CIV405 Final Report

Department of Civil and Structural Engineering, University of Sheffield, 2005

Optimal Design of Vertical Drains in Soft Ground

Sam Clarke

The University of Sheffield Department of Civil and Structural Engineering

CIV405 Final Report

Submitted:

06/05/2006

Candidate: MEng. Structural Engineering and Architecture

Supervisor: Dr C. C. Hird

i

Declaration statement

The author certifies that all material contained within this report is his own work except where it is clearly referenced to others.

Signed: ………………………………

Date: 06/05/2006

Acknowledgements

Mike Drew of Cofra UK Limited, for his guidelines on the costing of prefabricated vertical drains and their installation in practice.

ii

Abstract

This project is concerned with the design of a program to calculate the optimal solution for

geotechnical problems involving the consolidation of soft ground by the use of prefabricated

vertical drains. The effects being taken into account include smear, well resistance, ramped

loading and multiple layers. The program has allowed the author to complete a series of

parametric studies into the effects of the factors which contribute to the rate of consolidation

using vertical drains. The final solution is a distributable program that uses an intuitive

graphical user interface. This allows the user to input soil parameters and assumptions and

then run the program to find the optimal spacing of drains to achieve a given consolidation in

a given time. An element of probabilistic analysis has also been incorporated into the program

to allow the creation of risk to cost curves for any parameters. This allows the user to make an

educated decision based on the allowable cost and the degree of certainty in the soil

parameters.

iii

CONTENTS

Contents.....................................................................................................................................iii

List of Figures ............................................................................................................................ v

List of Tables............................................................................................................................. vi

List of Symbols ........................................................................................................................vii

1 Introduction .................................................................................................................... 1

2 Literature Review ........................................................................................................... 1

2.1 History of the Methods............................................................................................... 1

2.1.1 The Basic Method .............................................................................................. 1

2.1.2 Well Resistance .................................................................................................. 1

2.1.3 Smear.................................................................................................................. 2

2.1.4 Ramped Loading ................................................................................................ 2

2.1.5 Multi-Layered Systems ...................................................................................... 2

2.2 Assumptions and Values ............................................................................................ 3

2.3 Factors affecting Performance ................................................................................... 4

2.4 Practical Data ............................................................................................................. 4

3 Program Design.............................................................................................................. 4

3.1 Assumptions of the Program...................................................................................... 4

3.2 Development of the Model......................................................................................... 5

3.2.1 Well resistance and smear integration................................................................ 5

3.2.2 Ramped loading application............................................................................... 8

3.2.3 Multiple Layers ................................................................................................ 11

3.2.4 Vertical Drainage ............................................................................................. 13

3.3 Program Validation .................................................................................................. 15

3.4 Program Optimisation .............................................................................................. 17

3.4.1 Cost analysis of the problem ............................................................................ 17

3.4.2 Probabilistic analysis........................................................................................ 18

4 Parametric Studies ........................................................................................................ 22

5 Program Distribution.................................................................................................... 25

5.1 Worked Example using the Verticalc Program........................................................ 27

6 Conclusions .................................................................................................................. 28

6.1 Theoretical vs. Practical ........................................................................................... 28

6.2 Further Geotechnical Developments........................................................................ 29

6.3 Further Computing Developments ........................................................................... 29

7 References .................................................................................................................... 30

iv

8 Appendix 1: Program User’s Guide ............................................................................. 32

8.1 Installation of Verticalc............................................................................................ 32

8.2 The Graphical User Interface Explained. ................................................................. 33

8.3 Entering Values ........................................................................................................ 34

8.4 Using the Probabilistic Analysis Tools .................................................................... 35

8.5 Using the Calculate Function ................................................................................... 36

8.6 Accounting for multiple layers................................................................................. 37

8.7 Advanced Features ................................................................................................... 38

8.8 Troubleshooting ....................................................................................................... 39

9 Appendix 2: Useful data............................................................................................... 40

9.1 Mebradrain Specifications: 28-05-2005................................................................... 40

9.2 Equations.................................................................................................................. 41

9.2.1 Hansbo (1981):................................................................................................. 41

9.2.2 Olson (1977)..................................................................................................... 41

9.2.3 Carrillo (1942).................................................................................................. 42

v

List of Figures Figure 1: Consolidation curves as described by Hansbo (1981). n=25...................................... 5

Figure 2: Plot of the effect of Smear for varying depth and time .............................................. 6

Figure 3: Plot of the effect of Well Resistance for varying depth and time............................... 6

Figure 4: Effect of Smear in relation to depth and spacing of drains......................................... 7

Figure 5: Plot of variation of U with depth - double vs. single drained conditions. .................. 7

Figure 6: Comparison of the effect of ramped loading on the consolidation process................ 8

Figure 7: Internal workings of the matlab ramped loading script .............................................. 9

Figure 8: Comparison of ramped loading methods.................................................................. 10

Figure 9: Input matrix for the matlab working script............................................................... 10

Figure 10: Matlab script modification for multiple layers ....................................................... 10

Figure 11: Variation of the effect of Well Resistance in a multilayered soil........................... 13

Figure 12: Contribution to the consolidation process by vertical drainage.............................. 14

Figure 13: Relative Effect of Vertical Drainage ...................................................................... 15

Figure 14: Graphical comparison between Leo (2004) and the present paper......................... 17

Figure 15: Example of a Beta Distribution .............................................................................. 19

Figure 16: The updated matlab script for variable input values............................................... 20

Figure 17: PERT analysis matlab script ................................................................................... 21

Figure 18: Risk versus Cost curve from PERT analysis .......................................................... 22

Figure 19: The effects of well resistance on the consolidation process ................................... 23

Figure 20: The effects of smear on the consolidation process ................................................. 23

Figure 21: The effects of spacing on the consolidation process .............................................. 24

Figure 22: Principle of the Graphical User interface ............................................................... 25

Figure 23: GUI Initial Screen - Generic Version ..................................................................... 26

Figure 24: GUI Initial Screen - Cofra UK Version.................................................................. 27

Figure 25: GUI output showing Leo (2004) model.................................................................. 28

vi

List of Tables

Table 1: Values for the initial analyses ...................................................................................... 5

Table 2: Comparison of the superposition method to Olson (1977) ........................................ 11

Table 3: Assumptions and Values used in the Leo (2004) analysis......................................... 16

Table 4: Comparison of the Leo (2004) results to those of the author..................................... 16

Table 5: Installation Costs........................................................................................................ 18

Table 6: Z - values for beta distribution................................................................................... 19

Table 7: Converted parameters from the Leo (2004) Analysis................................................ 27

vii

List of Symbols

Symbol description ch Coefficient of consolidation

D Drainage boundary (m)

d Equivalent diameter of the drain (m)

H Total depth of the clay layer (m)

kc Horizontal permeability of the soil (m2/year)

M Oedometer compression modulus (MN/m2)

mv Compressibility coefficient of the soil (1/M)

qw Discharge capacity of the drain (m3/year)

S Drain spacing (m)

s Zone of smear = ds / d

t Time (years)

Th Time factor in radial consolidation

U Degree of consolidation

z Depth into the clay layer (m)

γw Unit weight of water (KPa)

μ Pore water pressure (KPa)

1

1 INTRODUCTION

The aim of this project is to provide a valuable tool for the design of vertical drains in

practice, with the emphasis being on the combination of an accurate prediction of

consolidation settlement and the ease and speed of use. In practice currently there are two

feasible approaches to the design of vertical drains. Firstly to use simple design tools and

methods based on basic assumptions (such as a single homogenous layer) to calculate the

consolidation then interpolating between different methods to gain a more informed estimate

of the settlements. The second option is reserved for projects that have little tolerance (such as

nuclear power plants), which involves using a finite element or difference program to

calculate the settlements. This second method is much more time consuming, and many more

of the soil parameters are required. In practice generally the first option is used. The aim of

this project is to fill the void between the two methods to give a quick estimate of the

settlements with the minimal amount of required information and time. The factors that are

going to be taken into account in this project are Ramped Loading, Smear, Well Resistance

and Multiple Layers.

2 LITERATURE REVIEW

2.1 History of the Methods

2.1.1 The Basic Method

One of the key principles in geotechnics is that of excess pore water pressure. When a load is

applied to a soil it induces an excess pore water pressure within the soil. As the soil dissipates

these pressures the amount of water within the soil body decreases and the volume of the soil

decreases. This process is called consolidation. The length of time required for a specific soil

to consolidate is dictated by the soil parameters and the drainage distance. The installation of

vertical drains effectively reduces the drainage distance and thus decreases the amount of time

required for consolidation.

One of the best known studies of Vertical Drains was conducted by Barron (1948.) He

assumed two types of vertical strain that might occur in the clay. Firstly free strain, resulting

from a uniform distribution of surface load and secondly equal strain, resulting from imposing

the same vertical deformation on the surface for uniform soil.

