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

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

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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.

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SLIDE 1

Analysis and 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 model Analysis and simulation 1/16

Scaling Up and Modeling for Transport and Flow in Porous Media Dubrovnik, Croatia, 13-16 October 2008

Analysis and simulation of a two phase flow model with phase apparition/disappearance

Application to gas migration in underground nuclear waste repository

  • F. Sma¨

ı1, A. Bourgeat1, M. Jurak2

1 Institut Camille Jordan, UMR 5208 du CNRS 2 Department of mathematics, University of Zagreb, Croatia

14 October 2008

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SLIDE 2

Analysis and 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 model Analysis and simulation 2/16

Context

◮ Production of hydrogen in the storage

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SLIDE 3

Analysis and 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 model Analysis and simulation 2/16

Context

◮ Production of hydrogen in the storage ◮ Problem of the gas phase apparition/disappearance

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SLIDE 4

Analysis and 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 model Analysis and simulation 2/16

Context

◮ Production of hydrogen in the storage ◮ Problem of the gas phase apparition/disappearance

◮ 2 kinds of flow : saturated (liquid) et unsaturated (liquid/gas)

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SLIDE 5

Analysis and 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 model Analysis and simulation 2/16

Context

◮ Production of hydrogen in the storage ◮ Problem of the gas phase apparition/disappearance

◮ 2 kinds of flow : saturated (liquid) et unsaturated (liquid/gas)

◮ Saturated/unsaturated global formulation ?

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SLIDE 6

Analysis and 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 model Analysis and simulation 3/16

Outline

Unsaturated two phase flow Physical assumptions of the model Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Choice of suitable variables Formulation in (pl, X) Analysis and simulation Existence of solutions Numerical test

description implementation results

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SLIDE 7

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible)

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SLIDE 8

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen

slide-9
SLIDE 9

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen ◮ Mass conservation for each component

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SLIDE 10

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen ◮ Mass conservation for each component ◮ Generalized Darcy law for each phase

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SLIDE 11

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen ◮ Mass conservation for each component ◮ Generalized Darcy law for each phase ◮ Diffusion of dissolved hydrogen in the liquid phase

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SLIDE 12

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen ◮ Mass conservation for each component ◮ Generalized Darcy law for each phase ◮ Diffusion of dissolved hydrogen in the liquid phase ◮ Capillary pressure law : pg − pl = pc(Sg)

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SLIDE 13

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen ◮ Mass conservation for each component ◮ Generalized Darcy law for each phase ◮ Diffusion of dissolved hydrogen in the liquid phase ◮ Capillary pressure law : pg − pl = pc(Sg) ◮ Thermodynamical equilibrium between liquid solution and

gaseous mixture (Henry and Raoult laws)

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SLIDE 14

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen ◮ Mass conservation for each component ◮ Generalized Darcy law for each phase ◮ Diffusion of dissolved hydrogen in the liquid phase ◮ Capillary pressure law : pg − pl = pc(Sg) ◮ Thermodynamical equilibrium between liquid solution and

gaseous mixture (Henry and Raoult laws)

◮ Ideal gas law

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SLIDE 15

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen ◮ Mass conservation for each component ◮ Generalized Darcy law for each phase ◮ Diffusion of dissolved hydrogen in the liquid phase ◮ Capillary pressure law : pg − pl = pc(Sg) ◮ Thermodynamical equilibrium between liquid solution and

gaseous mixture (Henry and Raoult laws)

◮ Ideal gas law ◮ Isothermal flow

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SLIDE 16

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 4/16

Unsaturated two phase flow

Physical assumptions of the model

◮ 2 phases : liquid (incompressible) and gas (compressible) ◮ 2 components : water and hydrogen ◮ Mass conservation for each component ◮ Generalized Darcy law for each phase ◮ Diffusion of dissolved hydrogen in the liquid phase ◮ Capillary pressure law : pg − pl = pc(Sg) ◮ Thermodynamical equilibrium between liquid solution and

gaseous mixture (Henry and Raoult laws)

◮ Ideal gas law ◮ Isothermal flow ◮ Additional assumption : no water vapor

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SLIDE 17

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 5/16

Unsaturated two phase flow

Unsaturated flow equations

The unsaturated flow is described by : Φ∂Sl ∂t + div

  • ql − 1

GJ

  • = Fw/ρstd

l

Φ ∂ ∂t(ChSlpg + CvpgSg) + div

  • Chpgql + Cvpgqg + J
  • = Fh/ρstd

g

with usual primary variables : (pl, Sl) ou (pl, pg).

