Internal Erosion and Dam StabilityAnalysis of the internal erosion effects on stability of an embankment dam

Arthur Jedenius

Civil Engineering, master's level

2018

Luleå University of Technology

Department of Civil, Environmental and Natural Resources Engineering

MASTER THESIS

Internal Erosion and Dam Stability Analysis of the internal erosion effects on stability of an embankment dam

Arthur Jedenius

Division of Mining and Geotechnical Engineering

Department of Civil, Environmental and Natural Resources

Luleå University of Technology

1

PREFACE

This thesis finalizes five years of Civil Engineering studies at Luleå University of

Technology.

The thesis has been performed as a contribution to the research on internal erosion in

embankment dams in development by LTU together with hydropower companies in Sweden

and financially supported by the Swedish Hydropower Centre (SVC). This work was a

proposed investigation suggested by Ingrid Silva and is based on the same dam she is

investigating.

I want to thank my supervisors Ingrid Silva and Jasmina Toromanovic for their invaluable

assistance and guidance throughout the semester and for the interest shown in the thesis. I

want to thank my examiner Jan Laue for his help in guiding me to interesting results during

the semester. I also want to thank Hans Mattsson for introducing me to geomechanics and

providing me with invaluable understanding of both geomechanics and the finite element

method (FEM).

I also want to thank my family and friends for all the support during my years of studying at

LTU.

Luleå, June 2017

Arthur Jedenius

ABSTRACT

Embankment dams encounter several problems in terms of dam safety. One of those problems

is called internal erosion. This phenomenon is induced by the movement of fine particles

within the dam due to seepage forces. If the dam is not able to self-heal, the eroded zones will

increase which will eventually cause the dam to fail. Thus standards have been created by

Svensk Energi and summarized in the Swedish dam safety guideline RIDAS 2012. These

standards are used as a basic in the risk analysis of existing dams and provide guidelines for

proper design of future dams.

A dam in Sweden has presented recurring incidents related to internal erosion within the core.

The impact of this internal erosion is analysed in this thesis with the use of Finite Element

Method/Analysis (FEM/A). FEA models simulate the in situ stresses in the dam and calculate

the strength. It also enables the analysis of changing hydraulic conductivity and its effect on

the overall effective strength due to changing pore pressure and seepage forces. The analysis

using numerical methods was performed in the program PLAXIS2D and SEEP/W while limit

equilibrium analysis was done in SLOPE/W.

The calculation in PLAXIS2D was performed by using the Mohr-Coulomb constitutive

model. The in situ stresses are initially calculated using gravity loading since this is the

preferred method on an uneven terrain instead of a K0-calculation. Then, through a set of

phases in the program, zones where erosion is assumed to have occurred are changed. These

zones have a higher permeability and will thus affect the pore pressures in the dam following

Darcy’s law with permeability through a set medium.

The increased permeability is set to follow an increased void ratio due to loss of fine material

in the core. How this increase of void ratio affects the permeability is investigated through

using Ren et al’s (2016) proposed equation for calculating permeability with a set void ratio.

Their equation, apart from the usually used Kozeny-Carman equation, considers both

effective and ineffective void ratio where the ineffective void ratios refers to the volume of

pores that is immobile when flow is considered.

The increased flow in the eroded zones of the core did not seem to impact the strength of the

dam in much regard. The phreatic surface and thus the pore pressure did not change enough to

influence the overall effective strength of the dam. It raises the question if the stability of an

earth-rock fill dam will be affected due to increased pore pressure at all due to its draining

properties and if it would rather fail due to increased seepage forces.

SAMMANFATTNING

Från utgångspunkten över dammsäkerhet innebär det att jordfyllnadsdammar står inför många

problem. Ett av dessa problem kallas inre erosion. Detta fenomen sker när jordpartiklar flyttas

på grund av flödena inne i dammen. Om dammen inte kan självläka så kommer de eroderade

zonerna bli större vilket kan leda till dammbrott. Därför har standarder gjorts utav Svensk

Energi som ska säkerställa att existerande och framtida dammar inte skall gå till brott, dessa

standarder är sammanfattade i verket RIDAS.

En damm i Sverige har haft återkommande problem med misstänkta eroderade zoner i dess

kärna. Den inverkan som zonerna har på dammens stabilitet analyseras med Finita Element

Metoden (FEM). FEM modellerar och simulerar in situ spänningarna i dammen och beräknar

dess hållfasthet. Det ger också möjligheten att analysera förändrat flöde när

permeabilitetsparametrarna förändras och hur det påverkar den effektiva hållfastheten då

portryck och flödeskrafter påverkar stabiliteten i dammen.

Analysen som gjorts har använts sig av den konstitutiva modellen Mohr-Coulomb. In situ

spänningarna beräknas initialt med ’gravity loading’ eftersom det är den rekommenderade

metoden då man har en ojämn ”terräng” så som en damm är istället för att utföra en K0-

beräkning. Sedan, genom bestämda steg, så får zoner med förväntad erosion sina

permeabilitets- och densitetsparametrar förändrade. Den ökade permeabiliteten kommer

genom Darcy’s lag att påverka dammen nedströms med ökade portryck.

Den ökade permeabiliteten är satt att följa bestämda portal då material succesivt eroderas bort

i kärnan. Den ökade permeabiliteten kopplat med det ökade portalet evalueras genom Ren et

als (2016) föreslagna ekvation som beaktar permeabilitet för ett bestämt portal. Deras

ekvation tar både i beaktning effektiv och ineffektivt portal, till skillnad från den vanligen

använda Kozeny-Carman ekvationen, där det ineffektiva portalet är den volymen av porer

som inte kan föra vidare vatten.

Det ökade flödet i de eroderade zonerna i kärnan verkar enligt analysen inte ha någon större

påverkan på dammens säkerhet med åtanke på stabilitet. Den ökade grundvattenytan och

portrycken förändrades inte tillräckligt för att påverka dammsäkerheten avsevärt. Resultatet

ställer frågan om en stenfyllnadsdamm är i riskzon för ökade portryck på grund av dess

dränerande egenskaper och snarare skulle gå till brott av ökade krafter från flödet.

TABLE OF CONTENTS

9

TABLE OF CONTENTS

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

1.1 Background ............................................................................................................................................... 1

1.2 Purpose ...................................................................................................................................................... 3

2 Dam Design ................................................................................................... 4

2.1 Introduction ............................................................................................................................................... 4

2.2 Case study ................................................................................................................................................. 4

2.2.1 Core ................................................................................................................................................. 4

2.2.2 Filter ................................................................................................................................................ 4

2.2.3 Shell ................................................................................................................................................. 5

2.3 Parameter evaluation ................................................................................................................................. 5

2.3.1 Introduction ..................................................................................................................................... 5

2.3.2 Density ............................................................................................................................................. 5

2.3.3 Poisson’s ratio and Young’s modulus ............................................................................................. 6

2.3.4 Dilatancy ......................................................................................................................................... 6

2.3.5 Hydraulic conductivity .................................................................................................................... 6

2.3.6 Strength parameters c and tan ϕ by iteration .................................................................................. 6

2.4 Other parameters ....................................................................................................................................... 8

2.5 Consideration for modelling...................................................................................................................... 9

3 Conductivity and void ratio ................................................................... 10

3.1 Void ratio, specific surface and permeability .......................................................................................... 10

3.2 Internal erosion ....................................................................................................................................... 14

4 Previous and current case studies ........................................................ 16

5 Geostudio ...................................................................................................... 17

5.1 SEEP/W .................................................................................................................................................. 17

5.1.1 Seepage behaviour ......................................................................................................................... 17

5.1.2 Boundary conditions ...................................................................................................................... 19

5.2 SLOPE/W ............................................................................................................................................... 19

5.3 Finished Model ....................................................................................................................................... 22

6 PLAXIS2D................................................................................................... 23

6.1 Mesh ........................................................................................................................................................ 23

