Sede Amministrativa: Università degli Studi di Padova Dipartimento di Ingegneria Civile, Edile e Ambientale (ICEA) SCUOLA DI DOTTORATO DI RICERCA IN: SCIENZE DELL’INGEGNERIA CIVILE ED AMBIENTALE XXVII CICLO MULTIPHASE MATERIAL MODELLING IN FINITE DEFORMATIONS: THEORETICAL ASPECTS, NUMERICAL IMPLEMENTATION AND APPLICATIONS Direttore della Scuola: Ch.mo Prof. Stefano Lanzoni Supervisori: Ch.mo Prof. C. Majorana Dott.ssa Valentina Salomoni Dottorando: Nicolò Spiezia 2 This doctoral thesis has been approved as follows: Prof. Lanzoni. Director of the PhD program Prof. Majorana . Thesis Supervisor Dr. Salomoni . Thesis Supervisor Nicolò Spiezia . PhD Candidate 4 Abstract Multiphase materials are of primary relevance in many disciplines in engineering and science. They consist of a porous solid skeleton, whose pores are filled by one or more fluid phases, suchas water, air, gas, oil etc. The relevant number of applications involving multiphase material motivated the development of several theoretical models and numerical procedures to describe the coupled behavior between the different phases. Most of these models, based on the formulation of thebalance laws for the coexisting phases, rely upon the assumption of small strains, which is a simplifying but restrictive hypothesis for several applications. The aim of this thesis is to investigate the theoretical aspects and the numerical solutions of a multiphase material undergoing large elastoplastic strains, taking into account the fully coupling between the solid and the fluid phases. The essential idea of the model consists in imposing the balance laws for the two (or more) phases in the current deformed configuration, and to solve it numerically with a finite element method. To deal with elastoplasticity at finite strains, the formulation adopts the multiplicative decomposition of the deformation gradient. The developed numerical model has been applied, in particular, to assess the stability of a horizontal wellbore drilled through a high porous rock formation, quantifying the stress and strain distribution, the evolution of the plastic deformations and the propagation of band of intense deformation. To capture both the shear-enhanced compaction and the shear-induced dilation characteristic of porous rock, an innovative elastoplastic constitutive model has been derived, endowed with a linear and elliptical yield surface that intersect smoothly. The results of the simulations show the capability of the finite deformations coupled approach to simulate the whole process. 5 6 Sommario I materiali multifase sono di primaria importanza in molte discipline dell’ingegneria e della scienza. Essi sono costituiti da uno scheletro solido poroso, i cui pori sono riempiti da una o più fluidi, ad esempio acqua, aria, gas, petrolio, etc. Il vasto campo di applicazione dei materiali multifase ha motivato lo sviluppo di diversi modelli teorici e procedure numeriche per descrivere il comportamento accoppiato fra le diverse fasi. La maggior parte di questi modelli, basati sulla formulazione delle leggi di bilancio per le differenti fasi coesistenti, si basano sul presupposto di pic- cole deformazioni, ipotesi che comporta una semplificazione dei modelli ma allo stesso tempo risulta essere restrittiva per diverse applicazioni. Lo scopo di questa tesi è indagare gli aspetti teorici e le soluzioni numeriche di un materiale multifase che subisce grandi deformazioni elastoplasticche, tenendo conto del completo accoppiamento tra la fase solida e la fase fluida. L’idea essenziale del modello consiste nell’imporre le leggi di bilancio per le due (o più) fasi nella configurazione corrente deformata, e poi risolvere tali equazioni numeri- camente con un metodo agli elementi finiti. Per quanto concerne l’elastoplasticità a deformazioni finite, la formulazione adotta la decomposizione moltiplicativa del gradiente di deformazione. Il modello numerico sviluppato è stato applicato, in particolare, per valutare la stabilità di un pozzo perforato orizzontalmente attraverso una formazione rocciosa altamente porosa, per quantificare la distribuzione delle tensioni e delle deformazioni, per descrivere l’evoluzione delle deformazioni plas- tiche e la propagazione di bande di deformazione. Per cogliere tanto il fenomeno di compattazione e di dilatazione plastica, caratteristico di rocce ad alta porosità, è stato sviluppato un innovativo modello costitutivo elastoplastico, dotato di una superficie lineare e di una superficie ellittica che si intersecano mantenendo la derivabilità in ogni punto. I risultati delle simulazioni mostrano la capacità dell’approccio a grandi deformazioni per simulare l’intero processo accoppiato. 