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where S S is the specific storativity, kxx , kyy , kzz are the permeability coefficient in the principal directions of anisotropy medium respectively, H is the water level at the pointat (x,y,z) the instant t , W is the source and sink terms , t is time, Ω is the computational domain.
2.2 Initial and boundary conditions
The initial and boundary conditions for the unsteady seepage in porous medium are given as follows.
(1) Initial conditions:
(2) Boundary conditions:
(a) The first kind of boundary condition:
(b) The second kind of boundary condition:
(c) Free surface boundary conditions :
where 0 0 H (x,y,z,t ) is the initial water-level at the point (x,y,z) , 1H (x,y,z,t) is the known
water-level at the boundary, q(x,y,z,t) is the recharge capacity per unit area for the second kind of the boundary conditions, cos(n,x) , cos(n,y) and cos(n,z) are the directional cosines for the normal of the body surface, μ is the saturation deficiency (free surface rise) or the specific yield (free surface drops), z n is the third component of the outward normal vector ={ , , } x y z n n n n at the free surface, 1 Γ , 2 Γ and 3 Γ are the first kind of the boundary, the second kind of the boundary and the free-water table boundary respectively.
3. Numerical simulation with finite element method
3.1 Finite element equation
The three-dimensional computational domain is pided into n units, and the interpolation function of the eight-node is oparametric element is utilized. According to the variational principles, the corresponding functional of the differential Eq.(1) is taken as zero, and time is applied in implicit difference. Then the finite element equation in the seepage domain is
where K is the general seepage matrix, S is the water storage matrix, G is the recharge matrix, F is the water quantity matrix, H is the water level vector of node to be calculated, t Δ is the time step.Let
then Eq.(7) can be taken as
where A is called the total stiffness matrix, and B is called the constant term.
3.2 Treatment of the first kind and free surface boundary conditions
In this article, the authors utilize the method of improved cut-off negative pressure[5,6], and select the improving element conduction matrix adjustment method[7,8] to deal with the free surface boundary conditions. The results prove to be satisfactory.
It is able to deal with the first kind of boundary conditions by using “putting large number” method, namely the principal diagonal elements of total stiffness matrix which sticks to known water head are put a large number, and then the corresponding elements on the right sides of equations are multiplied by the lager number[8].
3.3 Solution
In this work, the Preconditioned Conjugate Gradient (PCG) method[9,10] is used to solve the above finite element equations. The basic ideas are that the coefficient matrix of symmetric positive definite equations is pretreated in order to reduce the condition number of equivalence, and then the conjugate gradient method is used to enhance the convergence rate and to overcome the numerical instability[11,12]. The computation is translated to computer program with the Visual Fortran 90[13,14]. With the program, the computation was completed by PIV 3.0.
4. Engineering applications
4.1 Engineering situation
The forth subway of Dongjiadu tunnel repair foundation is located in the east of the South Zhongshan Road and in the south of Dongjiadu Ferry. The restoration project is located in the west of the Huangpu River and the 22-storey Linjiang Garden Building is in its north, and the ground elevation is about 3.5 m. The excavation is approximately 39.8 m in depth, the excavating scale on the plane is 236 m in length and 19 m-23 m in width and the distribution of the foundation is a circular arch,with the radius 350 m. Figure 1 shows the plane distribution of the foundation pit. The retaining wall of foundation pit shield is designed with a thickness of 1.2 m, and the weak phreatic aquifer is distributed below the miscellaneous fill in the upper part of the foundation. The groundwater level is from 0.5 m to1.2 m, and it is not rich in water quantity and not easy to cause dewatering. The confined aquifers of high head are buried below the base of foundation pit, which are classified as the upper Pleistocene confined aquifers Ι and ΙΙ, middle Pleistocene confined aquifer ΙΙΙ respectively. The three confined aquifers are connected with each other, and the static water level excavation is approximately –5 m. In order to ensure the smooth progress of the excavation, the water level in the foundation pit must be decreased to below the base of the foundation pit. To ensure the security of surrounding buildings, especially the security of the Linjiang Garden Building, the decrease of the underground water level around the foundation pit should be incapable of causing land subsidence and geologic disasters[15].