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 108 m (Wittke-Gattermann, 1998) and of less than 1012 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 108 m to about 103 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 m2 Sin1 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 References 1. 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. 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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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