Theoretical Solutions to The Problem of Seepage and Consolidation in Saturated Clay Based on The Spatial Axisymmetric Model

: The series solutions to the problem of spatial axisymmetric consolidation are deduced under non-homogeneous boundary conditions. Firstly, differentiable step function is introduced to construct the homogeneous operation function. Secondly, the operation function is used to superimpose the non-homogeneous boundaries to obtain homogeneous boundaries, non-homogeneous fundamental equation and new initial condition. Finally, the method of variables separation is used to construct the eigenfunction, and due to the mathematical justification of complete orthogonality of the eigenfunction, the series expansions of the fundamental equation and initial condition are carried out to obtain solutions for the seepage and consolidation in saturated clay with a borehole boundary. The correctness of the theoretical solutions are verified by the strict mathematical and mechanics derivation and the law of space-time variation in seepage flow.


Introduction
Surcharge preloading, vacuum preloading and pile compaction can produce excess pore water pressure in the saturated clay. The speed and degree of excessive-pore-water-pressure dissipation affect the construction progress, foundation treatment effects and the bearing capacity of pile foundation, and also reflect the average degree of consolidation and the law of variation in compression quantity with time of clay layer. The related seepage and consolidation research mainly includes the determination of the initial excess pore water pressure, the derivation of theoretical or numerical solutions to reflect the spatiotemporal variations of excess pore water pressure, and their applications in the engineering and environmental problems.
Piling mechanism and soil squeezing effects in saturated clay are described in the article [1]. In articles [2,3,4], the displacement, deformation and stability of soil under different drainage assumptions are researched. In the article [5], the 0 K consolidated drainage model is used to deduce the stress field of the pile squeezing in saturated clay, and the solution for the initial excess pore water pressure is obtained under the corresponding boundary conditions. In the article [6], the problem of pile compaction in saturated soft clay is studied, and logarithmic strain parameters are introduced to describe the characteristics of large deformation and softening. The stress field caused by piling compaction in soil is deduced, and the law of distribution of excess pore water pressure is described also. The excess pore water pressure caused by pile driving in saturated clay is estimated in the article [7], and a numerical model reflecting the consolidation of soil around the pile is established to calculate the average degree of consolidation. In articles [8,9], the additional displacement, strain, stress, excess pore water pressure caused by pile compression and its dissipation are studied, and the corresponding engineering problems are solved. However, the above-mentioned solutions to the consolidated seepage problem of soil are limited to the one-dimensional or homogeneous fundamental equation and boundary conditions basically.
For one-dimensional non-homogeneous boundary problems, the superposition constructor can be used to homogenize the boundary. However, for multi-dimensional problems, the homogeneous superposition operation in one boundary surface will superimpose non-zero values in the other boundary surface under normal conditions, making it difficult to realize the traditional homogeneous operation. To solve the conflict, we introduce the differentiable limit step function and construct the homogeneous operation function for multi-dimensional problems without values of interaction. The method of separation of variables is used to construct the eigenfunction of the inhomogeneous generalized equation, and the solution for the seepage and consolidation with a borehole boundary is obtained with the validation of the eigenfunction's completeness and orthogonality.

Mathematical model
Consolidation model is established as shown in Fig. 1, including the top surface at =0 z , bottom surface at z H  , drainage (or irrigation) well boundary at w r r  and the covering of the initial excess pore water pressure at e r r  . The consolidation differential equation [10] of spacial axis-symmetric problem is shown as Eq. (1) where u is excess pore water pressure, h C , v C are the horizontal and vertical consolidation coefficients and , r z are radial and vertical coordinates respectively. The initial condition is written as Eq. (2).
The boundary conditions are written as Eq.
where w r is the radius of borehole, pumping or irrigation well, e r is the influence radius of ( , ) f r z and H is the thickness of saturated clay layer.  To solve the above-mentioned problem, the differentiable Eq. (4) and (5) (4) and (5), and they can converge to the step function as shown in Fig. 2

Solution derivation
The homogeneous equation corresponding to Eq. (6) is written as Eq. (7).
Apply the separation of variables method, assuming         , , V r z t R r Z z T t  , and make d d By introducing parameters  and  , Eq. (9) can be obtained by separating variables from Eq. (8).
Eq. (9) The characteristic Eq. (12) can be obtained by combining the boundary condition w 0 r r R    and Eq. (11).
Eq. (16) is ordinary differential equation of order 1 with t as the variable and the right end of the equal sign is  

Computing arguments
The bearing capacity of a test pile and its converted average degree of consolidation [8,11] are shown in Table   1. Pile cross section is 0.45m ×0.45m, pile length is 30.9 m and the equivalent radius is The homogeneous operation function is constructed according to the specific boundary, and the inhomogeneous boundary of spatial axis-symmetric problem is superimposed to obtain the new homogeneous boundary, generalized equation and initial condition.
Using the method of variable separation to construct the complete orthogonal characteristic function, the series expansion of the generalized equation and initial condition are carried out, and the consolidation series solution to the spatial axis-symmetric problem with a borehole boundary is obtained.
Based on the theoretical formula obtained from the above-mentioned derivation, contour of excess pore water pressure is drawn by combining the calculated values of the solution with different boundary functions to verify the applicability of the solution to the corresponding engineering problem.