Flow Characteristics under Sheet Pile in Anisotropic Porous Soil
Associate Professor, Department of Civil Engineering,
Faculty of Engineering, King Mongkut’s University of Technology,
Thonburi, Bangkok, Thailand
Phanuwat.sur@kmutt.ac.th
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Flow Characteristics under Sheet Pile in Anisotropic Porous Soil
Associate Professor, Department of Civil Engineering, |
ABSTRACT
The objectives of this paper are to present the flow characteristics and to create normalized graphs for seepage analysis under a sheet pile in anisotropic soil layers by using finite element method. The model is simulated by using the varying properties of hydraulic conductivity (permeability) ratio extended to cover most general anisotropic porous soil and also by covering characteristics of Bangkok soil. To verify and accept the simulation of the case studies, the mesh density technique and minimum error setting are applied. Moreover, the suitable proportion of model dimensions which is 4:1, distance to depth from previous works is applied. The normalized graphs are presented. To utilize the graphs is shown through an example of the calculation the flow rate in anisotropic soil under sheet pile. Finally, limitations of this study are indicated and further study recommendation made.
Keywords: Anisotropic porous soil; Bangkok soil; Normalized Graph; Seepage; Sheet pile; Finite element method; FlexPDE.
INTRODUCTION
Traditionally, seepage problems are solved by flow nets. A number of relevant quantitis, including the rate of flow can be computed directly from the flow net. The major disadvantage in the flow net method is that it requires substantial effort in drawing a good flow net. In the preliminary design stages, where several alternatives are tried, drawing a separate flow net every individual configuration becomes a tedious task. This method was known to be limited to complication of boundary conditions such as configuration of the problem, many soil layers, different soil types, anisotropic permeability, transient conditions etc. Under such circumstances, it is very much desirable to have some methodology with sufficient accuracy.
Uses of the finite element method for geotechnical engineering and seepage analyses are reviewed in detail in previous work by Suriyachat and Yungyune (2003). Some recommendations from previous work are extended to present in this paper especially the flow of water in anisotropic soil.
The finite element method presented in this paper is still very simple and efficient technique. By this method, the quantity of flow can be estimated without drawing the flow nets and can be solved with all cases of situations mentioned above. The method provides quick and powerful solutions, with sufficient for all practical purpose.
THEORY
Flow of water in soil
The governing differential equation that expresses the flow water in two-dimensional case in Figure 1 is:
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(1) |
Where h is head loss or head difference (H1-H2), kx, kz are the hydraulic conductivity in the x and z-direction, respectively, Q is applied boundary flux or flow rate, 
is volumetric water content, and t is time.
Figure 1. (a) Single sheet pile driven into permeable layer; (b) flow at point A
This equation states that the difference between the flow (flux) the change in the volumetric water content. More fundamentally, it states entering and leaving an elemental volume at a point in time is equal to that the sum of the rates of change of flows in the x- and z-directions plus the external applied flux is equal to the rate of change of the volumetric water content with respect to time. Under steady-state conditions, the flux entering and leaving an elemental volume is the same at all times. The right side of the equation consequently vanishes and the equation reduces to:
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(2) |
The different value of hydraulic conductivity in x and z direction can present the anisotropic soil.
Finite Element Equations
The finite element equation that follows from applying the Galerkin method of weighed residual to the governing differential equation (Eq.2) become:
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(3) |
where
[B] = gradient matrix
[C] = element hydraulic conductivity matrix
{H} = vector of nodal heads
l = mw gw where mw is the slope of the storage curve,
gw = unit weight of water
<N>T <N> = [M] = mass matrix
{H},t = 
= change in head with time
q = unit flux across the side of an element
<N> = vector of interpolating function or shape function
For a two-dimensional analysis the thickness of the element is considered to be constant over the entire element. The finite element equation (3) can consequently be written as:
  |
(4) |
where t is the element thickness. When t is a constant the integral over the volume 
becomes the integral over the area 
and the integral over the area becomes the integral over the length 
from corner node to corner node. In abbreviated form, the finite element equation is
| [K]{H}+[M]{H},t = {Q} | (5) |
where
R is radial thickness in the axisymmetric case of three-dimension. Equation 5 is the general finite element equation including a transient seepage analysis. For a steady-state analysis, the head is not a function of time and consequently the term [M] {H},t vanishes, reducing the finite element equation to:
| [K] {H} = {Q} | (6) |
Gauss numerical integration is used to form the element characteristic matrix [K]. The integrals are sampled at specifically defined points in the elements and then summed for all the points. The following integral (from Equation 5)
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can be replaced by
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(7) |
where
j = integration point
n = number of integration points
det|Jj| = determinant of the Jacobian matrix
W1j, W2j = weighting factors
The number of integration points required in an element depends on the number of nodes and the shape of the elements and user define. Then, in Equation 6 nodal head vector can be solved by giving some boundary conditions.