6.2 Plane strain modelling ............................................................................................................................. 24

6.3 Stress field generation ............................................................................................................................. 25

6.3.1 K0 procedure .................................................................................................................................. 25

6.3.2 Gravity loadings ............................................................................................................................ 26

6.4 Boundary conditions and finding a solution ............................................................................................ 26

6.4.1 Calculating the Safety Factor ........................................................................................................ 27

6.5 Constitutive model .................................................................................................................................. 27

6.5.1 Choice of constitutive model .......................................................................................................... 27

6.5.2 Mohr Coulomb ............................................................................................................................... 27

6.6 Finished model ........................................................................................................................................ 30

6.7 Calculating phases ................................................................................................................................... 30

6.7.1 Initial phase ................................................................................................................................... 30

6.7.2 Internal erosion .............................................................................................................................. 30

6.7.3 Safety analysis................................................................................................................................ 31

7 Results ........................................................................................................... 31

7.1 Stability ................................................................................................................................................... 31

7.2 Displacement ........................................................................................................................................... 34

8 Discussion .................................................................................................... 36

8.1 Permeability and void ratio ..................................................................................................................... 36

8.2 Stability ................................................................................................................................................... 37

8.3 Displacement ........................................................................................................................................... 37

9 Concluding remarks ................................................................................. 39

9.1 Results and discussion ............................................................................................................................. 39

9.2 Further studies ......................................................................................................................................... 39

References ........................................................................................................... 40

APPENDIX A ................................................................................................. 42

APPENDIX B ................................................................................................... 43

APPENDIX C ................................................................................................... 44

TABLE OF CONTENTS

11

APPENDIX D .................................................................................................. 45

1

1 INTRODUCTION

As environmental concerns rise, a growing interest of sustainable power resources grows

with it. Hydroelectric power being such a resource brings a lot of attention to existing dams

built. Sweden has over 2000 dams where a majority are classified as embankment dams.

(ICOLD, 2017) These embankment dams can furtherly be designated as earthfill and

rockfill dams. By using an impermeable section in the construction named ‘core’ the

embankment can be sufficiently impermeable to withstand the stress brought by the

reservoirs hydrostatic pressure. This pressure, in combination with unstable gradation of

the core material and/or low stress in the soil matrix due to low compaction, has in some

instances triggered the initiation of a deterioration process called internal erosion. This

process involves the movement of fine particles in the dam. If the filter is poorly designed,

the dam core may selfheal. When the filter is not capable to arrest the migration of fine

particles, internal erosion may continue until the dam breaks, thus it is an important aspect

that is studied in dam safety.

This thesis presents the analysis of stability, in terms of safety factor, of a hydropower

embankment dam experiencing the internal erosion mechanism known as backward

erosion. The analysis was conducted using the finite element program PLAXIS2D by

applying a steady state analysis.

1.1 Background

The purpose of a dam being constructed can be for a single purpose or a combination of

several. The purposes varying between energy production, flood control, construction,

water reservoir, tailings storage, irrigation, recreational and even housing for various

animals. Summarised, they all store water (partly in tailings dams). This is achieved by

using different construction materials in the dams ranging from wood/timber, steel,

concrete and soil. In this thesis an embankment dam is analysed which consists of earth

and rock materials.

The choice of using embankment dams are often driven by an economic aspect but also

that it can be built on soil and other pervious foundations. Embankment dams are divided

into two categories; earth- and rockfill dams. There are an abundant variety of both types

but the most basic difference is that the fill material used is different.

According to ICOLD (International Commission On Large Dams), about 1% of the

world’s different dams have been prone to failure during the years where internal erosion is

a huge cause. Even though ICOLD’s measurement is based on high dams, meaning that

they are over 15 meters in height, it shows the importance of studying dam safety. One of

the main causes for dam failure is caused by internal erosion, which means that a transfer

of particles from the embankment dam has taken place and thus eroding the construction

internally. (ICOLD, 2017)

2

Also to mention is that according to ICOLD the number of failing dams has decreased by a

factor of four over the last 40 years which stems from increased surveillance and

technology used in calculating dam behaviour, thus encouraging the importance of this

field of study. (ICOLD, 2017)

The dam in this thesis is a Swedish earth- and rockfill dam and was built in the 1970’s. It

has four different materials where a well graded till has been used for the central low

permeability core. Surrounding this is a supporting sandy gravel material that is both a

filter and supporting material. Enclosing the filter is a macadam type of material used as a

coarse filter; which is used to gradually change the size of the materials while also

increasing permeability. The fourth material is the rock fill which consist of blasted rock

and has high strength.

Internal erosion is caused by the washout of fine particles from the soil matrix of a dam

due to seepage forces, thus increasing void ratio. This may cause an increase in

permeability which may follow a continuation of the erosion. If the core does not selfheal

the phenomenon piping might occur. As seen in Figure 1 there are different kinds of pipes

that might develop due to the internal erosion. The pipe that is analysed in this thesis will

be a concentrated leak pipe in the core.

Figure 1. Backward erosion and concentrated leak piping initiated by the internal erosion. (Foster & Fell,

1999)

3

1.2 Purpose

The purpose of this thesis is to analyse the impact of internal erosion in the stability of a

Swedish earth- and rockfill dam. There are several factors that have to be considered in

building the mode.

The impact of internal erosion on unit weight and hydraulic conductivity.

Variations in the stability of the dam due to internal erosion measured in terms of

safety factor.

4

2 DAM DESIGN

2.1 Introduction

The embankment dams can be classified, depending on the construction material, in:

earthfill dams, rockfill dams, and zoned dams. In the last case, both earth and rock are used

as construction material, and the dam cross section is typically dived into four zones: core,

filter, drainage (coarse filter), and fill.

2.2 Case study

The object of this study is an earth- and rockfill dam located in the northern part of

Sweden. The embankment was built using four different materials which are shown in

Figure 2 and where the parts are explained below.

Figure 2. A cross cut of the analysed dam with its respective parts.

1. Rock fill shells – Composed of blasted rock from a nearby constructed canal.

2. Coarse filter – Macadam material (gravel).

3. Filter – Sandy material

4. Core – Well graded till which follows the grading curve shown in Appendix A.

2.2.1 Core

The core is the impervious or near impervious (very low permeability) part of the dam. It is

created by heavy compaction during the layer by layer construction when using a soil

material. Silt, clay and well graded alluvial moraine are used. In rockfill dams, a layer of

concrete, geomembrane or even asphalt can also be used to guaranty a sufficiently low

permeability.

2.2.2 Filter

Filters are used when the core consists of soil material of low permeability. The function of

the filter is to gradually lower the permeability in the dam as a sort of barrier between the

5

fill and core in the upstream face. By being more permeable it also functions as a drain and

thus it can be used to lower pore pressure in the downstream face.

2.2.3 Shell

The shell is used to hold the impermeable part of the dam in place. It is often highly

permeable and has usually high strength and mostly envelops the majority of the dam’s

volume as seen in Figure 3.

Figure 3. Cross-cut of an embankment dam.

2.3 Parameter evaluation

2.3.1 Introduction

Not much data on the material parameters was available. Density and permeability of the

core and filter are known, in addition, the reports made during construction of the dam give

an indication of the type of materials used. Based on this information a literature was done

in order to obtain the additional geotechnical parameters needed to describe the mechanical

behaviour of the dam materials.

RIDAS guidelines also state that the dam should have a safety factor of 1.5 downstream

while having a filled reservoir and a safety factor of 1.3 upstream when a rapid drawdown

has been conducted (Svensk Energi, 2012). Based on the literature review regarding the

strength parameters of soil, a sensitivity analysis have been performed in order to find the

initial parameters that fit the two criteria indicated in RIDAS guidelines.

