MSc_Thesis_Report_-_Saeed_Hosseinzadeh.

MSc_Thesis_Report_-_Saeed_Hosseinzadeh.
AES/GE/12-41
Numerical assessment and validation of a
swelling rock model
October 2012
Saeed Hosseinzadeh
Title
:
Numerical assessment and validation of a swelling
rock model
Author(s)
:
Saeed Hosseinzadeh
Date
:
October 2012
Supervisor(s)
:
Dr. ir. R. B. J. Brinkgreve
Dr. ir. D. J. M. Ngan-Tillard
TA Report number
:
AES/GE/12-41
Postal Address
:
Telephone
Telefax
:
:
Section for Geoengineering
Department of Geoscience & Engineering
Delft University of Technology
P.O. Box 5028
The Netherlands
(31) 15 2781328 (secretary)
(31) 15 2781189
Copyright ©2012
Section for Geoengineering
All rights reserved.
No parts of this publication may be reproduced,
Stored in a retrieval system, or transmitted,
In any form or by any means, electronic,
Mechanical, photocopying, recording, or otherwise,
Without the prior written permission of the
Section for Geoengineering
Abstract
Abstract
In this research project, an assessment and validation of a swelling rock model (Benz, 2012),
which is applicable to anhydrite swelling rock, was carried out through the Soil Test Facility of
PLAXIS2D Finite Element Software. The validation process was conducted through the
simulation of different element tests including stress-controlled oedometer test, and straincontrolled uniaxial compression test. A sensitivity analysis and parameter variations were
carried out within the aforementioned tests.
The obtained results and recommendations from this study can be used for conducting a case
study of tunnelling within swelling rock. This will help to give a better understanding of swelling
deformation occurrence around an excavation leading to a better tunnel support design.
Furthermore, the way by which the swelling rock model parameters should be dealt with for a
practical application is provided.
I
Acknowledgments
Acknowledgments
I would like to thank my supervisor, Dr. Ronald Brinkgreve, for his continuous guidance and
encouragement. I also thank him for providing me with the PLAXIS Key, which was a significant
help to my work. I would like to thank my co-supervisor, Dr. Dominique Ngan-Tillard, for her
kindness and support during the entire time of my project.
I would like to thank Prof. Hans de Ruiter, Prof. Timo Heimovaara, and the board of EMMEP, for
organising such a splendid programme and for their sincere support during my entire Masters
study. I wish to express my gratitude to the European Commission, FEMP and its members for
the prestigious Erasmus Mundus scholarship, and for funding the programme.
I would like to thank Prof. Thomas Benz for providing me with the swelling rock model and Dr.
Bert Schaedlich for the useful personal communications I had with him and his sincere
assistance. I would also like to thank Karel Karsen, Ymke Verhoogh and John Stals for their
great help.
I would like to especially thank my friends: Dr. Sadegh Babaii Kocheksarii; Dr. Hadi Eghlidi;
Sanaz Saeid, Wojtek Alberski; Mahmood Jafari; Marjan Moghaddasi; Ebrahim Fathi Salmi,
Antonia Makra, Mohammadreza Barzegari, Swarna Kumarswamy; Stephen Thergesen; Julia
Ridder; Jack Pilkington and Jackson Kawala; for their sincere support and assistance during my
studies, for which I am always grateful.
Finally, I would like to thank my beloved parents, to whom I owe all the achievements and
success in my life and dedicate this work to them as well. I would like to thank categorically my
siblings, Leila and Hamid, who encouraged and supported me the most for continuing my
studies abroad. I am always grateful to their fruitful and sincere advice throughout my life.
II
Table of Contents
Table of Contents
Abstract .................................................................................................................................. I
Acknowledgments .................................................................................................................. II
Chapter 1: Introduction ........................................................................................................... 1
Chapter 2: Theory of swelling ................................................................................................. 3
2.1 Part I: Literature review on swelling of anhydrite bearing rocks ........................................... 4
2.1.1 General ......................................................................................................................... 4
2.1.2 Rock geology ................................................................................................................ 4
2.1.3 Causes of swelling......................................................................................................... 5
2.1.4 Laboratory testing of swelling ......................................................................................... 7
2.1.5 Laboratory and in situ observations as well as lining principles in swelling rock ................ 8
2.1.6 Constitutive formulations of swelling rock...................................................................... 11
2.2 Part II: Mathematical formulation of the swelling rock model & its concept ........................ 12
2.2.1 Elastic stress-strain behaviour...................................................................................... 12
2.2.2 Rock strength .............................................................................................................. 13
2.2.3 Visco-plastic stress-strain behaviour............................................................................. 14
2.2.4 Isotropic constitutive law for the stress -strain state due to final swelling ........................ 16
2.2.5 Time dependency of swelling ....................................................................................... 19
2.2.5.1 Dependency of swelling upon water access to the rock .............................................. 21
2.2.5.2 Development of the existing approach by Wittke-Gattermann (1998) .......................... 23
2.2.6 Complete stress strain behaviour ................................................................................. 24
2.2.7 Model‟s routines in the model under study defined by Benz (2012) ................................ 24
2.2.8 Conclusions ................................................................................................................ 25
Chapter 3: Soil Test Facility and layout of numerical simulations ............................................ 27
3.1 Introduction .................................................................................................................... 27
3.1.1 Implementation scheme ........................................................................................... 27
3.1.2 Time step ratio ......................................................................................................... 28
3.1.3 Sign convention in the Soil Test Facility .................................................................... 29
3.2 Element tests ................................................................................................................. 29
3.2.1 Oedometer test ........................................................................................................ 31
3.2.2 Uniaxial compression test ......................................................................................... 32
3.3 Simulation layout ............................................................................................................ 33
III
Table of Contents
Chapter 4: Results discussions and interpretations ................................................................ 35
4.1 Different time step ratios within implicit and explicit scheme ............................................. 35
4.1.1 Implicit scheme ........................................................................................................ 35
4.1.2 Explicit scheme ........................................................................................................ 37
4.1.3 Low applied loads .................................................................................................... 38
4.2 Influence of Poisson‟s ratio (ν) ........................................................................................ 39
4.3 Influence of swelling potential in horizontal direction (k q,t) ................................................. 40
4.4 Proposed critical time step ratio of 0.0526 or 1/19 ............................................................ 41
4.5 Influence of material stiffness (Young‟s modulus or E) ..................................................... 42
4.6 Influence of maximum swelling pressure in horizontal direction (σ0,t)................................. 44
4.7 Validation of swelling potential parameter (Kq) ................................................................ 46
4.8 Effect of A0 and Ael swelling time parameters.................................................................... 47
4.9 Model‟s stress path prediction in oedometer test.............................................................. 49
4.10 Evaluation of yield function ........................................................................................... 51
4.10.1 Uniaxial compression test via swelling rock model ................................................... 51
4.10.1.1 Elastic stress-strain behaviour .......................................................................... 52
4.10.1.2 Yielding ........................................................................................................... 52
4.10.2 Uniaxial compression test – Mohr-Coulomb material model ..................................... 52
4.11 Influence of cohesion (c‟) and internal friction angle (  ' ) ................................................ 53
4.12 Influence of dilatancy angle or  and Apl and Apl max swelling time parameters .............. 54
4.12.1 Dilatancy angle (  ) ............................................................................................... 54
4.12.2 Apl and Apl max influence on the swelling time parameter ......................................... 56
4.13 Conclusions ................................................................................................................. 57
Chapter 5: Conclusions and recommendations...................................................................... 59
5.1 Conclusions ................................................................................................................... 59
5.2 Recommendations for further studies .............................................................................. 61
5.2.1 Simulations in the Soil Test Facility ........................................................................... 61
5.2.2 A case study of tunnelling within anhydrite bearing rocks using PLAXIS2D................. 61
5.2.3 Parameter selection for a practical application ........................................................... 62
Nomenclature ...................................................................................................................... 66
References .......................................................................................................................... 67
Appendix A: List of results of the element tests‟ runs ............................................................. 69
IV
Table of Contents
Table of Figures
Figure 1: Transformation of anhydrite with orthorhombic crystal system into gypsum with monoclinic
crystal system – density of gypsum is approximately 2.32 g/cm3.............................................................. 6
Figure 2a) Experimental results (swelling strain vs. swelling pressure) for anhydrite claystones; (b)
Monitoring results from the test adit of Freudenstein tunnel, (1) Time development of the floor heave
for different support pressures; (2) floor heave dependency on support pressure ................................... 9
Figure 3: (a) Swelling strain and swelling pressure relation behaviour observed macroscopically in the
oedometer test; (b) Floor heave (u) and support pressure (Ps) relation (Anagnostou, 2007) .................. 10
Figure 4: The conventions used for transversely isotropic implementation in the model........................ 13
Figure 5: Mohr-Coulomb failure criterion shown in both σ1-σ3 and τ -σn diagrams (Wittke-Gattermann,
1998)..................................................................................................................................................... 13
Figure 6: One dimensional rheological model for showing elastic viscoplastic behaviour (Runesson, 2005)
.............................................................................................................................................................. 14
Figure 7: Schematic stress-strain and strain-time diagrams for elastic viscoplastic behaviour (WittkeGattermann, 1998) ................................................................................................................................ 15
Figure 8: Swelling tests after Huder and Amberg (1970) - Loading scheme in an oedometer .................. 16
Figure 9: Swelling strain against applied stress indicating swelling stress dependency–Huder and Amberg
test of mudstone samples containing anhydrite from the medium gypsum horizon in Stuttgart area,
Germany (Wittke-Gattermann, 1998) .................................................................................................... 17
Figure 10: Swelling strain versus time diagram indicating time dependency of swelling - Huder and
Amberg test results on a sample from Stuttgart area (Wittke-Gattermann, 2003) ................................. 20
Figure 11: Swelling time parameter as a measure of swelling rate ......................................................... 20
Figure 12: Change of discontinuity aperture width due to viscoplastic strains of joint surfaces (WittkeGattermann, 1998) ................................................................................................................................ 21
Figure 13: Possibilities of laboratory tests simulations in the Soil Test Facility of PLAXIS2D software...... 27
Figure 14: Loading conditions in a uniaxial compression test as an example for a vertical load which is
controlled with maximum strain of -0.1 % in the vertical direction and is applied in 100 steps ............... 33
Figure 15: Layout of the entire numerical simulations in order to assess the model under study ........... 34
Figure 16: Oedometer test R2 to R5 together - Implicit scheme - Time step ratio = 0.02 to 1 – Bias
between theoretical value and numerical results................................................................................... 36
Figure 17: Oedometer test's R17 - Explicit Scheme - Swelling strain decreased due to effect of tensile
strength of 1000 kPa ............................................................................................................................. 38
Figure 18: Effect of Poisson's ratio on lateral stressing and total vertical strains –Implicit schemeoedometer test – R23 ............................................................................................................................ 39
Figure 19: Influence of material stiffness on the lateral stress in oedometer runs - R45-R49 over the
same period of time .............................................................................................................................. 43
Figure 20: Influence of material stiffness on the final vertical swelling strain in oedometer runs – R50R54 over the same period of time – Kq,t=0 ............................................................................................ 44
Figure 21: Effect of maximum horizontal swelling pressure on the total vertical strain and lateral
stressing ................................................................................................................................................ 45
V
Table of Contents
Figure 22: Schematic diagram of the swelling strain versus logarithmic applied load diagram shown in
Figure 9 based on the experimental results obtained from S-Bahn Stuttgart project in Germany as well as
rough approximation read off data of swelling strains and the applied loads ......................................... 46
Figure 23: Influence of A0 swelling time parameter on the final swelling strain – vertical strain against
time over 100 days ................................................................................................................................ 47
Figure 24: Influence of Ael swelling time parameter on the final swelling strain - vertical strain against
time ...................................................................................................................................................... 48
Figure 25: (a) Major Principal Stress vs. vertical strain- (b) Major Principal Stress vs. Minor Principal
Stress -Model’s stress path prediction- load step of -130 kPa – Phase 1) Elastic response for which the
amount of strain is calculated through Eq.46; Phase 2) Both vertical and lateral stresses equalise each
other over a constant period of time; Phase 3) horizontal stresses keep on increasing while vertical
stress is constant (major and minor principal stresses swap) – R76 ........................................................ 50
Figure 26: (a) Horizontal stress vs. time- both σ3 and σxx against time curves together - Model’s stress
path prediction in the oedometer test – (b) Vertical (applied) stress vs. time - load step of -130 kPa – R76
.............................................................................................................................................................. 50
Figure 27: Vertical stress versus vertical and lateral strain – uniaxial compression test via swelling rock
model – R77 .......................................................................................................................................... 51
Figure 28: Vertical stress vs. vertical and lateral strain – uniaxial compression test via Mohr-Coulomb
material model – R78 ............................................................................................................................ 53
Figure 29: Influence of strength parameters – R79-R82 in uniaxial compression test ............................. 54
Figure 30: Schematic bi-linear curve of volumetric strain vs. vertical strain ............................................ 55
Figure 31: Effect of dilatancy angle on plastic volumetric strain - volumetric strain vs. vertical strain ..... 55
Figure 32: Oedometer test – R1, R3, R4, R5, R6, R7, and R8 – Implicit scheme – ԑyy & σxx & σyy vs. time
curves ................................................................................................................................................... 69
Figure 33: Oedometer test’s R9, R11, R12, R13, R14, R15, and R16 – Explicit scheme – ԑyy & σxx & σyy
vs. time curves ...................................................................................................................................... 70
Figure 34: Oedometer test’s R18, R19, R20, R21 and R22 – with zero tensile strength - Implicit scheme ԑyy & σxx & σyy vs. time curves ............................................................................................................. 71
Figure 35: Oedometer test’s R18’, R19’, R20’, R21’ and R22’ – with tensile strength of 100 kPa - Implicit
scheme - ԑyy & σxx & σyy vs. time curves .............................................................................................. 72
Figure 36: Oedometer test’s R23’ (a), R23’ (b), and R23’ (c) - Implicit scheme - ԑyy & σxx & σyy vs. time
curves ................................................................................................................................................... 73
Figure 37: Oedometer test’s R24, R25, R26, R27 and R28 – Implicit scheme –influence of horizontal
swelling potential - ԑyy & σxx & σyy vs. time curves............................................................................... 74
Figure 38: Oedometer test’s Ra, Rb, Rc, Rd, and Re – Implicit scheme – Time step ratio sensitivity analysis
with a low applied load- ԑyy & σxx & σyy vs. time curves ....................................................................... 75
Figure 39: Oedometer test’s Rf, Rg, Rh, Ri, and Rj – Implicit scheme – Time step ratio sensitivity analysis
with a high stiffness material - ԑyy & σxx & σyy vs. time curves ............................................................. 76
Figure 40: Oedometer test’s R29, R30, R31, R32, R33, R34, R35 and R36 – Implicit scheme – with
horizontal swelling potential - ԑyy & σxx & σyy vs. time curves .............................................................. 77
Figure 41: Oedometer test’s R37, R38, R39, R40, R41, R42, R43 and R44– Implicit scheme – without
horizontal swelling potential - ԑyy & σxx & σyy vs. time curves .............................................................. 78
VI
Table of Contents
Figure 42: Oedometer test’s R45, R46, R47, R48, R49 – Implicit scheme – Stiffness effect – Applied load
of -130 KPa - ԑyy & σxx & σyy vs. time curves ........................................................................................ 79
Figure 43: Oedometer test’s R50, R51, R52, R53, and R54 – Implicit scheme – Stiffness effect – Applied
load of -130 KPa - ԑyy & σxx & σyy vs. time curves ................................................................................. 80
Figure 44: Uniaxial compression test’s R88, R89, R90, R91, R92, R93 and R94 – Implicit scheme – Apl and
Apl max effect – Volumetric strain vs. vertical strain curves ................................................................... 81
List of Tables
Table 1: Gypsum Keuper formation geological layering............................................................................ 5
Table 2: Approximated values of swelling heave and swelling pressure reported from in situ and
laboratory results of different tunneling projects within gypsum Keuper ................................................. 9
Table 3: List of parameters of the model under study (Benz, 2012) – p and t used by Benz (2012)
indicates perpendicular and tangential directions in bedding plane respectively ................................... 30
Table 4: Physical and strength properties of the gypsum Keuper rock in gypsum horizon in Stuttgart area
in Germany – K and S are indices used in in Wittke-Gattermann’s model (1998) to indicate vertical joint
sets (perpendicular to beddings) and horizontal bedding, respectively .................................................. 30
Table 5: Some reasonable values of Wittke-Gattermann’s model swelling time parameters used in
numerical simulations – ‘a’ stands for annum (year) .............................................................................. 30
Table 6: Different units and symbols for parameters used in the Soil Test Facility .................................. 31
Table 7: Constant model parameters throughout oedometer runs unless otherwise specified –‘d’ stands
for day – Note: maximum swelling pressure should be input as a positive value in the model input ...... 31
Table 8: Loading conditions as an example for a vertical load of -130 kPa which is instantaneously
applied after which a swelling time of 1000 days in second phase in considered in 100 steps - Oedometer
test........................................................................................................................................................ 32
Table 9: Different time step ratios used throughout oedometer runs .................................................... 32
Table 10: Oedometer runs – Different time step ratios within implicit and explicit schemes – R1 to R16 35
Table 11: Increasing the stability of results within explicit scheme and large time step ratio by inserting a
large value of tensile strength– oedometer R17 .................................................................................... 37
Table 12: Oedometer test's runs – Implicit scheme – R18 to R22- dTime/Eta=1 ..................................... 38
Table 13: Effect of Poisson's ratio on the difference in final swelling strain (different between numerical
results and theoretical solution) ............................................................................................................ 40
Table 14: Oedometer runs – Influence of horizontal swelling potential Kq, t=0 - implicit scheme – R24 to
R28........................................................................................................................................................ 41
Table 15: Oedometer runs - Implicit scheme – R29-R44 – validation of the proposed dTime/Eta=0.0526
.............................................................................................................................................................. 42
Table 16: Oedometer runs - Implicit scheme – R45 to R54 - Time step ratio =0.0526 – Material stiffness
effect..................................................................................................................................................... 43
Table 17: Oedometer runs - Implicit scheme – dTime/Eta=0.0526 – Effect of maximum horizontal
swelling pressure – material stiffness of 4E+06 kPa used in all variations ............................................... 44
VII
Table of Contents
Table 18: Variation of swelling potential in vertical direction in oedometer runs - kq,t=0 and σ0=-750 kPa
- proposed ratio of 1/19 ........................................................................................................................ 46
Table 19: Influence of A0 swelling time parameter on the swelling time dependent behaviour over 100
days....................................................................................................................................................... 47
Table 20: Influence of Ael swelling time parameter on the swelling time dependent behaviour, A0=0.01,
time step = 500/100 .............................................................................................................................. 48
Table 21: Model parameters’ values used in Mohr-Coulomb material model through uniaxial
compression test – R78 ......................................................................................................................... 52
Table 22: Variation of cohesion and friction angle and their influence on the strength of material......... 53
Table 23: Variation of dilatancy angle and its influence on plastic volumetric strain............................... 55
Table 24: Variation of Apl swelling time parameter and its influence on plastic volumetric strain –
A0=0.001 and Ψ=1 deg .......................................................................................................................... 56
VIII
Chapter 1: Introduction
Chapter 1: Introduction
One of the challenges to tunnelling is the ground exhibiting swelling time dependent behaviour.
Swelling is a result of volume increase in ground in the presence of water causing inward
movement of the tunnel perimeter. If the increase in volume, for example, is prevented by the
tunnel lining, large compressive stresses occur, which are so-called swelling pressures. It has
been found that the unleached gypsum Keuper formation containing anhydrite (CaSO4) shows
very high swelling potential (Wittke-Gattermann, 1998). This formation largely consists of hard
shales with intermediate layers of marl and dolomite bedding planes with different sulphate
contents in the form of gypsum or anhydrite.
The design of tunnels in swelling rock formation including Gypsum Keuper has become an
important issue in recent years. This is because many tunnelling projects had to undergo
serious repair work either due to the tunnel floor heave or tunnel lining failure during and after
tunnel construction processes in such rocks as a result of swelling deformation. Example of
such projects includes Alder tunnel in Switzerland. Furthermore, many tunnelling projects have
had to be designed to sustain swelling. Example of such projects includes Stuttgart 21 project
in Germany which its tunnels have been planned for over a length of about 20 km in the
unleached gypsum Keuper (Wittke-Gattermann, 1998).
Several researchers have modelled swelling time dependent behaviour of such rocks using
constitutive laws and have implemented them into numerical methods since 1970s in order to
solve swelling problems (e.g. Wittke et al., 1976, cited in Wittke-Gattermann (1998); Gysel,
1987; Kiehl, 1990; Anagnostou, 1993; Wittke-Gattermann, 1998). However, there are still a lot
of uncertainties concerning tunnelling within anhydrite swelling rock. This is due to the long
duration of the swelling process, unknown or inadequate understanding of its mechanisms, and
also uncertainties regarding the tunnel lining principles within such rock. Moreover, the need to
implement the constitutive laws into numerical methods, in particular finite element method
(FEM) has increasingly been in demand. Numerical methods have had to overcome the
limitations of closed form solutions, to verify the experimental data and to predict swelling
deformation around excavations.
In this research project, an assessment and validation of a swelling rock model is carried out
through the Soil Test Facility of PLAXIS2D Finite Element Software (Brinkgreve et al., 2010).
The swelling rock model which is based on Wittke-Gattermann‟s model (1998) has been
implemented by Benz (2012) into PLAXIS2D as a user-defined model. The constitutive law of
the Wittke-Gattermann‟s model was calibrated based on the results of the monitoring
programme conducted in the experimental gallery of Freudenstein tunnel crossing gypsum
Keuper formation containing anhydrite of high swelling potential in Germany. Present study
therefore focuses on swelling time-dependent deformation of anhydrite bearing rock formations.
The implemented model is assessed and validated through the simulation of different element
tests including stress-controlled oedometer test and strain-controlled uniaxial compression test.
A sensitivity analysis and variation of individual parameters are conducted within the element
tests.
1
Chapter 1: Introduction
The objective of this research study is to investigate the meaning of the model parameters as
well as their influence on the test results. The obtained results and recommendations can be
used for conducting a case study of tunnelling within anhydrite swelling rock. This will help to
give a better understanding of swelling deformation occurrence around an excavation leading to
a better tunnel support design.
The thesis report begins with the literature review on swelling of anhydrite bearing rocks in
Chapter 2. Furthermore, the description of Wittke-Gattermann‟s model (1998), which its
mathematical formulation is the fundamental basis of the model under study is included.
Next, the way by which numerical results are obtained and the simulation plan is included in
Chapter 3. Besides numerical issues for simulation of the elements tests in the Soil Test Facility
of PLAXIS2D are explained.
All the numerical simulations of the aforementioned element tests and results discussions and
interpretations are included in Chapter 4.
A conclusion and the recommendations from the project are included in Chapter 5.
Appendix A includes the list of results of the different runs of the element tests.
2
Chapter 2: Theory of swelling
Chapter 2: Theory of swelling
The fundamental basis of the model under study is according to the swelling rock model
developed by Wittke-Gattermann (1998). The swelling rock model has been implemented into
PLAXIS2D by Benz (2012) which is applicable to anhydrite bearing rock formations including
Gypsum Keuper. Since this research study aims at evaluating the model through sensitivity
analyses and variation of individual parameters, this chapter is divided into two different parts;
First, the literature review which is related to anhydrite bearing rocks is explained. Second, the
mathematical formulation of the model and its concept is included.
First part of this chapter focuses on swelling of anhydrite bearing rocks. This consists of geology
of anhydrite bearing rocks including gypsum Keuper; causes of swelling in such rock formations;
laboratory testing and their limitations; in situ and laboratory observations from different projects
in gypsum Keuper; lining design concepts used in anhydrite swelling rock; and constitutive
formulations of swelling rock.
For description of the model under study, the concept behind Wittke-Gattermann‟s model (1998)
is explained. This comprises of elastic stress-strain behaviour, strength of the rock; visco-plastic
stress-strain behaviour; isotropic constitutive law for the stress-strain state due to final swelling;
time dependency of swelling; and complete stress-strain behaviour. Revised parameters as well
as the model‟s routines defined by Benz (2012) are also explained within this part.
3
Chapter 2: Theory of swelling - Part I: Literature review on swelling of anhydrite bearing rocks
2.1 Part I: Literature review on swelling of anhydrite bearing rocks
2.1.1 General
Based on the results of literature review on swelling rock, it has been revealed that there is often
some confusion in definitions of swelling and squeezing time dependent deformations as a
result of tunnelling. Therefore, it is of crucial importance to realise their difference before looking
into the swelling background.
Both squeezing (as a result of shearing of the ground) and swelling (as a result of volume
increase in the ground) cause inward movement of the tunnel perimeter with time development
(Gioda, 1982). In principle both processes and the corresponding deformations may occur
simultaneously. The deformation and its magnitude may vary depending on the geological
conditions, state of stress, the tunnel shape and geometry, etc. (Barla, 1999).
Rocks composed of particular minerals such as clay minerals exhibit swelling behaviour while
squeezing may occur in any type of material. Squeezing occurrence depends on rock strength
and overburden and may occur anywhere, even simultaneously with swelling in weak rock
(Anagnostou, 1993) especially in rock rich in clay minerals (Whittaker et al., 1990). Moreover,
time dependent deformation as a result of squeezing starts during excavation and can be
controlled by the support system employed; whereas swelling deformation may require
significant time to occur (Whittaker et al., 1990).
Several researchers conclude that a combination of high horizontal stresses and plastic
behaviour initiated by tunnel excavation and observed tunnel crown failure 1 can be the sign of
both squeezing and swelling occurrences (Lo et al. 1975, Vitale, 2004, cited in Moore et al.,
2005; Einstein et al., 1975).
2.1.2 Rock geology
The middle division (middle Triassic) of Keuper formation is so called gypsum Keuper occurring
mostly in South Germany and North Switzerland. Gypsum Keuper may have about 100 m
thickness and is mainly composed of gypsum, anhydrite, clay, silt, marl and carbonate layers.
Anhydrite is the most dominant mineral in such rock formation. From geological point of view,
this formation belongs to 225 to 230 million years ago (Rauh et al., 2007). From bottom to top,
this formation is divided into the base layers of gypsum plaster, the Bochinger horizon, the dark
1
Both squeezing and swelling make a distortion in the whole lining system which may cause a crown
failure. This is related to the tunnels where crown fall-out was observed; for example, shaly rocks of
Western Ontario region in Canada where squeezing and swelling concurred (Kramer et al., 2005).
Furthermore, the roof failure or any damage to the crown might also happen due to the presence of a clay
zone on the over layering of the tunnel face. Such zone which mainly consists of secondary or altered
minerals can swell by accessing to the water (such as seepage flowing from the other layers) resulting in
large stresses at the tunnel crown or on the walls and eventually causing collapse in some cases. Tunnel
collapse as a result of swelling of a clay zone on the overstrata was reported in the literature (e.g.
Seidenfuß, 2006). Having said that, most of the failure in tunnels within gypsum Keuper containing
anhydrite bearing rocks has been due to the noticeable swelling in the tunnel floor (floor heaving), which
has been reported in in situ observations (see 2.1.5).
4
Chapter 2: Theory of swelling - Part I: Literature review on swelling of anhydrite bearing rocks
red clay layer, middle gypsum horizon and Estheria beddings, where these layers contain
different amounts of sulphate rocks. The sulphate contents within the beddings and rock layers
may be massively present or finely scattered. The sulphate content and the approximate
thickness of different layers within gypsum Keuper formation are displayed in Table 1 (WittkeGattermann, 1998). It should be noted that the experimental and laboratory data which were
used for calibration of Wittke-Gattermann‟s model (1998) were obtained from the gypsum
horizon of the gypsum Keuper formation in Stuttgart area shown in Table 1.
The sulphate is found either in the form of anhydrite (CaS04) or gypsum (CaSO4·2H2O).
Anhydrite is a mineral with orthorhombic crystal system and its density varies from 2.89 to 2.98
g/cm3. In nature anhydrite is a rare mineral and anhydrite bearing rocks are more common
(Rauh et al., 2007). Literary definition of anhydrite is „without water‟ indicating it lacks water and
by absorbing water converts into gypsum with an increase in its initial volume (swelling or
expansive phenomenon).
Table 1: Gypsum Keuper formation geological layering
Type of layer
Sulphate content (%)
Thickness (m)
Estheria beddings (Top layers)
5-20
20
Galena (middle) gypsum horizon
30
30
Dark red clay
5-10
10
Bochinger horizon
-
5
Gypsum plaster (base layer)
50
30
2.1.3 Causes of swelling
According to ISRM (ISRM Committee, 1983) the swelling process in general is defined as a
combination of physico-chemical2 reaction involving water and stress relief where stress
changes usually have a significant effect. Swelling results in volume increase and takes place
only in the presence of water and of a particular mineralogical composition (Anagnostou, 1993).
This composition includes the minerals that are capable of a physico-chemical reaction with
water such as clay minerals and anhydrite.
There are two possible causes as origin for swelling occurrence in anhydrite bearing rocks, i.e.
Hydration and gypsum crystal growth (Berdugo, 2007; Alonso et al., 2008).
Hydration or intracrystalline swelling may vary depending on the different types of
ground involved. In the case of anhydrite, hydration is the transformation phase of
anhydrite to gypsum (so called anhydrite theory) either in a closed or in an open system.
In the case of a closed system where adequate amount of anhydrite and water are present,
anhydrite dissolves in water and gypsum precipitation occurs. In a complete conversion of
anhydrite to gypsum, the initial volume of anhydrite increases by about 61% (ΔV) (Figure 1). In
the case of an open system like in situ conditions, depending on the water circulation and the
2
Barla (1999) defined a physico-chemical process as involvement of a chemical reaction that may
develop between water and rock minerals.
5
Chapter 2: Theory of swelling - Part I: Literature review on swelling of anhydrite bearing rocks
reaction kinetics, hydration (Figure 1) or leaching of anhydrite may occur (Anagnostou, 1993
and 2007; Berdugo, 2007; Wittke-Gattermann, 2003). The chemical reaction of the conversion
of anhydrite into gypsum is also shown in Figure 1.
State of stress, water flow and pore pressure are among the most important controlling
factors of hydration mechanism. Claystones with finely distributed anhydrite exhibit a high
swelling potential while massive anhydrite with few fissures will not swell much because
hydration of anhydrite occurs on its surface (Einstein, 1996) in which active clay minerals
such as corrensite have tendency to swell when they are in contact with water (Berdugo, 2007).
In fact, Anhydrite swells only in conjunction with clay minerals and in a rock with clay content of
10-15%, the maximum swelling pressure occurs (Madsen et al., 1990 and 1995, cited in WittkeGattermann, 1998).
Figure 1: Transformation of anhydrite with orthorhombic crystal system into gypsum with monoclinic crystal system –
3
density of gypsum is approximately 2.32 g/cm
Besides hydration, swelling mechanism in anhydrite bearing rocks can also be originated from
gypsum crystal growth which requires two conditions (Berdugo, 2007). First one is
supersaturation of pore water for which evaporation of field waters rich in sulphates is the main
possible mechanism. Supersaturation of pore water is the state when pore water solution
contains more dissolved sulphates than its solubility threshold. Therefore, once the amount of
sulphates has increased in the pore water due to evaporation by changing in temperature,
gypsum crystals gradually start precipitating (crystallisation) until the saturation state is reached.
The second condition is the presence of some open spaces to allow for crystal development.
Despite the two aforementioned mechanisms for swelling of anhydrite bearing rock (hydration
and gypsum crystal growth) the governing mechanisms are still uncertain. This is mainly due to
the long duration of the swelling process of such rocks which may take up to several years even
under laboratory scales (Anagnostou, 2007).
6
Chapter 2: Theory of swelling - Part I: Literature review on swelling of anhydrite bearing rocks
It should be noted that, swelling may also occur during unloading process which is called
mechanical swelling. This mechanism is due to unloading and is caused by dissipation of
negative excess pore pressures (Barla, 1999), i.e. unloading through an excavation in low
permeable soil leads to a reduction in pore pressure. This mechanism mostly occurs in clays,
silt clays and corresponding rocks. This type of swelling is not the subject of this study.
2.1.4 Laboratory testing of swelling
Historically the oedometer test has been one of the most important swelling tests and is used in
tunnelling projects (Barla, 1999, Einstein, 1996). This is due to the fact that the test simulates
the tunnel invert in small scale conditions. However, there has been research into using triaxial
tests as this test allows a better understanding of swelling mechanisms which occur around an
excavation (Steiner, 1993). Triaxial tests map the entire three dimensional stress path, allowing
simulation of an excavation life cycle. In addition, through the triaxial test it is possible to study
the effect of pore pressure, lateral stressing, and changes in stiffness and strength of the
specimen under drained and undrained conditions (Barla, 1999). Despite the advantages of
triaxial tests, they are costly, time consuming and they are not easy to conduct in comparison to
one dimensional oedometer test.
Laboratory testing on rock samples from the Gypsum Keuper were carried out in different
projects (e.g. Grob, 1972; Wittke, 1978; Madsen, 1990, cited in Wittke-Gattermann, 1998). In
most cases, the experiments were either swelling pressure tests, in which the volume is kept
constant and only pressure is measured, or the swelling strain tests in which the load is kept
constant and only displacement is measured, or as Huder and Amberg experiments in which the
swelling strains of the sample at stepwise stress relief are measured (Paul, 1986).
According to the suggested methods given by International Society for Rock Mechanics, three
different tests which are used to measure swelling parameters are briefly explained (ISRM
committee, 1999). All the following laboratory techniques can be used for argillaceous 3 swelling
rock as well as swelling rock containing anhydrite and clay.
(1) In the first test, a conventional oedometer test is used to determine the axial swelling stress,
when the sample is laterally confined and is immersed in the water. The test continues until the
axial force reaches its maximum value or no further axial displacement occurs. Then, the axial
stress is calculated using the measured axial force divided by the cross section area of the
sample.
(2) The second test is used to determine the axial and radial free swelling strain. The test is
conducted in a simple free swell cell where the lateral deformation is allowed and the water can
be added to the cell. The axial strain is measured and recorded as a function of elapsed time
and therefore axial swelling strain versus time curve can be obtained through this test (Barla,
1999). The radial swelling strain is calculated using the measured radial displacement divided
by the initial specimen diameter.
3
Argillaceous swelling rocks do not contain anhydrite, e.g. marls.
7
Chapter 2: Theory of swelling - Part I: Literature review on swelling of anhydrite bearing rocks
(3) The third test which is an improved version of oedometer test originally proposed by Huder
and Amberg (1970)4 is employed to determine the axial swelling strain necessary to reduce
axial swelling stress. First of all, the sample is loaded in a conventional oedometer while it is
restricted laterally and is not immersed in the water. The load is increased up to a certain level
(desired level for the pertinent application) and then the water is added to the cell. The load is
reduced in a stepwise manner until there would be no displacement for any load increment
(Barla, 1999).
It should be noted that there is no adequate and reliable laboratory data regarding anhydrite
swelling rock due to long duration of its swelling process and inadequate understanding of the
dominant mechanism of swelling in such rocks (Anagnostou et al., 2010). Anagnostou et al.
(2010) note laboratory testing in such rocks are challenging due to the following reasons which
are still unanswered:

