The Drainage Consolidation Modeling of Sand Drain in Red Mud Tailing and Analysis on the Change Law of the Pore Water Pressure

Abstract

In order to prevent the occurring of dam failure and leakage, sand-well drainages systems were designed and constructed in red mud tailing. It is critical to focus on the change law of the pore water pressure. The calculation model of single well drainage pore water pressure was established. The pore water pressure differential equation was deduced and the analytical solution of differential equation using Bessel function and Laplace transform was given out. The impact of parameters such as diameter 𝑑, separation distance 𝑙, loading rate 𝑞, and coefficient of consolidation 𝐶_𝑣 in the function on the pore water pressure is analyzed by control variable method. This research is significant and has great reference for preventing red mud tailings leakage and the follow-up studies on the tailings stability.

1. Introduction

Red mud, a by-product of alumina refining, is produced in increasing quantities globally [1, 2]. Much of this red mud has traditionally been produced by methods which create a liquor of high moisture content [3, 4], which are deposited in impoundments that can reach depths of 20 m [5–7]. Red mud in this tailing is infiltrated by water and in a status of saturation, resulting in a low strength and high moisture content. As a consequence, the continuous stacking of Sintering red mud into this tailing might cause the dam failure and the leak-prone of the tailing when the original designed seepage prevention rank is low, especially for the tailing built in Karst area [8, 9]. As the mineral and chemical composition of red mud is very complex, red mud and its attached fluid have strong alkalinity and corrosivity [10, 11]. It is dangerous that the failure and leakage of this tailings dam occurs and further cause serious consequences that widespread environmental pollution [12, 13].

To prevent the occurring of dam failure and leakage, sand well drainage systems were designed and constructed in this red mud tailing, the lower part of which is mainly soft Bayer red mud [14]. With the assistance of sand well drainage systems, stacking of sintered dry red mud onto the tailings will promote the drainage of pore water and reinforcement of the basement of the tailings. Simultaneously, the pressure of pore water will change continuously as well [15]. Consequently, it is vital to focus on the law of pore water pressure distribution and change in the lower soft red mud during the stacking of Sintering red mud. There was a great deal of reports on the application of the sand well drainage method and one-dimensional drainage-consolidation calculation for saturated soils since 1930s [16–19]. However, as for the calculation on the drainage-consolidation by sand wells, the most well-known model is undoubtedly the axisymmetric radial drainage-consolidation equations of a single well, which was deduced and proposed by Barron in 1948 [20]. This equation not only gives us the analytical solution under both the conditions of isostrain and free strain, but also has considered the fluid seepage flowing-strain interaction.

In this theoretical research field of the in soft soil foundation, although a solution to this compound problem has been given with the help of the three-dimensional parabolic equation under the 2nd type boundary condition, yet, regretfully, it has no sufficient accuracy [21–23]. Therefore, it is important to give an accurate solution to this equation. This research aims at giving the analytical solution of hydrostatic pressure of the three-dimensional consolation utilizing the mathematical calculation methods such as Laplace transform. Crucially, this solution has a high applicability. This research is significantly important to the calculation of pore water pressure and the further control of leakage. In addition, it can theoretically support the follow-up studies on the tailings stability.

2. Physical Model of the Drainage Consolidation System

Because of the large area and complex shape of the red mud tailings researched in this project, the theoretical calculation is difficult to be carried out without reasonable simplification and the further construction of calculation model. The sand drains are designed and sited uniformly and equally, which means that these sand drains are alike in function and effect. Then, the drainage consolidation of the whole tailing red mud can be obtained through research on each sand drain and their superposition. The schematic diagram of two well-accepted distribution modes of sand drains are shown as in Figure 1. In addition, the smear zone formed as a result of construction disturbance when the sand drain separation distance is lower than 1.0 m [24]. It can be overlooked when the sand drain separation distance is larger than that of building and road foundation engineering [25]. In addition, seepage resistance caused by the material type of the sand drains, for the piling rate is low and the drainage time is long. As for the piling of the red mud, it can be assumed to be continuous and the production of aluminum as well.

(a)

(b)

Figure 1 

Two distribution modes of sand drains: (a) square; (b) triangle.