2.3.2 Density

The core and the support already have a predetermined density. The filter is made of

macadam-sized gravel and the rock fill of cobbles and, according to the documentation for

the dam, they are both made from blasted rock taken from the vicinity. The bedrock

material nearby is made of intrusive rock such as granite, thus 18kN/m3 dry unit weight for

the filter is used (Larsson, et al., 2007). RIDAS recommends 15kN/m3 for the rock fill

material if no other data is available. Thus 17kN/m3 is used instead as initial parameter

between the previous two values. For the filter, which is based on the same material but

6

with a lower void ratio, 18kN/m3 is assumed. These values were also used in the stability

analysis performed by the dam owner.

2.3.3 Poisson’s ratio and Young’s modulus

Poisson’s ratio and Young’s modulus (E) are the strain parameters in the stress-strain

relation used in FEM-calculations. The initial values used in the analysis are the same as

used in Vahdati’s work (2014). Even though strain is not being regarded in the stability

analysis, it still is an important factor in the gravity loading procedure for initial stresses in

PLAXIS2D. It also provides the opportunity to get an indication of main deformations in

the model, even though the Mohr-Coulomb constitutive model used in the analyses is not

recommended for this observation, (Brinkgreve, et al., 2016).

2.3.4 Dilatancy

The evaluation of the dilatancy angle follows the simple relation between the friction and

dilatancy angle as suggested by Plaxis, (Brinkgreve, et al., 2016).

𝜑 ≈ 𝜙 − 30 (1)

Thus the dilatancy angles are set as 5 degrees for the core and 2 degrees for support and

filter. The fill however follows a recommendation to be 15 degrees for angular (not

rounded alluvial) blasted rock material which has close to 40 degrees friction angle.

(Agahei Araei, et al., 2010)

2.3.5 Hydraulic conductivity

The hydraulic conductivity for the core was evaluated to be 1,2*10-7

m/s and for the filter

6*10-4

m/s during the construction of the dam and these values are used in the model. The

other two materials, filter and rock fill, are highly permeable material and was chosen from

Vahdati (2014).

2.3.6 Strength parameters c and tan ϕ by iteration

The strength parameters used for the analyses also had to be estimated by iterative process.

No cohesion in the dam is considered for any material since they are all based on frictional

soil (cohesionless). The base parameters for the friction angle were derived from the dam

stability report previously performed. The parameters are shown in Table 1.

7

Table 1. Parameters used in previous stability analysis.

Material Friction angle ϕ (˚)

Core 38

Filter 34

Coarse filter 34

Fill 42

The stability analysis performed was done to calculate if the dam, with an added berm, is

following the guidelines set by RIDAS as previously mentioned. During a rapid drawdown

the upstream safety factor must be at the least 1.3 and while being a full reservoir a safety

factor of 1.5 downstream needs to be achieved. Thus a model with this condition was

created, as shown in Figure 5. A similar one was created in Geostudio in order to compare

the results.

When the initial parameters are set an iterative lowering of the dam’s strength is performed

with the resulting safety factor shown in Figure 4. This is done with the aim of finding the

lowest possible strength parameters of the dam with the ratios between the different

parameters constant. The analysis was performed with a steady state analysis of a full

reservoir as well as a rapid draw down analysis (RDD). The lowered strength of the

parameters was performed following

𝑡𝑎𝑛−1(𝑡𝑎𝑛(𝜑) ∗ 𝑛) (2)

Where n is the percental of the lowered strength.

Figure 4. Iterative lowering of the overall strength in the dam.

1,886

1,532

1,390

1,108

1,882

1,325

70,00%75,00%80,00%85,00%90,00%95,00%100,00%

0,9

1,1

1,3

1,5

1,7

1,9

2,1

2,3

Strength lowered tan(φ) (%)

Safe

ty F

acto

r

SF Plaxis downstream SF Plaxis Upstream RDD

SF Slope/w Downstream SF Slope/W Upstream RDD

8

When the Geostudio analysis received a safety factor of 1.325 the strength parameters are

as shown in Table 2.

Table 2. Strength after iterative lowering of parameters.

Material Friction angle ϕ (˚)

Core 35

Filter 32

Coarse filter 32

Fill 39

2.4 Other parameters

The unit weight for the compact density for the soil material and water was set as 26kN/m3

and 9,81kN/m3. If these values has been set it means other parameters can be calculated.

For example the void ratio for the core which will be used in later calculations. Note that

these are not used as input in either SEEP/W, SLOPE/W or PLAXIS2D. To calculate the

void ratio Equation (3) is used.

𝑒 =𝛾𝑠 − 𝛾𝑑

𝛾𝑑 (3)

From this, porosity can also be calculated, as shown in Equation (4)

𝑛 =𝑒

1 + 𝑒 (4)

To calculate the saturated density, Equation (5) is used.

𝛾𝑠𝑎𝑡 = 𝛾𝑑 + 𝛾𝑤 ∗ 𝑛 (5)

The final parameter table used in the models are shown in Appendix C.

9

2.5 Consideration for modelling

The crosscut used for the finite element analysis was simplified in some regard. The

bedrock is set as near impermeable since the analysis of seepage is to focus on the dam and

not the bedrock, thus the blanket grout part below the core is disregarded. It is also

assumed to be a flat surface for easier modelling. The drain located at the toe of the dam is

also disregarded due to no given information about it was found.

The cross section shown in Appendix B has a very specific design. However, as also seen

on the same page is an overview of the dam. This shows an uneven terrain causing the

height of the dam construction to differ. Thus the plain-strain assumption made in

PLAXIS2D might be wrong and to assume a 3-dimensional design might be better. To get

a better comparison with the limit equilibrium analysis and the fact that embankment dams

are often analysed assuming plain-strain this is set to be an acceptable assumption. In

addition, PLAXIS3D solution would also require longer computation times.

It is also clearly shown in Figure 5 that there is an added berm to the dam which has been

added later on to strengthen the dam in accordance with the new regulation of Swedish

dam and safety guidelines created by the hydropower companies. This berm is assumed to

have the same parameters as the rockfill material as the one used in the shell.

Figure 5. The model in PLAXIS2D with the added berm.

Added berm

10

3 CONDUCTIVITY AND VOID RATIO

This chapter is largely influenced by the article ‘A relation of hydraulic conductivity – void

ratio for soils based on Kozeny-Carman equation’ by Ren et. al. (2016)

3.1 Void ratio, specific surface and permeability

Conductivity of soils is a fundamental subject when constructing an embankment dam,

thus it is important to understand the behaviour of seepage. Seepage is often explained

with Darcy’s law through a porous medium seen in

𝑞 = 𝐴𝑘𝑖. (6)

q is the measured water flow where A is the area of the cross cut where the water is being

moved through, k the permeability and i the hydraulic gradient. (Craig, 2004)

For a medium such as soil there is a close relation between the void ratio and the

permeability. The higher the porosity the higher the permeability is the general rule, which

is also the case for the Kozeny-Carman relation shown in

𝑘 = 𝐶𝐹

1

𝑆𝑆2

𝛾𝑤

𝜇𝜌𝑚2

𝑒3

1 + 𝑒. (7)

Where again k is the calculated coefficient for the permeability, Ss is the specific surface

for the material, 𝛾𝑤 the unit weight for the water, μ the coefficient for the fluid’s viscosity,

𝜌𝑚 the density of the soil, and e the void ratio. The void ratio in this case is the total void

ratio for the medium, meaning that there is the assumption of total use of the pores for fluid

flow. The fact is that the Kozeny-Carman equation works well for course materials like

sand and gravel but loses its coherency for finer soils such as silt and clay. (Ren, et al.,

2016)

Figure 6. An assumed 100% saturated soil with void ratio that is immobile due to electromechanical bonds

and mobile water that flows through the medium. (Ren, et al., 2016)

11

This is because of said assumption of total usage of the pores for fluid transportation,

which should be a linear function but is not the case for finer soils due to their

electromechanical bond between the particles. Thus there have been attempts to make a

better relation for soils such as clay and silt. (Ren, et al., 2016)

One interesting relationship between the logarithm of hydraulic conductivity and the void

ratio that was observed by Taylor (1948) and Lambe and Whitman (1969) can be seen in

𝑒 = 𝑒0 + 𝑐𝑘𝑙𝑜 𝑔 (𝑘

𝑘0). (8)

This can be rewritten as

𝑙𝑜 𝑔 (𝑘

𝑘0) =

𝑒 − 𝑒0

𝑐𝑘. (9)

Equation (9) is the equation used in PLAXIS2D for when soil is consolidated, meaning

that there is a change in void ratio and how that affects the permeability. ck is by default set

as the compression index given in constitutive models such as Hardening Soil and Soft

Soil. However, for the plastic calculation performed in this analysis this cannot be used.

