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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.

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 [

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 [

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.

According to Darcy’s law, the flow drag resistance of the

The negative sign on the right side of (

According to the mechanical equilibrium conditions, the flow drag resistance of the

Equations (

It is assumed that the water in porosity is incompressible under the consolidation process. Thus,

According to fractional derivative rule, (

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, (

Under the condition of 1D consolidation, the axial strain

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, (

Assuming that the soil profile is shown as Figure

Schematic diagram of stress in soil.

According to Figure

Substituting (

Take the derivative of (

According to the compression curve of 1D consolidation test, the volume compressibility factor

Substituting (

Substituting (

Equation (

Substituting (

Equation (

Tailings are mine wastes produced in the mining engineering, which are sent to tailings reservoir by pipe or flume. The tailings dam is an important part in mining engineering, which is consisted by the initial dam and fill dam. The initial dam is made by permeable rockfill generally, and the fill dam is formed by tailings. In general, the construction of a tailing dam takes many decades or even a century. According to construction method, the tailings dam can be divided into upstream tailings dam, downstream tailings dam, midline method tailings dam, and so on. The number of tailings ponds in China has reached more than 12,000 by statistics. Because the downstream of the tailings dam is residents living area or mining production area generally the social and people’s property damage a huge impact if the tailings dam is failed.

Because the upstream tailings dam has advantages of simple operation, low construction costs, the less need for coarse particles, and so on, according to statistics, 95% of the tailings dams are adopting the construction of the upstream tailings dam in China. On the other hand, the construction process of the upstream tailings dam cannot precisely control the shape of the tailings dam. The deposition structure of the tailings dam is very complex. The upstream tailings have shortcoming of long infiltration distant and poor stability. In view of the tailings slurry discharge and tailings particle deposition following the sediment mechanics, the section profile of the tailings dam has obvious regularity. The coarser the particle size is, the shorter the average distance migrates and vice versa. Generally, the deposition order of the tailings dam along the dry beach face is tailings fine sand, tailings silt, tailings sand, and tailings mud.

If particle size distribution of the tailings and the length of the dry beach face are kept constant, the interface of the tailings material should be substantially parallel to the surface of the dam slope, which is shown in Figure

Schematic diagram of the upstream tailings dam.

As can be seen from Figure

In this stage, the thickness of the tailings increases with time. The tailings material produces great compressive deformation under self-weight effect. The pore water pressure produced by the self-weight pressure at the early stage is partially dissipated. Nevertheless, the pore water pressure caused by self-weight pressure is increased with the tailings thickness at the later stage, which leads to the increase of the pore water pressure with time. Therefore, the net pore water pressure is accumulated during this process.

The self-deposition of the tailings mud continues to extend to interior of the tailings reservoir area in this stage. The self-deposition of the tailings layer has ended where the end of the tailings reservoir is near. It overlapped with particles coarse tailings and tailings mud. The tailings mud has to bear a growing load with time. The tailings mud continues to consolidate under the combined action of self-weight and additional load. At this stage, the pore pressure produced by the self-weight pressure dissipates a little part, and the more pore water pressure produced by additional load accumulates. The net pore water pressure tends to accumulate in this process.

For a specific part of the tailings dam, the upper boundary of the tailings dam has reached the designed elevation at a certain moment. Therefore, the additional load will remain unchanged from this moment. During this stage, the tailings mud will continue to consolidate under the combined action of self-weight and constant additional load. Thus, the pore water pressure will dissipate within the tailings dam. Obviously, the different parts of tailings mud have different consolidation stages. There is no additional load at the thickest tailings mud in the tailings reservoir until it is closed. Therefore, the tailings mud is always in the first stage and will never enter the second and third stages. From the above qualitative analysis, the pore water pressure, which is located at the slope dam segment, reaches the maximum value when the additional load just stops growing. The pore water pressure, which is located at dry beach segment and artificial lake segment, achieves the maximum value when the tailings dam reaches the maximum height. The numerical values of the pore water pressure at different locations and moments can be obtained by solving the partial differential equations of consolidation problems.

It is necessary to accurately calculate the accumulation and dissipation of the pore water pressure in tailings dam, of which a solution of 1D consolidation problem can be simplified. During the consolidation process, the soil parameters of tailings material (such as the change of bulk density, permeability coefficient, and consolidation coefficient) are changed with the consolidation process. The change law can be determined through many experiments. Under normal circumstances, it is difficult to obtain the analytical solution when the change law of soil parameters is considered. Therefore, the exact solution of the problem can only be depended on numerical calculation method.

As the horizontal length of the tailings reservoir is usually far greater than the vertical thickness (i.e., the horizontal length is 10 times more than the vertical thickness), the drainage consolidation effect of the horizontal direction can be ignored. Therefore, the vertical direction of drainage is needed to consider. Then, it is only a 1D consolidation problem. Based on the assumption of small-strain and constant of soil parameters, such as

The above three different stages are specifically discussed as follows.

