Multiphase Flow and Saturated - Unsaturated two phases flow Transport in saturated-unsaturated porous media Physical assumptions -(i) Alain Bourgeat, For simplicity, but consistant, with modelling gas migration in Universit´ e Claude Bernard Lyon 1 deep geological Nuclear Waste Repositories : Institut Camille Jordan-UMR 5208 ◮ 2 phases : liquid (incompressible), denoted l and gas Contributors: F.Sma¨ ı, IRSN (compressible)denoted g Fontenay aux Roses and ICJ-UCBLyon1; ◮ 2 components : water, w and and hydrogen, h M.Jurak, Univ. Zagreb ◮ Mass conservation for each component w, h Physical assumptions Unsaturated flow ◮ Generalized Darcy law for each phase , α ∈ { g, l } equations Saturated flow equations q α = − K k r,α ( S α ) ( ∇ p α − ρ α g ) (1) Construction of a µ α saturated/unsaturated model Three Numerical Tests ◮ local mechanical equilibrium of phases ≈ Capillary pressure Conclusions law : p g − p l = p c ( S g ) . ◮ 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 5/26
Multiphase Flow and Saturated - Unsaturated two phases flow Transport in saturated-unsaturated porous media Physical assumptions -(i) Alain Bourgeat, For simplicity, but consistant, with modelling gas migration in Universit´ e Claude Bernard Lyon 1 deep geological Nuclear Waste Repositories : Institut Camille Jordan-UMR 5208 ◮ 2 phases : liquid (incompressible), denoted l and gas Contributors: F.Sma¨ ı, IRSN (compressible)denoted g Fontenay aux Roses and ICJ-UCBLyon1; ◮ 2 components : water, w and and hydrogen, h M.Jurak, Univ. Zagreb ◮ Mass conservation for each component w, h Physical assumptions Unsaturated flow ◮ Generalized Darcy law for each phase , α ∈ { g, l } equations Saturated flow equations q α = − K k r,α ( S α ) ( ∇ p α − ρ α g ) (1) Construction of a µ α saturated/unsaturated model Three Numerical Tests ◮ local mechanical equilibrium of phases ≈ Capillary pressure Conclusions law : p g − p l = p c ( S g ) . ◮ 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 5/26
Multiphase Flow and Saturated - Unsaturated two phases flow Transport in saturated-unsaturated porous media Physical assumptions -(ii) Alain Bourgeat, Universit´ e Claude ◮ Diffusion of component i in phase α ( and infinite dilution ) Bernard Lyon 1 Institut Camille Jordan-UMR 5208 j h l = − Φ S l D ∇ ρ h j w l = − j h l , l , 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
Multiphase Flow and Saturated - Unsaturated two phases flow Transport in saturated-unsaturated porous media Physical assumptions -(ii) Alain Bourgeat, Universit´ e Claude ◮ Diffusion of component i in phase α ( and infinite dilution ) Bernard Lyon 1 Institut Camille Jordan-UMR 5208 j h l = − Φ S l D ∇ ρ h j w l = − j h l , l , Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; ◮ Local thermodynamical equilibrium liquid solution/ gas M.Jurak, Univ. Zagreb mixture Physical assumptions Unsaturated flow � ( p g − p c ) − p w � equations p w g = x w g p g = p w v x w v exp (2) RTρ w, ∗ l /M w Saturated flow l equations Construction of a � � p l − p 0 saturated/unsaturated p h g = x h g p g = K h x h model H exp l ; (3) RT/v h, ∞ Three Numerical Tests l Conclusions ∼ Raoult-Kelvin and Henry laws; with: x i α , the i -component molar concentration in the v the pure water saturated vapor pressure , ρ w, ∗ α -phase , p w l the pure liquid water mass density, K h H the Henry’s constant at p 0 (a reference pressure) , v h, ∞ the hydrogen molar l concentration at infinite dilution . 6/26
Multiphase Flow and Unsaturated two-phase flow Transport in saturated-unsaturated porous media An example of Unsaturated liquid+gas mixture flow (no vaporized water) Alain Bourgeat, Universit´ e Claude The components mass conservation reads : Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: Φ ∂ F.Sma¨ ı, IRSN ∂t ( S l ρ w l ) + div ( ρ w l ) = F w , l q l + j w Fontenay aux Roses (4) and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Φ ∂ S l ρ h ρ h = F h ; � � � l q l + ρ g q g + j h � l + S g ρ g + div (5) Physical assumptions l ∂t Unsaturated flow equations Phase diagram Usual primary variables: ( p l , S l ) , ( p l , p g ) , ( p l , p c ) . Saturated flow equations Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 7/26