The seepage quantity that flows across a section can be computed from the nodal heads and the coefficients of the finite element equation.
From Darcy’s Law, the total flow between two points Node i to Node j is:
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(8) |
where the coefficients c in Equation 8 are a representation of 
. The total flow quantity through any section can be computed.
METHODOLOGY
Procedure
This project is a study of the quantity of seepage under the influence of sheet piles depth and anisotropic property by finite element method. The project uses the information of soil that is in the past collected by many researchers and model verification from previous work by Suriyachat and Yungyune, 2003. The case study is specified to be a reference model which anisotropic hydraulic conductivity for soil layer in Table 1 is covered that mean kz/kx vary from 0.01 to 100 and ratio of sheet pile depth to soil layer depth vary from 0.1 to 0.9. In addition, non-dimensional values are calculated to create the normalized graphs.
Resources of information
This studies uses the information that is collected by varies researchers (MRTA, 1997) to study in each case as follows:
Table 1. Range of Hydraulic conductivity of each soil types.
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| Soil types | Hydraulic conductivity (m/s) |
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| Very soft silty clay Medium stiff silty clay Hard clay |
5 ´ 10-8 – 1 ´ 10-10 1 ´ 10-8 – 5 ´ 10-11 5 ´ 10-8 – 3 ´ 10-11 |
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From this information it is clear that the hydraulic conductivity ratio (kz/kx) varies from .01 to 100 applied to the simulation in this paper.
Discretization and Boundary condition
The finite element modeling techniques can be adopted for the analysis of the mentioned case studies in many difference ways. The steady state seepage of the each case study is analyzed by using FlexPDE software (PDE Solution Inc., 2005). In this study, the model idealizes the flow areas as triangular elements, and the mesh generation was performed with optimization. The discretization and boundary conditions in each case have details as follows.
CASE STUDY
Figure 2 shows the dimensions of the problem. A simplified geological configuration of a 80 m-Distance and the depth of 20 m meter (T) below ground surface is configured. The depth of sheet pile (S) will vary from 2 m to 18 m meter.
Figure 2. Schematic of the problem
The finite element mesh used for the analysis is shown in Figure 3. It is noted that the boundary node of the upstream and downstream surfaces are designed as difference head boundary with various total head (h). The hydraulic conductivity of anisotropic porous material applied can be varied in the range covering Bangkok soil properties.
Figure 3. Finite element mesh (a) Full scale (b) Zoom scale
RESULTS AND DISCUSSION
Preliminary study of appropriate model
A previous study (Suriyachat and Yungyune, 2003) of appropriate model was set and carried out in order to investigate the excellent results. In this analysis the ratio of distance per depth equal 4:1 is selected and adaptive mesh technique from FlexPDE is utilized.
Verification of Data Results
The values of quantity of seepage with S/T for these problems are computed and presented in Table 2. The percentage differences in results are compared to the values computed from the mathematical method as given by Harr (1962). It can be seen that the values of the quantity seepage showed an excellent agreement and acceptable results. The maximum differences were within 5.26%.
Table 2. Comparison the results from FlexPDE and given by Harr (1962).
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| S/T | Q/h/k | % Error | |
| FlexPDE | Harr | ||
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| 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 | 1.01 0.80 0.67 0.57 0.49 0.43 0.36 0.30 0.24 |
1.05 0.84 0.70 0.60 0.51 0.45 0.36 0.30 0.24 |
3.96 5.00 4.47 5.26 0.02 4.65 0.00 0.00 0.00 |
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Results of Flow Characteristics
Figure 4 represents an example of the complete flow net under sheet on isotropic porous flow where S/T = 0.5, k = kx = kz = 1 meter per day and head (H) = 100 meter. Figure 5 shows the different zoom flow net pattern where kz/kx is 0.1, 1.0, and 10 and S/T is 0.7.
