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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

Red mud, a by-product of alumina refining, is produced in increasing quantities globally [

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 [

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 [

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

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

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

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

the soil skeleton is isotropic material, regardless of the efficiency of the soil mass deformation and creep;

soil particle and pore water can not be compressed;

seepage flow both in vertical and horizontal radial directions obeys Darcy’s law and have the same permeability coefficient;

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

The diagram of an element in the seepage calculation.

Consider the seepage flow balance both in horizontal and vertical directions as follows:

And by the compression test curve, we have

Change the former equation to accord with the specific boundary of sand drain:

According to the single well drainage consolidation model shown in Figure

In this equation,

In addition,

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

Subproblem II is

Assuming the solution of subproblem I is

In (

Equation (

Bessel solution of (

The kinematic equation of subproblem I is

Then, we can put (

This is the final expression of the excess hydrostatic pressure changing along the value of pole diameter of the cylindrical polar coordinates

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

Scope of each calculation parameter.

Parameter and its extremum | Diameter of |
Separation distance |
Loading rate |
---|---|---|---|

Max | 0.35 | 5.0 | 45 |

Min | 0.10 | 1.0 | 25 |

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

On the basis of scope of each calculation parameter (Table

The initial value of each calculation parameter.

Diameter of sand drain |
Separation distance of sand drain |
Calculation depth |
Loading rate |
Coefficient of consolidation |
Action radius |
---|---|---|---|---|---|

0.30 m | 3.0 m | 20.0 m | 39 kPa/y | 0.002 | 1.575 m |

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

Under the initial parameters, the excess pore water pressure at different sites

From Figures

Under the initial parameters, change the value of separation distance of sand drain

From Figure

Under the initial parameters, change the value of separation distance of sand drain to see the influence of

From Figure

Under the initial parameters, change the value of loading rate to see the influence of

From Figure

Under the initial parameters, change the value of coefficient of consolidation to see the influence of

From Figure

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Pore pressure

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

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