Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

Multi lithology stratigraphic model under maximum erosion rate constraint

22 pages
Multi-lithology stratigraphic model under maximum erosion rate constraint R. Eymard‡ T. Gallouet D. Granjeon† R. Masson† Q.H. Tran† March 23, 2011 Abstract Non-linear single lithology or multi-lithology diffusion models have been widely used by sedimentologists and geomorphologists in the field of stratigraphic basin simulations to simulate the large scale depositional transport processes of sediments. Nevertheless, as noticed by many authors, erosion and sedimentation processes are non symmetric. Soil material must first be produced in situ by weathering processes prior to be transported by diffusion. This is usually taken into account through a prescribed maximum erosion rate of the sediments, but no mathematical description of the coupling with the diffusion model has been proposed so far. In this paper, we introduce a new mathematical formulation for the coupling of the weather limited erosion and the multi-lithology diffusion models, which appears as a non standard free boundary problem for a new variable acting as a limitor of the fluxes. One of the main advantages of this formulation, compared to existing discrete cou- pling models, is to enable the definition of efficient discretization schemes. A finite volume scheme with implicit time integration is introduced which is proved to be unconditionally stable in the l∞ norm for the sediment thickness, the sediment concentrations in the lithologies, and the flux limitor variables. A Newton algorithm with an iterative com- putation of the saturated constraints is used to solve efficiently the non-linear system resulting from the discretization.