2.1.2 Well Resistance

As the application of vertical drains became more widespread and deeper layers of clay were

involved other factors became apparent in the design. Originally the drains were created from

2

sand and had large diameters, 0.4 – 0.6m, but as the technology developed methods of

decreasing the construction costs led to the development of the band drain – that have a

comparatively small diameter. This led to the problem of well resistance, where by the drain

no longer has sufficient discharge capacity to cope with the volume of water drained from the

clay. The problem was first addressed from a design aspect by Yoshikuni & Nakanado (1974)

who arrived at a rigorous solution. Work since then has been carried out by Hansbo (1981)

who arrived at a simplified solution for the same case, which from testing provides a strong

correlation with the more rigorous Yoshikuni & Nakanado (1974) solution.

2.1.3 Smear

During the installation of vertical drains a mandrel is used to force the drain into the clay

layer. Although there are many ways of installing drains such as auger drilling, water-jetting,

the most common method is by the closed end mandrel. The mandrel causes disturbance to

the surrounding soil, and leads to a change in horizontal permeability in this area. The

disturbed area is known as the smear zone. Again Yoshikuni & Nakanado (1974) calculated a

rigorous solution and Hansbo (1981) the simplified version.

2.1.4 Ramped Loading

For all the solutions mentioned above the load is assumed to be placed instantaneously, in

practice this is impossible, and ramped loading needs to be accounted for. Olson (1977)

introduced a solution for a single ramped load, for both vertical and radial drainage separately

without account for the influence of Smear and Well Resistance. When used in conjunction

with Carrillo (1942) formula for combining vertical and radial flow, average flow for a whole

homogenous layer can be calculated. Through a process of superposition of the ramps,

variation in ramping can be taken into account using Olson’s (1977) solution. Zhu & Yin

(2004) also developed a solution for ramped loading for combined radial and vertical flow

independent of Carrillo (1942). The authors also extended their solution to cover the effects of

smear as well Zhu & Yin (2004), although well resistance is still not accounted for.

2.1.5 Multi-Layered Systems

Most solutions for problems based on the simplification that the clay layer is homogenous. In

practice this is never the case as even homogenous soils have varying values with depth (such

as the coefficient of compressibility). Very little research has been done to create a single

formula to take into account these variations. The layering of the soil dramatically reduces the

vertical permeability of the soil and means that assumptions such as those of Zhu & Yin

(2001, 2004) and Olson (1977) that combine vertical and radial drainage are void. Zhu & Yin

(1999) also formulated a solution for double soil layers under ramped loading, but again the

3

solution disregards the effects of smear and well resistance. Onoue (1988) developed a finite

difference solution that was able to take into account the vertical and horizontal flow from

each layer independently. These were combined with the effects of smear and well resistance,

although the actual formulae are not present in the paper. Onoue (1988) also developed a

method by which the effects of layering can be taken into account for simple analyses; this

method is recommended in Moseley and Kirsch (2004) for the practical design of drains.

2.2 Assumptions and Values

From the review above of the main work on calculating the settlements it can be seen that

only a select few have combined all of the major contributing factors into one solution, and no

author has created a way of calculating the solution quickly. (Finite difference programs do

take time to set up.) The author’s program will be based on the Hansbo (1981) method of

calculating consolidation settlements as this is a relatively quick and accurate method that

lends itself to manipulation for taking account of ramped loading. I will use Olson (1977) to

check my results for the case of ramped loading as this is an accepted standard for the

calculation of the ramped load.

Other than the actual formulae used for the calculations, thought also has to be given to the

values used within such formulae. Well resistance is dictated by the discharge capacity of the

drain, the length of the drain itself and whether the drain is single or double drained. The

drainage length is fixed for any given test as are the drainage conditions, but the discharge

capacity is a function of the lateral earth pressure at depth. The discharge capacity is a known

variable and the data is provided by the drain manufacturer following laboratory tests.

In the case of smear one can not be certain of the parameters for the diameter of the smear

zone and the reduction in permeability caused in such a zone, without experimental data to

back up such values. Hird & Moseley (2000) suggest values of 6.1=s and 3' =kckc for

heavily stratified clay this is backed up by small scale (Ø = 254mm) test models and pore

water pressure measurements. Hansbo (1981) also suggested similar values of 5.1=s and

3' =kckc but without any experimental data to back up the values. Sharma & Xiao (2000)

conducted a series of tests on a large scale (Ø = 1m) in a single homogenous clay layer with

values of 3.1' =kckc and the zone of smear being about 4 times the size of the mandrel. Note

this is not the same as stating 4=s , in reality s is now a function of the depth as the depth

increases so does the size of the mandrel required to penetrate to that depth. Indraratna &

Redana (2000) suggest a value of 43−=s , but give no numerical value to the reduction in

horizontal permeability apart from stating that the relationship is linear.

Although some authors Hansbo (2001), argue that the conditions for darcian flow are not

4

always valid, for the purposes of this project darcian flow is assumed as in Hansbo (1981) and

Yoshikuni & Nakanado (1974).

2.3 Factors affecting performance

As with all geotechnics there is an inherent uncertainty in the soil parameters used for design.

This includes such vital information as the horizontal permeability of the soil, to which the

whole radial consolidation process is linked. Chu et al. (2004) investigated the effects of

different factors on drainage, including guidance on the selection of PVDs and soil

parameters. Chai and Miura (1999) investigated the effects of the rectangular band drain

compared with circular wells in relation to the effects of smear, concluding that a circular

analysis agrees very closely with that of a rectangular analysis. The analytical model used has

an effect on the accuracy of the final solution Hawlader et al. (2002) compared Barron (1948)

with a finite difference analysis, showing a very high degree of agreement between the two.

Chu et al. (2004) also compared the Hansbo (1981) equation to a finite element analysis

concluding that for most cases Hansbo (1981) is good estimation for design purposes. There

is no real necessity to use a finite difference or element analysis to create an overly accurate

prediction based on uncertain parameters, when simple equations Hansbo (1981) can be used

to calculate equally valid predictions.

2.4 Practical Data

From contact with Cofra UK Ltd, the specifications for various drains were supplied giving

the analysis a realistic basis. The specifications for the Mebradrain series of drains are

contained within appendix 2. Chu et al. (2004) describe how the soil parameters can affect the

possible choice of drain by factors such as clogging and buckling of the drain.

3 PROGRAM DESIGN

3.1 Assumptions of the Program

From previous comparisons between analytical models (Hansbo (1981), Hawlader (2002)) it

has been decided to use the Hansbo (1981) method for the calculation of the consolidation.

Hansbo (1981) was chosen over Yoshikuni & Nakanado (1974) due to the shear simplicity of

the Hansbo (1981) equation given its close agreement with that of Yoshikuni & Nakanado

(1974). The program will be designed to take into account the effects of ramped loads,

multiple layers, well resistance and smear. For the values of smear zone and coefficient of

horizontal permeability the values from Hird & Moseley (2000) will be used: 6.1=s and

5

3' =kckc . For the purposes of this paper the values below have been used for all analyses

unless otherwise stated.

Table 1: Values for the initial analyses.

kc 5.5 m2/year H 20 m mv 0.25 m2/MN l 10 m D 1.5 m z 10 m d 0.06557 m qw 1736 m3/year Tc 0.5 years

3.2 Development of the model

3.2.1 Well resistance and smear integration.

The program was initiated by the use of the Hansbo (1981) equation to model the

consolidation curve for a given soil. This was quickly extended to encompass the effects of

smear and well as described by Hansbo (1981).

Figure 1: Consolidation curves as described by Hansbo (1981). n=25.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10Horizontal Time Factor (Th)

Ave

rage

Con

solid

atio

n (U

)

Simple Smear and Well Well

From figure 1 the effect of smear and well resistance can be clearly seen as having a

considerable effect on the rate of consolidation of the soil.

From the equations a series of studies were completed into the effects of each of the factors’

direct contribution to the retardation of the consolidation process with relation to depth. It can

be seen from the relative maximum magnitude of the graphs in figure 3 that well resistance

can have a very considerable effect on the process with up to a 0.37 reduction in average

6

consolidation, with the effect increasing with depth.

As well resistance is depth dependant and the effects of smear are related to the spacing of

the drains rather than the depth, smear is the predominant factor in shallow layers- this can be

seen in figures 2 and 3, with the effect of smear having a 0.095 influence compared with the

0.01 influence of well resistance in the same 5m deep case.

Figure 2: Plot of the effect of Smear for varying depth and time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.01 0.1 1 10 100Time (years)

Cha

nge

in D

egre

e of

Con

solid

atio

n

5m10m15m20m25m30m35m40m

Figure 3: Plot of the effect of Well Resistance for varying depth and time

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.01 0.1 1 10 100Time (years)

Cha

nge

in D

egre

e of

Con

solid

atio

n

5m10m15m20m25m30m35m40m

7

0

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1

Degree of Consolidation U

Dep

th D

(m)

Double Drained U Well U Smear + Well

Single Drained U Well U Smear + Well

Simple Consolidation U

A study was completed to see the effects of smear when varying the drain spacing. The results

prove interesting as the relative effect of smear varies only a little with increasing spacing, but

once again smear has the greatest effect in shallow soils, or more accurately well resistance

becomes the dominating factor in deeper soils.