7 8 Acknowledgments This work would not have been possible without the encouragement and help of so many people that may be difficult to count all of them in these acknowledgments. I would like to express my deep gratitude to Prof. Majorana and Dr. Salomoni, my research supervisors, for their patient guidance during the PhD, fundamental support and useful critiques of this research work. I would like to express my very great gratitude to Prof. Borja for his valuable and constructive suggestions during the nine months at Stanford University. His constant enthusiasm, his support in every situation and the willingness to give his time so generously has been very much appreciated. I would like also to thank Prof. Regueiro and Prof. Linder, for the constructive discussions we had about the topic of this work. I would like to thank the italian colleagues at Department ICEA: Dr. Mazzucco, for his humble and extremely precious advices, Dr. Xotta, Dr. Pomaro and Dr. Fincato, for the great time we spent together, and Dr. Muraro, for being the best office mate ever. My gratitude also goes toDr. Sanavia, for the inspiring discussions and his kindness. My grateful thanks are also extended to the colleagues at Stanford University, in particular to Dr. Song, Dr. Choo, Dr. Tjioe, Dr. Bennett and Dr. Semnani for their help end friendship, and all the mates at Moreno Avenue. Finally, I wish to thank my parents and friends for their support and encouragement throughout my study. This project would have been impossible without the support of the Gini Foundation, which provided financial support during the research period at Stanford University. 9 10 Contents 1 Introduction 19 1.1 Motivation and scope . 19 1.2 Outline of the thesis . 22 1.3 Notations and symbols . 23 2 Theoretical formulation of porous media at finite strains 25 2.1 Fundamentals of continuum mechanics . 25 2.2 Kinematics of multiphase continuum . 26 2.3 Transport equations . 29 2.4 Multiphase continuum mechanics . 29 2.4.1 Multiphase continuum body . 29 2.4.2 Balance laws . 31 3 Constitutive law for the solid and fluid phases 43 3.1 Introduction . 43 3.2 Constitutive law for the solid phase . 43 3.2.1 General framework for elastoplasticity at finite strain . 45 3.2.2 Hyperelastic law . 56 3.2.3 Modified Cam Clay . 58 3.2.4 Continuous Cap model . 71 3.3 Constitutive law for the fluid phase . 98 4 Numerical implementation of the model 101 4.1 Introduction . 101 4.2 Variational equations and linearization . 102 4.3 Finite element formulation . 105 4.4 Comparison with the small strains formulation . 109 4.5 Overview of the numerical code . 109 11 5 Numerical analysis and results 125 5.1 Introduction . 125 5.2 Consolidation process under a uniformed distributed load on soft clay . 125 5.2.1 One dimensional consolidation . 126 5.2.2 Two dimensional plane strain consolidation . 130 5.3 Drilling process of an horizontal wellbore through a porous rock formation . 135 5.3.1 Introduction . 135 5.3.2 Calibration of the constitutive model . 139 5.3.3 Description of the problem . 145 5.3.4 Numerical results and considerations . 149 6 Conclusions and future developments 171 A Tangent operator for the Continuos Cap model 173 A.1 Derivation of the Jacobian matrix A in eq. (3.81) . 173 A.2 Derivation of the consistent tangent operator a푒푝 .................... 174 12 List of Figures 2-1 Typical averaging volume 푑푉 of a porous medium consisting in two phases. 30 3-1 Convergence analysis with 10 increments (left) and 100 increments (right) imposing 휖푣 = 0 and 휖푠 = 0.05..................................... 72 3-2 Convergence analysis with 10 increments (left) and 100 increments (right) imposing 휖푣 = −0.05 and 휖푠 = 0.05.................................. 73 3-3 Hydrostatic pressure vs. volume strainvolume strain for high porosity rock. 74 3-4 Two-yield surface model for high porosity rock. Loading path A (low confining pres- sure) intersects the shear yield surface, with negative slope. Loading path B (high confining pressure) intersects the cap yield surface, with positive slope. 푃 : mean normal stress; 푄: norm of second invariant of deviatoric stress. 75 3-5 The two yield surfaces model. The elliptical yield surface is tangent to the linear surface in the point defined by (푃 *, 푄*).......................... 78 3-6 The two yield surfaces model . 80 3-7 Algorithm for selection of the correct Return-Mapping algorithm. 82 4-1 Element type and intrinsic node order for quadrilateral elements. 110 5-1 One dimensional consolidation: geometry and boundary conditions. 126 5-2 One dimensional consolidation: evolution with time [푠] of the displacement [푚푚] of the top node and the pore pressure [푀푃 푎] of the bottom node. 127 5-3 Nodal Cauchy pore pressure at different time steps. 128 5-4 Cauchy effective vertical stress at different time steps. 129 5-5 Isochrones of constant Cauchy pore pressure at different time [day]. 130 5-6 Comparison between the small strain and large strain formulation (1) for one dimen- sional elastic consolidation. 131 5-7 Normalized norm of force/fluxes residual vector versus iteration number. 131 5-8 Two dimensional consolidation: geometry and boundary conditions. 132 13 5-9 Initial configuration in terms of nodal pressure (a), equivalent Von Mises stress (b) and preconsolidation pressure (c) . 133 5-10 Nodal pore pressure distribution at different time steps. 134 5-11 Equivalent Von Mises Cauchy stress distribution at different time steps. 136 5-12 Deviatoric plastic deformation at different time steps.