Figure 4. Flow net under sheet (a) Full scale (b) Zoom scale
Figure 5. Comparison of flow net under sheet on anisotropic soil
Figure 6 shows the velocity vector field of water in the same above conditions and easily visualized the velocity of water flow under sheet pile.
Figure 6 Velocity vector field around sheet pile.
Figure 7 presents the normalized velocity (v/k) distribution along (a) under the sheet pile and (b) the back the sheet pile. As expected, the distribution graph shows the high velocity under the sheet pile and reduces when far from the sheet pile.
Figure 8 gives the normalized velocity at the points a, b, c, d, and e by varying the ratio of sheet pile depth to soil layer depth (S/T) from 0.1 to 0.9. At the point close to sheet pile (point a), the velocity gives high value and reduce like exponential curve when S/T ratio increases while at other points (b, c, d, and e) the graph reduces to an almost straight line when S/T ratio increases.
Figure 7. Normalized velocity distribution around sheet pile.
(a) below sheet pile
(b) behind sheet pile
Figure 8. Normalized velocity distribution along the back of sheet pile with S/T ratio.
Normalized Graphs
The normalized flowrate quantity of seepage (Q/h/k per m width) and ratio of sheet pile depth penetration to depth of soil layer (S/T) are calculated where k is the square root of (kz2+ kx2). The values of flowrate quantity with S/T for anisotropice soil problems is presented in Figure 9. For utilization and calculation, Figure 10 through Figure 13 are presented for covering in details.
Example
A soil layer in Figure 2 which has hydraulic conductivity kz = 1×10-6 m/sec, kx = 1×10-7 m/sec, difference in water head between the two sides of sheet pile equal to 6 m, if sheet piles are fixed down into soil layer 5 m from ground level and thickness soil layer is 20 m. What is the flow rate of water through sheet piles if sheet piles are 30 m long?
Given: kz = 1×10-6 m/sec, h = 6 m., kx = 1×10-7 m/sec, S = 4 m., T = 20 m. Sheet piles are 30 meters long.
Solution: from graph in Figure 10, for S/T = 0.2 and kz/kx = 10 obtained Q/h/k = 0.24. So, total flow rate through sheet piles
Q = 0.24 × h × k × 30m
= 4.3415×10-5 (m3/sec)
= 0.1563 (m3/hr) Ans
Figure 9. Normalized flowrate quantity and S/T (kz/kx from 0.01 to 100).
Figure 10. Normalized flowrate quantity and S/T (kz/kx from 1 to 10).
Figure 11. Normalized flowrate quantity and S/T (kz/kx from 10 to 100).
Figure 12. Normalized flowrate quantity and S/T (kz/kx from 0.01 to 0.10).
Figure 13. Normalized flowrate quantity and S/T. (kz/kx from 0.1 to 1.0)
CONCLUSION
The following conclusions can be drawn from the present study. A two-dimensional finite element analysis of the flowrate quantity of seepage can be using the computer package called FlexPDE. In the proposed finite element model of the problem, the flow regions are discretized by quadrilateral triangular elements. The case problem selected in the study consists of one anisotropic permeable soil layer. The studies confirm the successful numerical modeling of ground water flow through the single row of sheet pile. The quantity of seepage is verified by the quantity of flow curve obtained from the mathematical solution as given by Harr (1962). The flowrate quantity of seepage for different hydraulic conductivity in soil layer is presented in normalized graphs. The major advantage of the graphs is that it gives simple and quick solutions to the seepage problems without sacrificing the accuracy. The method is very useful in preliminary designs and feasibility studies.
Finally, the following are recommended for further study: the problems on two or more layered soil layers, the problems on three-dimensional case, both in steady state and transient state, with particular interest in boiling condition and strain energy distribution.
ACKNOWLEDGMENTS
The authors are grateful to the staff of civil engineering department, King Mongkut’s University of Technology, Thonburi who have kindly helped to collect the required data.
REFERENCES
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