Multiphase Flow and Unsaturated two-phase flow Transport in saturated-unsaturated porous media An example of Unsaturated liquid+gas mixture flow (no vaporized water) Alain Bourgeat, Universit´ e Claude The components mass conservation reads : Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: Φ ∂ F.Sma¨ ı, IRSN ∂t ( S l ρ w l ) + div ( ρ w l ) = F w , l q l + j w Fontenay aux Roses (4) and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Φ ∂ S l ρ h ρ h = F h ; � � � l q l + ρ g q g + j h � l + S g ρ g + div (5) Physical assumptions l ∂t Unsaturated flow equations Phase diagram Usual primary variables: ( p l , S l ) , ( p l , p g ) , ( p l , p c ) . Saturated flow equations ρ i l , the i-component mass concentration ; Construction of a Where : saturated/unsaturated model ρ g ≃ ρ h g = C v p g ( = ideal gas) ; p g = p c + p l ; Three Numerical Tests ρ h l = C h p g ( = Henry’s law) . Conclusions 7/26
Multiphase Flow and Unsaturated two-phase flow Transport in saturated-unsaturated porous media An example of Unsaturated liquid+gas mixture flow (no vaporized water) Alain Bourgeat, Universit´ e Claude The components mass conservation reads : Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: Φ ∂ F.Sma¨ ı, IRSN ∂t ( S l ρ w l ) + div ( ρ w l ) = F w , l q l + j w Fontenay aux Roses (4) and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Φ ∂ S l ρ h ρ h = F h ; � � � l q l + ρ g q g + j h � l + S g ρ g + div (5) Physical assumptions l ∂t Unsaturated flow equations Phase diagram Usual primary variables: ( p l , S l ) , ( p l , p g ) , ( p l , p c ) . Saturated flow equations ρ i l , the i-component mass concentration ; Construction of a Where : saturated/unsaturated model ρ g ≃ ρ h g = C v p g ( = ideal gas) ; p g = p c + p l ; Three Numerical Tests ρ h l = C h p g ( = Henry’s law) . Conclusions and constants : C h = M h , C v = M h RT ( M h = Hydrogen molar mass ) . K h H 7/26
Multiphase Flow and Unsaturated/Saturated flow Transport in saturated-unsaturated porous media Phase Diagram Alain Bourgeat, Universit´ e Claude Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: p l F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; S g = 0 M.Jurak, Univ. Zagreb ρ h l ≤ C h ( p l + p c (0)) l = C h ( p l + p c (0)) Physical assumptions Unsaturated flow equations S g > 0 ρ h Phase diagram ρ h l = C h p g Saturated flow equations p g = p l + p c ( S g ) Construction of a ≥ p l + p c (0) saturated/unsaturated model ρ h l Three Numerical Tests Conclusions Figure: Henry’s law: ρ h l = C h p g . Localization of the saturated state, S g = 0 , and the unsaturated state, S g > 0 . 8/26
Multiphase Flow and Saturated flow ( One phase Transport in saturated-unsaturated porous media Liquid saturated flow ◮ S l ≡ 1 , p g is then indeterminate 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
Multiphase Flow and Saturated flow ( One phase Transport in saturated-unsaturated porous media Liquid saturated flow ◮ S l ≡ 1 , p g is then indeterminate Alain Bourgeat, Universit´ e Claude ◮ Classical Darcy law for liquid flow (water + dissolved Bernard Lyon 1 Institut Camille hydrogen) 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
Multiphase Flow and Saturated flow ( One phase Transport in saturated-unsaturated porous media Liquid saturated flow ◮ S l ≡ 1 , p g is then indeterminate Alain Bourgeat, Universit´ e Claude ◮ Classical Darcy law for liquid flow (water + dissolved Bernard Lyon 1 Institut Camille hydrogen) Jordan-UMR 5208 Contributors: ◮ Dissolved hydrogen transported by diffusion and convection 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
Multiphase Flow and Saturated flow ( One phase Transport in saturated-unsaturated porous media Liquid saturated flow ◮ S l ≡ 1 , p g is then indeterminate Alain Bourgeat, Universit´ e Claude ◮ Classical Darcy law for liquid flow (water + dissolved Bernard Lyon 1 Institut Camille hydrogen) Jordan-UMR 5208 Contributors: ◮ Dissolved hydrogen transported by diffusion and convection F.Sma¨ ı, IRSN Fontenay aux Roses ◮ The and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb liquid solution flow (water + dissolved hydrogen) is described by : Physical assumptions Unsaturated flow = F w , Φ ∂ρ h equations � � ρ w l ρ h = F h ; � l q l − j h � l q l + j h div ∂t + div (6) Saturated flow l l equations ∇ p l − ( ρ w l + ρ h l = − Φ D ∇ ρ h � � j h q l = − K λ l (1) l ) g , l . (7) Construction of a saturated/unsaturated model Three Numerical Tests Conclusions 9/26
