Flow in porous media: physical, mathematical and numerical aspects

1. - Stavanger-CFD Workshop Peppino Terpolilli TFE-Pau Flow in porous media:physical, mathematical and numerical aspects

2. OUTLINE • Darcy law • Mathematical issues • Some models: Black-oil, Dead-oil,Buckley-Leverett……… • Numerical approach

3. Darcy law • Navier-Stokes equations: • Darcy law: • is the matrix of permeability: porous media characteristic

4. Darcy law • Continuum mechanics: at a REV located at : porosity: ratio of void to bulk volume permeability: Darcy law REV

5. Darcy law • Darcy law: empirical law (Darcy in 1856) • theoretical derivation: Scheidegger, King Hubbert, Matheron (heuristic) Tartar (homogeneization theory) Stokes Darcy law

6. Darcy law • Poiseuille flow in a tube: single-phase, horizontal flow steady and laminar no entrance and exit effects mean velocity radius length pressure gradient

7. Darcy law • Poiseuille flow in a tube: unit: darcy

8. Darcy law • Different scale: pore level: Stokes equations lab: measures numerical cell: upscaling field: heterogeneity Darcy law Darcy law

9. Black-oil model • Extended Darcy law: • relative permeability of phase p • the depth

10. Darcy law • Continuum mechanics: at a REV located at : saturation: fraction of pore volume relative permeability capillary pressure REV

11. Kr-pc

12. Kr-pc

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

14. Math issues • The mathematical model is a system of PDE with appropriate initial and boundary conditions • the coefficients of the equations are poorly known stochastic approach • geology + stochastic = geostatistic

15. A field….

16. Math issues Data: • wells : core, well-logging, well test • extension: geophysic, geology • scale problems and uncertainty (geostatistic)

17. Uncertainty • SPDE: • These problems are difficult: experimental design approach ‘ Grand projet incertitude ’ Industrial tools

18. Black-oil model • Hypotesis: three phases: 2 hydrocarbon phases and water hydrocarbon system: 2 components a non-volatile oil a volatile gas soluble in the oil phase

19. Black-oil model • Hypotesis: components phases oil oil gas oil gas gas water water

20. Black-oil model phases: water: wetting saturation oil : partially wetting saturation gas : non wetting saturation

21. Black-oil model • Validity of the hypothesis: dry gas depletion, immiscible water or gas injection oil with small volatility

22. Black-oil model • PVT behaviour: formation volume factor • where: volume of a fixed mass at reservoir conditions volume of a fixed mass at stock tank conditions

23. Black-oil model • Mass transfer between oil and gas phases: : gas component in the oil phase : oil component in the oil phase functions of the oil phase pressure

24. Black-oil model • Thermo functions for oil:

25. Black-oil model • Mass balance: water oil gas

26. Black-oil model • Extended Darcy law: • relative permeability of phase p • the depth

27. Black-oil model • Water: • oil: • gaz:

28. Black-oil model • saturation: • capillary pressures: • we obtain 3 equations with 3 unknowns:

29. Black-oil model:boundary conditions • Boundaries closed: no flux at the extreme cells aquifer: source term in corresponding cells • wells: Dirichlet condition: bottom pressure imposed Neumann condition: production rate imposed source terms for perforated cells (PI)

30. Black-oil model: initial conditions • capillary and gravity equilibrium • pressure imposed in oil zone at a given depth • oil pressure in all cells and then Pc curves

31. Black-oil model: theoretical results • Antonsev, Chavent, Gagneux: existence results for weak solutions • PME: porous media equation more resuts: Barenblatt, Zeldovich, Benedetti,…Vazquez.