Figure 4: Effect of Smear in relation to depth and spacing of drains

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1 1.5 2 2.5 3 3.5 4 4.5 5Spacing D (m)

Diff

eren

ce in

con

solid

atio

n ra

tio

5m10m15m20m25m30m35m40m

One thing to note is that the values for the average degree of consolidation used so far are for

mid-depth in the soil body. As Hansbo’s

(1981) equation is depth dependant the

end drainage conditions of the drain itself

have a large influence on the degree of

consolidation, as it doubles the drainage

distance within the drain body, thus

increasing the effect of well resistance.

Figure 5: Plot of variation of U with

depth - double vs. single drained

conditions.

As can be seen from figure 5 the effects of

well resistance are greatest at mid depth,

where the largest excess of pore water

pressures exist. The simple analysis

8

shown in blue is independent of depth. Smear in also relatively independent of depth showing

an almost direct relationship with well resistance. The figure also shows the massive effect of

the drainage condition on the drain itself, with the dashed lines denoting a single (top) drained

well. This has implications for the practical applications of vertical drains which will be

discussed later.

3.2.2 Ramped loading application

The next step was to introduce the idea of a ramped loading factor into the calculations.

Traditionally in practice this was done by applying the load half way through the construction

period, as this gives a reasonable degree of accuracy for long periods of time but in the short

term it is only a very rough estimate. To compare the effects of ramped loading the Olson

(1977) equation was plotted on the same axes as the Hansbo (1981) equations. From figure 6

it can be seen that the effects of ramped loading far outweigh those of well resistance and

smear combined. Also the method of placing the full load half way through the construction

period is shown to be inaccurate for short times (although it does provide a more informed

view than Hansbo’s (1981) simple equation alone.)

Figure 6: Comparison of the effect of ramped loading on the consolidation process.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10Time (Years)

Ave

rage

Con

solid

atio

n

Simple Well Well + Smear Ramping Half Time Ramp

The author then used Hansbo’s (1981) simple consolidation equation to model the Olson

(1977) curve, thus creating a link between the ramped loading consolidation and that of smear

and well resistance. This allows the combined analysis of smear, well resistance and ramped

loading in one program without the need for finite difference or element formulations. The

9

ramped loading was accounted for by the superposition of many small steps of loading at

equal time intervals, the smaller the step the more accurate the solution.

Figure 7: Internal workings of the matlab ramped loading script

consol.mprogram call

Step 1:Set Limits of

Accuracy

InitialValues drain

layertim

s

d

kckcdqw

D

tc

Discharge Capacity of Drain (m3/yr)

Time of Interest (years)

Matrix of Soil Parameters

End Conditions of Drain

Construction Period

Effect of Smear Zone

Reduction in permeability of Smear Zone

Radius of effect of the drain (m)

Drain diameter (m)

tmax

acc

Based on timvalue

This makes sure theprogram runs quickly

Step 2:Set Boundary

Conditions

If 'drain' =single

If 'drain' =doubleL=H L=H/2

Step 3:Calculate

Consolidationwithout Ramping

For AllTimes

For AllDepths

CALCULATECONSOLIDATIONHansbo Formula

U

Store U Values forall calculated

Depths

Steps of Depth dictatedby H / accuracy

Store average U forall times in matrix

'Uavg'average U

tmax is set in step 1tmin = 1 / accuracy

step length = 1 / accuracy

Uavg is divided byaccuracy to

account for thesuperposition

Time is also storedin Uavg to allow the

plot of aconsolidation curve

HansboAccounts for

Smear and WellResistance

Step 4:Uavg is

superimposed ontoits self to account

for ramped loading

The ramp load isapplied in very smallsteps giving a very

close approximation toa linear loading pattern

Uavg

The degree ofconsolidation associated

with the input 'tim' is foundwithin the Uavg matrix

If the degree of consolidation cannotbe found the program interpolates

between the nearest values to obtainthe value (when input is a fraction)

The Consolidation curveis plotted from the datawithin the Uavg matrix

Step 5:Output CalculatedValues and Curves

Average U Uavg

10

The program then was written as a matlab script, as represented in figure 7 to allow a more

flexible interface, smaller step sizes and the ability to easily change variables. Figure 7

represents the basic version of the script, taking into account well resistance, smear and

ramped loading.

By changing the version of the Hansbo (1981) formula used within step 3, the effects of

smear and well resistance can be removed to allow a direct comparison with the Olson (1977)

consolidation curve. In figure 8 the consolidations were plotted against time; it is obvious

from this comparison that the values are very similar, as little distinction can be made

visually. The values were analysed numerically to assess the accuracy of the superposition

method for changing step sizes.

Figure 8: Comparison of ramped loading methods

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10Time (years)

Ave

rage

Con

solid

atio

n

Olson (1977) Matlab Simple Matlab Smear and Well

It can be seen from table 2 that the superposition model agrees very closely with the results

directly given by Olson (1977), especially in the case of the 10,000 step model.

Even in the case of the 100 step model the prediction may start out with a large relative error

but in absolute terms the difference is insignificant. The superposition method allows the

effects of smear and well resistance to be accounted for at the same time as the ramped

loading. This can be seen in figure 8, showing how the effects of smear and well resistance

influence the emulation of the Olson (1977) curve. As the method is in script form it becomes

much simpler to perform parametric studies on the influence of individual factors.

11

layer=[0 0 0 0 1 6 5.5 0.20 2 12 5.0 0.22 3 20 2.0 0.25]

Table 2: Comparison of the superposition method to Olson (1977)

3.2.3 Multiple Layers

Taking into account the changing parameters with depth was done using the guidelines given

by Onoue (1988). This is done by calculating the consolidation at all depths within each layer

independently, but using the overall depth to calculate the effect of well resistance. This

method is valid for all combinations of layer depths and values but the degree of accuracy

becomes lower when the individual layers have greatly different properties.

Figure 9: Input matrix for the matlab working script.

Figure 9 shows an example of how the soil parameters are

entered into the matlab script. The 1st column denotes the

layer number, the 2nd column the depth the layer extends

to, the 3rd the associated kc value (m2/year) and the 4th the mv value (m2/MN). This allows the

script to process the different values as it moves down through the depth of the soil.

In the later versions of the script, step 3 was modified to account for the changes in soil

parameters with depth; the details of this change in step 3 have been demonstrated in figure

10. This modification allows multiple layers to be taken into account quickly and easily with

just a simple adjustment to the layer matrix seen in figure 9. The program assumes that the

assumptions made in the Hansbo (1981) model for well resistance are still valid for a multiple

layer analysis. This was investigated and proven by Onoue (1988) with the testing of a

rigorous finite difference model for multiple layers when compared with Hansbo (1981).

Time Olson Matlab superposition model 100 Steps 1000 Steps 10000 Steps

0.01 0.000165 0.000329 -49.72% 0.000182 -9.04% 0.000166 -0.69%0.02 0.000654 0.000975 -32.96% 0.000686 -4.71% 0.000655 -0.29%0.03 0.001454 0.001929 -24.59% 0.001502 -3.17% 0.001457 -0.19%0.04 0.002558 0.003180 -19.56% 0.002620 -2.39% 0.002562 -0.17%0.05 0.003953 0.004718 -16.21% 0.004030 -1.91% 0.003959 -0.15%0.06 0.005631 0.006535 -13.82% 0.005722 -1.59% 0.005639 -0.13%0.07 0.007583 0.008291 -8.54% 0.007687 -1.36% 0.007592 -0.11%0.08 0.009799 0.010648 -7.97% 0.009917 -1.18% 0.009809 -0.10%0.09 0.012271 0.013255 -7.42% 0.012401 -1.05% 0.012283 -0.10%0.10 0.014990 0.015777 -4.98% 0.015132 -0.94% 0.015003 -0.08%0.11 0.017949 0.018873 -4.90% 0.018103 -0.85% 0.017963 -0.08%0.12 0.021139 0.022196 -4.76% 0.021304 -0.77% 0.021154 -0.07%0.13 0.024553 0.025738 -4.61% 0.024729 -0.71% 0.024568 -0.06%0.14 0.028183 0.029493 -4.44% 0.028369 -0.66% 0.028198 -0.05%0.15 0.032022 0.033452 -4.28% 0.032219 -0.61% 0.032037 -0.05%0.16 0.036064 0.037611 -4.11% 0.036271 -0.57% 0.036079 -0.04%0.17 0.040301 0.041961 -3.95% 0.040518 -0.53% 0.040316 -0.04%0.18 0.044728 0.046497 -3.80% 0.044954 -0.50% 0.044743 -0.03%0.19 0.049338 0.051212 -3.66% 0.049573 -0.47% 0.049353 -0.03%0.20 0.054126 0.056101 -3.52% 0.054369 -0.45% 0.054140 -0.03%