There is no well-established model taking into consideration the different conditions in in
situ and in the laboratory and therefore the swelling tests are considered as index test to
show if a rock exhibits high or low swelling potential.

Swelling process lasts for a long time which cannot be simulated properly by laboratory
testing. For instance the lab tests for Freudenstein tunnel project in Germany which
started 20 years ago are still ongoing. Additionally, the results cannot be generalised for
all rocks exhibiting swelling potential.

The oedometer test condition prevents swelling in lateral direction resulting in an
overestimation of the swelling pressure. This can be due to the fact that swelling occurs
in nature in three different directions which is not the case in one dimensional oedometer
test. Besides in situ stresses are released when the sample is brought to the laboratory
(Al-Mhaidib, 1999).
2.1.5 Laboratory and in situ observations as well as lining principles in swelling rock
In the past, many tunnelling projects were excavated in Gypsum Keuper. Since this formation is
more common in Germany and Switzerland, many tunnels had to go through serious repair
work in these regions. This is because of either the significant floor heave of the tunnel or failure
in the lining system as a result of swelling deformation. Examples of such projects include
Kappelesberg tunnel, Schanz tunnel, Belchen tunnel and Wagenburg tunnel.
Based on the laboratory results from the oedometer tests, the swelling strain for anhydrite
claystones is largely independent of the pressure within the pressure range of up to 3-4 MPa
(see Figure 2a, Anagnostou et al., 2010). This laboratory observation is not in line with field
4
The difference between the Huder and Amberg test and its modified version goes back to the
compression apparatus used in the test (oedometer). In the third test suggested by ISRM (modified
version), the apparatus is a modified oedometer in which the steel rings with variety of diameters can be
used depending on the sample diameter. This reduces the time required for sample preparation, the risk
of disturbing the sample, etc. which are considered among the advantages of the modified version (ISRM
Committee, 1999).
8
Chapter 2: Theory of swelling - Part I: Literature review on swelling of anhydrite bearing rocks
observations since the in situ measurements indicate an obvious reduction of floor heave with
increasing support pressure (Figure 2b, Anagnostou, 2007). Figure 2b also displays the results
of monitoring measurements of the test adit of Freudenstein tunnel. The difference in rock
behavior observed in laboratory and in situ is due to difference in water circulation (Anagnostou,
1993).
Figure 2a) Experimental results (swelling strain vs. swelling pressure) for anhydrite claystones; (b) Monitoring results from
the test adit of Freudenstein tunnel, (1) Time development of the floor heave for different support pressures; (2) floor heave
dependency on support pressure
In the following, the reported in situ swelling heave and swelling pressure values of some
tunnelling projects in sulphate bearing rocks categorically in gypsum Keuper formation are
explained. Furthermore, the diameter for some of those projects is given and the tunnel strain is
calculated. This allows for a better comparison of the swelling deformation occurrence in those
projects (see Table 2).
Table 2: Approximated values of swelling heave and swelling pressure reported from in situ and laboratory results of
different tunneling projects within gypsum Keuper
Tunnel project
Tunnel
diameter (m)
Swelling
heave (mm)
Swelling pressure
(MPa)
Calculated
tunnel strain
(%)
Kappelesberg
tunnel
-
4700
-
-
Bözberg tunnel
11.8
300
-
≈ 2.5
Schanz tunnel
Wagenburg
tunnel
Freudenstein
tunnel
-
1500
6-9
-
10
340-1000
3.4
≈ 3.4-10
7.3 (section 22 of gallery)
220
1.3; 2.4 and 4.25 –
5
8 (laboratory )
≈3
Belchen tunnel
-
650
≈ 3.6-4.4
-
Heslach II
12.4
-
≈ 3.2; 6.8
(laboratory)
-
Reference
Hawlader et al.
(2003)
Wittke-Gattermann
(1998)
Raul et al. (2007)
Berdugo et al. (2009);
Raul et al. (2007)
Wittke et al. (2004);
Berdugo et al. (2009)
Steiner (1993); Grob
(1972)
Berdugo et al. (2009)
As shown in Table 2, based on field observations, the amount of maximum swelling pressure
varies depending on the host rock of different tunnelling projects. Moreover, the calculated
tunnel strains show the severe occurrence of swelling deformation in some projects including
5
The largest measured laboratory swelling pressures of approximately 8 MPa for the same rock
9
Chapter 2: Theory of swelling - Part I: Literature review on swelling of anhydrite bearing rocks
Wagenburg tunnel (strain ≈ 10%) which had to undergo serious repair work. It is also evident
that laboratory results overestimate the swelling pressure (see Heslach II and Freudenstein
laboratory swelling pressure values in Table 2).
It should be noted that the swelling pressure is linked to support pressure, and hence the tunnel
lining design. In principle, the tunnel lining design is governed by long term deformations and is
based on the ground in which the tunnel is driven. Therefore, swelling process should be fully
understood before designing the tunnel lining. This requires a proper understanding of rocksupport interaction, i.e. the relation between swelling pressure and the support pressure and
different lining principles used for tunnelling applications in swelling rock. However, there are
still a lot of uncertainties regarding different lining principles within such rock (Anagnostou,
2007).
Figure 3a displays that the swelling strain in claystones decreases with the logarithm of swelling
pressure (F). Figure 3b shows the relation between floor heave (u) (the outward movement of
the ground as a result of swelling) and support pressure (P s) depicting that floor heave rapidly
decreases with the exerted support pressure by means of an invert arch. Support pressure is
defined as the applied load to suppress swelling (Ps = F).
Figure 3: (a) Swelling strain and swelling pressure relation behaviour observed macroscopically in the oedometer test; (b)
Floor heave (u) and support pressure (Ps) relation (Anagnostou, 2007)
When designing lining for tunnels excavated through Gypsum Keuper, the resisting (stiff) and
yielding (flexible) support principles are applied (Wittke et al., 2004). In the case of stiff support,
the internal concrete lining of high bearing capacity is used resisting against the swelling. This
will limit the heave significantly. In the case of yielding support, construction of a yield zone
beneath the invert brings a reduction in the swelling pressure. The yield zone consists of a gap
space between the tunnel carriageway and the rock so that the tunnel floor is allowed to heave
without affecting the tunnel operation. A modern technique to allow floor heave to occur is to
have a deformable layer which is installed between the invert arc and the excavation boundary
(Anagnostou, 1993). The Freudenstein tunnel was the first tunnel where a flexible lining was
used in rocks containing anhydrite. The concept of the flexible lining was seriously debated at
the beginning; it is getting more accepted at least for shallow tunnels within the weak rock
(Wittke-Gattermann, 1998; Anagnostou, 2007 and 2010). Based on the experiences from
investigating tunnelling within anhydrite rock formations in Spain, resisting support pressure
stabilise the phenomenon while yielding support allow deformations to continue (Berdugo, 2007;
Ramon et al., 2011).
10
Chapter 2: Theory of swelling - Part I: Literature review on swelling of anhydrite bearing rocks
2.1.6 Constitutive formulations of swelling rock
The basic formulation of constitutive equations for swelling rock was based on the results of
Huder and Amberg tests (1970). The results of these experiments led to the first onedimensional formulation of the swelling law proposed by Grob (1972) and then continued to
become a three-dimensional extension of the constitutive law proposed by Einstein et al. (1972,
cited in Wittke-Gattermann, 1998). In this constitutive law, it was assumed that the volumetric
swelling strains depend only on the 1st Invariant of the stress tensor.
A similar three-dimensional swelling law was developed by Wittke et al. (1976, cited in WittkeGattermann, 1998) according to a numerical calculation method and implemented using the
Finite Element Method. More Calculation methods with three-dimensional swelling law were
developed from the results of Huder and Amberg tests by Gysel (1977 and 1987), and Fröhlich
(1989).
A three-dimensional constitutive law proposed by Kiehl (1990) which is an extension of the
constitutive law of Wittke et al. (1976, cited in Wittke-Gattermann, 1998), describes the timedependency and anisotropy of swelling. In addition, viscoplastic deformations with nonassociated flow rule are taken into account. In the constitutive law by Anagnostou (1993) taking
into account the anisotropy behaviour of swelling, effective stresses and pore water pressures
are coupled together (a hydraulic-mechanical model). In such models, the hydraulic diffusion
together with the stiffness matrix of the rock is taken into consideration (Anagnostou, 1993).
Barla (1999) notes both 1D and 3D relations of swelling law are restricted because first of all
swelling law is based on linear elastic assumption and only maximum swelling strain can be
obtained. Furthermore, the generalisation of the 3D relation is based on one dimensional
oedometer tests.
Different implementation of Wittke-Gattermann‟s model has been carried out by Heidkamp et al.
(2004) and later on by Benz (2012). A detailed overview of calculation methods for swelling rock
in the past can be found in a report published by ISRM Committee (1994).
11
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
2.2 Part II: Mathematical formulation of the swelling rock model & its
concept
As mentioned earlier, the fundamental basis of the model under study is according to the
swelling rock model developed by Wittke-Gattermann (1998). Hence, for description of the
model under study, the concept behind Wittke-Gattermann‟s model (1998) is explained in this
part; which begins with description of elastic stress-strain behaviour, strength of the rock and
visco-plastic stress-strain behaviour. Then, the isotropic constitutive law for the stress-strain
state due to final swelling, time dependency of swelling and complete stress-strain behaviour
are elaborated. Finally, the revised parameters as well as the model‟s routines defined by Benz
(2012) are explained.
2.2.1 Elastic stress-strain behaviour
It is assumed that unleached gypsum Keuper can be approximately described using a
homogeneous model. However, due to the joint pattern of its beddings, it must be expected that
the rock shows elastic transversely isotropic stress-strain behaviour (Wittke-Gattermann, 1998),
i.e. the same physical properties within a plane (e.g. x-y plane or isotropic plane) and different in
the axis normal to that plane (e.g. z axis or rotational symmetry axis).
Five independent elastic constants are employed to describe a transversely isotropic material
behaviour: the Young's modulus E 1 and E2; Poisson's ratio ν1 and ν2 and shear modulus G2.
Indices 1 and 2 represent for the directions parallel and perpendicular to the isotropic plane (e.g.
bedding plane) respectively. Assuming a Cartesian coordinate system {x, y, z} shown in Figure
4, with z axis of symmetry and x-y isotropic plane, the elastic constants resulting from the
stresses and strains in a rock element can be written using generalised Hooke‟s law as below.
1
E
 1
 1
 x   E1

