[PPT] - Modeling compressible Multiphase porous media Alain Bourgeat, PowerPoint Presentation

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 1/26

Modeling compressible Multiphase Flow and Transport in saturated-unsaturated porous media: Phase appearance-disappearance

Application to gas migration in underground nuclear waste repository Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb

RICAM; Linz- Austria; Oct3-7, 2011

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 2/26

Introduction

The Context

◮ in a deep geological Nuclear Waste Repository there could be

hydrogen gas generation due to corrosion of the steel engineered barriers

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 2/26

Introduction

The Context

◮ in a deep geological Nuclear Waste Repository there could be

hydrogen gas generation due to corrosion of the steel engineered barriers

◮ the flow could be saturated (≃ only the liquid phase) in

some regions and unsaturated (a ”liquid + gas” mixture) in other ones

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 2/26

Introduction

The Context

◮ in a deep geological Nuclear Waste Repository there could be

hydrogen gas generation due to corrosion of the steel engineered barriers

◮ the flow could be saturated (≃ only the liquid phase) in

some regions and unsaturated (a ”liquid + gas” mixture) in other ones

◮ Problem in simulations, using standard models, when there is

a phase appearance/disappearance

– - variable switching or variable substitution – - small artificial non-zero saturation

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 3/26

Introduction

A repository Figure: A Repository Zone; Waste Packages(containers sets); Storage Units

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 4/26

Introduction

Engineered Barriers Figure: LONG TERM RISKS, from pressure buid-up: hydraulic head pressure gradient ⇒RN transport enhancement; mechanical damaging

f the host rock and the barrier; perturbation of the seals resaturation;

...!

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 5/26

Saturated-Unsaturated two phases flow

Physical assumptions -(i)

For simplicity, but consistant, with modelling gas migration in deep geological Nuclear Waste Repositories :

◮ 2 phases : liquid (incompressible), denoted l and gas

◮ 2 components : water,w and and hydrogen,h ◮ Mass conservation for each component w, h ◮ Generalized Darcy law for each phase , α ∈ {g, l}

qα = −Kkr,α(Sα) µα (∇pα − ραg) (1)

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 5/26

Saturated-Unsaturated two phases flow

Physical assumptions -(i)

For simplicity, but consistant, with modelling gas migration in deep geological Nuclear Waste Repositories :

◮ 2 phases : liquid (incompressible), denoted l and gas

(compressible)denoted g

◮ 2 components : water,w and and hydrogen,h ◮ Mass conservation for each component w, h ◮ Generalized Darcy law for each phase , α ∈ {g, l}

qα = −Kkr,α(Sα) µα (∇pα − ραg) (1)

◮ local mechanical equilibrium of phases ≈ Capillary pressure

law : pg − pl = pc(Sg).

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 5/26

Saturated-Unsaturated two phases flow

Physical assumptions -(i)

For simplicity, but consistant, with modelling gas migration in deep geological Nuclear Waste Repositories :

◮ 2 phases : liquid (incompressible), denoted l and gas

(compressible)denoted g

◮ 2 components : water,w and and hydrogen,h ◮ Mass conservation for each component w, h ◮ Generalized Darcy law for each phase , α ∈ {g, l}

qα = −Kkr,α(Sα) µα (∇pα − ραg) (1)

◮ local mechanical equilibrium of phases ≈ Capillary pressure

law : pg − pl = pc(Sg).

◮ Ideal gaseous Mixture of Ideal gas,with Dalton’s law of partial

pressures, and diluted liquid solution

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 5/26

Saturated-Unsaturated two phases flow

Physical assumptions -(i)

For simplicity, but consistant, with modelling gas migration in deep geological Nuclear Waste Repositories :

◮ 2 phases : liquid (incompressible), denoted l and gas

(compressible)denoted g

◮ 2 components : water,w and and hydrogen,h ◮ Mass conservation for each component w, h ◮ Generalized Darcy law for each phase , α ∈ {g, l}

qα = −Kkr,α(Sα) µα (∇pα − ραg) (1)

◮ local mechanical equilibrium of phases ≈ Capillary pressure

law : pg − pl = pc(Sg).

