Incorporating one-dimensional bar elements with two-dimensional quadrilateral axisymmetrical elements to tackle problems of consolidation of clay with vertical drains.

1. The efficiency of vertical drainsThe efficiency of vertical drains
for accelerating consolidation – an appraisalfor accelerating consolidation – an appraisal
usingusing
finite element methodfinite element method
 Introduction
 Factors influencing the drain’s efficiency.
 Finite element models
 Finite element formulation
 Parameters and boundary conditions
 Results
 Conclusions
Dr Mazin Alhamrany
May 2017

2. IntroductionIntroduction
Consolidation problems involving vertical drains are conventionally solved by setting
the excess pore water pressure in the drains to zero; this implies that the drains are fully
efficient with no resistance to flow along the entire length of the drains (i.e. infinitely
pervious). Finite element technique, based on incorporating one-dimensional elements
into a computer program with compatible two-dimensional quadrilateral
axisymmetrical elements, has been used to investigate the efficiency of vertical drains
and the extent to which the assumption of full efficiency is justified.
The one-dimensional bar elements model the flow of water along the vertical drain and
solutions can, therefore, be obtained for specific values of coefficient of permeability
for the drain material.
The efficiency of the drain (η) has been defined by the expression: η = (1- us/uc).
Where us is the pore water pressure in the drain layer and uc is the pore water pressure
in the clay layer. The drain will be fully efficient (η = 1) when the water pressure in it
is zero. If, on the other hand, the drain is completely inefficient (η = 0), it will conduct
no water and the water pressure in it will rise rapidly to the value of uc in the adjacent
clay.
The different applications of incorporating one-dimensional bar elements with two-
dimensional elements are shown in Figure 1.
A paper has been published in “5th European Conference, Numerical Methods in Geotechnical Engineering, Paris, 4-6
September 2002; pages 777-782”. This work is part of my PhD research work at The University of Sheffield, supervided

3. Figure 1: Different applications of incorporating one-dimensional bar elements with
two-dimensional elements

4. Factors influencing the efficiency of verticalFactors influencing the efficiency of vertical
drainsdrains
 Coefficient of permeability of the drain material (ks),
 Coefficient of permeability of the parent clay (kc),
 Diameter of the drain (rw),
 Spacing of the drain (Length of radial drainage path; re),
 Thickness of the clay layer (H; Length of the drain), and
 Boundary condition.

5. Finite element modelsFinite element models
1. First Model (Conventional Method):
by setting the excess pore water pressure, at nodes representing the drain, to zero
(assuming the drains are fully efficient!); see Figure 2.
1. Second Model (Incorporating Bar-Elements):
One-dimensional bar-elements have been developed and successfully incorporated
with the two-dimensional quadrilateral axisymmetrical elements to tackle
consolidation problems involving vertical drains. Using this technique, the extent to
which the assumption of full drain efficiency is really justified has been examined
and quantified; see Figure 3.

6. Figure 2: First Model (Conventional Method)

7. Figure 3: Second Model (Incorporating Bar-Elements)

8. Finite element FormulationFinite element Formulation
The finite element formulation of Biot’s fully-coupled theory consists of two
simultaneous matrix equations;
1. the equilibrium equation:
[K1
] {δ } + [C ] {u} = F
2. the continuity equation:
[C] d{δ }/dt - [K2
] {u} = 0
Crank-Nicolson procedure has been used to approximate the time-derivative term in
equation 2; the procedure is efficient and maintains accuracy when a solution with a
variable time-step is used.

9. Parameters and boundary conditionsParameters and boundary conditions
 The efficiency of drains has been defined as:
η = (1- us/uc).
 The investigation is limited to H/re = 1.
 An isotropic soil, with krc = kvc = kc =1,0 unit.
 Solutions for ks/kc = 106
, 105
, 104
, 5x103
, 2x103
,
103
, 5x102
, 2x102
and 102
have been obtained.
 Cases for re/rw = 5, 10, 20 and 30 are considered.
 Impermeable surface and smooth impermeable
base.

10. Excess Pore Water Pressure Distribution
along the vertical drain and in the clay layer

11. Excess Pore Water Pressure Distribution
along the vertical drain and in the clay layer

12. Influence of drain efficiency on average degree of consolidation “Uav”

13. ResultsResults
The efficiency of the drain η has been drawn versus the time factor Tr (Tr = crc . t / re
2
).

14. re/rw = 5
0
0,2
0,4
0,6
0,8
1
0,001 0,01 0,1 1 10 100
Time factor
Drainefficiency
ks/kc=10000 ks/kc=5000 ks/kc=2000
ks/kc=1000 ks/kc=500 ks/kc=200
ks/kc=100

15. re/rw = 10
0
0,2
0,4
0,6
0,8
1
0,001 0,01 0,1 1 10 100
Time factor
Drainefficiency
ks/kc=10000 ks/kc=5000 ks/kc=2000
ks/kc=1000 ks/kc=500 ks/kc=200
ks/kc=100

16. re/rw = 20
0
0,2
0,4
0,6
0,8
1
0,001 0,01 0,1 1 10 100
Time factor
Drainefficiency
ks/kc=10000 ks/kc=5000 ks/kc=2000
ks/kc=1000 ks/kc=500 ks/kc=200
ks/kc=100

17. re/rw = 30
0
0,2
0,4
0,6
0,8
1
0,001 0,01 0,1 1 10 100
Time factor
Drainefficiency
ks/kc=10000 ks/kc=5000 ks/kc=2000
ks/kc=1000 ks/kc=500 ks/kc=200
ks/kc=100

18. ConclusionsConclusions

19. Questions?Questions?