Andrea Villa Three Dimensional Geophysical Modeling: from Physics to Numerical Simulation Ledizioni LediPublishing Andrea Villa Department of Mathematics University of Milan Via C. Saldini n.50 20133 Milan - Italy Series Editors: Vincenzo Capasso, Coordinator of the PhD Program in Mathematics and Statistics for Computational Sciences; Alfredo Lorenzi, Coordinator of the PhD Program in Mathematics; Bernhard Ruf, Director of the Graduate School. Editorial Assistant: Stefania Leonardi © 2010 Andrea Villa Open Access copy available at air.unimi.it First Edition: September 2010 ISBN 978-88-95994-14-7 All rights reserved for the print copy. For the digital copy rights please refer to air.unimi.it Ledizioni LediPublishing Via Alamanni 11 – 20141 Milano – Italy http://www.ledipublishing.com [email protected] Andrea Villa Three Dimensional Geophysical Modeling: from Physics to Numerical Simulation September15,2010 In memoryofFaustoSaleri Acknowledgements Iwishtoacknowledgetheactualandformerjuniormembersofthenumericalgeol- ogygroupatMOX:AnnaScotti,DavideSoresinetti,MatteoLongoni,FrancoDassi, CristianoMalossi,MarcoCuffaroandAlessioFumagalli. InamgratefultotheseniormembersofthenumericalgeologygroupatMOX:Prof. AlfioQuarteroni,Prof.LucaFormaggia,Dr.FabioNobile,Dr.EdieMiglio,Dr.Si- monaPerotto,Dr.CarloD’Angelo. ManythankstoP.RuffoandG.Scrofaniforhavingsupportedourresearchprogram. IwishalsotothankLauraBarbareschiandGabrieleVillafortheirlinguisticalhelp andadvices. vii Contents 1 Introduction................................................... 1 2 Anoverviewonbasingeology.................................... 5 2.1 Aninsightintosedimentarybasins ............................ 5 2.2 Sedimentarybasingeology .................................. 6 2.3 Motivationsfornumericalsimulations ......................... 9 2.4 Uncertainty................................................ 9 3 Mathematicalmodelsforbasinsimulationandcompaction ......... 11 3.1 Compactionmodels ........................................ 12 3.1.1 Thegeometryofthebasin............................. 12 3.1.2 Volumeaveragingtheorems ........................... 13 3.1.3 Momentumequationaverage .......................... 16 3.1.4 Massequationaveraging.............................. 21 3.1.5 Averagedenergybalance.............................. 22 3.1.6 Constitutiveequations ................................ 23 3.1.7 Thecompletemacroscopicmodels...................... 25 3.1.8 Approximatedmodels ................................ 27 3.2 Numericalsolutionforcompactionmodels ..................... 29 3.2.1 Timediscretizationalgorithmfor(3.50) ................. 29 3.2.2 Timediscretizationalgorithmfor(3.51) ................. 30 3.2.3 Algorithmanalysis................................... 31 3.3 Numericalsolutionforaone-dimensionalcase .................. 33 3.3.1 Theone-dimensionalcaseandnumericaltechniques....... 33 3.3.2 Numericalresults .................................... 34 3.4 Nomenclature.............................................. 39 4 Multi-fluidtracking ............................................ 41 4.1 Abriefreviewoftheliterature................................ 41 4.2 Theproposedmethod ....................................... 42 4.3 Analysisofthemethod...................................... 48 ix x Contents 4.4 Convergenceanalysisoftheonedimensionalcase ............... 52 4.5 Numericalresults........................................... 55 4.6 Results ................................................... 55 4.7 Nomenclature.............................................. 59 5 Numericalmodelsforbasicbasinsimulations ..................... 61 5.1 Physicalandmathematicalmodels ............................ 62 5.2 Timediscretization ......................................... 63 5.3 Velocityfieldsolver ........................................ 64 5.4 Tracking.................................................. 71 5.5 Algorithmworkflow ........................................ 76 5.6 Numericalresults........................................... 77 5.7 Nomenclature.............................................. 84 6 NumericalModelsforadvancedbasinsimulations ................. 87 6.1 Physicalandmathematicalmodels ............................ 89 6.1.1 Nomenclature ....................................... 89 6.1.2 Rheologicalmodels .................................. 90 6.1.3 Modelingthecompaction ............................. 91 6.1.4 Themodel.......................................... 93 6.2 Numericaldiscretization..................................... 93 6.3 Adaptivegridmovement .................................... 95 6.3.1 Gridmovementequations ............................. 95 6.3.2 Errorestimation ..................................... 98 6.3.3 Metricdefinition.....................................101 6.4 Thehandlingoffaults.......................................103 6.4.1 Afinitevolumescheme...............................103 6.4.2 Setreconstruction(continuouspart).....................104 6.4.3 Setreconstruction(discretepart) .......................106 6.4.4 Thereconstructionalgorithmappliedtoasimplecase......107 6.4.5 Localgridrefinement.................................108 6.5 Algorithmworkflow ........................................109 6.6 Numericalresults...........................................110 7 Conclusions ...................................................117 References.........................................................119