 
 y  -  2
   E2
z

 

 xy   0
  
 yz  
 zx   0


 0

-
1
E1
1
E1
-
2
E2
-
2
E2
2
E2
1
E2
0
0
0
0
0
0
2 1   1 
0
0
0
0
0
1
G2
0
0
0
0
E1
0

0 


0 
 
  x
 y
0  
 
 .  z 
  xy 
0  
  yz 
 
0   zx 


1 
G2 
Eq.1
Furthermore, the parameter rotation angle or  is defined to indicate the inclination angle of the
bedding plane (Benz, 2012). Bedding planes are mostly horizontal (  is equal to zero), even
though beddings can take any inclination as shown in Figure 4.
12
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
Figure 4: The conventions used for transversely isotropic implementation in the model
2.2.2 Rock strength
The strength for the rock in the model is described using the Mohr-Coulomb failure criterion.
Mohr-Coulomb Criterion describes the failure using a straightforward relation implying limiting
shearing stress  in a plane is only dependent on the normal stress  at a point on the same
plane as shown in Eq.2 (Edelbro, 2003).
  c   .tan 
Eq.2
In Figure 5, the failure criterion is shown both in    and σ1-σ3 diagrams.
Figure 5: Mohr-Coulomb failure criterion shown in both σ 1-σ3 and τ -σn diagrams (Wittke-Gattermann, 1998)
Where c and  are defined as cohesion and internal friction angle of the rock, respectively. Eq.2
corresponds to a failure envelope for Mohr‟s circles which is shown below (Eq.3).
F     .tan   c  0
Eq.3
By drawing the Mohr‟s circles, it is also possible to derive Eq.4 from Figure 5 as below which is
the failure criterion in terms of relationship between principal stresses.
13
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
1
1
F   1 1  sin     3 1  sin    c.cos   0
2
2
Eq.4
As illustrated in Figure 5, for F <0 only elastic strains occur and stress conditions that lead to
visco-plastic strain is F> 0 (see 2.2.3).
The Mohr-Coulomb failure criterion does not consider tensile failure and since the rock cannot
sustain large tensile stresses (Edelbro, 2003), a tension cut-off is usually included in this failure
criterion. Therefore, besides the aforementioned shear failure criterion, a tensile failure criterion
as shown in Eq.5 is defined (see Figure 5):
F   3   t
Eq.5
The uniaxial tensile strength  t  is considered positive and is obtained through Eq.6 as below:
t 
2c.cos 
1  sin 
Eq.6
The combination of Mohr-Coulomb failure criterion with a tension cut-off criterion is shown in
Figure 5. The Mohr-Coulomb failure criterion is also used for the description of strength of
discontinuity surfaces. The details of shear strength of discontinuities can be found in the
literature (e.g. Hoek, 2007).
2.2.3 Visco-plastic stress-strain behaviour
When the stress exceeds the strength of rock, visco-plastic stress-strain behaviour is assumed.
This behaviour can be illustrated via a one dimensional rheological model shown in Figure 6 in
which an elastic element is directly connected to the applied stress and swelling or sliding
device is in parallel to a dashpot (Wittke-Gattermann, 1998).
Figure 6: One dimensional rheological model for showing elastic viscoplastic behaviour (Runesson, 2005)
The applied stress or  shown in Figure 6 is carried by a spring which is responsible for the
elastic response and a dashpot and sliding member in parallel which is responsible for
viscoplastic behaviour. Under loading, first the spring deforms elastically until the yield stress σ y
of the sliding device is reached. As long as    y the sliding device will be inactive and once
the applied stress exceeds the yield stress, plastic strains also occur. This is because the stress
can be transferred to the viscous dashpot  when the frictional resistance of the sliding device
14
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
has been exhausted. Since in the one dimensional model apart from the constant shear
strength there are no stresses counteracting the sliding movement, the displacement increases,
and hence the strain of the system over all boundaries (see Figure 7).
Figure 7: Schematic stress-strain and strain-time diagrams for elastic viscoplastic behaviour (Wittke-Gattermann, 1998)
The total strain is therefore defined as the sum of elastic strain of the Hookean element with
Young‟s modulus of E and the viscoplastic strain in the dashpot and sliding device as shown in
Eq.7.
   el   vp
Eq.7
In this model, the already defined yield function F is used instead of limit (yield) stress σy in
which for F<=0 elastic strains occur and for F>0, viscoplastic strains also occur in which
stresses are redistributed; If stress redistribution be not possible, failure will occur.
The relationship between the viscoplastic strains and associated excess in the strength is
.
described using a flow rule; the viscoplastic strain rate  as a function of the plastic potentials
QG and QT of the rock and the bedding planes, the failure criteria F G and FT, the state of stress
  and the viscosity VPG and VPT (Wittke-Gattermann, 1998). If the strength in the rock and on
vp
the bedding planes T1, T2... is exceeded, it results in:
.


 VP

1
1
1
 Q 

 QT1 


FG .  G  
.FT1 
.FT2
 








 


VPG
  
 VPT1
  
 VPT2

 QT2 


  ...



  

Eq.8
If the plastic potential is described using the same function as the failure criterion, Eq.8 indicates
an associated flow rule. Such a flow rule often leads, especially when discontinuities are taken
into consideration, to volumetric strain rates, which are larger than the real case. If one uses the
same function for the plastic potential as for the failure criterion but replaces the respective
governing friction angle  with the dilatancy angle    , the volumetric strain rate can be
adopted to experimental results. If the plastic potential be modified in this manner, Eq.8
indicates a non-associated flow rule which is assumed in this model (Wittke-Gattermann, 1998).
If one chooses the coordinate system {x, y, z} such that σx, σy and σz corresponds to the
principal stresses σ1, σ2 and σ3, the above flow rule combines the principal viscoplastic strains
of the rock per time unit with the derivation of QG with respect to the principal stresses as below:
15
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
.
 VP

1G 
 . 
1
VP
 2G  
 .  VPG
 3VPG 
 
1

 2 1  sin  




.  1 1  sin    3 1  sin    c.cos   
0

2
2


1
  1  sin  
 2

.
 VP


 1G 
0
 . 
1
 
VP
In the case of tensile failure   2G  
.   3   t   0 
 .  VPG
1 
 
 3VPG 
 
Eq.9
Eq.10
It should be noted that, in combination of Mohr-coulomb with tension cut-off criteria, the MohrCoulomb part is taken as non-associated whereas for the tension cut off, an associated flow rule
is considered (Wittke-Gattermann, 1998).
From the viscoplastic strain rate, the viscoplastic strain is calculated by integrating over time.
2.2.4 Isotropic constitutive law for the stress -strain state due to final swelling
As mentioned earlier, based on the results of swelling tests after Huder and Amberg (1970),
Grob (1972) formulated a one dimensional swelling law. Huder and Amberg test is employed
to quantify expansive deformation as result of swelling (Serón et al., 2002). The loading
scheme after Huder and Amberg test is outlined according to Figure 9 (Wittke-Gattermann,
1998).
Figure 8: Swelling tests after Huder and Amberg (1970) - Loading scheme in an oedometer
16
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
In this test, an undisturbed sample is mounted in an oedometer (see Figure 8) and first
subjected to an initial loading (the compressive stress)  z   a ; a characteristic of such a stressstrain curve is shown in Figure 8 in a semi logarithmic scale as shown in curve 1. Subsequently,
unloading and reloading procedures are done  z   a (curve 2 and 3). Then the sample is
watered and the swelling process is initiated (4). Since the lateral strain is prevented, the
swelling (increase in volume) can only occur in z (vertical) direction and is expressed as  zq  a 
.This swelling strain is achieved only after a certain time (theoretically t   ). After the swelling
strain  zq  a  has stopped, the sample is gradually unloaded and corresponding swelling strain
at each load step is measured until swelling strain  zq  zi  is reached. This procedure leads to
curve 5 for which the swelling strain  zq , as a function of the applied compressive stress  z can
be read off. Extending the curves 3 and 5 in Figure 8, leads to a compressive stress  0 for
which greater than this stress, no swelling occurs. The same test results reported in semi
logarithmic scale shown in Figure 9.
Figure 9: Swelling strain against applied stress indicating swelling stress dependency–Huder and Amberg test of mudstone
samples containing anhydrite from the medium gypsum horizon in Stuttgart area, Germany (Wittke-Gattermann, 1998)
Grob (1972) formulated the following relationship between the axial swelling strain  t     zq
and the axial stress  z in the equilibrium state for evaluating and interpreting the Huder and
Amberg experiments as below:
 
 zq  k zq log  z 
z0
Eq.11
In this one-dimensional swelling law, the relationship between  zq and  z in the semi-logarithmic
scale can be described by a straight line. The slope of this straight line is determined by the
swelling potential parameter k zq ; the intersection of this line with the σ z-axis is equal to the
aforementioned stress  z 0 above which no swelling occurs (see Figure 9).
17
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
It should be noted that Eq.11 is valid only for compressive stresses  zc   z   z 0 .  zc is a
minimum stress, below this value no more swelling occurs. This was assumed through the
example in Figure 9 to  zc  10 kN 2 . Under this assumption, the maximum axial swelling strain
m
is equal to  zq  6.2 ‰. Furthermore, for the axial compressive stresses  z
 z 0 swelling strain
is equal to zero. The extended form of one-dimensional swelling law is written in the following
form (Eq.12) which is also valid for tensile stresses  z 0 .
 zq

0



 k zq log z
 z0


 zc
k zq log
 z0



,  z   z0


,  zc   z   z 0
Eq.12
,  z   zc
The three dimensional extension of Eq.12 formulation was proposed by Kiehl (1990) based on
the results of swelling tests by Pregl (1980, cited in Wittke-Gattermann, 1998), which showed
that the principal swelling strains approximately only depend on the principal stresses in those
directions.
The initial state of the swelling law is a state of stress  0  , above which no longer swelling
occurs (Kiehl, 1990).  0  as well as the state of stress   in rock can be described within any
Cartesian coordinate system {x, y, z} using six stress components as below.
 o    x0 , y 0 , z 0 , , ,
xy 0
yz 0
zx 0

&
    x , y , z , , ,
xy
yz
zx

Since the three-dimensional swelling tests by Pregl (1980) have shown that in the principal
stress directions i, the swelling strains  iq only depend on the stress  i , hence the stress tensor
 
is first transformed into the directions of principal stresses. Then, it is transformed to the
initial state of the swelling law
 0 
in the same direction, i.e. in the direction of principal
stresses   . The mathematical formulation of the three-dimensional swelling law in the
equilibrium state  t    is:


0
 


 iq  kq . log  i
   0i
 
log   ci
   0i
,  i   0i






,  ci   i   0i
,  i   ci
18
Eq.13
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
Where  iq are the normal strain components of swelling strains
  in
q

the direction of the
principal stresses  i .
Eq.13 is assumed for a material with isotropic swelling behaviour. However, anisotropy of
swelling behaviour in shales containing anhydrite was already confirmed (e.g. Anagnostou,
1992). Therefore, the following approach was considered by Wittke-Gattermann (1998) to
extend the swelling law for anisotropic behaviour.
Kiehl (1990) reported an analogy between elastic strains and swelling strains of a material.
According to the relations (Eq.1) stated for the elastic behaviour of a transversely isotropic
material, the relation between the elastic strains in a coordinate system {x‟, y‟, z‟} is described
as Eq.14. Eq.14 postulated stresses and deformations in the equilibrium state ( t   ) for the
anisotropic swelling behaviour. Then, it is assumed that a stress condition (  c     0 ) exists in
all principal stress directions. Because of the non-linear relationships between stress and
swelling strains a differential formulation is used as Eq.15. As an example, strain in x direction is
only shown.
 el x ' 


q
q
1
1
 x ' - 1  y ' - 2  z ' Eq.14  d  q x '   q d x '  - 1q d y '  - 2q d z ' 
E1
E1
E2
E1
E1
E2
Eq.15
The elastic constants due to the nonlinearity of the swelling stress-strain relationships are
introduced in Eq.15, which are functions of the state of stress.
Since according to the test results by Pregl (1980) the swelling strains are approximated only in
the direction of principal stresses, it can be expected that  1q and  2q be equal to zero (WittkeGattermann, 1998). This leads Eq.15 to Eq.16. Wittke-Gattermann (1998) proposed an
approach (Eq.17) with the same formulation framework as Eq.16 taking into consideration the
two already introduced assumptions including the one, which the condition for swelling is still
given by Eq.13 and also, the logarithmic relationship between swelling strains in the principal
directions and principal stresses holds true.
d  q x '   f  d x '   

3
1
 i
q
2
d

Eq.16
&


S
x '
x '
1  l 'i ln
 0i
E1q
i 1

Eq.17
li is the direction cosine of the angle between the directions of principal stresses and of the
coordinate axis x‟ in the considered coordinate system. The swelling deformation parameter S1
is defined as S1  kq .log e . In the case of a rock with anisotropic swelling properties, S 1 can be
determined experimentally through Huder and Amberg testing (Wittke-Gattermann, 1998).
2.2.5 Time dependency of swelling
According to the Huder and Amberg test results, the time dependency behaviour of swelling is
observed. This behaviour as an example is shown for a sample from the S-Bahn Stuttgart
project in Figure 10.
19
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
Figure 10: Swelling strain versus time diagram indicating time dependency of swelling - Huder and Amberg test results on a
sample from Stuttgart area (Wittke-Gattermann, 2003)
Figure 10 displays that the swelling strains develop quickly at first. Over time, however, the
curve flattens more and more. Kiehl (1990) described these results using following approach
(Eq.18) for the time dependency of swelling strains in isotropic materials.
 xq  t 
t
q
 xq  t    xalt  t  

q
x
 t  .dt
t
Eq.18a Where

q

1 
.  kq . li 2 .Li  i  t     xalt  t  
q  i



0
  t 


Li  i  t    log  i


  i0 
 
log   ic 
   i 0 
Eq.18b
,  i t    i0
,  ic   i  t    i 0
Eq.18c
,  i  t    ic
q
In Eq.18,  xalt is swelling strain occurring at the considered time t in the coordinate system. With
this approach, the results of laboratory experiments are well reproduced. The swelling time
parameter or  q is a measure of the rate of swelling (Wittke-Gattermann, 1998). This is
illustrated through the Figure 11 schematically.
Figure 11: Swelling time parameter as a measure of swelling rate
Time dependency of swelling in the case of anisotropy (transversely isotropic) behaviour, is
described using the following approach (Wittke-Gattermann, 1998):
20
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
 xq  t 
t



 q
1 
1
.  RG  S1  l 'i 2 .
Li  i  t      xalt  t  
q   i
log e


Eq.19
Relations stated in Eq.18a and Eq.18c still apply to this case. RG is the transformation function
of structure orientation in the global coordinate system.
2.2.5.1 Dependency of swelling upon water access to the rock
It is assumed that the conversion rate of the anhydrite into gypsum and thus the rate of swelling
are dependent on the rate of water entry. Laminar flow condition is also assumed for seepage
flow through rock so as Darcy‟s law is applied (Wittke-Gattermann, 1998).The pores in rock are
not initially water-saturated, but they fill up with water over the time. Accordingly, the rock
permeability depends on the degree of water saturation and thus not constant. Average
permeability coefficient of about 108 m (Wittke-Gattermann, 1998) and of less than 1012 m
s
s
(Vardar et al., 1984) were reported from in situ permeability tests in the unleached Gypsum
Keuper. The swellable rock in unleached Gypsum Keuper has very low permeability due to its
low clay content and porosity. Wittke (1984, cited in Wittke-Gattermann, 1998) proposed the
following relationship for the permeability of a rock crossed by a group of joint sets:
  2ai 3 g

 12. .d

3
kf  
 2ai  g

1.5
 
12. 1  8.8  k
 Dhy   .d


 


;k
;k
Dhy
Dhy
 0.032
0.032
Eq.20
In Eq.20, 2ai is the mean aperture width of discontinuities, d is the mean spacing of
discontinuities, Dhy  2.2ai denoting the hydraulic diameter and k is the roughness of the wall of
discontinuities. Moreover, g is the acceleration due to gravity and v is the kinematic viscosity.
This equation distinguishes between discontinuities surfaces which are hydraulically smooth
k

k
 Dhy  0.032  and rough  Dhy




0.032  .

