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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
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APPENDIX A
Silt Sand Cobbles 0,002 0,05 2 6,3 200 Gravel
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APPENDIX B
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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
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APPENDIX D