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SLIDE 18

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 5/16

Unsaturated two phase flow

Unsaturated flow equations

The unsaturated flow is described by : Φ∂Sl ∂t + div

  • ql − 1

GJ

  • = Fw/ρstd

l

Φ ∂ ∂t(ChSlpg + CvpgSg) + div

  • Chpgql + Cvpgqg + J
  • = Fh/ρstd

g

with usual primary variables : (pl, Sl) ou (pl, pg). Where we denote fluxes : ql = −Kkrl µl

  • ∇pl − (ρstd

l

+ Chρstd

g pg)g

  • ,

qg = −Kkrg µg

  • ∇pg − Cvρstd

g pgg

  • ,

J = − ΦSlF Chpg + F Dh

l Ch∇pg,

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SLIDE 19

Analysis and 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

  • f the model

Unsaturated flow equations Saturated flow Construction of a saturated/unsaturated model Analysis and simulation 5/16

Unsaturated two phase flow

Unsaturated flow equations

The unsaturated flow is described by : Φ∂Sl ∂t + div

  • ql − 1

GJ

  • = Fw/ρstd

l

Φ ∂ ∂t(ChSlpg + CvpgSg) + div

  • Chpgql + Cvpgqg + J
  • = Fh/ρstd

g

with usual primary variables : (pl, Sl) ou (pl, pg). Where we denote fluxes : ql = −Kkrl µl

  • ∇pl − (ρstd

l

+ Chρstd

g pg)g

  • ,

qg = −Kkrg µg

  • ∇pg − Cvρstd

g pgg

  • ,

J = − ΦSlF Chpg + F Dh

l Ch∇pg,

and constants : Ch = H(T)M h ρstd

g

, Cv = M h RTρstd

g

, G = ρstd

l

ρstd

g

, F = M hρstd

l

M wρstd

g

.

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SLIDE 20

Analysis and 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 model Analysis and simulation 6/16

Saturated flow

◮ Liquid saturated flow : Sl ≡ 1 and pg non definite

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SLIDE 21

Analysis and 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 model Analysis and simulation 6/16

Saturated flow

◮ Liquid saturated flow : Sl ≡ 1 and pg non definite

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

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SLIDE 22

Analysis and 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 model Analysis and simulation 6/16

Saturated flow

◮ Liquid saturated flow : Sl ≡ 1 and pg non definite

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

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SLIDE 23

Analysis and 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 model Analysis and simulation 6/16

Saturated flow

◮ Liquid saturated flow : Sl ≡ 1 and pg non definite

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

◮ We denote Rs = ρh

l

ρstd

g

, The (saturated) flow of the solution

(water + dissolved hydrogen) is described by :

div

  • ql − 1

GJ

  • = Fw/ρstd

l

Φ∂Rs ∂t +div

  • Rsql + J
  • = Fh/ρstd

g

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SLIDE 24

Analysis and 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 model Analysis and simulation 6/16

Saturated flow

◮ Liquid saturated flow : Sl ≡ 1 and pg non definite

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

◮ We denote Rs = ρh

l

ρstd

g

, The (saturated) flow of the solution

(water + dissolved hydrogen) is described by :

div

  • ql − 1

GJ

  • = Fw/ρstd

l

Φ∂Rs ∂t +div

  • Rsql + J
  • = Fh/ρstd

g ◮ Usual primary variables : (pl, Rs)

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SLIDE 25

Analysis and 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 model Analysis and simulation 6/16

Saturated flow

◮ Liquid saturated flow : Sl ≡ 1 and pg non definite

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

◮ We denote Rs = ρh

l

ρstd

g

, The (saturated) flow of the solution

(water + dissolved hydrogen) is described by :

div

  • ql − 1

GJ

  • = Fw/ρstd

l

Φ∂Rs ∂t +div

  • Rsql + J
  • = Fh/ρstd

g ◮ Usual primary variables : (pl, Rs)

How to globally describe saturated and unsaturated flows ?

slide-26
SLIDE 26

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 7/16

Construction of a saturated/unsaturated model

Choice of suitable variables

◮ Usual primary variables

slide-27
SLIDE 27

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 7/16

Construction of a saturated/unsaturated model

Choice of suitable variables

◮ Usual primary variables

◮ unsaturated : pressure/pressure or pressure/saturation

slide-28
SLIDE 28

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 7/16

Construction of a saturated/unsaturated model

Choice of suitable variables

◮ Usual primary variables

◮ unsaturated : pressure/pressure or pressure/saturation ◮ saturated : pressure/concentration

slide-29
SLIDE 29

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 7/16

Construction of a saturated/unsaturated model

Choice of suitable variables

◮ Usual primary variables

◮ unsaturated : pressure/pressure or pressure/saturation ◮ saturated : pressure/concentration