Figure 12: Change of discontinuity aperture width due to viscoplastic strains of joint surfaces (Wittke-Gattermann, 1998)
21
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
According to Eq.20, the permeability of the rock depends highly on the aperture width and the
spacing of discontinuities. For example, by the occurrence of visco-plastic deformations as a
result of exceeding the shear strength on the discontinuities‟ surfaces, the width of aperture can
increase to Δ2ai according to the viscoplastic stress-strain behaviour (see Figure 12). So the
rock permeability increases as a result of viscoplastic strain. The aperture width of 2aio prior to
the exceeding of the strength is determined as below.
2ai 0  3
12. .d
.k f 0
g
Eq.21
It was also found that the permeability even at small viscoplastic strain significantly increases
using the above determined assumptions. Assuming a discontinuity spacing of d=10 cm, the
permeability increases at a visco-plastic strain of 3 ‰ from 108 m to about 103 m (Wittkes
s
Gattermann, 1998).
The velocity of the water access to the rock also increases with increasing viscoplastic strain.
However, once the water has reached the rock, it must still penetrate through the joint surfaces.
The penetration process depends on the spacing of the hydraulically effective discontinuities. If
previously non-hydraulically effective discontinuities were opened as a result of exceeding the
strength in the rock, the thickness of the rock layer, which has to be moistened, would be
thinner and the time required for moistening the rock would decrease accordingly. In addition,
the rock permeability plays a role in speeding the moisture penetration velocity. Porosity is also
of an impact on the permeability (Terzaghi, 1925; cited in Wittke-Gattermann, 1998) which is
shown according to the empirical formula of Kozeny-Carman (Eq.22):
g.c
n3
kf = 2 .
v.S 1  n 2
Eq.22 where
 S : Specific surface area

n : Porosity
c : Empirical dimensionless constant

Assuming that the material is incompressible, following relationship between permeability and
elastic volume strain of the rock can be derived from Eq.22 (Wittke-Gattermann, 1998) as
follows:
kf
kf 0
=
1
  el 
1   vel 1  v 
n0 

3
Eq.23 where
 vel : Elastic volume strain

el
n0 : Porosity at  v  0

el
k f 0 : Permeability at  v  0
k : Permeability at  el
v
 f
According to Eq.23, the practical relationship between elastic volumetric strains  vel and
permeability k f is described. Based on the observations, the permeability due to an elastic
volume strain of 1 ‰ depending on the initial porosity either does not increase or becomes
almost 8 times larger. The size of the initial porosity is presently unknown. It is therefore difficult
to estimate the influence of the elastic strains on the swell rate. Wittke-Gattermann (1998)
22
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
concluded that the visco-plastic strains and possibly also the elastic strains share the influence
on the rate of water entry and hence on the swelling rate.
2.2.5.2 Development of the existing approach by Wittke-Gattermann (1998)
As the time course of swelling explained in Huder and Amberg test results, swelling was initially
strong and decreasing with time increase. Wittke-Gattermann (1998) explained the
phenomenon by assuming that the sample is initially dry, so the hydraulic gradient will be very
large at time t=0 and water flows through the rock very quickly, so that the anhydrite is soon
converted into gypsum. With time passing, the hydraulic gradient decreases and thus the
conversion rate.
Viscoplastic strains and elastic strains before the beginning of swelling affect the time course of
the results of Huder and Amberg tests and thus swelling rate. These results could be
reproduced very well with Kiehl‟s (1990) approach (Eq.19). Their effect was not included in the
approach stated in Eq19. Therefore, Wittke-Gattermann (1998) assumed that the swelling time
parameter in the general case depends on the elastic strains before beginning of the swelling
and the visco-plastic strains:
1
q  t 
=a0  ael 0 . vel0 +aVP . vVP  t 
Eq.24
In Eq.24, a 0 , a el 0 and aVP are constants. The parameter a0 is introduced as the initiating
parameter for swelling process. Furthermore, a0 takes into account the dependency of the
swelling rate on the distribution of the anhydrite in the rock mass, which also has an influence
on the development of strains with time due to swelling (Wittke et al., 2004).  vel0 considers the
elastic volumetric strains prior to the start of swelling;  vVP is viscoplastic volumetric strains.
As explained earlier, the permeability increases significantly with increasing viscoplastic strain.
The permeability coefficients can then become very large resulting in very fast water flow
through rock discontinuities. A further increase in the viscoplastic strain would then no longer
lead to an increase in the swelling rate, as this ingress of water into the rock no longer could be
accelerated. This can be taken into account in equation Eq.24 for η q which can be extended to
Eq.25:
1
=a0  ael 0 . vel0 +aVP . vVP  t 
for  vVP  max EVP
1
=a0  ael 0 . vel0 +aVP .max EVP  t 
for  vVP
q  t 
q  t 
max EVP
Eq.25
In Eq.25, max EVP is the viscoplastic volumetric strain, above which no further increase occurs
in the swelling rate.
The swelling time parameter in the model under study is defined as shown in Eq.26 (Benz,
2012), where viscoplastic volumetric strain  vVP is replaced with the absolute value of plastic
23
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
volumetric strain  vp . Hence, the difference between Eq.25 and Eq.26 goes back to time
dependency of deformations after the loads are applied in which in the case of viscoplastic
behaviour, in addition to irreversible deformations, the model undergoes a creep flow with time
development.
1
q
1
q

 


 A0  Ael   vel  Apl   vp

 A0  Ael   vel   Apl  Apl max 
for  
v
for  
v
p
p
Apl max

Apl max

Eq.26
Here Ael refers to the absolute value of current elastic volumetric strain. Hence, elastic strains
occurring within the swelling or plastic phase are also taken into account. Since an absolute
value for Ael is used as shown in Eq.26, there is no distinction between compression and
extension in elastic volumetric strains in the current version of the model under study. The
absolute values of Apl is also used as shown in Eq.26. This parameter refers to plastic strains.
Here Apl max is also a limit for the plastic strains in which no further change in swelling evolution
is triggered, i.e. it functions as a limit for the occurrence of plastic volumetric strains.
2.2.6 Complete stress strain behaviour
According to the superposition law, the total strain resulting from the sum of the elastic, the
visco-plastic and the swelling strain can be written as shown in Eq.27 in terms of the strain rate.
 . ges   . el   . VP   . q 
  t     t  +   t  +   t 

 
 
 

Eq.27
The following equations can be obtained for the total strain components in direction of principal
stresses from the previously derived relations in terms of the strain rate. As an example, total
strain rate in direction of major principal stress is shown.
. ges
1
t  
.
.
 t 
 1
1 .
 1 1 t 

t



t



.
.1  sin    3 . 1  sin    c.cos   . 1  sin 




1
2
3  t  +

E
2
 VP  2
 2
+


  t  
.   kq .log  1   1q  t  
q  t  
 0 

1
Eq.28
In the above equations, both the swelling as well as the visco-plastic strains depend on the
current state of stress.
2.2.7 Model’s routines in the model under study defined by Benz (2012)
Benz (2012) has defined three different routines (Model ID1; Model ID2 and Model ID3) as
solution procedures for swelling strain in the swelling rock model.
24
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept
In both Model ID 1 and Model ID 3, swelling pressures are assumed to be isotropic (to be the
same in all directions), i.e.  0, p   0,t . There is also no coupling term defined in any of these
models‟ routines, i.e. there is no relation between different principal directions and therefore
swelling strain is calculated separately in all directions.
In Model ID 1, the swelling potential parameters ( K q , p and K q ,t ); and maximum swelling
pressures in both perpendicular and tangential directions (  0, p and  0,t ) are first transformed
into Cartesian (xyz) coordinate system and then into the coordinate system of principal stresses‟
directions ( 1 , 2 , 3 ). The swelling strains are calculated in the coordinate system of principal
stresses. In fact, swelling strain is calculated separately in each direction toward the
corresponding principal stress direction. Then, the calculated swelling strains are transformed
back into the xyz coordinate system. While in Model ID 3 which is Wittke-Gattermann‟s model
on bedding plane, the maximum swelling pressures and swelling potential parameters are
projected into the bedding plane coordinate system. Then, swelling strains are obtained
according to the stresses acting perpendicular and tangential toward the bedding plane.
According to the general formulations for the calculation of swelling strain in the routines defined
by Benz (2012), swelling strain in the direction of major principal stress is defined as below.
d 1q  

 Max(10,  1 ) 
1 
q
.  kq1.(log 
  1a  t   dt
q 
0



This is true as long as ( kq1  0 ) and (  1
Eq.29
 0 ). Subscript 1 indicates the major principal stress
direction in which the Kq1 is transformed into. The numerical value of 10 is equivalent to  c . If
any of the abovementioned conditions is not met, the swelling strain is zero in the proposed axis
of the coordinate system. This formulation (Eq.29) is equivalent with the third term of total strain
rate relation introduced in Eq.28, where the swelling strain is calculated.
In Model ID 2 in which the coupling term is defined (representing Anagnostou‟s model), swelling
strains can also be evaluated in any coordinate system as well as bedding plane.
Therefore, in both Model ID 2 and Model ID 3 less transformation is processed. However, Benz
(2012) concluded that the results of swelling strains are not so different and in case Model ID 1
is used, still good results can be achieved.
2.2.8 Conclusions

Swelling time-dependent deformation is a result of volume increase in ground in the
presence of water. Swelling of anhydrite bearing rock formations including gypsum
Keuper is the subject of this study.

Gypsum Keuper includes different layers containing different amounts of sulphate rocks
either in the form of anhydrite or gypsum.
25
Chapter 2: Theory of swelling - Part II: Mathematical formulation of the swelling rock model and its concept

Anhydrite converts into gypsum by absorbing water causing an increase in its initial
volume (swelling). Two possible causes for swelling in such rocks are hydration and
gypsum crystal growth. If swelling is prevented by tunnel lining, swelling pressures are
induced.

The oedometer test has been vastly used in tunnelling projects since it simulates the
tunnel invert in small scale conditions. Despite the advantages of triaxial tests, they are
costly, time consuming and they are not easy to conduct in comparison to oedometer
test. Laboratory swelling tests on samples from gypsum Keuper included swelling
pressure tests, swelling strain tests and Huder and Amberg experiments. Laboratory
testing of anhydrite swelling rocks are challenging because (1) there is no wellestablished model taking into consideration the different conditions in in situ and in the
laboratory; (2) Swelling process lasts for a long time which cannot be simulated properly
by laboratory testing; (3) The oedometer test condition prevents swelling in lateral
direction resulting in an overestimation of the swelling pressure.

The final lining design is governed by the long term deformation as a result of swelling.
Based on field observations, the amount of maximum swelling pressure varies
depending on the host rock of different tunnelling projects. Furthermore, laboratory
results overestimate the swelling pressure. When designing lining for tunnels driven in
gypsum Keuper, the resisting (stiff) and yielding (flexible) support principles are applied.

Grob (1972) formulated the 1D swelling law between the axial swelling strain and the
axial stress based on the results of Huder and Amberg tests (1970). The 3D extension of
swelling law was proposed first by Einstein et al. (1972). A similar 3D swelling law was
developed by Wittke et al. (1976). Kiehl (1990) proposed the 3D extension of Wittke et
al. (1976) based on the results of swelling tests by Pregl (1980).The swelling tests
performed by Pregl (1980) showed that the principal swelling strains only depend on the
principal stresses in those directions. This was assumed for a material with isotropic
swelling behaviour. Wittke-Gattermann (1998) proposed an approach to extend the 3D
swelling law proposed by Kiehl (1990) for anisotropic behaviour. Specific implementation
of Wittke-Gattermann‟s model was done by Heidkamp et al. (2004) and Benz (2012).

The rock in unleached gypsum Keuper is assumed to show elastic transversely isotropic
stress-strain behaviour. The rock strength is described using the Mohr-Coulomb failure
criterion. When the rock strength is exceeded, visco-plastic behaviour is assumed which
can be explained via a rheological model, in which an elastic element is directly
connected to the applied stress and swelling or sliding device is in parallel to a dashpot.

It is assumed that the conversion rate of the anhydrite into gypsum and thus the rate of
swelling are dependent on the rate of water entry. The swellable rock in unleached
gypsum Keuper has very low permeability. The permeability even at small viscoplastic
strain significantly increases. The velocity of the water access to the rock also increases
with increasing viscoplastic strain. The visco-plastic strains and possibly also the elastic
strains share the influence on the rate of water entry and hence on the swelling rate.
26
Chapter 3: Soil Test Facility and layout of numerical simulations
Chapter 3: Soil Test Facility and layout of numerical simulations
3.1 Introduction
As mentioned earlier, the swelling rock model has been implemented into PLAXIS2D finite
element software by Benz (2012) as a user-defined model. The assessment and validation of
the user-defined model is carried out through the Soil Test Facility of PLAXIS2D Input. The Soil
Test Facility is an option in PLAXIS2D software allowing for convenient simulation of different
laboratory tests. All the available soil models as well as user-defined models in PLAXIS2D can
be run into the Soil Test Facility. Five different element test procedures can be simulated
through the Soil Test Facility as shown in Figure 13 consisting of a Triaxial test (1); an
Oedometer test (2); a Constant Rate of Strain (CRS) consolidation test (3); a Direct Simple
Shear (DSS) test (4); and a general test (5) in which different stress-strain conditions can be
simulated by user.
Figure 13: Possibilities of laboratory tests simulations in the Soil Test Facility of PLAXIS2D software
Before running the user-defined model (Benz, 2012) into the Soil Test Facility, it is necessary to
understand some important issues regarding numerical simulations as shown below.
3.1.1 Implementation scheme
A numerical simulation run can be carried out within two implementation schemes, i.e. implicit
and explicit. In the user-defined model written by Benz (2012), there is a numerical input
parameter that can be used to switch from implicit to explicit integration scheme and vice versa.
In fact, the scheme which is defined by the user indicates the way by which the numerical
results are obtained and is referred to the computation of swelling strain.
Implicit and explicit integration schemes can be explained using the elasto plastic stress-strain
incremental relation which is defined as Eq.30.
27
Chapter 3: Soil Test Facility and layout of numerical simulations
  De .(   p )
Eq.30
Where Δσ is the stress increment; De is the elastic material stiffness matrix (Hooke‟s law); Δԑ p is
plastic strain increment and the Δԑ is the total strain increment which are obtained using the
displacement increments in the system. For plastic behaviour, Vermeer (1979) describes plastic
strain increments as shown in Eq.31.
i 1
i

 g 
 g  
 p   1    





  
   

Eq.31
Where Δλ is the plastic multiplier increment, g is plastic potential function and ω indicates the
type of integration scheme. In the case of ω=1 the integration scheme is called implicit while
ω=0 represents explicit one. i  1 is related to the previous state of stress and i indicates the
current state of stress.
In both explicit and implicit integration schemes, at the end of each increment, the stiffness
matrix is updated for the next increment and then is applied to the system. Unlike explicit
scheme, implicit integration uses a Newton-type iteration procedure after each increment to
enforce equilibrium. This is considered among the implicit scheme advantages.
In the swelling rock model, in the case of using implicit scheme, the model parameter is kept
zero, otherwise, it would be defined as a value greater than zero indicating explicit scheme.
Stability and accuracy are the two key issues in numerical analysis. Model might be accurate
but not stable or vice versa. In fact, a model can be accurate to predict the value or results
exactly the same as theoretical values but it oscillates with time increasing which means that it
is accurate but instable. On the other hand, the model can be stable but result in a huge bias
but no oscillation which means that the model is not accurate but stable.
In the following, time step issue which is also related to numerical stability and accuracy is
introduced.
3.1.2 Time step ratio
Going back to Eq.18a (see Chapter 2) and substituting the
 xq  t 
t with the equivalent term
defined in Eq.18b, the swelling strain  xq  t  in direction of x-axis is written as below.
28
Chapter 3: Soil Test Facility and layout of numerical simulations
Where Li  i  t   is defined according to Eq.18c (see Chapter 2) and  xq  t    depicts the final
swelling strain at time equal to infinity (see Eq.11 in Chapter 2). Furthermore,
dt
is assumed as
q
time step ratio; In the Soil Test Facility, time step or dt is defined as duration of the test divided
by the number of steps in which the load is applied (Eq.33):
dt(Time step) =
Duration of Test
No. of Steps
Eq.33
Different time step ratios are tested with respect to stability and accuracy of the results in the
Soil Test Facility. Then, a critical time step ratio is introduced above which the accuracy and
stability in the results rapidly decrease.
3.1.3 Sign convention in the Soil Test Facility
In the Soil Test Facility, compression is always negative. Hence, Eq.13 (3D swelling law) and
Eq.4 (Yield function) are shown below taking into account negative sign in compression leading
to Eq.34 and Eq.35, respectively:


0
 


 iq  kq . log  i
   0i
 
log   ci
   0i
,  i   0i






,  ci   i   0i
Eq.34
,  i   ci
1
1
F    1 1  sin     3 1  sin    c.cos   0
2
2
Eq.35
3.2 Element tests
In order to assess and validate the model under study, some element tests consisting of stresscontrolled oedometer test and strain-controlled uniaxial compression test are simulated. In fact,
the sensitivity analyses and variations of model parameters (see Table 3) are carried out within
the element tests.
The required input data (including physical and strength properties of the rock in Gypsum
Keuper) for simulation in the Soil Test Facility were obtained from gypsum horizon in Stuttgart
area in Germany as shown in Table 4 (Wittke-Gattermann, 1998). Furthermore, the values of
maximum swelling pressure and swelling potential derived from Huder and Amberg testing on
the same rock are included. This table gives the suitable range for the aforementioned
parameters by which the simulations in the Soil Test Facility are conducted.
29
Chapter 3: Soil Test Facility and layout of numerical simulations
Table 3: List of parameters of the model under study (Benz, 2012) – p and t used by Benz (2012) indicates perpendicular and
tangential directions in bedding plane respectively
Parameter
Symbol
Unit
Parameter
Symbol
Unit
Friction angle
'
(°)
Swelling time parameter
A0
(1/d)
Cohesion
c'
(kPa)
Swelling time parameter
Ael
(1/d)
Dilatancy angle

(°)
Swelling time parameter
Apl
(1/d)
Tensile strength
 Ten
(kPa)
Maximum Plastic volumetric strain
Apl max
(-)
Young‟s modulus
E p , Et
(kPa)
Swelling potential
K q , p , K q ,t
(-)
Poisson‟s ratio
 p , t
(-)
Maximum swelling pressure
 0, p , 0,t
(kPa)
Shear modulus
G23
(kPa)
Bedding rotation angle (clockwise)

(°)
Table 4: Physical and strength properties of the gypsum Keuper rock in gypsum horizon in Stuttgart area in Germany – K and
S are indices used in in Wittke-Gattermann’s model (1998) to indicate vertical joint sets (perpendicular to beddings) and
horizontal bedding, respectively
Parameter
Unit
Value
Parameter
Unit
Value
k
(°)
35-40
s
(°)
20-30
ck
(kPa)
250-750
cs
(kPa)
0-100
 k , s
(°)
3-10

(-)
0.25-0.35
E
(MPa)
2000- 6000
Kq,s
(%)
0-2
K q,k
(%)
0-10
0
(kPa)
750 (up to 5000 in situ)
Table 5 shows some values of swelling time parameters in Wittke-Gattermann‟s model used in
numerical simulations of tunnelling applications within Gypsum Keuper formation (WittkeGattermann, 1998).
Table 5: Some reasonable values of Wittke-Gattermann’s model swelling time parameters used in numerical simulations – ‘a’
stands for annum (year)
Parameter
Unit
Value
Parameter
Unit
Value
A0
(1/a)
0-0.03
Ael 0
(1/a)
0-50
AVP
(1/a)
0-230
max EVP
(-)
0.0025-0.005
It should be noted that Wittke-Gattermann (1998) estimated „A‟ parameters‟ values after
simulations to reproduce the in situ measurements results of the exploration gallery in
Freudenstein tunnel in Germany. This is the main reason that „A‟ values is shown in a separate
Table (Table 5), indicating the other table (Table 4) is used for numerical simulations in the Soil
Test Facility.
30
Chapter 3: Soil Test Facility and layout of numerical simulations
In the following, the initial and boundary conditions for the above-mentioned element tests are
shown.
3.2.1 Oedometer test
Numerical simulation of the oedometer test is carried out in the Soil Test Facility using the
already available experimental data (Wittke-Gattermann, 2003). Different units for parameters
used in the Soil Test Facility compared to what was described before are shown in Table 6. The
constant parameters‟ values in oedometer runs are shown in Table 7. The parameters are kept
the same during the numerical runs unless otherwise specified
Table 6: Different units and symbols for parameters used in the Soil Test Facility
Parameter
Symbol
Unit
Friction angle
Phi
(°)
2
Parameter
Symbol
Unit
Swelling time parameter
A0
(1/d)
Cohesion
C
(kN/m )
Swelling time parameter
AEL
(1/d)
Dilatancy angle
Psi
(°)
Swelling time parameter
APL
(1/d)
Tensile strength
Tens
(kN/m )
2
Maximum Plastic volumetric strain
APLmax
(-)
Young‟s modulus
E p , Et
(kN/m )
2
Swelling potential
Kq p , Kqt
(-)
Poisson‟s ratio
nutp , nu p
(-)
Maximum swelling pressure
 0, p , 0,t
(kN/m )
Shear modulus
G 23
(kN/m )
Bedding rotation angle (clockwise)
Rot
(°)
2
2
Table 7: Constant model parameters throughout oedometer runs unless otherwise specified –‘d’ stands for day – Note:
maximum swelling pressure should be input as a positive value in the model input
Parameter
Unit
Value
Parameter
Unit
Value
Parameter
Unit
Value
'
(°)
35
A0
(1/d)
0.1
E p , Et
(kPa)
4E  06
c'
(kPa)
500
Ael
(1/d)
0
K q , p , K q ,t
(%)
0.33

(°)
0
Apl
(1/d)
0
 p , t
(-)
0.25
 Ten
(kPa)
0
Apl max
(-)
0.005
 0, p , 0,t
(kPa)
-750
G23
(kPa)
1.6E  06