As referred before, the affection region of each sand drain could be equivalent to a cylinder (Figure 2). The equivalent diameter can be deduced from the equation of 𝑑_𝑒=𝛼₁𝑙, where 𝑙 refers to the sand drain separation distance [26]. The value of 𝛼₁ would be 1.13 when it comes to square-like sand drain distribution and 1.05 for triangle-like one. We will also give the assumption that the seepage of pore water contained in the affection cylinder region would be along the radial direction towards the sand drain and then along the vertical direction towards the underground.

(a)

(b)

Figure 2 

The computational model of one sand drain: (a) the floor plan; (b) A-A section plan.

3. Derivation of Pore Pressure Function of the Red Mud Foundation with Sand Drain

In Barron’s theoretical research, although a solution to this compound problem has been given with the help of the three-dimensional parabolic equation under the 2nd type boundary condition, this solution is not accurate. We have tried to calculate the analytical solution of hydrostatic pressure of the three-dimensional consolation through the mathematical calculation methods such as Laplace transform.

Following Barron theory, we do the following basic assumptions for the consideration of pore water pressure under variable load change:(1)the soil skeleton is isotropic material, regardless of the efficiency of the soil mass deformation and creep;(2)soil particle and pore water can not be compressed;(3)seepage flow both in vertical and horizontal radial directions obeys Darcy’s law and have the same permeability coefficient;(4)soil particles move in the vertical direction; the applied load is a first-order function of time.

Take a cell cube from the computational domain as shown in Figure 3 and set that a unit of the 𝑥 direction is 𝑑𝑥 and a unit of the 𝑧 direction is 𝑑𝑧. So, the amount of soil volume change during the process of drainage consolidation is equal to the amount of pore water. Consider Δ𝑉=Δ𝑄. (1)

Figure 3 

The diagram of an element in the seepage calculation.

Consider the seepage flow balance both in horizontal and vertical directions as follows: 𝜕𝑉𝜕𝑡𝑑𝑡=𝜕𝑞_𝑧𝜕𝑧𝑑𝑥𝑑𝑧⋅𝑑𝑡+𝜕𝑞_𝑥𝜕𝑥𝑑𝑧𝑑𝑥⋅𝑑𝑡,1Δ𝑉=𝑒1+𝑒₀𝑑𝑥𝑑𝑧, (2)(3) where 𝑒is the void ratio of time 𝑡, and void ratio of time 𝑡=0 is 𝑒₀. Take this into (2) and then get 𝑒11+𝑒₀𝑑𝑥𝑑𝑧⋅𝑑𝑡=𝜕𝑞_𝑧+𝜕𝑧𝑑𝑥𝑑𝑧⋅𝑑𝑡𝜕𝑞_𝑥𝜕𝑥𝑑𝑧𝑑𝑥⋅𝑑𝑡. (4)

And by the compression test curve, we have 𝜕𝑒𝜕𝑡=𝑎𝜕𝑢,𝜕𝑡 (5) where 𝑎 is a compression coefficient. Putting (5) into (4) can deduce 𝛼𝑘1+𝑒₀⋅𝜕𝑢𝜕𝑡𝑑𝑥𝑑𝑧⋅𝑑𝑡=𝜕𝑞_𝑧+𝜕𝑧𝑑𝑥𝑑𝑧⋅𝑑𝑡𝜕𝑞_𝑥𝜕𝑥𝑑𝑧𝑑𝑥⋅𝑑𝑡, (6) where 𝑘 is osmotic coefficient, and 𝜕²𝑢𝜕𝑡²=𝜕𝑞_𝑥+𝜕𝑥𝜕𝑞_𝑦+𝜕𝑦𝜕𝑞_𝑧𝜕𝑧 (7) which can be put into (6) to get 𝐶_𝑣⋅𝜕𝑢=𝜕𝜕𝑡²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²+𝜕²𝑢𝜕𝑧², (8) where 𝐶_𝑣 represents the vertical coefficient of consolidation. It can be calculated from 𝛼 by the definition formula 𝐶_𝑣=((1+𝑒)/(𝜌_𝑤⋅𝛼))𝑘, where 𝜌_𝑤 is the density of water. That means 𝐶_𝑣 can be gotten from the data of both consolidation test and penetration test.

