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Ilya D. Mishev IMA Workshop on Compatible Discretizations May 11-16 14, 2004

Why Mixed Finite Elements are not used in the Petroleum Industry and what can we do about it ?. Ilya D. Mishev IMA Workshop on Compatible Discretizations May 11-16 14, 2004. Outline. Introduction to Petroleum Industry Compositional model (Black Oil model)

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Ilya D. Mishev IMA Workshop on Compatible Discretizations May 11-16 14, 2004

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  1. Why Mixed Finite Elements are not used in the Petroleum Industry and what can we do about it ? Ilya D. Mishev IMA Workshop on Compatible Discretizations May 11-16 14, 2004

  2. Outline • Introduction to Petroleum Industry • Compositional model (Black Oil model) • Black Oil is considered as a particular case of Compositional • General framework • Examples

  3. Introduction to Petroleum Industry • Seismic map • Geologic model • Analogs • interpretation • Reservoir model • Logs • Cores

  4. Compositional Model Phases - liquid (l), vapor (v), and aqueous (a) - components - methane, ethane, propane, etc., - phases, components n Conservation of mass Volume V overall concentration, component flow rate, sources and sinks, saturation of phase j. porosity, molar density of phase j, mole fraction of component i, mole fraction of component i in phase j

  5. Compositional Model - cont. • Darcy law (generalized) absolute permeability, relative permeability of phase j, mass density of phase j, gravitational acceleration, depth. phase velocity, phase pressure, capillary pressure, viscosity of phase j,

  6. Compositional Model - cont. Volume balance laws - total volume balance = pore volume - total volume of fluids - “pressure equation” moles of comp i

  7. Compositional Model - cont. Volume balance laws - liquid phase volume balance = volume of liquid - liquid saturation “saturation of oil equation” The equations for the other phase saturations are similar.

  8. Compositional Model - cont. Simplify - no capillary pressure, no saturation equations. Linearize (typically first discretize then linearize)

  9. Compositional Model - cont. x x We have to discretize Wang, Yotov, Wheeler, et. al introduced (possible problems for non smooth solution)

  10. Grids pinchouts

  11. General framework • Given general cell centered grid • build dual grid to approximate the fluxes, • choose approximation space for the pressure, • define local approximation of the flux on the dual volume, • exclude fluxes to get the finite volume method

  12. General framework • Model problem written as a system • Find such that (primal dual MFEM) Find such that

  13. Examples • Rectangular grid / full tensor (Ware, Parrott, and Rogers) • Dual grid - rectangles, • - piecewise constants, • - piecewise constant • vectors with continuous normals • Basis vectors M., “Analysis of a new Mixed Finite Volume Method”, Comp. Methods Appl. Math. V. 3, 2003

  14. Examples(Ware, Parrott, and Rogers)

  15. Examples(Ware, Parrott, and Rogers) • Theorem: • Numerical example:

  16. Examples(Ware, Parrott, and Rogers) • -error of the pressure, • - error of the pressure, • -error of the flux, • - error of the flux

  17. Examples • Voronoi/Donald mesh / full tensor

  18. Unstructured (Voronoi) grids - scalar coefficient (M. “Finite Volume Methods on Voronoi Meshes”,Numer. Meth. PDE, Herbin, et. al.) What about approximation

  19. Unstructured (Voronoi) grids For grids with extra regularity Hypothesis: The approximation of could be improved with post-processing.

  20. Examples(Voronoi/Donald mesh / full tensor) • Dual grid - triangles, • - linear piecewise continuous functions, • - piecewise constants on Voronoi volumes, • - piecewise constants on triangles • with continuousnormals M. “A New Flexible Mixed Finite Volume Method”, submitted

  21. Examples(Voronoi/Donald mesh / full tensor) • Discrete problem: • Findsuch that

  22. Error estimates • Findsuch that

  23. Extra

  24. Black Oil Model Phases Components + + Gas + Oil + + + Water Standard Conditions ReservoirConditions ReservoirConditions

  25. Black Oil Model Phases - liquid, vapor, aqueous components - oil, gas, water C/P l v a o x g x x w x

  26. Black Oil Model Phases - liquid, vapor, aqueous components - oil, gas, water C/P l v a o x w x No mass transfer between the phase If only 2 phases exist total velocity, global pressure (Chavent, Jaffre)

  27. Black Oil Model Phases - liquid, vapor, aqueous components - oil, gas, water C/P l v a o x x g x x x w x

  28. Examples • Quadrilateral mesh / full tensor (M. Edwards et. al.) pressure • Dual grid - cell-centers connected with the middles of the edges/faces • - piecewise linears • (nonconforming space) • - piecewise constants on • the cell • -piecewise constants with • continuous normals (4 dof), • - piecewise constants (8 dof) pressure pressure to be eliminated velocity

  29. Examples (quads)

  30. Example (quads)

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