The overburden load of the tailings mud at this stage is zero, that is,

If the bottom of the tailings mud is impermeable, the coordinates origin is taken as the impermeable bottom. The boundary conditions of the problem can be given as follows:

The initial condition of the problem is given by

According to Gibson’s [

It can be verified that (

For any function

Equation (

Equation (

Thence

Combining the solutions of (

It can be verified that (

During this stage, the thickness of tailings mud not only does not increase with time but also gradually decreases with the increase of effective stress. For the sake of simplicity, assuming that the thickness of tailings mud

The initial condition of the problem is given by

The boundary conditions of the problem can be given as

The following three questions are called problem

If the solution of problem

The solution of

For solution to the problem

When the tailings mud is below the groundwater level, the total stress

The ratio of pore water pressure to total stress can be easily determined as follows:

In this stage, the basic differential equation and the definite condition are problem

The following questions can be called problem

If the solution of problem

The definite condition of problem

Suppose that

Then

According to the orthogonal rule of the solution, the following relationship can be obtained:

Equation (

Substituting (

For the third stage, the analytical solution of pore water pressure is obtained.

Similar to (

During this stage,

The simplified calculation section of the representative section of a tailings dam is shown in Figure ^{3}, the saturation density of the tailings fine sand is 19.6 kN/m^{3}, the saturation density of the tailings sand is 19.1 kN/m^{3}, and the saturation density of the tailings mud is 18.8 kN/m^{3}. The permeability coefficient of gravel is 3 × 10^{−3} m/s, the permeability coefficient of the tailings fine sand is 5 × 10^{-6 }m/s, the permeability coefficient of the tailings sand is 2 × 10^{-6 }m/s, and the permeability coefficient of the tailings mud is 1.5 × 10^{-8 }m/s. The consolidation coefficient of gravel is 3.8 × 10^{-1 }cm^{2}/s, the consolidation coefficient of the tailings fine sand is 5.4 × 10^{-3 }cm^{2}/s, the consolidation coefficient of the tailings sand is 2.8 × 10^{-3 }cm^{2}/s, and the consolidation coefficient of the tailings mud is 1.2 × 10^{-5 }cm^{2}/s. Please determine the distribution of pore water pressure when the tailings dam reaches maximum height.

Simplified section for calculation of pore water pressure in a tailings dam.

The above problem can be solved through (

Distribution of pore water pressure at the moment when the dam rises to its maximum height.

Due to the similarity between the tailings dam and the reservoir dam in the geotechnical structure, many scholars directly introduce the calculation methods of the reservoir dam, whose theory is relatively mature, into the tailings dam. Considering the difference on construction cycle, construction materials between the tailings dam and reservoir dam, it makes the calculation results inconsistent with the actual.

Based on the Terzaghi consolidation theory of 1D, the tailings dam is divided into the slope dam segment, the dry beach segment, and the artificial lake segment. The solutions of the pore water pressure are derived, respectively. The analysis shows that the additional load of the slope dam segment is unchanged, which can be calculated using (

The theoretical derivation is based on 1D consolidation theory. It is only considering consolidation in the vertical direction. Since the horizontal scale of most tailings dams is much larger than the vertical direction, it has little influence to ignore the drainage of horizontal. From the point of engineering view, it is conservative to the stability of the tailings dam. Considering drainage of the horizontal, the pore water pressure will be lessened and the safety factor of the tailings dam will be greater.

It is assumed that the deformation of the tailings is small deformation during the consolidation process. If the actual tailings are loose relatively, the deformation of the tailings is large deformation. He et al. [

It is supposed that the mechanical parameters such as permeability coefficient and consolidation coefficient are constant during the consolidation process. Previous studies [

Tailings dam is a very important geotechnical structure of mine engineering. The calculation of pore water pressure has a great impact on the safety factor of tailings dam slope. How to accurately estimate pore water pressure is very difficult. Based on the assumption of 1D consolidation and small strain of tailings material, a general equation of the pore water pressure is proposed. According to dissipation and accumulation characteristics of the pore water pressure in the tailings dam, the tailings dam can be divided into the slope dam segment, the dry beach segment, and the artificial lake segment. The analytic solutions of the corresponding segment are obtained through solving the partial differential equation, which has great significance to the stability of the tailings dam.

Shear strength of the tailings material

Maximum increases in vertical total stress as a function of depth

A parameter introduced to transform an equation

The rate of the additional load on the tailings mud,

Coordinate of

A parameter introduced to transform an equation

Porosity of the tailings material

Counters, 1, 2, 3,…

A function of variable

A function introduced into solving partial differential equation

The rate of the thickness of the tailings mud,

A function introduced to transform an equation

Stress of the tailings material

Coordinate of

The pore water pressure

Coefficient of coefficient

The height of static water level

Thickness of tailings mud

Porosity ratio

Increment of stress

Termination pore water pressure as a function of depth

Initial pore water pressure as a function of depth

Termination time

Initial time

Coefficient of consolidation

Saturation

Coefficients to be determined

Coefficients to be determined

Initial total stress

Effective vertical stress

Strain in the

Bulk density of water

Saturated bulk density of tailings dam

Buoyancy unit weight of tailings material

The actual velocity along the flow direction in the tailings dam

Water pressure

Coefficient of volume compressibility

Hydraulic gradient in the

The total height of tailings mud plus static water level

The drag resistance force on the pore wall of a unit volume in the

The authors declare that they have no conflicts of interest.

This paper is supported by the Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, (Grant no. Z013009), the National Key Research and Development Program of China (Project no. 2017YFC0804601), the National Natural Science Foundation of China (Grant nos. 51764020; 51741410; 51234004), and the Natural Science Foundation of Yunnan Province (Grant no. 2015FB130). The authors would like to thank them for providing the financial support for conducting this research.