◮ Ideal gaseous Mixture of Ideal gas,with Dalton’s law of partial

pressures, and diluted liquid solution

◮ Isothermal flow, with the two phases locally at the same

temperature

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 5/26

Saturated-Unsaturated two phases flow

Physical assumptions -(i)

For simplicity, but consistant, with modelling gas migration in deep geological Nuclear Waste Repositories :

◮ 2 phases : liquid (incompressible), denoted l and gas

(compressible)denoted g

◮ 2 components : water,w and and hydrogen,h ◮ Mass conservation for each component w, h ◮ Generalized Darcy law for each phase , α ∈ {g, l}

qα = −Kkr,α(Sα) µα (∇pα − ραg) (1)

◮ local mechanical equilibrium of phases ≈ Capillary pressure

law : pg − pl = pc(Sg).

◮ Ideal gaseous Mixture of Ideal gas,with Dalton’s law of partial

pressures, and diluted liquid solution

◮ Isothermal flow, with the two phases locally at the same

temperature

◮ No chemical reactions

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 6/26

Saturated-Unsaturated two phases flow

Physical assumptions -(ii)

◮ Diffusion of component i in phase α ( and infinite dilution )

jh

l = −ΦSlD∇ρh l ,

jw

l = −jh l ,

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 6/26

Saturated-Unsaturated two phases flow

Physical assumptions -(ii)

◮ Diffusion of component i in phase α ( and infinite dilution )

jh

l = −ΦSlD∇ρh l ,

jw

l = −jh l , ◮ Local thermodynamical equilibrium liquid solution/ gas

mixture pw

g = xw g pg = pw v exp

(pg − pc) − pw

v

RTρw,∗

l

/M w

xw

l

(2) ph

g = xh gpg = Kh H exp

pl − p0

RT/vh,∞

l

xh

l ;

(3) ∼Raoult-Kelvin and Henry laws; with: xi

α, the i-component molar concentration in the

α-phase, pw

v the pure water saturated vapor pressure , ρw,∗ l

the pure liquid water mass density, Kh

H the Henry’s constant

at p0 (a reference pressure) , vh,∞

l

the hydrogen molar concentration at infinite dilution .

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Phase diagram Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 7/26

Unsaturated two-phase flow

An example of Unsaturated liquid+gas mixture flow (no vaporized water)

The components mass conservation reads : Φ ∂ ∂t (Slρw

l ) + div (ρw l ql + jw l ) = Fw,

(4) Φ ∂ ∂t

Slρh

l + Sgρg

+ div
ρh

l ql + ρgqg + jh l

= Fh;

(5) Usual primary variables: (pl, Sl), (pl, pg), (pl, pc).

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Phase diagram Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 7/26

Unsaturated two-phase flow

An example of Unsaturated liquid+gas mixture flow (no vaporized water)

The components mass conservation reads : Φ ∂ ∂t (Slρw

l ) + div (ρw l ql + jw l ) = Fw,

(4) Φ ∂ ∂t

Slρh

l + Sgρg

+ div
ρh

l ql + ρgqg + jh l

= Fh;

(5) Usual primary variables: (pl, Sl), (pl, pg), (pl, pc). Where : ρi

l, the i-component mass concentration;

ρg ≃ ρh

g = Cvpg( = ideal gas); pg = pc + pl;

ρh

l = Chpg( = Henry’s law) .