Change the former equation to accord with the specific boundary of sand drain: 𝐶_𝑣⋅𝜕𝑢=𝜕𝜕𝑡²𝑢𝜕𝑧²+1𝑟𝜕𝑢+𝜕𝜕𝑟²𝑢𝜕𝑟². (9)

According to the single well drainage consolidation model shown in Figure 2, the upper boundary has curtained pore pressure, the left one is water-proof, the right is sand drain, and the physical radius and action radius of which are separately 𝑟_𝑤 and 𝑟_𝑒. Then, the drainage consolidation with the addiction of boundary conditions can be modeled as the following mathematical physical equation: 𝜕𝑢𝜕𝑡=𝐶_𝑣𝜕²𝑢𝜕𝑟²+1𝑟𝜕𝑢+𝜕𝜕𝑟²𝑢𝜕𝑧²,𝑢(𝑧,𝑟,𝑡)|_𝑡=0=𝑝₀,𝑟_𝑤≤𝑟≤𝑟_𝑒𝑢(,0<𝑧<𝐻,𝑧,𝑟,𝑡)|_𝑧=0𝑢(=0,𝑧,𝑟,𝑡)|_{𝑟=𝑟𝑤}=0,𝑢_𝑟(𝑧,𝑟,𝑡)_{𝑟=𝑟𝑒}=0,lim_𝑡→∞𝑢(𝑧,𝑟,𝑡)=0. (10)

In this equation, 𝑢(𝑧,𝑟,𝑡) is excess hydrostatic pressure; 𝑟 is pole diameter of the cylindrical polar coordinates, flow coordinates; 𝑡 is consolidation time, flowing coordinates; 𝑧 is coordinates for vertical flow, calculation depth; 𝑝 is additional load, time function of 𝑡; 𝐶_𝑣 is the consolidation coefficient, determined by experiment; 𝐻 is soil drainage distance, soil thickness, which is constant.

In addition, 𝑝(𝑡)=𝑝₀+𝑞⋅𝑡, which refers to the additional load from the factory produced upper part of Bayer dry red mud, is linear. The 𝑞 is the loading velocity.

According to the theory of soil mechanics, the three-direction water flow in the soil is believed to be turbulent flow problem and can be converted into two parts, planar radial flow (or radiation) and vertical (or linear) laminar flow problem. Based on mathematical method, the total three direction water flow problem also can be transformed to subproblem I and subproblem II. Subproblem I is 𝜕𝑢₁−𝜕𝑡𝜕Θ₁𝜕𝑡=𝐶_𝑣𝑧𝜕²𝑢₁𝜕𝑧²;𝑢₁(𝑧,𝑡)_𝑡=0=𝑝₀;𝑢₁(𝑧,𝑡)_𝑧=0=0;lim_𝑡→∞𝑢₁(𝑧,𝑡)=0. (11) Subproblem II is 𝜕𝑢₂−1𝜕𝑡2𝜕Θ₂𝜕𝑡=𝐶_𝑣𝑟1𝑟𝜕𝑢₂+𝜕𝜕𝑟²𝑢₂𝜕𝑟²;𝑢₂(𝑟,𝑡)_𝑡=0=𝑝₀;𝑢₂(𝑟,𝑡)_{𝑟=𝑟𝑤}=0;𝜕𝑢₂𝜕𝑟_{𝑟=𝑟𝑒}=0;lim_𝑡→∞𝑢₂(𝑟,𝑡)=0. (12)

Assuming the solution of subproblem I is 𝑢₁(𝑧,𝑡) and solution of subproblem II is 𝑢₂(𝑟,𝑡), then the solution to the total question 𝑢(𝑧,𝑟,𝑡) must be confirmed to the form of 𝑢(𝑧,𝑟,𝑡)𝑢₀=𝑢₁(𝑧,𝑡)𝑢₀⋅𝑢₂(𝑟,𝑡)𝑢₀ (13) or 𝑢()=𝑢𝑧,𝑟,𝑡¹𝑢₂𝑢₀=𝑢₁𝑢₂𝑝₀. (14)

3.1. The Theoretical Solution of Subproblem II

In (12), there is Θ₂=𝜎_𝑥+𝜎_𝑦=2𝜐𝑝(𝑡),1−𝜐 (15) where 𝜐 is Poisson ratio of soil; 𝑟_𝑒 is the action radius of sand drain, half of the sand drain separation distance; 𝑟_𝑤 is the physical radius of sand drain; 𝐶_𝑣𝑟 is coefficient of consolidation in horizontal direction.