(Brinkgreve, et al., 2016)

So to conclude, there is a difference between void ratios or rather there is a need to

separate them. In the work of Ren et al. (2016) a new concept is introduced called effective

void ratio (ee) and ineffective void ratio (ei) which will separate the void volume into

different types. Effective void ratio being the parts of the pore which lets transport of water

to take place while the ineffective void ratio “breaks” the flow. The evident relation is

shown in (Ren, et al., 2016)

𝑒𝑡 = 𝑒𝑒 + 𝑒𝑖. (10)

Where et meaning the total void ratio which is simply noted as e in most literature. The

factor ei, meaning the void ratio of immobilized water, is strongly influenced by geometric

characteristics, salinity and cation (the charge). Usually these parameters usually can be

translated to specific surface (Ss), meaning the total surface area of the particles measured

in a medium. The general rule is that with decreased grain size a higher specific surface

needs to be regarded. How this affects the overall permeability with different void ratios

for certain materials can be shown in Appendix D. (Ren, et al., 2016) (Lambe & Whitman,

1969)

Lambe and Whitman observed that there was a logarithmic linear relation between void

ratio and permeability, as observed in the Appendix D for several materials so Equation (8)

can be used. Ren et al. suggest according to

12

𝑘 = 𝐶𝑒𝑡

3𝑚+3

(1 + 𝑒𝑡)53

𝑚+1 ∗ [(1 + 𝑒𝑡)𝑚+1 − 𝑒𝑡𝑚+1]

43

(11)

to remedy this. Where C is explained in

𝐶 =1

𝐶𝐹∗

𝛾𝑤

𝜇𝜌𝑚2

1

𝑆𝑠2

. (12)

In Figure 7 constant C is found to be 2,94𝐸−4 ∗ 𝑆𝑠−1,45

with use of linear regression for

different soils, thus this value is used.

Figure 7. Relation between different soils for parameter C. (Ren, et al., 2016)

The only parameter left is the constant m which refers to the difference in specific surface

between materials and was evaluated and shown in Figure 8.

13

Figure 8. Change of constant m to fit the results of different soils. (Ren, et al., 2016)

Figure 9. The results after using Equation(11) on different soils where the results tends to span between 3k

and 1/3k of measured permeability. (Ren, et al., 2016)

Thus concluded that the changes of permeability are caused by:

1. Particle size

2. Void ratio

3. Composition

4. Geometric arrangement

5. Degree of saturation

14

3.2 Internal erosion

When considering flow inside a medium like cohesionless soil, seepage forces will act on

the grains. If not well compacted or well graded a loss of fines will occur, meaning they

will follow the flow, thus causing suffusion. There are different types of internal erosion

where as in this study suffusion is being studied. (Sibille, et al., 2015)

Some assumptions are used:

1. The internal erosion mechanism is horizontal and is only localised in the core.

2. A plane strain assumption of the erosion is used.

3. No self-healing of the core.

4. No subsidence occurs when lowering the void ratio.

5. Only fine particles (silt) will erode from the soil matrix.

The first hurdle is classifying the material for constant m since it is not specified as clay,

silt or sandy material. For void ratio 0,215 which was calculated from in situ bulk density

and following the assumption of compact density being 2,6 t/m3. The specific surface was

assumed to be 0,1m2/g (silt). The constant m was iteratively found to be 1,175 when

finding the permeability 1,2E-7 m/s as measured for the core in its construction phase.

With the assumption that only silt will erode, the test is set to where the fines are totally

gone which is approximately 39% mass which coincidentally makes the void ratio about 1.

This enables the iterative process to have a goal where the m and specific surface has

sandy characteristics

The iterative process was started where the specific surface was decreased to resemble a

sandy material (mass has eroded and thus specific surface has decreased), going from 0,1

to 0,01m2/g. This was performed with the constant m as well going from 1,175 to 0 which

resembles sandy material. The results of this assumed study is shown in Figure 10.

15

Figure 10.Conductivity change with change of void ratio and how much mass has been eroded.

By regarding Figure 10 it is possible to extrapolate parameters for conductivity of the

eroded zones which are shown in Table 3.

Table 3. Parameters changed in the core for different void ratios.

e - 0,215 0,25 0,3 0,35 0,4 0,94 1

k m/s 1,20E-07 7,00E-07 1,07E-06 5,00E-06 1,21E-05 3,45E-02 1,17E-01

m - 1,175 1,15 1,07 0,95 0,84 0,090 0,000

mass eroded % 0,00 0,84 3,51 7,52 11,19 36,23 39,25

dry density ρ 21,4 20,8 20,0 19,3 18,6 13,4 13,0

wet density ρw 23,2 22,8 22,3 21,9 21,4 18,2 18,0

0,00,10,20,30,40,50,60,70,80,91,01,11,21,3

0,00%

5,00%

10,00%

15,00%

20,00%

25,00%

30,00%

35,00%

40,00%

1,0E-7

1,0E-6

1,0E-5

1,0E-4

1,0E-3

1,0E-2

1,0E-1

0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00

m constant

Mas

s er

od

ed (

%)

Hyd

rauli

c co

nd

uct

ivit

y (

m/s

)

Void ratio (e)

Simulated erosion in the core Conductivity/Change of void ratio Mass eroded/Change of constant m

16

4 PREVIOUS AND CURRENT CASE STUDIES

A previous study was performed which focused on an overall safety analysis of the dam.

The study concluded that the dam was assigned as class 1, which is the next to highest

classification (Svensk Energi, 2012). This means that the risk of loss of human lives or

damage to person cannot be neglected. This also means that the environmental impact will

be significant as well as having a substantial economic effect. Since internal erosion is one

of the most common failure modes for an embankment dam it proves the importance of

further analysing the effects on this dam, since it has a history of heightened turbidity

downstream, meaning a loss of finer material in the dam.

Since the construction of the dam there have been two sinkhole occurrences. In both

instances they were repaired with more material and grouting. They were also situated in

the vicinity of section 0+020 which is the cross section used in this report; the location is

also shown in (Silva, et al., 2017). In Silva’s article it is shown that that there is a clear

difference in terms of particles size distribution between the samples representative of the

current state and the design boundaries of the dam's core which could indicate that internal

erosion has occurred, as shown in Figure 11.

Figure 11. "Granulometric curves - Boreholes 0/045V, 0/013V and 0/090H.” (Silva, et al., 2017)

17

5 GEOSTUDIO

GeoStudio is a platform that enables the user to perform different types of analyses on a

soil structure. In this type of analysis it is necessary to perform a seepage and a stability

analysis, thus SEEP/W and SLOPE/W will be used.

5.1 SEEP/W

5.1.1 Seepage behaviour

SEEP/W is a numerical analysis program that is able to calculate a phreatic surface, flow

and pore pressure in a specified medium and geometry of the users choosing. Figure 12

shows the result of an experiment performed in 1985 by Jennifer Rulon and how it later on

could be calculated in SEEP/W (Geo-Slope, 2012).