Multiphase Flow and Saturated flow ( One phase Transport in saturated-unsaturated porous media Liquid saturated flow ◮ S l ≡ 1 , p g is then indeterminate Alain Bourgeat, Universit´ e Claude ◮ Classical Darcy law for liquid flow (water + dissolved Bernard Lyon 1 Institut Camille hydrogen) Jordan-UMR 5208 Contributors: ◮ Dissolved hydrogen transported by diffusion and convection F.Sma¨ ı, IRSN Fontenay aux Roses ◮ The and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb liquid solution flow (water + dissolved hydrogen) is described by : Physical assumptions Unsaturated flow = F w , Φ ∂ρ h equations � � ρ w l ρ h = F h ; � l q l − j h � l q l + j h div ∂t + div (6) Saturated flow l l equations ∇ p l − ( ρ w l + ρ h l = − Φ D ∇ ρ h � � j h q l = − K λ l (1) l ) g , l . (7) Construction of a saturated/unsaturated model Three Numerical Tests ◮ Primary variables are then ( p l , ρ h l ) ; the amount of dissolved Conclusions hydrogen, ρ h l ,is now an independent unknown( no more related to the p g ); and a gas phase cannot be taken in account. 9/26
Multiphase Flow and Saturated flow ( One phase Transport in saturated-unsaturated porous media Liquid saturated flow ◮ S l ≡ 1 , p g is then indeterminate Alain Bourgeat, Universit´ e Claude ◮ Classical Darcy law for liquid flow (water + dissolved Bernard Lyon 1 Institut Camille hydrogen) Jordan-UMR 5208 Contributors: ◮ Dissolved hydrogen transported by diffusion and convection F.Sma¨ ı, IRSN Fontenay aux Roses ◮ The and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb liquid solution flow (water + dissolved hydrogen) is described by : Physical assumptions Unsaturated flow = F w , Φ ∂ρ h equations � � ρ w l ρ h = F h ; � l q l − j h � l q l + j h div ∂t + div (6) Saturated flow l l equations ∇ p l − ( ρ w l + ρ h l = − Φ D ∇ ρ h � � j h q l = − K λ l (1) l ) g , l . (7) Construction of a saturated/unsaturated model Three Numerical Tests ◮ Primary variables are then ( p l , ρ h l ) ; the amount of dissolved Conclusions hydrogen, ρ h l ,is now an independent unknown( no more related to the p g ); 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 ). 9/26
Multiphase Flow and Saturated flow ( One phase Transport in saturated-unsaturated porous media Liquid saturated flow ◮ S l ≡ 1 , p g is then indeterminate Alain Bourgeat, Universit´ e Claude ◮ Classical Darcy law for liquid flow (water + dissolved Bernard Lyon 1 Institut Camille hydrogen) Jordan-UMR 5208 Contributors: ◮ Dissolved hydrogen transported by diffusion and convection F.Sma¨ ı, IRSN Fontenay aux Roses ◮ The and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb liquid solution flow (water + dissolved hydrogen) is described by : Physical assumptions Unsaturated flow = F w , Φ ∂ρ h equations � � ρ w l ρ h = F h ; � l q l − j h � l q l + j h div ∂t + div (6) Saturated flow l l equations ∇ p l − ( ρ w l + ρ h l = − Φ D ∇ ρ h � � j h q l = − K λ l (1) l ) g , l . (7) Construction of a saturated/unsaturated model Three Numerical Tests ◮ Primary variables are then ( p l , ρ h l ) ; the amount of dissolved Conclusions hydrogen, ρ h l ,is now an independent unknown( no more related to the p g ); 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 ? 9/26
Multiphase Flow and Construction of a Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: ◮ According to the phase state , the primary variables are: 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
Multiphase Flow and Construction of a Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: ◮ According to the phase state , the primary variables are: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ two-phase unsaturated : Liquid pressure p l /Phase Saturation and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb S l ; like in (4)-(5) 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
Multiphase Flow and Construction of a Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: ◮ According to the phase state , the primary variables are: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ two-phase unsaturated : Liquid pressure p l /Phase Saturation and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb S l ; like in (4)-(5) ◮ one-phase saturated : Liquid pressure p l /Hydrogen mass Physical assumptions concentration, ρ h l , like in (6) - (7) 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
Multiphase Flow and Construction of a Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: ◮ According to the phase state , the primary variables are: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ two-phase unsaturated : Liquid pressure p l /Phase Saturation and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb S l ; like in (4)-(5) ◮ one-phase saturated : Liquid pressure p l /Hydrogen mass Physical assumptions concentration, ρ h l , like in (6) - (7) Unsaturated flow equations ◮ Considering p l and ρ h l as main variables in both cases Saturated flow equations saturated and unsaturated , then eqs. (4)-(7) reduce to a Construction of a unique couple of equations: saturated/unsaturated model Choice of suitable variables Φ ∂ Choice i Model ∂t ( S l ρ w l ) + div ( ρ w l ) = F w , l q l + j w (8) Choice ii Model Capillary Pressure Curve, and Inverse Φ ∂ = F h ; ρ h ρ h l q l + ρ g q g + j h Three Numerical Tests � � � � + div (9) tot l ∂t Conclusions 10/26
Multiphase Flow and Construction of a Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Contributors: ◮ According to the phase state , the primary variables are: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ two-phase unsaturated : Liquid pressure p l /Phase Saturation and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb S l ; like in (4)-(5) ◮ one-phase saturated : Liquid pressure p l /Hydrogen mass Physical assumptions concentration, ρ h l , like in (6) - (7) Unsaturated flow equations ◮ Considering p l and ρ h l as main variables in both cases Saturated flow equations saturated and unsaturated , then eqs. (4)-(7) reduce to a Construction of a unique couple of equations: saturated/unsaturated model Choice of suitable variables Φ ∂ Choice i Model ∂t ( S l ρ w l ) + div ( ρ w l ) = F w , l q l + j w (8) Choice ii Model Capillary Pressure Curve, and Inverse Φ ∂ = F h ; ρ h ρ h l q l + ρ g q g + j h Three Numerical Tests � � � � + div (9) tot l ∂t Conclusions ◮ with ρ h tot = S l ρ h l + C v p g S g ; C v p h g = ρ h g ( ∼ ideal gas) 10/26
Multiphase Flow and Construction of a Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Choice of suitable variables Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ the Total Hydrogen mass concentration ρ h tot ≡ ρ h l S l + ρ h g S g , and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb is definite in both states of flow (according to the flow state): Physical assumptions ◮ two-phases unsaturated state : Unsaturated flow equations ρ h tot = a ( S g )( p l + p c ( S g )) ; S g > 0 (10) Saturated flow equations with: a ( S g ) = C h (1 − S g ) + C v S g ∈ [ C h , C v ] . (11) Construction of a saturated/unsaturated model ◮ one-phase saturated : Choice of suitable ρ h tot = ρ h variables l ; S g = 0 . Choice i Model Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 11/26
Multiphase Flow and Construction of a Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Choice of suitable variables Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ the Total Hydrogen mass concentration ρ h tot ≡ ρ h l S l + ρ h g S g , and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb is definite in both states of flow (according to the flow state): Physical assumptions ◮ two-phases unsaturated state : Unsaturated flow equations ρ h tot = a ( S g )( p l + p c ( S g )) ; S g > 0 (10) Saturated flow equations with: a ( S g ) = C h (1 − S g ) + C v S g ∈ [ C h , C v ] . (11) Construction of a saturated/unsaturated model ◮ one-phase saturated : Choice of suitable ρ h tot = ρ h variables l ; S g = 0 . Choice i Model Choice ii Model ◮ There is now two possible choices for the main variables in Capillary Pressure Curve, and Inverse eq. (8) and (9): Three Numerical Tests Conclusions Choice i : Liquid pressure, p l / Total Hydrogen concentration, ρ h tot Choice ii : Liquid pressure, p l / Hydrogen mass concentration, ρ h l 11/26