12

Figure 10: Matlab script modification for multiple layers

Step 3:Calculate

Consolidationwithout Ramping For All

Times

For AllDepths

CALCULATECONSOLIDATIONHansbo Formula

U

Store U Values forall calculated

Depths

Steps of Depth dictatedby H / accuracy

Store average Ufor all times inmatrix 'Uavg'

average U

tmax is set in step 1tmin = 1 / accuracy

step length = 1 / accuracy

Uavg is dividedby accuracy toaccount for thesuperposition

Time is also storedin Uavg to allow the

plot of aconsolidation curve

HansboAccounts for

Smear and WellResistance

Uavg

layer=[0 0 0 0 1 6 5.5 0.20 2 12 5.0 0.22 3 20 2.0 0.25]

The Layer input matrix nowhas values associated to more

than one layer to facilitate amulti layer analysis

LayerMonitor

LayerData

Counts up tothe number oflayers in the

system

Changes soilparameters dependanton depth / layer in soil

layermv

kc

Step 4

This assumption for well resistance holds well for small differences between the properties of

the layers (as can be seen in figure 11), but starts to become less accurate when the

differences become larger – in the region kc layer 1 / kc layer 2 = 400. The inaccuracies

created by the well resistance assumption lead to a more conservative result than that given by

the Onoue (1988) model. Figure 11 shows the variation in the degree of consolidation

according to depth, comparing the Hansbo (1981) models for smear and well resistance and

smear alone. In reality the pore water pressures created within the drain would show a

continuous distribution throughout the length of the drain, rather than the stepped distribution

13

shown in figure 11. For the purposes of the program, the degree of consolidation is calculated

at all depths, accounting for the variation in soil parameters as shown in figure 11. This

distribution is then averaged and stored in the matrix ‘Uavg’ with the corresponding time of

interest, to allow for the superposition influence of the ramped loading.

Figure 11: Variation of the effect of Well Resistance in a multilayered soil.

0

2

4

6

8

10

12

14

16

18

20

0 0.05 0.1 0.15 0.2 0.25 0.3

Degree of Consolidation U

Dep

th (m

)

Well Resistance Without Well Average

Calculation assumptions:Layer 1: Kc = 5.5 m2/yr mv = 0.20 m2/MNLayer 2: Kc = 4.5 m2/yr mv = 0.22 m2/MNLayer 3: Kc = 3.0 m2/yr mv = 0.25 m2/MNTime of interest 0.1 years

3.2.4 Vertical Drainage

One of the assumptions that has been made for the multiple layer case is that any vertical

drainage is ignored. This is due to the uncertainty of the coefficient of vertical consolidation

over the entire soil body, as it cannot simply be combined into a single average value for the

soil. Also there is no method of easily combining the effects of vertical and radial drainage for

multilayered soils as the Carrillo (1942) method can only be applied to homogenous cases.

Another reason is the minimal effect that vertical drainage has in most drained cases- the

14

radial drainage distance is massively shorter than the vertical drainage distance for 99% of

PVD problems. Again this assumption leads to a conservative solution, as any vertical

drainage will only add to the safety factor of the design. This is not the case however for a

homogenous soil where due to the nature of the soil, the coefficient of vertical consolidation

can be obtained with relative accuracy. The contribution made by vertical drainage can

therefore be calculated much more accurately, and thus its effect on the overall consolidation

can be taken into account with the aid of the Carrillo (1942) formula.

As there are no effects of smear and well resistance in the case of vertical drainage there is no

need to apply a version of the superposition method to this case. Instead, Olson’s (1977)

formula for ramped vertical drainage can be used to calculate directly the contribution to the

overall consolidation. Figure 12 shows an example of the contributions of both radial and

vertical drainage to the consolidation process.

Figure 12: Contribution to the consolidation process by vertical drainage.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (Years)

Deg

ree

of C

onso

lidat

ion

(U)

Average Consolidation Curve

Radial DrainageVertical DrainageCombined Drainage

Figure 12 shows the contribution of vertical drainage when the spacing of the wells is much

shorter than the vertical drainage distance. (Spacing (D) =1.5m, Depth (H) =10m). If the

spacing becomes greater then the effect of vertical drainage is much more pronounced, to the

point where the vertical drainage can have more of an effect than the radial drainage. In

reality this would never occur as there would be no realistic benefit from installing the drains

at such great centres. Figure 13 shows the effects of increasing the spacing of the drains and

15

the effect this has on the relationship between the vertical and horizontal drainage. So long as

the Uv/Uh value is less than 1 the effect of Radial Consolidation outweighs that of Vertical

Consolidation, and it is worth while implementing a drainage scheme. This equates to a

maximum viable spacing for a 10m deep soil of 5.2m and 6.4m in the case of the 20m deep

soil. Bearing in mind that these spacings correspond to n values of 79 and 98 respectively

they are way outside the normal constructional limits (for Cofra, the maximum viable

installation n value is 38). In the lower regions of figure 13 there is little difference between

the different depth plots; this adds weight to the argument that if the spacing can be

minimised the effects of vertical drainage can also be minimised.

Figure 13: Relative Effect of Vertical Drainage

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Drain Spacing - D (m)

Uv/

Uh

at T

=0.7

5 ye

ars

Depth = 20m

Depth = 10m

3.3 Program Validation

To make the program a valuable tool for use in practice it must be tested against current

models to check the validity of the analysis. To some extent individual sections have already

had to be validated to allow for the program to progress. In the initial stages to eliminate any

discrepancies within the Matlab script, the results output were verified by hand to check

accuracy and to prove the script was functioning as it was meant to. The modelling of Olson’s

(1977) equations allowed the testing of the superposition method for ramped loading; this

proved that the accuracy of the model was very high indeed with an average 1.5% error. This

can be seen in figure 8 and table 2. The next step in the validation process was to check the

16

finished model was accurate. Unfortunately there are few authors who have published work in

the same area; that of writing programs to model all the associated factors along with multiple

layers. Onoue (1988) published graphical outputs from a finite difference program, modelling

all the factors and more (such as vertical drainage in the individual layers), but these are

impossible to emulate as the input parameters and assumptions are unknown. Leo (2004)

wrote a spreadsheet program to model all the factors except for the consideration of multiple

layers. The input values and assumptions for the results calculated were given in the paper

and this allowed the direct comparison of the Leo (2004) model to that of the author.

Table 3: Assumptions and Values used in the Leo (2004) analysis

ch = 2.25 m2/year h = 4.5 m kh / ks = 3 re = 0.4725 m ch / cv = 3 rs = 0.075 m rw = 0.033 m s = 1.6

With the values in table 3 the author used the matlab script created for a single layer analysis

to model the results generated by Leo (2004). One thing to note is that as the soil is relatively

shallow (4.5m) Leo (2004) has not included the effects of well resistance in the analysis. To

this effect the author has used the ideal drain model (infinite drain permeability) to more

accurately compare the models.

Table 4: Comparison of the Leo (2004) results to those of the author

Ramp Ramp No Well Ramp Inc Well Time (yrs) Leo (2004) Present Relative Error Present % Difference 0.167 0.389 0.394 1.269% 0.386 -2.125% 0.250 0.632 0.634 0.284% 0.623 -1.734% 0.458 0.893 0.894 0.078% 0.886 -0.858% 0.708 0.975 0.976 0.062% 0.973 -0.308% 0.958 0.994 0.994 0.040% 0.993 -0.111% tc = 0.167 0.083 0.236 0.237 0.590% 0.232 -2.239% 0.292 0.783 0.784 0.166% 0.775 -1.265% 0.542 0.950 0.951 0.095% 0.946 -0.518% 0.792 0.989 0.989 -0.030% 0.987 -0.182% tc = .083 0.083 0.448 0.454 1.300% 0.445 -1.977% 0.292 0.876 0.878 0.182% 0.870 -0.920% 0.542 0.972 0.972 -0.010% 0.969 -0.341% tc = .208 Averages 0.335% -1.048%

Table 4 demonstrates the accuracy of the program, with the maximum relative error being

17

1.3% and the average error for all results being only 0.335%. The graphical check in figure 14

shows without a doubt the high degree of correlation between the models. Figure 14 also

demonstrates the validity of Leo (2001) in disregarding the effect of well resistance for the

analysis as it had a maximum influence of 2.1% and an average influence of only 1%. If the

soil in the analysis was deeper or the resistance of the drain very low then well resistance

would have a much more influential part to play.

Figure 14: Graphical comparison between Leo (2004) and the present paper

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

Time (years)

Deg

ree

of C

onso

lidat

ion

(%)

Leo 2004 Present Paper Present with Well

From the program checks done above it can be concluded that the single homogenous layer

soil analysis, taking into account well resistance, smear, ramped loading and vertical drainage,

is 99% accurate in testing. The expansion of the model for multiple layers is valid (Onoue

1988) and although it cannot claim the same degree of accuracy due to inherent errors in the

assumptions and the loss of the vertical drainage, it can claim to always give a conservative

result for the cases analysed. There are no precedents to test the multiple layer model against

to ascertain its exact accuracy.