Figure 12. Laboratory results (left) compared to SEEP/W results (right). (Geo-Slope, 2012)

SEEP/W uses numerical integration to calculate the seepage. The calculation process is

performed in the set of nodes and gauss points that are created in the mesh and with a set

of boundary conditions. The mesh can look differently but for this analysis a triangular

mesh has been chosen. The Gauss points inside the triangles calculate material behaviour

in terms of seepage and extrapolate that information to the nodes at the corner sides of the

triangles. The program then computes the information for each node in the whole model

and reveals if the node is either the head (phreatic surface) or flux (below the phreatic

surface) (Geo-Slope, 2012). Since the “correct” value is calculated in the Gauss points a

finer mesh can yield better results, however the more nodes and points used the longer time

the calculation process will take.

Since all four dam construction materials are both saturated and unsaturated at some point

in the geometry, a saturated/unsaturated model will be used for all materials. For regions

over the phreatic surface, as considered for the core and support, matric suction will be

considered. However, for the course filter and rockfill it is assumed to have no capillary

ability. The suction will create partially saturated zones which has a lower permeability

than fully saturated zones, as shown in Figure 13. For this a function is needed to simulate

the relation between soil water storage and the matric suction.

18

Figure 13. Different grades of saturation where water is held around the grains due to the capillary tension.

(Szymkiewicz, 2013)

The soil water storage considers the porosity (n) and the degree of saturation. This is

performed by using (Geo-Slope, 2012)

𝛩𝑤 = 𝑛𝑆 (13)

where 𝛩𝑤 is the volumetric water content, n is the porosity and S is the degree of saturation

that was set to 100%.

PLAXIS2D has a function that approximates the soil-water behavior using data from

Hydraulic Properties of European Soils (HYPRES) in combination with the calculation

model Van Genuchten. (Brinkgreve, et al., 2016) This approximation will be used in this

thesis in both the PLAXIS2D model as well as SEEP/W. The calculation model Van

Genuchten is an equation that takes into consideration the matric suctions resistance of the

water flow. This relation can be seen in Figure 14.

Figure 14. Relation between matric suction and permeability in course and fine grained soils. (Rahardjo, et

al., 2004)

What is not considered in the equation is the amount of compaction performed on the soil.

The rate of compaction has a direct correlation with void ratio which in itself will link to

19

the capillary rise. However, it is fairly mirrored in the permeability which is low for the

materials meaning that there is a low void ratio and thus highly compacted.

5.1.2 Boundary conditions

The last thing to do in the model is to set up boundary conditions for the seepage model.

This is to set “boundaries” for the analysis in where it is “allowed” to act and to set the

initial conditions, such as the reservoir head. What is also set is the seepage face on where

it is possible for seepage to exit the geometry created. Boundary conditions are used when

calculating differential equations, thus it is only used in SEEP/W and not SLOPE/W.

PLAXIS2D however will use it for both stability and seepage so it will be more discussed

in paragraph 5. PLAXIS2D.

5.2 SLOPE/W

After the SEEP/W analysis has been performed, GeoStudio is capable of using that data as

a ‘parent’ for the next analysis which in this case is a slope stability analysis. This analysis

is performed using SLOPE/W which uses the limit equilibrium method (Figure 15). This is

performed by slicing a slope into lamellas with a chosen circular shear surface and a point

of moment.

Figure 15. A slope with a visual representation of forces acting on a slice. (Geo-Slope, 2012)

Simply it is the acting unit weight in the lamella that will act on the bottom surface and

thus create shear strength. More modern ways of calculating limit equilibrium also

includes the action between the slices, so called interslice forces. The shear force and the

moment create a global factor of safety for the slope in that geometry. There are an

abundance of methods that can be used but in this thesis the Morgenstern-Price method is

used due to its inclusion of both force and moment equilibrium in the model. The equation

for the factor of safety in regards of moment equilibrium is shown in

𝐹𝑚 =∑(𝑐′𝛽𝑅 + (𝑁 − 𝑢𝛽)𝑅 ∗ tan (𝜙′))

∑ 𝑊𝑥 − ∑ 𝑁𝑓 ± ∑ 𝐷𝑑. (14)

The equation in regard of force equilibrium is shown in

𝐹𝑓 =∑(𝑐′𝛽 cos(𝛼) + (𝑁 − 𝑢𝛽) ∗ tan(𝜙′) ∗ cos (𝛼))

∑ 𝑁𝑠𝑖𝑛(𝛼) − ∑ 𝐷𝑐𝑜𝑠(𝜔). (15)

20

The terms for the equation are explained in Table 4. The effective strength parameters

cohesion and friction angle are used in the drained analysis where finding the critical slip

surface becomes harder to define. (Geo-Slope, 2012)

Table 4. Parameters used in the equations for moment and force equilibrium. (Geo-Slope, 2012)

c’ effective cohesion

𝜙′ effective angle of friction

u pore water pressure

N slice base normal force

W slice weight

D concentrated point load

𝛽, R, x, f, d, 𝜔 geometric parameters

𝛼 inclination of slice base

Morgenstern-Price calculates both the moment and force equilibrium and in SLOPE/W the

safety factor is chosen where 𝐹𝑓=𝐹𝑚 as seen in Figure 16.

Figure 16. Example of how the SF for Morgenstern-Price is chosen. (Geo-Slope, 2012)

Figure 17 shows the acting forces on a slice and how the forces are calculated in the terms

of vectors. Red being the residual normal force acting against the weight of the slice (blue)

and green the created shear strength. Black shows the action of the interslice forces and its

created shear strength. Adding these together with the action of the whole slip surface will

create the equilibrium equation. The interslice force and shear strength reaction is based on

𝑥 = 𝐸𝜆𝑓(𝑥) (16)

21

where E is the interslice normal force f(x) is a predetermined equation that can be changed

by the user of the program and λ is the percentage of said function. In this analysis the

half-sine equation is used. x is the calculated shear strength.

Figure 17. A visualisation of the acting forces used in the calculation of a slice using Morgenstern-Price

(left) and a sum of all acting forces with vectors (right). (Geo-Slope, 2012)

SLOPE/W also has the capability of creating none circular shear surfaces since a circular

shape does not always create the most critical surface by using the Optimized option which

is used in this analysis.

22

5.3 Finished Model

The finished model for the SEEP/W analysis is shown in Figure 18. The head of the

reservoir is set to the maximum allowed water level, which corresponds to the freeboard of

2,5 meters.

Figure 18. Initial conditions for the dam as it is designed in SEEP/W.

By using the option of ‘parent analysis’ the SLOPE/W model receives the same geometry

and pore pressures and also includes the calculated seepage head and flux as calculated in

the SEEP/W model.

23

6 PLAXIS2D

“PLAXIS2D is a two-dimensional finite element program, developed for the analysis of

deformation, stability and groundwater flow in geotechnical engineering.” (Brinkgreve, et

al., 2016)

That is the starting phrase describing PLAXIS2D which defines the relations between

different variables that are included in geomechanics. For example to describe stress and

failure criterion induced by external load has been used for a long time with the use of

empirical data, Boussineq to calculate in situ stresses caused by external load and

Terzhagi’s principle for one dimensional consolidation and ultimate bearing capacity are

all examples of these types of calculations that may be performed by hand calculations.

With increased computing power it has become possible to perform more advanced

calculations, for example by calculating with the definite element method and the finite

element method. The older ways of calculating is sufficient in many cases, as seen in

Figure 19 where the Prandtl and Rankine zones are visualised in PLAXIS2D. However for

more advanced constructions and for research FEM is a better tool to use due to its

capability to include more information in the design.

Figure 19. Figure of a typical failure zone in PLAXIS2D (left, compliments of Niclas Lindberg) compared to

empirically derived Prandtl and Rankine zones (right).

6.1 Mesh

After the geometry has been created the model is discretized into a mesh of triangles.