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Phase diagram Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 7/26

Unsaturated two-phase flow

An example of Unsaturated liquid+gas mixture flow (no vaporized water)

The components mass conservation reads : Φ ∂ ∂t (Slρw

l ) + div (ρw l ql + jw l ) = Fw,

(4) Φ ∂ ∂t

Slρh

l + Sgρg

+ div
ρh

l ql + ρgqg + jh l

= Fh;

(5) Usual primary variables: (pl, Sl), (pl, pg), (pl, pc). Where : ρi

l, the i-component mass concentration;

ρg ≃ ρh

g = Cvpg( = ideal gas); pg = pc + pl;

ρh

l = Chpg( = Henry’s law) .

and constants : Ch = M h Kh

H

, Cv = M h RT (M h = Hydrogen molar mass) .

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Phase diagram Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 8/26

Unsaturated/Saturated flow

Phase Diagram

ρh

l

pl ρh

l = Ch(pl + pc(0))

Sg = 0 ρh

l ≤ Ch(pl + pc(0))

Sg > 0 ρh

l = Chpg

≥ pl + pc(0) pg = pl + pc(Sg)

Figure: Henry’s law:ρh

l = Chpg. Localization of the saturated state,

Sg = 0, and the unsaturated state, Sg > 0 .

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Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 9/26

Saturated flow ( One phase

Liquid saturated flow

◮ Sl ≡ 1, pg is then indeterminate

SLIDE 22

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 9/26

Saturated flow ( One phase

Liquid saturated flow

◮ Sl ≡ 1, pg is then indeterminate

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

SLIDE 23

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 9/26

Saturated flow ( One phase

Liquid saturated flow

◮ Sl ≡ 1, pg is then indeterminate

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

SLIDE 24

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 9/26

Saturated flow ( One phase

Liquid saturated flow

◮ Sl ≡ 1, pg is then indeterminate

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

◮ The

liquid solution flow (water + dissolved hydrogen) is described by : div

ρw

l ql − jh l

= Fw, Φ∂ρh

l

∂t + div

ρh

l ql + jh l

= Fh;

(6) ql = −Kλl(1)

∇pl − (ρw

l + ρh l )g

,

jh

l = −ΦD∇ρh l .

(7)

SLIDE 25

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 9/26

Saturated flow ( One phase

Liquid saturated flow

◮ Sl ≡ 1, pg is then indeterminate

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

◮ The

liquid solution flow (water + dissolved hydrogen) is described by : div

ρw

l ql − jh l

= Fw, Φ∂ρh

l

∂t + div

ρh

l ql + jh l

= Fh;

(6) ql = −Kλl(1)

∇pl − (ρw

l + ρh l )g

,

jh

l = −ΦD∇ρh l .

(7)

◮ Primary variables are then (pl, ρh l ); the amount of dissolved

hydrogen, ρh

l ,is now an independent unknown( no more

related to the pg); and a gas phase cannot be taken in account.

SLIDE 26

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 9/26

Saturated flow ( One phase

Liquid saturated flow

◮ Sl ≡ 1, pg is then indeterminate

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

◮ The

liquid solution flow (water + dissolved hydrogen) is described by : div

ρw

l ql − jh l

= Fw, Φ∂ρh

l

∂t + div

ρh

l ql + jh l

= Fh;

(6) ql = −Kλl(1)

∇pl − (ρw

l + ρh l )g

,

jh

l = −ΦD∇ρh l .

(7)

◮ Primary variables are then (pl, ρh l ); the amount of dissolved

hydrogen, ρh

l ,is now an independent unknown( no more

related to the pg); and a gas phase cannot be taken in account.

◮ the couple (Saturation,Phase Pressure) cannot describe the

flow in such a liquid saturated region ( one-phase flow).

SLIDE 27

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 9/26

Saturated flow ( One phase

Liquid saturated flow

◮ Sl ≡ 1, pg is then indeterminate

◮ Classical Darcy law for liquid flow (water + dissolved

hydrogen)

◮ Dissolved hydrogen transported by diffusion and convection

◮ The

liquid solution flow (water + dissolved hydrogen) is described by : div

ρw

l ql − jh l

= Fw, Φ∂ρh

l

∂t + div

ρh

l ql + jh l

= Fh;

(6) ql = −Kλl(1)

∇pl − (ρw

l + ρh l )g

,

jh

l = −ΦD∇ρh l .