Equation (15) is the resolution of stress with the assumption that the soil is linear elasticity. Put (15) into (12) and then it can be deduced that 𝜕𝑢₂−𝜕𝑡𝑞𝜐1−𝜐=𝐶_𝑣𝑟1𝑟𝜕𝑢₂+𝜕𝜕𝑟²𝑢₂𝜕𝑟². (16) Equation (16) is the equilibrium equation in horizontal direction which can be rewritten as 𝜕𝑢₂𝜕𝑡−𝑄=𝐶_𝑣𝑟1𝑟𝜕𝑢₂+𝜕𝜕𝑟²𝑢₂𝜕𝑟², (17) where 𝑄=𝑞𝜐/(1−𝜐), 𝑞 is engineered loading velocity. Both 𝑄 and 𝑞 are constant. Equation (17) can be further deduced as follows by Laplace transformation: 𝑠𝑈₂−𝑄𝑠=𝐶_𝑣𝑟1𝑟𝜕𝑈₂+𝜕𝜕𝑟²𝑈₂𝜕𝑟². (18) Equation (18) can be solved as 1𝑟𝜕𝑈₂+𝜕𝜕𝑟²𝑈₂𝜕𝑟²−𝑠𝑈₂𝐶_𝑣𝑟+𝑄𝑠𝐶_𝑣𝑟=0. (19)