3.4 Program Optimisation

3.4.1 Cost analysis of the problem

The brief for this project is to design a program that optimises the use of vertical drains in a

practical installation. For this to work from a design point of view some extra data about the

installation is required, such as the area, target cost of the project and maximum time

18

allowable for the required degree of consolidation. The values for the installation costs in

table 5 were provided by Mike Drew of Cofra UK Ltd.

Table 5: Installation Costs N value Material Cost Installation Cost 17 – 20 £0.20 / m £0.34 / m 20 - 30 £0.20 / m £0.38 / m 30 - 38 £0.20 / m £0.48 / m Deep Layers of soil that require pre-drilling £0.95 / m Mobilisation of Equipment to site £8000

Knowing the area of the installation, the depth of the soil, the time allowed and the degree of

consolidation required, it is possible to calculate the spacing required to achieve the

associated degree of consolidation. From the spacing the number of drains required in the

installation can be calculated and using the installation costs in table 5 this can be turned into

a cost for the project. This approach enables the program to find the optimal spacing to

achieve the requirements provided, and thus the lowest cost solution to the problem. The only

problem with this approach is the variance in the soil parameters, as a small difference from

the calculated solution to those in practice would throw out the answers and not necessarily

lead to the project completing on time. For example a small decrease in Kc due to variation

across the site would lead to the required degree of consolidation not being reached in the

allotted time. The choice of the values for the soil parameters is left at the discretion of the

Geotechnical Engineer based upon the results of a ground investigation. Using a method of

probabilistic analysis the variation in soil parameter could be automatically accounted for,

thus giving the Engineer a method of backing up his instincts.

3.4.2 Probabilistic analysis

What was needed was a method of taking into account the variation in the values of the soil

parameters quickly and easily without needing any more data about the soil conditions

surrounding the installation. As the aim is to provide a design tool it was deemed by the

author unrealistic to expect the user to have to provide any amount of input data extra to that

required by the usual methods of ground investigation, as this would make the program more

complicated and require additional cost to implement at a design stage.

The Programme Evaluation and Review Technique or PERT is one such method. PERT was

originally devised to provide a time estimate based on best, expected and worst case scenarios

with no other inputs into the method, where a = minimum possible time, m = expected time

and b = maximum time. The PERT analysis assumes that the distribution of variables

corresponds to a beta distribution.

19

Expected mean duration (central tendency), 6

4 bmate++

=

Standard deviation (spread), 6

)( abte

−=σ

Variance, 2tev σ=

Figure 15: Example of a Beta Distribution

Figure 15 shows the standard form of the beta

distribution given the values of a, b and m.

This is unlike the standard normal distribution

as it does not have to be symmetric about the

mean, this allows it to take better account of

skewed input values.

To fully allow a probabilistic analysis a way

of transforming the input values into a

probability is required, in the case of the PERT analysis this is the ‘Z’ value, where:

and TS = Time required, TE = Time expected.

P is the probability that the time taken for completion of the consolidation will

be less than or equal to TS , these values are given in table 6.

Table 6: Z-values for beta distribution

This data allows the user to predict the spacing of

the drains required to be probabilistically sure that

the consolidation will complete in the allotted

time. A probability of 0.5 would mean that the

installation would be equally likely to over run as

it would to be on time, for this analysis the Z

value is 0, therefore TS = TE. This is the equivalent

of the standard analysis, with no probabilistic

element, as no direct account of the spread of the results is taken into account.

To allow the PERT analysis to be performed three estimates for the possible duration need to

be calculated - the best, expected and worst case scenarios. These can be calculated by

allowing the user to enter a spread of values for the soil parameter Kc - a maximum,

minimum and average value. When the consolidations are calculated using the different

values of Kc it will give a spread of U values corresponding to the best, expected and worst

case scenarios.

Z P 0.0 0.50 0.3 0.62 0.6 0.73 0.9 0.82 1.3 0.90 1.5 0.93 2.0 0.98 2.5 0.99 3.0 1.00

te

ES TTZσ−

=

20

Figure 16: The updated matlab script for variable input values

consolsimple_fast.mprogram call

Step 1:Set Limits of

Accuracy

InitialValues

supplied byrisk.m drain

layertim

s

d

kckcdqw

D

tc

Discharge Capacity of Drain (m3/yr)

Time of Interest (years)

Matrix of Soil Parameters

End Conditions of Drain

Construction Period

Effect of Smear Zone

Reduction in permeability of Smear Zone

Radius of effect of the drain (m)

Drain diameter (m)

tmax

acc

Based on timvalue

This makes sure theprogram runs quickly

Step 2:Set Boundary

Conditions

If 'drain' =single

If 'drain' =doubleL=H L=H/2

Step 4:Calculate

Consolidationwithout Ramping

For AllTimes

For AllDepths

CALCULATECONSOLIDATIONHansbo Formula

U

Store U Values forall calculated

Depths

Steps of Depth dictatedby H / accuracy

Store average U forall times in matrix

'Uavg'average U

tmax is set in step 1tmin = 1 / accuracy

step length = 1 / accuracy

Uavg is divided byaccuracy to

account for thesuperposition

Time is also storedin Uavg to allow the

plot of aconsolidation curve

HansboAccounts for

Smear and WellResistance

Step 5:Uavg is

superimposed ontoits self to account

for ramped loading

The ramp load isapplied in very smallsteps giving a very

close approximation toa linear loading pattern

Uavg

The degrees ofconsolidation associated

with the input 'tim' arefound within the Uavgmatrix for each case.

If the degree of consolidation cannotbe found the program interpolates

between the nearest values to obtainthe value (when input is a fraction)

Ouput returned torisk.m to allow

PERT analysis ofsoil data

averageU

Step 3:Set upper, lowerand average kc

values

layer

Layer contains newvalues relating to the

maximum, minimum andaverage kc values

Step 6:Olson's ramped

equation for verticaldrainage is used for

single layer

Step 7:Carrillo's formula is

applied to combine theeffects for vertical and

radial drainageavgUmin avgUmax

a

bm

consolmulti_fast.mincludes an extra loop toaccount for the changing

kc an mv values withdepth. Steps 6 and 7 are

also removed.

21

This was written as an extension to the matlab script to allow for the variation in the soil

parameters, which can be seen in figure 16. These values can then be entered into the PERT

analysis to give a degree of confidence in the end consolidation completing on time- fig 17.

Figure 17: PERT analysis matlab script

risk.mprogram call Initial

Values drainlayer

tim

s

d

kckcdqw

target U

tc

Discharge Capacity of Drain (m3/yr)

Time for Completion (years)

Matrix of Soil Parameters

End Conditions of Drain

Construction Period

Effect of Smear Zone

Reduction in permeability of Smear Zone

Required Degree of Consolidation

Drain diameter (m)

Iteration Loop whereNew D value is

calculated from thedifference between

expected U andtarget U

Arbitrary Dvalue

consolsingle_fast.mprogram call

consolmulti_fast.mprogram call

SINGLE LAYER MULTI LAYER'layers'==1 'layers'==2+

averageU avgUmaxavgUmin

Step1:Run scripts with

Arbitrary D value toestablish base points

Step 2:Use PERT analysis

to calculate theexpected U

Check: expected U =

target Uexpected U

New D value

No

Step 3:Use PERT P and Ztable combined withformula to changetarget U to account

for variable data

For Step 2 PERTanalysis P=0.5, Z=0therefore expected U

= target U

WarningMessageDisplayed

NotPossible

Yes

D

consolsingle_fast.mprogram call

consolmulti_fast.mprogram call

SINGLE LAYER MULTI LAYER'layers'==1 'layers'==2+

Step4:Run scripts with

Arbitrary D value toestablish base points

For All PValues

probable.mprogram calltarget U

Last D valuecalculated frovidesexcellent start pointfor further analysis

Step 5:Use PERT analysis

to calculateassociated U

Check: associated U =

target U

No WarningMessageDisplayed

Not possible

Step 6:U value is stored

along with spacing ina matrix

Yes

Output

Degree

Output stores targetU, spacing (D) and

Probability

Degree storesaverage U, avgUmax

and avgUmin

Step 7:Cost analysis is

run using Spacing

Plot of Cost versus Risk(Probability) is produced

22

The higher the probability of a timely completion, the closer the spacing needs to be and the

closer the spacing, the more drains are needed to cover the installation area. As the number of

drains goes up so does the cost associated. This link allows the plot of a risk (probability)

versus cost curve (figure 18) to demonstrate how much extra would need to be spent on a

project to greaten the chances of a timely completion.

