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1 Introduction to Finite Volume Scheme Roland Masson Universit de Nice Sophia Antipolis Dpartement de Mathmatiques J.A. Dieudonn Slide 2 Outline -Finite Volume Schemes for Elliptic Equation -1D case -Fluxes and discrete conservation equations -Discrete norms and Poincar inequality -A priori estimates -Error estimates -Extension to 2-3D case -Two Point Flux Approximation -Discrete norms and Poincar inequality -Error estimates -Convergence by compacity 2 Slide 3 Outline -Finite Volume Schemes for Parabolic equations -Euler implicit and explicit time integration schemes -A priori estimates -Error estimates -Finite Volume Schemes for hyperbolic equations -Two Point Flux Monotone schemes (1-2-3 D) -Discrete BV estimates (1D) -Discrete Entropy inequality (1D) -Convergence (1D) 3 Slide 4 Schedule -Courses: 14h-17h each Monday, room 2 -Theory: 09/12, 16/12, 06/01, 13/01, 20/01, 27/01, 3/02 -Numerical project: 10/02, 17/02, 03/03 -Final Exam: 10/03, 14h-17h 4 Slide 5 references -Webpage: http://math.unice.fr/~massonr/Master2/Master2.htmlhttp://math.unice.fr/~massonr/Master2/Master2.html -Finite Volume Methods, Eymard, Herbin Gallout, Handbook of Numerical Analysis -http://www.cmi.univ-mrs.fr/~herbin/PUBLI/bookevol.pdfhttp://www.cmi.univ-mrs.fr/~herbin/PUBLI/bookevol.pdf -Introduction to FV schemes for scalar hyperbolic equations by Jerome Droniou -http://users.monash.edu.au/~jdroniou/jaca_summer_school/poly_jaca_ droniou.cr.pdfhttp://users.monash.edu.au/~jdroniou/jaca_summer_school/poly_jaca_ droniou.cr.pdf -Numerical project (stratigraphic model) -http://math.unice.fr/~massonr/articles/CG2008_GM.pdfhttp://math.unice.fr/~massonr/articles/CG2008_GM.pdf -http://math.unice.fr/~massonr/articles/SINUM2005_EGGM.pdfhttp://math.unice.fr/~massonr/articles/SINUM2005_EGGM.pdf -http://math.unice.fr/~massonr/articles/M2AN2004_GM.pdf 5 Slide 6 6 Petroleum and sedimentary basins hydrocarbures Porous rock Petroleum = oil rock (latin petra and oleum ) Petroleum reservoir Modelisation of the formation of oil reservoirs in sedimentary basins Reservoir: oil trap in sedimentary basins Slide 7 7 Ex : paris basin Top view size : hundreds of km 1 color = 1 rock type Crote Vertical cut a few km width Age : roughly 300 millions years Slide 8 8 Where ? In sedimentary basins Off shore Inland Slide 9 Data exploration Cost of exploration seismic : 10 to 30 M$ Inland well drilling to 3000m : 2 to 10 M$, off shore : 15 to 30 M$, deep off shore (>500m) : 100M$ roughly, 1 exploration well out of 5 find oil in new exploration zones direct observations Seismic Well drilling Data acquisition Slide 10 1010 Modliser lhistoire dun bassin ptrolifre Dpt des sdiments Enfouissement - Compaction lvation de temprature Craquage expulsion - migration Pigeage dans des rservoirs Slide 11 1111 Cap rock ? Source rock ? ? Infill of sedimentary basins Slide 12 1212 Base Modelisation of the infill of sedimentary basins Slide 13 1313 Base (1)Accommodation = Tectonics - Eustasy = Tectonics - Eustasy Modelisation of the infill of sedimentary basins Slide 14 1414 Base (1) Accommodation Modelisation of the infill of sedimentary basins (2) Sediment fluxes Slide 15 1515 Base (1) Accommodation Modelisation of the infill of sedimentary basins Slide 16 1616 Base (2) Sediment fluxes + source terms (1) Accommodation (3) Transport Modelisation of the infill of sedimentary basins Slide 17 1717 (4) Simulation Base (1) Accommodation Modelisation of the infill of sedimentary basins (2) Sediment fluxes + source terms Slide 18 18 18 Inverse Problem Seismic + Wells Data Parameters Accommodation Inversion loop Model Transport laws Slide 19 19 Stratigraphic Model with a single lithology Transport law: q s = k(b) b (m 2 /s) Conservation of h(x,t) (sediment thickness): z x Sea level Sediment flux: g (m 2 /s) b(x,t) h(x,t) a(x,t) Slide 20 Multi-lithology Stratigraphic Model Transport law: q i,s = k i (b) C i s b x z b(x,t) C i (x,,t) C i s (x,t) Sdiments = mixture of L lithologies Slide 21 21 Multi-lithology stratigraphic model Conservation of Conservation of the c i inside the basin : Slide 22 22 Multi-lithology stratigraphic model Accumulation term: Change of coordinates: Slide 23 Modle multi-lithologique Slide 24 Example of a the progradation of a Delta 24 Slide 25 Example of the progradation of a Delta 25 Slide 26 Progradation of a Delta 26 Slide 27 Example of the Paris basin: 500x400 km, 40 My 27 Slide 28 Example of the Paris basin: sand + shale mixture 28