These triangles are made up of nodes and Gauss (stress) points where the numerical

calculations take place as seen in Figure 20. The more triangles there are in the mesh the

more precise the calculation will be performed and with a 15-noded triangle the calculation

will yield more realistic results. In this analysis 15-noded mesh triangles are used since the

matter of data and memory storage is not of an issue, and also for safety factor calculations

15-nodes are preferred due to that 6-noded triangles over predict failure loads. (Brinkgreve,

et al., 2016)

24

Figure 20. Nodes and stress points with their locations in a mesh triangle. (Brinkgreve, et al., 2016)

The stress-strain relation in the model is calculated in the Gauss points where after its

results are extrapolated to the nodes for calculating the deformation. Strain and

deformation are both variables for ‘movement’ of the material but since strains are unit-

less it enables you to put in the materials strain parameters whereas the deformation is the

actual length of deformation the user of the program sees. (Felippa, 2004) It is also

important to receive good quality in the mesh, as seen in Figure 21.

Figure 21. Examples for good and bad mesh quality with different kinds of mesh. (Felippa, 2004)

The integration calculation is subsequently looped until the mesh calculation finds

equilibrium in the model in an iterative process which thereafter produces the result of the

calculation. (Felippa, 2004)

6.2 Plane strain modelling

The assumption of plane strain is used when considering an elongated construction or

geometry. A dam construction could be seen as such, even with a few discrepancies like

depth to bedrock and geometry might differ some. It is up to the engineer to decide if the

plane strain assumption can be applied, where strains in z-direction are set as zero (stress is

however still considered). An example of a plane strain condition is shown in Figure 22.

25

Figure 22. An example of a footing using 3D-modeling where plane strain conditions could be assumed.

(Brinkgreve, et al., 2016)

6.3 Stress field generation

6.3.1 K0 procedure

For soil materials it is important to understand soil’s pressure at rest (lateral pressure),

meaning the ratio of how vertical pressures induced by gravity and external loads induce

horizontal stress, which is shown in the following expression:

𝐾0 =𝜎′𝑥

𝜎′𝑦. (17)

There is also a relation with the strain parameter Poisson’s ratio

𝐾0 =𝜈

𝜈 − 1, (18)

and also the angle of friction, as seen in (Jaky’s empiricial expression);

𝐾0 = 1 − 𝑠𝑖𝑛 (𝜙) (19)

For an overconsolidated soil, much like the highly compacted soil in a dam it is better to

add the over consolidation ratio as a component to receive the correct K0 as shown in

Equation (20) (Axelsson & Matsson, 2016).

𝐾0 = (1 − 𝑠𝑖𝑛 (𝜙))(𝑂𝐶𝑅)𝑠𝑖𝑛 (𝜙) (20)

PLAXIS2D K0-calculation procedure have a drawback when calculating for the geometry

that does not have a linear soil stratum and phreatic surface. This is because the full

equilibrium is only found for linear horizontal surfaces during the procedure. Examples of

such cases are shown in Figure 23. (Brinkgreve, et al., 2016)

26

Figure 23. 1. A vertical structure, possibly a lime cement column. 2. A non-horizontal soil strata. 3. A

lowering of the phreatic surface, possibly due to a draining well nearby. 4. A slope.

All of the examples observed in the figure can be seen in a dam and for the case of this

analysis, K0 procedure cannot be applied. Thus another type of calculation can be

performed called Gravity loading.

6.3.2 Gravity loadings

Gravity loading calculates the soil based on the materials volumetric weight as opposed to

the K0 procedure which adds the soil weight after the calculation is completed. Thus the

initial stresses will be dependent on the use of calculation model, such as Mohr-Coulomb.

The amount of stress thus becomes dependent on the used values in Poisson’s ratio which

has already been seen to have a relation with the K0. (Brinkgreve, et al., 2016)

6.4 Boundary conditions and finding a solution

A finite element analysis is a discretized partial differential equation that is solved by using

predetermined parameters, (previously mentioned in paragraph 5.1.2) boundary conditions

are the foundation for finding a solution for a finite element analysis (Geo-Slope, 2012). It

sets the boundaries for which the initial parameters can act. The program tries to solve the

partial differential equations by successive approximation of the solution for the iteration

as seen in Equation (21), where the global error is set in PLAXIS2D as a “tolerated error”.

Figure 24 illustrates how the iterative process finds the approximate solution. (Note:

Figure 24 illustrates the Newton-Raphson method to approximate the solution; Plaxis

however uses the Gaussian method. The figure does however still visualize the relation

with “Out of balance forces” well. The Newton-Cotes method is also used in some

elements such as geogrids and interfaces)

𝐺𝑙𝑜𝑏𝑎𝑙 𝑒𝑟𝑟𝑜𝑟 =∑ 𝑂𝑢𝑡 𝑜𝑓 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 𝑓𝑜𝑟𝑐𝑒𝑠

∑ 𝐴𝑐𝑡𝑖𝑣𝑒 𝑙𝑜𝑎𝑑𝑠 + ∫∆𝜀 ∗ ∆𝜎

∆𝜀 ∗ 𝐷𝑒∆𝜀 (𝐶𝑆𝑃) ∗ ∑ 𝐼𝑛𝑎𝑐𝑡𝑖𝑣𝑒 𝑙𝑜𝑎𝑑𝑠 (21)

27

Figure 24. Iterative process for finding the approximate nonlinear solution for a finite element analysis.

(Matsson, 2016)

6.4.1 Calculating the Safety Factor

Calculation of the safety factor in PLAXIS2D is performed by using a so called “phi/c-

reduction”. By dividing the initial strength parameters and with a gradually reduced

strength in the model, a final global safety can be measured in iterative steps. The equation

used for this is shown in (Brinkgreve, et al., 2016)

∑ 𝑀𝑠𝑓 =𝑡𝑎𝑛 (𝜙𝑖𝑛𝑝𝑢𝑡)

𝑡𝑎𝑛 (𝜙𝑟𝑒𝑑𝑢𝑐𝑒𝑑)=

𝑐𝑖𝑛𝑝𝑢𝑡

𝑐𝑟𝑒𝑑𝑢𝑐𝑒𝑑. (22)

Since PLAXIS2D finds the global safety factor a predetermined circular shear failure

surface is not possible to be created like it is possible in SLOPE/W and other programs

based on the limit equilibrium method. Thus only the most critical safety factor will be

calculated and shown in the model. If two or more surfaces are found it is an indication

that the model is yet to find a solution and more calculation steps may be needed.

(Brinkgreve, et al., 2016)

6.5 Constitutive model

6.5.1 Choice of constitutive model

Different types of soil behave differently than other materials such as steel and concrete,

which behaves linearly. To simulate soil material, models such as Hardening soil and Soft

soil can be used. These methods consider strain parameters that enable simulation of

deformation in a proper way. However, for a stability calculation, as the case of this study

they all follow the Mohr Coulomb failure criterion and thus there is no need to use another

model. (Brinkgreve, et al., 2016)

6.5.2 Mohr Coulomb

Mohr Coulomb is a linear elastic perfectly plastic constitutive model. The soil will behave

according to Hooke’s law in the elastic face while behaving perfectly plastic in the plastic

28

phase according to the Mohr-Coulomb failure criterion. An illustration of the behaviour is

shown in Figure 25.

Figure 25. Behaviour of a linear elastic perfectly plastic model and how unloading will perform (left)

(Brinkgreve, et al., 2016) The figure to the right compares the Mohr Coulomb model to real soil behaviour.

(Gouw, 2014)

A perfectly plastic model such as Mohr-Coulomb has a theoretical fixed yield surface thus

a constant stiffness. Mohr-Coulomb is usually used for engineers to give a rough

estimation of a deformation problem which will be considered in this study. (Brinkgreve,

et al., 2016)

The Mohr-Coulomb yield surface is described with six functions that include the strength

parameters of the soil (cohesion and angle of friction) as well as the effective principal

stresses (pore pressure is included, thus effective parameters). These equations visualises

the yield surfaces shown in Figure 26. (Brinkgreve, et al., 2016)

29

Figure 26. Yield surface for the material model Mohr-Coulomb, note that Plaxis programs use negative

values for compressive stresses. (Brinkgreve, et al., 2016)

For positive values on strains another strength parameter is introduced, dilatancy angle (φ).