(°)
0
Model ID
(-)
1
6
As displayed in Table 7, the swelling time parameters related to the plasticity including A pl, Apl
max as well as dilatancy angle (  ) are set to zero during oedometer runs and shall be
6
A0 is outside the range as given in Table 5. As mentioned earlier, Wittke-Gattermann (1998) used „A‟
parameters‟ values only in numerical simulations for reproducing the in situ measurement results in the
exploration gallery of Freudenstein tunnel. Hence, in the oedometer numerical simulation runs of the Soil
Test facility, a large value of A 0 value was used to accelerate the swelling process and reach the final
theoretical state in less duration.
31
Chapter 3: Soil Test Facility and layout of numerical simulations
investigated through uniaxial compression test 7. Ael is also set to zero in the model input and as
a result swelling time parameter (Eq.26, see Chapter 2) is leading to Eq.36.
q 
1
A0
Eq.36
Furthermore, the applied load of -130 kPa is inserted and theoretical final swelling strain ( t   )
is calculated according to Eq.34 as below:
 130 
-750  130  10   zq  0.0033.log 
  2.51‰
 750 
The loading conditions are shown (Table 8) as an example for a vertical load of -130 kPa which
is instantaneously applied after which a swelling time of 1000 days is considered in 100 steps.
Table 8: Loading conditions as an example for a vertical load of -130 kPa which is instantaneously applied after which a
swelling time of 1000 days in second phase in considered in 100 steps - Oedometer test
Table 9 shows different time step ratios used during the oedometer runs. This allows for
checking different time step ratios and corresponding model‟s responses.
Table 9: Different time step ratios used throughout oedometer runs
No. of
Run
R1
R2
R3
R4
Duration
(day)
No. of
Steps
q
20
263
400
800
100
500
100
100
10
10
10
10
Time step
ratio
0.02
0.0526
0.4
0.8
No. of
Run
R5
R6
R7
R8
Duration
(day)
No. of
Steps
q
1000
1400
2000
20000
100
100
100
500
10
10
10
10
Time step
ratio
1
1.4
2
4
3.2.2 Uniaxial compression test
The uniaxial loading is used to assess and validate the strength and plasticity parameters as
well as yield function. Uniaxial compression test is simulated through the General Lab of the Soil
Test Facility using the same sample properties as used in the oedometer test (Table 7). The
boundary conditions and initial stress state is shown below. This test is strain controlled (the
formulations related to elastic stress-strain behaviour can be derived from the generalised
Hooke‟s law shown in Eq.1, see Chapter 2):
7
This is because during the oedometer runs, yield conditions are not reached and hence the plasticity
parameters cannot influence the results and are not checked.
32
Chapter 3: Soil Test Facility and layout of numerical simulations
Figure 14: Loading conditions in a uniaxial compression test as an example for a vertical load which is controlled with
maximum strain of -0.1 % in the vertical direction and is applied in 100 steps
Stress Conditions   yy


1
 xx   zz    yy ;  yy  yy


E
E
 0;  xx   zz  0  
(1  2 )
  2
 yy
v
xx ( zz )   yy   v 

E

Eq.37
Eq.38
3.3 Simulation layout
A flowchart shown in Figure 15 illustrates the entire layout of simulation and validation process
carried out in the Soil Test Facility of PLAXIS2D. All the sensitivity analyses and parameter
variations are conducted within the aforementioned element tests to observe the model
parameters‟ influence on the results.
Through the oedometer runs 8, the suitable implementation scheme and a critical time step ratio
regarding the accuracy and stability of the results and different applied loads are selected. The
latter case was in fact verified after realising the effect of lateral stressing on the swelling strain
time curve. Hence, the influence of Poisson‟s ratio and swelling potential in horizontal direction
are discussed within the initial part. Having proposed a critical time step ratio of 1/19, the effect
of material stiffness (E) on the smoothness of the swelling curve and lateral stressing are
discussed. Subsequently, the variation of maximum swelling pressure  0,t  is shown.
Thereafter, swelling potential parameter  kq , p  is validated using a back analysis of experimental
data from S-Bahn Stuttgart project in Germany (Wittke-Gattermann, 1998). Then, the influence
of A0 and Ael swelling time parameters are illustrated. Finally, the model‟s stress path through
oedometer test within a certain period of time is discussed.
8
It should be noted that it is necessary to conduct the simulations in oedometer test for the evaluation of
some parameters since the one dimensional swelling law was obtained through the results of Huder and
Amberg oedometer testing. Hence, the oedometer run is the only way to simulate the same procedure
and check the maximum swelling pressure (  0 ) influence as well as swelling potential parameter ( k q ).
Furthermore, final theoretical swelling strain versus time curve in Huder and Amberg test can be wellreproduced in oedometer and hence the effect of A0 , Ael are investigated.
33
Chapter 3: Soil Test Facility and layout of numerical simulations
Uniaxial loading runs 9 are simulated firstly to evaluate the proper implementation of the yield
function (Mohr-Coulomb failure criterion).Then the effect of strength parameters including
cohesion (c‟) and friction angle  ' are explained. After that, by considering the state of failure,
the effect of plasticity parameters including A pl, Apl max and dilatancy angle   are discussed.
Figure 15: Layout of the entire numerical simulations in order to assess the model under study
9
The reason to conduct uniaxial compression test is to study the influence of plasticity parameters.
34
Chapter 4: Results discussions and interpretations
Chapter 4: Results discussions and interpretations
In the following, the element tests‟ runs are shown and discussed which are based on the
flowchart illustrated in Figure 15.
4.1 Different time step ratios within implicit and explicit scheme
Different time step ratios shown in Table 9 are checked within both implicit and explicit
schemes. This sensitivity analysis helps to determine the suitable implementation scheme as
well as a critical time step ratio for calculation of the swelling strain. Table 10 shows the results
of the oedometer runs with different time step ratios within the aforementioned schemes.
Table 10: Oedometer runs – Different time step ratios within implicit and explicit schemes – R1 to R16
Implicit Scheme
Parameter
R1
R2
R3
R4
R5
R6
R7
R8
Time step ratio
0.02
0.0526
0.4
0.8
1
1.4
2
4
0.00226
0.00259
0.00259
0.00259
0.00259
0.00259
0.00259
0.01474
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Vertical swelling strain
(numerical)
F<0
Explicit Scheme
Parameter
R9
R10
R11
R12
R13
R14
R15
R16
Time step ratio
0.02
0.0526
0.4
0.8
1
1.4
2
4
0.00223
0.00262
0.00516
0.01056
0.1082
0.5902
∞
∞
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Vertical swelling strain
(numerical)
F<0
4.1.1 Implicit scheme
The first 8 runs (R1 to R8) shown in Table 10 are simulated within implicit scheme. As displayed
in R1, the obtained numerical value does not match the theoretical value. This is because the
duration has not been enough so that swelling strain has not reached the plateau yet (see
Appendix A.1). The numerical values obtained from R2 to R7 are quite close to the
corresponding theoretical values but not the same, i.e. one can notice that there is a bias
between the numerical results obtained from the Soil Test Facility and the theoretical value. This
is due to the assumption in the theoretical calculation of one dimensional swelling law (Eq.11)
indicating that the whole vertical strain is due to the swelling in the vertical direction.
This assumption does not hold true since there is swelling in horizontal direction as well. If the
sample is allowed to deform freely in all directions, the total vertical and horizontal strains can
be written as Eq.39 and Eq.40.
 Total vertical   Elastic   q , p ( Swelling )
Eq.39
 Total  horizontal   Elastic   q ,t
Eq.40
( Swelling )
35
Chapter 4: Results discussions and interpretations
Where
 q , p ( Swelling ) : Swelling strain in vertical direction which is equal to the sum of equivalent elastic
strain (when material is back from the elastic response to its original position, which is
equivalent to the amount of elastic strain) and plastic strain starting from the origin.
 q ,t ( Swelling ) : Swelling strain in horizontal direction.
In the standard oedometer apparatus, the lateral strains are prevented; hence the total strain in
horizontal direction is zero. According to oedometric boundary conditions, Eq.40 must be zero
leading to Eq.41.
Oedometric boundary conditions  Total horizontal  0   q ,t
( Swelling )
  Elastic
Eq.41
This causes an increase in lateral stress by time increasing. Such increase produces vertical
elastic extension according to Poisson‟s ratio (  ) contributing to the total vertical strains.
Therefore, the numerical results are always greater than the theoretical values (Figure 16). This
difference (0.00008) between theoretical solution (0.00251) and numerical results (0.00259)
shown in Figure 16 which is indeed an elastic strain (Eq.41) is validated by reducing the
Poisson‟s ratio value indicating the difference value is reduced (see 4.2).
Figure 16: Oedometer test R2 to R5 together - Implicit scheme - Time step ratio = 0.02 to 1 – Bias between theoretical value
and numerical results
Based on the numerical results from the R5 to R8 through the oedometer test, there is a
problem with increasing the time step ratio. Up to the time step ratio of 1 (see Figure 16), the
swelling strain ceases at the same amount as in R2-R4. After this ratio, the oscillation is
observed both in the” applied load against time curve” and “vertical strain versus time curve”
(see Appendix A.1). If time step ratio increases more, the swelling suddenly keeps on increasing
36
Chapter 4: Results discussions and interpretations
and it does not stop at all, i.e. swelling approaches infinity. The critical time step ratio seems to
be 1 in the Soil Test Facility which is defined as below using Eq.32 leading to Eq.42.
using Eq.31
Critical time step ratio  1 
dTime
q
=1  dTime  q
Eq.42
Eq.42 indicates that any time step greater than swelling time parameter causes troubles in
calculation steps (see related simulations‟ results in Appendix A.1). As an example, considering
A0=0.1 (1/d) results in ηq equal to 10 days, hence dTime  10 days
4.1.2 Explicit scheme
The next runs (R9-R16) shown on the bottom row of Table 10 are simulated within explicit
scheme. This helps to determine whether or not increasing the time step ratio as observed
within implicit scheme causes a problem.
As it can be seen from the resulting values (R9-R16), the problem occurred with increasing the
time step arises, i.e. swelling keeps increasing by increasing the time step ratio within explicit
scheme. There is also a perturbation in the “applied load against time” curve while the load
should be constant (see Appendix A.2). As shown in R13, one can notice that even with time
step ratio of 1 within explicit scheme, the accurate and stable results are not obtained
contradicting critical time step ratio defined in Eq.42.
There is a small different in R9 and R10 with the corresponding numerical results within implicit
scheme. Significant increase in swelling strain and massive instability in both „horizontal stress
versus time‟ and „vertical stress versus time‟ curves are observed in R12 to R16 (see Appendix
A.2). On the contrary to the results shown with implicit scheme, it is obvious that explicit scheme
overestimate swelling strain.
It should be noted that the problem with massive instability can be decreased by defining a very
large value of tensile strength (e.g. σten > 1000 kPa). For example, R17 is simulated by means of
the same parameters‟ values in R16 except defining an input of tensile strength equal to 1000
kPa (Table 11). As shown in Figure 17, swelling strain decreased one order of magnitude but is
still very inaccurate in comparison to the theoretical value, indicating that oedometer testing
using large time step ratio and in particular within explicit scheme does not result in the desired
outcomes.
Table 11: Increasing the stability of results within explicit scheme and large time step ratio by inserting a large value of
tensile strength– oedometer R17
Parameter
R17
Time step ratio
4
σten (kPa)
1000
Vertical swelling strain (numerical)
0.01474
F<0
Yes
37
Chapter 4: Results discussions and interpretations
Figure 17: Oedometer test's R17 - Explicit Scheme - Swelling strain decreased due to effect of tensile strength of 1000 kPa
In a short conclusion, due to the aforementioned drawbacks to the explicit scheme and the fact
that more accurate results can be obtained through implicit scheme rather than explicit one, all
parameters‟ variations are conducted within implicit scheme in the next sections.
4.1.3 Low applied loads
Besides the aforementioned problem with time step ratio of 1 (instability and inaccuracy due to
time step increase within explicit scheme as observed in R13), the results of sensitivity analyses
shown in Table 12 (R18 to R22 simulated runs with time step ratio of 1) indicates that in the
case of applying low loads within implicit scheme, inaccurate and instable results are obtained
(see Appendix A.3a).
Table 12: Oedometer test's runs – Implicit scheme – R18 to R22- dTime/Eta=1
Parameter
R18
R19
R20
R21
R22
Applied load (kPa)
-2
-5
-10
-20
-70
Theoretical value of swelling strain
0.00619
0.00619
0.00619
0.00519
0.00339
Vertical swelling strain (numerical)
0.12432
0.04668
0.00248
0.01236
0.00353
F<0
Yes
Yes
Yes
Yes
Yes
The values shown in the second row of Table 12 are the theoretical results with different applied
loads obtained using Eq.34. The results obtained though R18 to R22 clearly show a huge
swelling increase in the case of applying low applied loads, which occurs with the time step ratio
of 1. Furthermore, by looking at R22 results it can be understood that still there is a huge
perturbation in the results. Although, a small value of tensile strength (e.g.100 kPa) can fix the
problem with the applied load of equal to or less than -10 kPa, there will be a problem with low
applied loads such as -70 kPa (see Appendix A.3b for simulated R18‟, R19‟, R20‟, R21‟, R22‟
with the same data as used in R18-R22 shown in Table 12, where tensile strength effect is
observed on the final swelling strain).
One can conclude that there should be a limit on time step ratio at which the swelling ceases
avoiding the occurrence of the observed oscillation and instability in the results (see 4.4). Before
introducing the proposed critical time step ratio in this project, the effect of Poisson‟s ratio (  )
38
Chapter 4: Results discussions and interpretations
and horizontal swelling potential ( kq ,t ) on lateral stressing which were referred in the abovementioned results are discussed.
4.2 Influence of Poisson’s ratio (ν)
As mentioned earlier, due to the oedometric boundary conditions, when permanent swelling
strain cannot develop in horizontal directions, it causes an increase in the lateral stresses. Such
increase causes vertical elastic extensions according to Poisson‟s ratio contributing to the total
vertical strain. The effect of Poisson‟s ratio on lateral stressing can be investigated by setting
  0 where the stresses can only develop in the loading or vertical direction. Hence, it is
expected that the theoretical solution (Eq.11) can be reproduced. To this end, R23 is simulated
with the same parameters‟ values in R2 with the exception of  p  0 and  t  0 .
Additionally, a very small value of tensile strength (σ Ten) (e.g. 0.001 kPa) needs to be inserted.
Figure 18 shows that indeed the expectation is met and the total vertical swelling strain is
exactly the same as the theoretical value.
Figure 18: Effect of Poisson's ratio on lateral stressing and total vertical strains –Implicit scheme- oedometer test – R23
As shown earlier, the vertical swelling strain (  q , p ) is equal to sum of equivalent elastic strain
and the plastic strain starting from the original position of the sample. The equivalent elastic
strain can be calculated using the oedometer stiffness and the applied load as follows:
EOedometer 
E 1  
1  1  2 
39
Eq.43
Chapter 4: Results discussions and interpretations
The straightforward linear stress-strain relationship is defined as in Eq.44:


EOedometer
Eq.44
Therefore, in the case of applied load (σ) of -130 KPa and   0 , the elastic strain is calculated
as shown below.
Substituted in Eq .43
  0 & E  4GPa 
 EOedometer 
Substituted in Eq .44

 
4(1  0)
 4 GPa
(1  0)(1  0)
130
 0.0000325
4000000
The obtained resulting value for elastic strain validates the obtained value from numerical
simulations illustrated in Figure 18.
It should be noted that three different runs (R23‟ (a), R23‟ (b) and R23‟ (c)) showing the
difference between numerical results and theoretical value discussed in 4.1.1 are simulated by
reducing the Poisson‟s ratio value with the same parameters‟ values in R2 (Table 13) and are
shown in Appendix A.4a.
Table 13: Effect of Poisson's ratio on the difference in final swelling strain (different between numerical results and
theoretical solution)
Parameter
Theoretical solution
Vertical swelling strain (numerical)
Poisson’s ratio
Difference (Numerical – theoretical)
F<0
R23’ (a)
0.00251
0.00257
0.20
0.00006
Yes
R23’ (b)
0.00251
0.00256
0.15
0.00004
Yes
R23’ (c)
0.00251
0.00253
0.07
0.00002
Yes
As shown in Table 13, this indeed reduces the elastic strain contributing to the total vertical
strain in swelling strain time curve in oedometric boundary conditions and as explained earlier,
with zero Poisson‟s ratio the difference is disappeared.
4.3 Influence of swelling potential in horizontal direction (kq,t)
The contribution of horizontal stresses to the total vertical strains does not occur unless the
maximum horizontal stress is less than the maximum swelling pressure in horizontal direction
(σ0,t). This indicates that there is still swelling in horizontal direction. For instance, even if a
vertical load equal to maximum swelling pressure is applied, there is still vertical strain observed
in the results because the horizontal stress is less than the maximum swelling pressure input.
Besides Poisson‟s ratio (see 4.2), swelling potential parameter in horizontal direction (K q,t) is of
influence on the lateral stressing. It is expected that the theoretical solution (Eq.11) can be
reproduced by defining Kq,t=0 so that the bias between numerical results and theoretical value is
40
Chapter 4: Results discussions and interpretations
disappeared. To this end, the first five conducted runs within implicit scheme in section 4.1.1 are
simulated through R24-R28 with different time step ratios as shown in Table 14.
Table 14: Oedometer runs – Influence of horizontal swelling potential Kq, t=0 - implicit scheme – R24 to R28
Parameter
Time step ratio
Vertical swelling strain (numerical)
F<0
R24
0.02
0.00215
Yes
R25
0.0526
0.00251
Yes
R26
0.4
0.00251
Yes
R27
0.8
0.00251
Yes
R28
1
0.00251
Yes
This shows indeed the effect of horizontal stresses‟ contribution to the total vertical strain in
oedometric boundary conditions. In R24, the time step has not been adequate for reaching the
plateau or maximum swelling strain (cf. R1). As shown in Table 14, from R25 to R28 by putting
swelling potential in horizontal direction equal to zero ( kq ,t  0 ), the simulated swelling strain and
the theoretical one are identical and the expectation is met (see Appendix A.4b).
The influence and variations of material stiffness (E) and maximum swelling pressure on lateral
stressing shall also be shown later on within the proposed critical time step ratio and implicit
scheme.
4.4 Proposed critical time step ratio of 0.0526 or 1/19
As mentioned earlier, due to drawbacks to the model implementation including the problem with
time increasing after certain amount of time step ratio; oscillations in the resulting curves within
explicit scheme, etc. it is necessary to define a limit on time step ratio. Based on the obtained
results in the previous sections and of the sensitivity analysis regarding different time step ratios
(see Ra-Rj in Appendix A.5a and A.5b), the time step ratio of 0.0526 seems to be working pretty
well in implicit scheme. Therefore, a sensitivity analysis of this ratio toward different load steps
are carried out as shown in Table 15 to determine whether it can be considered as the critical
time step ratio. The considered ratio of 0.0526 can be defined as a factor of 1 over 19 with the
maximum number of steps as follows:


 dTime
1
1
 0.0526   dTime  

Duration
263 

19
19
dTime 
; e.g.
Steps
500 
19
 Steps  Duration
Eq.45
Eq.45 postulates a critical time step ratio in which the time steps smaller than a factor of 1 over
19 of swelling time parameter (  ) do not give troubles in the calculation steps and avoid
oscillation and inaccuracy in the results.
Table 15 shows the results of the simulated runs with the proposed critical time step ratio of
1/19. The results of R29 to R36 (see Appendix A.5c) show the contribution of the horizontal
stresses to the total vertical strain. The proposed ratio responses well for the low applied
stresses including -10 kPa and lower ones, i.e. As it can be seen in R29, R30 and R31, the
41
Chapter 4: Results discussions and interpretations
applied loads of less than or equal to -10 kPa were used and all resulted in the same amount of
swelling strain (cf. R18-R20 in 4.1.3).
Table 15: Oedometer runs - Implicit scheme – R29-R44 – validation of the proposed dTime/Eta=0.0526
Parameter
R29
R30
R31
R32
R33
R34
R35
R36
Applied load (kPa)
-2
-5
-10
-20
-70
-280
-480
-750
Theoretical value
0.00619
0.00619
0.00619
0.00519
0.00340
0.00141
0.00064
0.00000
0.00627
0.00627
0.00627
0.00527
0.00347
0.00148
0.00070
0.00020
F<0
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Parameter
R37
R38
R39
R40
R41
R42
R43
R44
0.00619
0.00619
0.00619
0.00519
0.00340
0.00141
0.00064
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Vertical swelling strain
(numerical)
Vertical swelling strain
(numerical) (kq,t=0)
F<0
10
0.00015
Yes
R37 to R44 shown in Table 15, are the same runs as illustrated in R29 to R36 except that the
effect of swelling potential in horizontal direction is removed, i.e. Kq,t=0. In all runs, the same
results as the theoretical solution (Eq.11) are obtained. As it can be seen in R36 (see Appendix
A.5c), there is a little bias in the results where it is expected to be zero as the applied load is the
same as maximum swelling pressure. In fact, this is the contribution of horizontal stresses only.
According to the R36 related figures (see Appendix A.5c), the maximum horizontal stress (once
the strain ceases) is -706.5 KPa which is still less than -750 KPa (maximum swelling pressure in
horizontal direction) indicating there is still swelling in horizontal direction.
The value of 0.00015 shown in R44 (see Appendix A.5d) is only the elastic strain (which can be
calculated through Eq.43 and Eq.44) since no horizontal swelling occurs by defining zero
swelling potential in lateral direction.
In conclusion, the proposed critical time step ratio of 0.0526 works well with different applied
loads within implicit scheme. Further investigations on the model shall be done with/within the
proposed critical time step ratio.
4.5 Influence of material stiffness (Young’s modulus or E)
In this section, a sensitivity analysis regarding the effect of material stiffness on the smoothness
of the final swelling results as well as the lateral stressing is tested with the proposed time step
ratio of 0.0526. The applied load of -130 kPa is used in the oedometer in all runs.
The result of sensitivity analyses has revealed that material stiffness is of a significant influence
on the smoothness of final vertical swelling strain as well as the lateral stressing. It is expected
to observe lesser increase in lateral stresses in the case of testing a flexible material in
comparison to a stiff material where the elastic response is a very little value. Table 16 shows
10
Only elastic response
42
Chapter 4: Results discussions and interpretations
the variation of material stiffness (E) parameter used in the sensitivity analysis and the
corresponding results.
Table 16: Oedometer runs - Implicit scheme – R45 to R54 - Time step ratio =0.0526 – Material stiffness effect
Parameter
R45
R46
R47
R48
R49
Ep=Et (kPa)
1E+04
1E+05
5E+05
10E+06
100E+06
G23 (kPa)
4E+03
4E+04
20E+04
40E+05
20E+06
σH-Horizontal stress (kPa)
-84.95
-251.85
-476.57
-723.76
-747.24
Vertical swelling strain (numerical)
0.0046
0.003554
0.00294
0.00254
0.00251
F<0
Yes
Yes
Yes
Yes
Yes
Parameter
R50
R51
R52
R53
R54
Vertical swelling strain (numerical) (kq,t=0)
0.00251
0.00251
0.00251
0.00251
0.00251
F<0
Yes
Yes
Yes
Yes
Yes
σH is
the horizontal stress which is reached during loading simulation of different samples shown
in Table 16. The maximum swelling pressure is the same as used previously, i.e. σ0,t = -750 kPa.
Figure 19 displays the influence of material stiffness on the lateral stressing. The stiffer the
material is the larger horizontal stresses are reached and hence the expectation is met. This is
coherent in R48 and R49 where the difference between the horizontal stress and maximum
swelling pressure is very small and thus the contribution of lateral stressing to the swelling
strain.
Figure 19: Influence of material stiffness on the lateral stress in oedometer runs - R45-R49 over the same period of time
Figure 20 illustrates the impact of stiffness variation on the smoothness of the swelling strain
curve versus time for which the swelling potential in horizontal direction has been set to zero.
The amount of total vertical strain in all cases either with flexible or stiff materials reach the
same value as the theoretical values. As it can be seen in R50, there is a huge value for elastic
43
Chapter 4: Results discussions and interpretations
response indicating a very flexible material; as the material gets stiffer the elastic portion is
almost negligible (see R51-R54). For the related Figures refer to Appendix A.6a and A.6b.
Figure 20: Influence of material stiffness on the final vertical swelling strain in oedometer runs – R50-R54 over the same
period of time – Kq,t=0
Despite testing very stiff materials in the above-mentioned runs, yield condition was never
reached (this is because of the cohesion and friction angle (strength properties of rock)) and is
highly influenced by the value used for maximum swelling pressure in the horizontal direction
(σ0,t=-750 kPa) which shall be discussed in next section.
4.6 Influence of maximum swelling pressure in horizontal direction (σ 0,t)
Besides material stiffness, Poisson‟ ratio and swelling potential in horizontal direction, maximum
horizontal swelling pressure can be considered as another parameter relating to the horizontal
stress, i.e. depending on the defined value of the maximum swelling pressure by the user
(which varies in different rock samples), it can whether or not bring the horizontal stress to the
failure. To this end, maximum swelling pressure in horizontal direction variations are made with
the applied load of -130 kPa as shown in Table 17.
Table 17: Oedometer runs - Implicit scheme – dTime/Eta=0.0526 – Effect of maximum horizontal swelling pressure – material
stiffness of 4E+06 kPa used in all variations
Parameter
R55
R56
R57
R58
σ0,t (kPa)
-750
-1500
-2400
-3281
σH-Horizontal stress (kPa)
-689.2
-1276.5
-1885.9
-2400.5
Vertical swelling strain (numerical)
0.00259
0.00363
0.00441
0.00491
F<0
Yes
Yes
Yes
No
44
Chapter 4: Results discussions and interpretations
Figure 21: Effect of maximum horizontal swelling pressure on the total vertical strain and lateral stressing
With the strength properties consisting of cohesion  c '  of 500 kPa and friction angle  '  of 35 ,
the material will be failed once the horizontal stress reaches -2400.5 kPa (=σ3) as shown below
(using Eq.35).
1
1
F  . 130  .1  sin 35   .(2400.5). 1  sin 35   500.cos35  F  0
2
2
Table 17 shows that the failure state is never achieved when maximum swelling pressure in
horizontal direction (  0,t ) is less than -3281 kPa (see R55, R56 and R57). R58 brings the
horizontal stress up to -2400.5 kPa and the material is failed (Figure 21). If swelling potential in
horizontal direction is set to zero, then increasing the maximum horizontal swelling pressure will
have no impact on the lateral stressing.
It should be noted that, maximum swelling pressure differs depending on the rock properties
including stiffness, i.e. maximum swelling pressure is not a physical properties of rock but on the
contrary in practice, it can be obtained from laboratory Huder and Amberg testing „based on the
physical properties of rock‟ as discussed in Chapter 2. Since in the laboratory testing such as
Huder and Amberg test, maximum swelling pressure of -750 kPa is obtained (the maximum
applied load above which no swelling occurred) reveals that not high stiffness values were used
in those examples and hence, lower horizontal stress was caused and plasticity was not
initiated during the test. This is also the reason here to investigate the plasticity and strength
parameters in uniaxial compression test.
In a short conclusion, one can notice that the greater the stiffness is used, the larger horizontal
stress will be reached. Upon the defined amount of swelling potential and corresponding
maximum swelling pressure, the plasticity may set in.
45
Chapter 4: Results discussions and interpretations
4.7 Validation of swelling potential parameter (Kq)
As discussed in Chapter 2, swelling potential is the slope of a straight line in a plot of swelling
strain against the logarithmic applied stress. This indicates the trend in which the reduction in
the applied stress leads to an increase in swelling strain. Since the effect of swelling potential in
horizontal direction is already understood, this parameter is set to zero, i.e. kq ,t  0 and a
variation of swelling potential in vertical direction is conducted.
In order to validate the swelling potential parameter, a back-analysis using the experimental
data shown in Figure 9 (see Chapter 2) from S-Bahn Stuttgart project is carried out. To this end,
the data related to the applied loads and their corresponding final swelling strains are (in a
rough approximation) read off manually from the right-hand side diagram of Figure 9 which
illustrated schematically in Figure 22. Figure 22 also displays the read off data.
Figure 22: Schematic diagram of the swelling strain versus logarithmic applied load diagram shown in Figure 9 based on the
experimental results obtained from S-Bahn Stuttgart project in Germany as well as rough approximation read off data of
swelling strains and the applied loads
Table 18 shows the simulated runs consisting of R59-R65.The calculated swelling potential
values from theoretical formulation of swelling law (Eq.11) are used in the simulation to obtain
the numerical values of final swelling strain.
Table 18: Variation of swelling potential in vertical direction in oedometer runs - kq,t=0 and σ0=-750 kPa - proposed ratio of
1/19
Parameter
Vertical swelling strain (‰) – read off from Figure22
σ - applied load (kPa) – read off from Figure 22
Theoretical kq,p (%) – calculated from Eq.11
Vertical swelling strain (‰) (Numerical – Soil Test Facility)
F<0
11
Only elastic response as explained previously
46
R59
4.60
-17
0.28
4.60
Yes
R60
4.36
-36
0.33
4.35
Yes
R61
3.27
-66
0.31
3.27
Yes
R62
2.51
-130
0.33
2.51
Yes
R63
1.32
-279
0.31
1.33
Yes
R64
0.31
-552
0.23
0.31
Yes
R65
0.00
-750
0.00
11
0.15
Yes
Chapter 4: Results discussions and interpretations
Since the runs are simulated for the same sample, the maximum swelling pressure is kept as 750 kPa; but different load steps are used according to the experimental data shown in Figure
22.
As shown in an example in Figure 9 (see Chapter 2), the slope of the best fitting line of the
swelling strain versus logarithmic of applied load corresponding to the one dimensional swelling
law formula (Eq.11) is 3.3‰ (0.0033); which is considered as kq,p.
As it can be seen in Table 18, the obtained numerical swelling strains correspond to the values
read off from the experimental results. This indicates the same K q,p are obtained for each load
steps as well. Hence, this validates the swelling potential parameters and thus the
implementation of one dimensional swelling law in the model‟s routine.
4.8 Effect of A0 and Ael swelling time parameters
Having known the effect of time step, i.e. influence of the time steps and duration of the test, it
can be straightforward to notice the influence of A 0 swelling time parameter as the main
component on time dependent behaviour. Any increase in A0 value compensates the smaller
time steps to reach the final swelling strain, i.e. it can be considered as an accelerator for
reaching the final state.
To evaluate the influence of A0 parameter, five successive runs are simulated. It should be
noted that the time step for each run is selected based on the A0 value in the same run within
the proposed critical time step ratio. Table 19 shows the variation of A0 value and chosen time
steps accordingly.
Table 19: Influence of A0 swelling time parameter on the swelling time dependent behaviour over 100 days
Parameter
A0 (1/day)
Time step (day)
R66
0.007
100/13
R67
0.01
100/19
R68
0.03
100/57
R69
0.06
100/114
R70
0.1
100/190
Figure 23: Influence of A0 swelling time parameter on the final swelling strain – vertical strain against time over 100 days
47
Chapter 4: Results discussions and interpretations
The diagrams shown in Figure 23, demonstrate the influence of this value sharply in five
successive sequences. In fact, increasing the A 0 value accelerates the swelling process over
the same period of time (e.g. 100 days).
On the other hand, the Ael swelling time parameter which is related to the absolute value of
current elastic volumetric strain has minor effect on the time dependent behaviour. Swelling time
parameter formula (Eq.26, see Chapter 2) considering Ael and A0 together is rewritten as below:
q 

1
A0  Ael   vel
Eq.46

The variation of Ael swelling time parameter at each run is displayed in Table 20. Throughout the
simulations (R71-R75) the swelling potential in horizontal direction is set to zero, i.e. kq ,t  0 and
hence the final swelling strain is the total vertical strain. This indicates elastic volumetric strain is
equal to the vertical strain according to oedometric boundary conditions.
Table 20: Influence of Ael swelling time parameter on the swelling time dependent behaviour, A0=0.01, time step = 500/100
Parameter
Ael (1/day)
R71
1
R72
10
R73
100
R74
500
R75
1000
The contribution of Ael to the swelling time parameter and thus swelling rate is displayed in
Figure 24. It is coherent that in comparison to A0 parameter, Ael is less influential and in fact the
greater the A0 value is, the larger Ael values are required as the input parameter to observe its
impact on the results of swelling strain time curve.
Figure 24: Influence of Ael swelling time parameter on the final swelling strain - vertical strain against time
48
Chapter 4: Results discussions and interpretations
4.9 Model’s stress path prediction in oedometer test
As explained earlier, in order to perform oedometer test, first the desired applied stress is
applied instantaneously and then the load is maintained constant over a certain amount of time
to observe the swelling effect of the tested sample. In fact, this section aims at observing the
model‟s response while the swelling has not ceased yet and also whether or not its behaviour
within the loading conditions as well as its stress path can be predicted. To this end, a very
small time step ratio of 0.000012 (which is less than the proposed critical time step ratio and
hence is stable) is selected as below and the run is simulated (R76):
A 0  0.00001  q  100000 
1.2
 dTime

 0.000012

600

100000
dTime 
 1.2

500

Eq.47
The other model parameters are the same as used in the previous oedometer runs. In this
example (R76), the results for the load step of -130 kPa is shown (Figures 25 and 26).
When the material is compressed and the load is applied instantaneously in a very short
duration (Phase 1, shown in Figure 25), there will be an elastic response for which the amount
of strain can be calculated through Eq.43. The initial horizontal stress is normally less than the
vertical stress. Over the elastic range, it is possible to calculate the initial horizontal stress
based on the vertical stress and the Poisson‟s ratio (Eq.48).
 
 3/ h  
 1 

 1/ v

Eq.48
Where 1 and 3 represent major and minor principal directions and are equivalent to vertical and
horizontal stresses‟ directions respectively. In the current case, with an applied stress of -130
kPa and a Poisson‟s ratio of 0.25, the minor principal stress (horizontal stress) is become equal
to -43.3 kPa.
After the application of the load, it is maintained constant over nearly 600 days in 500 steps. In
terms of principal stresses, the lateral and axial stresses equalise each other so that both
vertical and lateral stresses become equal at σ 1=σ3=-130 kPa (Phase 2, see diagrams shown in
Figure 25). It should be noted that, the plotted results in terms of principal stresses in the Soil
Test Facility are ordered in such a way that in absolute sense, σ1 is the largest and σ3 is the
smallest principal stresses and after increase in lateral stress  k0  1 , σ1 and σ3 swap. In
engineering practice, this can be explained according to Heim‟s suggestion (1912, Hoek et al.,
1990) stating the inability of rock to support stress difference and the effects of time dependent
deformation of the rock mass can cause lateral and vertical stresses to equalise over certain
period of time. The difference between the initial stresses triggers a plastic flow in the rock and
they act in a way to equalise each other and as a result the K 0 must be reached 1.
49
Chapter 4: Results discussions and interpretations
In Phase 3, lateral stresses keep on increasing significantly while the axial load is constant (see
Figure 25 and Figure 26b). This is due to the oedometric boundary conditions causing an
increase in lateral stresses with time development.
Figure 25: (a) Major Principal Stress vs. vertical strain- (b) Major Principal Stress vs. Minor Principal Stress -Model’s stress
path prediction- load step of -130 kPa – Phase 1) Elastic response for which the amount of strain is calculated through Eq.46;
Phase 2) Both vertical and lateral stresses equalise each other over a constant period of time; Phase 3) horizontal stresses
keep on increasing while vertical stress is constant (major and minor principal stresses swap) – R76
Figure 26: (a) Horizontal stress vs. time- both σ3 and σxx against time curves together - Model’s stress path prediction in the
oedometer test – (b) Vertical (applied) stress vs. time - load step of -130 kPa – R76
By looking at horizontal stress versus time diagram shown in Figure 26a, it t can be seen that
σxx becomes equal to -130 kPa after 526.8 days. On the same diagram, it is observed that σ3
reach -130 at the same time (and then σ3 and σ1 swap as explained earlier). By time
development, the horizontal stress increases and becomes larger than the vertical stress and
reach -140 kPa after 600 days (the same amount as shown in Figure 25 for σ1).
Hence, according to the obtained results, the model‟s stress path can be predicted through the
oedometer test.
50
Chapter 4: Results discussions and interpretations
4.10 Evaluation of yield function
As mentioned earlier (see Chapter 2), Mohr-Coulomb failure criterion has been implemented as
the yield surface in the current model. To evaluate the proper implementation of yield function,
uniaxial compression test is simulated in the general lab facility. First, the test is simulated using
the swelling rock model and then the results are compared with the results of the same test
using Mohr-coulomb material model. This also allows for validation of the strains within the
same test. The initial and boundary conditions for simulation of the uniaxial compression test in
general lab facility is shown in 3.2.2 (see Chapter 3).
4.10.1 Uniaxial compression test via swelling rock model
The uniaxial compression test is simulated in strain-controlled conditions in the vertical direction
using the same parameters‟ values employed in oedometer runs. The maximum strain in y
direction is selected as -0.1% (see Chapter 3). The obtained results from R77, which is the
uniaxial compression test via swelling rock model, are illustrated in Figure 27.
Figure 27 shows that the maximum vertical stress reached during loading of the sample is -1921
kPa. This can be validated through yield function described in Eq.35 (see Chapter 3).
The obtained numerical results (shown in Figure 27) for the elastic strain in vertical and
horizontal direction before yielding are -0.00048 and +0.00012 respectively. Furthermore, the
obtained numerical value for elastic volumetric strain is -0.00024. These values regarding
elastic strains can be validated through theoretical formula (generalised Hooke‟s law) described
Eq.37 and Eq.38 (see Eq.1 in Chapter 2 and Eq.37 and Eq.38 in Chapter 3).
Figure 27: Vertical stress versus vertical and lateral strain – uniaxial compression test via swelling rock model – R77
Hence, the validation is carried out for elastic stress-strain behaviour and after yielding of the
sample as below.
51
Chapter 4: Results discussions and interpretations
4.10.1.1 Elastic stress-strain behaviour
Using Eq.37 and Eq.38, the elastic strains both in vertical and lateral directions as well as
elastic volumetric strains are calculated. As shown below, the theoretical results in essence
verify the obtained numerical values.
1
1

 xx   E  yy  4 E  06  0.25  1921  0.00012

 yy
1921

Using Eq.37  yy 

 0.00048
E
4 E  06

(1  2 )

 v  2 xx   yy  E  yy  0.00024

4.10.1.2 Yielding
To verify the yield function itself and validate the above-mentioned numerical results, yield
function described in Eq.35 is employed. In Eq.35, σ 1 and σ3 are replaced with the maximum
vertical stress of -1921 kPa observed in the simulation (see Figure 27) and zero lateral stress
respectively. The material cohesion (c‟=500 kPa) and internal friction angle (  '  35 ) are the
same as used previously. The failure criterion is shown as below:
1
1
Using Eq.35  F   (1921) 1  sin 35  (0) 1  sin 35   500.cos35  0
2
2
The yield function becomes zero with the defined input strength parameters indicating that the
material is in the state of failure. This not only validates the obtained numerical results, but also
indicates that the yield function has been implemented properly.
4.10.2 Uniaxial compression test – Mohr-Coulomb material model
The same run as R77 is simulated by means of Mohr-Coulomb material model to verify the
obtained results in the swelling rock model and the yield function as well. As displayed in Table
21, the values used as input parameters of Mohr-Coulomb material model are the same as used
previously. This allows for validation of strains as well. The obtained result for R78, which is the
uniaxial compression test via Mohr-Coulomb material model, is shown in Figure 28.
Table 21: Model parameters’ values used in Mohr-Coulomb material model through uniaxial compression test – R78
Parameter Value Parameter Value Parameter Value Parameter Value
 ' (◦)
35
c’ (kPa)
500
E (kPa)
4E+06
ν (-)
0.25
The obtained numerical results with this model are exactly the same as observed results with
the swelling rock model. Hence, one can conclude that the yield function has been implemented
properly. Furthermore, the elastic strains which are obtained theoretically through generalised
Hooke‟s law have been validated.
52
Chapter 4: Results discussions and interpretations
Figure 28: Vertical stress vs. vertical and lateral strain – uniaxial compression test via Mohr-Coulomb material model – R78
4.11 Influence of cohesion (c’) and internal friction angle (  ' )
The strength parameters including cohesion (c‟) and friction angle (  ' ) are of influence on yield
function. To observe the influence of strength parameters some runs with a successive
sequence in increasing the strength of the material are carried out. This is achieved by
increasing the cohesion and friction angle in different runs and the following results are
obtained. Therefore, four different runs‟ data including the variation of strength parameters as
well as maximum vertical stress shown in Table 22 are simulated.
Table 22: Variation of cohesion and friction angle and their influence on the strength of material
Parameter
c’ (kPa)
 ' (◦)
Maximum vertical stress (kPa) (numerical)
F<0
R79
R80
R81
R82
500
800
1200 2000
20
25
30
35
-1428.15 -2511.5 -4000 -4000
No
No
Yes
Yes
As it can be seen in Figure 29, after a certain increase in the cohesion and friction angle values,
the vertical stress at failure does not increase and hence bi-linearity is no longer observed in the
vertical stress-vertical strain (see R81 and R82).This is because the yield condition is already
met and hence increasing the strength parameters will not lead to any more increase in the
stress at failure. The effect of strength parameters can be shown through yield function (Eq.35)
as below (R79 as an example of state of failure (F=0) and R81 for F<0 where no failure occurs):
1
1
R79  F   (1438.15) 1  sin 20   (0) 1  sin 20   500.cos 20  0  Failed
2
2
1
1
R81  F   (4000) 1  sin 30   (0) 1  sin 30   500.cos30  39.2  0  Not failed
2
2
53
Chapter 4: Results discussions and interpretations
Figure 29: Influence of strength parameters – R79-R82 in uniaxial compression test
4.12 Influence of dilatancy angle or  and Apl and Apl max swelling time parameters
In this section the influence of parameters related to plasticity are validated, i.e. Apl and Apl
max
swelling time parameters and dilatancy angle (  ). Dilatancy angle ( ) affects only plastic
volumetric strain; Apl swelling time parameter affect plastic strains and Apl
max
swelling time
parameter is considered as a limit for plastic strains. Hence, these parameters influence the
results while the sample is in the state of failure. First dilatancy angle is validated as below.
4.12.1 Dilatancy angle ( )
According to the results of triaxial test, the slope of ascending line in the bi-linear curve of
2sin
volumetric strain against vertical strain could be approximated by m 
(as shown
1  sin
schematically in Figure 30). This is in fact used in triaxial tests, however, in the uniaxial loading
run, a triaxial test with lateral stresses of zero is assumed.
Hence, the following runs (R83-R87) are simulated to validate the effect of dilatancy angle on
the plastic volumetric strain quantitatively though the considered approximation (Figure 30). This
is conducted within the uniaxial compression test with the same data used previously (R77
where the sample failed) as shown in Table 23. In Table 23, first row indicates different values
of dilatancy angle, which are used in the swelling rock model in the Soil Test Facility. The
second row indicates the slope (m) of the curve from the results of Soil Test Facility, which is
calculated through two different points on the ascending line. Finally, the third row is dilatancy
angle, which is calculated using Eq.49.
 m 

