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1 Introduction to Finite Volume Scheme Roland Masson Université de Nice Sophia Antipolis Département de Mathématiques J.A. Dieudonné

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Presentation on theme: "1 Introduction to Finite Volume Scheme Roland Masson Université de Nice Sophia Antipolis Département de Mathématiques J.A. Dieudonné"— Presentation transcript:

1 1 Introduction to Finite Volume Scheme Roland Masson Université de Nice Sophia Antipolis Département de Mathématiques J.A. Dieudonné

2 Outline -Finite Volume Schemes for Elliptic Equation -1D case -Fluxes and discrete conservation equations -Discrete norms and Poincaré inequality -A priori estimates -Error estimates -Extension to 2-3D case -Two Point Flux Approximation -Discrete norms and Poincaré inequality -Error estimates -Convergence by compacity 2

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

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

5 references -Webpage: http://math.unice.fr/~massonr/Master2/Master2.htmlhttp://math.unice.fr/~massonr/Master2/Master2.html -Finite Volume Methods, Eymard, Herbin Gallouët, 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

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

7 7 Ex : paris basin Top view size : hundreds of km 1 color = 1 rock type Croûte Vertical cut a few km width Age : roughly 300 millions years

8 8 Where ? In sedimentary basins Off shore Inland

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

10 1010 Modéliser lhistoire dun bassin pétrolifère Dépôt des sédiments Enfouissement - Compaction – élévation de température Craquage – expulsion - migration Piégeage dans des réservoirs

11 1111 Cap rock ? Source rock ? ? Infill of sedimentary basins

12 1212 Base Modelisation of the infill of sedimentary basins

13 1313 Base (1)Accommodation = Tectonics - Eustasy = Tectonics - Eustasy Modelisation of the infill of sedimentary basins

14 1414 Base (1) Accommodation Modelisation of the infill of sedimentary basins (2) Sediment fluxes

15 1515 Base (1) Accommodation Modelisation of the infill of sedimentary basins

16 1616 Base (2) Sediment fluxes + source terms (1) Accommodation (3) Transport Modelisation of the infill of sedimentary basins

17 1717 (4) Simulation Base (1) Accommodation Modelisation of the infill of sedimentary basins (2) Sediment fluxes + source terms

18 18 18 Inverse Problem Seismic + Wells Data Parameters Accommodation Inversion loop Model Transport laws

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)

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) Sédiments = mixture of L lithologies

21 21 Multi-lithology stratigraphic model Conservation of Conservation of the c i inside the basin :

22 22 Multi-lithology stratigraphic model Accumulation term: Change of coordinates:

23 Modèle multi-lithologique

24 Example of a the progradation of a Delta 24

25 Example of the progradation of a Delta 25

26 Progradation of a Delta 26

27 Example of the Paris basin: 500x400 km, 40 My 27

28 Example of the Paris basin: sand + shale mixture 28


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