)
- The fluid is incompressible.
- (4)
- The water table above the ground surface is horizontal and remains unchanged.

_{w}is the unit weight of water, and y is the elevation head.

## 3. Analytical Solutions

_{1}, r

_{2}represent the distance from the point M(x,y) to the centers of tunnels 1 and 2, respectively; and r

_{3}, r

_{4}represent the distance from the point M(x,y) to the centers of tunnels 3 and 4, respectively. According to geometric properties, r

_{1}, r

_{2}, r

_{3}, and r

_{4}can be denoted as:

_{M}of point M(x,y) in the infinite aquifer composed of four tunnels can be written as

_{1}= r

_{3}and r

_{2}= r

_{4}), the total hydraulic head is equal to h

_{w}. Substituting this boundary condition into Equation (10), the integral constant C can be obtained as C = h

_{w}.

_{1}= r

_{0}), r

_{2}, r

_{3}, and r

_{4}can be denoted as ${r}_{2}=\sqrt{{r}_{0}^{2}-4bx}$, ${r}_{3}=\sqrt{{r}_{0}^{2}-4hy}$, and ${r}_{4}=\sqrt{{r}_{0}^{2}-4bx-4hy}$. Here, x and y represent the abscissa and ordinate values at the perimeter of tunnel 1. Thus, x and y follow the relationships $-b-{r}_{0}\le x\le -b+{r}_{0}$ and $-h-{r}_{0}\le y\le -h+{r}_{0}$. If r

_{0}is small enough relative to b and r

_{0}is small enough relative to h, x and y can be approximately denoted as x = −b and y = −h. r

_{2}, r

_{3}, and r

_{4}can be approximately rewritten as ${r}_{2}=\sqrt{{r}_{0}^{2}+4{b}^{2}}$, ${r}_{3}=\sqrt{{r}_{0}^{2}+4{h}^{2}}$, and ${r}_{4}=\sqrt{{r}_{0}^{2}+4{b}^{2}+4{h}^{2}}$.

_{2}= r

_{0}), r

_{1}, r

_{3}, and r

_{4}can be denoted as ${r}_{1}=\sqrt{{r}_{0}^{2}+4bx}$, ${r}_{3}=\sqrt{{r}_{0}^{2}+4bx-4hy}$, and ${r}_{4}=\sqrt{{r}_{0}^{2}-4hy}$. Here, x and y represent the abscissa and ordinate values at the perimeter of tunnel 2. Thus, x and y follow the relationships $b-{r}_{0}\le x\le b+{r}_{0}$ and $-h-{r}_{0}\le y\le -h+{r}_{0}$. If r

_{0}is small enough relative to b and r

_{0}is small enough relative to h, x and y can be approximately denoted as x = b and y = −h. r

_{1}, r

_{3}, and r

_{4}can be approximately rewritten as ${r}_{1}=\sqrt{{r}_{0}^{2}+4{b}^{2}}$, ${r}_{3}=\sqrt{{r}_{0}^{2}+4{b}^{2}+4{h}^{2}}$, and ${r}_{4}=\sqrt{{r}_{0}^{2}+4{h}^{2}}$.

_{0}at the perimeter of the tunnel and (2) constant water pressure p

_{0}at the perimeter of the tunnel.

#### 3.1. Considering Constant Total Hydraulic Head H_{0} at the Perimeter of a Symmetrical Tunnel

_{0}can be written as

_{M}of point M(x,y) in the semi-infinite aquifer is denoted as

_{M}in the semi-infinite aquifer can be written as

#### 3.2. Considering Constant Water Pressure p_{0} at the Perimeter of a Symmetrical Tunnel

_{0}at the perimeter of a symmetrical tunnel is described as p

_{0}= γ

_{w}H

_{0}, and the total hydraulic head at the perimeter of the tunnel is constant. Therefore, according to the derived equation in Section 3.1, the pore pressure p

