Mathematical Problems in Engineering

Volume 2018, Article ID 4573780, 9 pages

https://doi.org/10.1155/2018/4573780

## Analytical Solution and Application for One-Dimensional Consolidation of Tailings Dam

^{1}Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan 650224, China^{2}State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China

Correspondence should be addressed to Hai-ming Liu; moc.361@1780gnimiah

Received 10 November 2017; Revised 16 January 2018; Accepted 12 February 2018; Published 19 March 2018

Academic Editor: Shuo Wang

Copyright © 2018 Hai-ming Liu 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

The pore water pressure of tailings dam has a very great influence on the stability of tailings dam. Based on the assumption of one-dimensional consolidation and small strain, the partial differential equation of pore water pressure is deduced. The obtained differential equation can be simplified based on the parameters which are constants. According to the characteristics of the tailings dam, the pore water pressure of the tailings dam can be divided into the slope dam segment, dry beach segment, and artificial lake segment. The pore water pressure is obtained through solving the partial differential equation by separation variable method. On this basis, the dissipation and accumulation of pore water pressure of the upstream tailings dam are analyzed. The example of typical tailings is introduced to elaborate the applicability of the analytic solution. What is more, the application of pore water pressure in tailings dam is discussed. The research results have important scientific and engineering application value for the stability of tailings dam.

#### 1. Introduction

Based on the assumption that the soil is isotropic and uniform, an external surface load is instantaneously applied and is held constant; a classical one-dimensional (1D) consolidation theory was proposed by Terzaghi [1]. In order to analyze time-dependent loading, an analytical solution to the layered consolidation problem for a general set of boundary conditions and an arbitrary load history was presented by Schiffman and Stein [2]. The 1D consolidation analytical solutions considering ramp loading were presented by Olson [3]. A simple semianalytical method to solve the problem of 1D consolidation by taking into account the varied compressibility of soil under cyclic loadings was brought up by Cai et al. [4]. A rigorous solution of the conventional Terzaghi 1D consolidation under haversine cyclic loading with any rest period was proposed by Müthing et al. [5], which is achieved using Fourier harmonic analysis for the periodic function representing the rate of imposition of excess pore water pressure. A semianalytical solution to 1D consolidation of viscoelastic unsaturated soils with a finite thickness under oedometric conditions and subjected to a sudden loading was put forward by Qin et al. [6]. A semianalytical solution to 1D consolidation equation of fractional derivative Kelvin-Voigt viscoelastic saturated soils subjected to different time-dependent loadings was presented by Wang et al. [7]. Under the condition of the increasing weight of superincumbent material and the length of the drainage path varies, a solution for the 1D consolidation of a clay layer whose thickness increases with time was proposed by Gibson [8]. An exact analytical solution of the nonhomogeneous partial differential equation governing the conventional 1D consolidation under haversine repeated loading was derived and discussed by Razouki et al. [9, 10].

In order to analyze different boundary conditions, single drainage solutions for several specific variations of the permeability and shear modulus were given by Mahmoud and Deresiewicz [11]. Several analytical solutions for the consolidation analysis of a soil layer with fairly general laws of variation of permeability and compressibility for both a single-drained condition and a double-drained condition were proposed by Zhu and Yin [12]. Tang et al. [13] propose a closed-form solution for consolidation of three-layered soil with a vertical drain system and a method to solve a convergence solution by enlarging the precision for the entire thickness of the foundation and adding controlling precision for the overall average degree of consolidation of each soil layer. The solution of Terzaghi 1D consolidation equation with general boundary conditions is proposed by Mei and Chen [14], whose solution is validated by comparing it to the classical solution. Hawlader et al. [15] develop a new constitutive model for the compressibility behavior of soft clay sediments at low effective stress level and the model is used to solve finite strain 1D consolidation with pertinent initial and boundary conditions. A solution to the consolidation equation with boundary conditions that are cyclic with time is given by Rahalt and Vuez [16]. A nonlinear theory of consolidation has been developed for an ideal normally consolidated soil by Davis and Raymond [17]. A simple calculation procedure to analyze the one-dimensional response of saturated soil layers to pore pressure variations at the boundary described by a general time-dependent function is developed by Conte and Troncone [18].

The tailings dam is an important geotechnical structure in mining engineering. For a long time, the theory of reservoir dam is applied to tailings dam without any modification. However, there are many differences between tailings dam and reservoir dam, which lead to inaccurate calculation results of the pore water pressure. According to the classical Terzaghi consolidation theory, the analytical solution of the pore water pressure is discussed in this paper.

#### 2. The General Equation of 1D Consolidation

According to Darcy’s law, the flow drag resistance of the -direction is as follows:

The negative sign on the right side of (1) indicates that the drag resistance is opposite to the direction of flow velocity.

According to the mechanical equilibrium conditions, the flow drag resistance of the -direction can also be expressed as follows:

Equations (1) and (2) can be combined as follows:

It is assumed that the water in porosity is incompressible under the consolidation process. Thus, is a constant. During the process of soil consolidation, because the porosity of soil particles is continuously compressed, the permeability coefficient of soil is decreased continuously and the permeability coefficient of soil is changed with depth, that is, . Taking the derivative of (3), one can have

According to fractional derivative rule, (4) can be expanded as follows:

Assume that the horizontal direction consolidation of the soil can be neglected; that is, the consolidation is under confined compression condition. According to the 1D consolidation condition, (6) can be deduced

Under the condition of 1D consolidation, the axial strain is equal to the volume strain of the soil. According to Pane and Schiffman [19] research, (4) and (5) can be rewritten as follows:

In the consolidation problem, it is of great significance to study the change of pore water pressure with time and position under external load. Therefore, (8) can be expanded according to pore water pressure.

Assuming that the soil profile is shown as Figure 1, the pore water pressure of a point located under load and is . According to the Terzaghi effective stress principle, the effective stress could be given as follows under consolidation: