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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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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) • Black Oil is considered as a particular case of Compositional • General framework • Examples
Introduction to Petroleum Industry • Seismic map • Geologic model • Analogs • interpretation • Reservoir model • Logs • Cores
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
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,
Compositional Model - cont. Volume balance laws - total volume balance = pore volume - total volume of fluids - “pressure equation” moles of comp i
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.
Compositional Model - cont. Simplify - no capillary pressure, no saturation equations. Linearize (typically first discretize then linearize)
Compositional Model - cont. x x We have to discretize Wang, Yotov, Wheeler, et. al introduced (possible problems for non smooth solution)
Grids pinchouts
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
General framework • Model problem written as a system • Find such that (primal dual MFEM) Find such that
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
Examples(Ware, Parrott, and Rogers) • Theorem: • Numerical example:
Examples(Ware, Parrott, and Rogers) • -error of the pressure, • - error of the pressure, • -error of the flux, • - error of the flux
Examples • Voronoi/Donald mesh / full tensor
Unstructured (Voronoi) grids - scalar coefficient (M. “Finite Volume Methods on Voronoi Meshes”,Numer. Meth. PDE, Herbin, et. al.) What about approximation
Unstructured (Voronoi) grids For grids with extra regularity Hypothesis: The approximation of could be improved with post-processing.
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
Examples(Voronoi/Donald mesh / full tensor) • Discrete problem: • Findsuch that
Error estimates • Findsuch that
Black Oil Model Phases Components + + Gas + Oil + + + Water Standard Conditions ReservoirConditions ReservoirConditions
Black Oil Model Phases - liquid, vapor, aqueous components - oil, gas, water C/P l v a o x g x x w x
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)
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
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