Consolidación de suelos blandos en presencia de drenes de banda

UNIVERSITY OF SEVILLE DEPARTMENT OF BUILDING STRUCTURES AND GROUND ENGINEERING CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS ELISA MARIA DE JESUS DA SILVA JUNE 2013 UNIVERSITY OF SEVILLE DEPARTMENT OF BUILDING STRUCTURES AND GROUND ENGINEERING CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS Thesis submitted by Elisa Maria de Jesus da Silva to accede to the degree of Doctor by the University of Seville (Doctora por la Universidad de Sevilla) Under the supervision and guidance of: Director: José Luis de Justo Alpañés (Doctor Ingeniero de Caminos, Canales y Puertos, Professor Emeritus by the University of Seville) Co-director: Percy Durand Neyra (Doctor Ingeniero de Caminos, Canales y Puertos, Professor at University of Seville) Seville, June 2013 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE “THERE IS ONLY ONE THING ABOUT WHICH I AM CERTAIN, AND THIS IS THAT THERE IS VERY LITTLE ABOUT WHICH ONE CAN BE CERTAIN.” WILLIAM SOMERSET MAUGHAM (1874-1965) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE i TO This thesis is dedicated to my mother and to my late father, as a token of my appreciation for their valuable lessons and the spirit of determination that they instilled in me. They helped me in the pursuit of my dreams and in achieving my goals. I also would like to share some special words that my father once wrote to me, which touched me, and that I keep always in my mind. “Dreaming is good Dreams should lead our lives Unfortunately, life is not only made of dreams… Nevertheless, you are always in mine!” (Augusto José Ascensão da Silva, 27-Maio-1999) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS ACKNOWLEDGMENTS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE ii ACKNOWLEDGMENTS I would like to express my sincere gratitude and appreciation to Professor José Luís de Justo Alpañés from the University of Seville, Spain, for his valuable guidance, inspiration and continuous support throughout the researching of this work and for providing the data of the case study analysed in this thesis. Valuable suggestions and advice were also given by Professor Percy Durand Neyra from the University of Seville and by Professor Mário Jesus from the University of Algarve, Portugal. Their contributions were crucial to the development, organization and drafting of this document and I would like to thank them both. I must address a special word of thanks to my former Professor, Pedro Guedes de Melo, from “Instituto Superior Técnico” of the Technical University of Lisbon, Portugal, for so skilfully transmitting his knowledge of soil mechanics and kindling my fascination for the field of geotechnics. I am grateful to the University of Algarve for providing the financial support to acquire the PLAXIS software, which was vital for the development of this thesis, and to the PLAXIS Support Team, from Delft University, Netherlands, for their assistance with the numerical analysis. Special thanks go to the “Confederación Hidrográfica del Guadalquivir” for allowing the publication of the monitoring results for Dikes 1 and 3 of the Lebrija Pond. To the enterprise "INPOLUKIS, Traduções" a word of appreciation for the effort of translation and revising this thesis from Portuguese to English, in particular to Marta Aragão and Sónia Costa; These acknowledgements would not be complete without a word of appreciation to my colleague David Pereira for his helpful advice on the graphics and for his friendship over these last years, as well as to Ângela Vairinhos for being always available in the difficult moments. To my friends and colleagues Carolina Sousa, Paula Ribeiro and Sara Madeira, a special word of thanks for their friendship, encouragement and moral support. Finally, with heartfelt love and an enormous debt of gratitude, I wish to thank my beloved mother, daughter Mariana and husband Fernando for their affection, support and encouragement throughout the time I devoted to this special and personal project. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS ABSTRACT UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE iii ABSTRACT This thesis focuses on the study of the consolidation of soft soil with vertical band-shaped drains, comparing real data collected in a case study (Dikes No.1 and No.3 of the Lebrija Pond) with those obtained via numerical simulation. The comparison between these results leads to the perception and understanding of the phenomena associated to these types of work and soils, also providing the possibility of intervention in equivalent scenarios. In view of the high compressibility of the soil foundation of these dikes, they were treated with prefabricated vertical band-shaped drains and monitored since construction, allowing the gathering of relevant data. Later, the monitoring results were confronted with numerical simulations. Finite element software, which is recognized internationally among the geotechnical engineers, is used to this end. This programme offers several soil material models. In the case study, three models were applied to define the behaviour of the soft soil foundation, namely the Mohr-Coulomb, Soft Soil and Soft Soil Creep models. Special attention was paid to the last model, since it allows considering the viscosity (creep) of soft soil. A sensitivity analysis of the secondary compressibility index was also performed. The analysis of the consolidation phenomenon, in the areas where the vertical band-shaped drains are used, is a problem of axisymmetric nature (tridimensional-3D), nevertheless it is performed in the plane strain state (two dimensional-2D). Therefore, the spatial pattern adopted in the design had to be converted into several parallel drainage screens. An equivalent degree of consolidation in both systems must be ensured. New horizontal permeability coefficients for this zone are estimated, according to several hypotheses for the diameter and reduction of the permeability in the disturbed zone around the drains (smear effect). The flow was also incorporated in these analyses, with the purpose of introducing more realism to the numerical simulations, and with this option a better approximation between the numerical results and the monitoring data was achieved. However, a better agreement is only obtained when the numerical analyses include, in addition to the flow, the correct magnitude of the smear effect, the viscosity creep (creep) in the behaviour of the soft soil and appropriate secondary compressibility indexes. Therefore, the confrontation of the monitoring data associated to a real case work, with numerical results allowed for the validation and gauging of methods, models and geotechnical parameters for the soft soil when prefabricated vertical band-shaped drains were used. KEY-WORDS: Soft Soil, Soil Improvement, Vertical Band-shaped Drains, Consolidation Analysis, Material Models, Creep Behaviour, Smear Effect, Lebrija Pond, PLAXIS software. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS RESUMEN UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE iv RESUMEN Esta tesis se centra en el estudio de la consolidación de suelos blandos tratados con drenes verticales prefabricados de banda, comparando datos reales recogidos en un caso de obra (Diques Nº1 y Nº3 del Embalse de Lebrija), con los obtenidos por simulación numérica. La comparación entre estos resultados conduce a la percepción y comprensión de los fenómenos asociados a este tipo de obras, proporcionando de igual modo la posibilidad de intervención en escenarios equivalentes. Frente a la elevada compresibilidad de los suelos de fundación de estos diques, los mismos fueron tratados con drenes verticales prefabricados de banda y monitorizados desde su construcción, conduciendo a la obtención de datos relevantes sobre de su comportamiento. Posteriormente se validaron los resultados de la monitorización recurriendo a la simulación numérica. En las simulaciones numéricas se utilizó un programa de elementos finitos reconocido internacionalmente en el seno de la geotecnia, el cual facilita varios modelos de comportamiento para suelos. En el caso de estudio, se aplicaron tres modelos para definir el comportamiento de los suelos blandos de la fundación, concretamente el modelo Mohr-Coulomb, Soft Soil y Soft Soil Creep. Especial atención recibe el modelo Soft Soil Creep, dado que el mismo permite traducir la fluencia viscosa (creep) de los suelos blandos a muy blandos, habiéndose realizado también un análisis paramétrico del índice de compresibilidad secundario. El análisis del fenómeno de la consolidación en las zonas tratadas con drenes verticales prefabricados de banda se realiza en el estado plano de deformación (bidimensional -2D), a pesar de que se trata de un problema de naturaleza axisimétrica (tridimensional-3D). Como tal, el sistema espacial de la malla triangular se ha convertido en una pared de drenes verticales paralelos entre sí, siempre garantizando la igualdad del grado de consolidación en ambos sistemas. Se han calculado nuevos coeficientes de permeabilidad horizontal para esta zona, analizándose diversas hipótesis para el diámetro y reducción de la permeabilidad en la zona perturbada alrededor de los drenes (efecto smear). Con el objetivo de dotar de mayor realismo a las simulaciones numéricas, se incorporó el flujo en estos análisis, consiguiéndose una mejor aproximación entre los resultados numéricos y los datos de la monitorización. Asimismo, tales resultados sólo se alcanzan cuando en los análisis numéricos se integran, además del flujo, la magnitud de la perturbación asociada a la ejecución de los drenes en la zona tratada (efecto smear), la fluencia viscosa (creep) en el comportamiento de los suelos blandos de fundación e índices de compresibilidad secundaria adecuados a los mismos. De este modo, la confrontación de los datos asociados a un caso real con los resultados numéricos permitió la validación y contraste de los métodos, modelos y parámetros geotécnicos de los suelos blandos de la fundación cuando se tratan con drenes verticales de banda. PALABRAS-LLAVE: Suelos Blandos, Mejora del Terreno, Drenes de Banda, Análisis de Consolidación, Modelos de Comportamiento, Fluencia Viscosa, Smear, Balsa de Lebrija, PLAXIS. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS NOTATION AND ACRONYMS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE v NOTATION AND ACRONYMS The symbols and parameters referred here, are explain throughout the text, however it is convenient to present a list of the most important. Some of the following symbols have more than one meaning, so one should see the context in which they are used. Nevertheless, the author tried to keep the notation and terminology used in the bibliography of the specialties covered. The symbols are organized by alphabetic order, beginning with Latin capital and then lowercase letters, followed by Greek capital and lowercase letters. Finally the meanings of some mathematical symbols and abbreviations are presented. Latin Symbols Nomenclature A Parameter representing the recompressibility on logarithmic strain formulation of secondary consolidation; Parameter used on the parabolic model for the permeability on smear zone, by Walker & Indraratna (2006) Aʹ Parameter used on the linear model for the permeability on smear zone, by Walker & Indraratna (2007) Aw Cross section of the drain B Equivalent half with consecutive parallel drains in plane strain; Parameter representing the part correspondent to the subtraction between compressibility and recompressibility on logarithmic strain formulation of secondary consolidation; Parameter used on the parabolic model for the permeability on smear zone, by Walker & Indraratna (2006) Bʹ Parameter used on the linear model for the permeability on smear zone, by Walker & Indraratna (2007) C Parameter representing the creep, on logarithmic strain formulation, of secondary consolidation, C Slope representing the linear variation of smear permeability on Chai et al. (1997) model; Parameter used on the parabolic mode, for the permeability on smear zone, by Walker & Indraratna (2006) C1; C2 Slopes representing the bi-linear variation of smear permeability on Chai et al. (1997) model CB Secondary consolidation coefficient Cc Compression index Cd Deviatoric compression coefficient in (e; log(1+(  2+M2))) plane Cf Ratio of field and laboratory coefficient of permeability Ch Coefficient of horizontal consolidation CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS NOTATION AND ACRONYMS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE vi ChAx;hPD Axisymmetric and plane strain equivalent horizontal coefficient of consolidation Cijkl Elasticity Tensor Cr Recompression index Cs Swelling index Cu Uniformity coefficient Cv Coefficient of vertical consolidation C Secondary compression index D85;60;50;15;10 Diameter of soil particles corresponding to 85%, 60%, 50%, 15% and 10% passing E Young’s modulus EDMT Marchetti dilatometer modulus Eref Young’s reference modulus on PLAXIS for Mohr-Coulomb model Eoed Oedometer modulus Es Soil stiffness modulus Eu Undrained strain modulus Eur Unload-reload strain modulus Fc Influence factor on discharge capacity of prefabricated vertical drain due to drain deformation Ffc Influence factor on discharge capacity of prefabricated vertical drain due to clogging Ft Influence factor on discharge capacity of prefabricated vertical drain due to time F(n) Drain spacing factor on consolidation analysis F(r) Drain discharge capacity factor or drain hydraulic resistance factor on consolidation analysis F(s) Drain smear factor on consolidation analysis F () Yield surface function F (,) Hardening or softening law G Shear modulus Gs Specific gravity of soil solid H Soil thickness IP Plasticity index K0 Coefficient of lateral earth pressure at rest K0NC Coefficient of lateral earth pressure at rest for normally consolidated soils K0OC Coefficient of lateral earth pressure at rest for overconsolidated soils Ld Drain length CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS NOTATION AND ACRONYMS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xiii  Effective friction angle cri Effective friction angle on critical state peak Maximum effective friction angle pp Effective friction angle on perfect plasticity u Undrained friction angle  Slope of unloading-reloading line or recompressibility index in (e; lnp  ) plane * Slope of unloading-reloading line or modified recompressibility index in (  v; lnp  ) plane  Slope of normal compression line or compressibility index in (e; lnp  ) plane * Slope of normal compression line or modified compressibility index in (  v; lnp  ) plane d Deviatoric compressibility index in (e; log(1+(  2+M2))) plane  Factor incorporating spacing, smear and drain hydraulic resistance for radial flow Ax Axisymmetric factor incorporating spacing, smear and drain hydraulic resistance  PD Plane strain factor incorporating spacing, smear and drain hydraulic resistance * Creep modified compression index in (  v; lnp  ) plane * Parameter representing hydraulic resistance of the drain in plane strain by Indraratna & Redana (1997) r Residual water content s Saturated water content  Poisson’s ratio ur Unload-reload Poisson’s ratio  Stress 1, 2, 3 Axial, intermediate and minor principle stresses f Effective final vertical stress ij Stress Tensor 0ij Initial stress Tensor i ou0 In situ effective vertical stress or initial effective vertical stress h Total horizontal stress h Effective horizontal stress v Total vertical stress v Effective vertical stress CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS NOTATION AND ACRONYMS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xiv p Effective preconsolidation vertical stress p0 Initial effective preconsolidation vertical stress pc Effective preconsolidation vertical stress at the end of primary consolidation s Principal stress in plane strain t Maximum shear stress in plane strain y Yielding stress xx; yy; zz Stress on x, y, z directions =xy Shear strength c Time giving to the instant line on Bjerrum graphical representation of oedometer test f Shear strength on partly saturated soils  Dilatancy angle; Soil suction  Water content L Liquid limit nat Natural moisture content P Plastic limit  Ratio between secondary compressibility index and compressibility index or recompressibility index Mathematical Symbols Nomenclature ln (…) Natural logarithm log (…) Decimal logarithm  Variation or change of…  Some of…  Increment of….  PI number  Partial derivative or derivative in order to…  Matrix Units Nomenclature m Meter km Kilometer cm Centimeter mm Millimeter CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS NOTATION AND ACRONYMS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xv m2 Square meter m3 Cubic meter Hm3 Cubic hectometer N Newton kN Kilo Newton MN Mega Newton kg Kilogram kPa Kilopascal % Percentage Acronyms Nomenclature AOS Apparent Opening Size ASCE American Society of Civil Engineering ASTM American Standard Testing Materials CC Cam-Clay CPT Cone Penetration Test CPTU Cone Penetration Test with pore pressure measurements CPVD Cap Prefabricated Vertical Drain CSL Critical State Line DP Dynamic probe DTM Marchetti Dilatometer EVD Electric Prefabricated Vertical Drain EXPO’98 1998 Lisbon World Exposition HS Hardening Soil HSsmall Hardening Soil with small-strain stiffness I Inclinometer ISO Isotropic Consolidation Line LNEC Laboratório Nacional de Engenharia Civil MC Mohr-Coulomb MCC Modified Cam-Clay NC Normally consolidated soil OC Overconsolidated soil PL Settlement plate PP Perfect plasticity PVD Prefabricated Vertical Drain CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS NOTATION AND ACRONYMS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xvi PZ Piezometer SC Sand columns SCP Stone columns SD Sand drains Probing or survey SPT Standard Penetration Test SS Soft Soil SSC Soft Soil Creep SWCC Soil-Water Characteristic Curve TS Topographic stake UC Underconsolidated soil USCS Unified Soil Classification System VCL Virgin Compression Line VT Vane-Test CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS TABLE OF CONTENTS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xvii TABLE OF CONTENTS I. Introduction .................................................................................................................................- 1 - I.1. Background and outline of the thesis ...................................................................................- 1 - I.2. Structure of the thesis ..........................................................................................................- 4 - II. Consolidation ..............................................................................................................................- 7 - II.1. Introduction ........................................................................................................................- 7 - II.2. Analytical solution – unidirectional vertical flow ..................................................................- 9 - II.3. Analytical solution - radial flow (axisymmetric analysis) ..................................................... - 15 - II.4. Analytical solution - horizontal drainage (plane strain analysis) ......................................... - 20 - II.5. Analytical solution - combined drainage ............................................................................ - 25 - II.6. Viscosity of fine soils ......................................................................................................... - 25 - III. Material models ....................................................................................................................... - 29 - III.1. Introduction ..................................................................................................................... - 30 - III.2. Behaviour models............................................................................................................. - 30 - III.2.1. Elastic behaviour ...................................................................................................... - 30 - III.2.2. Plastic behaviour ...................................................................................................... - 31 - III.2.2.1. Yielding and yield function ................................................................................ - 32 - III.2.2.2. Hardening law .................................................................................................. - 34 - III.2.2.3. Flow rule........................................................................................................... - 35 - III.2.2.4. Equilibrium: stability and instability .................................................................. - 36 - III.2.3. Elastoplastic behaviour ............................................................................................. - 37 - III.2.4. Critical state model ................................................................................................... - 40 - III.2.4.1. Original Cam-Clay Model .................................................................................. - 47 - III.2.4.2. Modified Cam-Clay Model ................................................................................ - 48 - III.3. PLAXIS material models .................................................................................................... - 50 - III.3.1. Soft Soil model ......................................................................................................... - 53 - III.3.1.1. Stress and strain isotropic state (’1=’2=’3 ) ................................................... - 53 - III.3.1.2. Yield function for the compression triaxial state (’2=’3 ) ................................. - 54 - III.3.1.3. Soft Soil model parameters ............................................................................... - 55 - III.3.2. Soft Soil Creep model ............................................................................................... - 57 - III.3.2.1. One-dimensional creep ..................................................................................... - 58 - III.3.2.2. Introduction of creep in the model ................................................................... - 61 - III.3.2.3. Soft Soil Creep model parameters ..................................................................... - 63 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS TABLE OF CONTENTS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xviii IV. Vertical drains .......................................................................................................................... - 65 - IV.1. Historical synopsis............................................................................................................ - 66 - IV.2. Types of drains ................................................................................................................. - 67 - IV.2.1. Circular drains .......................................................................................................... - 68 - IV.2.2. Prefabricated band-shaped vertical drains ............................................................... - 69 - IV.3. Installation methods ........................................................................................................ - 70 - IV.4. Constructive sequence of the foundation treatment with band-shaped drains ................. - 72 - IV.4.1. Band-shaped drains with vacuum (CPVDs)................................................................ - 77 - IV.4.2. Electrical band-shaped drains (EVDs) ........................................................................ - 79 - IV.5. Spacing between drains and equivalent diameter ............................................................ - 79 - IV.6. Factors which influence drains performance .................................................................... - 81 - IV.6.1. Permeability of the drainage blanket ........................................................................ - 81 - IV.6.2. Limitation of the discharge capacity of the drains ..................................................... - 81 - IV.7. Case studies ..................................................................................................................... - 85 - IV.7.1. Works cases in Portugal ........................................................................................... - 89 - IV.7.1.1. Dock No.4, Leixões Port .................................................................................... - 89 - IV.7.1.2. East expansion of the Beirolas landfill ............................................................... - 90 - IV.7.1.3. Pedestrian footpath on the Tagus waterfront, EXPO’98 .................................... - 92 - IV.7.1.4. Caldas da Rainha / Marinha Grande ................................................................. - 94 - IV.7.1.5. Study for the design project of new Ota Airport ............................................... - 94 - IV.7.1.6. Lezíria Park, Forte da Casa ................................................................................ - 96 - V. Smear ...................................................................................................................................... - 99 - V.1. Introduction ...................................................................................................................... - 99 - V.2. Diameter of the disturbed zone ...................................................................................... - 100 - V.2.1. Overlapping of disturbed zone diameters ................................................................ - 101 - V.3. Horizontal permeability in the disturbed zone................................................................. - 102 - V.4. Sensitivity analysis on the horizontal permeability in the disturbed zone ........................ - 109 - V.5. Sensitivity analysis of the consolidation degree with prefabricated vertical drains .......... - 114 - VI. Dikes No.1 and No.3 of the Lebrija Pond ................................................................................ - 121 - VI.1. Geographical location .................................................................................................... - 121 - VI.2. Geological-geotechnical characterisation ....................................................................... - 124 - VI.2.1. Geotechnical characterisation of the soils foundation of Dike No.1 ........................ - 126 - VI.2.2. Geotechnical characterisation of the soils foundation of Dike No.3 ........................ - 134 - VI.2.3. Analysis and comparison of the geotechnical parameter results ............................. - 137 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS TABLE OF CONTENTS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xix VI.2.4. Geotechnical characterisation of the dikes ............................................................. - 142 - VI.3. Dikes standard cross-section profile ............................................................................... - 143 - VI.3.1. Initial design ........................................................................................................... - 143 - VI.3.2. Revision of Dikes No.1 and No.3 ............................................................................. - 144 - VI.4. Instrumentation of the Lebrija Pond dikes ...................................................................... - 148 - VI.4.1. Dike No.1 instrumentation ..................................................................................... - 149 - VI.4.2. Dike No.3 instrumentation ..................................................................................... - 151 - VI.5. Construction timeline of dikes and reservoir filling ......................................................... - 161 - VI.5.1. Dike No.1 construction stages ................................................................................ - 162 - VI.5.2. Dike No.3 construction stages ................................................................................ - 163 - VI.6. Monitoring results of Dikes No.1 and No.3 ..................................................................... - 166 - VI.6.1. Pore pressures in the foundation and in Dike No.1 ................................................. - 166 - VI.6.2. Pore pressures in the Dike No.3 foundation ............................................................ - 173 - VI.6.3. Settlements in the foundation and in Dike No.1 ..................................................... - 175 - VI.6.4. Settlements in the foundation and in Dike No.3 ..................................................... - 181 - VI.6.5. Horizontal displacements on Dike No.1 .................................................................. - 184 - VI.6.6. Horizontal displacements on Dike No.3 .................................................................. - 188 - VI.6.7. Summary of monitoring results and recommendations .......................................... - 191 - VII. Modelling of Dikes No.1 and No.3 of the Lebrija Pond ........................................................... - 195 - VII.1. PLAXIS Programme – general features ........................................................................... - 197 - VII.2. Modelling the dikes with PLAXIS .................................................................................... - 199 - VII.2.1. Definition of the dikes geometry ........................................................................... - 199 - VII.2.2. Definition of the materials characteristics and properties ...................................... - 203 - VII.2.2.1. Soils used on Dikes No.1 and No.3 ................................................................. - 204 - VII.2.2.2. Drains and geotextile reinforcement.............................................................. - 205 - VII.2.2.3. Soil foundation .............................................................................................. - 205 - VII.2.3. Definition of the initial stress state and initial hydrostatic pressures ...................... - 215 - VII.2.4. Definition of the construction stages ..................................................................... - 218 - VII.2.5. Definition of the control points.............................................................................. - 226 - VII.3. Results of the numerical simulation and comparison with monitoring ........................... - 229 - VII.3.1. Numerical analysis without flow ............................................................................ - 230 - VII.3.1.1. Sensitivity analysis and Dike No.1 settlements ............................................... - 233 - VII.3.1.2. Dike No.3 settlements ................................................................................... - 244 - VII.3.1.3. Pore pressures in Dike No.1 ........................................................................... - 251 - VII.3.1.4. Pore pressures in Dike No.3 ........................................................................... - 264 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS TABLE OF CONTENTS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xx VII.3.2. Results from numerical simulation considering the flow ........................................ - 265 - VII.3.2.1. Dike No.1 and Dike No.3 settlements with flow ............................................. - 266 - VII.3.2.2. Pore pressures in Dike No.1 and Dike No.3 with flow ..................................... - 273 - VII.3.3 Horizontal displacements in Dike No.1 with flow .................................................... - 286 - VII.4. Slope stability analysis for Dikes No.1 and No.3 ............................................................. - 290 - VIII. Conclusions and future developments ................................................................................. - 295 - VIII.1. Conclusions .................................................................................................................. - 295 - VIII.1.1. Conclusions associated to the monitoring results of Dikes No.1 and No.3 of the Lebrija Pond ............................................................................................................ - 296 - VIII.1.2. Conclusions associated with the parametric and sensitivity analysis ..................... - 298 - VIII.1.3. Conclusions associated with the comparison of numerical and monitoring results ..................................................................................................................... - 300 - VIII.2. Future developments ................................................................................................... - 303 - Bibliography ............................................................................................................................... - 305 - Appendix I: Data referent to Dike No.1 and Dike No.3 of Lebrija Pond......................................... - 325 - Appendix II: Prefabricated Colbonddrain CX1000 technical data sheet ........................................ - 337 - Appendix III: PLAXIS simulation results of Lebrija Pond Dike No.1................................................ - 339 - Appendix IV: PLAXIS simulation results of Lebrija Pond Dike No.3 ............................................... - 343 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF TABLES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxi LIST OF TABLES Table II.1 – Parameter for several types of soil (Terzaghi et al., 1996) ........................................ - 27 - Table III.1 – Equations from the Original and Modified Cam-Clay Model ....................................... - 49 - Table III.2 – Values of *,*,  and ratio*-*/ (Manivannan, 2005) ..................................... - 63 - Table III.3 – Ratio (*/*) andC values for several highly compressible soils ............................... - 64 - Table IV.1 – Characteristics supplied by the manufacturer of band-shaped prefabricated vertical drains, according to McGown & Hughes (1981)....................................................... - 69 - Table IV.2 – Methods currently used for the installation of vertical drains, according to McGown & Hughes (1981) .................................................................................................. - 72 - Table IV.3 – Criteria proposed for PVD geotextile filters (Bergado et al., 1996c) ............................ - 82 - Table IV.4 – General characteristics of the Ports where vertical drains were applied, according to the Port Expert Group - APEC 20th TPT-WG (2002) ........................................................... - 87 - Table IV.5a – Geotechnical characteristics of the soil foundations, according to the Port Expert Group - APEC 20th TPT-WG (2002) ....................................................................................... - 87 - Table IV.5b – Geotechnical characteristics of the soil foundations, according to the Port Expert Group - APEC 20th TPT-WG (2002) ....................................................................................... - 88 - Table IV.6 – Type of vertical drains used, area of intervention and height of the embankment, according to the Port Expert Group - APEC 20th TPT-WG (2002) ........................................... - 88 - Table IV.7 – Average characteristics of the silty layer of the Dock No. 4, Leixões Port (Furtado, 1995) ................................................................................................................... - 90 - Table IV.8 – Average characteristics of the soil foundation layer on the south zone of the Tagus waterfront, EXPO’98 (Cristóvão, 1997; Pinto & Da Silva, 2000) ............................................ - 93 - Table V.1 - Triangular pattern: Influence of the diameter of the disturbed zone ds ...................... - 111 - Table V.2 - Quadrangular pattern: Influence of the diameter of the disturbed zone ds ................ - 111 - Table V.3 - Triangular pattern: Influence of the permeability reduction in the disturbed zone khsAx .......................................................................................................................... - 111 - Table V.4 - Quadrangular pattern: Influence of the permeability reduction in the disturbed zone khsAx .......................................................................................................................... - 111 - Table V.5 - Triangular pattern: Influence of the spacing between the parallel drains S* ............... - 111 - Table V.6 - Quadrangular pattern: Influence of the spacing between the parallel drains S* .......... - 112 - Table V.7 - Triangular pattern: Influence of the discharge capacity of the drains and smear effect, considering the analytical formulation of Indraratna & Redana (1997) ................... - 113 - Table V.8 – Properties of the clay soil for the analysis in question ............................................... - 114 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF TABLES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxii Table VI.1 – Dike No.1: Average test results: Grain-size analysis, Atterberg limits, moist unit weight, specific gravity of soil solid and unified soil classification system (USCS) ............... - 126 - Table VI.2 – Dike No.1: Average results of dynamic probe tests (DP), standard penetration test (SPT), direct shear tests and unconfined compression tests .............................................. - 131 - Table VI.3 – Dike No.1: Average results of piezocono tests (PZC) and Vane-Test .......................... - 131 - Table VI.4 – Dike No.1: Average results of Marchetti dilatometer tests and triaxial test .............. - 131 - Table VI.5 – Dike No.1: Initial void ratio, compressibility index, recompressibility index and effective preconsolidation stress (Soriano, 2005); Initial effective vertical stress and OCR value (Da Silva, 2009) ....................................................................................................... - 132 - Table VI.6 – Dike No.1: Secondary compressibility index, consolidation coefficient and permeability coefficients (vertical and horizontal) ............................................................. - 133 - Table VI.7 – Dike No.1: Initial relationship between */*and */* based on edometer test results ............................................................................................................................... - 134 - Table VI.8 – Dike No.1: New C, considering the expression proposed by Terzaghi et al. (1996), and new values for the ratio*/* ................................................................................... - 134 - Table VI.9 – Dike No.1: Average test results: Grain-size analysis, Atterberg limits, moist unit weight, specific gravity of soil solid and unified soil classification system (USCS) ............... - 135 - Table VI.10 – Dike No.3: Average results of dynamic probe tests (DP), standard penetration test (SPT), direct shear tests and unconfined compression tests .............................................. - 135 - Table VI.11 – Dike No.3: Average results of Marchetti dilatometer tests (DMT) and Vane-Test ... - 135 - Table VI.12 – Dike No.3: Initial void ratio, compressibility index, recompressibility index, effective preconsolidation stress, initial effective vertical stress and OCR value (Da Silva, 2009) ................................................................................................................. - 136 - Table VI.13 – Dike No.3: Secondary compressibility index, consolidation coefficient and permeability coefficients (vertical and horizontal) ............................................................. - 136 - Table VI.14 – Dike No.3: Initial relationship between */*and */* based on edometer test results ............................................................................................................................... - 136 - Table VI.15 – Dike No.3: New C, considering the expression proposed by Terzaghi et al. (1996), and new values for the ratio*/* ................................................................................... - 136 - Table VI.16 – Dike No.1: Maximum undrained shear strength correction for the Vane-Test ........ - 137 - Table VI.17 – Dike No.3: Maximum undrained shear strength correction for the Vane-Test ........ - 138 - Table VI.18 – Dike No.1 e No.3: Comparison between SPT and DP results ................................... - 138 - Table VI.19 – Dike No.1: Summary of several undrained shear strength results ........................... - 139 - Table VI.20 – Dike No.3: Summary of several undrained shear strength results ........................... - 139 - Table VI.21 – Dike No.1: Overconsolidation ratio obtained from PZC tests, according to Hansbo (1957) and Mayne & Kemper (1988) ..................................................................... - 141 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF FIGURES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxix Figure IV.4 – Prefabricated vertical band-shaped drains (images token from http://www.geosko.com/en/sub2/wick-drain.php, in January of 2013) ............................ - 70 - Figure IV.5 – Schemes of prefabricated vertical band-shaped drains (images token from http://www.geosko.com/en/sub2/wick-drain.php, in January of 2013) ............................ - 70 - Figure IV.6 – Drainage of prefabricated vertical band-shaped drains (images token from http://www.gobizkorea.com/blog/..., in January of 2013)................................................. - 70 - Figure IV.7 – Sand drainage blanket sand placed above the intervened area (image taken from http://www.americanwick.com/products/..., in January of 2013) ............................. - 72 - Figure IV.8 – Layer of sand placed in a restricted area, forming a drainage mat or mattress above the area treated with drains (image taken from http://www.americanwick.com/products/..., in January of 2013) ............................. - 73 - Figure IV.9 – Connection of the vertical band-shaped drains to the Stripdrain type drains 9a) image taken from http://www.ceteau.com/Products and Services /CeTeau Stripdrain, in January of 2013; 9b) image taken from http://www.geotechnics.com/vertical.htm, in January of 2013 ................................................................................................................ - 73 - Figure IV.10 - Steel mandrel with specific formats for driving PVD drains ...................................... - 74 - Figure IV.11 – Steel hollow mandrel where the band is placed, and anchor steel plate 11a) image taken from http://www.fundesp.com.br/..., in January of 2013; 11b) image taken from http://www.solotrat.com.br/..., in January of 2013; 11c) image taken from http://www.geosyntheticsworld.com, in January of 2013 ................................................. - 74 - Figure IV. 12 – Anchoring the band-shaped drain to the ground (images token from http://www.fundesp.com.br/..., in January of 2013) ......................................................... - 74 - Figure IV.13 – Cutting of the prefabricated band-shaped drain, after their complete installation (images token from http://www.ceteau.com/..., in January of 2013) ................................ - 75 - Figure IV.14 – Execution methodology of the prefabricated vertical band-shaped drains (image taken from http://www.vibromenard.co.uk/techniques/vertical-drains, in January of 2013) ................................................................................................................................ - 75 - Figure IV.15 – Equipment used for the installation of prefabricated vertical band-shaped drains (image taken from http://www.fundesp.com.br/..., in January of 2013) ........................... - 75 - Figure IV.16 – Panoramic view of a zone treated with prefabricated vertical band-shaped drains 16a) image taken of http://www.layfieldenvironmental.com/pages/Products/..., in January of 2013; 16b) image taken from http://www.admirgeo.com/Consolidation_Acceleration-Vertical_Drains, in January of 2013 ........................ - 76 - Figure IV.17 – Preload embankment combined with vertical drains and stabilising berms (Alves, 2011) ..................................................................................................................... - 76 - Figure IV.18 – Application of the vacuum to the clay deposit layer ................................................ - 77 - Figure IV.19 – Methodology applied to the Cap-drains (image taken from http://www.pentaocean.co.jp/english//business/civil/vacuum_drain.html, in January of 2013) .................... - 78 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF FIGURES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxx Figure IV.20 – Pressure envelop for single and double drainage boundary (Chai et al., 2010) ........ - 78 - Figure IV.21 – Vertical drain patterns adopted in the design ......................................................... - 79 - Figure IV.22 – Dimensions of a prefabricated band-shaped drain and equivalent diameter of the drain .......................................................................................................................... - 80 - Figure IV.23 – Deformation of the geotextile filter due to lateral earth pressure ........................... - 83 - Figure IV.24 – Possible deformation modes of PVD drains, due to ground movements, for homogeneous soils (Lawrence & Koerner, 1988) .............................................................. - 83 - Figure IV.25 – Possible deformation modes of PVD drains, due to ground movements, for stratified foundations (Lawrence & Koerner, 1988) .......................................................... - 84 - Figure IV.26 – Container terminal of Dock No.4, Leixões Port (image taken from http://www.bing.com/maps/..., in November of 2012) ............................................ - 89 - Figure IV.27 – Area of the EXPO 98 where the contaminated soil were removed, and the cells of the Beirolas landfill where they were deposited (images taken from http://www.google.com) .................................................................................................. - 91 - Figure IV.28 – Sealed landfill located near the Beirolas water waste powerplant treatment (image taken from http://www.google.com, in March of 2013) ........................................ - 92 - Figure IV.29 – Pedestrian footpath along the right bank of the River Tagus, EXPO’98 (image taken from http://es.wikipedia.org, in March of 2013) ..................................................... - 92 - Figure IV.30 – Industrial area Forte da Casa - Lezíria Park (image taken from http://www.bing.com/maps, in March of 2013) ....................................................... - 96 - Figure V.1 – Dimensions of the mandrel/drain and disturbed zone ............................................. - 100 - Figure V.2 – Overlapping of the disturbed zone of the drains (Walker & Indraratna, 2007).......... - 101 - Figure V.3 – Model proposed by Barron (1948) and Hansbo (1979) ............................................. - 103 - Figure V.4 – Non-linear model proposed by Onoue et al. (1991) ................................................. - 103 - Figure V.5 – Linear model proposed by Chai et al. (1997) ............................................................ - 104 - Figure V.6 – Bi-linear model proposed by Chai et al. (1997) ......................................................... - 105 - Figure V.7 – Hawlader et al. (2002) model, with constant permeability in zone I and II................ - 105 - Figure V.8 – Hawlader et al. (2002) model, with constant permeability in the disturbed zone, although khsII > khsI .......................................................................................................... - 106 - Figure V.9 – Linear model proposed by Hawlader et al. (2002) .................................................... - 106 - Figure V.10 – Parabolic model proposed by Walker & Indraratna (2006) ..................................... - 107 - Figure V.11 – Linear model proposed by Walker & Indraratna (2007) .......................................... - 107 - Figure V.12 – Linear model for the transition zone proposed by Basu & Prezzi (2007) ................. - 108 - Figure VI.1 – Flyer presentation of Lebrija or Don Melendon Pond .............................................. - 122 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF FIGURES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxxi Figure VI.2-a) Ducks and other birds in the Lebrija Pond (Photo of Paco Infante, in 2007); b) Reservoir of Lebrija Pond (Photo of Elisa Silva in February 2013) .................................... - 122 - Figure VI.3 - Aerial view of Lebrija Pond - location of the four dikes (image taken from Google Maps in January 2013) .................................................................................................... - 123 - Figure VI.4 – Topographic implantation of the Lebrija dikes (Photo of Elisa Silva in February 2013) .............................................................................................................................. - 123 - Figure VI.5 – Plan showing the geotechnical site investigation made in Dike No.1. ...................... - 127 - Figure VI.6 – Plan showing the geotechnical site investigation made in Dike No.3. ...................... - 128 - Figure VI.7 – Geotechnical foundation profile on Dike No.1, including level of initial ground surface, crest dike, and cut-off. ....................................................................................... - 129 - Figure VI.8 – Geotechnical foundation profile on Dike No.3, including level of initial ground surface, crest dike, and cut-off. ....................................................................................... - 130 - Figure VI.9 – Standard initial design of the Lebrija Pond dikes (Saura Martínez et al., 2004) ........ - 144 - Figure VI.10 – Final design of Dike No.1, showing reinforcement geotextile, band-shaped drains and clay layers. ............................................................................................................... - 145 - Figure VI.11 – Final design of Dike No.3, showing band-shaped drains and clay layers. ................ - 146 - Figure VI.12 – a) Dike No.1 of Lebrija Pond (Photo of Elisa Silva, in February of 2013) b) Dike No.3 of Lebrija Pond (Photo of Elisa Silva, in February of 2013) ....................................... - 148 - Figure VI.13 – Plan showing all the instrumentation in Dike No.1. ............................................... - 153 - Figure VI.14 – Final design of Dike No.1 (PK0+240), showing settlement plates, topographic stake marks, piezometers and inclinometers. ................................................................. - 154 - Figure VI.15 – Plan showing all the instrumentation in Dike No.3. ............................................... - 155 - Figure VI.16 – Location of the settlement plates PL 140-1 and PL 140-2 on Dike No.3 ................. - 156 - Figure VI.17 – Location of the settlement plates PL 170-1 and PL 170-2 on Dike No.3 ................. - 157 - Figure VI.18 – Location of the piezometers PZ 180-1 and PZ 180-2 on Dike No.3 ......................... - 158 - Figure VI.19 – Location of the settlement plates PL 200-1 and PL 200-2 on Dike No.3 ................. - 159 - Figure VI.20 – Location of the inclinometers I 220-1 and I 220-2 on Dike No.3 ............................. - 160 - Figure VI.21 – Instrumentation devices installed on the Lebrija Pond Dike No.1: a) Topographic stake mark on the upstream side of the crest; b) Topographic stake mark on the downstream side of the crest; c) Piezometer; d) Inclinometer (Photos of Elisa Silva, in February of 2013) ........................................................................................................... - 161 - Figure VII.1 – Triangular elements with 6 nodes (1a) and 15 nodes (1b) ...................................... - 197 - Figure VII.2 – Stress points for triangular elements with 6 nodes (2a) and 15 nodes (2b) ............. - 198 - Figure VII.3 – Geometric model used for the cross-section PK0+240 of Dike No.1, including drains and geotextile ...................................................................................................... - 200 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF FIGURES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxxii Figure VII.4 – Geometric model used for the cross-section PK0+140 of Dike No.3, including drains ............................................................................................................................. - 200 - Figure VII.5 – Geometric model used for the cross-section PK0+170 of Dike No.3, including drains ............................................................................................................................. - 201 - Figure VII.6 – Geometric model used for the cross-section PK0+180 of Dike No.3, including drains ............................................................................................................................. - 201 - Figure VII.7 – Geometric model used for the cross-section PK0+200 of Dike No.3 ....................... - 201 - Figure VII.8 – Finite element mesh used on modeling the cross-section PK0+240 of Dike No.1 .... - 202 - Figure VII.9 – Finite element mesh quality - cross-section PK0+240 of Dike No.1 ......................... - 202 - Figure VII.10 – Geometry used on the generation of the initial stress state - PK0+240 of Dike No.1 ....................................................................................................................... - 216 - Figure VII.11 – Initial horizontal and vertical stresses on PK0+240 of Dike No.1 ........................... - 216 - Figure VII.12 – Initial position of the water table on the ground foundation - PK0+240 of Dike No.1 ....................................................................................................................... - 217 - Figure VII.13 – Initial pore pressures on PK0+240 of Dike No.1 .................................................... - 217 - Figure VII.14 – Ground excavation up to the level of the embankment foundation and cut–off - PK0+240 of Dike No.1 ..................................................................................................... - 219 - Figure VII.15 – Cut–off compaction - PK0+240 of Dike No.1......................................................... - 219 - Figure VII.16– Drains activation - PK0+240 of Dike No.1 .............................................................. - 220 - Figure VII.17 – Change of the horizontal permeability coefficients in the parallel drainage screens. Geotextile activation - PK0+240 of Dike No.1 ................................................... - 220 - Figure VII.18 – Conclusion of embankment construction - PK0+240 of Dike No.1 ........................ - 221 - Figure VII.19 – Position of the water table on the ground foundation before water raise up in reservoir - PK0+240 of Dike No.1 .................................................................................... - 221 - Figure VII.20 – Final water level in reservoir - PK0+240 of Dike No.1 ........................................... - 222 - Figure VII.21 – Modification of the conditions in the embankment soils due to water raise up in reservoir - PK0+240 of Dike No.1 .................................................................................... - 222 - Figure VII.22 – Soil-water Characteristic Curve (SWCC) (Fredlund, 2005) ..................................... - 224 - Figure VII.23 – Permeability decrease on partly saturated soils due to suction (Fredlund, 2005) . - 224 - Figure VII.24 – Relationship between stress and volume-mass ratio for partly saturated soils (Fredlund, 2005) ............................................................................................................. - 226 - Figure VII.25 – Nodes used on the control of the settlements during simulation analysis - PK0+240 of Dike No.1 ..................................................................................................... - 227 - Figure VII.26 – Stress points used to control the pore pressures during simulation analysis - PK0+240 of Dike No.1 ..................................................................................................... - 227 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF FIGURES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxxiii Figure VII.27 – Nodes used on the control of the settlements in PK0+140 of Dike No.3 ............... - 228 - Figure VII.28 – Nodes used on the control of the settlements in PK0+170 of Dike No.3 ............... - 228 - Figure VII.29 – Nodes used on the control of the settlements in PK0+200 of Dike No.3 ............... - 228 - Figure VII.30 – Stress points used to control the pore pressures in PK0+180 of Dike No.3............ - 228 - Figure VII.31 – PK0+240 of Dike No.1: Deformed mesh associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ................................. - 243 - Figure VII.32 – PK0+240 of Dike No.1: Settlements associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw .......................................... - 243 - Figure VII.33 – PK0+140 of Dike No.3: Settlements associated to the final calculation step, using the SSC model, and smear effect given by KhsAx=0.10Kh0 and ds=3dw ................................ - 250 - Figure VII.34 – PK0+170 of Dike No.3: Settlements associated to the final calculation step, using the SSC model, and smear effect given by KhsAx=0.10Kh0 and ds=3dw ................................ - 250 - Figure VII.35 – PK0+200 of Dike No.3: Settlements associated to the final calculation step, using the SSC model, and smear effect given by KhsAx=0.10Kh0 and ds=3dw ................................ - 250 - Figure VII.36 – PK0+240 of Dike No.1: Grandwater head associated to the conclusion of dike, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ................................. - 261 - Figure VII.37 – PK0+240 of Dike No.1: Grandwater head associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ................................. - 261 - Figure VII.38 – PK0+240 of Dike No.1: Pore pressures associated to the conclusion of dike, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ................................. - 262 - Figure VII.39 – PK0+240 of Dike No.1: Pore pressures associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ................................. - 262 - Figure VII.40 – PK0+240 of Dike No.1: Excess pore pressures associated to the conclusion of dike, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ......................... - 263 - Figure VII.41 – PK0+240 of Dike No.1: Excess pore pressures associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw......................... - 263 - Figure VII.42 – PK0+240 of Dike No.1: Final settlements obtained incorporating flow, calculated with SSC model, with CPLAXIS, and smear effect given by KhsAx=0.10Kh0 and ds=3dw........... - 272 - Figure VII.43 – PK0+140 of Dike No.3: Final settlements obtained incorporating flow, calculated with SSC model, with CPLAXIS, and smear effect given by KhsAx=0.10Kh0 and ds=3dw........... - 272 - Figure VII.44 – PK0+170 of Dike No.3: Final settlements obtained incorporating flow, calculated with SSC model, with CPLAXIS, and smear effect given by KhsAx=0.10Kh0 and ds=3dw........... - 273 - Figure VII.45 – PK0+200 of Dike No.3: Final settlements obtained incorporating flow, calculated with SSC model, with CPLAXIS, and smear effect given by KhsAx=0.10Kh0 and ds=3dw........... - 273 - Figure VII.46 – PK0+240 of Dike No.1: Excess pore pressures, without flow, associated to the dike conclusion, with the SSC model, and smear (KhsAx=0.10Kh0 + ds=3dw) ....................... - 278 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF FIGURES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxxiv Figure VII.47– PK0+240 of Dike No.1: Excess pore pressures, with flow, associated to the dike conclusion, with the SSC model, and smear (KhsAx=0.10Kh0 + ds=3dw) ............................... - 278 - Figure VII.48 – PK0+240 of Dike No.1: Excess pore pressures, without flow, associated to the final step, with the SSC model, and smear (KhsAx=0.10Kh0 + ds=3dw) ................................. - 279 - Figure VII.49 – PK0+240 of Dike No.1: Excess pore pressures, with flow, associated to the final step, with the SSC model, and smear (KhsAx=0.10Kh0 + ds=3dw) ........................................ - 279 - Figure VII.50 – PK0+240 of Dike No.1: Pore pressures, with flow, associated to the dike conclusion, with the SSC model, and smear (KhsAx=0.10Kh0 + ds=3dw) ............................... - 281 - Figure VII.51 – PK0+240 of Dike No.1: Pore pressures, with flow, associated to the final step, with the SSC model, and smear (KhsAx=0.10Kh0 + ds=3dw) ................................................. - 281 - Figure VII.52 – PK0+240 of Dike No.1: Soil suction associated to the dike conclusion, with the SSC model, and smear (KhsAx=0.10Kh0 + ds=3dw) ............................................................... - 282 - Figure VII.53 – PK0+240 of Dike No.1: Soil suction associated to the final step, with the SSC model, and smear (KhsAx=0.10Kh0 + ds=3dw) ..................................................................... - 282 - Figure VII.54 – PK0+180 of Dike No.3: Excess pore pressures, with and without flow, associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ............................................................................................................................ - 283 - Figure VII.55 – PK0+180 of Dike No.3: Pore pressures, with and without flow, associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ............................................................................................................................ - 284 - Figure VII.56 – PK0+180 of Dike No.3: Soil suction correspondent to the conclusion of the dike and to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ..................................................................................................................... - 284 - Figure VII.57 – PK0+200 of Dike No.3: Pore pressures, with and without flow, associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ............................................................................................................................ - 285 - Figure VII.58 – PK0+200 of Dike No.3: Excess pore pressures, with and without flow, associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ............................................................................................................................ - 285 - Figure VII.59 – PK0+200 of Dike No.3: Soil suction associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw .......................................... - 286 - Figure VII.60 – PK0+240 of Dike No.1: Horizontal displacements, without flow, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ..................................................... - 288 - Figure VII.61 – PK0+240 of Dike No.1: Horizontal displacements, with flow, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw ..................................................... - 289 - Figure VII.62 – PK0+140 of Dike No.3: Slope stability surface at the end of dike construction, with SSC model and smear given by KhsAx=0.10Kh0 + ds=3dw (with PVD drains downstream of the cut-off trench) ................................................................................. - 292 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF FIGURES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxxv Figure VII.63 – PK0+140 of Dike No.3: Slope stability surface for the final calculation step, with SSC model and smear given by KhsAx=0.10Kh0 + ds=3dw (with PVD drains downstream of the cut-off trench) .......................................................................................................... - 292 - Figure VII.64 – PK0+200 of Dike No.3: Slope stability surface at the end of dike construction, with SSC model and smear given by KhsAx=0.10Kh0 + ds=3dw (without PVD drains) ............ - 293 - Figure VII.65 – PK0+200 of Dike No.3: Slope stability surface for the final calculation step, with SSC model and smear given by KhsAx=0.10Kh0 + ds=3dw (without PVD drains) .................... - 293 - Figure VII.66 – PK0+240 of Dike No.1: Slope stability surface at the end of dike construction, with SSC model and smear given by KhsAx=0.10Kh0 + ds=3dw (with PVD drains on both sides of the cut-off trench) ............................................................................................. - 293 - Figure VII.67 – PK0+240 of Dike No.1: Slope stability surface for the final calculation step, with SSC model and smear given by KhsAx=0.10Kh0 + ds=3dw (with PVD drains on both sides of the cut-off trench) .......................................................................................................... - 293 - Figure AP I.1 – Plan showing the geotechnical prospecting made in Dike No.1. ........................... - 326 - Figure AP I.2 – Plan showing the geotechnical prospecting made in Dike No.3. ........................... - 327 - Figure AP III.1 – PK0+240 of Dike No.1: Deformed mesh at the final calculation step (considering flow) ........................................................................................................... - 341 - Figure AP III.2 – PK0+240 of Dike No.1: Groundwater head at the conclusion of the dike construction (considering flow) ...................................................................................... - 341 - Figure AP III.3 – PK0+240 of Dike No.1: Groundwater head at the final calculation step (considering flow) ........................................................................................................... - 341 - Figure AP III.4 – PK0+240 of Dike No.1: Horizontal groundwater flow at the conclusion of the dike construction ............................................................................................................ - 342 - Figure AP III.5 – PK0+240 of Dike No.1: Combined groundwater flow at the conclusion of the dike construction ............................................................................................................ - 342 - Figure AP III.6 – PK0+240 of Dike No.1: Horizontal groundwater flow at the final calculation step ................................................................................................................................ - 342 - Figure AP III.7 – PK0+240 of Dike No.1: Combined groundwater flow at the final calculation step ................................................................................................................................ - 342 - Figure AP IV.1 – PK0+180 of Dike No.3: Groundwater head at the conclusion of the dike construction (considering flow) ...................................................................................... - 345 - Figure AP IV.2 – PK0+180 of Dike No.3: Groundwater head at the final calculation step (considering flow) ........................................................................................................... - 345 - Figure AP IV.3 – PK0+180 of Dike No.3: Horizontal groundwater flow at the conclusion of the dike construction ............................................................................................................ - 345 - Figure AP IV.4 – PK0+180 of Dike No.3: Combined groundwater flow at the conclusion of the dike construction ............................................................................................................ - 346 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF FIGURES UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxxvi Figure AP IV.5 – PK0+180 of Dike No.3: Horizontal groundwater flow at the final calculation step ................................................................................................................................ - 346 - Figure AP IV.6 – PK0+180 of Dike No.3: Combined groundwater flow at the final calculation step ................................................................................................................................ - 346 - Figure AP IV.7 – PK0+200 of Dike No.3: Groundwater head at the conclusion of the dike construction (considering flow) ...................................................................................... - 347 - Figure AP IV.8 – PK0+200 of Dike No.3: Groundwater head at the final calculation step (considering flow) ........................................................................................................... - 347 - Figure AP IV.9 – PK0+200 of Dike No.3: Combined groundwater flow at the final calculation step ................................................................................................................................ - 347 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF GRAPHS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxxv ii LIST OF GRAPHS Graph V.1 - Rate of the consolidation degree, in axisymmetric analysis, with S=1.5m, khsAx=0.10kh0 and varying ds (ds=1.5dw; ds=3dw; ds=6dw) ................................................... - 115 - Graph V.2 - Rate of the consolidation degree, in axisymmetric analysis, with S=1.5m, khsAx=0.33kh0 and varying ds (ds=1.5dw; ds=3dw; ds=6dw) ................................................... - 115 - Graph V.3 - Rate of the consolidation degree, in axisymmetric analysis, with S=1.5m, khsAx=0.66kh0 and varying ds (ds=1.5dw; ds=3dw; ds=6dw) ................................................... - 116 - Graph V.4 - Rate of the consolidation degree, in axisymmetric analysis, with S=1.5m, ds=1.5dw and varying khsAx (khsAx=0.10kh0; khsAx=0.33kh0; khsAx=0.66kh0) ............................................ - 116 - Graph V.5 - Rate of the consolidation degree, in axisymmetric analysis, with S=1.5m, ds=3dw and varying khsAx (khsAx=0.10kh0; khsAx=0.33kh0; khsAx=0.66kh0) ............................................ - 117 - Graph V.6 - Rate of the consolidation degree, in axisymmetric analysis, with S=1.5m, ds=6dw and varying khsAx (khsAx=0.10kh0; khsAx=0.33kh0; khsAx=0.66kh0) ............................................ - 117 - Graph V.7 - Rate of the consolidation degree, in axisymmetric analysis, varying S, without smear and with smear (ds=3dw y KhsAx=0.33Kh0) ............................................................... - 118 - Graph V.8 - Rate of the consolidation degree, in axisymmetric analysis, with ideal drain and non ideal drain, considering the smear effect (ds=3dw y KhsAx=0.33Kh0) ............................ - 118 - Graph V.9 - Rate of the consolidation degree, in axisymmetric analysis, for various drainage conditions (vertical, radial and combined) ...................................................................... - 119 - Graph V.10 - Rate of the consolidation degree, in plane strain analysis, for Hird et al. (1992) and Indraratna & Redana (1997) analytical solutions, with smear and hydraulic resistance of the drain, considering the distance between drains equal in both systems (S*=S=3m) ...................................................................................................................... - 120 - Graph VI.1 – Dike No.1: Pore pressure readings in piezometers PZ 180-1, PZ 240-1 and PZ 260-1 ......................................................................................................................... - 167 - Graph VI.2 – Dike No.1: Pore pressure readings in piezometers PZ 135-1, PZ 300-1 and PZ 340-1 ......................................................................................................................... - 168 - Graph VI.3 – Dike No.1: Pore pressure readings in piezometers PZ 180-2, PZ 240-2 and PZ 260-2 ......................................................................................................................... - 168 - Graph VI.4 – Dike No.1: Pore pressure readings in piezometers PZ 222-A/B and PZ 258-A/B ....... - 170 - Graph VI.5 – Dike No.1: Pore pressure readings in piezometers PZ 240-3, PZ 240-4 and PZ 240-5 ......................................................................................................................... - 171 - Graph VI.6 – Dike No.1: Comparison of pore pressure readings in piezometers PZ 240- 6A/6B/6C and PZ 240-7A/7B/7C ..................................................................................... - 172 - Graph VI.7 – Dike No.1: Pore pressure readings in piezometers PZ 240-6A, PZ 240-6B and PZ 240-6C ....................................................................................................................... - 173 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS LIST OF GRAPHS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE xxxviii Graph VI.8 – Dike No.3: Pore pressure readings in piezometers PZ 180-1A and PZ 180-1B .......... - 174 - Graph VI.9 – Measurements of all settlements plates of Dike No.1 ............................................. - 176 - Graph VI.10 – Dike No.1: Ratio between settlements and embankment height above the correspondent plate (in percentage) .............................................................................. - 177 - Graph VI.11 – Dike No.1: Relationship between construction stages of embankment and water level of reservoir, with the settlement rates observed in plates PL240-1 and PL 240-2 .... - 178 - Graph VI.12 – Dike No.1: Readings of settlement plates PL 240-1, PL 240-2 and topographic stakes TS 205, TS 240-1, TS 240-2, TS 250-1, TS 250-2, TS 260-1, TS 260-2 and TS 270 ..... - 179 - Graph VI.13 – Dike No.1: Readings of settlement plates PL 300-1, PL 300-2 and topographic stake TS 310 ................................................................................................................... - 179 - Graph VI.14 – Dike No.1: Measurements on topographic stakes TS 240-1 to TS 240-6 ................. - 180 - Graph VI.15 – Dike No.1: Measurements on topographic stakes TS 250-1 to TS 250-6 ................. - 180 - Graph VI.16 – Dike No.1: Measurements on topographic stakes TS 260-1 to TS 260-6 ................. - 180 - Graph VI.17 – Settlements measured on all the plates of Dike No.3 ............................................ - 181 - Graph VI.18 – Dike No.3: Ratio between settlements and embankment height above the level plate ............................................................................................................................... - 182 - Graph VI.19 – Dike No.3: Relationship between construction stages of embankment and water level of reservoir, with the settlement rates observed in plates PL140-1 and PL 200-2 .... - 183 - Graph VI.20 – Dike No.3: Readings of settlement plates PL 140-1, PL 140-2, PL 170-1, PL 170-2 and topographic stakes TS 135 and TS 170 ..................................................................... - 183 - Graph VI.21 – Dike No.1: Horizontal displacements measured on I 220-1 - axis A0(+)-A180(-) ..... - 185 - Graph VI.22 – Dike No.1: Horizontal displacements measured on I 260-1 – axis A0(+)-A180(-) .... - 185 - Graph VI.23 – Dike No.1: Horizontal displacements measured on I 220-2 – axis A0(+)-A180(-) .... - 186 - Graph VI.24 – Dike No.1: Horizontal displacements measured on I 260-2 - axis A0(+)-A180(-) ..... - 186 - Graph VI.25 – Dike No.1: Horizontal displacements measured on I 240-1 - axis A0(+)-A180(-) ..... - 187 - Graph VI.26 – Dike No.1: Maximum horizontal displacements on all inclinometers - axis A0(+)-A180(-) ........................................................................................................... - 187 - Graph VI.27 – Dike No.3: Horizontal displacements measured on I 220-1 - axis A0(+)-A180(-) ..... - 189 - Graph VI.28 – Dike No.3: Horizontal displacements measured on I 220-2 - axis A0(+)-A180(-) ..... - 189 - Graph VI.29 – Dike No.3: Horizontal displacements measured on I 220-1 - axis B0(+)-B180(-) ..... - 189 - Graph VI.30 – Dike No.3: Representation of maximum horizontal displacements measurements, with time, for I 220-1 and I 2002-2 - axis A0(+)-A180(-) .......................... - 190 - Graph VII.1 – PL 240-1 of Dike No.1: Settlements calculated with the Morh-Coulomb model for the soil foundation, in undrained conditions (Method A and B) ...................................... - 234 - CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS I. INTRODUCTION University of Seville – Superior Technical School of Architecture - 3 - When considering the smear effect the problem lies on the determination of these values and in their correspondence between theory and practice. It has been noted that the extent and permeability of the disturbed area depends on the installation process, the size and the shape of the mandrel and the soil ground characteristics. As these parameters cannot be determined accurately, they are estimated based on field experiences, laboratory tests and theoretical considerations, or by retro-analysis of real case study results. This research project focuses the consolidation of soft soil in the presence of prefabricated vertical band-shaped drains, and their application to the analysis of an actual worksite, namely in Dikes No.1 and No.3 of the Lebrija Pond, located in southern Seville, Spain. Dikes No.1 and No.3 of the Lebrija Pond were founded in soils classified as clay or silty soils, with high plasticity and compressibility. Due to the reduced geotechnical properties of these soils, the initial design was revised and prefabricated vertical band-shaped drains, arranged in a triangular pattern, were applied to the soil foundation in both dikes. In the case of Dike No.1, it was also decided to place geotextile reinforcement between the dike embankment and the foundation. In addition, the dikes were properly instrumented, being subjected to monitoring from the beginning of their construction in order to control their stability and safety, as well as the efficacy of the prefabricated vertical drains. Based on data from settlement plates, piezometers and inclinometers installed in the body of the embankments and foundations of the dike, it was found that the settlements slightly exceeds those estimated in the design. Settlements are now practically stabilised, but the order of magnitude of the maximum settlements is around 7% to 11% of the total height of the dikes, indicating significant secondary settlements. The finite element method (FEM) used in the simulation of consolidation and stability of soils treated with prefabricated vertical drain, presents a high potential, as it allows the incorporation of not only the construction phases but also reinforcement elements, when necessary. However, while the analysis of the phenomenon of soil consolidation around a drain is a problem of axisymmetric nature (3D-tridimensional), real geotechnical consolidation problems are mostly analysed in plane strain (two-dimensional-2D). So, in order to perform the simulations, some adaptations are needed. In this case, the system of the drain arrangement needs to be converted, on a wall of parallel vertical drains. The spacing of the drains and the coefficient of the permeability of the treated area must be adjusted along with the three-dimensional system, always ensuring the equality of the average degree of consolidation in both systems. The solutions currently used are those of Hird et al. (1992), and Indraratna & Redana (1997). The Plaxis, a finite element method (FEM) software, was used to model and to simulate the consolidation process and several built-in material models, including the Mohr-Coulomb, Soft Soil and Soft Soil Creep were considered. Based on these numerical simulations, it was possible to evaluate the response of the soil foundation and the dikes, compare the adequacy of the material models adopted to characterise them, as well as the mechanical and permeability soil parameters. Furthermore, and by retro-analysis, and estimation for the parameters associated with the smear effect and some considerations regarding the viscosity of the soil are also made. Additionally, the program also allows accounts the flow. In this case, the soils are saturated or partially saturated depending on the location of the groundwater head, drainage conditions and time. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS I. INTRODUCTION University of Seville – Superior Technical School of Architecture - 4 - The most significant contribution of this thesis is focused on the analysis of consolidation of soft soils, considering simultaneously: a) the influence of the smear effect of soil treated with prefabricated vertical band-shaped drains; b) the viscosity effect (creep) of clay soils with high compressibility; c) the influence of the flow net on the foundation and dikes. The previous paragraph raises the following important questions:  "What is the interest in combining such diverse phenomena, as the smear, the creep and the flow, in the consolidation of soft soil in the presence of band-shaped drains?"  "Why is it so important to incorporate a real case work on the research?" The answer to these questions will be answer in the conclusions of this thesis, which structure will be presented in the following section. I.2. STRUCTURE OF THE THESIS This thesis consists of eight chapters and four appendices, being the sequence and preview of the themes, presented in the paragraphs below. Chapter I (INTRODUCTION) has the goal of introducing the reader to the topic covered throughout the thesis, the motivations that led to its study and the way in which it is structured. In Chapter II (CONSOLIDATION), the consolidation phenomenon will be covered, with the presentation of several analytical formulations. The introduction to the subject starts with the Terzaghi’s unidirectional analytical consolidation solution, followed by the solutions considering vertical drains, and more specifically prefabricated vertical drains. Therefore, the Hansbo formulation (1979) will initially be presented for radial flow (axisymmetric problem), and subsequently its conversion to the plane strain. In the last case two analytical solutions are presented, namely those of Hird et al. (1992) and Indraratna & Redana (1997). Note that in the analytical solutions for vertical drains, both the discharge capacity of the drain and the effect of the disturbed area around it (smear effect) can be taken into account. In addition, a first discussion of the secondary consolidation is considered. This phenomenon occurs on fine soils at constant pressure, depends on time and on the soil viscosity (creep). This settlement is quite significant when in the presence of compressible or soft soil. This chapter ends with the presentation of the combined solution of Carrillo (1942) witch considerer the vertical and horizontal flow. Chapter III (MATERIAL MODELS) concerns the constitutive laws that describe the soil behaviour and their response in different contexts. Therefore, and according to the previous chapter, firstly it will be presented some simple material models, progressing to ones that are more complex. The model behaviour of soft soils that considerers the viscosity (creep) will merit a special emphasis. The mathematical modelling of the behaviour of these soils is complex, yet it is essential for the correct analysis of the associated strain throughout the design. Lastly, the Mohr-Coulomb, Soft Soil and Soft Soil Creep models of Plaxis software will be presented since they were used in the modelling of dikes of the Lebrija Pond. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS I. INTRODUCTION University of Seville – Superior Technical School of Architecture - 5 - Chapter IV (VERTICAL DRAINS) is completely focused on vertical drains, their origins and how they have evolved until today. References are made to their installation process, as well as the equipment and modus operandi. Special attention is given to the prefabricated vertical band-shaped drains, their advantages and drawbacks. We will also present some work cases where this technique has been applied to the treatment of soil foundations. Chapter V (SMEAR) is completely dedicated to the smear effect and to the characterisation of the parameters, which allows its consideration in the consolidation analyses. A state-of-the-art on the topic in question is presented, where it can be seen the importance of the smear in the consolidation process delay. Next, a comparative study is made between the analytical solutions of Hird et al. (1992) and Indraratna & Redana (1997), both for plane strain. In this stage, the intention is to demonstrate not only the differences between the two solutions, but also the influence of some parameters in the estimation of the horizontal permeability coefficient on the disturbed area (smear). This is a key point of the thesis since the case work is analysed in plane strain, and this study allows to decide which analytical solution should be adopted in the future simulations. The chapter ends with a sensitivity analysis of the smear parameters and their influence on the evolution of the degree of consolidation. The theoretical concepts previously presented will be applied to the analysis of a specific casework, introduced in Chapter VI (DIKES NO.1 AND NO.3 OF THE LEBRIJA POND). In this phase, the geographic location of the dikes , it’s purpose and it’s integration considering the environmental and socio-economic of the region, is presented, as well as: a) the initial design project and modifications suffered during construction; b) the prospecting campaigns performed at each stage; c) geotechnical characterisation of the soil foundation, based on the results of the field and laboratory tests; d) geological-geotechnical profiles for the cross sections analysed; d) geotechnical characterisation of the materials which form the dike embankments; e) instrumentation applied for control measurements; f) analysis of the monitoring results. In chapter VII (MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND), the results of numerical simulations performed are showed and confronted to the monitoring data. However, a first step is necessary, which correspond to the estimation of the horizontal permeability coefficients to be adopted in the areas treated with PVD. In the suite, modelling of the cross sections are presented, including a description of the calculation procedure, the boundary conditions for displacements and drainage in each phase, and the generation of the initial stress state. The simulations are performed with and without the consideration of a flow net in the foundation as well as in the embankment dikes. Once this option is adopted, the software considers the soil located above the saturation streamline as partly saturated. In order to analyse the importance of material models in the soil response, the Mohr-Coulomb model is used in the initial simulations for the soil foundations; Lately, Soft Soil and Soft Soil Creep models will be given preference. A sensitivity analysis is also performed for the smear and creep parameters. Based on the results of the numerical simulations and by comparing them with the monitoring ones, the permeability coefficient will be estimated along with the diameter of the disturbed zone (smear), together with the secondary compressibility index (C) for the soil foundation, and the importance of considering the flow net. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS I. INTRODUCTION University of Seville – Superior Technical School of Architecture - 6 - Finally, the safety coefficients for the slope stability of the dikes are determined, for the phases associated to the end of the construction and the last step for the raised up of the water level in the reservoir. The objective will be to establish the efficiency of the treatment in the global stability of the dikes. In chapter VIII the conclusions are presented, as well as some research lines to be followed in the future, regarding the topic analysed. Finally, in Appendices I, II, III and IV, the following results will be presented: APPENDIX I: DATA REFERENT TO DIKE NO.1 AND DIKE NO.3 OF LEBRIJA POND. APPENDIX II: PREFABRICATED COLBONDDRAIN CX1000 TECHNICAL DATA SHEET. APPENDIX III: PLAXIS SIMULATION RESULTS OF LEBRIJA POND DIKE NO.1. APPENDIX IV: PLAXIS SIMULATION RESULTS OF LEBRIJA POND DIKE NO.3. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 7 - II. CONSOLIDATION This chapter is dedicated to the consolidation theory, which represents an essential and fundamental theme in this thesis, and therefore it will be the first to be covered. As mentioned in Chapter I, the vertical strains observed in fine soils, when subject to a load, are due to the consolidation phenomenon, and are prolonged over time. The factors, which control the time associated with the phenomenon in question, depend on both the location of the drainage boundaries and the permeability of the soil. In normal conditions, without the use of artificial techniques, the consolidation is treated as a vertical unidirectional problem, since the closest boundary drainage conditions are located at the top and/or the base of the stratum, and the vertical flow depends on the permeability in that direction. When vertical drains are used in the ground, the predominant flow changes since the boundary drainage conditions also changes. In this case, the consolidation around the drain is formulated as an axisymmetric problem since the dominant flow is radial and the associated parameter will be the horizontal permeability of the soil. The distance that a water particle has to travel to arrive at the vicinity of the drain is substantially reduced and consequently the acceleration of the consolidation phenomenon occurs. In the analysis of a real case, where vertical drains are used, the consolidation will be a combination of the two situations. As the thesis includes the study of a real case work, in which the application of vertical drains was used, the analytical formulations associated with consolidation will be presented. The introduction to the theme begins with the analytical formulation of Terzaghi’s unidirectional vertical consolidation, followed by the consolidation formulations taking into account the vertical drains, and more specifically the prefabricated vertical drains. Therefore, the Hansbo solutions (1979) will initially be presented for radial drainage (axisymmetric problem) followed by the solutions for plane strain analysis, namely those of Hird et al. (1992) and Indraratna & Redana (1997). Beware that the consolidation analytical solutions for the drains allow both the effect of the disturbed zone around it (smear effect), as well as the limitations of the discharge capacity of the drains. Lastly, the combined solution of Carrillo (1942), taking into consideration the vertical and horizontal flow, is also presented. At the end of this chapter, a first discussion of the secondary consolidation is considered. This phenomenon occurs on fine soils at constant pressure, depends on time and on the soil viscosity (creep). This settlement is quite significant when in the presence of compressible or soft soil. II.1. INTRODUCTION When soil is submitted to a load that is different to that of its normal state, a vertical strain occurs, called settlement. This settlement may be significant or not; however, it must be estimated. The settlements in the soil are not elastic as they are mainly due to the reorganisation of the soil, due to both the displacement of particles and void reduction, and the fracturing of other particles at the contact points. Therefore, most settlements in soil are not recoverable after the removal of the load. The vertical displacement of the particles often implies a reduction in the voids, and this CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 8 - causes a temporary increase in the pore pressure (u), if the soil is saturated, and negligible when dry. Theoretically, the settlement of soil is defined as being the vertical displacement (s) which occurs due to a load (), in a soil stratum with a thickness H and with a strain module (Es), i.e.: H H s 0 0 s dz dz E        (II.1) If the thickness of the layer is relatively small, and the strain module is considered constant, then the integration will be: s H    , being  the vertical strain (II.2) However, the parameter Es is not simple to obtain, as it depends on the type of soil, it’s conditions on ground, the confinement to which it is subjected and also the depth. The increase in stress () at depth is also difficult to quantify, and so approximations are usually used in both cases. When the soil is saturated or partially saturated, then the complexity of the problem increases as the settlement will not be complete until the total dissipation of the excess pore pressures has occurred. In certain soils, this process may take quite some time as the dissipation depends on the permeability of the soil. In granulated saturated soil, especially when poorly graded, drainage is practically instantaneous and so it is usual to apply the equation (II.1). In fine saturated soil, the settlements can be rather significant and delayed on time due to the low permeability coefficients (k). These types of vertical displacements, which depend on the time factor, are designated as consolidation settlements. In fine, partly saturated soils, drainage also does not occur instantaneously, and depends not only on time but also on their degree of saturation (Sr) (Bowles, 1996). Therefore, the total settlement of the foundations are composed by three distinct parts, namely the immediate settlement (si), the primary consolidation settlement (spc) and the secondary or secular consolidation settlement(ssc). The immediate settlement, which is low in clayed soils and higher in sandy soils, is the result of the strain of the soil at constant volume. Generally, and depending on the type of soil, it is common to have the following relations between total and immediate settlement (Soriano, 2005): si  (10% a 15%) stot for soft and very soft clays, si  (60%) stot for overconsolidated clays, si  (80%) stot for silts, si  (100%) stot for granular soils (gravels and sand). Primary consolidation settlements are related to the dissipation of excess pore pressure generated by the application of a load, and secondary consolidation to the settlements of the soil under constant stress. Even though both phenomena are associated to time, the settlement that corresponds to the primary consolidation ends when pore pressure dissipates; however, secondary consolidation continues to occur, even though it evolves more slowly. In reality, both consolidations occur simultaneously, which complicates the analysis of the settlement results. However, it is usual to consider that secondary consolidation only occurs when the dissipation of the pore pressure has CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 9 - been practically concluded (Bardet, 1997). This theme will be covered in chapter II.6 and later in chapter III.3.2, in terms of mathematical formulation. II.2. ANALYTICAL SOLUTION – UNIDIRECTIONAL VERTICAL FLOW As already mentioned, consolidation is a phenomenon, which is associated with fine saturated soil when submitted to a load higher than the actual stress state condition, and the effect corresponds to the vertical strain, or settlement, in time. Since the water and solid particles present reduced or no compressibility al all, then the overburden stress applied to the soil will be supported by the water and not by the solid skeleton of the soil. Only as the increased pressure in the water dissipates, which occurs due to the drainage of the water from the soil, is the load transferred to the solid skeleton of the soil. As the permeability of this soil are low, the drainage of the water particles take time, and the settlements which correspond to the dissipation of the pore pressure excess also occurs very slowly, (Coelho, 1996). These corresponds the primary consolidation or hydrodynamic settlements. The consolidation theory was developed by Terzaghi in 1925, and is based on the following assumptions: a) the layer of the soil is homogeneous; b) the soil is completely saturated (Sr=100%); c) the compressibility of the water and particles which form the soil is negligible; d) the strains or extensions () are infinitesimal, meaning that an element of the dimensions dx, dy and dz, have the same response as an element with the dimensions x, y and z; e) the draining and compression are one-dimensional; f) Darcy's Law is valid; g) the properties of the soil at depth are constant; h) the relation between the void index (e) and the stress applied () is linear. Therefore, and in accordance with this theory, the variation of volume V of a mass of soil, in a three dimensional medium and with drainage in the three directions will be: 2 2 2 x y z 2 2 2 h h h V k k k dx dy dz x y z t                  (II.3) When h is the hydraulic head and kx, ky and kz the permeability coefficients on the x, y and z directions. This expression depends on assumptions a), d) and f) of Terzaghi’s theory. For one-dimensional flow (in the vertical direction z), the two first terms disappear, and in this case, it can be assumed that k=kv=kZ and the volumetric draining is defined as: 2 2 h V k dx dy dz z t      (II.4) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 10 - The volumetric element is dx dy dz, and the volume of pores will be   e dx dy dz 1 e         , with (e) being the void ratio. Based on assumption c) of Terzaghi’s theory, the variation of volume with time will be: e V dx dy dz t 1 e t             (II.5) in addition, as (dx dy dz)/(1+e) is the constant volume of a solid; the previous equation may be re-written as: dx dy dz e V 1 e t t             (II.6) Applying this in equation II.4, and cancelling the terms dx, dy and dz, we obtain: 2 2 h 1 e k z 1 e t       (II.7) Only excess pore pressure (u) will cause drainage and a variation in volume, and as the hydraulic head is h=z0+u/w, where w is the water volume weight, then the previous expression becomes: 2 2 w k u 1 e z 1 e t        (II.8) In order to continue with the theory, it is now necessary to define the compressibility coefficient av and the rate of compressibility a’v. These parameters taken from the linear part of a graphic, relates the void ratio (e), or the strain () to the applied pressure or strain (), being: v e e a       (II.9) v a'       (II.10) Before the occurrence of the pore pressure dissipation, the stress increment equals the pore pressure (=u), and therefore, e=av u. Substituting this expression in equation II.8, ones obtain: 2 2 v w k (1 e) u u a z t              (II.11) The expression, which appears in square brackets, is defined as being the consolidation coefficient Cv, or in other words: v v w k (1 e) Ca    (II.12) One should now introduce the volumetric compressibility coefficient mv, in which the void ratio (e) is substituted by the initial void ratio (e0). This coefficient is defined as: v v v 0 a m a' 1 e    (II.13) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 11 - Another way to present the consolidation coefficient, and normally used in numerical method applications, is the following: 0 v v w v w k (1 e ) k Ca m      (II.14) in this case, equation II.11 can be re-written as: 2 v2 u u C z t      (II.15) The analytical solution of the previous equation is not trivial, and uses the expansion of the Taylor series. This solution will be: 22v 2 n C t 2H n4H i n 1 0 1 n z n z u u sin dz sin dz e H 2H 2H                          (II.16) being H the thickness of the layer. This is the general solution and is applied to any initial hydrostatic pressure ui in the stratum, thickness of the compressible stratum, and for any depth z. This last parameter is measured from the top to bottom. As the consolidation coefficient Cv is considered constant, and time t is a multiple of CvH, then one can introduce a time factor, dimensionless, defined as: v 2 C t Tv d  (II.17) where d is the drainage length. Substituting the previous expression in equation II.16, then it becomes: 22 n Tv 2H n4 i n 1 0 1 n z n z u u sin dz sin dz e H 2H 2H                          (II.18) If the fine soil is between two permeable layers, such as two layers of sand (Figure II.1a), then the drainage length will be d=H/2. Otherwise, the drainage is only processed in one direction (Figure II.1b), and then the drainage distance will be equal to H. The progress of consolidation, both time t and depth z, is given by the equation: i (z,t) (z,t) (z,t) i u u ' Uu      (II.19) where U(z,t=0)=0 at the start of the consolidation and U(z,t=)=1 at the end of the consolidation. However, the most common procedure to determine the average degree of consolidation of a soil layer, for a given time t, is based on the expression: H z 0 1 U U dz H   (II.20) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 12 - 1a) 1b) Figure II.1 – Unidirectional vertical flow: 1a) in both ways; 1b) only in one way Substituting the expression II.18 in this last, we obtain: 22 n Tv 4 2 2 n 1 8 e U 1 n                (II.21) Based on approximations and simplifications, equation II.21 will be sub-divided into two, depending on the time factor (Bardet, 1997). The most common are those proposed by Fox, in 1948, being: (Tv) 4Tv U  , for Tv < 0.2827 (II.22) 2 Tv 4 (Tv) 2 8 U 1 e             , for Tv  0.2827 (II.23 In turn, the variation of the time factor Tv with the average degree of consolidation may also be approximately determined, based on the following expressions (Das, 1999): 2 [%] U Tv 4 100         , for U < 60% (II.24) [%] Tv 1,781 0,933ln(100 U )    , for U  60% (II.25) Sivaram & Swamee (1977) developed an empirical relationship (equation II.26) which relates these two parameters, and this is valid for any value of the average degree of consolidation. 2 [%] 0,357 5,6 [%] U 4 100 Tv U 1100                     for 0%  U < 100% (II.26) Vertical load increment Permeable layer d=H Clayed soil Impermeable layer z Flow direction Flow direction Vertical load increment Permeable layer d=H/2 Clayed soil Permeable layer z CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 19 -   (r) d h w F L z q z k    (II.46) z is the distance considered from the top of the drain, Ld is the length of the drain and qw is the drain discharge capacity for a unitary hydraulic gradient (i=1), given by: w w w w w q k iA k A   and 2 w w A r   (II.47) and (II.48) where kw is the coefficient of permeability of the drain and Aw the cross-section of the drain. Finally, considering the sum of the three previous factors, and for n≥20, the  will be:   d e s h h s hs w w 3 ln l d d k k d k d n L z q z 4                          (II.49) yet:     h 2 w d h hs w k k ns s k k 3 ln ln 4 L d z z 4                   (II.50) where the parameter (s) is the ratio between the diameter of the smear zone (ds) and the diameter of the drain (dw): s w d s d  (II.51) If the drain flow is limited at the end, then Ld=H, becoming only half of this value (Ld=H/2) if the drainage can occur at both ends of the drains. The Hansbo (1979) solution is generalised, and is the most used, due to its simplicity. The results obtained based on this solution do not differ much from the results obtained using more sophisticated solutions, determined using the finite element method (Onoue, 1988; Lo, 1991). Figure II.7 – Exemplifying outline of the drain, of the disturbed zone (smear) and radial drainage L d - drain length dw - drain diameter ds - diameter of the disturbed zone de - equivalent diameter kh - horizontal permeability on the undisturbed zone khs - horizontal permeability on the disturbed zone d e d s dw Prefabricated drain Disturbed zone Undisturbed zone Vertical load increment Permeable layer d w de L d Vertical discharge capacity of the drain d s k hs kh z q w Radial flow CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 20 - The validity of Darcy’s law was questioned by several researchers, namely Silfverberg (1947), Hansbo (1960 and 1997a), Miller & Low (1963), and Dubin & Moulin (1986). In 1960, Hansbo adopted an exponential flow law in the theory for the consolidation of soil treated with vertical drains, and later, in 1986, Dubin & Moulin solved the solution for the vertical consolidation of a one-dimensional problem. Later, based on real scale field tests, performed in Sweden in 1960, Hansbo observed that the settlements calculated with an exponential flow law gave closer results to the measured data, although alerting that this may not be the only explanation, also referring the possible reduction in the consolidation coefficient throughout the consolidation process (Hansbo, 1997a). II.4. ANALYTICAL SOLUTION - HORIZONTAL DRAINAGE (PLANE STRAIN ANALYSIS) The finite element method (FEM) used for the analysis of soil consolidation including prefabricated vertical drains, with or without the application of preload embankments or others type of loads, are an unquestionable powerful tool if well employed. It allows the incorporation, not only of the construction phases gradually, but also the reinforcement elements if needed (Hird et al., 1992). Its application in this type of analysis was study and investigated by Zeng et al. (1987 and 1989), Kumamoto et al. (1988), Sanchez & Sagaseta (1990), Cheung et al. (1990), Hird et al. (1992), followed by Indraratna & Redana (1997) and Chai & Miura (1999), among others. The analysis of the phenomenon of the consolidation of the soil around a drain is a problem of an axisymmetric nature (three-dimensional 3D), however in real case works, and using two-dimensional (2D) software, the problem is considered in the plane strain state. This is an extremely important topic for designers and researchers, since appropriate adaptations are necessary in order to obtain an equivalent average degree of consolidations in both systems. In a two-dimensional analysis, the water flow in the soil is considered horizontal and not radial, and so it is necessary to convert the spatial system of the drains, on a wall of parallel vertical drains (Figure II.8) (Shinsha et al., 1982). So, the spacing of the drains and the permeability coefficient in the area, where the drains were incorporated, must be adjusted for the plane strain. In a very simplified manner, a ratio between the two systems can be expressed by the equation II.52; however, the solutions currently used are those of Hird et al. (1992) and Indraratna & Redana (1997), which will also be presented in detail. 2 hPD 2 hAx e k B k r   (II.52) where: khPD - Permeability coefficient in the plane strain; khAx - Axisymmetric permeability coefficient;  - Conversion coefficient between the two systems; B - Half the width between two parallel and consecutive drains (axis to axis); re - Equivalent radius of the drain influence diameter (related to the original pattern). In 1992, Hird et al., proposed a method to incorporate the vertical drains in the analysis of problems in plane strain and, in 1995, presented new studies in which they confirmed the adequacy of this method. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 21 - Figure II.8 – Conversion of the radial flow in a vertical drain for a wall of vertical drains, with geometric consideration of a disturbed zone, only in the axisymmetric system. These researchers presented an analytical solution for the plane strain identical to Hansbo’s solution for the axisymmetric consolidation of a soil cylinder in the surrounding of a vertical drain. This solution considerers both the effect of the disturbance of the ground due to the drain installation process, but also the discharge limitation of the drains. According to them, the average degree of consolidation in both system, plane strain (PD) and axisymmetric state (Ax), as to be equal, at any time and for each load level (Hird et al., 1995). Therefore: hAx hPD U U  (II.53) with: hAx Ax 8T hAx r 1 eU U           and hPD PD 8T hPD 1 eU           (II.54) and (II.55) consequently, for the same time t, the match is obtained if: hAx hPD hAx hPD hAx hPD hAx hPD 2 *2 2 2 2 2 PD e Ax PAx AxD e PD v w e Ax v w PD 8T 8T C t C t C C k k   d S  4r 4B m r m B                    2 hPD PD 2 hAx e Ax kB k r     (II.56) Once again, the parameters PD and Ax express the effects associated with the spacing between the drains F(n), the smear effect F(s) and the hydraulic resistance of the drain F(r), for both plane strain and an axisymmetric state. The parameter Ax will be given by the expression II.49 or II.50 (Hansbo, 1979), and the PD parameter will be given by the equivalent expression, but for the plane strain (equation II.57).   2 hPD PD d wPD k 22 L z z 3 Bq    (II.57) Since no delimitation of the smear zone was considered in plane strain scheme, between the parallel drains, this means that, even though smear effect is considered in both systems, it is not geometrically represented in the drain parallel wall (Figure II.8). d e d s dw Radial flow (3D) Parallel drain wall (2D) S*=2B b w B CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 22 - Consequently the new horizontal coefficient of permeability calculated is extends to the entire area between parallel drains. Therefore, substituting the parameters Ax andPD in equation II.56, the relation between permeability coefficients is given by:     2hPD d 2wPD hPD 2 hAx e 2 e shAx hAx d s hsAx w wAx k 22 L z z 3 Bq k B k r d d k k3 ln ln π L z z d k d 4 q                                     (II.58) which is equivalent to:     2 2 2 2 e s hAx hAx hPD hPD e d hAx d s hsAx w wAx wPD d dk k k3 2 k r ln ln π L z z k B 2 L z z d k d 4 q 3 Bq                                        (II.59) alternatively, even to:     2 2 2 2 hAx hAx hPD hPD e d hAx d hsAx wAx wPD k k kn 3 2 k r ln ln(s) π L z z k B 2 L z z s k 4 q 3 Bq                                 (II.60) being: qwPD - Discharge capacity of a drain in plane strain; qwAx - Discharge capacity of a drain in axisymmetric state; khsAx - Permeability coefficient in the disturbed zone for axisymmetric state. The compatibility between both the systems is ensured, as long as there is equivalence or correspondence, either in the discharge capacity of the drains (II.61) and in the soil consolidation (II.62). 2 22 hPD e hAx e hAx hPD wAx wPD wAx wPD k r k rk 2B k 2B q Bq q q      (II.61) 2 2hAx hPD e hAx hsAx kn 3 2B k r ln ln(s)- k s k 4 3                      (II.62) Replacing, in the equation II.61, the drain discharge capacities for a unitary hydraulic gradient, and adopting identical permeability in both systems (kwPD=kwAx), then bw is given by: 2 2 2 * * 2 * 2 e e e wPD wAx w wPD w wAx w wPD w wAx d d d q S q  2b .i.k  S r .i.k  b k  S r k   4 4 2          * 2 * 2 w wAx w w w 2 2 e wPD e 2S r k 2S r b  b  d k d     (II.63) when particularizing for square and triangular patterns, the values are estimated by: square pattern: * 2 * 2 * 2 w w w w 2 2 2 2S r 2S r S r b    1,57 (1,128S) 1,2724S S    (II.64) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 23 - triangular pattern: * 2 * 2 * 2 w w w w 2 2 2 2S r 2S r S r b   1,814 (1,05S) 1,1025S S    (II.65) To satisfy the equation II.62, three distinct methods can be applied, corresponding to: a) 1st Hypothesis: Geometric correspondence - This is determined by assuming that the permeability coefficients in both systems are equal (khPD=khAx), and so the distance between drains on the plane strain depends on the real distance between the drains (II.66): 2 2 hAx e hsAx k 3 n 3 B r ln ln(s) 2 s k 4                        (II.66) b) 2nd Hypothesis: Permeability correspondence - This is determined by assuming that the spacing between drains, in both systems are equal (2B=de), and the permeability coefficient in plane strain is (II.67): hPD hAx hAx hsAx 2 k k k n 3 3 ln ln(s) s k 4                     (II.67) c) 3rd Hypothesis: Combination of the two previous hypotheses - In this case a given spacing between drains is allowed for the plane strain and then the corresponding coefficient of permeability is calculated so that the equality of the systems is ensured (II.68). 2 hPD hAx 2 ehAx hsAx B 2 k k rkn 3 3 ln ln(s) s k 4                    (II.68) This last hypothesis is a more complex solution, nevertheless it guarantees greater equality between the two systems. In 1997, Indraratna & Redana, incorporated the smear zone geometrically in the plane strain (Figure II.9), as well as also in the analytical solution. Figure II.9 – Conversion of the radial flow in a vertical drain for a wall of vertical drains, with geometric consideration of a disturbed zone, in both systems. Parallel drain wall (2D) S*=2B b S b w B d e d s dw Radial flow (3D) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 24 - The base assumptions are kept, however the PD parameter will be defined by the following equation (Indraratna & Redana, 1997):   * * * 2 hPD PD d hsPD k α β θ L z z k            (II.69) with: 3 2 * s s s 3 2 4b 2b 2b 2 α 3 3B B B     (II.70) 3 2 23 2 * s w s s w s w w 3 3 3 2 2 2 b b b b 2b b 2b b β 3B 3B B B B B        (II.71) 2 2 * hPD w hPD 2 wPD wPD 2k 2b k θBq B q   (II.72) moreover, khsPD is the coefficient of permeability in the disturbed zone, in plane strain conditions. Considering the PD factor, defined by Indraratna & Redana (equation II.69), in equation II.56, then the general analytical solution to the problem, according to these researchers, will be:     * * * 2 hPD d 2 hsPD hPD hAx 2 e 2 e shAx hAx d s hsAx w wAx k α β θ L z z k B k  k  r d d k k3 ln ln π L z z d k d 4 q                                              (II.73) which is equivalent to:       * * * 2 hPD d 2 hsPD hPD hAx 2 e hAx hAx 2 hsAx wAx w k α β θ L z z k B k  k  r k kn 3 ln ln s 4z H z s k 4 k d                                         (II.74) on the other hand, even to:       2 2 2 * * * 2 hAx hAx hPD hPD e d hAx d hsAx wAx hsPD k k kn 3 k r ln ln s π L z z k B α β θ L z z s k 4 q k                                           (II.75) If the discharge capacity is high, this parameter can be disregarded and the solution is simplified to:   * * hPD 2 hsPD hPD hAx 2 e hAx hsAx k α β k B k  k  r kn 3 ln ln s s k 4                                     (II.76) On the other hand, ignoring the smear effect but including the discharge capacity, the solutions turns out: CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 25 -       3 2 * 2 w w w d 3 2 2 hPD hAx 2 hAx e 2 wAx w 4b 2b 2b2 θ L z z 3 3B B B B k  k  k3 r ln n 4z H z 4 k d                              (II.77) as khsAx=khAx, khsPD=khPD, bs=bw, rs=rw, which implies that s=1 and ln(s)=0. When none of the previous effects are taken into account, then estimations of the plane strain permeability coefficient is considerably simplified (II.78).     2 2 hAx hPD hAx hPD 2 2 e e 2 2k B B 3 k  k  k  3 r r 3 ln n 0,75 ln n 4                            (II.78) II.5. ANALYTICAL SOLUTION - COMBINED DRAINAGE When vertical drains are used, one must consider the vertical drainage direction in addition to the horizontal drainage direction. The simultaneous occurrence of the two drainages is called combined flow. This phenomenon was studied by Carrillo (1942), whom applied the separation of variables method. Therefore, the average degree of consolidation of a soil layer will be given by: v r (1 U) (1 U )(1 U )     (II.79) The contribution of vertical drainage in this type of solution is significant for compressible soil layers with thicknesses of less than 10 metres (Vertematti et al., 2004; Vairinhos, 2013). II.6. VISCOSITY OF FINE SOILS The viscosity of fine soils (creep) is a factor, which must be considered in the analysis of consolidation over time, and the corresponding strains are designated as secondary consolidation settlements. Generally, it is assumed that secondary consolidation settlement may correspond to a percentage of the primary consolidation settlement. Therefore, soil that has high values for hydrodynamic consolidation will have significant settlements associated to creep, hence, it is extraordinarily important to consider this aspect in the consolidation analyses. This phenomenon may be observed in oedometer tests, as long as the evolution of the settlements during the 24 hours, in which a given increase in stress is applied, is carefully measured. Given the small thickness of the samples, it is usual that almost all of the primary consolidation occurs in the first hours, even in soil with very low permeability. Therefore, during the remaining time, the settlements correspond to phenomenon of consolidation under creep effect, even if one does not have this perception. In these tests, the relation of the strain with time only becomes obvious when the primary consolidation ends (Figure II.10); however, this does not mean that the phenomenon has not initiated before the conclusion of the primary consolidation (Bjerrum, 1967). CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 26 - According to Mesri (1973), secondary consolidation settlement ssc may be estimated based on the expression: sc 0 c H t' s C log 1 e t           (II.80) where: tC - is the time that corresponds to the end of the primary consolidation; t’- is the time that corresponds to real time of the secondary consolidation; C- is the secondary compressibility index, obtained based on the oedometer test. The C parameter represents the slope of the curve on the graph (e; log t) for a given load increment, once it has been considered that the “total” dissipation of the excess pore pressure has occurred (primary consolidation), and given by:       E F E F F F E E e e e e e C t log t log t log t log t          (II.81) The magnitude of this settlement will depend on C and also on the ratio between t’/ tC. Figure II.10 – Definition of instantaneous settlement, primary settlement and secondary settlement, for a constant load (Bjerrum,1967) Mesri & Godlewski (1977) observed that, for any loading stage, the ratio between the secondary compressibility index (Cα) and compressibility index (Cc), as well as between secondary compressibility (Cα) and recompressibility (Cr), were constant. According to them, the magnitude and the soil behaviour in the secondary consolidation phase are directly related to the magnitude of Cc and Cr, and with the ratio between the final effective vertical stress and the effective preconsolidation stress. Therefore, Cα remains constant, decreases or increases with time, to a given CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 27 - final effective vertical stress value (’f), in accordance with its positioning on the graph (e; log  ’), which means that there is a dependency or a ratio between these parameters. The positioning of the final effective vertical stress, in relation to the effective preconsolidation stress (’p), is crucial and determines this evolution. In consequence, the three scenarios can occur: a) When ’f is located on the recompression curve (’f /’p<0,7), the time of the primary consolidation (tc) is small, as it is Cr, and the secondary settlements will not be significant, being Cα=Cr; b) When ’f is close to the effective preconsolidation stress ’p, the time of the primary consolidation may also be small, but in this case there is a transition from the recompressibility curve to the primary compression curve (passage from Cr to Cc), and so there are two secondary compressibility indexes, with Cα1=Cr and Cα2=Cc. As a result the settlements for secondary compression become significant (Mesri et al., 1997); c) When ’f is on the primary compression curve, the secondary consolidation settlements will be very significant, since Cα will also be very high since it only depends on the value of the compressibility index Cc, or in other words Cα=Cc. The ratios between the secondary compressibility index and the compressibility or recompressibility indexes, for geotechnical materials, are given in table II.1. Table II.1 – Parameter for several types of soil (Terzaghi et al., 1996) Type of soil  = Ca/Cr = Ca/Cc Gravels and sand 0.02± 0.01 Schist and mudstone 0.03± 0.01 Inorganic clays and silts 0.04± 0.01 Organic clays and silts 0.05± 0.01 Peat 0.06± 0.01 The compressibility concept described above is, for simplification purposes, denominated by ratio Cα versus Cc or Cr, as these ratios are constant, both in the compression phase and in the recompression phase. In 1992, Fox et al. also analysed this concept, applying it to peat. Special attention must be given to secondary settlements in the analysis of dams and dikes founded in this type of soft soil, and also in foundations. Even in overconsolidated soils, once the final effective vertical stress exceeds the preconsolidation vertical stress, then the situation described above occurs. The primary consolidation settlement value is small, but the secondary consolidation settlement may be high (Soriano, 2005; PLAXIS 2D-V11.02, 2012). Such cases are very dangerous as there is no significant primary consolidation settlement before, which might indicate the settlement, which follows (PLAXIS 2D-V11.02, 2012). Buisman (1936) was the first researcher to propose a constitutive law, taking into account the creep phenomenon of soft clay soils, after observing that the settlements of compressible soils were not fully explained by the classic consolidation theory. This law was developed for problems with CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS II. CONSOLIDATION UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 28 - unidirectional flow, such as the Terzaghi’s theory, being continued by Bjerrum (1967), Garlanger (1972), Mesri & Godlewski (1977) and Leroueil (1977) some decades later. Mesri & Godlewshi, in 1977, mentioned that there is no logical reason to separate the consolidation phenomena into the mechanisms of volume variation, due to the change in the effective vertical stress, from the volume variations at constant load over time. According to Hawlader et al. (2002), primary consolidation should be viewed as the load phase during which the hydrodynamic effect overlaps the creep effect. In a second phase, consolidation due to the hydrodynamic effect decreases substantially, and settlements are due to the creep effect with time. Hansbo (1997b) also recommends that the creep effect be considered in the primary consolidation phase. In 1981, Pilot mentioned that the creep effect in clayed soil was often ignored in the analyses of consolidation incorporating vertical drains, in spite of the fact that it was a phenomenon already well known. In 2002, Hawlader et al. led a study on soil treated with prefabricated vertical drains, considering both the creep and the smear effect, concluding that the influence of creep in the consolidation of clayed soil has greater significance when the drains are quite far apart, and is much lower when the drains are relatively close. However, during this study, it was observed that the consideration of the creep effect is significant even if the drains are relatively close. Constitutive models, which include creep in the primary consolidation, were developed by (Kabbaj et al., 1986; Murakani, 1988; Yin & Graham, 1989; Kutter & Sathialingam, 1992). Some were formulated based on time and others on the strain rate. The PLAXIS V02.2011 software also provides a material model that considers the viscosity or creep of the soft soil, which will be presented in the following chapter. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 35 - Figure III.6 – Representation of the variation of the yield stress with plastic strain When a material as associated a yield function of the hardening type, then there is an increase in the zone where the response is elastic; the materials with a softening yield function presents the opposite behaviour. In both situations, this kind of yield functions implies a modification in the material behaviour, increasing or reducing the stress state associated to the yielding. III.2.2.3. FLOW RULE The flow rule for the plastic material gives the ratios of the plastic strain increments when the material is yielding in a particular stress state, as well as theirs direction or orientation for each stage. Therefore, the calculation of plastic strain increments depends on this rule, which is given by the mathematical equation III.13.   p Q ,        , (III.13) where p represents the plastic strain increments, Q is the plastic potential function and  the parameter, which controls the size or dimension of the surface of the plastic potential. If the previous equation is plotted in the principal stress space, it can be seen that the vector, which defines the direction of these increments in plastic strain, is normal to the plastic potential surface (Brito & Gunn, 1987). For many materials, it is observed that the yield function and the plastic potential function are equal, being the direction of the plastic strain increments also normal to the yield function. When this happens then it is often said that the “normality condition” holds, and the flow rule is called as an “associated” (Figure III.7a). In this case, the expression of the flow rule becomes:   p F ,        , (III.14) If the yield function and the plastic potential function are not the same, then the flow rule is called as a “non-associated” flow rule, and the plastic strains increments are just normal to the plastic potential function (Figure III.7b). y   p Hardening Perfect = Ideal Softening CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 36 - 7.a) 7.b) Figure III.7a) – Representation of an “associated” flow rule and the “normality condition”; III.7b) – Representation of a “non-associated” flow rule and the “normality condition”. III.2.2.4. EQUILIBRIUM: STABILITY AND INSTABILITY It is extremely important not to confuse Drucker’s principle of normality with Drucker’s principle of stability (1954, 1966) which says that, for a material subjected to a given state of stress, imposed by an external agent, the work produced by this agent, in the corresponding displacement (d), must be positive or equal to zero. However, it must be noted that the principle of stability implies the existence of normality, even though the opposite is no longer true (Wood, 1990). Taking, for example, a saturated soil sample, subject to a load-unload cycle, and observing the relation between the load increments and the respective displacements for different conditions, the following situations may occur: a) The OP path described in figure III.8a represents the slow application of a load increment i X  , which leads to a slow compression, thereby allowing the absence of excessive pore pressure of the soil sample. The OP path represents, in turn, the slow removal of the same load, and the occurrence of a slow expansion of the soil is observed. It is clear that in P, the external agent slowly transferred a quantity of work equal to i 1 X d 2   to the system, and that at the end of the OPO cycle this work was fully recovered; b) In contrast, the OQ path indicated in figure III.8b represents the fast application of a load increment i X  , thereby an excess of pore pressure (u) is registered. This excess gradually dissipates and when it reaches R, it is null. During the QR transition, the work transferred by the system is i X d   , with half being dissipated in the system by trying to accelerate the outflow of water from the soil (OQR area) and the other half is stored in the sample (ORS area). The RS path represents a fast removal of the load increment. In this phase a negative gradient pore pressure is observed, which quickly sucks the remaining water from the sample and, at the end of the cycle, the work which was temporarily stored has been completely dissipated; 2; 2 Yield surface F(,) = 0 Stress state  3;  3  1;  1  ij p Plastic Potential surface Q(,) = 0  2;  2 Yield surface and Plastic Potential Q(,) = F(,) = 0 Stress state  3;  3  1;  1  ij p CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 37 - c) The most common case is represented in figure III.8c, in which permanent strains occur. In this case, the total displacement is separated in an elastic component (de), which corresponds to the part that is recovered after the removal of the load, and in a plastic component (dp), which corresponds to the permanent displacement. If the soil presents a rigid behaviour, then de=dp=0 and no change occurs. If the soil presents an elastic behaviour, then dp=0 and all the displacements are recoverable without the occurrence of work transference to the system. If the soil does not completely recover from the displacements suffered at the start, or in other words dp>0 (in the case of material with plastic behaviour), then the quantity of the work transferred to the system is given by the OUT area (Schofield & Wroth, 1968). 8a) 8b) 8c) Figure III.8 - Work carried out during the load-unload cycle for saturated soil (Schofield & Wroth, 1968) In any of these three hypotheses, the system always verified the stability criterion, or in other words, Drucker’s stability principle (1954). This is mathematically represented by the expression: p i i X d 0     (III.15) in which dip is the increment in plastic strain, associated with the respective load increment i X  . Therefore, it is possible to define three distinct types of responses, respectively: a) Stable System - whenever the condition p i i X d 0     is satisfied; b) Unstable System - whenever the condition indicated in equation III.15 is not verified; c) Neutral System - whenever the condition p i i X d 0     is assured; III.2.3. ELASTOPLASTIC BEHAVIOUR The soil generally presents behaviours that correspond, in good measures, to the standards described by the elastoplastic models. To fully describe the stress-strain relations for an elastoplastic material, it is necessary to consider and specify the four conditions previously introduced. Now it is only necessary to group them in order to obtain the relation between the stresses and strains, or between their increments: a) The elastic behaviour, i.e., the elastic parameters which describe the elastic relation between stress and strain (Wood, 1990); d Load increment   Xi O  displacement P O R Q S O T Q U d e d p d d d  Xi  Xi CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 38 - b) The yield function and the definition of the stress state correspondent to the beginning of plastic strains ; c) The flow rule; d) The hardening law. The strains corresponding to this type of material behaviour, are separate into elastic plastic terms, meaning: e p kl kl kl      (III.16) Considering that the formulation is made from the point of view of the incremental strains, then the anterior expression is formulated as: e p kl kl kl     (III.17) with the elastic and plastic component of the strain given, respectively, by: e e kl ijkl ij C    (III.18) p p kl ijkl ij C    (III.19) The determination of the elastic strains increments are performed based on the elasticity theory, already described, and the strain corresponding to the plastic part is obtained in accordance with the expression III.13, which can be re-written as:   p kl ij Q ,       , (III.20) being necessary to known the hardening or softening parameter value. The determination of  is done for the yield condition, taking into consideration expression III.17 and III.20, and that the stress increment ij must always result in the condition of F()=0, as the stress state cannot be exterior to the yield surface. According to the procedure indicated by Naylor et al. (1981) and by Mateus da Silva (1996), the expression that allows one to know the value of the scalar with the strain increment kl, is: ij ij ij ij kl mn ij mn ij ij ij F 3K 2G F 6G 3 2G Q F Q F 6G                                                         (III.21) Now introducing the concept of mean effective stress or isotropic stress (p) and distortional or deviator stress (q), which represents the stress invariants and are given by the expressions: 1 2 3 p 3        (III.22) 1 3 q     (III.23) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 39 - and applying this same procedure, but in terms of the stress invariants increments p and q, it is demonstrate bellow, how to obtain the plastic stress-strain increments, dependent of the plastic potential and yield functions. The corresponding irreversible strains or plastic strains increments, will be designated by p p  and p q  . Assuming that the yield function is described by:   0 F p ,q,p 0    (III.24) where pʹ0 is the parameter that indicates the increase of the yield surface. If represented in a differential form, it becomes: 0 0 F F F p q p 0 p q p                 (III.25) If it is assumed that the size of the yield surface, i.e., the increase of pʹ0, is related to both plastic strain increments, in accordance with the hardening law: p p 0 0 0 p q p p p q p p p             (III.26) that the plastic potential function is defined by:   Q p ,q, 0    (III.27) and that the flow rule may be represented as: p p Q p       and p q Q q      (III.28) and (III.29) in which  is the parameter that expresses the hardening or softening of the soil, given by the expression: 0 0 p p 0 p q F F p q p q p p F Q Q p p q                                 (III.30) then, once the parameter  has been found, the determination of the partial plastic strain increments and consequently, the total plastic strain, is obtained by the sum of equations (III.28) and (III.29). Therefore, mathematically the relation between the mean effective stress (p) and deviator stress (q) increments, with the plastic strain increments, can be described, in a matrix, in the following manner: p p p q0 0 p pp o0 p q F Q F Q p p p q p 1 F Q F Q q p p F Q Q p q q q p p q                                                                              (III.31) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 40 - In the case of an associated flow rule, then the plastic potential function Q() coincides with the yield function F(), and so the determination of the plastic multiplier becomes simpler and the matrix is symmetrical. III.2.4. CRITICAL STATE MODEL The unification of the deformation and strength concepts is due largely to the work undertaken by the University of Cambridge, and from which the formulation of a model that incorporated the concept of critical states was derived. The model was initially used to predict the response of remoulded saturated clay, when subjected to triaxial compression tests, since this one of the most widely used test for obtaining the soil strength parameters. As such, this model incorporates the principal soils parameters used to describe the soil resistance of these types of soils, but it can also be applied to other types of materials. This model was originally developed under the leadership of Roscoe, between the 50’s and the 60’s, and was presented in 1958 by Roscoe et al., and later in 1968 by Roscoe & Burland. However, only in 1968 did Schofield & Wroth make the theory accessible with the launching of their book; even today this book is a reference for the understanding of the model. The model unifies different concepts accepted in soil mechanics that are usually seen in a separated way, this being one of the main innovations introduced by the method, which combines the concepts of failure criterion, load and plastic potential, volume variation as well as its relation with effective stresses (Maranha das Neves & Guedes de Melo, 1975). The presentation of this concept is important, since the Soft Soil and Soft Soil Creep models of PLAXIS, used in the numerical simulations, are somewhat based on the Modified Cam Clay model. Therefore, there will be a first approach to the critical states model, followed by a brief reference to the Original Cam-Clay model and lastly the Modified Cam-Clay. The description of soil behaviour in the critical state model is done in a three-dimensional space, in which two axes represents the stress invariants, isotropic (p) and distortional (q) stresses, and the other represents the variations in the specific volume (v ), i.e. the overall volume occupied by the unit volume of solids, or the void ratio. The stress invariants were already defined, but the specific volume still needs to be specified (v). This last parameter will be given by: v 1 e   (III.32) The critical state concept postulates the existence of an ideal material, which behaves as a frictional fluid at a specific constant volume, when p’ and q satisfies the equations: q M p    (III.33) e v lnp v log p           (III.34) where M,  and  represents basic soil properties. In fact, when the planes defined by these two equations intersect, it results in a line that represents the failure at the triaxial tests, called as the critical state line (CSL) (Figure III.9). Therefore, regardless of the stress trajectory, once the critical CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 41 - state line is reached, the material will suffers distorcional strains at p, q and v constants in drained conditions, or in the case of undrained conditions to pore pressure (u). Mathematically, these conditions can be described in the following manner: q v 0    q q 0    and q p 0     (III.35) (III.36) and (III.37) being the volumetric strain and the distortional strain, respectively, equal to: v 1 2 3     (III.38)   q 1 3 2 3     (III.39) Figure III.9 – Spatial representation of the critical state line (Mateus da Silva, 1996) However, and in order to better understand the concept associated with critical states, a brief description of the stresses trajectories or paths for the triaxial compression tests, both drained and undrained, will be given. When a triaxial compression test is conducted, a stress-strain relationship is obtained. Based on these curves, it is possible to determine the parameters of the shear strength, using Mohr circle representation. With three different test results, associated to three different stress confinement of the specimen, the Mohr-Coulomb failure criterion or yield surface can be estimated. The correspondent procedure is plotted in figure III.10 for normally consolidated soil, and in figure III.11 for over-consolidated soil. The stresses used to obtain the Mohr-Coulomb envelope correspond to the yielding stress state in the triaxial tests. When in the presence of overconsolidated soil, it presents a maximum stress value or a peak value, then reducing to a residual value, which is designated as the critical state baseline or perfect plasticity. Critical state line (CSL) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 42 - Figure III.10 – Shear strength parameters and Mohr-Coulomb envelope of normally consolidated soils Figure III.11 – Shear strength parameters and Mohr-Coulomb envelope of overconsolidated soils The critical state can also be represented in a (p  ; q) plot as a straight line q=Mp, being the yielding stress state obtained by the interception of this same line with the one which defines the stress path of the triaxial test, both drained or undrained. Associated to these yielding stress states there is also a void ratio, designated as critical void ratio (Biarez & Hicher, 1994). The M parameter is determined based on equation III.40. pp pp 6 sin M3 sin        (III.40) However, the representation of the stress paths in the (p  -q) plot is not unique, depending on the type of triaxial test performed (Da Silva, 1999). Therefore:  For the isotropic consolidation phase, in which 1=2=3 and 1=2=3, then p>0 and q=0, and the corresponding stress trajectory is a horizontal line which coincides with the x-axis (Figure III.12). Figure III.12- Representation of the stress path in the isotropic consolidation phase of the triaxial test, in the (p’; q) plot (Da Silva, 1999) '  1  Perfect Plasticity or Critical State Normally consolidated soil q/2 'pp or  'cri q/2 '1 '3  1   ' Perfect Plasticity or Critical State Overconsolidated soil q/2  'pp or  'cri 'peak q /2 q q = Mp' Isotropic consolidation p' p'0 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 43 -  For the conventional triaxial test, where 2=3 and 2=3, the slope of the stress path is of 1/3, starting at p0; so, the trajectory equation will be p=p0 +q/3 (Figure III.13). Figure III.13 - Representation of the stress path in the compression phase of the triaxial test, in the (p’; q) plot (Da Silva, 1999)  For the triaxial test with 1+2+3=cte, which means that 1+2+3=0, or in other words p=cte and q>0, the stress path is described by a vertical line which starts at p0 (Figure III.14). Figure III.14 - Representation of the stress path of the triaxial test with 1+2+3=cte, in the (p’; q) plot (Da Silva, 1999)  In the case of undrained tests, the trajectory of the total stresses will be equal to that of the drained test; however, in order to obtain the mean effective stress path, the pore pressure must be subtracted to the total stress, with the distortional stress remaining constant, or in other words (p=p–u and q=q) (Figure III.13) and (Figure III.14). In addition, it is also necessary to observe the rate of the specific volume v in the (p  ; v) plot, in accordance with the Cambridge notation (Britto & Gunn, 1987). Following the same procedure as mentioned for the (p  ; q) plot, current triaxial tests (2=3) performed on remoulded clay, begins with an isotropic stress state (1=2=3) or (p>0; q=0), which is characterised in the (e; ln p) plot by the isotropic consolidation line (ISO) and whose slope is given by . This course is similar to the virgin compression line (VCL) obtained in the oedometer test in the (e; log p) plot, being the slope characterised by the compressibility index Cc. The same type of similarity in the progress of the curves or lines is observed when there is a release of the stress on the soil followed by a new load. In this case we observe that the expansibility or recompressibility index Cs obtained in the oedometer test is replaced by the  parameter (Figure III.15). p' p'0 q q = Mp' 1 3 Drained test Undrained test uf q q = Mp' Drained test Undrained test uf p' p'0 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 44 - Figure III.15 – Representation of the variation of the void ratio with mean effective stress in the (e; p’), (e; ln p’) and (e; log p’) plots (Da Silva, 1999) The relation between the parameters  and Cc, as well as between  and Cs, are given by: c c C 1C ln10 2,3    (III.41) s s C 1C ln10 2,3    (III.42) The rate of the specific volume with the mean effective stress, in isotropic consolidation phase, for load and unload-reload cycles, can be described, respectively, by the equations III.43 and III.44. 