Flow in porous media: physical, mathematical and numerical aspects 13/04/201 - Stavanger-CFD Workshop 5 Peppino Terpolilli TFE-Pau OUTLINE • Darcy law • Mathematical issues • Some models: Black-oil, Dead-oil,BuckleyLeverett……… • Numerical approach 13/04/ - CFD Stavanger 2 Darcy law • Navier-Stokes equations: v 1 vv p v f t • Darcy law: q • K 13/04/ - CFD Stavanger K p is the matrix of permeability: porous media characteristic 3 Darcy law • Continuum mechanics: at a REV located at x : ( x) K ( x) 13/04/ - CFD Stavanger porosity: ratio of void to bulk volume permeability: Darcy law REV 4 Darcy law • Darcy law: empirical law (Darcy in 1856) • theoretical derivation: Scheidegger, King Hubbert, Matheron (heuristic) Tartar (homogeneization theory) Stokes 13/04/ - CFD Stavanger G Darcy law 5 Darcy law • Poiseuille flow in a tube: single-phase, horizontal flow steady and laminar no entrance and exit effects R 2 p v 8 L v R L p 13/04/ - CFD Stavanger mean velocity radius length pressure gradient 6 Darcy law • Poiseuille flow in a tube: K R2 8 unit: darcy m2 1012 m2 13/04/ - CFD Stavanger 7 Darcy law • Different scale: pore level: Stokes equations lab: measures numerical cell: upscaling field: heterogeneity Darcy law 13/04/ - CFD Stavanger G Darcy law 8 Black-oil model • Extended Darcy law: qp Kkrp p ( p p p gD) • krp relative permeability of phase p • D the depth 13/04/ - CFD Stavanger 9 Darcy law • Continuum mechanics: at a REV located at x : So , w, g kr (S ) pc (S ) 13/04/ - CFD Stavanger saturation: fraction of pore volume relative permeability capillary pressure REV 10 Kr-pc 13/04/ - CFD Stavanger 11 Kr-pc 13/04/ - CFD Stavanger 12 Math issues • For single-phase flows Darcy law leads to linear equation: C p div ( K ( x).p) f t • For multi-phase flow we recover nonlinear equtions: hyperbolic, degenerate parabolic etc….. 13/04/ - CFD Stavanger 13 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 K ( x, ) 13/04/ - CFD Stavanger 14 A field…. 13/04/ - CFD Stavanger 15 Math issues Data: • wells : core, well-logging, well test • extension: geophysic, geology • scale problems and uncertainty (geostatistic) 13/04/ - CFD Stavanger 16 Uncertainty • SPDE: p div ( K ( x, ).p ) f t • These problems are difficult: experimental design approach ‘ Grand projet incertitude ’ Industrial tools 13/04/ - CFD Stavanger 17 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 13/04/ - CFD Stavanger 18 Black-oil model • Hypotesis: components oil oil gas water oil gas gas water 13/04/ - CFD Stavanger phases 19 Black-oil model phases: water: wetting oil : partially wetting gas : non wetting 13/04/ - CFD Stavanger 20 Sw So Sg saturation saturation saturation Black-oil model • Validity of the hypothesis: dry gas depletion, immiscible water or gas injection oil with small volatility 13/04/ - CFD Stavanger 21 Black-oil model • PVT behaviour: formation volume factor Bo Vo Vdg RC Vo STC ; Vg RC Bg Vg STC ; VW RC Bw VW STC • where: volume of a fixed mass at reservoir VRC conditions VSTC 13/04/ - CFD Stavanger volume of a fixed mass at stock tank conditions 22 Black-oil model • Mass transfer between oil and gas phases: Vdg RS V o STC Vdg : gas component in the oil phase Vo : oil component in the oil phase functions of the oil phase pressure 13/04/ - CFD Stavanger 23 Black-oil model • Thermo functions for oil: 160 140 120 1,35 1,3 1,1 1,25 1 1,2 0,9 1,15 0,8 0,7 1,1 1,05 0 100 200 300 400 P Bo 0,6 muo 0,5 1 0 100 200 P (bars) 24 300 0,4 400 muo (cP) 100 80 60 40 - CFD Stavanger 1,4 1,2 20 0 13/04/ 1,4 1,3 Bo (Sm3/m3) Rs(m3/m3) 200 180 Black-oil model • Mass balance: water oil gas 13/04/ - CFD Stavanger (S w w ) div( w qw ) Qw t (So o ) div ( o qo ) Qo t (S g g So dg ) div ( g q g dg qo ) Qg Qdg t 25 Black-oil model • Extended Darcy law: qp Kkrp p ( p p p gD) • krp relative permeability of phase p • D the depth 13/04/ - CFD Stavanger 26 Black-oil model • Water: • oil: • gaz: Kkrw Sw div Pw wgZ 0 t Bw Bw w Kkro So div Po ogZ 0 t Bo Bo o Kkrg Sg So Rs Pg ggZ div t Bg Bo Bg g KkroRs div Po ogZ 0 Bo o 13/04/ - CFD Stavanger 27 Black-oil model • saturation: So Sw Sg 1 • capillary pressures: pw po pcow p g po pcog • we obtain 3 equations with 3 unknowns: 13/04/ - CFD Stavanger po , Sw , Sg if po pb po , Sw , Rs if po pb 28 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) 13/04/ - CFD Stavanger 29 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 13/04/ - CFD Stavanger 30 Black-oil model: theoretical results • Antonsev, Chavent, Gagneux: existence results for weak solutions • PME: porous media equation more resuts: Barenblatt, Zeldovich, Benedetti,…Vazquez. 13/04/ - CFD Stavanger 31