Multiphase Flow and Choice i : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Liquid pressure, p l / Total Hydrogen concentration, ρ h Contributors: tot F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; ◮ In system (8)-(9) we compute S g = S g ( p l , ρ h tot ) , (from M.Jurak, Univ. Zagreb (11)); S l = 1 − S g , and Physical assumptions ρ h l = ρ h l ( p l , ρ h tot ) = min( C h p g ( p l , ρ h tot ) , ρ h p g ( p l , ρ h tot ) , tot ) = Unsaturated flow p l + p c ( S g ( p l , ρ h equations tot )) . 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
Multiphase Flow and Choice i : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Liquid pressure, p l / Total Hydrogen concentration, ρ h Contributors: tot F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; ◮ In system (8)-(9) we compute S g = S g ( p l , ρ h tot ) , (from M.Jurak, Univ. Zagreb (11)); S l = 1 − S g , and Physical assumptions ρ h l = ρ h l ( p l , ρ h tot ) = min( C h p g ( p l , ρ h tot ) , ρ h p g ( p l , ρ h tot ) , tot ) = Unsaturated flow p l + p c ( S g ( p l , ρ h equations tot )) . Saturated flow ◮ Noticing a () is > C h , increasing and p g > p l + p c (0) ; the equations Construction of a State of flow is then characterized by: saturated/unsaturated unsaturated : ρ h model tot > C h ( p l + p c (0)) Choice of suitable variables Choice i Model saturated : ρ h tot ≤ C h ( p l + p c (0)) Choice ii Model Capillary Pressure Curve, and Inverse Three Numerical Tests Conclusions 12/26
Multiphase Flow and Choice i : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Liquid pressure, p l / Total Hydrogen concentration, ρ h Contributors: tot F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; ◮ In system (8)-(9) we compute S g = S g ( p l , ρ h tot ) , (from M.Jurak, Univ. Zagreb (11)); S l = 1 − S g , and Physical assumptions ρ h l = ρ h l ( p l , ρ h tot ) = min( C h p g ( p l , ρ h tot ) , ρ h p g ( p l , ρ h tot ) , tot ) = Unsaturated flow p l + p c ( S g ( p l , ρ h equations tot )) . Saturated flow ◮ Noticing a () is > C h , increasing and p g > p l + p c (0) ; the equations Construction of a State of flow is then characterized by: saturated/unsaturated unsaturated : ρ h model tot > C h ( p l + p c (0)) Choice of suitable variables Choice i Model saturated : ρ h tot ≤ C h ( p l + p c (0)) Choice ii Model Capillary Pressure ρ h tot ≡ ρ h Curve, and Inverse l Three Numerical Tests Conclusions 12/26
Multiphase Flow and Choice i : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Liquid pressure, p l / Total Hydrogen concentration, ρ h Contributors: tot F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; ◮ In system (8)-(9) we compute S g = S g ( p l , ρ h tot ) , (from M.Jurak, Univ. Zagreb (11)); S l = 1 − S g , and Physical assumptions ρ h l = ρ h l ( p l , ρ h tot ) = min( C h p g ( p l , ρ h tot ) , ρ h p g ( p l , ρ h tot ) , tot ) = Unsaturated flow p l + p c ( S g ( p l , ρ h equations tot )) . Saturated flow ◮ Noticing a () is > C h , increasing and p g > p l + p c (0) ; the equations Construction of a State of flow is then characterized by: saturated/unsaturated unsaturated : ρ h model tot > C h ( p l + p c (0)) Choice of suitable variables Choice i Model saturated : ρ h tot ≤ C h ( p l + p c (0)) Choice ii Model Capillary Pressure ρ h tot ≡ ρ h Curve, and Inverse l Three Numerical Tests ◮ Then: Conclusions ◮ 1 st equation is parabolic(unsaturated)/elliptic(saturated) in p l ◮ 2 nde equation is parabolic in ρ h tot 12/26
Multiphase Flow and Choice i : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Existence of solutions Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ After an ad hoc variable change, the Alt-Luckhaus theorem and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb applies, and the existence of a solution could be proved ( F. Physical assumptions ı, PhD Thesis ) . Sma¨ 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 13/26