0 p v v ln p            (III.43) 0 p v v ln p            (III.44) The value of vis usually substituted by the parameter . Conventionally, this parameter is attributed with the value of the specific volume, which corresponds to the mean effective stress equal to the unit v(p’=1), hence, the value of p0 is omitted, and then equation III.43 can be re-write as: v lnp    (III.45) Once the isotropic consolidation of the soil has been concluded, a load increment is applied to the sample, and the soil yields once it reaches the yield surface, correspondent to the critical state line (CSL) or “perfect plasticity” (PP). In the (e; ln p) plot the critical state line is parallel to the virgin compression line. It is important to indicate that during the undrained tests, the soil sample does not suffer volumetric variation, which is why the specific volume value of perfect plasticity is equal to the specific volume after the isotropic consolidation phase. e n Log p' e n p' Elastic limit p'p Elastic “OC” Plastic “NC” C s C c e n Ln p'    p'p VCL ISO p'p CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 51 -  Mohr-Coulomb model (MC) – This is the well-known linear elastic perfectly plastic model, with non-associated flow rule. This model is recommended as a first approximation of soil the behaviour, and so is more suitable for a preliminary analysis, regardless the problem in question. Average constant soil stiffness is adopted, and, as such, the computations tend to be much quicker in comparison to others. In addition to this, a first order of magnitude, associated to the deformations expected, can be obtained.  Hardening Soil model (HS) - The current HS model is an advanced model applicable to various type of soils, ranging from stiff soils such as gravels and sand, to very soft soil, such as clay and silt. It is an elastoplastic model of the hyperbolic type, formulated in the framework of shear hardening plasticity. When the soil is under primary compression conditions, the model calculates irreversible plastic strains and hardening compression is considered. Therefore there is an increment in the stiffness of the medium according to the stress state. The HS model, although suitable for any type of soil, does not consider the creep effect.  Hardening Soil with small-strain stiffness model (HSsmall) - This model results from a modification to the HS model, incorporating dependency between stiffness modulus with strains, thereby allowing the simulation of soil reactions for different states of strain (from low strain levels (<10-5), usually associated with vibrations, to large strain levels (>10-3), typically imposed by engineering works). When applied to dynamic situations, it can also model the hysteresis of the material damping.  Soft Soil model (SS) – This is a Cam-Clay type model, suitable to simulate the behaviour of normally consolidated clay soil or soils with degrees of consolidation ranging one (OCR=1), in primary compression situations. Even though the Hardening Soil model has surpassed this one, it continues to be used quite often, as it is a reliable model, one that is easy to understand and its entry parameters are obtained based on well-known tests. For all these reasons, PLAXIS users are comfortable using the SS model, for soils with the previously mentioned characteristics.  Modified Cam-Clay model (MCC) - Based on the Cam-Clay critical states model which is well known, this model arose from the original, with the objective of overcoming some limitations and insufficiencies which existed in the base model. The model assumes a logarithmic relationship between volumetric strains and the mean effective stress. This model, according to the PLAXIS manual, sometimes has convergence problems in some stress trajectories, and so it is not recommended for some practical applications.  Soft Soil Creep model (SSC) - This is a model formulated on the framework of soil viscoplasticity. It allows the simulation of the behaviour of compressible soil, such as very soft to soft soils (clay soil and peats) or normally consolidated soils, accounting for primary and secondary soil consolidation or compression, being this last one due to creep phenomena. This model is relatively recent and was specially developed to analyse the foundation settlement of very compressible soils, embankments and others. However, it should be noted that, for analyses involving unloading, or the removal of loads, such as tunnels or large excavations, it is preferable to use the MC model. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 52 - All these models are elastoplastic. The interesting particularity of some of these elastoplastic models is the consideration of the dilatancy angle () as one of the soil behaviour parameters. This parameter is very important in sandy soil or medium to dense soils, as well as on overconsolidated clayed soils, since it allows the description of the dilatancy phenomenon and therefore the progress of the void ratio with the strain more realistic. Regarding the initial stress state of the soil, which plays a fundamental role in the study of geotechnical problems, this one is calculated based on the consideration of a suitable coefficient of earth pressure at rest (K0). Despite the attempts to improve, in each version, both the code associated with the computational methods of the finite elements, as well as the material models which simulates the soil behaviour, there are still some limitations. Although some of them are of analytical nature, others are related to models themselves. Thus, for each of the models mentioned before, and in accordance with the PLAXIS 2D-V11.02 (2012) manual, the limitations associated to the models are now described:  Mohr-Coulomb model (MC) – This model includes only a limited number of features that soil behaviour shows in reality. Although the increase of stiffness with depth can be taking into account, the model does not incorporate either the stress-dependency, or the stress-path dependency of stiffness, or the anisotropy of the soil. In general, the Mohr-Coulomb failure criterion describes well the behaviour in drained conditions, considering the effective stress parameters. However, in undrained conditions, and since the effective stress path followed in the model may not be realistic, the resulting shear stress may be incorrect. This situation is worse in soft soil such as normally consolidated clay, as well as very dense, stiff, or highly overconsolidated soils.  Hardening Soil model (HS) - This model does not account the softening of the soil due to the dilatancy phenomenon. It is an isotropic hardening model that does not consider the hysteretic and cyclic loading or anisotropy behaviour. Its greatest disadvantage is that the calculation time is excessive as the stiffness matrix is formed at each step of the calculation.  Hardening Soil with small-strain stiffness model (HSsmall) – This model incorporates the loading history of the soil and a strain-dependent stiffness. It can be used to simulate cyclic loads, but is not suitable for cyclic problems in which softening plays an important role. As the anterior model, this model does not account the softening of the soil due to the dilatancy phenomenon. The HSsmall model presents a calculation time even greater than that of the HS model, thus only advisable for special cases.  Soft Soil Creep model (SSC) - The limitations of the HS model are also applied in this model. Furthermore the range of elastic soil behaviour is over-estimated in this model, especially when in the presence of excavation problems, including tunnelling.  Soft Soil model (SS) - Not recommended for problems with large excavations, and should be limited to problems which involve primary consolidation settlements.  Modified Cam-Clay model (MCC) - Due to the characteristics of this model, the shear stresses for overconsolidated soils are very high and unrealistic, if the stress path crosses the critical state line. Furthermore, the model may give softening behaviour for such stress trajectories. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 53 - Still, in accordance to the PLAXIS manual, both the Hardening Soil and the Soft Soil models estimate relatively well, the primary consolidation settlements, however, they do not consider the creep effect and so are not suitable for soils with high compressibility. III.3.1. SOFT SOIL MODEL Soft Soil is a behaviour model adequate for soil which has considerable compressibility values, such as clays, clayey silts and peats. According to the PLAXIS 2D-V11.02 (2012) manual, some features of this model are: a) Stress dependent stiffness (compression behaviour of the logarithmic type); b) Distinction between the primary loading and unloading-reloading; c) Memory of the pre-consolidation stress; d) Failure behaviour according to the Mohr-Coulomb criterion. III.3.1.1. STRESS AND STRAIN ISOTROPIC STATE (’1=’2=’3 ) This model assumes the existence of a logarithmic relation between the volumetric strain (v) and the mean effective stress (p’), according to expression III.63. f v v0 p p * ln p                , with f p p p    (III.63) During isotropic unloading-reloading, different path is followed, and the elastic volumetric strains can be formulated as: e e v v0 0 p * ln p             , with p p p    (III.64) These equations differ from those defined in the Cam-Clay models since the soil compressibility of is defined in terms of the volumetric strain and not the void ratio (Figure III.20). Therefore, * and * will be, respectively, the modified compressibility index and the modified recompressibility or expansibility index. The relationship between these modified parameters to those defined by Burland (1967), can be expressed as:   * 1 e     and   * 1 e     (III.65) and (III.66) However, the ratio between */*, is the same as the ratio /. The elastic behaviour is simulated by Hooke’s model and equation III.64 implies a linear dependency between the stress and the volumetric strain module (ur), i.e.:   ur ur ur E p' * 3 1 2       (III.67) in which “ur” means unload-reload. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 54 - Figure III.20 – Logarithmic relation between volumetric strain and mean effective stress for the SS model (PLAXIS 2D-V11.02, 2012) It should be emphasised that the soil parameters are defined in terms of effective stress instead of total stresses (undrained), therefore neither the strain modulus Eur, nor the volumetric strain modulus ur, are considered. In contrast, and to define the elastic component of the soil, ur and * parameters are required. The pre-consolidation stress p’p defines the stress state value from which the irreversible volumetric strains start to occur (plastic state). Thus, in an unload-reload phase and while the pre-consolidation stress is not reached, the strains will only be elastic, but as soon as they exceed this stress state, the soil follows the virgin consolidation line and so, for each load increase the pre-consolidation stress will increase, leading to an increment in the plastic strains. III.3.1.2. YIELD FUNCTION FOR THE COMPRESSION TRIAXIAL STATE (’2=’3 ) The SS model is able to simulate the behaviour of the soil for several stress states; however, and only to obtain a better understanding of the model, it will presented the case of the triaxial compression test, where ’2=’3. For this stress state, the yield function is defined as:   2 p 2 q F p' p' M p' c cotg               (III.68) being the pre-consolidation stress dependent of the plastic strain, and given by the equation: p v p p0 p' p' exp * *           (III.69) This yield function describes an ellipse in the (p’; q) plot, as shown in figure III.21. This ellipse, where M indicates the height of the ellipse, has the same format as the Modified Cam-Clay model, and is responsible for the relationship between the vertical stress and the horizontal stress in primary compression, meaning that the M parameter is related to the value of K0NC. This assumptions differs  v * Ln p' 1 LCV p'p 1  * CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 55 - slightly from the Cam-Clay model, since it is different from the original concept of the critical state line and does not only depend on the friction angle of the soil, but also of the soil cohesion; however, it ensures that the value assumed for K0NC is correct. Therefore, in the SS model, the failure is not necessarily related to the concept of the critical state model, being the failure criterion defined by the Mohr-Coulomb criterion, and by the parameters (c’ e ’). The slope of the failure line in the SS model is smaller than the slope of the critical state line or M-line. Although the failure line is fixed, the cap may increase in primary compression. Figure III.21 – Yield function for the Soft-Soil model in the (p’; q) plot (PLAXIS 2D-V11.02, 2012) The isotropic stress of pre-consolidation p’p, indicates the length of the ellipse along the p’ axis. During the load, there is an infinite number of ellipses, each one corresponding to a given value of p’p. In traction, i.e.: for p’<0, the ellipse extends to c.cotg(). In order to make sure that the hand side of the ellipse (i.e. the “cap”) is always located on the compressed side (p’>0), then a minimum value for c.cotg() is adopted. Hence there is a “threshold” ellipse as illustrated in figure III.21. In summary, this model considers a yield function which is defined by equation III.68, which describes the irreversible volumetric strains (plastic) in primary compression, and which delimits superiorly the yield zone (the top=cap). The Mohr-Coulomb failure criterion is adopted, being noted that this line is below the M-line in the (p’; q) plot (Figure III.21). The final contour of the yield or yield limit, in the elastic state, will then be given by the “cap” lines, being variable and increasing with primary compression, even though the failure line is fixed (Mohr-Coulomb). Stress paths within these limits only cause elastic strains, but as soon as the limits are crossed the soil presents both plastic and elastic strains. The final contour, described in the space of the principal stresses, is given in Figure III.22. III.3.1.3. SOFT SOIL MODEL PARAMETERS In addition to the initial void ratio (e0), the model needs another 5 basic parameters, namely: a) Modified compressibility index (*); b) Modified recompressibility index (*); c) Cohesion (c); d) Internal effective friction angle (’); e) Dilatancy angle . q p' p'p M “ Cap ” c.cotg  1 Mohr - Coulomb failure criterion p'p0 “Threshold” ellipse CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 56 - Figure III.22 – Representation of the yield surface of the Soft Soil model, in the principal stress space (PLAXIS 2D-V11.02, 2012) The first two parameters will define the soil stiffness, which and can be determined based on the isotropic compression test. The ratios between *and , as well as between * and  have already been indicated in equations III.65 and III.66. The ratio between */* will be equal to /, and usually range from 2.5 to 7, for most soils. Alternatively, the compressibility Cc and recompressibility or expansibility Cs indexes may also be used, if the one-dimensional compression test is performed (oedometer test). If the latter is considered, then the ratios between * and *, with the internationally standardised and most used parameters will be:   c c 0 0 C loge C * 1 e 2,3 1 e      and   s s 0 0 2C loge 2C * 1 e 2,3 1 e      (III.70) and (III.71) Regarding these parameters, the following comments are yet to be made: a) It is assumed that the void ratio used on the previous equations are constant, however, they change in the course of a compression test. Nevertheless this variation is considered very small and, so the initial void ratio, or the average void ratio after the test has been concluded, may be considered. The first situation is more commonly used; b) The ratio between the modified recompressibility index* and the recompressibility index Cr is not exact, as the ratio between the horizontal and vertical stress changes in the unloading of the one-dimensional test. By approximation, it is considered that the average stress state during the unloading will be an isotropic stress state, meaning that the horizontal and vertical stresses will then be the same. “ Cap ” Failure surface CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 57 - In addition to these, there are also advanced parameters, namely: a) The Unload-reload Poisson’s ratio (ur). In this model, ur is a purely elastic constant, in contrast to the pseudo-elastic behaviour used in linear elastic perfectly plastic models. Their value may vary between 0.1 and 0.2, and 0.15 is the value adopted by default. This parameter has little expression in normally consolidated soil, in compression. However, in the unloading phase, and since this value is very small, the reduction in the lateral stress will be less than the vertical stress applied in this phase, leading to an increase of the ratio between the horizontal and vertical stress; this phenomenon is associated with overconsolidated soils. For this reason, it is advisable to determine this value based on oedometer tests, in the unloading and reloading phase and not on the value of K0NC. b) The lateral earth pressure coefficient for the normally consolidated state (K0NC); c) The “M” parameter. This is automatically calculated by the programme after entering the value of the coefficient of lateral earth pressure for the normally consolidated state K0NC, depending also on the unload-reload Poisson’s ratio ur, and also on the ratio */*, even though the first parameter is dominant. According to Brinkgreve, in 1994, it will be given by:                 NC 2 NC 0 ur 0 2 NC NC NC 0 0 ur 0 ur * 1 k 1 2 1 1 k * M 3 * 1 2k 1 2k 1 2 1 k 1 *                               (III.72a) or more simply, by NC 0 M 3 2.8k   (III.72b) For these parameters, it is recommended to use the default values assumed in the PLAXIS manual. The Soft Soil parameters are introduced in effective stresses, even though undrained type analyses may also be performed. In this case, the programme transforms the data entered into undrained parameters and performs the calculations in total stress, but always considering the excess pore pressure. III.3.2. SOFT SOIL CREEP MODEL The Soft Soil Creep model is suitable to simulate the behaviour of soft to very soft soil, with a high degree of compressibility, taking in consideration the plastic strains due to creep, which depends on time, designated by secondary or secular consolidation. As already mentioned, secondary consolidation corresponds to the re-arranging of the soil particles at a constant pressure. This value may be negligible in stiff and dense soil, bust usually very high for soft to very soft soils. This last situation is generally associated with soil that also presents high settlements due to primary consolidation. The model’s features are the same of the SS model, including the consideration of the secondary compression (time dependent behaviour). CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 58 - This model is also an elastoplastic model, with un-associated flow law. One of the surfaces is defined by the Mohr-Coulomb rupture criterion and fixed in the stress space, and another is variable depending on the isotropic stress of pre-consolidation, which delimits superiorly the yield zone (the top=cap), as treated in the SS model. The representation in the main stress space is equal to that of figure III.22. The mathematical modelling of Soft Soil Creep is duly detailed and described in the PLAXIS manual, however, for a better understanding of the work carried out in this study, it will be presented. III.3.2.1. ONE-DIMENSIONAL CREEP Normally, on oedometer tests, one is focused on the soil response related to step loading, even thought natural loading processes tend to continue in nature. Buisman, in 1936, was probably the first to considerer such phenomenon, associated to the creep behaviour. According to Buisman’s formulation, the strain of soil under constant effective stress is given by: pc B c t C log t           , for t>tC (III.73) where: pc - is the strain at the end of the primary consolidation; t - is the time measured from the beginning of loading, and must be greater than tC; tC - is the time correspondent to the end of the primary consolidation; CB - is a constant of the soil, determined in the phase correspondent to creep behaviour, and it is given by: 1 2 1 B tot 2 s(t ) s(t ) t C log s t          (III.74) As PLAXIS does not follow the conventional methodology for soil mechanics in which the compressions are considered positive, then the equation III.73 must be re-written in the following manner: c pc B c t t' C log t            , for t’>0 (III.75) with t’= t-tC, and corresponding to time in which secondary compression occurs, also designated as the effective creep time. In 1972, Garlanger, based on the works of Bjerrum from 1967 related to secondary consolidation, proposed that this should be a function of the void ratio, as indicated: c c c t' e e C log              (III.76) where Crepresents the secondary compressibility index. This last is related to CB by:   B 0 C C 1 e   ,for t’>0 (III.77) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 59 - The differences between both formulations are modest, with strain  being substituted by the void ratio, and time tc by parameter C. It should be noted that C is not the time of the primary consolidation tc. Time C is not a parameter of the material but instead a factor which does not only depend on consolidation, but also on the geometry of the samples tested (Alves, 2011). If it is considered that C=tc then the equations III.75 and III.76 will be entirely identical. The exact value ofC is difficult to calculate; however, the difference between these last two parameters disappears as the time of creep t’ increases, it then being possible to consider equality between both. According to PLAXIS 2D-V11.02 (2012), the determination of tc and C will depend on the evolution of the response from the soil, and is distinct for each soil type. Another way to describe secondary consolidation was given by Butterfield, in 1979, using the logarithmic strain H, as it is showed in equation III.78. H H c pc c t' C ln              (III.78) The logarithmic strain is not currently used in terms of engineering, and was originally adopted by Hencky. This strain is defined by: H 0 0 V 1 e ln ln V 1 e                  (III.79) For small displacements, the logarithmic strains H are approximately equal to conventional strains , and thus it results: B 0 C C C (1 e )ln10 ln10     (III.80) Janbu, in 1969, has developed a graphical construction to evaluate the parameters C and c, having separated equation III.78 in terms of time and dropping the term (H) to simplify the notation (Figure III.23b). Both the traditional method represented in figure III.23a), and the Janbu method can be used to determine parameter C. Figure III.23 – Representation of consolidation in a standard oedometer test C 1 ln (t) tc c - C 1 -1/ tc c t’ t’=tc-t t Creep behaviour Consolidation a) b) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS III. MATERIAL MODELS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 60 - However, the Janbu method is more attractive, as it also allows the obtaining of parameter c, which corresponds to the interception of the straight creep line with the time axis (non-logarithmic). The linear curve of the graph, for t<tc, is due to the primary consolidation of the sample (Alves, 2011). According to the same author, primary consolidation strains are calculated based on the expression: pc H He Hcr f pc pc pc 0 p0 ' ' A ln B ln ' '                            (III.81) meaning that the elastic and viscoplastic strains are equal to: He f pc 0 ' A ln '            - elastic strains at the end of primary consolidation; pc Hcr pc p0 ' B ln '              - viscoplastic strains at the end of primary compression; where: ’f – final effective vertical stress; ’0 – initial effective vertical stress; ’p0 – initial effective vertical pre-consolidation stress; ’pc – effective vertical pre-consolidation stress at the end of primary consolidation. The total strain is obtained by adding secondary consolidation in equation III.81, resulting: pc HC f 0 p0 C ' t t' ' A ln B ln C ln ' ' t                                 (III.82) With the last term corresponding to the viscoplastic strains during the secondary consolidation. Parameters A, B and C are related to parameters Cc, Cr and C in the following manner: r 0 C A (1 e ) ln10  (III.83)   c r 0 C C B (1 e ) ln10   (III.84) 0 C C (1 e ) ln10   (III.85) The strains associated with the three terms of equation III.82 are plotted in figure III.24. Figure III.24 – Elastic and plastic strains, in primary and secondary consolidation, with effective vertical stress (PLAXIS 2D-V11.02, 2012) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 67 - of prefabricated vertical, appear on the market, designated as band-shaped drain. According to some specialists, the relative efficiency of the prefabricated drains and the sand drains, as long as the latter ones are properly executed, are equivalent, although some see advantages for those made of sand, in spite of the associated high cost and greater complexity in execution, while others have the opposite opinion. Today, polymer-based drains are the most commonly used and there is a great variety available. There are the band-shaped or wick type drains with a rectangular or circular cross-section, although this last one is less used. These drains can reach a depth of 20m to 30m. Regarding the historical evolution of design and consolidation analysis of compressible soil under the influence of these elements, we have included a very brief summary below since this theme was covered in chapter II. Thus, in 1935 Rendulic was the first researcher to analyse the influence of vertical drains in the consolidation of clay, based on the simplified one-dimensional consolidation Terzaghi’s theory in 1925, with the assumption that Darcy’s law was valid, and considering a sand drain with cylindrical geometry. Later, in 1944, Barron continues with this theme, and only in 1948 considers, for the first time, the effects of the soil disturbance near the drains, caused by their installation and known as the smear effect. At the same time he also regards the limitations of the drain’s discharge capacity (hydraulic resistance), on his solution. In 1974, Yoshikuni & Nakamodo presented a thorough solution for the consolidation problem with vertical drains considering both vertical and radial flows, and the hydraulic resistance of the drain, excluding the smear effect. In 1979, Hansbo introduced some approximations and modified the equations developed by Barron (1948) for prefabricated drains. The Hansbo solution is widespread, and it is the most used, due to its simplicity. The results obtained based on this solution do not differ much from the results obtained using more sophisticated solutions, determined using the finite element method (Onoue, 1988; Lo, 1991). In 1991, Cheung et al. presented an analytical solution to determine the average degree of consolidation considering the problem in plane strain, i.e. transposing the radial drainage solution (three-dimensional) to horizontal drainage (two-dimensional), and taking in consideration only the limitation of the drain discharge capacity. Hird et al., in 1992, and Indraratna & Redana, in 1997, presented, respectively, analytical solutions for the analysis of consolidation in plane strain, incorporating the smear effect and the hydraulic resistance of the drain. Although the Indraratna & Redana (1997) analytical solution is more complete, it is also more complex; however, its results do not present significant differences compared to the Hird et al. (1992) solution, as will be shown in the course of the present work. IV.2. TYPES OF DRAINS Vertical drains can be divided into two large groups, namely vertical circular drains and vertical wick or wick drains, commonly designated by band-shaped drains. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 68 - Within the group of circular drains, there are the traditional sand drains, Sandwicks drains, and Wrapped flexible drains. The prefabricated drains are included on the second group, whose shapes are usually rectangular and can be made from various materials (McGown & Hughes, 1981). IV.2.1. CIRCULAR DRAINS Traditional circular drains are made by opening a borehole on the ground and fill it with sand (Sand Drain), with minimum and maximum diameters of 15cm and 45cm. Usually the diameters varies between 18cm and 30cm, being noted that smaller diameters may lead to continuity problems in the sand drains, and therefore are not advisable. Larger diameters do not present additional advantages that justify the higher cost. Spacing between sand drains go from 2.5m to 5m, approximately (axes to axes) (Purushothama Raj, 1999). In 1948, Stanton mentioned the importance and care that should be taken in selecting the material used to fill the sand drains. Whenever possible, the filter conditions must be respected so that the water is drained without dragging fine particles into the drain, thereby avoiding clogging. When this situation is not possible, the use of clean, poorly graded sand is recommended. The vertical permeability of the sand drains should be around 10-2cm/s (Coelho, 1996). The Sandwick drains consist on a filter, set into a previously opened borehole, or placed in the ground using a mandrel, and then filled with sand (Hughes & Chalmers, 1972). These combine the technologies of prefabricated drains with those of sand drains, hence the reason for being called also as prefabricated sand drains (Figure IV.2). These drains, in addition to being more economical in terms of the quantity of the sand needed, may be installed in two ways: driven or using vibration, with a continuous auger or hollow continuous auger. They can support significant ground movements, due to the flexibility of the geotextile sleeve (Bell, 2004). Figure IV.2 – Sand drain, Sandwick or prefabricated vertical drain and band-shaped drain (image taken from http://www.barriergeocon.com/services drains.htm, in January of 2013) 4ʹʹx 1ʹʹ Level of water in the ground Sandwick drain Sand Sand drain Band - shaped drain PVD Sand PVD Diameter 10ʹʹ Diameter 4ʹʹ CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 69 - Wrapped flexible drains are made up of flexible, circular corrugated plastic tubes, which are wrapped by both a layer of natural fibre and a geotextile sleeve similar to those in prefabricated sand drains (Figure IV.3). These can be installed using several methods; however, they are usually driven with a circular cross-section mandrel. Figure IV.3 – Wrapped flexible drains (image taken from http://cleanculverts.com/ products.htm, in January 2013) IV.2.2. PREFABRICATED BAND-SHAPED VERTICAL DRAINS The group of wick drains or band-shaped prefabricated drains includes several types (Table IV.1). Since all of them as is origins from the cardboard drain developed by Kjellman, their sizes are practically similar. For greater longevity of the exterior filters, when made from paper, they are soaked in chemical products in order to prevent their deterioration. Table IV.1 – Characteristics supplied by the manufacturer of band-shaped prefabricated vertical drains, according to McGown & Hughes (1981) Type of drain Approximate sizes Nominal diameter Materials Permeability of the filter Width Thickness Core Filter [cm/s] Kjellman 100 mm 3.0 mm 66 mm card card 1 x 10 - 5 Mebra Paper filter Polypropylene Polyester 95 mm 95 mm 95 mm 3.2mm 3.4mm 3.4mm 63 mm 63 mm 63 mm polyethylene polyethylene polyethylene treated paper polypropylene polyester 6 x 10-7 2 x 10-2 2 x 10-2 Geodrain 95 mm 4.0 mm 63 mm polyethylene treated paper 6 x 10 - 7 Colbond Initial 300 mm 4.0 mm 194 mm non-woven polyester KF650 3 x 10-2 Alidrain 100 mm 7.0 mm 68 mm plastic cellulosic 3 x 10 - 4 Castle Drain Boards 94 mm 2.6 mm 62 mm polyolefin non - woven 2 x 10 - 2 Based on table IV.1, only the initial Colbond drain presents a larger width compared to the others; Nowadays, the Colbond drain width is similar to the other prefabricated vertical band-shaped drains for issues related to the drilling equipment. The prefabricated vertical drains (PVDs) are currently made up of a plastic core, wrapped in a geotextile filter, as it can be seen in the images (Figures IV.4a, IV.4b, IV.4c, IV.4d) or schemes (Figures IV.5a, IV.5b), and they presents a high discharge capacity which corresponds to 70% of the total area of the drain (Figures IV.6a, IV.6b). Exterior geotextile filter Sand Perforated PVC tube Strip of polymer CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 70 - 4a) 4b) 4c) 4d) Figure IV.4 – Prefabricated vertical band-shaped drains (images token from http://www.geosko.com/en/sub2/wick-drain.php, in January of 2013) Figure IV.5 – Schemes of prefabricated vertical band-shaped drains (images token from http://www.geosko.com/en/sub2/wick-drain.php, in January of 2013) 6a) 6b) Figure IV.6 – Drainage of prefabricated vertical band-shaped drains (images token from http://www.gobizkorea.com/blog/..., in January of 2013) Although the PVD drains present some advantages over the sand drains, they do not withstand large ground deformations. These two issues are covered in more detail in the following subchapters. IV.3. INSTALLATION METHODS The main recommendations for the use of prefabricated band-shaped drains concern their installation method. For sand drains and Sandwick drains, the main issue if related to the drilling method used to open the borehole, or the driven method used to install the tube or pipe into the ground. The installation of the drains has a huge influence on the future behaviour of the set, due to the disturbances caused around the drain (Calderon, 1992). According to Coelho (1996) and Purushothama Raj (1999), the boreholes made for the installation of the sand drains are relatively cheap, but the drain itself is expensive because they need sandy material with specific hydraulic and filter properties. Although the procedure associated with the execution of this type of drain is always the introduction of a steel tube into the ground, which is later removed as the sand is inserted from the top of the hole, the methods used to do this may be distinct, namely: 5a) 5 b) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 71 - a) Driving a tube closed at the end, with a mandrel, using a cover-hinge-valve, with vibration or percussion (dynamic), being the sand introduced into the borehole as the tube is extracted. This method has the inconvenience of causing a high disturbance to the adjacent soil, with a subsequent increase in pore pressure, and reduction in the permeability in the zone surrounding the drain, in addition to a decrease in the shear strength; b) Sinking of the steel tube on the soil due to its own proper weight. This is possible when water is injected under pressure (“lançage”) into the ground, causing the disaggregation of the soil. The soil comes out of the borehole with the flow of ascending water, and filling is made at the same time as the equipment is removed, already with a very low injection pressure. The inconvenience of this method is the high quantity of water needed to put it into practice, and the pollution of the work platform due to the disaggregated material returning to the surface. This method causes less disturbance to the adjacent soil; c) Rotary drilling with continuous flight auger, if the soil near the borehole can stand vertically during the extraction of the equipment and also the filling with sand, or with a hollow auger for the supply of sand, thereby allowing the self-sustaining of the borehole walls to be dispensed. This method has the same inconvenience regarding the disturbance of the soil in the zone around the borehole; d) Opening of the borehole only by washing. This method may form boreholes with an irregular format and doubts or uncertainties may arise regarding their diameter. Often the diameter is larger than the expected and its performance exceeds the estimated results. In addition to this, sand columns with a large diameter tend to increase the resistance of the soil and the settlements of the surrounding soil change. The costs associated with this process are high due to the large volumes of sand needed to fill the borehole, as well as the possibility of cavities forming during the filling stage, and their later collapse. It is a very slow method and so is not suitable when it involves large areas, or when the lengths of the drains are extensive. Regarding prefabricated vertical drains, these are installed using the static driving methods. The vibrating or dynamic driving of the soil must be avoided. The drive of the mandrel always causes displacements and disturbances to the surrounding and adjacent soil; however, there is less displacement with this methods and type of drains due to their reduced volume. Their installation is very simple and quick, with reduced costs, making it a more desirable technique than vertical sand drains (Rathmayer & Saari, 1983). According to Purushothama Raj (1999), these may be inserted up to a depth of 45m with the application of a static force of 200kN and with speeds that varies between 0.3 and 0.6m/sec. However, in 1987, Folque indicated that the drains might be installed up to a depth of 60m and with an installation speed of 1m/sec. In table IV.2, and according to McGown & Hughes (1981), the installation methods and processes of the vertical drains are summarised. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 72 - Table IV.2 – Methods currently used for the installation of vertical drains, according to McGown & Hughes (1981) Group description Specific methods Remarks Displacement methods  Dynamic driving;  Vibration;  Static driving;  Washing;  Combination of several. For any of the methods, a mandrel is used, with or without a disposal shoe at the end. Drilling or perforation methods  Rotary drill with or without a casing;  Rotary auger, including continuous standard and hollow flight augers;  Percussion (shell and auger) methods, with or without casing;  Hand auger. Washing methods  Rotary wash jet;  Washed open ended casing;  Weighted wash jet head on flexible hose. Methods in which sand is washed through the jet pipe are not suitable for prefabricated drains. IV.4. CONSTRUCTIVE SEQUENCE OF THE FOUNDATION TREATMENT WITH BAND-SHAPED DRAINS Since vertical drains are an efficient and very common method applied to highly compressible soils, then their bearing capacity is often insufficient to support the equipment used for their installation. In that case, and if this type of soil is located very close the surface, then the application of a reinforcement geotextile is recommended, before the application of the drainage sand blanket. This sand layer will allow the flow of the water to outside of the area under intervention (Figure IV.7). The global aim is to ensure the local stability of the foundation soil under surcharge, whether caused by the machinery during the installation phase of the drains, or by the overloads to which it will later be subjected (Lorenzo et al. 2004; Vertematti et al. 2004). Figure IV.7 – Sand drainage blanket sand placed above the intervened area (image taken from http://www.americanwick.com/products/..., in January of 2013) Embankment Saturated soil Hard plastic core Drainage blanket CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 73 - Another possibility is the construction of drainage mats forming strips or bands of sand, if the drains are concentrated in specific zones, as indicated in figure IV.8. Figure IV.8 – Layer of sand placed in a restricted area, forming a drainage mat or mattress above the area treated with drains (image taken from http://www.americanwick.com/products/..., in January of 2013) When the vertical band-shaped drains are distributed according to straight-line alignments, or are not far apart, as an alternative, they can be connected at the top to horizontal drains designated as Stripdrains (Figure IV.9a) and redirected to drainage trenches (Figure IV.9b). 9a) 9b) Figure IV.9 – Connection of the vertical band-shaped drains to the Stripdrain type drains 9a) image taken from http://www.ceteau.com/Products and Services /CeTeau Stripdrain, in January of 2013 9b) image taken from http://www.geotechnics.com/vertical.htm, in January of 2013 After the sand blanket is concluded, the next step corresponds to the drain installation. They should cross, preferably, the entire thickness of the compressible layer. In stratified foundations is advisable to cross at least the most compressible soil. In 1985, Runesson et al., performed studies on the efficiency of the drains when they do not completely cross the compressible layers. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 74 - The placing of the prefabricated drains in the ground is done through the driving of a steel mandrel, with a hollow cross-section, where the draining wick or band is placed. This should present a crosssectional as small as possible to minimize the disturbance to the surrounding soil, but at the same time, it must be able to withstand the driving forces. For prefabricated drains, driving mandrels were developed with specific formats and sizes (Figures IV.10a and IV.10b). 10a) 10b) Figure IV.10 - Steel mandrel with specific formats for driving PVD drains At the end of the driving equipment, the band-shaped drain is connect it to a steel plate, called an anchoring or fixing plate (Figures IV.11a, IV.11b and IV.11c), which will anchor the drain to the ground. This has the double role of preventing the entry of material into the equipment during the driving, and, maintaining the drain at the depth desired during withdraws of the mandrel (Figures IV.12a and IV.12b). 11a) 11b) 11c) Figure IV.11 – Steel hollow mandrel where the band is placed, and anchor steel plate 11a) image taken from http://www.fundesp.com.br/..., in January of 2013 11b) image taken from http://www.solotrat.com.br/..., in January of 2013 11c) image taken from http://www.geosyntheticsworld.com, in January of 2013 12a) 12b) Figure IV. 12 – Anchoring the band-shaped drain to the ground (images token from http://www.fundesp.com.br/..., in January of 2013) Band - shape d vertical drain Steel mandrel Steel anchor plate CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 75 - Once the total withdraws of the mandrel is complete, the band-shaped drain is cut about 15cm to 20cm above the ground surface (McGown & Hughes, 1981) (Figures IV.13a and IV.13b). 13a) 13b) Figure IV.13 – Cutting of the prefabricated band-shaped drain, after their complete installation (images token from http://www.ceteau.com/..., in January of 2013) The methodology of the prefabricated vertical band-shaped drains execution is outlined in Figure IV.14. Figure IV.14 – Execution methodology of the prefabricated vertical band-shaped drains (image taken from http://www.vibromenard.co.uk/techniques/vertical-drains, in January of 2013) Figures IV.15a and IV.15b present the machinery used for the execution of this type of drains. 15a) 15b) Figure IV.15 – Equipment used for the installation of prefabricated vertical band-shaped drains (image taken from http://www.fundesp.com.br/..., in January of 2013) Excavator Drain Mandrel Drain roll Anchor plate Drain cut Anchored drain Compressible soft soil to be consolidated Driving mandrel + plate Anchoring drain and extracting mandrel CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 76 - Generally, this equipment can cross sandy layers, superficial or deep, with some consistency (Coelho, 1996). Figure IV.16a shows a panoramic sky view of a zone treated with prefabricated vertical drains, and figure IV.16b is a 3D representative scheme of it. 16a) 16b) Figure IV.16 – Panoramic view of a zone treated with prefabricated vertical band-shaped drains 16a) image taken from http://www.layfieldenvironmental.com/pages/Products/..., in January of 2013 16b) image taken from http://www.admir-geo.com/Consolidation_Acceleration- Vertical_Drains, in January of 2013 As already mentioned, this technique is often combined with the construction of temporary preload embankments (Figure IV.17). The combination of the embankment surcharge with the drains, allows the anticipation of the consolidation, and a substantial part of the foundation settlements take place during the construction and the time just after it (Holtz, 1987). At the toe of the slope embankment is usual to place stabilising berms, especially when soil foundation stability is a major concern (Alves, 2011). Figure IV.17 – Preload embankment combined with vertical drains and stabilising berms (Alves, 2011) Preload embankment Soft Clay Drainage blanket Stabilising berm Drains PVD CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 83 - The ratio O50/D50 ensures that the seepage forces in the filter are relatively small. The maximum value of 12 attributed to the ratio O50/D50 is to prevent the infiltration of fine particles into the drain core (Bergado et al., 1993a). O950.0075mm is often specified for prefabricated drains (Chu et al., 2004).  Regarding the geotextile filter permeability, this value must always be greater than the soil permeability, namely kgeotextile 10ks (Holtz, 1987). Bergado et al. (1996c) suggest, for the Bangkok soft soil clays, the ratio of kgeotextile 2ks. b) Reduction of the drain core area available for flow, sometimes termed as the free volume, due to the infiltration of fine particles of soil to the drainage system; c) Deformation of the drain with depth, due to the increase of lateral earth pressure. In reality, it is the exterior geotextile sleeve that suffers the deformation, leading to the reduction of the drainage area in the drain (Figure IV.23); Figure IV.23 – Deformation of the geotextile filter due to lateral earth pressure d) Possible folding, bending and crimping of the drain, due to large consolidation settlements or slope instability. The soils where the PVD drains are applied generally exhibit high settlements, and consequently the drains are subjected to these deformations. If the thickness of the layer is relatively large and uniform, then the deformations of the drain will present a less pronounced curvature. On the other hand, on the presence of stratified foundation, some layers may present weaker strength characteristics and, subsequently drain deformation are higher in these zones, so the angles formed are more accentuated and may cause a considerable decrease in the drain discharge capacity (Bergado et al. 1996b). Possible deformation modes of band-shaped drains, as the result of ground movements, is presented in figures IV.24a, IV.24b for homogeneous soils, and in figures IV.25a, IV.25b, and IV.25c for stratified foundations, according to Lawrence & Koerner (1988). 24a) 24b) Figure IV.24 – Possible deformation modes of PVD drains, due to ground movements, for homogeneous soils (Lawrence & Koerner, 1988) Plastic core Filter - final position Filter - initial position s s Initial ground surface Ground surface after settlement CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 84 - 25a) 25b) 25c) Figure IV.25 – Possible deformation modes of PVD drains, due to ground movements, for stratified foundations (Lawrence & Koerner, 1988) Kremer (1983) concluded that the bending of the drains, associated to the foundation settlements may reduce the drainage capacity of a PVD drain to zero if the settlements reach 15% of the thickness of the compressible layer. Thus, in the design phase, the required value for the discharge capacity should be evaluated with more care, since the type of drain will be chosen based on this result. Therefore, the theoretical discharge capacity required (qreq) for a prefabricated band-shaped drain, considering the consolidation of a soil cylinder mass with a diameter de, and drain length Ld, can be estimated using the expression defined by Kamon et al. (1984). h 10 req f d h C q s .U .L  4T   in m3/year (IV.5) where: sf - Final settlement of the soft clay layer, equivalent to 25% of the total depth of the layer to be improved with PVD drains; 10 U  - Time associated to 10% of the average degree of consolidation; Ch - Horizontal consolidation coefficient of the soil; Th - Time factor for the horizontal consolidation. Furthermore, in order to consider the factors that might affect and reduce the discharge capacity, this parameter must be increased by three reduction factors (Bergado et al., 1996b). Therefore, the proposed minimum drain discharge capacity is given by: w,min t c fc req q F .F .F .q  in m3/year (IV.6) where: Ft - Reduction factor associated to time; Fc - Reduction factor associated to drain deformation; Ffc - Reduction factor associated to drain and filter clogging, in the long term. According to the authors, the values associated with these reduction factors are approximately 1.5, 2.0 and 3.5 respectively, thus: w,min req req 1.5x2.0x3q q 10. x ,5q 5   in m3/year (IV.7) In 1985, Fellenius & Castonguay published the following results related to the drain deformation factor (Fc): s s s Initial ground surface Ground surface after settlements Soil with weak characteristics CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 85 - a) Drains with a hard plastic core presents reduction factors in the drain discharge capacity of around 5 to 15; b) Drains with more flexible plastic cores present more moderate reduction factors, ranging between 1.5 and 2. According to Hansbo (1997b), the prefabricated vertical drains present a higher drainage capacity when compared to sand drains; therefore, the hydraulic resistance of the drain may be neglected in consolidation analyses, as long as the previous conditions are respected. IV.7. CASE STUDIES There are many reports from work cases performed with sand drains and PVD drains. The majority of them are extremely extensive, since they were used as case studies, on researches developed for Masters and PhD theses. However, the compilation of the results that Justo presented in 1966, in his research study, concerning the application of sand drains is highlighted. He detailed, with great precision, 116 work cases, indicating: a) Name of the project and construction start date; b) Characteristics of the improved soil, including soil consolidation coefficient and thickness of the layer; c) Characteristics of the sand drains (diameter, depth, spacing and pattern adopted) and installation methods; d) Thickness of the drainage blanket; e) Characteristics of the preload embankment (height of the embankment, % of the load applied in comparison to the definitive load and application time); f) Settlements obtained; g) Excess pore pressure dissipation over time. Among these cases we have works in Brazil, Italy, England, France, India, Sweden, New Zealand, Norway, Germany, Japan, Belgium, Puerto Rico and America (Connecticut, Los Angeles, Texas, Illinois, Washington, Salt Lake City, New Jersey, Chicago, California, Ontario, Oregon, Hawaii, New York, Norfolk, Seattle, Boston, San Francisco), and others (Justo, 1966). Although the sand drains are not the scope of the research in question, it is always interesting to consult and analyse the characteristics of these works, especially given their detail. From the bibliography consulted, the works with greatest projection in terms of international publications and references, which subsequently led to many research studies, are those where the field tests were conducted in a real scale. Among them, we have the following works: I. Ska-Edeby airport, west of Stockholm, Sweden (Hansbo et al., 1981); II. Motorway connecting Örebro to Gothenburg, Sweden (Hansbo et al., 1981); III. Refinery expansion in Trieste, Italy (Hansbo et al., 1981); CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 86 - IV. New industrial zone in the west Port of Amsterdam in Aziëhaven, Netherlands (Hansbo et al., 1981); V. Elevation of the railway line in Hemspoor, near Amsterdam, Netherlands (Hansbo et al., 1981); VI. Thermo-electric plant in Porto Tolle, Italy (Hansbo et al., 1981); Pothiraksanon et al., 2010); VII. Tha Chang Bridge in Bangkok, Thailand (Bergado et al., 1993a); VIII. Asian Institute of Technology in Bangkok, Thailand (Bergado et al., 1993a; Bergado & Long, 1994b); IX. Pom Prachul Naval Dock, Thailand (Bergado et al., 1993a); X. New Bangkok International Airport, Thailand (Bergado et al., 1993a; Balasubramanian et al., 1997; Bergado et al., 2002); XI. Storage tanks for petroleum products in Valencia, Spain (Calderon & Romana, 1997); XII. Highway in Belfast, Northern Ireland (Davies & Humpheson, 1981); XIII. Road and rail conjunction in Queenborough, Isle of Sheppey, UK (Nicholson & Jardine, 1981). Many of the results observed in these case studies and sites, were used by researchers all over the world, and some of the theories and empirical expressions presented previously are based, tested and compared to them. Field tests in real scale are extremely important, since various scenarios can be experimented, namely different types of drains, with different spacing and patterns, applying several construction methods, and so on. It is also important to point out that the foundation soil, in these case studies, is generally extremely well characterised, based on “in-situ” and laboratory tests results. Additionally, all of these experimental real case studies were monitored, at each phase of the construction. All the cases mentioned above are available in the bibliography of this thesis. The use of vertical drains are often used in port construction sites, and are mostly combined with preload embankments as the loads supported are usually very high (heavy cranes, warehouse zones, container loading and unloading zones, among others), and so, minimum deformation in the work platform is required. The choice of the improvement methods to be applied on these areas depends greatly on the characteristics of the ground foundation, the magnitude of the loads, which the soil will be subjected, the dimension of the area where the intervention is needed and finally the cost associated to each scenario. To give an idea of the importance of the vertical drains in these type of construction work, in the next tables are presented the compilation results of several works cases, made by the Group of Port Experts from the Republic of Korea, in 2002. They studied three port construction sites located in their own country, as well as three other cases in Australia and five in Japan (Tables IV.4, IV.5a, IV.5b, and IV.6). In all of them, the soil foundation was improved using sand drains (SD), either Sandwick drains, prefabricated band-shaped drains (PVD), sand columns (SC) or stone columns (SCP). CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 87 - Table IV.4 – General characteristics of the Ports where vertical drains were applied, according to the Port Expert Group - APEC 20th TPT-WG (2002) Name of the Port Type of Port Location Total Area of the Port [m2] Construction Period Start Conclusion Port of Brisbane ---- Bishop Island, Australia 5 000 000 1980 ---- Sakai-Senboku Industrial Port Sakai-2, Japan 3 500 000 1997 2008 Amagasaki - Nishinomiya - Ashiya Industrial Port Amagasaki, Japan 40 000 1987 2003 Sakai Commercial Port Gaiko -7.5G, Japan 3 900 1986 1995 Sakai Commercial Port Gaiko -14G, Japan 12 900 1995 2003 Hiroshima Commercial Port Dejima -14G, Japan 26 400 1993 2002 Gamman Commercial Port Busan, Korea 310 000 1997 2002 Kwangyang-1 Commercial Port Kwangyang, Korea 596 664 1996 2001 Kwangyang-2 Commercial Port Kwangyang, Korea 597 910 1999 2003 Table IV.5a – Geotechnical characteristics of the soil foundations, according to the Port Expert Group - APEC 20th TPT-WG (2002) Name of the Port Type of soil and thickness Strength Characteristics of consolidation Description Thickness of the layer [m] OCR Miscellaneous properties Port of Brisbane Marine sand 7.1 cu = 32° to 34° Marine clay 6.0 su = 6 kPa Sandy clay 1.0 su = 80 kPa Sandy silt 9.0 cu = 33° Silty clay 18.0 su = 150 kPa Sakai-Senboku Clay 14.0 cu=1.8z-12.7(kN/m2) z:depth(m) 1 mv=0.22P-1.04 Cv=40(cm2/ day)  t =1.5tf/m3 Sand 0.6 Clay 4.0 cu=17.8z-365.7(kN/m2) z:depth(m) mv=0.22 P-0.04 Cv=70(cm2/day)  t =1.7 tf/m3 Amagasaki- Nishinomiya-Ashiy Sand 10-11 NSPT = 4-13 Clay 14-19 qu=-0.7+0.2z (tf/m2) z: depth(m) Cc=0.58-1.17 Cv=50-1200(cm2/ day) w n = 59.1-95.6 % Sakai-7.5G Clay 30.0 su=2.35+0.13z(tf/m2) z:depth(m) 1 Cc=0.49-2.42 Cv=232(cm2/ day) w n =40-85% Sakai-14G Clay 25.0 su=2.73+0.16z(tf/m2) (z=0.0 up to 13m) 1 Cc=0.25-1.01 Cv=518(cm2/ day) w n =30-90% Hiroshima-14G Clay 20.0 su=0.18+0.14z(tf/m2) (z = 0.0 up to 12 m) 1 Cc= 0.89-1.80 Cv = 167 (cm2/ day) w n = 40-140 % Gamman Clay 17.1 su=0.40 kgf/cm2 Cc= 0.71 Cv = 43 (cm2/ day) w n = 62 % CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 88 - Table IV.5b – Geotechnical characteristics of the soil foundations, according to the Port Expert Group - APEC 20th TPT-WG (2002) Name of the Port Type of soil and thickness Strength Characteristics of consolidation Description Thickness of the layer [m] OCR Miscellaneous properties Kwangyang-1 Clay 14.0 su=0.48 kgf/cm2 1 Cc= 0.71 Cv = 43 (cm2/ day)  n = 62 % Clay 21.0 su=0.78 kgf/cm2 1 Cc= 0.71 Cv = 43 (cm2/day)  n = 62 % Kwangyang-2 Clay 3-14 su=0.15 kgf/cm2 1 Cc= 0.71 Cv = 43 (cm2/ day)  n = 62 % Clay 6.5-18.5 su=0.13 kgf/cm2 1 Cc= 0.71 Cv = 43 (cm2/ day)  n = 62 % su - undrained shear strength obtained from the Vane-Test; qu – unconfined compression strength; c - cohesion; cu - friction angle obtained from the triaxial (Consolidated-Undrained) test; Cc - compressibility index; Cv - vertical consolidation coefficient; mv - volumetric compressibility coefficient; n - natural water content; t - total volumetric weight; NSPT – number of blows from the Standard Penetration Test Table IV.6 – Type of vertical drains used, area of intervention and height of the embankment, according to the Port Expert Group - APEC 20th TPT-WG (2002) Name of the Port Type of drain Improved area [m2] Embankment height [m] Thickness of the layer [m] Length of the vertical drains [m] Degree of consolidation required [%] Consolidation time Estimated Real Port of Brisbane SCP 80 000 4 (approx.) SC - - 6.0 6.0 - 3 months 3 months Sakai-Senboku SD 44 517 10.5-17.2 14.2 14.2-16.1 90 % 12 months ?? Amagasaki- Nishinomiya-Ashiy SCP/SD 484 3.8-4.2 30.6 SCP = 18.4 SD = 12.2 90 % 7 months 31 months PVD 792 4.2-5.1 32.2 32.2(media) 90 % 7 months 31 months Sakai-7.5G SD 5 200 1.9 15.5 14.5 90 % 33 months 89 months Sakai-14G SD 4 600 1.9 22.6 10.0 80 % 1 month ?? Hiroshima-14G SD 8 900 4.8 15.3 12.0 90 % 6 months ?? Gamman Sandwicks/ PVD/SD 227 000 9.4 17.1 28.0 96 % 8 months 7.7 months Kwangyang-1 Sandwicks 255 000 8.0-8.3 14.0 19.5 90 % 12 months 12 months Sandwicks 330 500 2.0-5.5 21.0 23.8 90 % 15 months 15 months Kwangyang-2 Sandwicks (1.2×1.2) 182 066 1.5-9.5 3.0-14.0 7.0-18.0 92.5-96.9% 287 days 293 days Sandwicks (1.2×0.8) 156 182 6.0-6.4 13.5-18.5 16.5-21.5 84.0-92.7% 353 days 357 days Sandwicks (0.9×0.9) 179 851 2.8-3.8 6.5-13.5 9.5-16.5 88.0-96.0% 355 days 306-333 days CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 89 - Based on the data presented above, in table IV.6, most of the improvement technique used the sand drains and the Sandwicks drains. Although these methods have associated higher costs, it is believed that they were chosen due to the dimensions of the improved areas, and in some cases the depth. Furthermore, since the soils foundations presented high compressibility and the expected settlements were significant, then the risk of prefabricated drains suffered relevant deformations was a factor to be considered. In addition to this, the sand drains, sand columns and stone columns, brings additional resistance to the soil foundation, since the weaker clay soils are substituted by materials with better characteristics. In order to add new work cases to those already referred, five recent works undertaken in Portugal will be presented. These had less international visibility and projection, but were published in congresses and specialised Portuguese magazines. Finally, some aspects related to the case study for the new Ota airport are indicated. They are presented in chronological order. IV.7.1. WORKS CASES IN PORTUGAL IV.7.1.1. DOCK NO.4, LEIXÕES PORT According to Furtado (1995), the Administration of the Douro River and Leixões Ports decided, in mid-1987, to install a container terminal on an embankment adjacent to the designated South Quay of Dock No.4 of the Port of Leixões, located in the North of Portugal, near the River Leça (Figure IV.26). Figure IV.26 – Container terminal of Dock No.4, Leixões Port (image taken from http://www.bing.com/maps/..., in November of 2012) In the geological characterisation of the site, a thick layer of alluvial silt, filling the fossil valley of the River Leça, was observed. Therefore, the soil was characterised as having low shear strength, high compressibility and consequently low permeability. The combination of these factors resulted in the prediction of major settlements in long term, and so, the Port Authorities decided to undertake works that promoted the preconsolidation of the silty soils. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 90 - The area to be consolidated was divided into four areas, each one of them treated with vertical prefabricated drains and using a preload embankment. The embankment surcharge represented the future demands of the project and its construction was carried out in steps. The goal was to obtain, in three months, 90% of the estimated total settlement, which was achieved satisfactorily. The characteristics of the clay stratum profile considered in the analyses represented the average soils conditions. The in-situ test included 28 CPT tests and 24 boreholes, in which SPT and Vane Tests were performed, with re-mixed samples being collected. Laboratory tests were also performed, including Atterberg limits, grain-size distribution analysis, specific soil weight, water content, triaxial compression and oedometer tests with vertical drainage and radial drainage. The estimated values for the silty layer, based on the results from the tests performed, were (Table IV.7): Table IV.7 – Average characteristics of the silty layer of the Dock no. 4, Leixões Port (Furtado, 1995) Parameters West Area East Area mv 3x10 - 2 cm 2 /kg 5x10 - 2 cm 2 /kg Cv 4x10 - 4 cm 2 /s 4x10 - 4 cm 2 /s Cr 16x10 - 4 cm 2 /s 25x10 - 4 cm 2 /s Based on these values, and only in three months, 80% to 90% of the estimated settlements were achieved, correspondent to a value of approximately 80cm. The prefabricated vertical drains used were the MEBRADRAIN 7407 type, with an equivalent diameter of 5cm, for a triangular pattern with a spacing of 1.40m. During the execution of the drains and of the loading embankment, their behaviour was constantly monitored by settlements plates, piezometers and inclinometers. The control and measurement continued until the complete stabilisation of the settlements. IV.7.1.2. EAST EXPANSION OF THE BEIROLAS LANDFILL The 1998 World Exposition, also known as EXPO ' 98, or, by 1998 Lisbon International Exhibition, took place in Lisbon-Portugal. The area chosen for the construction of this important exposition covered a big area at the eastern boundary of the city of Lisbon, near the River Tagus, currently designated as Parque das Nações (Figure IV.27a). Studies conducted in the southern part of Parque das Nações, identified the presence of high hydrocarbon levels, above the accepted values for residential land, as indicated in the approved Urbanisation Plan. However, the results of these studies were encouraging, as it was possible to confirm that the geological characteristics of the site (a layer of low permeability clay was identified) served as a natural barrier to the migration of the contaminants in depth. The depth up to which the contamination of the soil was identified did not exceed 2 metres, estimating approximately 250 000m3 of the soil volume needed to be treated and/or removed. Therefore, most of this contaminated soil was removed and sent to the Beirolas landfill, located at the far North end of Parque das Nações, between the River Trancão and the Beirolas water waste power plant treatment. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 91 - In order to house all the contaminated soil, removed from the construction site, three additional cells were built (Figure IV.27b), and occupying a rectangular area of approximately 7.5 hectares and stretching to the side adjacent to the east slope of the landfill. In June 1995, a large landslide affected this east slope. 27a) 27b) Figure IV.27 – Area of the EXPO 98 where the contaminated soil were removed, and the cells of the Beirolas landfill where they were deposited (images taken from http://www.google.com) At the request of Parque EXPO'98, the National Laboratory of Civil Engineering (LNEC) of Portugal analysed the causes of the landslide and concluded that the insufficient resistance of the silty soil of the foundation had caused the accident. Based on tests, it was observed that, as an effect of the landslide, the silt suffer an important loss of strength, and so it was decided to carry out a treatment of the foundation ground with prefabricated vertical drains in the zone, together with the application of an overload platform. With this solution, the shear strength of the soil foundation should increase due to the acceleration of the consolidation process, under the load of the existing embankment and the new overload. The zone for which the treatment with vertical drains was planned was located between the cells for the contaminated soil at the East extension of the landfill and the bank of the River Tagus, forming a strip of about 60m wide, in front of cells 2 and 3, and 40m in front of cell 1. Drains could not be installed inside the cells because of the impermeable geomembranes located at the base of the cells. They cannot be perforated in order to ensure that no leakage occurs. The plastic vertical drains (geodrains) were installed between December 1996 and January 1997, forming a triangular pattern with spacing of 2m (area per drain of 3.46m2), and with a length of between 30 and 35m from the ground surface. The installation of drains in the entire area was not possible due to the thickness and consistency of the superficial layers in some locations. Prior to the installation of the vertical drains, in the first months of 1996, re-profiling of the slope in the improved area was carried out, as a corrective measure of the June 1995 landslide. With this reprofiling, which covered the zone of cells 2 and 3, from the slope of the landfill up to the river, and also the area between cell 1 (empty) and the river, the ground surface was also levelled and its inclination was softened. The sunken areas, located between the East berm of the cells and the river, were filled with sand. A sand blanket with a minimum thickness of 50cm was also applied in the areas to be improved with PVD drains. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS IV. VERTICAL DRAINS UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 92 - After the installation of the drains, an overload embankment was built with a maximum height of approximately 1.75m. A geotextile was placed between the sand blanket and the embankment. Considering the information available of the soil characteristics, with a horizontal consolidation coefficient of Ch of 7x10-8m2/s, 75% of the consolidation would be achieved within 11 months and 90% in 19 months. Once again, the area was instrumented in order to control the evolution of the consolidation degree of the soil foundation. It was necessary to considered that, between the date of installation of the drains and the placement of the embankment, some important settlements possibly occurred because the soils were underconsolidated. The final consolidation settlement values were estimated, and corresponded to 0.69m and 1.14m for the zones where the settlement plates were applied. From the analysis of the settlement measurement plates, it was possible to verify the efficacy of these drains in the increment of the shear strength of the silty soil. The cells are currently sealed, and its final appearance is that presented in figure IV.28. Figure IV.28 – Sealed landfill located near the Beirolas water waste powerplant treatment (image taken from http://www.google.com, in March of 2013) All the information referred here was taken from De Santayanna & Fernando (1997). IV.7.1.3. PEDESTRIAN FOOTPATH ON THE TAGUS WATERFRONT, EXPO’98 In approximately 1996, it was decided to build a pedestrian footpath with about 700m in length along the right bank of the River Tagus, in the EXPO’98 intervention zone, north of the Vasco da Gama panoramic tower (Figure IV.29). Figure IV.29 – Pedestrian footpath along the right bank of the River Tagus, EXPO’98 (image taken from http://es.wikipedia.org, in March of 2013) CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 99 - V. SMEAR Chapter V is entirely dedicated to the smear effect and to the characterisation of the parameters that allow it to be accounted in the consolidation analyses. It is presented an extensive summary of the studies performed by several researchers, related to this subject, due to the importance of the phenomenon on the consolidation process. Immediately thereafter, a comparative study is made between the solutions of Hird et al. (1992) and Indraratna & Redana (1997), both for plane strain conditions, in order to demonstrate not only the differences between them, but also the influence of some parameters on the estimation of the horizontal permeability coefficient of the disturbed area (smear). This is a key point of the work undertaken since the practical work case is analysed in plane strain and this comparison allowed to decide which solution should be adopted in future analyses. The chapter ends with a sensitivity analysis of the smear parameters and their influence on the evolution of the consolidation degree. In this phase, the consolidation analytical formulations will be applied taking into account the prefabricated vertical drains, already described in chapter II, for both the axisymmetric strain and the plane strain conditions. V.1. INTRODUCTION As has already been mentioned, the drains installation in the ground creates a disturbance in the surrounding soil adjacent to them, designated by disturbed zone. Usually, PVD drains are installed using mandrels closed at the base and driven in the ground, thereby causing an increase in pore pressure and a reduction in both the strength and permeability of the soil in the disturbed zone, usually known as smear effect (Holtz & Holm, 1973). The most significant consequence will be the decrease in the water flow to the drain and a delay in the consolidation process of the improved soil. This effect can be considered in the consolidation analyses in two distinct manners: a) Through the global reduction in the consolidation coefficient (Ch and Cv); b) Through the introduction of a disturbed area around the drain, with a reduction in the coefficient of permeability in this zone. Adopting the second hypothesis, two parameters will still be needed to be able to characterise the smear effect properly, these being: a) The diameter of the disturbed zone (ds); b) The horizontal permeability of the disturbed zone (khs). Regardless of the method used, the problem lies in the determination of these parameters and in their correspondence between theory and practice. It has been noted that the extent and permeability of the disturbed zone vary depending on the installation process, the size and the shape of the mandrel and the characteristics of the soils. As these parameters cannot be determined accurately, they are attributed based on field experiences, laboratory tests, and theoretical considerations, or by means of retro-analyses of real case results (Hird & Moseley, 2000). CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 100 - V.2. DIAMETER OF THE DISTURBED ZONE The extension of the disturbed zone or width of the smear was studied by several researchers and numerous relationships were suggested between the diameter of the disturbed zone (ds) and the diameter of the drain (dw), or between the corresponding areas, including: a) ds(1 to 4)dw, by Hotz & Holm (1973), Jamiolkowski et al. (1983), Hansbo (1987, 1997b), Bergado et al. (1991, 1993b), Onoue et al. (1991), Holtz (1991), Mesri et al. (1994), Madhav et al. (1993), Chai & Miura (1999), Indraratna & Redana (1998); b) Other studies led by Akagi (1979), Jamiolkowski & Lancelota (1981), Bergado et al. (1994a and 1996a), Hansbo (1987, 1997b), Chai & Miura (1999), and also Hird & Moseley (2000), recommend that ds(1.6 to 3)dw; c) Chai & Miura (1999), advise the use of ds3dw if there are no elements/tests which allow for the evaluation of the extension of the disturbance; d) McDonal (1985), Aboshi & Inoue (1986) suggest the adoption of AsAm,eq; e) Basu & Prezzi, in 2007, considered that the extension or equivalent diameter of the disturbed zone is given by: sx sy s s,eq 4(d d ) d d     (V.1) where dsx and dsy correspond to the dimensions of the disturbed zone (Figure V.1) for a PVD drain, fixed by a rectangular mandrel. These dimensions are determined in accordance with expressions V.2 and V.3: sx sy d w ( ρ 1)yandd ρy     , with 2≤ρ≤3 (V.2) and (V.3) Figure V.1 – Dimensions of the mandrel/drain and disturbed zone In addition to this disturbed zone, some researchers also consider a transition zone between the smear zone and the undisturbed zone. The diameter of this transition zone (dt) is also given according to the diameter of the drain (dw), or the dimensions of the mandrel. Therefore, we have: a) Onoue et al. (1991) and Indraratna & Redana (1998), based on experimental results concluded that dt(6 to 7)dw; b) Xiao (2001) obtained dt(10 to 15)dw; c) Madhav et al. (1993) and Jamiolkowski et al. (1983), reached for dt values expressed on the basis of the maximum dimension of the mandrel, obtaining respectively, 12w and 20w; Disturbed zone y w dsx dsy CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 101 - d) Bazu & Prezzi (2007), followed the same principle as they used to defined the smear zone, adopted the following expressions for the transition zone: tx ty t t,eq 4(d d ) d d    , admitting that: (V.4) tx ty d w ( ρ 1)yandd ρy     , with 6≤ρ≤12 (V.5) and (V.6) Hawlader et al., in 2002, also considered the existence of two distinct zones; however, instead of calling them smear zone and transition zone, they named them smear zone I (of a smaller size and immediately adjacent to the drain) and smear zone II (larger and surrounding the previous zone). These same researchers refer that it is the disturbed zone closer to the drain, which controls and dominates the progress of the consolidation process. Even zones with small smear diameter may significantly delay consolidation, being the degree of disturbance far more important than the extension of this zone, as other researchers had already reported. In this case, the degree of disturbance refers to the reduction in the horizontal coefficient of permeability in the disturbed zone. In fact, this is the conditioning parameter, as it will be seen in the sensitivity analysis presented in subchapters V.4 and V.5. However, this finding is hardly surprising as consolidation is extraordinarily dependant on the soil permeability. V.2.1. OVERLAPPING OF DISTURBED ZONE DIAMETERS As the technology for soil foundation improvement, based on PVD drains becomes economically more accessible, it has been observed a tendency to place the drains increasingly closer, or in other words, with less distance between axes, always with the objective of accelerating the consolidation process of fine soils (Chu et al., 2004). However, based on experimental field tests, Saye (2001) has noticed that there is a minimum limit for the distance between drains; once this limit is attained, there is no advantage on the consolidation rate. Walker & Indraratna (2007) explained this phenomenon with the possible overlapping of the disturbed zone of the drains (Figure V.2). Additionally, taking into account only one smear zone with a linear variation of khs, they concluded that there is no overlapping of the disturbed diameter, if the ratio e s r r  0.6 is respected. Figure V.2 – Overlapping of the disturbed zone of the drains (Walker & Indraratna, 2007) disturbed zone 2 r e Distance to the drain axis r w rs1 k h disturbed zone 1 Drain 1 Drain 2 k s0 rs2 r w kh k s0 Disturbed zone Drain Undisturbed zone CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 102 - V.3. HORIZONTAL PERMEABILITY IN THE DISTURBED ZONE The degree of disturbance is expressed by the ratio between the horizontal permeability in the disturbed (khs) and undisturbed (kh0=kh) zones, being this value inferior to one   hs h0 k k 1   . Several researchers have studied the theme, having proposed the following values for the ratio khs/kh0: a) Bergado et al. (1991) and Onoue et al. (1991), based on laboratory tests, estimate that this value varies between 0.5 and 0.66; b) Madhav et al. (1993), from samples collected in the field, obtain a ratio equal to 0.2; c) Later, Bergado et al. (1993a, 1993b), Hansbo (1987, 1997b), Madhav et al. (1993) and Hird & Moseley (2000), based on field data, laboratory tests and numerical simulations, obtained, for this ratio, values ranging between 0.1 and 0.33. If the ratio is calculated with the horizontal coefficient of permeability in the disturbed zone khs and the undisturbed vertical coefficient of permeability (kv), then the values varied between: a) hs v 0.9 k k 1.42   for natural clayed soils, where kv is determined with the conventional oedometer test, and hs v 1.2 k k 1.3   for the “Matagani Varved clay”, according to Tavenas et al. (1983); b) Hansbo, in 1987, suggests that hs v k k 1  for relatively large disturbed areas; c) Bergado et al. (1991), concluded that khskv for the Bangkok soft clay; d) hs v 0.9 k k 1.3   , with an average value of 1.15, according to Indraratna & Redana (1998); e) Chai & Miura (1999), propose that   h hs h s f L k k k k .C , where khL and ksL are the horizontal coefficients of permeability determined in the laboratory, associated with the undisturbed and disturbed zone, and Cf a dimensionless parameter that relates the permeability in the field and in the laboratory, this latter being, generally, greater than the unit. This correction factor is related to the disturbance of the sample and with the scale effect, and it may be considered, in some cases, ksL=kvL. It should be point out that the soil samples collected in the field, in order to determine their permeability in the laboratory, must be collected on the drain border. Even though the analytical solutions of consolidation with drains proposed by Barron (1948), Yoshikuni (1979) and Hansbo (1979) consider a constant coefficient of permeability in the disturbed zone, experimental tests confirm a variation in this parameter with the radial distance from the drain (Onoue et al., 1991; Madhav et al., 1993; Bergado et al., 1996a; Indraratna & Redana, 1998). Next, an overview of the proposals for the evolution of this parameter in the disturbed zone, and their respective equations, are presented. However, due to the complexity of implementing them in the numerical simulations, the engineer should verified if the gain in precision, on the rate of the consolidation is significant, or if it would be preferable to maintain more simplified solutions. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 103 - a) Barron (1948) and Hansbo (1979) - Scheme for khs constant Figure V.3 – Model proposed by Barron (1948) and Hansbo (1979) b) Onoue et al. (1991) - Scheme for khs with non-linear variation Figure V.4 – Non-linear model proposed by Onoue et al. (1991) For rw ≤r≤rs: h0 hs tMAX w k r r k r r                , if r=rw  khs=kA (V.7) For rs ≤r≤rt: ht h0 t r k k r         , if r=rs  khs= kht =kB and if r=rt  kht=kh0 (V.8) where  is a function of the void ratio, MAX represents the remoulded effect and  is given by: MAX s w log r log r            (V.9) Distance to the drain axis r w r s k hs Drain k h0 undisturbed zone disturbed zone kh [m 2 /s] 10 - 8 10-9 10-10 Distance to the drain axis k h [m 2 /s] 10 - 8 10-9 10-10 r w r s Drain k h0 undisturbed zone disturbed zone transition zone r t k A k B CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 104 - c) Chai et al. (1997) - Scheme for khs with linear variation and khs with bi-linear variation Figure V.5 – Linear model proposed by Chai et al. (1997) For rw ≤r≤rs, the variation of the permeability coefficient in the disturbed zone, will be given by:   hs hs0 k k C r   , if r=rw  khs=khrw and if r=rs  khs=kh (V.10) being ksh0 the imaginary permeability at the centre of the drain and C a positive constant. Although the model is linear, these researchers defined three distinct constants values for khs, in order to compare their results with Hansbo’s solution (1979). The equations correspondents to these three distinct hypotheses are: 1st hypothesis: Permeability coefficient in the disturbed zone given by the average weighted value defined by the area: 2h0 hs1 hs0 hrw 2hrw k 2 s s 1 k k k -1 3 s -1 k           (V.11) 2nd hypothesis: Permeability coefficient in the disturbed zone defined by the simple average value: hrw h0 hs2 k k k 2   (V.12) 3rd hypothesis: Permeability coefficient in the disturbed zone which gives the same smear result as the linear variation hypothesis:   hs0 hs3 2 2 2 k ln(s) k s 1 s-1 ln s -(1-m ) ln m m m (m -2) n n n                       (V.13) where the parameter s, n and m are defined as: s w s r r  , w n r r  and   hs0 m k C r   (V.14), (V.15) and (V.16) k hs0 r w r s Drain k h0 undisturbed zone C disturbed zone k hrw 1 Distance to the drain axis k h [m 2 /s] 10 - 8 10-9 10-10 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 105 - Figure V.6 – Bi-linear model proposed by Chai et al. (1997) The bi-linear model will be identical to the previous model, with the slope of disturbed zone 1 being more accentuated than that of zone 2, which means that C1>C2, both being positive constants. Therefore, we have for rw ≤r≤rs1 and for rs1 ≤r≤rs2, respectively: hs1 hs01 1 k k (C r)   , if r=rw  khs1=khrw (V.17) hs2 hs02 2 k k (C r )   , if r=rs1  khs2=khs1 and if r=rs2  khs2=kh0 (V.18) Developing these equations to take into account the evolution of the average degree of consolidation, the dimensionless parameters s1, s2, m1 and m2 will be needed, being determined by the following equations (Chai et al., 1997): 1 s1 w s r r  and 2 s2 w s r r  (V.19) and (V.20)   1 hs01 1 m k C r   and   2 hs02 2 m k C r   (V.21) and (V.22) d) Hawlader et al. (2002) - Scheme for khs constant or linear These researchers used the non-linear model by Onoue et al. (1991) as the base of their formulation; however, instead of considering two zones with a non-linear variation of permeability, they admitted two distinct disturbed zones, designating the zone adjacent to the drain as interior disturbed zone (zone I), and the transition zone as exterior disturbed zone (zone II). In addition to this, they considered four distinct scenarios for the permeability coefficients for each zone, three of them with khs constant, and one with a linear variation. Scenario 1 Scenario 2 Figure V.7 – Hawlader et al. (2002) model, with constant permeability in zone I and II Distance to the drain axis r w r s khs2 undisturbed zone r t k A k B I II k h0 kh [m 2 /s] 10 - 8 10-9 10-10 r w r s khs1 k h0 undisturbed zone r t k A k B I II k h [m 2 /s] 10 - 8 10-9 10-10 Distance to the drain axis r w r s2 Drain k h0 undisturbed zone C 1 disturbed zone 1 k hrw 1 1 C2 r s1 disturbed zone 2 k hs1 kh [m 2 /s] 10 - 8 10-9 10-10 10-11 k hs02 k hs01 k hs2 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 106 - For scenario 1, and with rw ≤r≤rt, the permeability is given by:   s hs1 t w k r kr -r     (V.23) For scenario 2, and with rw ≤r≤rt, the permeability coefficient changes to: w t r r r r hs2 k k 2 k     (V.24) Scenario 3 Figure V.8 – Hawlader et al. (2002) model, with constant permeability in the disturbed zone, although khsII > khsI For scenario 3, the permeability value in the disturbed areas remains constant, although different from each other. So, and for the zone adjacent to the drain (zone 1), with rw≤r≤rs, the permeability coefficient will be: w s r r r r hs3 k k k 2     (V.25) In the exterior disturbed zone (zone II), with the radius r varying between rs and rt, the permeability coefficient will correspond to: s t r r r r hs3 k k k' 2     (V.26) Scenario 4 Figure V.9 – Linear model proposed by Hawlader et al. (2002) Distance to the drain axis k h [m 2 /s] 10 - 8 10-9 10-10 r w r s k hs3 k h0 undisturbed zone r t k A k B I II k’hs3 kh [m 2 /s] 10 - 8 10-9 10-10 k hs4 Distance to the drain axis r w r s k h0 undisturbed zone r t k A I II CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 107 - For the fourth scenario, corresponding to the linear variation, when r=rw, than the permeability coefficient will be kA value, indicated by Onoue et al. (1991), in their non-linear model. For r=rs, the permeability coefficient will corresponds to the undisturbed soil (kh0), and it will remains constant and equal to kh0 from there on. e) Walker & Indraratna (2006) - Scheme for khs with parabolic variation Figure V.10 – Parabolic model proposed by Walker & Indraratna (2006) For rw ≤r≤rs, the variation of the permeability coefficient in the smear zone for this model is given by:   hs hs0 w w r r k k k -1 A -B C A B-C r r             (V.27) being parameters k, A, B and C obtained from the expressions presented below and s from equation V.19: h0 0 hs k k k , k A k-1 , s B s-1  and 1 C s-1  (V.28), (V.29), (V.30) and (V.31) f) Walker & Indraratna (2007) - Scheme for khs with linear variation Figure V.11 – Linear model proposed by Walker & Indraratna (2007) Distance to the drain axis r w r s Drain k h0 undisturbed zone K hs0 disturbed zone kh [m 2 /s] 10 - 8 10-9 10 - 10 Distance to the drain axis r w r s Drain kh0 undisturbed zone K hs0 disturbed zone kh [m 2 /s] 10 - 8 10-9 10-10 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS V. SMEAR UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 108 - In this case, and for rw ≤r≤rs, the evolution of the permeability coefficient in the smear zone will be defined by the equation: hs hs0 w A k k r B r           (V.32) In which the dimensionless parameter k is given by the equation V. 28, being AʹandBʹ equal to: k-1 A s-1  and s-k B s-1  (V.33) and (V.34) g) Basu & Prezzi (2007) - Scheme for khs constant and kht with linear variation Figure V.12 – Linear model for the transition zone proposed by Basu & Prezzi (2007) Lastly, the Basu & Prezzi (2007) model is presented, which considers a constant permeability coefficient in the disturbed zone and equal to khs, and a linear variation in the transition zone expressed by:   s ht hs h0 hs t s 2r-d k k k -k d -d         (V.35) This equation, contrary to the previous ones, is expressed according to the diameters of the smear zone and the transition zone, not the radius. All the permeability models briefly presented in this chapter are well detailed in the correspondent bibliography. As a closure, the most important considerations and remarks made by several researchers, regarding this subject, are now presented: a) Bergado et al. (1996a) reported that the area of the disturbed zone, in the field, is greater than that obtained in laboratory tests; k h [m 2 /s] 10 - 8 10-9 10-10 Distance to the drain axis rw rs Drain k h0 undisturbed zone disturbed zone r t k hs transition zone CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 211 - Table VII.11 – Dike No.1: Horizontal permeability coefficients in the improved area with drains (KhPD), considering Indraratna & Redana (1997) analytical solution and KhsAx= KhsPD =0.33Kh0 Kh0=KhAx [m/day] KV0 [m/day] KhPD(1) [m/day] KhPD(2) [m/day] KhPD(3) [m/day] KhPD (i) /Kh0 KhPD(i)/Kv0 i=2 KhsPD(i)/Kv0 i=2 i=1 i=2 i=3 Soils KhsAx=KhsPD=0.33Kh0 ; ds=2dw B’ (S=1.5m) 6.06x10-5 2.32x10-5 3.75x10 - 5 3.83x10 - 5 6.05x10 - 5 0.62 0.63 1.00 1.65 0.78 B1 (S=1.5m) 6.19x10-5 1.58x10-5 3.83x10 - 5 3.91x10 - 5 6.19x10 - 5 0.62 0.63 1.00 2.47 1.17 B2 (S=1.5m) 1.04x10 - 4 3.53x10 - 5 6.45x10 - 5 6.59x10 - 5 1.04x10 - 4 0.62 0.63 1.00 1.87 0.89 B3 (S=1.5m) 5.00x10 - 5 1.29x10 - 5 3.10x10 - 5 3.16x10 - 5 5.00x10 - 5 0.62 0.63 1.00 2.46 1.17 B’ (S=1.75m) 6.06x10-5 2.32x10-5 2.66x10 - 5 2.70x10 - 5 4.18x10 - 5 0.44 0.45 0.69 1.17 0.78 B1 (S=1.75m) 6.19x10-5 1.58x10-5 2.72x10 - 5 2.76x10 - 5 4.27x10 - 5 0.44 0.45 0.69 1.75 1.17 B2 (S=1.75m) 1.04x10 - 4 3.53x10 - 5 4.58x10 - 5 4.65x10 - 5 7.20x10 - 5 0.44 0.45 0.69 1.32 0.89 B3 (S=1.75m) 5.00x10 - 5 1.29x10 - 5 2.20x10 - 5 2.23x10 - 5 3.45x10 - 5 0.44 0.45 0.69 1.73 1.17 B’ (S=2.0m) 6.06x10-5 2.32x10-5 1.98x10 - 5 2.00x10 - 5 3.04x10 - 5 0.33 0.33 0.50 0.86 0.78 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 2.02x10 - 5 2.05x10 - 5 3.11x10 - 5 0.33 0.33 0.50 1.29 1.17 B2 (S=2.0m) 1.04x10 - 4 3.53x10 - 5 3.40x10 - 5 3.45x10 - 5 5.24x10 - 5 0.33 0.33 0.50 0.98 0.89 B3 (S=2.0m) 5.00x10 - 5 1.29x10 - 5 1.63x10 - 5 1.65x10 - 5 2.51x10 - 5 0.33 0.33 0.50 1.28 1.17 Soils KhsAx=KhsPD=0.33Kh0 ; ds=3dw B’ (S=1.5m) 6.06x10-5 2.32x10-5 3.00x10 - 5 3.15x10 - 5 6.05x10 - 5 0.50 0.52 1.00 1.36 0.78 B1 (S=1.5m) 6.19x10-5 1.58x10-5 3.07x10 - 5 3.22x10 - 5 6.19x10 - 5 0.50 0.52 1.00 2.04 1.17 B2 (S=1.5m) 1.04x10 - 4 3.53x10 - 5 5.17x10 - 5 5.42x10 - 5 1.04x10 - 4 0.50 0.52 1.00 1.54 0.89 B3 (S=1.5m) 5.00x10 - 5 1.29x10 - 5 2.48x10 - 5 2.60x10 - 5 5.00x10 - 5 0.50 0.52 1.00 2.02 1.17 B’ (S=1.75m) 6.06x10-5 2.32x10-5 2.16x10 - 5 2.24x10 - 5 4.18x10 - 5 0.36 0.37 0.69 0.97 0.78 B1 (S=1.75m) 6.19x10-5 1.58x10-5 2.21x10 - 5 2.29x10 - 5 4.27x10 - 5 0.36 0.37 0.69 1.45 1.17 B2 (S=1.75m) 1.04x10 - 4 3.53x10 - 5 3.72x10 - 5 3.85x10 - 5 7.20x10 - 5 0.36 0.37 0.69 1.09 0.89 B3 (S=1.75m) 5.00x10 - 5 1.29x10 - 5 1.79x10 - 5 1.85x10 - 5 3.45x10 - 5 0.36 0.37 0.69 1.44 1.17 B’ (S=2.0m) 6.06x10-5 2.32x10-5 1.62x10 - 5 1.67x10 - 5 3.04x10 - 5 0.27 0.28 0.50 0.72 0.78 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 1.66x10 - 5 1.71x10 - 5 3.11x10 - 5 0.27 0.28 0.50 1.08 1.17 B2 (S=2.0m) 1.04x10 - 4 3.53x10 - 5 2.80x10 - 5 2.87x10 - 5 5.24x10 - 5 0.27 0.28 0.50 0.81 0.89 B3 (S=2.0m) 5.00x10 - 5 1.29x10 - 5 1.34x10 - 5 1.38x10 - 5 2.51x10 - 5 0.27 0.28 0.50 1.07 1.17 Soils KhsAx=KhsPD=0.33Kh0 ; ds=4dw B’ (S=1.5m) 6.06x10-5 2.32x10-5 2.57x10 - 5 2.80x10 - 5 6.05x10 - 5 0.42 0.46 1.00 1.21 0.78 B1 (S=1.5m) 6.19x10-5 1.58x10-5 2.63x10 - 5 2.86x10 - 5 6.19x10 - 5 0.42 0.46 1.00 1.81 1.17 B2 (S=1.5m) 1.04x10 - 4 3.53x10 - 5 4.44x10 - 5 4.83x10 - 5 1.04x10 - 4 0.43 0.46 1.00 1.37 0.89 B3 (S=1.5m) 5.00x10 - 5 1.29x10 - 5 2.12x10 - 5 2.31x10 - 5 5.00x10 - 5 0.42 0.46 1.00 1.80 1.17 B’ (S=1.75m) 6.06x10-5 2.32x10-5 1.88x10 - 5 2.00x10 - 5 4.18x10 - 5 0.31 0.33 0.69 0.86 0.78 B1 (S=1.75m) 6.19x10-5 1.58x10-5 1.92x10 - 5 2.04x10 - 5 4.27x10 - 5 0.31 0.33 0.69 1.29 1.17 B2 (S=1.75m) 1.04x10 - 4 3.53x10 - 5 3.24x10 - 5 3.45x10 - 5 7.20x10 - 5 0.31 0.33 0.69 0.98 0.89 B3 (S=1.75m) 5.00x10 - 5 1.29x10 - 5 1.55x10 - 5 1.65x10 - 5 3.45x10 - 5 0.31 0.33 0.69 1.28 1.17 B’ (S=2.0m) 6.06x10-5 2.32x10-5 1.42x10 - 5 1.49x10 - 5 3.04x10 - 5 0.23 0.25 0.50 0.64 0.78 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 1.45x10 - 5 1.53x10 - 5 3.11x10 - 5 0.23 0.25 0.50 0.97 1.17 B2 (S=2.0m) 1.04x10 - 4 3.53x10 - 5 2.46x10 - 5 2.58x10 - 5 5.24x10 - 5 0.24 0.25 0.50 0.73 0.89 B3 (S=2.0m) 5.00x10 - 5 1.29x10 - 5 1.17x10 - 5 1.23x10 - 5 2.51x10 - 5 0.23 0.25 0.50 0.96 1.17 (1) Considering the smear diameter and reduced permeability on this zone, as well as drain resistance in both systems; (2) There is no delimitation of smear in plane strain, even though it exists in axisymmetric, being considered drain resistance in both systems (comparable to the solution of Hird et al. (1992); (3) Without considering smear and without drain resistance. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 212 - Table VII.12 – Dike No.1: Horizontal permeability coefficients in the improved area with drains (KhPD), considering Indraratna & Redana (1997) analytical solution and KhsAx= KhsPD =0.66Kh0 Kh0=KhAx [m/day] KV0 [m/day] KhPD(1) [m/day] KhPD(2) [m/day] KhPD(3) [m/day] KhPD (i) /Kh0 KhPD(i)/Kv0 i=2 KhsPD(i)/Kv0 i=2 i=1 i=2 i=3 Soils K hsAx = K hsPD =0.66K h0 ; d s =2d w B’ (S=1.5m) 6.06x10-5 2.32x10-5 5.15x10 - 5 5.27x10 - 5 6.05x10 - 5 0.85 0.87 1.00 2.27 1.72 B1 (S=1.5m) 6.19x10-5 1.58x10-5 5.28x10 - 5 5.39x10 - 5 6.19x10 - 5 0.85 0.87 1.00 3.41 2.59 B2 (S=1.5m) 1.04x10 - 4 3.53x10 - 5 8.89x10 - 5 9.08x10 - 5 1.04x10 - 4 0.85 0.87 1.00 2.57 1.96 B3 (S=1.5m) 5.00x10 - 5 1.29x10 - 5 4.27x10 - 5 4.36x10 - 5 5.00x10 - 5 0.85 0.87 1.00 3.38 2.56 B’ (S=1.75m) 6.06x10-5 2.32x10-5 3.61x10 - 5 3.67x10 - 5 4.18x10 - 5 0.60 0.61 0.69 1.58 1.72 B1 (S=1.75m) 6.19x10-5 1.58x10-5 3.70x10 - 5 3.75x10 - 5 4.27x10 - 5 0.60 0.61 0.69 2.37 2.59 B2 (S=1.75m) 1.04x10 - 4 3.53x10 - 5 6.22x10 - 5 6.32x10 - 5 7.20x10 - 5 0.60 0.61 0.69 1.79 1.96 B3 (S=1.75m) 5.00x10 - 5 1.29x10 - 5 2.99x10 - 5 3.03x10 - 5 3.45x10 - 5 0.60 0.61 0.69 2.36 2.56 B’ (S=2.0m) 6.06x10-5 2.32x10-5 2.66x10 - 5 2.69x10 - 5 3.04x10 - 5 0.44 0.44 0.50 1.16 1.72 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 2.72x10 - 5 2.75x10 - 5 3.11x10 - 5 0.44 0.44 0.50 1.74 2.59 B2 (S=2.0m) 1.04x10 - 4 3.53x10 - 5 4.59x10 - 5 4.64x10 - 5 5.24x10 - 5 0.44 0.45 0.50 1.32 1.96 B3 (S=2.0m) 5.00x10 - 5 1.29x10 - 5 2.19x10 - 5 2.22x10 - 5 2.51x10 - 5 0.44 0.44 0.50 1.73 2.56 Soils K hsAx = K hsPD =0.66K h0 ; d s =3d w B’ (S=1.5m) 6.06x10-5 2.32x10-5 4.68x10 - 5 4.91x10 - 5 6.05x10 - 5 0.77 0.81 1.00 2.11 1.72 B1 (S=1.5m) 6.19x10-5 1.58x10-5 4.78x10 - 5 5.02x10 - 5 6.19x10 - 5 0.77 0.81 1.00 3.17 2.59 B2 (S=1.5m) 1.04x10 - 4 3.53x10 - 5 8.07x10 - 5 8.47x10 - 5 1.04x10 - 4 0.77 0.81 1.00 2.40 1.96 B3 (S=1.5m) 5.00x10 - 5 1.29x10 - 5 3.86x10 - 5 4.05x10 - 5 5.00x10 - 5 0.77 0.81 1.00 3.15 2.56 B’ (S=1.75m) 6.06x10-5 2.32x10-5 3.31x10 - 5 3.43x10 - 5 4.18x10 - 5 0.55 0.57 0.69 1.48 1.72 B1 (S=1.75m) 6.19x10-5 1.58x10-5 3.38x10 - 5 3.51x10 - 5 4.27x10 - 5 0.55 0.57 0.69 2.22 2.59 B2 (S=1.75m) 1.04x10 - 4 3.53x10 - 5 5.71x10 - 5 5.92x10 - 5 7.20x10 - 5 0.55 0.57 0.69 1.68 1.96 B3 (S=1.75m) 5.00x10 - 5 1.29x10 - 5 2.73x10 - 5 2.83x10 - 5 3.45x10 - 5 0.55 0.57 0.69 2.20 2.56 B’ (S=2.0m) 6.06x10-5 2.32x10-5 2.45x10 - 5 2.52x10 - 5 3.04x10 - 5 0.40 0.42 0.50 1.09 1.72 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 2.51x10 - 5 2.57x10 - 5 3.11x10 - 5 0.40 0.42 0.50 1.63 2.59 B2 (S=2.0m) 1.04x10 - 4 3.53x10 - 5 4.23x10 - 5 4.34x10 - 5 5.24x10 - 5 0.41 0.42 0.50 1.23 1.96 B3 (S=2.0m) 5.00x10 - 5 1.29x10 - 5 2.02x10 - 5 2.08x10 - 5 2.51x10 - 5 0.40 0.42 0.50 1.61 2.56 Soils KhsAx=KhsPD=0.66Kh0 ; ds=4dw B’ (S=1.5m) 6.06x10-5 2.32x10-5 4.29x10 - 5 4.67x10 - 5 6.05x10 - 5 0.71 0.77 1.00 2.01 1.72 B1 (S=1.5m) 6.19x10-5 1.58x10-5 4.39x10 - 5 4.78x10 - 5 6.19x10 - 5 0.71 0.77 1.00 3.02 2.59 B2 (S=1.5m) 1.04x10 - 4 3.53x10 - 5 7.41x10 - 5 8.07x10 - 5 1.04x10 - 4 0.71 0.77 1.00 2.29 1.96 B3 (S=1.5m) 5.00x10 - 5 1.29x10 - 5 3.55x10 - 5 3.86x10 - 5 5.00x10 - 5 0.71 0.77 1.00 3.00 2.56 B’ (S=1.75m) 6.06x10-5 2.32x10-5 3.07x10 - 5 3.27x10 - 5 4.18x10 - 5 0.51 0.54 0.69 1.41 1.72 B1 (S=1.75m) 6.19x10-5 1.58x10-5 3.14x10 - 5 3.35x10 - 5 4.27x10 - 5 0.51 0.54 0.69 2.12 2.59 B2 (S=1.75m) 1.04x10 - 4 3.53x10 - 5 5.31x10 - 5 5.65x10 - 5 7.20x10 - 5 0.51 0.54 0.69 1.60 1.96 B3 (S=1.75m) 5.00x10 - 5 1.29x10 - 5 2.54x10 - 5 2.70x10 - 5 3.45x10 - 5 0.51 0.54 0.69 2.10 2.56 B’ (S=2.0m) 6.06x10-5 2.32x10-5 2.30x10 - 5 2.41x10 - 5 3.04x10 - 5 0.38 0.40 0.50 1.04 1.72 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 2.35x10 - 5 2.46x10 - 5 3.11x10 - 5 0.38 0.40 0.50 1.56 2.59 B2 (S=2.0m) 1.04x10 - 4 3.53x10 - 5 3.96x10 - 5 4.16x10 - 5 5.24x10 - 5 0.38 0.40 0.50 1.18 1.96 B3 (S=2.0m) 5.00x10 - 5 1.29x10 - 5 1.90x10 - 5 1.99x10 - 5 2.51x10 - 5 0.38 0.40 0.50 1.55 2.56 (1) Considering the smear diameter and reduced permeability on this zone, as well as drain resistance in both systems; (2) There is no delimitation of smear in plane strain, even though it exists in axisymmetric, being considered drain resistance in both systems (comparable to the solution of Hird et al. (1992); (3) Without considering smear and without drain resistance. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 213 - In light of results and observations mentioned for Dike No.1, concerning this issue, it was decided to reduce the scenarios for Dike No.3. In this case, the horizontal permeability coefficients in plane strain (KhPD) were determined according to the Indraratna & Redana (1997) analytical solution, only for the scenarios: a) Not considering the smear diameter around the drain, in the plane strain geometry model, even though it was considered in axisymmetric, and drain hydraulic resistance in both systems (comparable to the solution of Hird et al. (1992); b) Without considering smear and without drain resistance. The remaining conditions are the same for Dike No.3, with the results of KhPD being indicated in tables VII.13 and VII.15. Analysing the values obtained, it can be seen: a) For B’ and B1 soils, the ratio KhsPD/Kv0 is very low for KhsPD =0.10Kh0, ranging between 0.28 and 0.39, and very high for KhsPD =0.66Kh0, as it varies from 1.82 to 2.58. For KhsPD=0.33Kh0 the results obtained are comprehended between 0.91 and 1.29. These results are identical to those found for Dike No.1; however there is an exception for the B2 soil, as the value for the ratio KhsPD/Kv0 range from 0.97 to 6.41. This is due to the fact that ratio Kh0/Kv0 for this soils, is approximately three times greater than the others (see Table VI.13 of chapter VI); b) For the ratio KhPD/Kv0 the soil B2 presents, once more, bigger results than the others. For soils B’ and B1 the values are similar to those obtained for Dike No.1, being: KhsAx=0.10Kh0 and ds=2dw: B’ and B1  0.42  KhPD/Kv0  0.60; B2  KhPD/Kv0=1.48; KhsAx=0.10Kh0 and ds=3dw: B’ and B1  0.30  KhPD/Kv0  0.42; B2  KhPD/Kv0=1.05; KhsAx=0.10Kh0 and ds=4dw: B’ and B1  0.25  KhPD/Kv0  0.35; B2  KhPD/Kv0=0.87; KhsAx=0.33Kh0 and ds=2dw: B’ and B1  0.91  KhPD/Kv0  1.29; B2  KhPD/Kv0=3.21; KhsAx=0.33Kh0 and ds=3dw: B’ and B1  0.76  KhPD/Kv0  1.08; B2  KhPD/Kv0=2.68; KhsAx=0.33Kh0 and ds=4dw: B’ and B1  0.68  KhPD/Kv0  0.96; B2  KhPD/Kv0=2.39; KhsAx=0.66Kh0 and ds=2dw: B’ and B1  1.22  KhPD/Kv0  1.74; B2  KhPD/Kv0=4.31; KhsAx=0.66Kh0 and ds=3dw: B’ and B1  1.15  KhPD/Kv0  1.63; B2  KhPD/Kv0=4.04; KhsAx=0.66Kh0 and ds=4dw: B’ and B1  1.10  KhPD/Kv0  1.56; B2  KhPD/Kv0=3.86. Of all the smear possibilities studied, and based on the ratios presented, it can be immediately point it out that the influence of reducing the horizontal permeability coefficient in the smear zone is greater than the increment of the smear diameter, as it was already been seen in the analyses performed in subchapter V.4. In addition to this, the new horizontal permeability coefficients for the plane strain (KhPD) analysis, in the improved zone with drains, are very low when KhsPD=0.10Kh0 and very high for KhsPD=0.66Kh0. According to the suggestions made by various researchers, the situation which would be, probably, the most appropriate, is the one which corresponds to the scenario where KhsPD=0.33Kh0. However, this conclusion can only be drawn after observing the results from the numerical simulations. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 214 - Table VII.13 – Dike No.3: Horizontal permeability coefficients in the improved area with drains (KhPD), considering Indraratna & Redana (1997) analytical solution and KhsAx= KhsPD =0.10Kh0 Kh0=KhAx [m/day] KV0 [m/day] KhPD(2) [m/day] KhPD(3) [m/day] KhPD(i)/ Kh0 KhPD(i)/Kv0 i=2 KhsPD(i)/Kv0 i=2 i=2 i=3 Soils KhsAx=KhsPD=0.10Kh0 ; ds=2dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 6.55x10 - 6 2.17x10 - 5 0.15 0.50 0.42 0.28 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 9.42x10 - 6 3.11x10 - 5 0.15 0.50 0.60 0.39 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 2.27x10 - 5 7.51x10 - 5 0.15 0.50 1.48 0.97 Soils KhsAx=KhsPD=0.10Kh0 ; ds=3dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 4.65x10 - 6 2.17x10 - 5 0.11 0.50 0.30 0.28 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 6.69x10 - 6 3.11x10 - 5 0.11 0.50 0.42 0.39 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 1.61x10 - 5 7.51x10 - 5 0.11 0.50 1.05 0.97 Soils KhsAx=KhsPD=0.10Kh0 ; ds=4dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 3.86x10 - 6 2.17x10 - 5 0.09 0.50 0.25 0.28 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 5.55x10 - 6 3.11x10 - 5 0.09 0.50 0.35 0.39 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 1.34x10 - 5 7.51x10 - 5 0.09 0.50 0.87 0.97 (2) There is no delimitation of smear in plane strain, even though it exists in axisymmetric, being considered drain resistance in both systems (comparable to the solution of Hird et al. (1992); (3) Without considering smear and without drain resistance. Table VII.14 – Dike No.3: Horizontal permeability coefficients in the improved area with drains (KhPD), considering Indraratna & Redana (1997) analytical solution and KhsAx= KhsPD =0.33Kh0 Kh0=KhAx [m/day] KV0 [m/day] KhPD(2) [m/day] KhPD(3) [m/day] KhPD(i)/ Kh0 KhPD(i)/Kv0 i=2 KhsPD(i)/Kv0 i=2 i=2 i=3 Soils KhsAx=KhsPD=0.33Kh0 ; ds=2dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 1.42x10 - 6 2.17x10 - 5 0.33 0.50 0.91 0.91 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 2.05x10 - 6 3.11x10 - 5 0.33 0.50 1.29 1.29 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 4.94x10 - 5 7.51x10 - 5 0.33 0.50 3.21 3.21 Soils KhsAx=KhsPD=0.33Kh0 ; ds=3dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 1.19x10 - 6 2.17x10 - 5 0.28 0.50 0.76 0.91 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 1.71x10 - 6 3.11x10 - 5 0.28 0.50 1.08 1.29 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 4.12x10 - 5 7.51x10 - 5 0.28 0.50 2.68 3.21 Soils KhsAx=KhsPD=0.33Kh0 ; ds=4dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 1.06x10 - 6 2.17x10 - 5 0.25 0.50 0.68 0.91 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 1.53x10 - 6 3.11x10 - 5 0.25 0.50 0.96 1.29 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 3.68x10 - 5 7.51x10 - 5 0.25 0.50 2.39 3.21 (2) There is no delimitation of smear in plane strain, even though it exists in axisymmetric, being considered drain resistance in both systems (comparable to the solution of Hird et al. (1992); (3) Without considering smear and without drain resistance. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 215 - Table VII.15 – Dike No.3: Horizontal permeability coefficients in the improved area with drains (KhPD), considering Indraratna & Redana (1997) solution and KhsAx= KhsPD =0.66Kh0 Kh0=KhAx [m/day] KV0 [m/day] KhPD(2) [m/day] KhPD(3) [m/day] KhPD (i) / Kh0 KhPD(i)/Kv0 i=2 KhsPD(i)/Kv0 i=2 i=2 i=3 Soils KhsAx=K hsPD =0.66Kh0 ; ds=2dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 1.91x10 - 6 2.17x10 - 5 0.44 0.50 1.22 1.82 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 2.75x10 - 6 3.11x10 - 5 0.44 0.50 1.74 2.58 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 6.63x10 - 5 7.51x10 - 5 0.44 0.50 4.31 6.41 Soils KhsAx=K hsPD =0.66Kh0 ; ds=3dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 1.79x10 - 6 2.17x10 - 5 0.42 0.50 1.15 1.82 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 2.57x10 - 6 3.11x10 - 5 0.42 0.50 1.63 2.58 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 6.21x10 - 5 7.51x10 - 5 0.42 0.50 4.04 6.41 Soils KhsAx=K hsPD =0.66Kh0 ; ds=4dw B’ (S=2.0m) 4.31x10-5 1.56x10-5 1.71x10 - 6 2.17x10 - 5 0.40 0.50 1.10 1.82 B1 (S=2.0m)) 6.19x10-5 1.58x10-5 2.46x10 - 6 3.11x10 - 5 0.40 0.50 1.56 2.58 B2 (S=2.0m) 1.49x10 - 4 1.54x10 - 5 5.94x10 - 5 7.51x10 - 5 0.40 0.50 3.86 6.41 (2) There is no delimitation of smear in plane strain, even though it exists in axisymmetric, being considered drain resistance in both systems (comparable to the solution of Hird et al. (1992); (3) Without considering smear and without drain resistance. VII.2.3. DEFINITION OF THE INITIAL STRESS STATE AND INITIAL HYDROSTATIC PRESSURES The initial stress state and the initial hydrostatic pressures are generated in the first step of the calculation procedure, with no time being associated to it. For the case studied, the initial vertical and horizontal stresses are calculated based on the lateral earth pressure coefficient at rest (K0 procedure). In this phase, the geometry of the model corresponds only to the initial ground level and to the soil layers of the foundation, as can be seen in figure VII.10. Simultaneously with the definition of the initial stress state, the initial hydrostatic pressures must be also generated (Generate by phreatic level). For the latter step, the initial water table is defined, as well as the boundary conditions associated to permeability (Figure VII.12). Figures VII.11 and VII.13 represents the total stresses (0) and the initial hydrostatic pressures (u0) for section PK0+240 of Dike No.1. For profiles PK0+140, Pk0+170, PK0+180 and PK0+200 of Dike No.3, the same procedure was performed. The effective stresses are calculated by Terzaghi’s expressions for saturated soils (ʹ=- u). After this initial procedure, the following calculation steps are defined according to the construction rate and phases, being also necessary to indicate the timeline associated to the installation of the instrumentation, as well as their level and starting date of readings, and finally the raise up of the water level in the pond. However, between the K0 procedure and the subsequent steps, a plastic “Nil-step” calculation is advisable, in order to re-establish the equilibrium of the field stress, and then all stresses will obey the failure condition. In this phase, no additional load is applied, and time is only necessary for the Soft Soil Creep model, although very small. The geometry remains identical to the initial situation, as does the position of the water table, and the plastic calculation option should be considered (Plastic calculation). CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 216 - Figure VII.10 – Geometry used on the generation of the initial stress state - PK0+240 of Dike No.1 Figure VII.11 – Initial ho rizontal and vertical stress es on PK0+240 of Dike No.1 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 217 - Figure VII.12 – Initial position of the w ater t able on the ground f oundation - PK0+240 of Dike No.1 Figure VII.13 – Initial pore pressures on PK0+240 of Dike No.1 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 218 - VII.2.4. DEFINITION OF THE CONSTRUCTION STAGES Once the initial conditions of the problem are generated, and the “nil-step” performed, the displacements are set to zero. From this moment on, all the stages associated with to the case study can be discriminated, including: a) Ground excavation; b) Cut-off compaction; c) Activation of the band-shaped drains and modification of the horizontal permeability coefficients in the zones where the drains are installed (when applicable); d) Activation of the geotextile reinforcement (when applicable); e) Application of all monitoring devices associated with each cross-section; f) Embankment compaction over time; g) Rising of reservoir water level over time. In all these phases, the time associated to each step must be indicated, since the consolidation phenomenon of the soil is in question (Consolidation). The chronological order of construction and the timelines associated to each steps were defined previously in subchapter VI.5. So, it is advisable to consult table VI.29 for the cross-section PK0+240 of Dike No.1, and tables VI.30 to VI.33 for the four cross-sections of Dike No.3. It should be remembered that, in addition to the properties of the material and the geometry, the correct definition of the construction stages and their chronological order, is also fundamental to achieve good results on the simulations analysis. Some of these phases are represented in figures VII.14 to VII.21 for section PK0+240 of Dike No.1. The procedure for the other three profiles of Dike No.3 is identical to that listed for Dike No.1 with the exception of the activation of the geotextile reinforcement. In addition to this, the section Pk0+200 of Dike No.3 have no drains installed on the ground foundation. In addition to the consolidation analyses, the safety factors correspondent to the conclusion of the dikes constructions, and to the last readings of the monitoring devices, after the water in the reservoir stabilised at level 7.80m. These calculations were only performed after the consolidation analyses were concluded, and to perform it, a new step was created, being chosen the option “Safety”. Since this is a geotechnical problem facing extremely high displacements, then the finite element mesh is updated at the end of each calculation stage (Update mesh), as is the water pressure (Update water pressure). When not flow is not considered, the pore pressures are determined based on the position of the water level at each stage, considering also the excess pore pressure associated to the surcharge of the embankment dike, for the saturated clayed and silty soils. The pressures calculated at each new step take into account the previous values (From previous calculated step). CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 219 - Figure VII.14 – Ground excavation up to the level of the embankment foundation and cut–off - PK0+240 of Dike No.1 Figure VII.15 – Cut – off compaction - PK0+240 of Dike No. 1 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 220 - Figure VII.17 – Change of the horizontal p ermeability coefficients in the parallel drainage screens. Geotextile activation - PK0+240 of Dike No. 1 Figure VII.16 – Drains activation - PK0+240 of Dike No. 1 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 227 - Figure VII.25 shows the nodes chosen for profile PK0+240 of Dike No.1, including the two settlement plates located inside the embankment of the dike and all the topographic stake marks. The stress points for the pore pressures are either located in the embankment or in the foundation (Figure VII.26). Figure VII.25 – Nodes used on the control of the settlements during simulation analysis - PK0+240 of Dike No.1 Figure VII.26 – Stress points used to control the pore pressures during simulation analysis - PK0+240 of Dike No.1 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 228 - The control nodes correspondent to the settlement plates and topographic stake marks of profiles PK0+140, PK0+170 and PK0+200 of Dike No.3 are represented in figures VII.27, VII.28 and VII.29. The only two stress points of the pore pressures of Dike No.3 are located in section PK0+180, in the foundation below the upstream berm (Figure VII.30). In this case, stress point of PZ 180-1A was placed at level 0.50m and PZ 180-1B at level -4.50m, considering a difference of only 4 metres between them (see subchapter VI.6.2). Figure VII.27 – Nodes used on the control of the settlements in PK0+140 of Dike No.3 Figure VII.28 – Nodes used on the control of the settlements in PK0+170 of Dike No.3 Figure VII.29 – Nodes used on the control of the settlements in PK0+200 of Dike No.3 Figure VII.30 – Stress points used to control the pore pressures in PK0+180 of Dike No.3 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 229 - Finally, when all the procedures, conditions and steps have been defined, the numerical calculation is initiated (Calculate current project). The results associated with each of the stages can be viewed once the simulations have been concluded. VII.3. RESULTS OF THE NUMERICAL SIMULATION AND COMPARISON WITH MONITORING After the numerical calculations are completed, for any of the calculation steps, the following results can be plotted in the “Output” sub-programme of PLAXIS: a) Deformed mesh; b) Vertical, horizontal or combined displacements; c) Vertical, horizontal or combined strains; d) Vertical, horizontal or combined total stress; e) Vertical, horizontal or combined effective stress; f) Pore pressures; g) Pore pressures considering suction, if the flow is considered; h) Excess pore pressures; i) Percolated flow through the foundation and dike, if the flow is considered; j) Groundwater head; k) Saturated zones; l) Zones with suction and the correspondent value, if the flow is considered; m) The stress points that are in a plastic state (Plastic points). In PLAXIS, the red squares, called the “Mohr-Coulomb points”, indicates that the stress state is located on the surface of Mohr-Coulomb failure envelope. These points are also useful to check whether the size of the mesh is sufficient. When these appear near the mesh boundaries, it means the mesh is too small and must, if possible, be enlarged. The blue inverted triangle, designated as “Cap points”, represents a state of normal consolidation (primary compression), where the actual stress state is equal to the pre-consolidation stress. The evolution of some of these results can also be analysed in depth, if the “Cross section” option is used. Additionally, the programme also allows plotting curves for the control points defined, when using the “Curves” option. It was based on this option, that the settlement and pore pressure curves associated to the control points selected before were outlined. In an initial phase, the simulations for all the sections of Dikes No.1 and No.3 of the Lebrija Pond were performed without considering the flow. Due to the reduced computational time associated to the calculations without taking the flow in consideration, when compared with the simulations with flow, some sensibility analyses were performed in this phase. Later, in the modelling of the dikes considering the flow, the number of simulations was substantially reduced and only those considered most appropriate to the problem were performed. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 230 - In this chapter, the order followed for the presentation of the numerical simulations results will be different of the monitoring results, already discussed in chapter VI. In this case, the settlements precede the pore pressures results, being the latter useful to understand some of the settlement values obtained in the numerical analysis. Additionally, firstly will be presented the graphs for the PK0+240 of Dike No.1, and secondly the graphs for the cross-sections of Dike No.3. VII.3.1. NUMERICAL ANALYSIS WITHOUT FLOW Since the cross-section PK0+240 of Dike No.1 is the more instrumented, the quantity of data associated with the monitoring results is higher. In addition to this, geotextile reinforcement at the base of the foundation was installed, and the ground foundation was improved with band-shaped drains, both upstream and downstream of the cut-off trench. Consequently, a larger number of distinct analyses can be performed, allowing to observe the importance of these elements in the settlements, and compared them to the readings of the monitoring equipments. For the smear values, the Indraratna & Redana (1997) solution was considered, although without the delimitation of the disturbed zone in the geometric modelling of the dikes. If this zone were discredited in the modelling, elements with extremely small dimensions would appear next to the drains leading to numerical problems in the calculation phase. In light of this, the horizontal permeability coefficients between the drains, in plane strain correspond to the values of KhPD(2) indicated in tables VII.10 to VII.12 for Dike No.1, and in tables VII.13 to VII.15 for Dike No.3. When the smear effect is not considered, then the values KhPD(3) are applied. Therefore, the simulations performed for profile PK0+240 in Dike No.1 include the following analyses: a) Consideration of the Mohr-Coulomb (MC) model for the soil foundation in undrained conditions, applying both method A and method B. In method A the stiffness modulus Eref was determined either based on the oedometer modulus, on the initial undrained cohesion or on the reduced undrained cohesion (Table VII.4). In method B, only these two last hypotheses were used (Table VII.5). The smear conditions adopted were KhsAx=KhsPD=0.33Kh0 and ds=3dw. In this case, the ratio Khs/Kv0 varies between 0.78 and 1.17 and the value of KhPD ranges between: KhsAx=0.33Kh0 and ds=3dw: S=1.5m  1.36Kv0  KhPD  2.04Kv0; S=1.75m  0.97Kv0  KhPD  1.45Kv0; S=2.0m  0.72Kv0  KhPD  1.08Kv0; b) Consideration of the Soft Soil (SS) and Soft Soil Creep (SSC) models, in which the secondary compressibility index used in the SSC model was the one given by the Terzaghi et al. (1996) relation (Table VII.8). For these models, a sensitivity analysis was simultaneously performed for the horizontal permeability coefficient in the area with drains (KhPD). CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 231 - In a first phase, the influence of the variation of the smear diameter was studied, (ds=2dw; ds=3dw; ds=4dw), and the smear horizontal permeability coefficient was kept constant and equal to KhsAx=KhsPD=0.33Kh0. In addition to the values of KhPD already presented, with Khs=0.33Kh0 and ds=3dw, the others range between: KhsAx=0.33Kh0 and ds=2dw: S=1.5m  1.65Kv0  KhPD  2.47Kv0; S=1.75m  1.17Kv0  KhPD  1.75Kv0; S=2.0m  0.86Kv0  KhPD  1.29Kv0; KhsAx=0.33Kh0 and ds=4dw: S=1.5m  1.21Kv0  KhPD  1.81Kv0; S=1.75m  0.86Kv0  KhPD  1.29Kv0; S=2.0m  0.64Kv0  KhPD  0.97Kv0. In a second phase, it was analysed the effect of the variation of the smear horizontal permeability coefficient (Khs=0.10Kh0; Khs=0.33Kh0; Khs=0.66Kh0), maintaining the smear diameter around the drain constant and equal to ds=3dw. The values of KhPD associated with the two new smear situations are: KhsAx=0.10Kh0 and ds=3dw: S=1.5m  0.51Kv0  KhPD  0.77Kv0; S=1.75m  0.37Kv0  KhPD  0.56Kv0; S=2.0m  0.28Kv0  KhPD  0.42Kv0; KhsAx=0.66Kh0 and ds=3dw: S=1.5m  2.11Kv0  KhPD  3.17Kv0; S=1.75m  1.48Kv0  KhPD  2.22Kv0; S=2.0m  1.09Kv0  KhPD  1.63Kv0; In these analyses, some previously calculated smear conditions were excluded, since they were not relevant for the parametric study. c) The influence of the geotextile reinforcement, as well as the band-shaped drains, on the settlements, having simulated several scenarios, including:  Foundation soil without drains and without geotextile reinforcement;  Foundation soil without drains and with geotextile reinforcement;  Foundation soil with drains, but without smear, and without geotextile reinforcement;  Foundation soil with drains, but without smear, and with geotextile reinforcement; In these analyses, only the SS and SSC models were used. Once again, the secondary compressibility index used in this last model was estimated according Terzaghi et al. (1996) expression; d) Sensitivity study associated to the secondary compressibility index (C), which can only be performed when the SSC material model is adopted. Three distinct values were used for this parameter, namely for foundation soils B’, B1, B2 and B3. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 232 - In the first hypothesis, C correspond to the values obtained directly from the oedometer test, secondly, C were estimated by the expression of the Terzaghi et al. (1996), and lastly, these indices were calculated based on the ratios */* proposed in the PLAXIS manual (PLAXIS 2D-V11.02, 2012). Calculating */* from the laboratory results, extraordinarily high values are obtained, while for C/Cc=0.045 proposed by Terzaghi et al. (1996) this ratio is reduced to 22.2. According to the PLAXIS manual, the ratio between */* should be between 15 and 25. As 22.2 is close to the upper limit indicated by PLAXIS, then the third value of Cused for the parametric analysis corresponds to the lower limit, i.e. */*=15. Using this limit, higher values are obtained for this index (Table VII.16), implying higher creep settlements. The ratio between the parameters * and *, with the compressibility indexes Cc, Cr and C, are indicated in subchapters III.3.1 and III.3.2 (MATERIAL MODELS). For the smear effect it was considered KhsAx=KhsPD=0.33Kh0 and ds=3dw. Table VII.16 – Dike No.1: Values tested in the sensitivity analysis for the secondary compressibility index (C)  C(Oedometer) C(Terzaghi) C(PLAXIS) Soils * C * */* C * */* C * */* B’ 0,0975 0.00476 0,00214 45.5 0.02511 0.00439 22.2 0.03627 0.00650 15.0 B1 0,1295 0.00694 0,00163 79.7 0.03366 0.00583 22.2 0.04987 0.00864 15.0 B2 0,1094 0.00400 0,00067 164.3 0.02475 0.00492 22.2 0.04273 0.00729 15.0 B3 0,0803 0.00766 0,00218 36.9 0.02579 0.00361 22.2 0.03153 0.00535 15.0 e) Finally, it was performed a simulation combining the SSC model with the secondary compressibility index estimated from the PLAXIS relations, and with smear corresponding to KhsAx=KhsPD=0.10Kh0 and ds=3dw. The simulations for the various profiles of dike No.3 were performed after the analysis of the results from Dike No.1. Therefore, it was considered unnecessary to repeat all the parametric analyses. Hence, in Dike No.3, the simulations for each section are restricted to the conditions and hypotheses which are considered reasonable to traduce the soil behaviour. Thus, only the Soft Soil Creep model was used with the secondary compressibility index (C) being estimated according to Terzaghi et al. (1996) and PLAXIS (Table VII.7). Table VII.17 – Dike No.3: Values tested in the sensitivity analysis for the secondary compressibility index (C)  C(Terzaghi) C(PLAXIS) Soils * C * */* C * */* B’ 0.0925 0.02511 0.00416 22.2 0.03720 0.006164 15.0 B1 0,1295 0.03366 0.00583 22.2 0.04987 0.00864 15.0 B2 0.0947 0.02475 0.00426 22.2 0.03667 0.00632 15.0 B3 0.1005 0.02579 0.00452 22.2 0.03820 0.00670 15.0 CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 233 - In sections PK0+140, PK0+170 and PK0+180 of Dike No.3, where drains were installed to the foundation soil, it was assumed the following smear conditions: KhsAx=KhsPD=0.10Kh0 and ds=3dw. Associated to them, the value KhPD varied between:  0.30Kv0  KhPD  0.42Kv0 for B’ and B1 ;  KhPD=1.05Kv0 for B2 The ratio KhsPD/Kv0 for soils B’ and B1 is also low, ranging between 0.28 and 0.39. Soil B2 is the exception as the value obtained for this ratio is 0.97. This is due to the fact that ratio Kh0/Kv0 is approximately three times bigger than the others, as previously mentioned. In light of this, an additional analysis was performed, where the horizontal permeability coefficient value of Dike No.1 was applied to soil B2 of Dike No.3. This analysis was extended to section PK0+200 of Dike No.3. Once the conditions associated with the different simulations conducted for Dikes No.1 and No.3 of the Lebrija Pond have been indicated, the results and the graphs obtained for the different scenarios can be presented, without considering the flow. In the graphs, for simplicity, the horizontal permeability coefficient in the disturbed zone is indicated only as Khs, instead of KhsAx or KhsPD, as the value is always the same. Also, the initial horizontal permeability coefficient kh0 becomes Kh. VII.3.1.1. SENSITIVITY ANALYSIS AND DIKE NO.1 SETTLEMENTS In the next graphs, the curves of the numerical simulation results, for the several sensitivity analyses performed, without considering the flow, are plotted. These are always compared to the values measured in plates PL 240-1 and PL 240-2, or in the topographic stake marks TS 240-1 and TS 240-2. a) Results considering the Mohr-Coulomb (MC) model for the soil foundation, in undrained conditions, applying both method A and method B: Based on the curves shown in graphs VII.1 and VII.2, it is observed that the Mohr-Coulomb model cannot translate the response of the soil foundation reasonably. The model permits to obtain just an order of magnitude for the settlements, since it is not a model adequate neither to consolidation analysis nor to this type of clayed soils. The numerical settlements during the construction phase are bigger than the measured, decreasing a little after the conclusion of the dike with method A, and then increasing just a little during the reservoir filling phase. In method B, and since this cannot traduce the increment of shear strength associated to the soil consolidation, the curves are almost flat after the conclusion of the embankment. The settlements with the stiffness estimated based on the oedometer modulus gave extremely high values, being more reasonable to use the undrained shear strength empirical relations of Justo et al. (2006). Besides, the settlements calculated with the initial undrained shear strength are smaller, since the stiffness modulus is bigger. Nevertheless, none of the results correspondent to the settlements observed, being the curves far below of the real values, once the construction is concluded. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 234 - Graph VII.1 – PL 240-1 of Dike No.1: Settlements calculated with the Morh-Coulomb model for the soil foundation, in undrained conditions (Method A and B) Graph VII.2 – PL 240-2 of Dike No.1: Settlements calculated with the Morh-Coulomb model for the soils foundation, in undrained conditions (Method A and B) 0 10 20 30 40 50 60 70 80 90 100 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Time [days] Dike No.1 - PL 240-1 Comparison between measured settlements and calculated with PLAXIS (Mohr-Coulomb model: MC) With geotextil, drains and considering smear (Khs=0.33Kh; ds=3dw) PL 240-1 (settlement plate readings) MC_Met.A (Eref=75cu_initial) MC_Met.B (Eref=170cu_initial) MC_Met.A (Eref=fEoed) MC_Met.A (Eref=75cu*) MC_Met.B (Eref=170cu*) 11-07-2001 01-04-2000 Settlements [cm] Conclusion of Dike No.1 28-06-2002 -818 days Start of the raising water in reservoir 24-07-2002 -844 days Last plate measurements 31-03-2008 -2921 days 0 10 20 30 40 50 60 70 80 90 100 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Time [days] Dike No.1 - PL 240-2 Comparison between measured settlements and calculated with PLAXIS (Mohr-Coulomb model: MC) With geotextil, drains and considering smear (Khs=0.33Kh; ds=3dw) PL 240-2 (settlement plate readings) MC_Met.A (Eref=75cu_initial) MC_Met.B (Eref=170cu_initial) MC_Met.A (Eref=fEoed) MC_Met.A (Eref=75cu*) MC_Met.B (Eref=170cu*) 11-07-2001 01-04-2000 Settlements [cm] Conclusion of Dike No.1 28-06-2002 -818 days Start of the raising water in reservoir 24-07-2002 -844 days Last plate measurements 31-03-2008 -2921 days CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 235 - b) Results considering the Soft Soil and Soft Soil Creep models for the soil foundation, including sensitivity analysis of the smear effect:  Varying diameter of the smear zone (ds), and keeping smear horizontal permeability coefficient (Khs) constant and equal to 0.33Kh0.  Varying smear horizontal permeability coefficient (Khs), and keeping diameter of the zone (ds) constant and equal to 3dw. According to results plotted from graphs VII.3 to VII.6, the following statements may be made:  The Soft Soil and Soft Soil Creep models settlements curves are quite similar the construction phase; however, in the post-construction phase and in the filling of the pond, only the SSC model can traduced relatively well the settlements increments with time. This is precisely the expected result, as this last model can calculate the settlements associated with secondary consolidation, which are quite significant in soft soils, such as the case of B1, B2 and B3;  The variation of the diameter of the disturbed zone around the drains, known as smear diameter (ds), has smaller influence on the settlements, when compared with the variation of the disturbed horizontal permeability coefficient or smear permeability coefficient (Khs). Again, these results are consistent with the parametric analyses already performed in subchapters V.4 and V.5. The values of the horizontal permeability coefficient for plane strain analysis (KhPD), in the zones with band-shaped drains, determined with the Indraratna & Redana (1997) analytical solution, gives exactly these same indication. Therefore, the graphical numerical results of these two sensitivity studies reinforce the analytical results obtained before.  In addition, greater reduction in the horizontal permeability coefficient in the disturbed zone, implicates smaller settlement values. However, this does not mean that the drains are not working properly, just indicating that the drainage will take longer, which is why the greatest difference in the numerical results occurs precisely in the initial phase. Analysing graphs VI.5 and VII.6, it can be seen that the difference between the settlements for the different values of Khs smooth out with time;  From the various scenarios considered for the smear, the curves correspondent to the smear conditions KhsAx=0.10Kh0 and ds=3dw, are almost coincident with the measured in the plates, especially for the construction phase. In fact, although this hypothesis presents relatively low values for the ratio KhsPD/Kv0, it is the one which traduces the progress of the settlements fairly well, although in the post-construction phase they are still slightly below the values measured. This is already a reasonable approximation to the monitoring data, and the result is common to both settlement plates. It is important to indicate that the statement above can only be applied for the SSC model. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 236 - Graph VII.3 – PL 240-1 of Dike No.1: Settlements calculated with the SS and SSC models, considering several diameters for (ds), keeping permeability in smear zone constant and equal to Khs=0.33Kh Graph VII.4 – PL 240-2 of Dike No.1: Settlements calculated with the SS and SSC models, considering several diameters for (ds), keeping permeability in smear zone constant and equal to Khs=0.33Kh 0 10 20 30 40 50 60 70 80 90 100 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Time [days] Dike No.1 - PL 240-1 Comparison between measured settlements and calculated with PLAXIS With drain, considering smear (Khs=0,33Kh; varying ds) and with geotextil PL 240-1 (settlement plate readings) SS (khs=0.33kh; ds=3dw) SS (Khs=0.33Kh; ds=2dw) SS (Khs=0.33Kh; ds=4dw) SSC-Csec_Terzaghi (Khs=0.33Kh; ds=3dw) SSC-Csec_Terzaghi (Khs=0.33Kh; ds=2dw) SSC-Csec_Terzaghi (Khs=0.33Kh; ds=4dw) 11-07-2001 01-04-2000 11-07-2001 01-04-2000 Settlements [cm] Conclusion of Dike No.1 28-06-2002 -818 days Start of the raising water in reservoir 24-07-2002 -844 days Last plate measurements 31-03-2008 -2921 days 0 10 20 30 40 50 60 70 80 90 100 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Time [days] Dike No.1 - PL 240-2 Comparison between measured settlements and calculated with PLAXIS With drain, considering smear (Khs=0,33Kh; varying ds) and with geotextil PL 240-2 (settlement plate readings) SS (khs=0.33kh; ds=3dw) SS (Khs=0.33Kh; ds=2dw) SS (Khs=0.33Kh; ds=4dw) SSC-Csec_Terzaghi (Khs=0.33Kh; ds=3dw) SSC-Csec_Terzaghi (Khs=0.33Kh; ds=2dw) SSC-Csec_Terzaghi (Khs=0.33Kh; ds=4dw) 11-07-2001 01-04-2000 Settlements [cm] Conclusion of Dike No.1 28-06-2002 -818 days Start of the raising water in reservoir 24-07-2002 -844 days Last plate measurements 31-03-2008 -2921 days CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 243 - Figure VII.31 – PK0+240 of Dike No.1: Deformed mesh associated to the final calculation step, using the SSC model, and smear given by KhsAx=0.10Kh0 and ds=3dw Figure VII.32 – PK0+240 of Dike No.1: Settlements associated to the final calculation step, using the SSC model, and smear giv en by K hsAx =0.10K h0 and d s =3d w CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 244 - VII.3.1.2. DIKE NO.3 SETTLEMENTS In Dike No.3, and concerning the settlements analysis, the number of cross-sections modelled and studied is superior then Dike No.1. The study of the profiles PK0+140, PK0+170 and PK0+200 of Dike No.3 are of great interest, as it allows the comparison of the numerical results with and without drains. While no drains were applied to section PK0+200, the other sections of the foundation soil were treated with drains, although only downstream of the cut-off trench. By contrast, in Dike No.1, drains were applied both upstream and downstream of the cut-off trench. These differences allow the analyses to be diversified and the influence of these elements in the evolution of the consolidation and the settlements to be determined. The analysis of the settlements readings have been already conducted in subchapter VI.6.4, however some of the results were difficult to interpret. In this chapter an effort will be made in order to understand the differences of the measured results, by comparing them to the numerical results. This can be done, since Dike No.1 gave an extremely good match between both results, and so, an order of the magnitude of the smear effect, as well as the secondary compressibility index, for the ground foundation, is already expected. Thus, and due to the considerations taken for Dike No.1, for the analysis performed in Dike No.3 only the Soft Soil Creep model will be considered to simulate the behaviour of the soil foundation, with C being estimated based on the Terzaghi et al. (1996) expression or on the relations proposed by PLAXIS. With respect to the smear, two analyses are held, one ignoring the smear effect and the other with smear equal to KhsAx=0.10Kh0 and ds=3dw. The analysis without smear permits to attain an idea of the maximum settlements for each cross-section, thought this is not a credible hypothesis. Graphs VII.13 to VII.20 represent the settlement curves associated with the simulations for Dike No.3, and respective profiles. From comparing the results of the calculated and measured settlements, it is difficult to find a reasonable concordance for all the sections of Dike No.3. In fact:  When the final settlements calculated come close to those measured, there is a significant difference in the initial phase;  When the settlements have similar agreement in the construction phase, then the numerical results of the pos-constructive stage are relatively lower than the readings from the plates. Looking at the different sections individually, it can be seen that: In section PK0+140, both the plates present settlements substantially lower than those calculated, not only on the construction phase but also on the filling of the reservoir. This result occurs even when considering C estimated with the PLAXIS relations, and for smear conditions given by KhsAx=0.10Kh0 and ds=3dw. The difference between the results is more significant in plate PL 140-2 as the curves come closer 3 years (1175 days = 3.2 years) after the dike construction was concluded. This time is reduced to 1 year and 3 months (475 days =1.3 years) for plate PL 140-1. Additionally, while very good concordance of the settlements in the post-construction phase is observed for the latter, on the other plate these are always slightly above the values measured (Graphs VII.13 and VII.14); CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 245 - Graph VII.13 – PL 140-1 of Dike No.3: Settlements calculated with the SSC model, considering two different values for the secondary compressibility index (C), with and without smear effect Graph VII.14 – PL 140-2 of Dike No.3: Settlements calculated with the SSC model, considering two different values for the secondary compressibility index (C), with and without smear effect 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Time [days] Dike No.3 - PL 140-1 Comparison between measured settlements and calculated with PLAXIS (Soft Soil Creep model: SSC) With geotextil, drains and considering smear (Khs=0.10Kh; ds=3dw) PL 140-1 (settlement plate readings) SSC-Csec_Terzaghi (Khs=0.10Kh; ds=3dw) SSC-Csec_Terzaghi (without smear) SSC-Csec_PLAXIS (Khs=0.10Kh; ds=3dw) SSC-Csec_PLAXIS (without smear) SSC-Csec_PLAXIS (Khs=0.10Kh; ds=3dw; B2mod) 01-05-2001 20-06-2001 Settlements [cm] Conclusion of Dike No.3 10-076-2002 -435 days Start of the raising water in reservoir 24-07-2002 -449 days Last plate measurements 31-03-2008 -2526 days 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Time [days] Dike No.3 - PL 140-2 Comparison between measured settlements and calculated with PLAXIS (Soft Soil Creep model: SSC) With geotextil, drains and considering smear (Khs=0.10Kh; ds=3dw) PL 140-2 (settlement plate readings) SSC-Csec_Terzaghi (Khs=0.10Kh; ds=3dw) SSC-Csec_Terzaghi (without smear) SSC-Csec_PLAXIS (Khs=0.10Kh; ds=3dw) SSC-Csec_PLAXIS (without smear) SSC-Csec_PLAXIS (Khs=0.10Kh; ds=3dw; B2mod) 01-05-2001 21-11-2001 Settlements [cm] Conclusion of Dike No.3 10-076-2002 -435 days Start of the raising water in reservoir 24-07-2002 -449 days Last plate measurements 31-03-2008 -2526 days CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 246 -  In section PK0+170, the results from the simulations are coincident in the construction phase for plate PL 170-1, being much lower than the values measured in the post-construction phase. In turn, in plate PL 170-2 , it is observed precisely the opposite, meaning that there is concordance in the progress of the settlements for the post-construction phase, but in the initial phase the values calculated slightly exceed the real values. Discarding the simulation in which the smear effect is not considered, then, in general, the results which are closer to the plates readings correspond to C based on PLAXIS and smear equal to KhsAx=0.10Kh0 and ds=3dw (Graphs VII.15 and VII.16);  In section PK0+200, excellent concordance is once again seen between the settlements calculated and measured for plate PL200-1 in the construction phase, although they are unable to traduce the evolution of the real settlements afterwards. In plate PL 200-2, again the settlements calculated are greater than those monitored in the construction phase, although this difference tends to decrease over time (Graphs VII.17 and VII.18). From the comparison of the plate settlements numerical results, the fact that they present identical behaviour for plates PL170-1 and PL200-1 stands out, with the same occurring with plates PL 140-2, PL 170-2 and PL 200-2, although some slight differences are noted for these last plates. This would not be surprising, as long as these remarks were similar in all the plates, allowing conclusions to be withdrawn regarding to the parameters of the soils foundation in Dike No.3. As this is not the case, the following questions are immediately raised: a) Are these plates exactly placed in the coordinates indicated before, which were considered in the simulations? b) Are the compressibility parameters and permeability coefficients adopted for the foundation soil of Dike No.3 the most suitable? These doubts are related more to plates PL 140-2, PL 170-2 and PL 200-2, which are located near the chimney drain and the horizontal downstream drain, and near the area improved with band-shaped drains. Looking at the results from the simulations, it is noticed that for these plates they always exceed the plate’s readings, unlike to what happens with the other plates. This means, that the drainage in the simulation is occurring much faster than it happens in reality, or that the compressibility estimated for the soil foundation is too high. This second hypothesis is set aside, since the settlements for the plates located in the centre of the cut-off are quite good in the construction phase. Consequently, there is still the issue of the location of these plates and the soil horizontal permeability. To clarify the last issue, and since the horizontal permeability coefficient for stratum B2 of Dike No.3 is three times higher than the one adopted for Dike No.1, an additional calculation was made. For this calculation, the same horizontal permeability of Dike No.1 was considered for B2. The results obtained are shown in graphs VII.13 to VII.18. From their analysis and comparison, it can be seen that the decrease of settlements is of little significance for the sections treated with drains, and even less in the PK0+200, where there is no drains. Thus, given the results, it seems that these differences can be related to the location of the plates. If this is the case, they should be at a higher level and farther from the draining elements. In any case, it is very premature to take conclusions at this moment, as the results with the flow are yet to be analysed. CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 247 - Graph VII.15 – PL 170-1 of Dike No.3: Settlements calculated with the SSC model, considering two different values for the secondary compressibility index (C), with and without smear effect Graph VII.16 – PL 170-2 of Dike No.3: Settlements calculated with the SSC model, considering two different values for the secondary compressibility index (C), with and without smear effect 0 5 10 15 20 25 30 35 40 45 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Time [days] Dike No.3 - PL 170-1 Comparison between measured settlements and calculated with PLAXIS (Soft Soil Creep model: SSC) With geotextil, drains and considering smear (Khs=0.10Kh; ds=3dw) PL 170-1 (settlement plate readings) SSC-Csec_Terzaghi (Khs=0.10Kh; ds=3dw) SSC-Csec_Terzaghi (without smear) SSC-Csec_PLAXIS (Khs=0.10Kh; ds=3dw) SSC-Csec_PLAXIS (without smear) SSC-Csec_PLAXIS (Khs=0.10Kh; ds=3dw; B2mod) 01-05-2001 28-06-2001 Settlements [cm] Conclusion of Dike No.3 10-076-2002 -435 days Start of the raising water in reservoir 24-07-2002 -449 days Last plate measurements 31-03-2008 -2526 days 0 5 10 15 20 25 30 35 40 45 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Time [days] Dike No.3 - PL 170-2 Comparison between measured settlements and calculated with PLAXIS (Soft Soil Creep model: SSC) With geotextil, drains and considering smear (Khs=0.10Kh; ds=3dw) PL 170-2 (settlement plate readings) SSC-Csec_Terzaghi (Khs=0.10Kh; ds=3dw) SSC-Csec_Terzaghi (without smear) SSC-Csec_PLAXIS (Khs=0.10Kh; ds=3dw) SSC-Csec_PLAXIS (without smear) SSC-Csec_PLAXIS (Khs=0.10Kh; ds=3dw; B2mod) 01-05-2001 21-11-2001 Settlements [cm]QQ Conclusion of Dike No.3 10-076-2002 -435 days Start of the raising water in reservoir 24-07-2002 -449 days Last plate measurements 31-03-2008 -2526 days CONSOLIDATION OF SOFT SOILS IN THE PRESENCE OF BAND-SHAPED DRAINS VII. MODELLING OF DIKES NO.1 AND NO.3 OF THE LEBRIJA POND UNIVERSITY OF SEVILLE – SUPERIOR TECHNICAL SCHOOL OF ARCHITECTURE - 248 - Graph VII.17 – PL 200-1 of Dike No.3: Settlements calculated with the SSC model, considering two different values for the secondary compressibility index (C) Graph VII.18 – PL 200-2 of Dike No.3: Settlements calculated with the SSC model, considering two different values for the secondary compressibility index (C) 0 5 10 15 20 25 30 35 40 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Time [days] Dike No.3 - PL 200-1 Comparison between measured settlements and calculated with PLAXIS (Soft Soil Creep model: SSC) Without drains PL 200-1 (settlement plate readings) SSC-Csec_PLAXIS (without drains) SSC-Csec_Terzaghi (without drains) SSC-Csec_PLAXIS (without drains; KhB2mod) 01-05-2001 27-05-2001 Settlements [cm] Conclusion of Dike No.3 10-076-2002 -435 days Start of the raising water in reservoir 24-07-2002 -449 days Last plate measurements 31-03-2008 -2526 days 0 5 10 15 20 25 30 35 40 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Time [days] Dike No.3 - PL 200-1 Comparison between measured settlements and calculated with PLAXIS (Soft Soil Creep model: SSC) Without drains PL 200-2 (settlement plate readings) SSC-Csec_PLAXIS (without drains) SSC-Csec_Terzaghi (without drains) SSC-Csec_PLAXIS (without drains; KhB2mod) 01-05-2001 27-05-2001 Settlements [cm] Conclusion of Dike No.3 10-076-2002 -435 days Start of the raising water in reservoir 24-07-2002 -449 days Last plate measurements 31-03-2008 -2526 days [Document text truncated for crawler view.]