Bessel solution of (19) can be calculated as 𝑈₂()=𝑄𝑠,𝑟𝑠²+𝐶₁𝐽₀𝑖𝑟√𝑠𝐶_𝑣𝑟+𝐶₂𝑌₀𝑖𝑟√𝑠𝐶_𝑣𝑟. (20) Equation (20) can be transformed into 𝑈₂()=𝑄𝑠,𝑟𝑠²+𝐶₁𝐽₀𝑟√−𝑠𝐶_𝑣𝑟+𝐶₂𝑌₀𝑟√−𝑠𝐶_𝑣𝑟. (21) Assume 𝜏=−𝑠, and then it can be deduced to 𝑈₂()=𝑄𝜏,𝑟𝑠²+𝐶₁𝐽₀𝑟√𝜏𝐶_𝑣𝑟+𝐶₂𝑌₀𝑟√𝜏𝐶_𝑣𝑟. (22) The 𝐶₁ and 𝐶₂ can be calculated by the boundary conditions in (12), 𝐶₁=𝑄𝑌₁𝐴𝑟_𝑒𝑠²𝐽₁𝐴𝑟_𝑒𝑌₀𝐴𝑟_𝑤−𝐽₀𝐴𝑟_𝑤𝑌₁𝐴𝑟_𝑒,𝐶₂=𝑄𝐽₁𝐴𝑟_𝑒−𝑠²𝐽₁𝐴𝑟_𝑒𝑌₀𝐴𝑟_𝑤+𝐽₀𝐴𝑟_𝑤𝑌₁𝐴𝑟_𝑒, (23)(24) where 𝐴=𝜏/𝐶_𝑣𝑟 and (24) can be deduced as 𝑈₂()=𝑄𝜏,𝑟𝜏²+𝑄𝑌₁𝐴𝑟_𝑒𝐽₀(𝐴𝑟)𝜏²𝐽₁𝐴𝑟_𝑒𝑌₀𝐴𝑟_𝑤−𝐽₀𝐴𝑟_𝑤𝑌₁𝐴𝑟_𝑒+𝑄𝐽₁𝐴𝑟_𝑒𝑌₀(𝐴𝑟)−𝜏²𝐽₁𝐴𝑟_𝑒𝑌₀𝐴𝑟_𝑤+𝐽₀𝐴𝑟_𝑤𝑌₁𝐴𝑟_𝑒. (25) Equation (25) can be simplified as 𝑈₂()=(𝜏)𝜏,𝑟𝑄𝐺+𝑄𝑌₁𝐴𝑟_𝑒𝐽₀(𝐴𝑟)−𝑄𝐽₁𝐴𝑟_𝑒𝑌₀(𝐴𝑟)𝜏²𝐺(𝜏). (26) In this equation, 𝐺(𝜏)=(𝐽₁(𝐴𝑟_𝑒)𝑌₀(𝐴𝑟_𝑤)−𝐽₀(𝐴𝑟_𝑤)𝑌₁(𝐴𝑟_𝑒)), (26) can be further deduced as following by anti-Laplace transformation: 𝑢₂()=1𝑡,𝑟2𝜋𝑖^{−𝜎−𝑖∞}_{−𝜎+𝑖∞}𝑈(𝜏,𝑟)𝑒^−𝜏𝑡𝑑𝜏(𝑡≥0). (27) Equation (27) can be written as 𝑢₂(𝑡,𝑟)=Residues. (28) The singularity of (28) is 𝜏=0, and all the null points of 𝐺(𝜏)=0, and the residue of singularity of 𝜏=0 is (𝜏)Re𝑠𝑄𝐺+𝑄𝑌₁𝐴𝑟_𝑒𝐽₀(𝐴𝑟)−𝑄𝐽₁𝐴𝑟_𝑒𝑌₀(𝐴𝑟)𝑒^−𝜏𝑡𝜏²𝐺(𝜏),0=𝑄𝐺(0)+𝑄𝑌₁(0)𝐽₀(0)−𝑄𝐽₁(0)𝑌₀(0)(0)=12𝐺2. (29) As for the root of 𝐺(𝜏)=0, 𝜏 can be replaced by 𝛽 and can certified by 𝐺(𝛽)=0. Then, we have 𝐺𝛽=𝐽₁𝐵𝑟_𝑒𝑌₀𝐵𝑟_𝑤−𝐽₀𝐵𝑟_𝑤𝑌₁𝐵𝑟_𝑒=0. (30) In the equation, we have 𝐵=𝛽/𝐶_𝑣𝑟. The root of formulas 𝐺(𝛽)=0, 𝛽_𝑖(𝑖=1,2,3,…) can be natural frequency and all are of arithmetic number. Residue, when 𝑠=𝛽_𝑖, is (𝜏)Re𝑠𝑄𝐺+𝑄𝑌₁𝐴𝑟_𝑒𝐽₀(𝐴𝑟)−𝑄𝐽₁𝐴𝑟_𝑒𝑌₀(𝐴𝑟)𝑒^−𝜏𝑡𝜏²𝐺(𝜏),𝛽_𝑖𝑌=𝑄¹𝐵_𝑖𝑟_𝑒𝐽₀𝐵_𝑖𝑟−𝐽₁𝐵_𝑖𝑟_𝑒𝑌₀𝐵_𝑖𝑟𝑒^{−𝛽𝑖𝑡}𝛽²_𝑖(𝜕𝐺(𝜏)/𝜕𝜏)_{𝜏=𝛽𝑖}, (31) where 𝐵_𝑖 is 𝐵_𝑖=𝛽_𝑖/𝐶_𝑣𝑟. According to the character of Bessel function, it can be deduced that (𝜏)Re𝑠𝑄𝐺+𝑄𝑌₁𝐴𝑟_𝑒𝐽₀(𝐴𝑟)−𝑄𝐽₁𝐴𝑟_𝑒𝑌₀(𝐴𝑟)𝑒^−𝜏𝑡𝜏²𝐺(𝜏),𝛽_𝑖=𝑄𝑌₁𝐵_𝑖𝑟_𝑒𝐽₀𝐵_𝑖𝑟−𝐽₁𝐵_𝑖𝑟_𝑒𝑌₀𝐵_𝑖𝑟𝑒^{−𝛽𝑖𝑡}×𝛽²_𝑖𝑟_𝑒𝐽₀𝑟_𝑒𝐵_𝑖−𝐽₂𝑟_𝑒𝐵_𝑖𝑌₀𝑟_𝑤𝐵_𝑖4𝐶_𝑣𝑟𝐵_𝑖−𝛽²_𝑖𝐽₀𝑟_𝑤𝐵_𝑖𝑌₁𝑟_𝑤𝐵_𝑖⁻¹. (32) Then, the excess hydrostatic pressure can be 𝑢₂()=1𝑡,𝑟2+^∞_𝑖=1𝑄𝑌₁𝐵_𝑖𝑟_𝑒𝐽₀𝐵_𝑖𝑟−𝐽₁𝐵_𝑖𝑟_𝑒𝑌₀𝐵_𝑖𝑟𝑒^{−𝛽𝑖𝑡}×𝛽²_𝑖𝑟_𝑒𝐽₀𝑟_𝑒𝐵_𝑖−𝐽₂𝑟_𝑒𝐵_𝑖𝑌₀𝑟_𝑤𝐵_𝑖4𝐶_𝑣𝑟𝐵_𝑖−𝛽²_𝑖𝐽₀𝑟_𝑤𝐵_𝑖𝑌₁𝑟_𝑤𝐵_𝑖⁻¹. (33)