_{M}of point M(x,y) in the semi-infinite aquifer can be written as

_{M}should be assumed as

_{M}= p

_{0}; when the point M(x,y) is on the ground surface, ${p}_{M}={\gamma}_{w}{h}_{w}-{\gamma}_{w}y$. Substituting the boundary conditions above into Equation (16), the solutions to X(x,y) and Y(x,y) can be found with

_{M}of point M(x,y) in the semi-infinite aquifer can be written as

_{M}of point M(x,y) in the semi-infinite aquifer is denoted as

## 4. Comparison of Analytical Solutions and Numerical Solutions

#### 4.1. Numerical Model

^{3D}(3.00-261, Itasca, Minneapolis, MN, USA). Taking the F4 weathered slot of the Xiamen Xiang’an subsea tunnel in China as an example, the calculating parameters are shown in Table 1. According to the research results by Li et al. [14], the lateral boundary should be at least nine times the tunnel diameter when analyzing seepage field by numerical simulation. Figure 3 shows a schematic diagram of the numerical model. The numerical model measures 377.6 m in width, 1 m in length, and 207.8 m in height. The lateral displacement boundaries are fixed in the normal direction and the displacement boundaries at the bFottom are fixed in both the horizontal and vertical directions. The ground surface boundaries are free and permeable.

#### 4.2. Pore Pressure Distribution

^{3D}(3.00-261, Itasca, Minneapolis, MN, USA), which indicates that the analytical solutions derived in this paper can accurately predict the pore pressure distribution of symmetrical underwater tunnels in semi-infinite aquifers. Note that in Figure 4b,d, the most significant deviations between analytical solutions and numerical solutions occur at the perimeter of symmetrical tunnels. In addition, the analytical solutions of pore pressure at the perimeter of symmetrical tunnels are not equal to the assumed constant water pressure p

_{0}. This is the result that approximate solutions adopt when substituting the boundary conditions at the perimeter of symmetrical tunnels into Equation (10). However, the deviations are small and in an acceptable range, especially when the tunnel radius r

_{0}is small enough relative to the distance b and h. Figure 4f describes a comparison of pore pressure contours between analytical solutions and numerical solutions. It is clear that the analytical solutions match well with the numerical solutions for points near the tunnels. The most significant deviation occurs at the value of 0.16 MPa, but in an acceptable range.

#### 4.3. Water Inflow

^{3D}(3.00-261, Itasca, Minneapolis, MN, USA). It provides a safe prediction for an engineering application. Therefore, the analytical solution derived in this paper can accurately predict the water inflow of symmetrical underwater tunnels.

## 5. Conclusions

^{3D}(3.00-261, Itasca, Minneapolis, MN, USA). The results show that the analytical solutions derived in this paper can accurately predict pore pressure distribution and water inflow for symmetrical underwater tunnels.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Comparisons of pore pressure distribution between analytical solutions and numerical solutions: (

**a**) 1–2; (

**b**) 3–4; (

**c**) 5–6; (

**d**) 7–8; (

**e**) 9–10; (

**f**) pore pressure (MPa) contours near the tunnel by this study (left-hand side) and FLAC

^{3D}(right-hand side).

r_{0} (m) | b (m) | h (m) | h_{w} (m) | k_{r} (m/s) | H_{0} (m) | p_{0} (Pa) |
---|---|---|---|---|---|---|

7.4 | 33.4 | 52.4 | 20 | 5.0 × 10^{−6} | 5 | 0 |

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**MDPI and ACS Style**

Wang, F.; Li, P.
An Analytical Model of Seepage Field for Symmetrical Underwater Tunnels. *Symmetry* **2018**, *10*, 273.
https://doi.org/10.3390/sym10070273

**AMA Style**

Wang F, Li P.
An Analytical Model of Seepage Field for Symmetrical Underwater Tunnels. *Symmetry*. 2018; 10(7):273.
https://doi.org/10.3390/sym10070273

**Chicago/Turabian Style**

Wang, Fan, and Pengfei Li.
2018. "An Analytical Model of Seepage Field for Symmetrical Underwater Tunnels" *Symmetry* 10, no. 7: 273.
https://doi.org/10.3390/sym10070273