m2
  Sin1 
54
Eq.49
Chapter 4: Results discussions and interpretations
Figure 30: Schematic bi-linear curve of volumetric strain vs. vertical strain
Table 23: Variation of dilatancy angle and its influence on plastic volumetric strain
Parameter
R83 R84
R85 R86 R87
 (◦)
0
9
18
26
35
m (from the Soil Test Facility) 0
0.373 0.895 1.56 2.69
 (◦) (from Eq.49)
0
9.04
18
26
35
The results shown in Table 23 indicate that the numerical results (see the pertinent diagrams in
Figure 31) can be validated through theoretical approximation.
Figure 31: Effect of dilatancy angle on plastic volumetric strain - volumetric strain vs. vertical strain
As Figure 31 shows, any increase in the dilatancy angle value, once the material is in the plastic
region, leads to a larger portion of plastic strain. It should be noted that the amount of dilatancy
effect on the results may vary depending on the physical properties of the sample including
material stiffness and hence on the parameters used in the test.
55
Chapter 4: Results discussions and interpretations
4.12.2 Apl and Apl max influence on the swelling time parameter
In this section, the influence of Apl and Apl max parameters on the swelling time parameter is
investigated. Going back to Eq.26 (see Chapter 2), the swelling time parameter (η) for the case
of occurrence of plastic strains by assuming Ael=0, is leading to Eq.50:
1


p
 A0  Apl   v
q  
1

A A  A
pl
pl max 
 0


for  
for  
p
v
p
v
Apl max
Apl max

Eq.50

For quantitative validation of Apl and Apl max , seven different runs are simulated within the
uniaxial compression test with the same data used previously (R77 where the sample failed) as
shown in Table 24. Throughout the simulations, dilatancy angle (  ) is set to 1  only in order to
observe the dilatancy effect in plastic volumetric strain, otherwise much larger value of Apl
should be employed. Elastic volumetric strain is already known (which can be calculated from
Eq.38) and used in order to determine the increase in plastic volumetric strain (  v   vel   vp ) in
volumetric strain versus vertical strain curve. A0 is set to 0.001 value in all runs.
Table 24: Variation of Apl swelling time parameter and its influence on plastic volumetric strain – A0=0.001 and Ψ=1 deg
Parameter
Apl
R88
R89
R90
R91
R92
R93
R94
1
500
2000
10000
50000
60000
60000
Max value reached for
-0.000172
0.0000503
0.0006976
0.0036145
0.0139499
0.0189053
0.0189053
 vp
 vel
Apl max
 vp
 vp  Apl max
q  day 
-0.00024
-0.00024
-0.00024
-0.00024
-0.00024
-0.00024
-0.00024
0.005
0.005
0.005
0.005
0.005
0.005
0.04
0.000067
0.000290
0.000937
0.003854
0.014189
0.019145
0.019145
-0.004932
-0.004709
-0.004062
-0.001145
0.009189
0.014145
-0.020854
936.346
6.84093
0.53296
0.02594
0.00399
0.003333
0.000870
First, Apl max is set to 5‰. At each run, the absolute value of plastic volumetric strain is
calculated. Then, it is compared to the value of Apl max . Thereafter, swelling time parameter
q  day  is obtained through Eq.50. In the beginning, where the amount of plastic volumetric
strain is small, it can be seen that the rate of swelling is very slow (e.g. see R88). Increasing Apl
leads to a sharp increase in plastic volumetric strain and thus a reduction in swelling time
parameter indicating the lesser time for reaching the final theoretical state (e.g. R89-R91).
It should be noted that in R92, when Apl max is less than the current plastic volumetric strain,
model‟s routine takes into account the Apl max and hence no further increase in swelling rate
occurs, despite further increase in plastic volumetric strain (see R92 and R93). In the last run
(R94), Apl max is increased and hence swelling time parameter increases, and thus the rate of
swelling. The related Figures to R88-R94 are shown in Appendix A.7.
56
Chapter 4: Results discussions and interpretations
In a practical application, it is recommended to select this parameter  Apl max  based on the lining
principle used for tunnel support; since in the resisting principle, swelling is suppressed and
hence the plastic strain is always low and thus the parameter is of no influence on swelling time
parameter; whereas in the yielding principle, it is important to define a reasonable value for this
parameter in accordance with lining characteristics.
4.13 Conclusions
The current model under study was assessed according to the layout shown in Figure 15.
Based on the results of sensitivity analyses and variation of parameters, one can conclude the
following points:
In oedometer test conditions:

The explicit scheme overestimates swelling strain and also causes trouble in large time
steps. Hence, it is recommended to use implicit scheme for all the simulations via the
swelling rock model to have more accurate and stable results.

Due to instability and oscillation observed in the results with different loading conditions,
in particular low applied lows and within different implementation scheme, a critical time
step ratio of 1/19 was proposed to be used for the numerical simulations.

A bias between obtained numerical results and theoretical formulation of one
dimensional swelling law was noted, which was due to the lateral stressing according to
Poisson‟s ratio. Hence, it was illustrated that by setting Poisson‟s ratio equal to zero,
where no stress development occurs in horizontal direction, the same numerical results
are obtained. Furthermore, this effect was shown by setting the swelling potential in
horizontal direction equal to zero indicating no lateral stressing, where the same results
were obtained.

It was found that in the case of high stiffness value, the difference between the
horizontal stress and maximum swelling pressure is very small and thus the contribution
of lateral stressing to the swelling strain. The materials with lower values of stiffness
showed a greater elastic response, as expected. Furthermore, the theoretical results
were reproduced in both high and low stiffness values within the implicit scheme and
proposed critical time step ratio. It was found out that depending on the defined value of
the maximum swelling pressure by the user, it may bring the horizontal stress to the
failure. It was concluded that the greater the stiffness value is used, the larger horizontal
stress will be reached. Then, upon the defined amount of swelling potential and
corresponding maximum swelling pressure, the plasticity may set in.

Swelling potential parameter (the slope of swelling strain versus logarithmic applied
stress) was validated through a back-analysis using the experimental results obtained
from S-Bahn Stuttgart project. This has also validated the proper implementation of
57
Chapter 4: Results discussions and interpretations
swelling law; since the model routines defined by Benz (2012) go back to one
dimensional swelling law formula (Eq.11).

It was found that A0 is the main swelling time parameter component affecting time
dependent behaviour. Increasing the A0 value accelerates the swelling process over the
same period of time. Ael relating to the current elastic volumetric strains was found less
influential and hence large values of Ael needed to be input to observe its impact on the
results, especially when a large A0 value is employed.

It was demonstrated that the model‟s stress path through oedometer test can be
predicted properly via swelling rock model.
In uniaxial compression test conditions:

It was proved that the Mohr-Coulomb failure criterion has been implemented properly in
the model under study. It was validated through strain-controlled uniaxial compression
test via swelling rock model and Mohr-Coulomb material model. Hence, the elastic
stress-strain behaviour was also validated.

It was found that after a certain increase in the strength parameters (cohesion and
friction angle), the vertical stress at failure does not increase when the yield condition is
met and hence bi-linearity is no longer observed in the stress-strain diagram.

It was shown that plasticity parameters including dilatancy angle, Apl and Apl max
influence the results once the plasticity has already started. It was also displayed that
any increase in the dilatancy angle, once the material is in the plastic region, leads to a
larger portion of plastic strains. Furthermore, increasing Apl causes a sharp reduction in
swelling time parameter or η indicating the lesser time for reaching the final theoretical
state. Apl max as a limit for plastic volumetric strains, should be defined based on the
lining principle used in the practical applications.
58
Chapter 5: Conclusions and recommendations
Chapter 5: Conclusions and recommendations
5.1 Conclusions

Swelling time-dependent deformation is a result of volume increase in ground in the
presence of water. Swelling of anhydrite bearing rock formations including gypsum
Keuper is the subject of this study. Gypsum Keuper includes different layers containing
different amounts of sulphate rocks either in the form of anhydrite or gypsum. Anhydrite
converts into gypsum by absorbing water causing an increase in its initial volume
(swelling). Two possible causes for swelling in such rocks are hydration and gypsum
crystal growth. If swelling is prevented by tunnel lining, swelling pressures are induced.

The oedometer test has been vastly used in tunnelling projects since it simulates the
tunnel invert in small scale conditions. Despite the advantages of triaxial tests, they are
costly, time consuming and they are not easy to conduct in comparison to oedometer
test. Laboratory swelling tests on samples from gypsum Keuper included swelling
pressure tests, swelling strain tests and Huder and Amberg experiments. Laboratory
testing of anhydrite swelling rocks are challenging because (1) there is no wellestablished model taking into consideration the different conditions in in situ and in the
laboratory; (2) Swelling process lasts for a long time which cannot be simulated properly
by laboratory testing; (3) The oedometer test condition prevents swelling in lateral
direction resulting in an overestimation of the swelling pressure.

The final lining design is governed by the long term deformation as a result of swelling.
Based on field observations, the amount of maximum swelling pressure varies
depending on the host rock of different tunnelling projects. Furthermore, laboratory
results overestimate the swelling pressure. When designing lining for tunnels driven in
gypsum Keuper, the resisting (stiff) and yielding (flexible) support principles are applied.

Grob (1972) formulated the 1D swelling law between the axial swelling strain and the
axial stress based on the results of Huder and Amberg tests (1970). The 3D extension of
swelling law was proposed first by Einstein et al. (1972). A similar 3D swelling law was
developed by Wittke et al. (1976). Kiehl (1990) proposed the 3D extension of Wittke et
al. (1976) based on the results of swelling tests by Pregl (1980).The swelling tests
performed by Pregl (1980) showed that the principal swelling strains only depend on the
principal stresses in those directions. This was assumed for a material with isotropic
swelling behaviour. Wittke-Gattermann (1998) proposed an approach to extend the 3D
swelling law proposed by Kiehl (1990) for anisotropic behaviour. Specific implementation
of Wittke-Gattermann‟s model was done by Heidkamp et al. (2004) and Benz (2012).

The rock in unleached gypsum Keuper is assumed to show elastic transversely isotropic
stress-strain behaviour. The rock strength is described using the Mohr-Coulomb failure
criterion. When the rock strength is exceeded, visco-plastic behaviour is assumed which
can be explained via a rheological model, in which an elastic element is directly
connected to the applied stress and swelling or sliding device is in parallel to a dashpot.
59
Chapter 5: Conclusions and recommendations

It is assumed that the conversion rate of the anhydrite into gypsum and thus the rate of
swelling are dependent on the rate of water entry. The swellable rock in unleached
gypsum Keuper has very low permeability. The permeability even at small viscoplastic
strain significantly increases. The velocity of the water access to the rock also increases
with increasing viscoplastic strain. The visco-plastic strains and possibly also the elastic
strains share the influence on the rate of water entry and hence on the swelling rate.
Based on the results of sensitivity analyses and variation of parameters in oedometer test
conditions:

A bias between obtained numerical results and theoretical formulation of one
dimensional swelling law was noted, which was due to the lateral stressing according to
Poisson‟s ratio. Hence, it was illustrated that by setting Poisson‟s ratio equal to zero,
where no stress development occurs in horizontal direction, the same numerical results
are obtained. Furthermore, this effect was shown by setting the swelling potential in
horizontal direction equal to zero indicating no lateral stressing, where the same results
were obtained.

It was found that in the case of high stiffness value, the difference between the
horizontal stress and maximum swelling pressure is very small and thus the contribution
of lateral stressing to the swelling strain. The materials with lower values of stiffness
showed a greater elastic response, as expected. Furthermore, the theoretical results
were reproduced in both high and low stiffness values within the implicit scheme and
proposed critical time step ratio. It was found out that depending on the defined value of
the maximum swelling pressure by the user, it may bring the horizontal stress to the
failure. It was concluded that the greater the stiffness value is used, the larger horizontal
stress will be reached. Then, upon the defined amount of swelling potential and
corresponding maximum swelling pressure, the plasticity may set in.

Swelling potential parameter (the slope of swelling strain versus logarithmic applied
stress) was validated through a back-analysis using the experimental results obtained
from S-Bahn Stuttgart project. This has also validated the proper implementation of
swelling law; since the model routines defined by Benz (2012) go back to one
dimensional swelling law formula (Eq.11).

It was found that A0 is the main swelling time parameter component affecting time
dependent behaviour. Increasing the A0 value accelerates the swelling process over the
same period of time. Ael relating to the current elastic volumetric strains was found less
influential and hence large values of Ael needed to be input to observe its impact on the
results, especially when a large A0 value is employed.

It was demonstrated that the model‟s stress path through oedometer test can be
predicted properly via swelling rock model.
60
Chapter 5: Conclusions and recommendations
Based on the results of sensitivity analyses and variation of parameters in uniaxial compression
test conditions:

It was proved that the Mohr-Coulomb failure criterion has been implemented properly in
the model under study. It was validated through strain-controlled uniaxial compression
test via swelling rock model and Mohr-Coulomb material model. Hence, the elastic
stress-strain behaviour was also validated.

It was found that after a certain increase in the strength parameters (cohesion and
friction angle), the vertical stress at failure does not increase when the yield condition is
met and hence bi-linearity is no longer observed in the stress-strain diagram.

It was shown that plasticity parameters including dilatancy angle, Apl and Apl max
influence the results once the plasticity has already started. It was also displayed that
any increase in the dilatancy angle, once the material is in the plastic region, leads to a
larger portion of plastic strains. Furthermore, increasing Apl causes a sharp reduction in
swelling time parameter or η indicating the lesser time for reaching the final theoretical
state. Apl max as a limit for plastic volumetric strains, should be defined based on the
lining principle used in the practical applications.
5.2 Recommendations for further studies
Recommendations from the project regarding numerical simulations in the Soil Test Facility,
conducting a case study of tunnelling within an anhydrite bearing rock formation in PLAXIS2D
and the way by which the swelling rock model parameters should be selected in general, are
explained, which can be used for further studies.
5.2.1 Simulations in the Soil Test Facility

It is recommended to use the implicit scheme for all the simulations via the swelling rock
model to avoid overestimation of swelling strain.

It is highly recommended to employ the proposed time step ratio of 1/19 for the
numerical simulations in the Soil Test Facility. This avoids instability and oscillation in the
results.
5.2.2 A case study of tunnelling within anhydrite bearing rocks using PLAXIS2D
knowing the influence of the parameters, a case study of tunnelling regarding the exploration
gallery in the Freudenstein tunnel project simulated by Wittke et al. (2004) within an anhydrite
bearing rock formation (see the Wittke et al. (2004) and Wittke-Gattermann (1998 and 2003) for
details) is recommended. Hence, the following questions/actions are suggested for the pertinent
investigations:

How far is the plastic zone due to swelling? To this end, the different size of the rock
zone beneath the tunnel invert can be simulated as a swelling zone (by activating
swelling properties through construction stages). This will help to determine whether or
61
Chapter 5: Conclusions and recommendations
not the resulting deformations go as far as the plastic zone defined by the user. The
aforementioned question can also be answered through simulation of an imaginary
tunnel in PLAXIS2D in which running the swelling rock model in the boundary value
conditions can be tested.

Measurement of displacements and structural forces: Measuring vertical displacements
under invert and at the crown and on the wall of the simulated tunnel. This will help
determine which part of the tunnel (invert, the wall or the crown) is more influenced by
swelling deformation. This can also be observed by looking at the resulting bending
moments on the lining segment depicting how the tunnel is distorted after the occurrence
of swelling deformation.

Measurement of maximum pressures reached after construction of lining: This will help
to determine the role of maximum swelling pressure defined by the user.
5.2.3 Parameter selection for a practical application
The selection of the swelling rock model parameters should be based on the real behaviour of
the ground (rock/soil) of the practical application. Here, the practical application is considered as
a tunnel excavation. There are different ways of estimating the model parameters. First, the
required data related to the type of the ground should be determined using geological
investigations. Furthermore, laboratory and field testing can be used for rock/soil
characterisation. Data from similar projects, engineering judgement and rules of thumb are also
employed when there is not sufficient data available. Tunnel excavation disturbs the in situ
stresses, and hence the stress development as a result of tunnel advancement is of importance,
i.e., the stress path around the tunnel excavation. In the following, the way by which every
single parameter should be dealt with is discussed:
From the results of Huder and Amberg testing, the maximum swelling pressure (  0, p , 0,t ),
above which no swelling occurs, can be obtained. Furthermore, from the slope of the final
swelling strain versus time curve, swelling potential parameter ( K q , p , K q ,t ) is obtained. Selecting
these parameters are somewhat debatable, since the greater the swelling potential and
maximum swelling pressure is selected, the sooner plasticity may be reached. Swelling potential
parameter in the vertical direction should be selected as a greater value than the horizontal one,
i.e. Kq , p  Kq ,t (see Moore et al. for further details). There are also some uncertainties in the
literature about how maximum swelling pressure should be taken into account. For instance,
Anagnostou considers the maximum swelling pressure as a hydrostatic pressure (further details
can be found in Anagnostou‟s papers, 1992, 1993, 2007 and 2010). Wittke Gattermann (1998)
considered the value of maximum swelling pressure as twice as the vertical in situ stress at the
tunnel invert level for simulating the exploration gallery of Freudenstein tunnel in Germany. This
was based on the results of in situ measurements. She also considered 10% and 2% as
maximum swelling potential in vertical and horizontal directions, respectively.
62
Chapter 5: Conclusions and recommendations
Bedding rotation angle (  ) can be selected as zero, since the beddings are mostly horizontal,
unless there is another situation in the application. Then it can be obtained through the results
of rock mass mapping, which can be done either manually or through Dips software, in which all
the joints sets and beddings (discontinuities) orientations can be determined.
Dilatancy angle (  ), which comes in only once the plasticity behaviour begins, can be
approximated through the resulting curve of volumetric strain versus axial strain in a triaxial test.
It should be noted that based on the stress level of the practical application, dilatancy might be
affected; for instance, it might be suppressed in high stress levels. Wittke-Gattermann (1998)
considers the same dilatancy angle value for both tangential and perpendicular directions.
Friction angle (  ' ) and cohesion ( c ' ) as strength parameters can be approximated at the peak
strength of the resulting curve of deviatoric stress versus axial strain in a triaxial test. In fact, the
results of triaxial tests can be interpreted using both Mohr-Coulomb and Hoek and Brown
strength criteria to determine cohesion and friction angle.
Poisson‟s ratio ( p , t ) can be approximated through the resulting curve of volumetric strain
versus axial strain in a triaxial test. It can also be approximated through the coefficient of earth