◮ Introduction of a new variable

X = RsSl + CvpgSg

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SLIDE 30

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 7/16

Construction of a saturated/unsaturated model

Choice of suitable variables

◮ Usual primary variables

◮ unsaturated : pressure/pressure or pressure/saturation ◮ saturated : pressure/concentration

◮ Introduction of a new variable

X = RsSl + CvpgSg

◮ (pl, X) is well definite in the 2 states of flow

(saturated/unsaturated)

slide-31
SLIDE 31

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 7/16

Construction of a saturated/unsaturated model

Choice of suitable variables

◮ Usual primary variables

◮ unsaturated : pressure/pressure or pressure/saturation ◮ saturated : pressure/concentration

◮ Introduction of a new variable

X = RsSl + CvpgSg

◮ (pl, X) is well definite in the 2 states of flow

(saturated/unsaturated)

◮ State of flow characterization

unsaturated : X > Ch(pl + pc(0)) saturated : X ≤ Ch(pl + pc(0))

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SLIDE 32

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 7/16

Construction of a saturated/unsaturated model

Choice of suitable variables

◮ Usual primary variables

◮ unsaturated : pressure/pressure or pressure/saturation ◮ saturated : pressure/concentration

◮ Introduction of a new variable

X = RsSl + CvpgSg

◮ (pl, X) is well definite in the 2 states of flow

(saturated/unsaturated)

◮ State of flow characterization

unsaturated : X > Ch(pl + pc(0)) saturated : X ≤ Ch(pl + pc(0)) X ≡ Rs

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SLIDE 33

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 8/16

Construction of a saturated/unsaturated model

Choice of suitable variables

Unsaturated :

usual primary variables (pl, Sl) and (pl, pg)

Φ∂Sl ∂t +div

  • ql − 1

GJ

  • = Fw/ρstd

l

Φ∂X ∂t +div

  • Rsql + Cvpgqg + J
  • = Fh/ρstd

g

Saturated :

usual primary variables (pl, Rs)

div

  • ql − 1

GJ

  • = Fw/ρstd

l

Φ∂Rs ∂t +div

  • Rsql + J
  • = Fh/ρstd

g

slide-34
SLIDE 34

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 9/16

Construction of a saturated/unsaturated model

Formulation in (pl, X)

◮ The saturated/unsaturated flow may be described with

primary variables (pl, X) by only one couple of equations : Φ ∂ ∂t(X − GSg) + div

  • (G + Rs)ql + Cvpgqg
  • = G Fw

ρstd

l

+ Fh ρstd

g

Φ∂X ∂t + div

  • Rsql + Cvpgqg + J
  • = Fh

ρstd

g

slide-35
SLIDE 35

Analysis and 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 model Choice of suitable variables Formulation in (pl, X) Analysis and simulation 9/16

Construction of a saturated/unsaturated model

Formulation in (pl, X)

◮ The saturated/unsaturated flow may be described with

primary variables (pl, X) by only one couple of equations : Φ ∂ ∂t(X − GSg) + div

  • (G + Rs)ql + Cvpgqg
  • = G Fw

ρstd

l

+ Fh ρstd

g

Φ∂X ∂t + div

  • Rsql + Cvpgqg + J
  • = Fh

ρstd

g ◮ 1st equation is parabolic/elliptic in pl,

2nde equation is parabolic in X.

slide-36
SLIDE 36

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 10/16

Analysis and simulation

Existence of solutions

◮ We consider the simplified formulation :

         Φ∂Sl ∂t + div

  • ql + 0
  • = Fw/ρstd

l

Φ∂X ∂t + div

  • Rsql + Cvpgqg + J
  • = Fh/ρstd

g

slide-37
SLIDE 37

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 10/16

Analysis and simulation

Existence of solutions

◮ We consider the simplified formulation :

         Φ∂Sl ∂t + div

  • ql + 0
  • = Fw/ρstd

l

Φ∂X ∂t + div

  • Rsql + Cvpgqg + J
  • = Fh/ρstd

g ◮ We can show the following existence result :

slide-38
SLIDE 38

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 10/16

Analysis and simulation

Existence of solutions

◮ We consider the simplified formulation :

         Φ∂Sl ∂t + div

  • ql + 0
  • = Fw/ρstd

l

Φ∂X ∂t + div

  • Rsql + Cvpgqg + J
  • = Fh/ρstd

g ◮ We can show the following existence result :

Suppose rmin ≤ Rs ≤ rmax and pl ≥ 0 and assume that initial and Dirichlet conditions are enough regular. Then there is a weak solution to the simplified formulation.

slide-39
SLIDE 39

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 10/16

Analysis and simulation

Existence of solutions

◮ We consider the simplified formulation :

         Φ∂Sl ∂t + div

  • ql + 0
  • = Fw/ρstd

l

Φ∂X ∂t + div

  • Rsql + Cvpgqg + J
  • = Fh/ρstd

g ◮ We can show the following existence result :

Suppose rmin ≤ Rs ≤ rmax and pl ≥ 0 and assume that initial and Dirichlet conditions are enough regular. Then there is a weak solution to the simplified formulation.

◮ A well chosen variable change allows to apply the

Alt-Luckhaus theorem in order to prove existence of a solution.

slide-40
SLIDE 40

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 11/16

Analysis and simulation

Numerical test : a simple configuration

slide-41
SLIDE 41

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 11/16

Analysis and simulation

Numerical test : a simple configuration

◮ Boundary conditions :

◮ Injection of pure gas on left side ◮ Impervious condition on top and bottom side ◮ Pure water (Xout = 0) at fixed pressure on right side

slide-42
SLIDE 42

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 11/16

Analysis and simulation

Numerical test : a simple configuration

◮ Boundary conditions :

◮ Injection of pure gas on left side ◮ Impervious condition on top and bottom side ◮ Pure water (Xout = 0) at fixed pressure on right side

◮ Initial conditions :

stationary state without injection (Qh

in = 0)

slide-43
SLIDE 43

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 12/16

Analysis and simulation

Numerical test : a simple configuration

◮ Van Genuchten-Mualem model for capillary pressure and

relative permeabilities

◮ Fixed temperature, T = 303 K Porous medium parameters Fluid characteristics Parameter Value Parameter Value k 5 10−20 m2 Dh

l

3 10−9 m2/s Φ 0.15 (−) µl 1 10−3 P a.s Pr 2 106 P a µg 9 10−6 P a.s n 1.49 (−) H(T = 303K) 7.65 10−6 mol/P a/m3 Slr 0.4 (−) Ml 10−2 kg/mol Sgr (−) Mg 2 10−3 kg/mol ρstd

l

103 kg/m3 ρstd

g

8 10−2 kg/m3 Parameter Value Lx 200 m Ly 20 m Qh 1.5 10−5 m/year pl,out 106 P a Tsimul 5 105 years

slide-44
SLIDE 44

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 13/16

Analysis and simulation

Numerical test : implementation

◮ Fully implicit time discretization of the space/time pde system

slide-45
SLIDE 45

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 13/16

Analysis and simulation

Numerical test : implementation

◮ Fully implicit time discretization of the space/time pde system ◮ Fixed point iteration to solve nonlinearities of the space pde

system

slide-46
SLIDE 46

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 13/16

Analysis and simulation

Numerical test : implementation

◮ Fully implicit time discretization of the space/time pde system ◮ Fixed point iteration to solve nonlinearities of the space pde

system

◮ Spatial discretization of the linear pde with a MHFE method

slide-47
SLIDE 47

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 13/16

Analysis and simulation

Numerical test : implementation

◮ Fully implicit time discretization of the space/time pde system ◮ Fixed point iteration to solve nonlinearities of the space pde

system

◮ Spatial discretization of the linear pde with a MHFE method ◮ Blockwise Gauss-Seidel method to solve the 2 coupled linear

systems

slide-48
SLIDE 48

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 13/16

Analysis and simulation

Numerical test : implementation

◮ Fully implicit time discretization of the space/time pde system ◮ Fixed point iteration to solve nonlinearities of the space pde

system

◮ Spatial discretization of the linear pde with a MHFE method ◮ Blockwise Gauss-Seidel method to solve the 2 coupled linear

systems

◮ Bi-conjugate gradient stabilized method to inverse each block

slide-49
SLIDE 49

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 13/16

Analysis and simulation

Numerical test : implementation

◮ Fully implicit time discretization of the space/time pde system ◮ Fixed point iteration to solve nonlinearities of the space pde

system

◮ Spatial discretization of the linear pde with a MHFE method ◮ Blockwise Gauss-Seidel method to solve the 2 coupled linear

systems

◮ Bi-conjugate gradient stabilized method to inverse each block ◮ Implementation with the modular code Cast3m