Figure 18: Risk versus Cost curve from PERT analysis

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 11.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8x 105

Probability

Cos

t (£)

Target U = 0.95Completion Time = 1 yearArea of Project = 30000m2Kc (m2/yr) min= 4, avg= 5.5, max= 6Calculated Necessary Spacing = 1.20 m

4 PARAMETRIC STUDIES

To some degree during the writing of the program some parametric studies have been

completed to assess the individual effects of factors affecting the consolidation process. The

full possibilities of using parametric studies could not be fully investigated until the matlab

script for the program was fully completed. Here the author has used parametric studies to

further explore the factors affecting the consolidation process.

Figure 19 shows the effects of well resistance with increasing depth by comparing the time to

reach 90% consolidation between an ideal drain (no resistance) and one with well resistance.

The figure demonstrates the massive effect that well resistance can have on the consolidation

process with a maximum of a 600% increase in the time required (this is the most extreme

case with the drain being 60m in length). What is interesting is the exponential behaviour the

curve demonstrates with the effects being much less noticeable with drains up to a length of

10m (effect at 10m = 15% increase).

23

0 10 20 30 40 50 601

2

3

4

5

6

7

Drain Length (m)

Th90

(fin

ite d

rain

per

mea

bilit

y) /

Th90

(ide

al d

rain

)

n = 25n = 50

qw/kh = 400 m2

1

1.2

1.4

1.6

1.8

2

2.2

2.4

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Smear Ratio (rs/rw)

Th90

(sm

ear)

/ Th9

0(no

sm

ear)

Kc/Kc' = 2

Kc/Kc' = 3

Kc/Kc' = 4

Hansbo (1981)

Hird & Moseley (2000)

Figure 19: The effects of well resistance on the consolidation process

Many authors choose to disregard well

resistance for shallow soils. Figure 19

demonstrates the significance of this, for

example in the Leo (2004) paper well

resistance was ignored in a soil of depth

4.5m. The effects of the well resistance

in a soil so shallow are minimal as

demonstrated by figure 19 and by the

author again in table 4.

While the influence of smear is

universally accepted, the magnitude of

its defining factors S and Kc/Kc’ are not.

Figure 20 shows the effect that changing

the smear parameters has on the overall

rate of consolidation. A visual

comparison has been drawn between the

parameters recommended by Hansbo

(1981) and Hird & Moseley (2000).

Figure 20: The effects of smear on the consolidation process

24

Figure 20 demonstrates the point that a small change in the parameters can have a large effect

on the rate of consolidation, with a 7% difference between the Hansbo (1981) and Hird &

Moseley (2000) factors.

Vertical drains rely on the principle that if you decrease the spacing of the drains then the

consolidation process will accelerate. This said there is a limit to how many drains can be

installed into a soil before it becomes the horizontal permeability of the soil itself that

becomes the limiting factor and not the spacing of the drains. The author conducted a

parametric study into the spacing (n value) of drains and its effect on the rate of consolidation.

Figure 21: Effect of spacing on the consolidation process

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60 70 80 90 100

n (d/D)

Effe

ct o

n C

onso

lidat

ion

(U/U

n)

Figure 21 shows a flat section in the curve between n=1 and n=17. In this area any decrease in

the spacing has a minimal effect on the rate of consolidation, and thus is uneconomical to

install. Of course figure 21 is only valid for a specific set of initial conditions set in table 1,

although the general trend will always be the same. Although technically it would be possible

to do parametric studies into the effect of ramping and multiple layers the outcomes are much

more dependant on the input parameters provided. For example we know the effect of ramped

loading is directly proportional to the construction period as they are intrinsically linked. This

makes their value as an educational tool much less than the parametric studies in figures 19-

21.

25

5 PROGRAM DISTRIBUTION

Previously the matlab scripts that have been used to model the problem have been command

line based, that is there is no visualisation to the script. This approach is used for the

development stages of the program as it allows the author to rapidly change the scenarios and

calculation methods utilised within the scripts. Users other than the author may have problems

in understanding the notation and methodology of this approach, which is where a Graphical

User Interface (GUI) comes in. The GUI allows the input to have a visual element to it,

making the use of the program much simpler and more intuitive. As the task is to create a

design tool the GUI plays an important part in the ease and speed of the use of the program.

To accompany the program a manual would also be written to aid in the rapid use of the built-

in features.

Figure 22: Principle of the Graphical User interface

verticalc.mprogram call

Graphical UserInterface

InitialValues

Valuesinput

Assumptionsinput

"OPTIMISE"button

Switches:'Drainage''Spacing''Layers'

"CALCULATE"button

risk.mprogram call

SINGLE LAYER MULTI LAYER

consolsingle_fast.mprogram call

consolmulti_fast.mprogram call

olsonvert.mprogram call

probable.mprogram call

OPTIMUM SPACING +ASSOCIATED U VALUES FOR

PROBABILITIES

costing.mprogram call

TRI SQUARE

Cost vs RiskPlot

OPTIMAL VALUES FORANALYSIS

chosen by probability

Optimum SolutionCreation

In depth Analysis

'layers'==1 'layers'==2+

'spacing'=='tri' 'spacing'==0

MULTI LAYER'layers'==2+

SINGLE LAYER'layers'==1

'acc'=100

consolsingle.mprogram call

consolmulti.mprogram call

olsonvert.mprogram call

Input ValueCheck

DIFFERENT SAME

speedcalc.mprogram call

OUTPUT CURVES ANDVALUES

ConsolidationCurve Plot

26

Figure 22 demonstrates the role that the GUI plays in the interaction with the programmed

matlab scripts. The two main functions can be seen clearly, with the ‘optimise’ function on

the left which acts to calculate the most suitable calculation parameters for the in-depth

‘calculate’ function on the right of the diagram. The ‘calculate’ function can also be used

separately to analyse predictions and other case histories. Figure 23 shows a screenshot of

how the GUI looks when run under Windows XP, with the main area for the generated plots

to the top right, and the area for input parameters around to the bottom and left of the screen.

Figure 23: GUI Initial Screen – Generic Version

Two versions of the GUI were created. Figure 23 is the generic version, which has no

assumed values within its programming, so can be used to analyse any combination of

parameters. Figure 24 is designed for use by Cofra UK, with the values for the drain

parameters pre-programmed into the software, along with Cofra’s own costing data from table

5. From a design point of view customised versions of the software can be created specifically

for a company with their own values inserted into the program, as this makes the program

easier to use for employees as the parameters for the companies’ products do not need to be

looked up. The differences between figure 23 and 24 occur in the ‘Drain Properties’ box with

the installation costs being removed from figure 23 and the drop down menu for drain choice

being inserted into figure 24.

27

Figure 24: GUI Initial Screen – Cofra UK Version

5.1 Worked Example using the Verticalc Program

For the purposes of this paper the author has decided to insert a worked example showing the

capabilities of the program at modelling case histories. In this case the author has decided to

verify the results for the Leo (2004) case, shown in table 4.

Table 6: Converted parameters from the Leo (2004) Analysis

Kc = 5.0 m2/year Depth = 4.5 m Mv = 0.22652 m2/MN D = 0.945 m ch / cv = 3 kh / ks = 3 d = 0.066 m s = 2.273 t = 0.25 years Tc = 0.167 years S = 0.9m Arrangement = Triangular Drainage = Single qw = inf

As the Leo (2004) analysis does not account for well resistance, putting in an infinite value

into the drain permeability (qw) accounts for this. To model this analysis the Calculate

function will have to be used, as the parameters are already defined and are in no need of

optimising.

28

Figure 25: GUI output showing Leo (2004) model

The program returns an average consolidation of 0.634 which is the same as the value in table

4, column 3, row 4.

For the full details of how to use the program please consult the program user’s guide in the

appendix. For the purposes of program user’s guide, the working of the generic version will

be described as this involves more steps than the Cofra UK version.

6 CONCLUSIONS

6.1 Theoretical vs. Practical

The aim of this project was to create a design tool to speed up the calculation of vertical drain

installations in practice. In this requirement it was a complete success, as the program has

developed its own intuitive interface (the GUI) rather than using the usual confines of an

excel spreadsheet. While the theory behind the model is relatively simple, it uses Hansbo

(1981) and Olson (1977) methods, which are well known and well used benchmarks for the

industry. Cofra’s own in-house software for calculating vertical drains solely depends on

these same equations, but without any attempt to manipulate them to account for more than

one factor at any one time, i.e. it uses Olson (1977) for ramped loading and Hansbo (1981) for

29

the effects of smear and well resistance.

What makes the author’s program different to any before it is its ability to combine the effects

of many different factors into a quick and easy to use analysis, with the added feature of being

able to calculate the probability of a timely completion of the consolidation process.

The method does have theoretical drawbacks, for example the uneven distribution of well

resistance pressures through multiple layers. However the author has backed up the

assumptions used with results from more rigorous models such as Onoue (1988), who has

modelled these assumptions and compared them with the more basic solutions (Hansbo,

1981) used in the author’s model.

