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**Flow in porous media: physical, mathematical and numerical aspects**

01/04/201701/04/2017 - Stavanger-CFD Workshop Peppino Terpolilli TFE-Pau

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**OUTLINE Darcy law Mathematical issues**

Some models: Black-oil, Dead-oil,Buckley-Leverett……… Numerical approach 01/04/201701/04/2017

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**Darcy law Navier-Stokes equations: Darcy law:**

is the matrix of permeability: porous media characteristic 01/04/201701/04/2017

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**Darcy law Continuum mechanics: at a REV located at :**

porosity: ratio of void to bulk volume permeability: Darcy law REV 01/04/201701/04/2017

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**Darcy law Darcy law: empirical law (Darcy in 1856)**

theoretical derivation: Scheidegger, King Hubbert, Matheron (heuristic) Tartar (homogeneization theory) Stokes Darcy law 01/04/201701/04/2017

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**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 01/04/201701/04/2017

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Darcy law Poiseuille flow in a tube: unit: darcy 01/04/201701/04/2017

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**Darcy law Different scale: pore level: Stokes equations lab: measures**

numerical cell: upscaling field: heterogeneity Darcy law Darcy law 01/04/201701/04/2017

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**Black-oil model Extended Darcy law: relative permeability of phase p**

the depth 01/04/201701/04/2017

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**Darcy law Continuum mechanics: at a REV located at :**

saturation: fraction of pore volume relative permeability capillary pressure REV 01/04/201701/04/2017

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Kr-pc 01/04/201701/04/2017

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Kr-pc 01/04/201701/04/2017

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**Math issues For single-phase flows Darcy law leads to linear equation:**

For multi-phase flow we recover nonlinear equtions: hyperbolic, degenerate parabolic etc….. 01/04/201701/04/2017

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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 01/04/201701/04/2017

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A field…. 01/04/201701/04/2017

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**Math issues Data: wells : core, well-logging, well test**

extension: geophysic, geology scale problems and uncertainty (geostatistic) 01/04/201701/04/2017

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**Uncertainty SPDE: These problems are difficult:**

experimental design approach ‘ Grand projet incertitude ’ Industrial tools 01/04/201701/04/2017

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**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 01/04/201701/04/2017

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**Black-oil model Hypotesis: components phases oil oil gas oil gas gas**

water water 01/04/201701/04/2017

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**Black-oil model phases: water: wetting saturation**

oil : partially wetting saturation gas : non wetting saturation 01/04/201701/04/2017

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**Black-oil model Validity of the hypothesis: dry gas**

depletion, immiscible water or gas injection oil with small volatility 01/04/201701/04/2017

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**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 01/04/201701/04/2017

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**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 01/04/201701/04/2017

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Black-oil model Thermo functions for oil: 01/04/201701/04/2017

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Black-oil model Mass balance: water oil gas 01/04/201701/04/2017

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**Black-oil model Extended Darcy law: relative permeability of phase p**

the depth 01/04/201701/04/2017

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Black-oil model Water: oil: gaz: 01/04/201701/04/2017

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**Black-oil model saturation: capillary pressures:**

we obtain 3 equations with 3 unknowns: 01/04/201701/04/2017

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**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 source terms for perforated cells (PI) 01/04/201701/04/2017

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**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 01/04/201701/04/2017

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**Black-oil model: theoretical results**

Antonsev, Chavent, Gagneux: existence results for weak solutions PME: porous media equation more resuts: Barenblatt, Zeldovich, Benedetti,…Vazquez. 01/04/201701/04/2017

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