Multiphase Flow and Choice i : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Existence of solutions Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ After an ad hoc variable change, the Alt-Luckhaus theorem and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb applies, and the existence of a solution could be proved ( F. Physical assumptions ı, PhD Thesis ) . Sma¨ Unsaturated flow equations Suppose r min ≤ ρ h tot ≤ r max and p l ≥ 0 and assume that Saturated flow initial and Dirichlet conditions are enough regular. equations Then there is a weak solution to the simplified formulation. 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 13/26
Multiphase Flow and Choice i : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Existence of solutions Contributors: F.Sma¨ ı, IRSN Fontenay aux Roses ◮ After an ad hoc variable change, the Alt-Luckhaus theorem and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb applies, and the existence of a solution could be proved ( F. Physical assumptions ı, PhD Thesis ) . Sma¨ Unsaturated flow equations Suppose r min ≤ ρ h tot ≤ r max and p l ≥ 0 and assume that Saturated flow initial and Dirichlet conditions are enough regular. equations Then there is a weak solution to the simplified formulation. Construction of a saturated/unsaturated model ◮ Could also certainly be investigated using ”Entropy weak Choice of suitable variables solutions”, defined by Kruzkov (hyperbolic) and extended by Choice i Model Choice ii Model Carillo (parabolic). Capillary Pressure Curve, and Inverse Remarks: Three Numerical Tests ◮ no need of capillary pressure for this formulation Conclusions ( h -component eq. in (6) Parabolic − → Hyperbolic) ◮ Coefficients in the div operators and in the ∂ ∂t operators could become discontinuous 13/26
Multiphase Flow and Choice ii : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude Liquid pressure, p l / Hydrogen mass concentration, ρ h Bernard Lyon 1 l Institut Camille ◮ Introducing in system (8)-(9), from ( p c ) − 1 , the extended Jordan-UMR 5208 Contributors: gas-phase Pressure p ∗ g = π + p l ; and the extended saturation F.Sma¨ ı, IRSN Fontenay aux Roses � ρ h � and ICJ-UCBLyon1; S ∗ g = f C h − p l . l M.Jurak, Univ. Zagreb Physical assumptions f ( π ) p c ( S g ) Unsaturated flow equations 1 Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model 0 S g 0 Choice ii Model π = ρ h Capillary Pressure 0 1 0 Curve, and Inverse l C h − p l Three Numerical Tests Conclusions ρ h Figure: p c = p g − p l ; p ∗ g = C h . l 14/26
Multiphase Flow and Choice ii : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases ) Alain Bourgeat, Universit´ e Claude Liquid pressure, p l / Hydrogen mass concentration, ρ h Bernard Lyon 1 l Institut Camille ◮ Introducing in system (8)-(9), from ( p c ) − 1 , the extended Jordan-UMR 5208 Contributors: gas-phase Pressure p ∗ g = π + p l ; and the extended saturation F.Sma¨ ı, IRSN Fontenay aux Roses � ρ h � and ICJ-UCBLyon1; S ∗ g = f C h − p l . l M.Jurak, Univ. Zagreb Physical assumptions f ( π ) p c ( S g ) Unsaturated flow equations 1 Saturated flow equations Construction of a saturated/unsaturated model Choice of suitable variables Choice i Model 0 S g 0 Choice ii Model π = ρ h Capillary Pressure 0 1 0 Curve, and Inverse l C h − p l Three Numerical Tests Conclusions ρ h Figure: p c = p g − p l ; p ∗ g = C h . l ◮ leads to a system with the main variables p l and ρ h l 14/26
Multiphase Flow and Choice ii : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases Alain Bourgeat, Universit´ e Claude )model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Liquid pressure, p l / Hydrogen mass concentration, ρ h Contributors: l F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions ◮ 1 st equation is parabolic (unsaturated)/elliptic(saturated) in Unsaturated flow equations p l , non uniformly ( coefficient of ∇ p l → 0 as s g → 1 ) Saturated flow 2 nde equation is strictly parabolic in ρ h equations l 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 15/26