6.2 Further Geotechnical Developments

The author has taken into account the main factors in the designing of a vertical drain

installation. This said there are many other factors that can affect the consolidation process,

such as creep and the inclusion of vertical drainage in multiple layers. With time these could

be included into the program to allow a more in-depth analysis of the soil, although these

were seen as being outside of the scope of the current project. Another possibility would be

for the program to predict the increase in strength in the soil from the associated

consolidation, although this would require many more input parameters to predict accurately.

To allow for a degree of confidence to be placed in the program, a series of tests could be run

on case histories to compare the results predicted by the program to those in practice.

6.3 Further Computing Developments

The author can see a near infinite amount of scope for enhancements to the user interface and

possibilities associated with it. For example a second GUI could be implemented to deal with

the data directly from the ground investigation – positions of the samples across the site and

their associated values. This would allow the model to begin to appreciate the three

dimensional nature of the problem and be able to suggest different schemes for different areas

on the site where the parameters vastly differ. The ramped loading could also be expanded to

a number of smaller ramps, rather than a single one, with options for small steps or a linear

approach to the load increments.

30

7 REFERENCES

Almeida, M. S. S., Santa Maria, P. E. L., Martins, I. S. M., Spotti, A. P., Coelho, L. B. M.

(2000). Consolidation of a very soft clay with vertical drains. Geotechnique 50, No. 6,

633-643.

Barron, R. A. (1948). Consolidation of fine-grained soils by drain wells. Trans. ASCE, Vol.

113, Paper No. 2346.

Bergado, D. T., Balasubramaniam, A. S., Fannin, R. J., Holtz, R. D. (2002). Prefabricated

vertical drains (PVDs) in soft Bangkok clay: a case study of the new Bangkok

International Airport project. Can. Geotech. J. 39. 304-315.

Borges, J. L., (2004). Three-dimensional analysis of embankments on soft soils incorporating

vertical drains by finite element method. Computers and Geotechnics 31, 665-676.

Chai, J. C., Miura, N. (1999). Investigation of Factors Affecting Vertical Drain Behaviour. J.

Geotechnical and Environmental Engineering, March, 216-226

Chu, J., Bo, M. W., Choa, V. (2004). Practical considerations for using vertical drains in soil

improvement projects. Geotextiles and Geomembranes, 22, 101-117.

Hansbo, S. (1981). Consolidation of Fine-Grained Soils by Prefabricated Drains. Proc. 10th

Int. Conf. Soil Mechanics Found. Engng, Stockholm 3, 677-682.

Hansbo, S. (2001) Consolidation equation valid for both Darcian and non-Darcian flow.

Geotechnique 51, No. 1, 51-54.

Hawlader, B. C., Imai, G., Muhunthan, B. (2002). Numerical study of the factors affecting the

consolidation of clay with vertical drains. Geotextiles and Geomembranes 20, 213-239.

Hird, C. C., Sangtian, N. (2002). Model study of seepage in smear zones around vertical

drains in layered soil: further results. Geotechnique 52, No. 5, 375-378.

Hird, C.C., Moseley, V. J. (2000). Model study of seepage in smear zones around vertical

drains in layered soil. Geotechnique 50, No1, 89-97.

Holtz, R. D., Jamiolkowski, M. B., Lancellotta, R., Pedroni, R. (1991). Perfabricated vertical

drains - design and performance. Construction Industry Research ad Information

Association Ground Engineering Report. Oxford: Butterworth-Heinemann.

Hong, H. P., Shang, J. Q. (1998). Probabilistic analysis of consolidation with prefabricated

vertical drains for soil improvement. Can. Geotech. J. 35, 666-677.

Horn, R. (2005). CIV 371 Construction Management Course Notes - Construction

Administration and Planning Handout 2, 1.

Indraratna, B., Redana, I. W. (2000) Numerical modelling of vertical drains with smear and

well resistance installed in soft clay. Can. Geotech. J. 37, 132-145.

31

Leo, C. J. (2004) Equal Strain Consolidation by Vertical Drains. J. Geotechnical and

Geoenvironmental Engineering, March, 316-327.

Moseley, M. P., Kirsch, K. (2004) Ground Improvement 2nd Edition. Spoon Press. Chapter 1,

4-56. (Contributed by Hansbo, S.)

Nash, D. F. T., Ryde, S. J. (2001). Modelling consolidation accelerated by vertical drains in

soils subject to creep. Geotechnique 51, No 3, 257-273.

Olson, R. E. (1977). Consolidation under Time Dependant Loading. J. Geotechnical

Engineering Division, GT1, 55-60.

Onoue, A. (1988). Consolidation of multilayered anisotropic soils by vertical drains with well

resistance. Soils and Foundations, Vol. 28, No. 3, 75-90.

Seah, T. H., Tangthansup, B., Wongsatian, P. (2004). Horizontal Coefficient of Consolidation

of Soft Bangkok Clay. Geotechnical testing Journal, Vol. 27, No. 5, 1-11.

Sharma, J. S., Xiao, D. (2000). Characterisation of smear zone around vertical drains by large

scale laboratory tests. Can. Geotech. J. 37, 1265-1271.

Tang, M., Shang, J. Q., et al., Almeida, M. S. S., et al. (2002) Discussion Vacuum preloading

consolidation of Yaoqiang Airport runway. Geotechnique 52, No. 2, 148-154.

Yeung, A. T. (1997). Design Curves for Prefabricated Vertical Drains. J. Geotechnical and

Environmental Engineering, August, 755-759.

Yoshikuni, H., Nakanado, H. (1974). Consolidation of soils by vertical drain wells with finite

permeability. Soils and Foundations, Vol. 14, No. 2, 35-46.

Zhu, G., Yin, J. H. (2001). Consolidation of with vertical and horizontal drainage under ramp

load. Geotechnique 51, No. 4, 361-367

Zhu, G., Yin, J. H. (2001). Design charts for vertical drains considering construction time.

Can. Geotech. J. 38, 1142-1148.

Zhu, G., Yin, J. H. (2004). Consolidation analysis of soil with vertical and horizontal drainage

under ramp loading considering smear effects. Geotextiles and Geomembranes 22, 63-74.

32

8 APPENDIX 1: PROGRAM USER’S GUIDE

8.1 Installation of Verticalc

This guide will demonstrate how to run the program under Windows XP, please contact the

author if you require a version of the program to run under any other operating system.

Once the install has completed you are now ready to use the Verticalc program.

Figure 8.1: Start-up Screen

Once the program has run successfully you should be looking at a screen that looks like that

in figure 8.1. This is the main screen from where all the facilities within the Verticalc program

are available.

Step 1: Run ‘MCRinstaller.exe’ – This installs the matlab component runtimes

which allows you to run the Verticalc program.

Step 2: Run ‘verticalc_genericm.exe’ – This runs the Verticalc program on your

system. During future uses of the program only Step 2 needs to be

followed.

33

8.2 The Graphical User Interface Explained.

The Graphical User Interface or GUI is the heart of the use of the program. This section of the

manual will hopefully explain the basic capabilities and how to go about using Verticalc.

Figure 8.2: The GUI breakdown

Figure 8.2 shows the different components of the GUI:

• Radio Buttons allow exclusive choices of variables- an either, or scenario.

• Input Boxes allow the input of data through selecting the box and typing in the new

value.

• Option Panel refers to the sections of the GUI, this allows for better referencing of the

individual components within the GUI.

• Chart Plot Area is where the program generates its visual plots of the consolidation

curves and the cost versus risk curve, more on these later.

• Drop Down Menus allow the choice of many different factors from within the

programmed choice, but only one may be selected at any one time.

• Execute Buttons – click on these to run sections of the analysis suite of the program.

The program itself has two main functions accessed by the Execute Buttons in the Output

Option Panel: Optimise and Calculate.

Chart Plot Area

Input Boxes

Drop Down Menu

Radio Button

Execute Buttons Option

Panel

34

8.3 Entering Values

By selecting any white textbox with the mouse the initial values of the program can be

changed. All the parameters are self explanatory, although it should be noted that the reason

for the 3 values for Kc is for use in the probabilistic analysis. The maximum, minimum and

average values from the site investigation should be input.

Figure 8.3: Optimisation Output

For users who wish to disregard the probabilistic element of the program, input the same

value in all three boxes, and the probabilistic analysis will be disabled.

The Optimise Function.

This function allows the user to find the optimal spacing for the arrangement of

vertical drains, given the soil parameters are input into the program. The Optimise

function is capable of generating parameters to allow the Calculate function to run

and incorporates an element of Probabilistic analysis.

The Calculate Function.

This function allows the user to plot the consolidation curve for a given set of soil

parameters, these can be manually input, or the user can use the Optimise function

to find the best parameters for the soil.

35

Once all the values are in place press the Optimise button to generate an output, as seen in

figure 8.3. By comparing the Spacing Option Panels of figures 8.2 and 8.3 you can see that

the optimise function has created a new set of values for the drain spacing, now based on the

soil parameters and the probabilistic analysis.