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SLIDE 50

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

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SLIDE 51

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

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SLIDE 52

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-53
SLIDE 53

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-54
SLIDE 54

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-55
SLIDE 55

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-56
SLIDE 56

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-57
SLIDE 57

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-58
SLIDE 58

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-59
SLIDE 59

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-60
SLIDE 60

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-61
SLIDE 61

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-62
SLIDE 62

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-63
SLIDE 63

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-64
SLIDE 64

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-65
SLIDE 65

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-66
SLIDE 66

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-67
SLIDE 67

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-68
SLIDE 68

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-69
SLIDE 69

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-70
SLIDE 70

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-71
SLIDE 71

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-72
SLIDE 72

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-73
SLIDE 73

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-74
SLIDE 74

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-75
SLIDE 75

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-76
SLIDE 76

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-77
SLIDE 77

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-78
SLIDE 78

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-79
SLIDE 79

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-80
SLIDE 80

Analysis and 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 model Analysis and simulation Existence of solutions Numerical test description implementation results 14/16

Analysis and simulation

Numerical test : results

abscissa (m) total H2 density (mol/m3)

40 80 120 160 200 4 8 12 16 20

abscissa (m) gas saturation (%)

40 80 120 160 200 0.2 0.4 0.6 0.8 1 1.2 1.4

abscissa (m) liquid pressure (MPa)

40 80 120 160 200 1 1.02 1.04 1.06 1.08 1.1 1.12

slide-81
SLIDE 81

Analysis and 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 model Analysis and simulation 15/16

Conclusions

◮ A good choice of variables allows to write a

saturated/unsaturated model

slide-82
SLIDE 82

Analysis and 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 model Analysis and simulation 15/16

Conclusions

◮ A good choice of variables allows to write a

saturated/unsaturated model

◮ We have an existence result for a simplified formulation

slide-83
SLIDE 83

Analysis and 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 model Analysis and simulation 15/16

Conclusions

◮ A good choice of variables allows to write a

saturated/unsaturated model

◮ We have an existence result for a simplified formulation ◮ A simple numerical test shows the ability of the model to

simulate the appearance and the evolution of the unsaturated area

slide-84
SLIDE 84

Analysis and 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 model Analysis and simulation 15/16

Conclusions

◮ A good choice of variables allows to write a

saturated/unsaturated model

◮ We have an existence result for a simplified formulation ◮ A simple numerical test shows the ability of the model to

simulate the appearance and the evolution of the unsaturated area Perspectives :

slide-85
SLIDE 85

Analysis and 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 model Analysis and simulation 15/16

Conclusions

◮ A good choice of variables allows to write a

saturated/unsaturated model

◮ We have an existence result for a simplified formulation ◮ A simple numerical test shows the ability of the model to

simulate the appearance and the evolution of the unsaturated area Perspectives :

◮ Numerical testing of the validity limits (mathematical and

physical)

slide-86
SLIDE 86

Analysis and 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 model Analysis and simulation 15/16

Conclusions

◮ A good choice of variables allows to write a

saturated/unsaturated model

◮ We have an existence result for a simplified formulation ◮ A simple numerical test shows the ability of the model to

simulate the appearance and the evolution of the unsaturated area Perspectives :

◮ Numerical testing of the validity limits (mathematical and

physical)

◮ Taking in account the rock change

slide-87
SLIDE 87

Analysis and 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 model Analysis and simulation 15/16

Conclusions

◮ A good choice of variables allows to write a

saturated/unsaturated model

◮ We have an existence result for a simplified formulation ◮ A simple numerical test shows the ability of the model to

simulate the appearance and the evolution of the unsaturated area Perspectives :

◮ Numerical testing of the validity limits (mathematical and

physical)

◮ Taking in account the rock change ◮ Homogenization on the storage geometry

slide-88
SLIDE 88

Analysis and 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 model Analysis and simulation 16/16

References

◮ Alt, H. W. and Luckhaus, S. (1983). Quasilinear elliptic-parabolic

differential equations. Math. Z., 183, 311-341.

◮ Bourgeat, A. and Jurak, M. and Sma¨

ı, F. (2008) Two partially miscible flow and transport modeling in porous media ; application to gas migration in a nuclear waste repository. Computational Geosciences .

◮ Sma¨

ı, F. (-) A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository. submitted