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08/06/ Stavanger-CFD Workshop Peppino Terpolilli TFE-Pau Flow in porous media: physical, mathematical and numerical aspects

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- CFD Stavanger 2 08/06/2014 OUTLINE Darcy lawDarcy law Mathematical issuesMathematical issues Some models: Black-oil, Dead-oil,Buckley- Leverett………Some models: Black-oil, Dead-oil,Buckley- Leverett……… Numerical approachNumerical approach

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- CFD Stavanger 3 08/06/2014 Darcy law Navier-Stokes equations:Navier-Stokes equations: Darcy law:Darcy law: is the matrix of permeability: porous is the matrix of permeability: porous media characteristic media characteristic

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- CFD Stavanger 4 08/06/2014 Darcy law Continuum mechanics:Continuum mechanics: at a REV located at : at a REV located at : porosity: ratio of void to bulk volume porosity: ratio of void to bulk volume permeability: Darcy law permeability: Darcy law REV REV

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- CFD Stavanger 5 08/06/2014 Darcy law Darcy law:Darcy law: empirical law (Darcy in 1856) empirical law (Darcy in 1856) theoretical derivation:theoretical derivation: Scheidegger, King Hubbert, Matheron (heuristic) Scheidegger, King Hubbert, Matheron (heuristic) Tartar (homogeneization theory) Tartar (homogeneization theory) Stokes Darcy law Stokes Darcy law

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- CFD Stavanger 6 08/06/2014 Darcy law Poiseuille flow in a tube:Poiseuille flow in a tube: single-phase, horizontal flow single-phase, horizontal flow steady and laminar steady and laminar no entrance and exit effects no entrance and exit effects mean velocity mean velocity radius radius length length pressure gradient pressure gradient

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- CFD Stavanger 7 08/06/2014 Darcy law Poiseuille flow in a tube:Poiseuille flow in a tube: unit: darcy unit: darcy

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- CFD Stavanger 8 08/06/2014 Darcy law Different scale:Different scale: pore level: Stokes equations pore level: Stokes equations lab: measures lab: measures numerical cell: upscaling numerical cell: upscaling field: heterogeneity field: heterogeneity Darcy law Darcy law Darcy law Darcy law

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- CFD Stavanger 9 08/06/2014 Black-oil model Extended Darcy law:Extended Darcy law: relative permeability of phase p relative permeability of phase p the depth the depth

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- CFD Stavanger 10 08/06/2014 Darcy law Continuum mechanics:Continuum mechanics: at a REV located at : at a REV located at : saturation: fraction of pore volume saturation: fraction of pore volume relative permeability relative permeability capillary pressure capillary pressure REV REV

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- CFD Stavanger 11 08/06/2014 Kr-pc

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- CFD Stavanger 12 08/06/2014 Kr-pc

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- CFD Stavanger 13 08/06/2014 Math issues For single-phase flows Darcy law leads to linear equation:For single-phase flows Darcy law leads to linear equation: For multi-phase flow we recover nonlinear equtions: hyperbolic, degenerate parabolic etc…..For multi-phase flow we recover nonlinear equtions: hyperbolic, degenerate parabolic etc…..

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- CFD Stavanger 14 08/06/2014 Math issues The mathematical model is a system of PDE with appropriate initial and boundary conditionsThe mathematical model is a system of PDE with appropriate initial and boundary conditions the coefficients of the equations are poorly known stochastic approachthe coefficients of the equations are poorly known stochastic approach geology + stochastic = geostatisticgeology + stochastic = geostatistic

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- CFD Stavanger 15 08/06/2014 A field….

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- CFD Stavanger 16 08/06/2014 Math issuesData: wells : core, well-logging, well test wells : core, well-logging, well test extension: geophysic, geologyextension: geophysic, geology scale problems and uncertaintyscale problems and uncertainty (geostatistic) (geostatistic)

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- CFD Stavanger 17 08/06/2014 Uncertainty SPDE:SPDE: These problems are difficult:These problems are difficult: experimental design approach experimental design approach Grand projet incertitude Grand projet incertitude Industrial tools Industrial tools

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- CFD Stavanger 18 08/06/2014 Black-oil model Hypotesis:Hypotesis: three phases: 2 hydrocarbon phases and three phases: 2 hydrocarbon phases and water water hydrocarbon system: 2 components hydrocarbon system: 2 components a non-volatile oil a non-volatile oil a volatile gas soluble in the oil phase a volatile gas soluble in the oil phase

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- CFD Stavanger 19 08/06/2014 Black-oil model Hypotesis:Hypotesis: components phases components phases oil oil oil oil gas oil gas oil gas gas gas gas water water water water

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- CFD Stavanger 20 08/06/2014 Black-oil model phases: phases: water: wetting saturation water: wetting saturation oil : partially wetting saturation oil : partially wetting saturation gas : non wetting saturation gas : non wetting saturation

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- CFD Stavanger 21 08/06/2014 Black-oil model Validity of the hypothesis:Validity of the hypothesis: dry gas dry gas depletion, immiscible water or gas injection depletion, immiscible water or gas injection oil with small volatility oil with small volatility

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- CFD Stavanger 22 08/06/2014 Black-oil model PVT behaviour: formation volume factorPVT behaviour: formation volume factor where:where: volume of a fixed mass at reservoir volume of a fixed mass at reservoir conditions conditions volume of a fixed mass at stock tank volume of a fixed mass at stock tank conditions conditions

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- CFD Stavanger 23 08/06/2014 Black-oil model Mass transfer between oil and gas phases:Mass transfer between oil and gas phases: : gas component in the oil phase : gas component in the oil phase : oil component in the oil phase : oil component in the oil phase functions of the oil phase pressure functions of the oil phase pressure

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- CFD Stavanger 24 08/06/2014 Black-oil model Thermo functions for oil:Thermo functions for oil:

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- CFD Stavanger 25 08/06/2014 Black-oil model Mass balance:Mass balance: water water oil oil gas gas

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- CFD Stavanger 26 08/06/2014 Black-oil model Extended Darcy law:Extended Darcy law: relative permeability of phase p relative permeability of phase p the depth the depth

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- CFD Stavanger 27 08/06/2014 Black-oil model Water:Water: oil:oil: gaz:gaz:

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- CFD Stavanger 28 08/06/2014 Black-oil model saturation:saturation: capillary pressures:capillary pressures: we obtain 3 equations with 3 unknowns:we obtain 3 equations with 3 unknowns:

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- CFD Stavanger 29 08/06/2014 Black-oil model:boundary conditions BoundariesBoundaries closed: no flux at the extreme cells closed: no flux at the extreme cells aquifer: source term in corresponding cells aquifer: source term in corresponding cells wells:wells: Dirichlet condition: bottom pressure Dirichlet condition: bottom pressure imposed imposed Neumann condition: production rate Neumann condition: production rate imposed imposed source terms for perforated cells (PI) source terms for perforated cells (PI)

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- CFD Stavanger 30 08/06/2014 Black-oil model: initial conditions capillary and gravity equilibriumcapillary and gravity equilibrium pressure imposed in oil zone at a given depthpressure imposed in oil zone at a given depth oil pressure in all cells and then Pc curvesoil pressure in all cells and then Pc curves

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- CFD Stavanger 31 08/06/2014 Black-oil model: theoretical results Antonsev, Chavent, Gagneux:Antonsev, Chavent, Gagneux: existence results for weak solutions PME: porous media equationPME: porous media equation more resuts: Barenblatt, Zeldovich, Benedetti,…Vazquez.

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