3.2. The Theoretical Solution of Subproblem I

The kinematic equation of subproblem I is 𝐶_𝑣𝑧𝜕²𝑢₁𝜕𝑧²=𝜕𝑢₁−𝜕𝑡𝜕𝜎_𝑧=𝜕𝑡𝜕𝑢₁𝜕𝑡−𝑞. (34) The boundary conditions are 𝑢₁(𝑧,𝑡)_𝑡=0=𝑝₀,𝑢₁(𝑧,𝑡)_𝑧=0=0,𝜕𝑢₁(𝑧,𝑡)𝜕𝑧_𝑧=𝐻0=0. (35) Equation (34) can be further deduced as follows by Laplace transformation: 𝑈₁(𝑧,𝑠)_𝑧=0=0,𝜕𝑈₁(𝑧,𝑠)𝜕𝑧_𝑧=𝐻0=0. (36) Solving the upper equations, 𝑈₁(𝑧,𝑠)=𝐶₃sinh𝑠𝐶_𝑣𝑧𝑧+𝐶₄cosh𝑠𝐶_𝑣𝑧𝑧+𝑞+𝑝₀𝑠𝑠². (37) Putting the boundary conditions can give the value that 𝐶₃=𝑞+𝑝₀𝑠tanh𝑠/𝑎𝐻₀𝑠²,𝐶₄=−𝑞+𝑝₀𝑠𝑠². (38) Equation (37) can be deduced as 𝑈₁(𝑧,𝑠)=−𝑞+𝑝₀𝑠𝑠²cosh𝑠/𝐶_𝑣𝑧𝐻₀×cosh𝑠𝐶_𝑣𝑧𝐻₀−cosh𝑠𝐶_𝑣𝑧𝐻₀×cosh𝑠𝐶_𝑣𝑧𝑧+sinh𝑠𝐶_𝑣𝑧𝐻₀×sinh𝑠𝐶_𝑣𝑧𝑧. (39) As for the 𝑠²cosh(𝑠/𝐶_𝑣𝑧𝐻₀), null point of 𝑠 is ()𝑠=0,𝑠=2𝑘−1²𝜋²𝐶_𝑣𝑧4𝐻²₀(𝑘=1,2,3,…). (40) Besides, 𝑠=0 is the moving singularity of 𝑈₁(𝑧,𝑠); other points are all first grade singularity of 𝑈₁(𝑧,𝑠). The it can be deduced by the expansion of residue theorem:𝑢₁(𝑧,𝑡)=𝐿⁻¹𝑈₁(=𝑧,𝑠)Re𝑠𝑞+𝑝₀𝑠𝑒^𝑠𝑡𝑠²cosh𝑠/𝐶_𝑣𝑧𝐻₀×cosh𝑠𝐶_𝑣𝑧𝐻₀−cosh𝑠𝐶_𝑣𝑧𝐻₀×cosh𝑠𝐶_𝑣𝑧𝑧+sinh𝑠𝐶_𝑣𝑧𝐻₀×sinh𝑠𝐶_𝑣𝑧𝑧=^∞_𝑘=1−𝑞+𝑝₀𝑠cosh𝑠𝐶_𝑣𝑧𝐻₀−cosh𝑠𝐶_𝑣𝑧𝐻₀×cosh𝑠𝐶_𝑣𝑧𝑧+sinh𝑠𝐶_𝑣𝑧𝐻₀×sinh𝑠𝐶_𝑣𝑧𝑧×𝑠²cosh^′𝑠𝐶_𝑣𝑧𝐻₀⁻¹𝑒^𝑠𝑡_{𝑠=−((2𝑘−1)2𝜋2𝐶𝑣/4𝐻20)}=^∞_𝑘=12𝑠𝐶_𝑣𝑧𝑞+𝑝₀𝑠cosh𝑠𝐶_𝑣𝑧𝐻₀−cosh𝑠𝐶_𝑣𝑧𝐻₀×cosh𝑠𝐶_𝑣𝑧𝑧+sinh𝑠𝐶_𝑣𝑧𝐻₀×sinh𝑠𝐶_𝑣𝑧𝑧×𝑠²𝐻₀sinh𝑠𝐶_𝑣𝑧𝐻₀⁻¹𝑒^𝑠𝑡_{𝑠=−((2𝑘−1)2𝜋2𝐶𝑣/4𝐻20)}=^∞_𝑘=14−4𝐻²₀𝑞+(2𝑘−1)²𝜋²𝐶_𝑣𝑧𝑝₀sinh(2𝑘−1)/2𝐻₀𝜋𝑧(2𝑘−1)³𝜋³𝐶_𝑣𝑧×𝑒^{−((2𝑘−1)2𝜋2𝐶𝑣𝑧/4𝐻20)𝑡}. (41)Then,𝑢₁()=𝑧,𝑡_∞𝑘=14−4𝐻²₀𝑞+(2𝑘−1)²𝜋²𝐶_𝑣𝑧𝑝₀sinh(2𝑘−1)/2𝐻₀𝜋𝑧(2𝑘−1)³𝜋³𝐶_𝑣𝑧×𝑒^{−((2𝑘−1)2𝜋2𝐶𝑣𝑧/4𝐻20)𝑡}. (42)