pressure k0 
which is obtained using known vertical stress  v and horizontal stress  H ;
1 
H
.  v is obtained as  v   .h , where is  unit weight of the rock/soil and h is the
v
height of overburden.
i.e., k0 
Young‟s modulus ( E p , Et ), which indicates the stiffness of the material, can be obtained in
different ways including laboratory testing results. The laboratory tests include the oedometer
test, uniaxial compressive strength (UCS) test and triaxial test. In the oedometer test, Young‟s
modulus cannot be obtained directly, since the obtained stiffness from the stress strain curve is
the oedometer stiffness. Hence, knowing the Poisson‟s ratio and already obtained oedometer
stiffness, the Young‟s modulus can be obtained. Through the triaxial test and UCS test, E can
be obtained directly from the results. For instance, the slope of the linear part of the stress-strain
curve in UCS test leads to stiffness. Similarly, a tangent of deviatoric stress versus vertical
strain curve in the triaxial test at 50% of the peak stress can lead to E 50.
It should be noted that stiffness is not a constant parameter and is in fact stress dependent. So
the range of stress for a practical application influences the determination of a reliable value for
stiffness. Furthermore, the strength of the rock mass, which is influenced by discontinuities‟
conditions, affects the choices. If a rock mass is terribly fractured, the quality of the rock
decreases, and hence its strength, so that the stiffness will be lower. It should be also noted that
cell pressure in the triaxial test influences the results affecting the choices not only for stiffness
or strength parameters but also for dilatancy angle. This is because the real horizontal stresses
might not be well-produced by a cell pressure in a triaxial test (for some applications such as
deep underground applications where the horizontal pressure is very large).
63
Chapter 5: Conclusions and recommendations
It should be also noted that since the excavation process is an unloading process, it is better to
use the unloading path of the stress strain curve to obtain the stiffness of the material.
Shear modulus ( G23 ) can be straightforwardly obtained once Young‟s modulus and Poisson‟s
ratio are known.
Tensile strength (  Ten ) is obtained through a Brazilian test in the laboratory. This should be
considered as a very small value. For instance, a value of 200 kPa was used for simulation a
tunnel within a swelling clay layer in Switzerland through the specific implementation of WittkeGattermann‟s model (see the Heidkamp et al. papers, 2003 and 2004 for further details). As a
rule of thumb in rock mechanics, uniaxial tensile strength is assumed as 0.1 time of uniaxial
compressive strength, which can be determined from a uniaxial compression test. WittkeGattermann (1998) considered a zero value of tensile strength in the simulations.
( Apl max ) parameter, which functions as a limit on the swelling rate, depends on the type of
lining principle used for the practical application. For instance, in the case of using the resisting
principle, which avoids the occurrence of plastic strains (swelling deformation), the amount of
plastic strains will be always low, and hence the parameter is irrelevant and only A0 , Ael and Apl
are taken into account for swelling time dependency and thus swelling rate. Therefore, it also
depends on the type of application and the pertinent strain level. Wittke-Gattermann (1998)
ignored the parameter max EVP when simulating the exploration gallery of the Freudenstein
tunnel in Germany via the resisting principle.
( A0 ) parameter, which accelerates the swelling process over the same period of time, should be
chosen according to the zone in the vicinity of tunnel cross section which is simulated in
engineering practice. The greater the distance from the tunnel cross section, the smaller strains
will be and hence the effect of swelling in those areas should be very small. It is therefore
recommended to use A0  0 . This means that the user should consider a zone based on
engineering judgement, for which beyond that area swelling is not accelerated. Furthermore,
Wittke-Gattermann (1998) concluded that the maximum value of 0.008 per annum correlated
well with the measurement results. However, she came to this conclusion after testing different
simulation runs to reproduce the measurement results of the exploration gallery of the
Freudenstein tunnel in Germany. ( Ael ) parameter, which refers to the value of elastic
volumetric strains, is less influential and can be neglected if high values of A 0 are employed.
( Apl ) parameter, which refers to the plastic volumetric strain, contributes to the dilatancy effect.
As mentioned earlier, once the plasticity begins, dilatancy plays a role and hence Apl parameter
can cause increase in the plastic volumetric strain. Similar to what was mentioned earlier
regarding Apl max parameter, depending on the strain level and stress level of the application,
the parameter value should be selected. This also indicates the importance of how far a plastic
zone should be in the application. There are some empirical methods allowing for determination
of plastic zone using characteristic curves. There is no data available regarding this parameter.
64
Chapter 5: Conclusions and recommendations
Wittke-Gattermann defined a maximum value of 230 per annum for the „visco-plastic‟ swelling
time parameter after testing different simulation runs to reproduce the measurement results of
the exploration gallery of the Freudenstein tunnel in Germany.
In conclusion, it should be noted that, depending on the stress level which the application deals
with, the parameters should be selected. For instance, excavation of a tunnel in a deep
underground mine would lead to significant strains as a result of unloading, which is much
smaller in a shallow excavation. Furthermore, the discontinuity conditions, permeability, porosity
and water accessibility to the rock are all of significant importance to the swelling rate and
hence parameters selection. Hence, such a selection process requires an engineering
improvisation by making a compromise between the rock/soil investigation data and their real
behaviour.
65
Nomenclature
Nomenclature
All quantities are considered in SI units.
No.
Symbol
Definition
Volume of the crystallised
gypsum
No.
Symbol
Definition
1
V
34
i
Principal stresses‟ direction
2

Strain
35
S1
Swelling deformation parameter
3
F
Load
36
li
Direction cosine of the angle between the directions
of principal stress and the coordinate axis
4
P  Ps
Support pressure
37
Swelling strains occurring at the considered time t
5
u
Floor heave
38
 xalt
q
q
Swelling time parameter
6
E
Young‟s modulus
39
RG
Transformation function of structure orientation in the
global coordinate system
7

Poisson‟s ratio
40
kf
Rock permeability
Mean aperture width of discontinuities
8
9
10
11
12
13
14
G



c

F  Fy
Shear modulus
41
Bedding rotation angle
42
Normal stress
43
2ai
v
g
Shear stress
44
Dhy
Hydraulic diameter
Roughness of the wall of discontinuities
Kinematic viscosity
Acceleration due to gravity
Cohesion
45
Friction angle
46
Yield function
47
k
d
S
Major principal stress
48
n
Porosity
Minor principal stress
49
c
Empirical dimensionless constant
Mean spacing of discontinuities
Specific surface area
17
1  1/v
 3   3/h
t
Uniaxial tensile strength
50

18
y
Yield stress
51
n0
Porosity at
Viscoplastic strain
52
kf0
Permeability at
Elastic strain
53
a0 , ael 0 ,aVP
Swelling time parameters constants
Viscoplastic strain rate
54
A0 , Ael , Apl
Swelling time parameters constants
55
max EVP
Limit on viscoplastic volumetric strain
56
Apl max
Limit on plastic volumetric strain
57
 ges  t 
Total strain
58
ModelID
Routines No. for computation of swelling strain
Dilatancy angle
59
 0, p , 0,t
Maximum swelling pressure
60
p, t
Perpendicular and tangential to bedding defined by
Benz (2012)
61
K q , p , K q ,t
Swelling potential
62
dt / q
Time step ratio
63
k, s
Vertical and horizontal directions in bedding defined
by Wittke-Gattermann (1998)
64
H
Horizontal stress
65
k0
Earth pressure coefficient
66
v
Volumetric strain
15
16
19
20
 vp
 el
. vp
22

QG , QT
23
FG , FT
24
VPG ,VPT
21
Plastic potential of the
rock and bedding
Failure criterion of the
rock and bedding
Rock and bedding
viscosity
Bedding plane
26
T

27
t
Time
28
z
29
 t     zq
30
k zq
31
 z0
Applied compressive
stress
Final theoretical swelling
strain
Swelling potential
parameter
Maximum compressive
stress
Minimum compressive
stress
25
32
33
 zc

Unit weight
66
el
v
Elastic volumetric strain
 vel  0
 vel  0
References
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Wittke-Gattermann, P., 1998. Verfahren zur Berechnung von Tunnels in quellfähigem Gebirge und Kalibrierung an einem
Versuchsbauwerk. Geotechnik in Forschung und Praxis, WBI-PRINT 1, Verlag Glückauf, Essen.
2.
Gysel, M., 1987. Design of Tunnels in Swelling Rock. Journal of Rock Mechanics and Rock Engineering 20, pp. 219-242.
3.
Anagnostou, G., 1993. A model for swelling rock in tunnelling. Swiss federal institute of technology, Zurich, Switzerland,
Journal of Rock Mechanics and Rock Engineering, pp. 307-3.
4.
Kiehl, J. R., 1990. Ein dreidimensionales Quellgesetz und seine Anwendung auf den Felshohlraumbau. Proc. 9. Nat. Rock
Mech. Symp., Aachen 1990, pp. 185 - 207.
5.
Seidenfuß, T., 2006. Collapses in tunnelling. Fondation Engineering and Tunneling, Stuttgart, Germany, MSc thesis, pp.
194.
6.
International Society for Rock Mechanics (ISRM), 1983. Characterisation of swelling rock. ISRM Report.
7.
Hawlader, B.C., Lee, Y.N., Lo, K.Y., 2003. Three-dimensional stress effects on time-dependent
swelling behavior of
shaly rocks. NRC Research Press Website, Canadian Geotechnical
Journal 40, Canada, pp. 501-511.
8.
Wittke, W., 1978. Fundamentals for the design and construction of tunnels located in swelling rock and their use during
construction of the turning loop of the subway Stuttgart. Publ. of the Institute for Foundation Engineering, Soil Mechanics,
Rock Mechanics and Water Ways Construction RWTH (University) Aachen, Vol. 6.
9.
Benz, T., 2012. Preliminary documentation of swelling rock model routines for the use of TU Delft only (not publicly
published). Norwegian University of Science and Technology (NTNU), pp. 11.
10. Steiner, W. 1993. Swelling rock in tunnels: rock characterisation, effect of horizontal stresses and construction
procedures. Int. J. Rock Mech. Min. Sci. Vol. 30, No. 4, pp. 361-380.
11. Gioda, G., 1982. On the non-linear „squeezing‟ effects around circular tunnels. International journal of for numerical and
analytical methods in geomechanics, vol. 6, pp. 21-46.
12. Barla, M., 1999. Tunnels in swelling ground - Simulation of 3D stress paths by triaxial laboratory testing. Politecnico di
Torino, Italy, PhD Thesis, pp. 189.
13. Rauh, F., Thuro, K., 2007. Investigations on the swelling behaviour of the pure anhydrite. Engineering geology, Technical
University of Munich, Germany, pp. 7.
14. Whittaker, B.N. and Frith, R.C., 1990. Tunnelling: Design, Stability and Construction. The Institution of Mining and
Metallurgy, London, England, pp. 460.
15. Einstein, H.H. and Bischoff, N., 1975. Design of Tunnels in Swelling Rocks, 16th Symposium on Rock Mechanics,
University of Minnesota, Minneapolis, MN, pp. 185-195.
16. Kramer, G.J.E., Moore, I.D., 2005. Finite element modelling of tunnels in swelling rock. K. Y. Lo Symposium, Technical
session D, The Geo-Engineering Centre, The University of Western Ontario, Canada, pp. 37.
17. Berdugo, I.R., 2007. Lessons learned from tunnelling in sulphate-bearing rocks. Department of Geotechnical and
Geosciences, Technical University of Catalonia, Barcelona, Spain, Presentation, pp. 77.
18. Alonso, E. E., Olivella, S., 2008. Modelling tunnel performance in expansive gypsum claystone. 12 th International
Conference of IACMAG, Goa, India, pp. 891-910.
19. Wittke-Gattermann, P., 2003. Dimensioning of tunnels in swelling rock. ISRM 2003–Technology roadmap for rock
mechanics, South African Institute of Mining and Metallurgy, pp. 8.
20. Anagnostou, G., 2007. Design uncertainties in tunnelling through anhydrite swelling rocks. Journal of Rock and Soil
Engineering, Vol. 25, No. 4, pp. 48-54.
21. Grob, H. 1972. Schwelldruc im Belchentunnel. In Proceedings of the International Symposium on Underground Openings,
Lucerne, Switzerland, pp. 99– 119.
22. Madsen, F.T., 1999. International society for rock mechanics commission on swelling rocks and commission on testing
methods: Suggested methods for laboratory testing of swelling rocks. International Journal of Rock Mechanics and Mining
Sciences 36 (1999) 291-306.
67
References
23. Paul, A., 1986. Empfehlung Nr. 11 des Arbeitskreises Versuchstechnik Fels (DGEG): Quellversuche an Gesteinsprobe n.
In: Bautechnik Nr. 3, Ernst & Sohn, pp. 100 - 104.
24. Anagnostou, G., Pimentel, E., Serafeimidis, K., 2010. Swelling of sulphatic claystones – some fundamental questions and
their practical relevance. 59th Geomechanics Colloquy 2010, Session on “Tunnel construction in swelling ground”,
Switzerland, pp. 12.
25. ISRM, 1994. Commission on Swelling Rock: Comments on Design and Analysis Procedures for Structures in
Argillaceous Swelling Rock. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts,
Vol. 3, No. 5, pp. 547 - 546.
26. Brinkgreve, R.B.J., Swolfs, W.M., Engin E. PLAXIS2D-Version 2010.01. Delft University of Technology & Plaxis b.v., the
Netherlands.
27. Al-mhaidib, A.I., 1999. Swelling Behavior of Expansive Shales from the Middle Region of Saudi Arabia. Journal of
Geotechnical and Geological Engineering 16, pp. 291-307.
28. Gysel, M., 1977. A contribution to the design of a tunnel lining in swelling rock. Journal of Rock Mechanics and
Rock Engineering, by Springer-Verlag, pp. 55-71.
29. Heidkamp, H., Katz, C., 2004. The swelling phenomenon of soils: Proposal of an efficient
approach. Germany, pp. 6.
continuum modelling
30. Fröhlich, B., 1989. A summary of Anisotropic Swelling Behavior of Diagenetic Consolidated Claystone. Published by
International Society of Rock Mechanics (ISRM), pp. 12.
31. Alonso, E., Ramon, A., 2011. Lilla Tunnel. Department of Geotechnical and Geosciences,
Catalonia, Barcelona, Spain, Presentation, pp. 47.
Technical University of
32. Wittke-Gattermann, P., Wittke, M., 2004. Computation of strains and pressures for tunnels in swelling rocks. Foundation
Engineering and Construction in Rock, Aachen / Stuttgart, Germany, pp. 8.
33. Edelbro, C., 2003. Rock Mass Strength: a review. A technical report published by Lulea University of Technology,
Sweden, pp. 160.
34. Vermeer, P.A. 1979. A modified initial strain method for plasticity problems. Proc. 3rd Int. Conf. on Numerical Methods in
Geomechanics, Rotterdam: Balkema, 337-387.
35. Hoek, E., 2007. Practical Rock Engineering. Canada, Rocscience, E-book.
36. Runesson, K., 2005. Constitutive modeling of engineering materials - theory and computation. Lecture Notes, Dept. of
Applied Mechanics, Chalmers University of Technology, pp. 227.
37. Franklin, J., Chandra, R., 1972. The slake durability test. International Journal of Rock Mechanics and Mining Sciences &
Geomechanics Abstracts, 9(3), 325-328.
38. Vardar, M., Fecker, E., 1984. Theorie und Praxis der Beherrschung löslicher und quellender Gesteine im Felsbau.
Essen, Germany.
39. Serón, J., Garrido, E., Romana, M., 2002. Characterization of swelling rocks by Huder-Amberg oedometric test.
Paramètres de calcul géotechnique. Magnan (ed.) 2002, Presses de l‟ENPC/LCPC, Paris, pp. 161-166.
40. Hoek, E., Brown, E.T., 1990. Underground excavations in rocks. The Institution of Mining and Metallurgy. pp. 536.
41. Anagnostou, G., 1992. Importance of Unsaturated Flow in Predicting the Deformation around Tunnels in Swelling Rock.
Scientific colloquium “porous or fractured unsaturated media. Transport sand behavior”, Monte Verita, Centro Stefano
Franscini, pp. 17.
42. Einstein, H.H., 1996. Tunnelling in Difficult Ground - Swelling Behaviour and Identification of Swelling Rocks. Journal of
Rock Mechanics and Rock Engineering 29 (3), pp. 113-124.
68
Appendix A: List of results of the element tests’ runs
Appendix A: List of results of the element tests’ runs
A.1 Different time step ratios within implicit scheme
Figure 32: Oedometer test – R1, R3, R4, R5, R6, R7, and R8 – Implicit scheme – ԑyy & σxx & σyy vs. time curves
69
Appendix A: List of results of the element tests’ runs
A.2 Different time step ratios within explicit scheme
Figure 33: Oedometer test’s R9, R11, R12, R13, R14, R15, and R16 – Explicit scheme – ԑyy & σxx & σyy vs. time curves
70
Appendix A: List of results of the element tests’ runs
A.3a Low applied loads with time step ratio of 1, with zero tensile strength
Figure 34: Oedometer test’s R18, R19, R20, R21 and R22 – with zero tensile strength - Implicit scheme - ԑyy & σxx & σyy vs. time curves
71
Appendix A: List of results of the element tests’ runs
A.3b Low applied loads with time step ratio of 1, with tensile strength of 100 kPa
Figure 35: Oedometer test’s R18’, R19’, R20’, R21’ and R22’ – with tensile strength of 100 kPa - Implicit scheme - ԑyy & σxx & σyy vs. time curves
72
Appendix A: List of results of the element tests’ runs
A.4a Effect of Poisson’s ratio on final vertical swelling strain (the difference between numerical results and theoretical
value)
Figure 36: Oedometer test’s R23’ (a), R23’ (b), and R23’ (c) - Implicit scheme - ԑyy & σxx & σyy vs. time curves
73
Appendix A: List of results of the element tests’ runs
A.4b Influence of horizontal swelling potential (kq,t)
Figure 37: Oedometer test’s R24, R25, R26, R27 and R28 – Implicit scheme –influence of horizontal swelling potential - ԑyy & σxx & σyy vs. time curves
74
Appendix A: List of results of the element tests’ runs
A.5a Sensitivity analysis regarding different time step ratios (with low applied loads)
Runs Ra-Re are simulated runs with different time step ratios with a low applied load. The obtained results show an increase in both instability and i naccuracy after
the ratio of 0.0526 in the Soil Test Facility.
Figure 38: Oedometer test’s Ra, Rb, Rc, Rd, and Re – Implicit scheme – Time step ratio sensitivity analysis with a low applied load- ԑyy & σxx & σyy vs. time curves
Parameter
Time step ratio
Applied load
(kPa)
Theoretical
value
Comment on the
results
Ra
0.0526
Rb
0.1
Rc
0.2
Rd
0.3
Re
0.4
-2
-2
-2
-2
-2
0.00619
0.00619
0.00619
0.00619
0.00619
Stable and
accurate (with
kq,t=0)
Both inaccuracy and instability increase
75
Appendix A: List of results of the element tests’ runs
A.5b Sensitivity analysis regarding different time step ratios (with high stiffness in the Soil Test Facility)
Runs Rf-Rj are simulated runs with different time step ratios with a high stiffness material. The obtained results show an increase in both instability and inaccuracy
after the ratio of 0.0526 in the Soil Test Facility.
Figure 39: Oedometer test’s Rf, Rg, Rh, Ri, and Rj – Implicit scheme – Time step ratio sensitivity analysis with a high stiffness material - ԑyy & σxx & σyy vs. time curves
Parameter
Time step ratio
Applied load (kPa)
Young’s modulus
(kPa)
Theoretical value
Comment on the
results
Rf
0.0526
-130
Rg
0.1
-130
Rh
0.2
-130
Ri
0.3
-130
Rj
0.4
-130
100E+06
100E+06
100E+06
100E+06
100E+06
0.00251
Stable and accurate (with
kq,t=0)
0.00251
0.00251
0.00251
0.00251
Both inaccuracy and instability increase
76
Appendix A: List of results of the element tests’ runs
A.5c Sensitivity analysis of the proposed critical time step ratio of 0.0526 within implicit scheme with different load steps –
Oedometer test - with lateral swelling potential
Figure 40: Oedometer test’s R29, R30, R31, R32, R33, R34, R35 and R36 – Implicit scheme – with horizontal swelling potential - ԑyy & σxx & σyy vs. time curves
77
Appendix A: List of results of the element tests’ runs
A.5d Sensitivity analysis of the proposed critical time step ratio of 0.0526 within implicit scheme with different load steps –
Oedometer test - without lateral swelling potential
Figure 41: Oedometer test’s R37, R38, R39, R40, R41, R42, R43 and R44– Implicit scheme – without horizontal swelling potential - ԑyy & σxx & σyy vs. time curves
78
Appendix A: List of results of the element tests’ runs
A.6a Material stiffness Effect on the results (within implicit scheme and critical time step ratio of 0.0526) with horizontal
swelling potential
Figure 42: Oedometer test’s R45, R46, R47, R48, R49 – Implicit scheme – Stiffness effect – Applied load of -130 KPa - ԑyy & σxx & σyy vs. time curves
79
Appendix A: List of results of the element tests’ runs
A.6b Material stiffness Effect on the results (within implicit scheme and critical time step ratio of 0.0526) without horizontal
swelling potential
Figure 43: Oedometer test’s R50, R51, R52, R53, and R54 – Implicit scheme – Stiffness effect – Applied load of -130 KPa - ԑyy & σxx & σyy vs. time curves
80
Appendix A: List of results of the element tests’ runs
A.7 Effect of Apl and Apl max swelling time parameters on the plastic volumetric strain
Figure 44: Uniaxial compression test’s R88, R89, R90, R91, R92, R93 and R94 – Implicit scheme – Apl and Apl max effect – Volumetric strain vs. vertical strain curves
81
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