8.4 Using the Probabilistic Analysis Tools

The values created in the Spacing Option Panel are only valid for a specific Probability,

which is defined in the Risk Option Panel; the default setting is a 50% probability.

The plot of probability versus cost allows you to decide on the correct balance between

financial cost and the element of risk. To be 100% sure of the consolidation occurring in the

stated amount of time you will need to spend more capital than if you only needed to be 62%

sure. By changing the probability in the Risk Option Panel, the spacing details are updated

accordingly, as is the cost and degree of consolidation in the Output Option Panel.

Figure 8.4: Using the Probabilistic Analysis

Figure 8.4 shows the Probability options in the drop down menu, here we can see by selecting

the 93% probablilty the Spacing values have changed, as has the information in the Output

Option Panel. If any of the details are changed in the input parameters the Optimise function

needs to be re-run to take into account the changes.

36

Once the Probability has been chosen, the Calculate function can be run to give the

consolidation curve for given soil parameters and generated drain spacing.

8.5 Using the Calculate Function

The Calculate function can be used in two different ways:

Once the input parameters have been decided, run the calculate function to give an output like

that in figure 8.5.

Figure 8.5: The Calculate Function Output

The Stand-Alone Method:

When you have a pre-defined spacing and know the soil parameters. This method

can be used to compare results from different approaches with those of the

program.

The Optimised Method:

After the Optimise function has been run you can use the Calculate function to run

an in depth analysis on the solution giving the consolidation curves and a higher

degree of accuracy.

37

A common mistake is to forget to select the drain arrangement; in this case the program will

pop up a warning like the one in figure 8.6 to ask you for the additional data.

Figure 8.6: Error Message

Pressing OK then selecting a drain spacing

will solve the problem.

Once the Calculate function has run you can

see the consolidation curve in the plot area of

the program as in figure 8.5.

8.6 Accounting for multiple layers

The program easily accounts for layered soils, by just inputting the values for the different

layers into the layers option panel, the program will automatically take the different values

into account. The resulting output is much the same as figure 8.5 but as the program no longer

accounts for vertical drainage there is only a single plot in the chart plot area.

Figure 8.7: Multiple Layer Output

As you can see in figure 8.7 the Layers Input Panel now has an extra set of data for the 2nd

soil layer, extending down from 10-15m depth. The Optimise function also works for multiple

layers using the same method as mentioned here.

38

8.7 Advanced Features

The advanced features are contained within the Plot View input panel. The Cost and Consol

buttons allow you to switch between the consolidation plot and the cost versus probability

plot, this is very useful if you need to change the probability used once a calculation has been

run without having to re-run the Optimise function.

The other feature in the Plot View input panel is the Comparison drop down menu. This

allows you to compare the effects of changing the input parameters or the probability visually

in the chart plot area.

Figure 8.8: Comparison Feature

Figure 8.8 is comparing the effects of changing the drainage conditions from ‘single’ drainage

conditions to ‘double’, this has the effect of speeding up the consolidation as the vertical

drainage has much more of an effect in the double drainage conditions.

To compare two sets of Parameters:

Run the Calculate function on the first set of soil parameters, then change the

parameters and re-run the Calculate function taking care to make sure the drop

down menu is set to Comparison.

39

This will generate a plot much like that in figure 8.8.

To generate a clean plot without comparing result to the previous plot, ensure that the

‘Individual’ option is selected from the drop down menu in the Plot View option panel.

8.8 Troubleshooting

The only real problems can arise in the Optimise function where there is no solution for the

set of parameters chosen. Or the wrong kind of inputs are used in the program. Examples of

the error messages are shown in figures 8.9 and 8.10.

Figure 8.9 Optimisation Error Message

For example, if you accidentally set the ramped

loading for a longer period than the target

completion time. In such cases the program will

pop up an error message like that in figure 8.9.

Figure 8.10: Alphanumeric Error

If you accidentally input a non-numeric

character into an input box that cannot handle it

a error message like that in figure 8.10 will

appear. The title ‘Construction Period’ refers to

the input box that needs its value retyping.

To compare two Probabilities:

Run the Optimise function to generate the cost versus probability curve and

choose a probability of the initial calculation. Run the Calculate function then

switch back to the probability curve by pressing the ‘Cost’ button. Select a new

probability and then switch again to the Consolidation plot by pressing the

‘Consol’ button. Run the Calculate function again, ensuring that ‘Comparison’ is

displayed in the drop down menu window.

40

9 APPENDIX 2: USEFUL DATA

9.1 Mebradrain Specifications: 28-05-2005

Physical Properties Standard Unit MD7007 MD88M MD88H MD88HD

Configuration

Channels 38 44 44 44 Material PP PP PP PP Weight g/m 75 70 85 110 Width mm 100 100 100 100 Thickness mm 3.0 3.0 3.5 5 Mechanical properties Tensile Strength EN 10319 kN 2.2 1.8 2.2 4.2 Elongation EN 10319 % 60 40 60 60 Elongation at 0.5 kN EN 10319 % 2 2 2 1.5 Grab strength ASTM D4632 N 970 580 970 970 Bursting Strength ASTM D3785 kPa 1000 900 1000 1000 Tear Strength ASTM D4533 N 270 180 270 270 Hydraulic properties drain In-plane flow cap. qp(10/1.0) EN 12958 l/m.s 1.1 0.8 2.7 In-plane flow cap. qp(100/1.0) EN 12958 l/m.s 0.75 0.68 2.5 In-plane flow cap. qp(350/1.0) EN 12958 l/m.s 0.59 0.57 1.8 Discharge cap. qw(300/0.1) EN 12958 10-6 m3/s 49 50 70 155 Discharge cap. qw(500/0.1) EN 12958 10-6 m3/s 1 14 20 25 D.C. buckled qwb(200/0.1) EN 12958 10-6 m3/s 60 55 82 130 Transmissivity θ(10/0.1) ASTM D4716 10-3 m2/s 1.2 0.6 0.94 2.5 Transmissivity θ(200/0.1) ASTM D4716 10-3 m2/s 0.7 0.5 0.9 2.2 Discharge cap. qw(200/0.1) ASTM D4716 10-6 m3/s 87 94 Discharge cap. qw(300/0.1) ASTM D4716 10-6 m3/s 55 55 76 175 Discharge cap. qw(500/0.1) ASTM D4716 10-6 m3/s 14 25 42 80 D.C. buckled θ ASTM D6918 % 37 22 18 28 Hydraulic properties filter Velocity Index vh50 EN 11058 mm/s 16 5 16 16 Permittivity ψ ASTM D4491 s-1 0.3 0.3 0.3 0.3 Permeability k ASTM D4491 10-4 m/s 1.3 0.3 1.3 1.3 Pore Size O95 ASTM D4751 µm 75 75 75 75 Transport Details Roll length m 300 300 250 200 Roll diameter m 1.1 1.1 1.2 1.2 Inside diameter m 0.15 0.15 0.23 0.23 Weight Roll kg 22 22 25 25 40 ft container km 160 160 130 80

All mechanical Properties are average values.Standard variations in mechanical strength of 10% andin hydraulic flow and pore size of 20% have to beallowed for.

All EN/ISO 12958 tests are made with apparatus 2 (sample in pressure cell wrapped in latex membrane) ASTM D4716 test is equal to EN/ISO 12958 (apparatus 1)

41

9.2 Equations

9.2.1 Hansbo (1981):

Average degree of consolidation uTh

heU 81 −−= In which,

dDn = ,

w

ch

Mkcγ

= and 2DtcT h

h =

For simple analysis,

⎟⎠⎞

⎜⎝⎛ −+−

−= 422

2

411

43ln

1 nnn

nnu

Including the effects of well resistance,

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+= )2()1(

2

2

zlzqn

knuuw

cr π

For the combined effects of smear and well resistance,

⎟⎠⎞

⎜⎝⎛ −−+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

−−

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

−= 2

22

4

22

2

2

2

2

2 11)2(14

11

1'4

114

3ln'

ln1 nq

kzlzsn

snk

kns

nss

kk

sn

nnu

w

c

c

c

c

cs π

9.2.2 Olson (1977)

9.2.2.1 Horizontal Equations

2

2

22

413

1ln)(

NN

NNNFn

−−

−= ,

w

e

rrN = and

nFA 2=

For rcr TT ≤ ,

[ ]

rc

rr

r T

ATA

TU

)exp(11−−−

=

And when rcr TT ≥ ,

[ ] )exp(1)exp(11 rrcrc

r ATATT

U −−−=

9.2.2.2 Vertical Equations

For rcr TT ≤ ,

[ ]⎭⎬⎫

⎩⎨⎧ −−−= ∑ )exp(1121 2

4 TMMTT

TUc

v

42

And when rcr TT ≥ ,

[ ] )exp(1)exp(121 224 TMTM

MTU c

cv −−−= ∑

9.2.3 Carrillo (1942)

For the combination of vertical and horizontal consolidation in homogenous soils.

)1)(1(1 vr UUU −−−=