Then, we can put (33) and (42) into (14) of total question to get the final solution. Consider𝑢()=𝑢𝑧,𝑟,𝑡₁(𝑧,𝑡)𝑢₂(𝑟,𝑡)𝑝₀=^∞_𝑘=12−4𝐻²₀𝑞+(2𝑘−1)²𝜋²𝐶_𝑣𝑧𝑝₀sinh(2𝑘−1)/2𝐻₀𝜋𝑧𝑝₀(2𝑘−1)³𝜋³𝐶_𝑣𝑧×𝑒^{−((2𝑘−1)2𝜋2𝐶𝑣𝑧/4𝐻20)𝑡}+^∞∞_𝑘=1𝑖=14𝑄𝑌₁𝐵_𝑖𝑟_𝑒𝐽₀𝐵_𝑖𝑟−𝐽₁𝐵_𝑖𝑟_𝑒𝑌₀𝐵_𝑖𝑟×𝛽²_𝑖𝑟_𝑒𝐽₀𝑟_𝑒𝐵_𝑖−𝐽₂𝑟_𝑒𝐵_𝑖𝑌₀𝑟_𝑤𝐵_𝑖4𝐶_𝑣𝑟𝐵_𝑖−𝛽²_𝑖𝐽₀𝑟_𝑤𝐵_𝑖𝑌₁𝑟_𝑤𝐵_𝑖⁻¹⋅−4𝐻²₀𝑞+(2𝑘−1)²𝜋²𝐶_𝑣𝑧𝑝₀sinh(2𝑘−1)/2𝐻₀𝜋𝑧𝑝₀(2𝑘−1)³𝜋³𝐶_𝑣𝑧×𝑒^{−((2𝑘−1)2𝜋2𝐶𝑣𝑧/4𝐻20)𝑡−𝛽𝑖𝑡}. (43)

This is the final expression of the excess hydrostatic pressure changing along the value of pole diameter of the cylindrical polar coordinates (𝑟), consolidation time (𝑡), and calculation depth (𝑧).

4. Analysis on the Change Law of the Pore Water Pressure

From the derivation of former part of the research, the function of the excess hydrostatic pressure is influenced by multiple variables, whose variation characteristic cannot be vividly expressed in a three-dimensional space coordinate system. For the purpose of understanding the variation rule of 𝑢 with the time and space, we use MATLAB as a tool, by examining a parameter and fixing other parameters, to get the affects way of each variable effects on the value of 𝑢. Based on the previous engineering experience and based on the survey and design of tailings and actual production, scope of calculation parameter is as shown in Table 1.

Table 1 

Scope of each calculation parameter.

Control variate method is employed to conclude the influence of various parameters on the law of drainage. Concretely, what this means is to study the effect of target variable from the four variables, diameter 𝑑, separation distance 𝑙, loading rate 𝑞, and coefficient of consolidation 𝐶_𝑣, and fix the other three ones as initial value.

On the basis of scope of each calculation parameter (Table 1), the initial values are set as Table 2.

