Prediction of tunnel localized water leakage influences on adjacent lateral pile responses in saturated clay

Abstract

In the realm of constructing urban underground spaces, it is imperative to address the impact of tunnel leakage on the surrounding environment. This paper introduced a theoretical analysis to investigate the localized water leakage's influence on adjacent pile foundations. A pore pressure distribution function, accounting for localized leakage water, was formulated. Integrated with the seepage control equation, this function facilitated the calculation of additional stress imposed on piles due to tunnel localized water leakage. Employing the Pasternak foundation model, an analytical solution was developed to assess the lateral performance of adjacent piles under localized water leakage conditions. This approach was compared with numerical simulations to validate the reliability of soil seepage fields and pile lateral performance resulting from localized tunnel leakage at different positions. Through comprehensive parameter analysis, it was observed that the width of the leakage joint significantly influenced pile lateral responses, manifesting in three distinct stages: linear increase, nonlinear gradual augmentation, and stabilization. Different positions of the lining leakage joint yielded varying effects on adjacent piles' lateral responses, with closer proximity intensifying the impact on the pile. When leakage joints were situated near the pile toe, a pronounced negative bending moment was generated. Furthermore, this study summarized the influence range of tunnel localized leakage adjacent to piles. It established that the maximum pile-tunnel horizontal distance inducing lateral pile responses due to tunnel localized leakage was set at 8 times the pile diameter (8 D_p). Additionally, tunnel leakage influences should be considered when the pile length exceeded 0.6 times the depth of the tunnel axis.

Appendix A

The traditional conformal mapping method of tunnel leakage model is shown in Fig.

Fig. 18

Conformal mapping of tunnel leakage model a Cartesian coordinate system for the tunnel, b Polar coordinate system for the tunnel

18. The outer diameter of the tunnel is r₂ and the inner diameter is r₁. h is the cover depth C plus the outer diameter of the tunnel r₂, and the tunnel is buried in saturated soil. To simplify the solution to the problem, the following assumptions are put forward: (1) the soil mass is homogeneous, continuous and isotropic; (2) Both soil and water are incompressible, and the water level remains constant at the surface; (3) The seepage is irrotational and stable, and the movement law follows Darcy’ s law; (4) The pore water pressure at the inner boundary of tunnel lining is zero. The water head at the outer boundary of the lining is H_g, which is a variable studied in this paper.

According to Darcy’s law and conservation of mass, the two-dimensional fluid around the tunnel can be expressed in polar coordinates of Laplace equation as follows:

$$\frac{{\partial^{2} \varphi }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial \varphi }{{\partial r}} + \frac{1}{{r^{2} }}\frac{{\partial^{2} \varphi }}{\partial \alpha } = 0$$

(A-1)

where φ is the total water head at any point in the seepage field, equal to the pore water head plus the potential head in Eq. (A-2)

$$\varphi =z + \frac{p}{{\gamma_{w} }}$$

(A-2)

where p is pore pressure, γ_w is the weight of water and z is the potential head.

The total head acting on the position with radius r can be obtained by solving the general solution of Eq. (A-1) [31].

$$\varphi = C_{1} + C_{2} \ln \left( r \right) + \sum\limits_{n = 1}^{\infty } {\left( {C_{3} r^{n} + C_{4} r^{ - n} } \right)\cos \left( {n\alpha } \right)}$$

(A-3)

where C₁, C₂, C₃ and C₄ are constants, which are determined by the drainage boundary conditions at the water level and the outer radius of the tunnel.

The boundary conditions of polar coordinate plane water head are as follows

$$\left\{ \begin{aligned}& \varphi \left( {r = 1} \right) = 0 \hfill \\& \varphi \left( {r = \alpha_{s} } \right) = H_{g} \hfill \\ \end{aligned} \right.$$

(A-4)

where

$$\alpha_{s} = \frac{h}{{r_{2} }} + \sqrt {\frac{{h^{2} - r_{2}^{2} }}{{r_{2}^{2} }}}$$

(A-5)

Taking the boundary conditions into the Eq. (A-3), the total water head can be obtained

$$\varphi = \frac{{H_{g} }}{{\ln \left( {\alpha_{s} } \right)}}\ln r$$

(A-6)