Analysis and simulation of a two phase flow F. Sma , A. Bourgeat, - PowerPoint PPT Presentation

Scaling Up and Modeling for Transport and Flow in Porous Media Analysis and simulation of a two Dubrovnik, Croatia, 13-16 October 2008 phase flow model with phase appari- tion/disappearance Analysis and simulation of a two phase flow F. Sma¨ ı, A. Bourgeat, M. Jurak model with phase apparition/disappearance Unsaturated two phase flow Application to gas migration in underground nuclear waste Saturated flow repository Construction of a saturated/unsaturated model Analysis and simulation ı 1 , A. Bourgeat 1 , M. Jurak 2 F. Sma¨ 1 Institut Camille Jordan, UMR 5208 du CNRS 2 Department of mathematics, University of Zagreb, Croatia 14 October 2008 1/16

Analysis and Context simulation of a two phase flow model with phase appari- tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase flow Saturated flow Construction of a saturated/unsaturated ◮ Production of hydrogen in the storage model Analysis and simulation 2/16

Analysis and Context simulation of a two phase flow model with phase appari- tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase flow Saturated flow Construction of a saturated/unsaturated ◮ Production of hydrogen in the storage model Analysis and ◮ Problem of the gas phase apparition/disappearance simulation 2/16

Analysis and Context simulation of a two phase flow model with phase appari- tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase flow Saturated flow Construction of a saturated/unsaturated ◮ Production of hydrogen in the storage model Analysis and ◮ Problem of the gas phase apparition/disappearance simulation ◮ 2 kinds of flow : saturated (liquid) et unsaturated (liquid/gas) 2/16

Analysis and Context simulation of a two phase flow model with phase appari- tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase flow Saturated flow Construction of a saturated/unsaturated ◮ Production of hydrogen in the storage model Analysis and ◮ Problem of the gas phase apparition/disappearance simulation ◮ 2 kinds of flow : saturated (liquid) et unsaturated (liquid/gas) ◮ Saturated/unsaturated global formulation ? 2/16

Analysis and Outline simulation of a two phase flow model with phase appari- tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase flow Physical assumptions of the model Unsaturated two phase flow Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated Saturated flow model Analysis and simulation Construction of a saturated/unsaturated model Choice of suitable variables Formulation in ( p l , X ) Analysis and simulation Existence of solutions Numerical test description implementation results 3/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions of the model Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations ◮ Mass conservation for each component Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations ◮ Mass conservation for each component Saturated flow ◮ Generalized Darcy law for each phase Construction of a saturated/unsaturated model Analysis and simulation 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations ◮ Mass conservation for each component Saturated flow ◮ Generalized Darcy law for each phase Construction of a saturated/unsaturated ◮ Diffusion of dissolved hydrogen in the liquid phase model Analysis and simulation 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations ◮ Mass conservation for each component Saturated flow ◮ Generalized Darcy law for each phase Construction of a saturated/unsaturated ◮ Diffusion of dissolved hydrogen in the liquid phase model Analysis and ◮ Capillary pressure law : p g − p l = p c ( S g ) simulation 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations ◮ Mass conservation for each component Saturated flow ◮ Generalized Darcy law for each phase Construction of a saturated/unsaturated ◮ Diffusion of dissolved hydrogen in the liquid phase model Analysis and ◮ Capillary pressure law : p g − p l = p c ( S g ) simulation ◮ Thermodynamical equilibrium between liquid solution and gaseous mixture (Henry and Raoult laws) 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations ◮ Mass conservation for each component Saturated flow ◮ Generalized Darcy law for each phase Construction of a saturated/unsaturated ◮ Diffusion of dissolved hydrogen in the liquid phase model Analysis and ◮ Capillary pressure law : p g − p l = p c ( S g ) simulation ◮ Thermodynamical equilibrium between liquid solution and gaseous mixture (Henry and Raoult laws) ◮ Ideal gas law 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations ◮ Mass conservation for each component Saturated flow ◮ Generalized Darcy law for each phase Construction of a saturated/unsaturated ◮ Diffusion of dissolved hydrogen in the liquid phase model Analysis and ◮ Capillary pressure law : p g − p l = p c ( S g ) simulation ◮ Thermodynamical equilibrium between liquid solution and gaseous mixture (Henry and Raoult laws) ◮ Ideal gas law ◮ Isothermal flow 4/16

Analysis and Unsaturated two phase flow simulation of a two phase flow model with phase appari- Physical assumptions of the model tion/disappearance F. Sma¨ ı, A. Bourgeat, M. Jurak Unsaturated two phase ◮ 2 phases : liquid (incompressible) and gas (compressible) flow Physical assumptions ◮ 2 components : water and hydrogen of the model Unsaturated flow equations ◮ Mass conservation for each component Saturated flow ◮ Generalized Darcy law for each phase Construction of a saturated/unsaturated ◮ Diffusion of dissolved hydrogen in the liquid phase model Analysis and ◮ Capillary pressure law : p g − p l = p c ( S g ) simulation ◮ Thermodynamical equilibrium between liquid solution and gaseous mixture (Henry and Raoult laws) ◮ Ideal gas law ◮ Isothermal flow ◮ Additional assumption : no water vapor 4/16