Table 2 

The initial value of each calculation parameter.

A serious of relation curves between the excess pore water pressure and time can be obtained by successively changing the above calculation parameters in proportion. Due to the fact that objective continuous loading time is less than 8 years, the maximum value of excess pore water pressure will emerge when the time value ranges in 0~8. The variation diagrams of excess pore water pressure can be calculated and plotted via the MATLAB drawing routine PLOT and calculating routine FindRoots. All the figures are about the calculated point on the action radius and also the border of the single sand drain modeling, except Figure 5. The abscissa axis unit is year. All the curves results and discussions are as following.(1)Under the initial parameters, the excess pore water pressure at different sites (𝑥,𝑧) of the sand drain modeling is compared and shown as in Figures 4 and 5. And the pressure 𝑢’s change curves with depth 𝑧 and position 𝑥 are also given as insets. From Figures 4 and 5, it can be known that the excess pore water pressure in the depth vertical direction is nonlinear, and in the shade is of low change and gradient. While, in the horizontal direction, the static pore water pressure will increase slowly and have a lower final value when the distance between two sand drains is wide. Inversely, it will increase faster and have a larger final value when the distance is narrow. What is more is that the trade is nonlinear.(2)Under the initial parameters, change the value of separation distance of sand drain 𝑑 to see the influence of 𝑑 on static pore water pressure. From Figure 6, it can be indicated that the conditions of all the other parameters are the same and diameter variation of sand drain can make no difference on the reduction of excess pore water pressure, which demonstrate the rationality of former assumption on a certain extent. This has come from the low permeability and loading rate of red mud.(3)Under the initial parameters, change the value of separation distance of sand drain to see the influence of 𝑙 on excess pore water pressure. From Figure 7, it can be obtained that the conditions of all the other parameters are the same and the change of the separation distance can significantly affect the change speed and the final value of excess pore water pressure. The influence of reducing the distance is more obvious than enlarging the diameter on the pore water pressure. The final excess pore water pressure can only be one quarter of the initial value when 𝑙 is just half of the initial one.(4)Under the initial parameters, change the value of loading rate to see the influence of 𝑞 on excess pore water pressure. From Figure 8, it can be deduced that the conditions of all the other parameters are the same, and larger loading rate can result in larger pore water pressure. However, even if the loading rate is identical, different reduction can lead to different value of final value of excess pore water pressure. A modest reduction has a small effect and continuous reduction has an obvious effect.(5)Under the initial parameters, change the value of coefficient of consolidation to see the influence of 𝐶_𝑣 on static pore water pressure. From Figure 9, it can be deduced that the conditions of all the other parameters are the same; the larger the coefficient of consolidation, the smaller the final value of excess pore water pressure, or vice versa. Consequently, applying sand wells drainage consolidation method on different pore medium, only more drainage channel or shorter drainage distance can support the consolidation effect when the consolidation time is the same, while 𝐶_𝑣 is smaller.

Figure 4 

Pore pressure 𝑢’s change curve with depth 𝑧.

Figure 5 

Pore pressure 𝑢’s change curve with position 𝑥.

Figure 6 

Pore pressure 𝑢’s change curve with sand well’s diameter 𝑑.

Figure 7 

Pore pressure 𝑢’s change curve with sand well’s spacing 𝑙.

Figure 8 

Pore pressure 𝑢’s change curve with loading rate 𝑞.

Figure 9 

Pore pressure 𝑢’s change curve with coefficient of consolidation 𝐶𝑣.

5. Conclusion

This research has established the calculation model of single well drainage pore water pressure, deduced pore water pressure differential equation based on the three-dimensional consolidation, and get the analytical solution of differential equation using Bessel function and Laplace transform. The impact of parameters in the function on the pore water pressure is analyzed by control variable method. Changes of excess pore water pressure on four variables such as diameter 𝑑, separation distance 𝑙, loading rate 𝑞, and coefficient of consolidation 𝐶_𝑣 are analyzed and proved to be nonlinear. The calculation results help to further set up a depository stability evaluation model to prevent loading exceeding design capacity. And together with inhomogeneous deformation data of the field, it is promising to create warning systems for site managers based merely on the field logging data of pore pressure data. Therefore, this research has a guiding significance on preventing red mud tailings leakage of continuous loading pile exceeding design capacity.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.