The dilatancy angle is a required parameter if positive values for volumetric strain are

considered both for dense or very coarse angular soil. In Table 5 the basic parameters used

in the Mohr-Coulomb model are shown. (Brinkgreve, et al., 2016)

Table 5. Parameters used to create Mohr-Coulombs yield surface. (Brinkgreve, et al., 2016)

c Cohesion

ϕ Angle of friction

φ Angle of dilatancy

E Young’s modulus

ν Poisson’s ratio

30

6.6 Finished model

The finished model for the PLAXIS2D analysis is shown in Figure 27. Note that the

phreatic surface inside of the geometry does not matter how it is drawn for a steady state

calculation. This is because of input permeabilities where the model tries to find the head

using Darcy’s law when a gradient is considered. This enables the model to change its

ground water head in different phases. (Brinkgreve, et al., 2016)

Figure 27.Initial conditions as they are designed in PLAXIS2D.

6.7 Calculating phases

6.7.1 Initial phase

By using ‘gravity loading’ in the initial phase the in situ stresses are calculated at the first

phase instead of conventionally build the dam in segments.

6.7.2 Internal erosion

The internal erosion phase was calculated using the plastic calculation method where the

internal erosion analysis was performed on three different locations in the core where

suffusion has occurred. The material loss was done in regards to loss of mass with

increasing void ratio. By using the different permeabilities shown in Table 3 the increased

permeability in that part of the core will simulate the internal erosion. The volume of the

area affected by suffusion was also increased ranging from 50-500 mm in diameter. The

results from these are shown in Table 6.

As the analysis progressed it was seen that the pore pressures downstream was not affected

in much regard due to the piping first specified. Because of this a couple of extreme cases

were performed with the results shown in Table 7.

31

6.7.3 Safety analysis

New pore pressures has been calculated in the internal erosion phase and thus the last step

is to calculate the safety factor by using the “phi-c-reduction”-method on the new stress

situation.

7 RESULTS

7.1 Stability

The overall stability was investigated for the dam with different dimensions on the pipe

shown in Figure 28 with altered void ratio e=1 and two pipes. The results for the global

stability calculations performed in PLAXIS2D are shown in Table 6.

Figure 28. Example of piping occurring in the dam and thus increasing the total flow downstream.

32

Table 6. Results of the safety factor calculation performed in PLAXIS2D.

Safety Calculation Void ratio (e)

0,25 0,3 0,35 0,4 1

Depth from

crest (m)

Diameter

(mm) Safety Factor (ΣMsf)

5 50 1,365 1,365 - - -

100 1,369 - - 1,38 -

200 - - - - -

300 1,37 1,37 - 1,37 -

400 1,375 1,375 - - -

500 1,376 1,373 1,374 1,375 -

8 50 1,378 1,376 1,377 1,377 1,384

100 1,377 1,373 1,378 1,375 1,395

200 1,379 - - - 1,389

300 - - 1,425 1,45 -

400 1,381 1,382 1,382 1,378 1,388

500 1,375 1,375 1,376 1,377 1,391

11 50 1,381 1,376 1,381 1,376 1,386

100 1,381 1,38 1,381 1,381 1,39

200 - 1,369 1,373 1,377 1,38

300 1,375 1,377 1,376 1,377 1,388

400 - - - - 1,379

500* - 1,373 1,373 1,373 1,373

*Note: 500mm erosion was remeshed for 11m depth

The results in Table 6 found the most critical failure surface to be on the upstream side of

the dam, as seen in Figure 29.

Figure 29. Upstream failure surface found in the model.

33

Some extreme cases with higher permeability and void ratio were also analysed in order to

see if the safety factor will present significant difference would be seen. These results are

shown in Table 7.

Table 7. Results of the safety factor calculation performed in PLAXIS2D.

Safety Calculation Void ratio (e)

0,94 1 2

Depth (m) Diameter (mm) Safety Factor (ΣMsf)

5 1000 1,42 1,42 1,424

5 and 8 1000 x 2 1,384 1,394 1,383

5 and 8 connected 4000 1,377 1,376 1.392

In these results the most critical surface was found on the downstream side, as shown in

Figure 30.

Figure 30. Downstream failure surface found in the model.

34

7.2 Displacement

Some deformation, were found when piping occurs in the dam, as shown in Figure 31.

Even though the displacement is observed to be found in micrometer it is still interesting to

see what initiates the “swelling” phenomena observed.

Figure 31. Total displacement in the dam when piping is initiated on a 1000mm pipe at 5m depth with void

ratio e=1. Note the scaled up deformation (over 3000 times).

A cross cut was made through the centre of the embankment dam where the primary

effective stress was observed through four different analyses. The first analysis is

performed on an eroded pipe at 5 meters depth where both the density and permeability is

changed. The second analysis is the same but where the permeability is unchanged. The

third analysis only changes the permeability and keeps the density the same. The fourth

one is the dam with no change in both permeability and density. This was done to show the

effects on the results due to changes of the mentioned parameters. The results from the four

analyses are shown in Figure 32.

35

Figure 32. Primary effective stress at different levels in the core with changes to density and flow.

-120,00

-110,00

-100,00

-90,00

-80,00

-70,00

-60,00

-50,00

-40,00

4,05,06,07,08,09,010,0

Effe

ctiv

e p

rim

ary

stre

ss (σ

')

Level (m)

Effective primary stress change with pipe at 5m depth

e=1 e=1, Only change of density e=1, Only change flow No erosion

36

8 DISCUSSION

8.1 Permeability and void ratio

The high permeability where e=1 is rather unrealistic, if the water would be able to flow at

1 m/s in a medium such as soil it would probably bring with all material due to the seepage

forces affecting the soil, it does however work in a theoretical calculation such as this. The

changed permeability, connected to the void ratio, was also tested for a lower void ratio

(e=0,35), if the pipe would theoretically have undergone settlements, this result is shown in

Figure 33. This figure shows that the permeability is within the area of sands and gravel

(k≈0,01m/s) which makes the chosen permeability in the analysis plausible.

Figure 33. The results from the internal erosion analysis put in Lambe and Whitman’s diagram of

permeability changes due to different void ratios. (Lambe & Whitman, 1969)

The small change in pore pressure may be due to several reasons. The first reason is that

boundary conditions do not change because of the steady state calculation. If boundary

conditions would change the calculation would have to be performed by using a transient

flow analysis.

Due to only the core being affected by piping it will not be able to create a great flow in the

pipe due to the surrounding soil with lower permeability. Thus if the pipe would be

extended through the filters the results might have become a bit different.

37

It should also be noted that this analysis was performed on an earth and rock fill dam

which has a highly permeable shell, thus a creation of a higher phreatic surface and thus

pore pressure is harder due to its draining capabilities.

8.2 Stability

As observed in Table 6 there are several points where the safety factor is not indicated.

This is due to the program failing to converge, meaning that equilibrium for the flow could

not be found (Ultimate state not reached, Error 34 in PLAXIS2D) (Brinkgreve, et al.,

2016). The solutions to bypass this problem is to either create a new mesh in the model by

making the mesh finer or courser since the mesh might be appropiate for the calculation to

converge, or higher the tolerated error described in Equation (21) and Figure 24. Creating

a new mesh does however create new locations for stress points where the calibration takes

place, and the model has proven to be sensitive to mesh changes due to soil body collapse

if the mesh is “too fine”. Also if the mesh is changed it is harder to compare the results due

to different stress fields created. Therefore the solution was to change the tolerated error

for the out of balance flux which uses the same equation for the forces but for the flow

equilibrium equation instead and keep the mesh coarseness factor same in all models. The

test with 500 mm pipe at 11 meters depth is a special case which was performed to see if

the mesh was changed (coarser mesh) and how this affected the safety factor. The results

show that there is no change in safety factor with increased permeability in the pipe.