(7)

◮ Primary variables are then (pl, ρh l ); the amount of dissolved

hydrogen, ρh

l ,is now an independent unknown( no more

related to the pg); and a gas phase cannot be taken in account.

◮ the couple (Saturation,Phase Pressure) cannot describe the

flow in such a liquid saturated region ( one-phase flow). How to have a unique model for both saturated and unsaturated flows ?

SLIDE 28

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 10/26

Construction of a saturated(1-phase)/unsaturated(2-phases ) model

◮ According to the phase state , the primary variables are:

SLIDE 29

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 10/26

Construction of a saturated(1-phase)/unsaturated(2-phases ) model

◮ According to the phase state , the primary variables are:

◮ two-phase unsaturated : Liquid pressure pl/Phase Saturation

Sl; like in (4)-(5)

SLIDE 30

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 10/26

Construction of a saturated(1-phase)/unsaturated(2-phases ) model

◮ According to the phase state , the primary variables are:

◮ two-phase unsaturated : Liquid pressure pl/Phase Saturation

Sl; like in (4)-(5)

◮ one-phase saturated : Liquid pressure pl/Hydrogen mass

concentration, ρh

l , like in (6) - (7)

SLIDE 31

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 10/26

Construction of a saturated(1-phase)/unsaturated(2-phases ) model

◮ According to the phase state , the primary variables are:

◮ two-phase unsaturated : Liquid pressure pl/Phase Saturation

Sl; like in (4)-(5)

◮ one-phase saturated : Liquid pressure pl/Hydrogen mass

concentration, ρh

l , like in (6) - (7)

◮ Considering pl and ρh l as main variables in both cases

saturated and unsaturated , then eqs. (4)-(7) reduce to a unique couple of equations: Φ ∂ ∂t (Slρw

l ) + div (ρw l ql + jw l ) = Fw,

(8) Φ ∂ ∂t

ρh

tot

+ div
ρh

l ql + ρgqg + jh l

= Fh ;

(9)

SLIDE 32

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 10/26

Construction of a saturated(1-phase)/unsaturated(2-phases ) model

◮ According to the phase state , the primary variables are:

◮ two-phase unsaturated : Liquid pressure pl/Phase Saturation

Sl; like in (4)-(5)

◮ one-phase saturated : Liquid pressure pl/Hydrogen mass

concentration, ρh

l , like in (6) - (7)

◮ Considering pl and ρh l as main variables in both cases

saturated and unsaturated , then eqs. (4)-(7) reduce to a unique couple of equations: Φ ∂ ∂t (Slρw

l ) + div (ρw l ql + jw l ) = Fw,

(8) Φ ∂ ∂t

ρh

tot

+ div
ρh

l ql + ρgqg + jh l

= Fh ;

(9)

◮ with ρh tot = Slρh l + CvpgSg ; Cvph g = ρh g (∼ ideal gas)

SLIDE 33

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 11/26

Construction of a saturated(1-phase)/unsaturated(2-phases) model

Choice of suitable variables

◮ the Total Hydrogen mass concentration ρh tot ≡ ρh l Sl + ρh gSg ,

is definite in both states of flow (according to the flow state):

◮ two-phases unsaturated state :

ρh

tot = a(Sg)(pl + pc(Sg)) ; Sg > 0

(10) with: a(Sg) = Ch(1 − Sg) + CvSg ∈ [Ch, Cv]. (11)

◮ one-phase saturated :

ρh

tot = ρh l ; Sg = 0.