  • maximum erosion

  • single lithology

  • time integration

  • rate constraint

  • diffusion model

  • multi-lithology diffusion

  • diffusion

  • dissymmetry between


Voir plus Voir moins
Multi-lithologystratigraphicmodelundermaximumerosionrateconstraintR.EymardT.Galloue¨t§D.GranjeonR.MassonQ.H.TranMarch23,2011AbstractNon-linearsinglelithologyormulti-lithologydiffusionmodelshavebeenwidelyusedbysedimentologistsandgeomorphologistsinthefieldofstratigraphicbasinsimulationstosimulatethelargescaledepositionaltransportprocessesofsediments.Nevertheless,asnoticedbymanyauthors,erosionandsedimentationprocessesarenonsymmetric.Soilmaterialmustfirstbeproducedinsitubyweatheringprocessespriortobetransportedbydiffusion.Thisisusuallytakenintoaccountthroughaprescribedmaximumerosionrateofthesediments,butnomathematicaldescriptionofthecouplingwiththediffusionmodelhasbeenproposedsofar.Inthispaper,weintroduceanewmathematicalformulationforthecouplingoftheweatherlimitederosionandthemulti-lithologydiffusionmodels,whichappearsasanonstandardfreeboundaryproblemforanewvariableactingasalimitorofthefluxes.Oneofthemainadvantagesofthisformulation,comparedtoexistingdiscretecou-plingmodels,istoenablethedefinitionofefficientdiscretizationschemes.Afinitevolumeschemewithimplicittimeintegrationisintroducedwhichisprovedtobeunconditionallystableinthelnormforthesedimentthickness,thesedimentconcentrationsinthelithologies,andthefluxlimitorvariables.ANewtonalgorithmwithaniterativecom-putationofthesaturatedconstraintsisusedtosolveefficientlythenon-linearsystemresultingfromthediscretization.Theefficiencyofthemodelandthenumericalschemeisillustratedon2Dand3Dbasinsimulationexamples.1IntroductionInrecentyears,therehasbeenagrowinginterestinthedevelopmentofmathematicalandnumericalmodelsinstratigraphyandsedimentologyinresponsetotheneedforquantitativemodelinginatraditionallyqualitativescience.Inthefieldofsequencestratigraphy,forwardnumericalmodelshaveshowntobeusefultoolstostudytheeffectsofeustasy,tectonics,andsedimentsupplyonfaciesdistributionandstratalgeometryofbasins.Oneofthemostimportantprocessinbasinevolutionistheerosionanddepositionmecha-nismofsediments.Authorsusuallydistinguishbetweenfluidflowanddynamic-slopemodelsofthesedimentationerosionprocesses(see[R92],[R97]).Thefirstonesusefluidflowequa-tionsandempiricalalgorithmstosimulatethetransportofsedimentsinthehydrodynamicflowfield(seee.g.[TH89]).Theyprovideanaccuratedescriptionofdepositionalprocessesforsmallscalesintimeandspace.Atlargerscalesuchasbasinscales,fluidflowmodelsareInstitutFranc¸aisduPe´trole,1et4av.deBoisPre´au92852RueilMalmaisonCedexDe´partementdeMathe´matiques,Universite´deMarnelaValle´e,5boulevardDescartes,ChampssurMarne,F-77454,Marneslavalle´e,Cedex2.§LATP,Universite´deProvence,39rueFre´de´ricJoliotCurie,13453Marseillecedex13.1
computationallytooexpensiveanddynamic-slopeapproachesareusuallypreferred.Theselattermodelsactatamoremacroscopicscaleandusemassconservationequationsofsedimentscombinedwithdiffusivetransportlawsthataverageoverseveralprocesses(suchasrivertransport,creep,slumpsandsmallslides).Theyprovideagooddescriptionofdepositionalprocessesfortimescaleslargerthan,say,105yrandbasinspacescales(see[AH89],[R92],[TS94],[R97],[G97],[GJD98],[GJ99]).Inmulti-lithologyapproaches,sedimentsaremodeledasamixtureofseverallithologiescharacterizedbydifferentgrainsizepopulations.Thesetofequationsaccountsforthemassconservationofeachlithologyinthebasin,knowingthesurficialfluxes,averticalcompactionmodel(usuallygivenbydepth-porosityempiricallawsforthelithologies),andanisostaticmodelforthelithosphereflexure(usuallytakenasabeamequation).Themainsurficialtransportprocessisamulti-lithologydiffusionmodelintroducedby[R92]forwhichfluxesareproportionaltotheslopeofthetopographyaswellasalithologyfractiondefinedatthesurfaceofbasin(seealso[G97],[GJ99],[QAD00]).Diffusioncoeffi-cientsarenon-linearfunctionsoftheelevation(orequivalentlythebathymetry)tomodelthetransitionfromnon-marinetomarinediffusionthatcandifferfromuptooneortwoorderofmagnitude.However,itiswellknownthatsedimentationanderosionarenonsymmetricprocesses.Tobetransportedbysurficialprocesses,asdescribedby[AH89],materialmustfirstbeproducedinsitubyweatheringprocessesdependingonclimate,elevation,compactionofsediments,...Thisproductionismodeledin[AH89]byaweatheringratealsocalledsoilproductionratedependingonthesedimentdepthwithparametersfunctionofclimate.Then,oneachcellofthediscretemodel,thesedimentfluxesattheedgesareconstrainedsuchthaterosioncannotexceedtheavailablesoilthickness.In[G97]or[GJ99],themodelisslightlydifferentinthesensethatitdirectlyprescribesamaximumerosionrate(definedasthepartialtimederivativeofthesedimentthickness)dependingonclimate,elevation,composition,andtheageofthesediment.Asillustratedby[AH89],iftheweatheringrateislow,itcanbecomethedominantpro-cessforerosionwhichisnolongergovernedbyadiffusiontransportwhilesedimentationisstilldiffusiontransportlimited.Itappearsthatthecouplingoftheweatherlimitedmodelandthediffusiontransportmodelisanessentialissueformodelingdepositionalprocesses.Nevertheless,thisquestionisnotclearlyaddressedin[AH89]andisonlyreferredtoas“adiffusivetransportwithfluxeslimitedbyavailablesediment”withoutadetaileddescriptionofthecouplingbetweenbothprocesses.Sofar,wearenotawareofanymathematicaldescriptionofthecouplingbetweenweatheringlimitedanddiffusivetransportmodels.Themainobjectiveofthispaperistoproposesuchamathematicalmodelfromwhichweareabletoderivemoreefficientnumericalalgorithmsforstratigraphicsimulations.Amodeltakingintoaccountthedissymmetrybetweenerosionandsedimentationisalsoproposedin[R97].Theauthorusesacorrectionofthediffusioncoefficientbyaporosityratio(sedimentporosityoverthedepositionalporosity)equaltooneforsedimentationandlowerthanoneforerosion.Thisratioactsasafluxlimitor,but,beingindependentonthefluxes(ortheslope),itwillfailtocorrectlyfreezetheerosionatthemaximumsoilproductionrate.Similarlyapointwisedependencyofthediffusioncoefficientontheerosionratewillalsofailsince,asweshallseeinthesubsequentdevelopment,thefluxlimitorsatisfyingtheerosion2
Un pour Un
Permettre à tous d'accéder à la lecture
Pour chaque accès à la bibliothèque, YouScribe donne un accès à une personne dans le besoin