All tests in Table 6 has had their critical shear surface upstream which would not be a

concern to the guidelines set by RIDAS since this consider as critical condition for a full

reservoir in the downstream slope and minimum safety factor required equal to 1.5. It is

also observed that the shear surface tend to be located on materials with less strength than

the rockfill (the filters), similar to SLOPE/W’s results.

It is observed that the safety factor gradually becomes slightly higher in almost all

simulations with higher void ratio. This can be due to several factors but the most probable

cause is the lowering of the pore pressure upstream which yields higher effective stresses.

This is confirmed by the extra analysis performed and showed in some extreme cases with

higher permeability and void ratio (Table 7). The most critical surface changed to be

downstream where a higher pore pressure is built as well as seepage forces affecting the

slope.

8.3 Displacement

Since PLAXIS2D calculates with FEM and thus calculates the stress strain relation of the

model it is good to analyse the stresses that has created the displacement due to set strain

parameters.

The results from the four different cases show that the increased flow and density affects

the primary effective stress as seen in Figure 32. The flow change does not directly affect

the stresses but with increased flow there is a lowering of pore pressure that increases the

effective stress above the pipe. When the density is lowered due to material being eroded

the primary effective stress is decreased below the pipe. With both combined, in the first

38

analysis it is observed that both affect the dam in this way but since the dam seem to

‘swell’ with increased erosion it means that the loss of density in the pipe affects the

primary stresses more than the decrease of pore pressure.

This displacement is however minimal and might not be able to be observed if such a

method would be used when looking at the dam in terms of dam safety and see if internal

erosion has occurred. Also to note is that the Mohr-Coulomb model is used and thus the

displacement may be observed but should not be seen as a good presentation for real in

movement.

39

9 CONCLUDING REMARKS

9.1 Results and discussion

The viability of using a FEM program in internal erosion analyses.

Even though some calculating problems were encountered during the work, the results are

still plausible during set circumstances. It should still be considered that FEM is incapable

of creating an eroded zone itself due to the reliance of continuum mechanics.

The impact of internal erosion on unit weight and conductivity.

The results show that the approach made to calculate the different changes in permeability

was a plausible method. Even though the permeability at higher void ratios is unrealistic it

is already unrealistic to have void ratios at that scale in a soil medium and the soil would

probably have undergone settlement before.

Variations in the stability of the dam due to internal erosion in terms of safety

factor.

The calculated factor of safety evaluated in the model at different sizes of pipes and

locations is changing but only to a minimal degree. This is due to only a slight change of

the phreatic surface that affects the effective strength in the slopes both downstream and

upstream of the dam.

9.2 Further studies

1. A similar study could be conducted but on an earth embankment dam instead of a

rock fill dam, in the case that the shell in the embankment dam would have a lower

permeability.

2. The flow in the toe of the dam could be studied and may be used to evaluate with a

rising water level in that calculated phase to change the boundary condition.

3. How much the seepage forces affect the dam is unknown and would be interesting

to analyse.

4. The analysis performed in this report only had a pipe in the core, it would be

interesting to see what a continuation of the pipe into the filter(s) would affect the

dam.

5. A laboratory experiment of the calculated permeability with different void ratios.

40

REFERENCES

Agahei Araei, A., Soroush, A. & Rayhani, M., 2010. Large-Scale Triaxial Testing and

Numerical Modeling of Rounded and Angular Rockfill Materials, Sharif: Sharif University

of Technology.

Axelsson, M. & Matsson, H., 2016. Geoteknik. 1:1 ed. Lund: Studentliterratur AB.

Brinkgreve, R., Kumarswamy, S. & Swolfs, W., 2016. Plaxis material model manual,

Delft: Plaxis.

Brinkgreve, R., Kumarswamy, S. & Swolfs, W., 2016. Plaxis reference manual, Delft:

Plaxis.

Craig, R. F., 2004. Craig's Soil Mechanics. 7th ed. Oxford: Taylor & Francis.

Felippa, C., 2004. Introduction to Finite Element Method. Boulder: University of

Colorado.

Foster, M. & Fell, R., 1999. Assessing embankment dam filters which do not satisfy design

criteria. Sydney: University of New South Wales.

Geo-Slope, 2012. SEEP/W. Calgary: s.n.

Geo-Slope, 2012. SLOPE/W. Calgary: s.n.

Gouw, T.-L., 2014. Common mistakes on the Application of Plaxis 2D in Analyzing

Excavation Problems, Jakarta: Research India.

ICOLD, 2017. ICOLD. [Online]

Available at: http://www.icold-cigb.net/GB/world_register/database_presentation.asp

[Accessed 23 05 2017].

Kovacevic, N., 1994. Numerical analyses of rockfill dams, cut slopes and road

embankments, London: University of Lonfon.

Kumarswamy, S. & Swolfs, W., 2016. PLAXIS 2016, Delft: PLAXIS.

Kunitomo, N., 2000. Design and Construction of Embankment Dams, Nagoya: Department

of civil engineering, Aichi Institute of Technology.

Lambe, T. W. & Whitman, R. V., 1969. Soil Mechanics - SI version. New York: John

Whiley & Sons.

Larsson, R. et al., 2007. Skjuvhållfasthet, Linköping: Swedish Geotechnical Institute.

Matsson, H., 2016. An overview of the mathematical background to the nonlinear finite

element method. [Sound Recording] (Luleå University of Technology).

Rahardjo, H., Eng Choon, L. & Tami, D., 2004. Capillary Barrier for Slope Stabilisation,

Nanyang: Nanyang Technological University.

Ren, X. et al., 2016. A relation of hydraulic conductivity - void ratio for soils based on the

Kozeny-Carman equation. Science Direct, 31 August, p. 9.

41

Rick, A. & Wahlqvist, P., 2010. Säkerhetsanalys av befintliga fyllningsdammar, Lund:

Faculty of Engineering at Lund University .

Sibille, L., Marot, D. & Sail, Y., 2015. A description of internal erosion by suffusion and

induced settlements on cohesionless granualar matter., s.l.: Acta Geotechnica.

Silva, I., Lindbom, J., Vikander, P. & Laue, J., 2017. Assesment of internal erosion in the

glacial till core of a swedish dam.. Prauge, Czech Republic, 85th Annual Meeting of

ICOLD.

Svensk Energi, 2012. RIDAS. s.l.:Svensk Energi.

Szymkiewicz, A., 2013. Modelling Water Flow in Unsaturated Porous Media Accounting

for Nonlinear Permability and material Heterogeneity, s.l.: Springer.

Vahdati, P., 2014. Identification of Soil Parameters in an Embankment dam by

Mathematical Optimizations, Luleå: Luleå University of Technology.

42

APPENDIX A

Silt Sand Cobbles 0,002 0,05 2 6,3 200 Gravel

43

APPENDIX B

44

APPENDIX C

INITIAL PARAMETERS Name Unit Rock fill Course filter Sandy gravel Core (moraine)

Drainage type - - Drained Drained Drained Drained

Strength Parameters

Cohesion c kN/m2 1 1 1 1

Friction angle ϕ ˚ 39 32 32 35

Dilatancy angle φ ˚ 15 2 2 5

Unit Weight Parameters

Dry unit weight γ kN/m3 17 18 22 21,4

Saturated unit weight γsat kN/m3 20,4 21,0 23,5 23,1

Strain parameters

Young's modulus E' kN/m2 120000 170240 170240 83700

Poisson's ratio ν - 0,33 0,33 0,33 0,35

Pore parameters

Initial void ratio e - 0,529 0,444 0,182 0,215

Porosity n - 0,346 0,308 0,154 0,177

Groundwater flow

Model used - - Van Gen. Van Gen. Van Gen. Van Gen.

Conductivity kx/ky m/s 0,01 0,005 0,0006 1,20E-07

Conductivity kx/ky m/day 864 432 51,84 0,010368

45

APPENDIX D