Geofluids

Volume 2018 (2018), Article ID 1859285, 14 pages

https://doi.org/10.1155/2018/1859285

## A Simplified Solution for Calculating the Phreatic Line and Slope Stability during a Sudden Drawdown of the Reservoir Water Level

Correspondence should be addressed to Yongtao Yang; moc.621@chhucs

Received 21 June 2017; Revised 6 September 2017; Accepted 8 January 2018; Published 6 February 2018

Academic Editor: Qinghui Jiang

Copyright © 2018 Guanhua Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

On the basis of the Boussinesq unsteady seepage differential equation, a new simplified formula for the phreatic line of slopes under the condition of decreasing reservoir water level is derived by means of the Laplacian matrix and its inverse transform. In this context, the expression of normal stress on the slip surface under seepage forces is deduced, and a procedure for obtaining the safety factors under hydrodynamic forces is proposed. A case study of the Three Gorges Reservoir is used to analyze the influences of the water level, decreasing velocity and the permeability coefficient on slope stability.

#### 1. Introduction

During the normal operation of a dam, a decrease in water level is the main cause of bank slope failure [1]. There are many potential landslide sites that exist in the region of the Three Gorges Reservoir, and these are generally triggered by changes in the water level. Landslides cause huge losses both in terms of property and life. For example, the landslide that occurred on 13 July 2003 at Qiangjiangping on the Three Gorges Reservoir claimed many lives and was responsible for considerable infrastructure losses. The accident serves as a profound warning [2, 3] and strongly motivates further study of this type of natural disaster. Many studies have confirmed that changing water levels are the main cause of landslide events [2–9].

The Chinese government has invested considerable resources to ensure slope stability under normal operation and to prevent natural disasters such as landslides. During the landslide survey and design review of the Three Gorges Reservoir, some problems were identified and require further clarification. First, there is no scientific basis for the determination of phreatic lines. A decrease in water level is the most unfavorable situation in terms of slope stability and usually leads to landslides. Furthermore, a decrease in water level can cause seepage problems related to the rate of water level decrease and the slope’s permeability coefficient. Therefore, the correct approach should focus on the above-mentioned factors to determine the phreatic line and on the seepage pressure to analyze the slope stability. However, most of the determination of phreatic lines is based on the designer’s experience, which may lead to incorrect stability calculations and safety risk assessment. The second problem is related to the calculation of water forces, which is ambiguous. For example, both the seepage force and the surrounding hydrostatic pressure are considered in calculations, which cause repetition of the water forces in the equations and hence introduce more uncertainty. The third problem is related to the velocity of the decreasing water level, which is again not clearly defined in the slope stability calculation. In the case of power generation or flood control, the Three Gorges Reservoir releases water in a very short span of time. Such action leads to a rapid decrease in the water level and in some cases triggers landslides.

The determination of the phreatic line constitutes a free surface (unconfined) seepage problem in rock and soil mechanics. In such a case, the key calculation is to determine the free surface that delimits the flow boundaries. The free surface can be found using nonlinear numerical techniques, such as the finite difference method with adaptive mesh [10] and the finite element method with adaptive mesh [11] or fixed mesh [12]. Among the proposed methods, the extended pressure (EP) method proposed by Brezis and Kinderlehrer [13] and further simplified by Bardet and Tobita [14] using finite differences is considered the simplest and most efficient for free surface calculation. Through an extension of Darcy’s law, the EP method substantially reduces the variational inequalities, which can then be applied for computation of the whole domain. By applying variational inequalities, Zheng et al. [15] and Chen et al. [16] made significant contributions in solving the slope and dam free surface seepage problems. To improve the accuracy and computational efficiency of the EP method, an iterative error analysis is used with the finite difference equations [17]. Considerable progress has been made in calculating the slope phreatic line, especially through numerical analysis using the finite difference and finite element methods. However, such numerical methods are not commonly used in engineering practice and are usually ignored in soil mechanics, as they are obtained through complicated derivations and are challenging to implement. Therefore, there is a need for a simple and efficient method for practical engineering applications and educational training.

To explain the effects of water load on the slope stability to engineers and technicians, the surrounding water pressure is introduced by the Swedish Slice method calculated from the drift properties. Based on these experiments, it is concluded that the seepage force, water weight on the soil slices, and the surrounding hydrostatic pressure act as balancing forces for each other [18].

In slope stability analysis under a changing water level, the conventional slice method is the limit equilibrium method. The uncertainty is usually applied to the interslice forces after eliminating the slice bottom normal compressive force by the two equilibrium equations of a single slice [19]. Afterwards, the statically indeterminate problem of the slope limit safety factor can be solved using methods such as the Morgenstern-Price method [20], Spencer’s method [21], the Swedish method [22], the Bishop simplified method [19], or the Janbu simplified method [23]. Many slice methods have demonstrated that the interslice force is important in calculating the slope stability in statically indeterminate problems [24]. Such traditional slice methods are categorized as local analysis methods.

Alternatively, other limit equilibrium methods can be categorized as global analysis methods, including the graphic method [25], the variational method [26], and Bell’s global analysis method [24]. In contrast to the other limit equilibrium methods, Bell’s method considers the entire system instead of using a single slide bar as a research object. Therefore, the interslice force is not required in this method, which offers a new way to solve the strict slice method. Theoretically, the distribution of the slip surface normal stress in global analysis methods should be easier to understand than the distribution of interslice force in the Morgenstern-Price method. However, this method has not received sufficient attention, even in the summary provided by Duncan [27]. Until 2002, researchers in many studies [4, 28–30] have used similar methods. For example, in Bell’s derivation process, Zhu et al. [29] and Zhu and Lee [28] used quadratic interpolation to solve the slip surface normal stress distribution and derive one variable cubic equation with an unknown safety factor. Zheng and Tham [30] used the Green formula to convert the relevant domain integral calculus to boundary integral calculus.

Firstly, this paper applies the Boussinesq unsteady seepage basic differential equations and boundary conditions for derivation of the expression for the phreatic line during decreasing water level conditions. The polynomial fitting method is used to simplify the formula because of its lower complexity. For solving the effects of water load on the slope stability, the surrounding water pressure is calculated from the drift property. Finally, the global analysis method is proposed to analyze the slope stability under seepage forces. A typical landslide in the Three Georges Reservoir is used as a case study to analyze the influential factors for slope stability during conditions of decreasing water level.

#### 2. Phreatic Line Calculations

##### 2.1. Fundamental Assumptions

(1)The aquifer is homogeneous and isotropic with infinite lateral extension;(2)The phreatic flow parallel to slope surface is caused by water level fluctuation and the rainfall infiltration causes the phreatic flow to be perpendicular to slope surface;(3)The reservoir water level is decreasing at a constant speed of ;(4)The reservoir bank is considered as a vertical slope. The reservoir bank within the declining amplitude is much smaller than the ground, and in order to simplify this, it is considered as vertical reservoir bank [5].

As shown in Figure 1, based on the Boussinesq equation and following the above-mentioned assumptions, the differential equation of diving unsteady motion can be expressed aswhere and are two directions of local coordinates, as shown in Figure 1; is the aquifer thickness; is time; is the permeability coefficient; and is the specific yield, which is defined as the amount of water released as a result of gravity in a unit volume of saturated rock or soil and can be expressed as the ratio between the water volume released by the gravity and the rock or soil volume in saturated conditions.