SLIDE 34

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 11/26

Construction of a saturated(1-phase)/unsaturated(2-phases) model

Choice of suitable variables

◮ the Total Hydrogen mass concentration ρh tot ≡ ρh l Sl + ρh gSg ,

is definite in both states of flow (according to the flow state):

◮ two-phases unsaturated state :

ρh

tot = a(Sg)(pl + pc(Sg)) ; Sg > 0

(10) with: a(Sg) = Ch(1 − Sg) + CvSg ∈ [Ch, Cv]. (11)

◮ one-phase saturated :

ρh

tot = ρh l ; Sg = 0.

◮ There is now two possible choices for the main variables in

eq. (8) and (9):

tot

◮ In system (8)-(9) we compute Sg = Sg(pl, ρh tot), (from

(11)); Sl = 1 − Sg, and ρh

l = ρh l (pl, ρh tot) = min(Chpg(pl, ρh tot), ρh tot),

pg(pl, ρh

tot) =

pl + pc(Sg(pl, ρh

tot)). ◮ Noticing a() is > Ch, increasing and pg > pl + pc(0); the

State of flow is then characterized by: unsaturated : ρh

tot > Ch(pl + pc(0))

saturated : ρh

tot ≤ Ch(pl + pc(0))

ρh

tot ≡ ρh l

SLIDE 38

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 12/26

Choice i: saturated(1-phase)/unsaturated(2-phases ) model

Liquid pressure, pl / Total Hydrogen concentration, ρh

tot

◮ In system (8)-(9) we compute Sg = Sg(pl, ρh tot), (from

(11)); Sl = 1 − Sg, and ρh

l = ρh l (pl, ρh tot) = min(Chpg(pl, ρh tot), ρh tot),

pg(pl, ρh

tot) =

pl + pc(Sg(pl, ρh

tot)). ◮ Noticing a() is > Ch, increasing and pg > pl + pc(0); the

State of flow is then characterized by: unsaturated : ρh

tot > Ch(pl + pc(0))

saturated : ρh

tot ≤ Ch(pl + pc(0))

ρh

tot ≡ ρh l ◮ Then:

◮ 1st equation is parabolic(unsaturated)/elliptic(saturated) in pl ◮ 2nde equation is parabolic in ρh

tot

SLIDE 39

applies, and the existence of a solution could be proved (F.

Sma¨ ı, PhD Thesis) .

Suppose rmin ≤ ρh

tot ≤ rmax and pl ≥ 0 and assume that

initial and Dirichlet conditions are enough regular. Then there is a weak solution to the simplified formulation.

◮ Could also certainly be investigated using ”Entropy weak

solutions”, defined by Kruzkov (hyperbolic) and extended by Carillo (parabolic). Remarks:

◮ no need of capillary pressure for this formulation

(h-component eq. in (6) Parabolic − → Hyperbolic)

◮ Coefficients in the div operators and in the

∂ ∂t operators

could become discontinuous

SLIDE 42

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 14/26

Choice ii: saturated(1-phase)/unsaturated(2-phases )

Liquid pressure, pl / Hydrogen mass concentration, ρh

l

◮ Introducing in system (8)-(9), from (pc)−1, the extended

gas-phase Pressure p∗

g = π + pl; and the extended saturation

S∗

g = f

ρh

l

Ch − pl

.

Sg pc(Sg) 1 π = ρh

l

Ch − pl

f(π) 1

Figure: pc = pg − pl; p∗

g = ρh

l

Ch .

SLIDE 43

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 14/26

Choice ii: saturated(1-phase)/unsaturated(2-phases )

Liquid pressure, pl / Hydrogen mass concentration, ρh

l

◮ Introducing in system (8)-(9), from (pc)−1, the extended

gas-phase Pressure p∗

g = π + pl; and the extended saturation

S∗

g = f

ρh

l

Ch − pl

.

Sg pc(Sg) 1 π = ρh

l

Ch − pl

f(π) 1

Figure: pc = pg − pl; p∗

g = ρh

l

Ch .

Analysis and simulation; Numerical Tests

Advertising

Ongoing benchmark on:

”Modelling Multiphase Flows”

http://www.gdrmomas.org/Benchmark/multiphase/ multiphasique.html

SLIDE 47

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 17/26

Analysis and simulation; Quasi-1D scale field numerical simulations

Pb 1, Pb 2, Pb 3 in Numerical Test Data Base

Total Hydrogen concentration, ρh

tot is denoted X in the following

SLIDE 48

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 17/26

Analysis and simulation; Quasi-1D scale field numerical simulations

Pb 1, Pb 2, Pb 3 in Numerical Test Data Base

Total Hydrogen concentration, ρh

tot is denoted X in the following ◮ Boundary conditions :

◮ Injection of pure gas on left side ◮ Impervious condition on top and bottom side ◮ Pure water (Xout = 0) (Test 1)

r

Two-phases(Xout = 0)(Test 2) and a fixed pressure, on the right side

SLIDE 49

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 17/26

Analysis and simulation; Quasi-1D scale field numerical simulations

Pb 1, Pb 2, Pb 3 in Numerical Test Data Base

Total Hydrogen concentration, ρh

tot is denoted X in the following ◮ Boundary conditions :

◮ Injection of pure gas on left side ◮ Impervious condition on top and bottom side ◮ Pure water (Xout = 0) (Test 1)

r

Two-phases(Xout = 0)(Test 2) and a fixed pressure, on the right side

◮ Initial conditions :

stationary state without injection (Qh

in = 0)

SLIDE 50

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 18/26

Analysis and simulation

Quasi-1D scale field numerical simulations

◮ 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

For more, see : http://sources.univ-lyon1.fr/cas test.html

SLIDE 51

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 19/26

Analysis and simulation

Numerical test : Implementation

◮ Fully implicit time discretization of the space/time p.d.e.

system

SLIDE 52

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 19/26

Analysis and simulation

abscissa (m) gas saturation (%)

0.2 0.4 0.6 0.8 1 4 8 12 16

abscissa (m) liquid pressure (MPa)

0.2 0.4 0.6 0.8 1 1 1.5 2

SLIDE 158

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 23/26 t = 10 s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 t = 10 x (m) pression du liquide (rouge) et du gaz (vert) (MPa)

liquid and gas Pressure profiles

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 t = 10 x (m) saturation de gaz (%)

Gas Saturation profiles t = 5000 s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 t = 5000 x (m) pression du liquide (rouge) et du gaz (vert) (MPa)

liquid and gas Pressure profiles

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 t = 5000 x (m) saturation de gaz (%)

Gas Saturation profiles t = 100000 s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 t = 100000 x (m) pression du liquide (rouge) et du gaz (vert) (MPa)

liquid and gas Pressure profiles

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 t = 100000 x (m) saturation de gaz (%)

Gas Saturation profiles

SLIDE 159

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 24/26 t = 200000 s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 t = 200000 x (m) pression du liquide (rouge) et du gaz (vert) (MPa)

liquid and gas Pressure profiles

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 t = 200000 x (m) saturation de gaz (%)

Gas Saturation profiles t = 1000000 s

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 t = 1000000 x (m) pression du liquide (rouge) et du gaz (vert) (MPa)

◮ Ongoing implementation:

◮ 2-phases, N + 1 components (1 solvent and N solutes) ◮ Thermal flows (Energy equation)

◮ Modelling in progress:

◮ Chemical reactions (... CO2 sequestration)

SLIDE 163

Multiphase Flow and Transport in saturated-unsaturated porous media Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions Unsaturated flow equations Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 26/26

References

◮ Bourgeat, A. and Jurak, M. and Sma¨

ı, F. Two-phase, partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository. Computational Geosciences,Volume 13, Number 1, mars 2009 .

◮ Bourgeat, A. and Jurak, M. and Sma¨

ı, F. Modelling and Numerical Simulation of Gas Migration in a Nuclear Waste Repository . http://arxiv.org/abs/1006.2914, June 2010