Multiphase Flow and Choice ii : Transport in saturated-unsaturated porous media saturated (1-phase)/ unsaturated (2-phases Alain Bourgeat, Universit´ e Claude )model Bernard Lyon 1 Institut Camille Jordan-UMR 5208 Liquid pressure, p l / Hydrogen mass concentration, ρ h Contributors: l F.Sma¨ ı, IRSN Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Physical assumptions ◮ 1 st equation is parabolic (unsaturated)/elliptic(saturated) in Unsaturated flow equations p l , non uniformly ( coefficient of ∇ p l → 0 as s g → 1 ) Saturated flow 2 nde equation is strictly parabolic in ρ h equations l Construction of a ◮ Remarks: saturated/unsaturated model ◮ capillary pressure is absolutely necessary for this formulation Choice of suitable ◮ p l and ρ h variables l are continuous no matter the discontinuity of the Choice i Model Saturations ( porous media highly heterogeneous) Choice ii Model Capillary Pressure ◮ The coefficients in all the div and ∂ ∂t operators are continuous Curve, and Inverse Three Numerical Tests Conclusions 15/26
Multiphase Flow and Analysis and simulation; Numerical Tests Transport in saturated-unsaturated porous media Advertising 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 Ongoing benchmark on: Unsaturated flow equations ”Modelling Multiphase Flows” Saturated flow equations http://www.gdrmomas.org/Benchmark/multiphase/ Construction of a multiphasique.html saturated/unsaturated model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 16/26
Multiphase Flow and Analysis and simulation; Quasi-1D scale field Transport in saturated-unsaturated porous media numerical simulations Alain Bourgeat, Universit´ e Claude Pb 1, Pb 2, Pb 3 in Numerical Test Data Base Bernard Lyon 1 Institut Camille Total Hydrogen concentration, ρ h Jordan-UMR 5208 tot is denoted X in the following 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
Multiphase Flow and Analysis and simulation; Quasi-1D scale field Transport in saturated-unsaturated porous media numerical simulations Alain Bourgeat, Universit´ e Claude Pb 1, Pb 2, Pb 3 in Numerical Test Data Base Bernard Lyon 1 Institut Camille Total Hydrogen concentration, ρ h Jordan-UMR 5208 tot is denoted X in the following 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 ◮ Boundary conditions : Setting Simulations ◮ Injection of pure gas on left side Implementation in the IRSN code: ◮ Impervious condition on top and bottom side DIPHPOM Numerical Test I ◮ Pure water ( X out = 0 ) (Test 1) or Numerical Test II Numerical Test III Two-phases( X out � = 0 )(Test 2) and a fixed pressure, on the Conclusions right side 17/26
Multiphase Flow and Analysis and simulation; Quasi-1D scale field Transport in saturated-unsaturated porous media numerical simulations Alain Bourgeat, Universit´ e Claude Pb 1, Pb 2, Pb 3 in Numerical Test Data Base Bernard Lyon 1 Institut Camille Total Hydrogen concentration, ρ h Jordan-UMR 5208 tot is denoted X in the following 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 ◮ Boundary conditions : Setting Simulations ◮ Injection of pure gas on left side Implementation in the IRSN code: ◮ Impervious condition on top and bottom side DIPHPOM Numerical Test I ◮ Pure water ( X out = 0 ) (Test 1) or Numerical Test II Numerical Test III Two-phases( X out � = 0 )(Test 2) and a fixed pressure, on the Conclusions right side ◮ Initial conditions : stationary state without injection ( Q h in = 0 ) 17/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Quasi-1D scale field numerical simulations Alain Bourgeat, Universit´ e Claude ◮ Van Genuchten-Mualem model for capillary pressure and Bernard Lyon 1 Institut Camille Jordan-UMR 5208 relative permeabilities Contributors: F.Sma¨ ı, IRSN ◮ Fixed temperature, T = 303 K Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb Porous medium parameters Fluid characteristics Physical assumptions Parameter Value Parameter Value 5 10 − 20 m 2 D h 3 10 − 9 m 2 /s k Unsaturated flow l equations 1 10 − 3 Φ 0 . 15 ( − ) µ l P a.s 2 10 6 9 10 − 6 Saturated flow P r P a µ g P a.s equations 7 . 65 10 − 6 mol/P a/m 3 n 1 . 49 ( − ) H ( T = 303 K ) Construction of a 10 − 2 S lr 0 . 4 ( − ) M l kg/mol saturated/unsaturated 2 10 − 3 S gr 0 ( − ) M g kg/mol model ρ std 10 3 kg/m 3 l Three Numerical Tests ρ std 8 10 − 2 kg/m 3 Setting g Simulations Implementation in Parameter Value the IRSN code: L x 200 m DIPHPOM Numerical Test I L y 20 m For more, see : Numerical Test II Q h 1 . 5 10 − 5 m/year Numerical Test III 10 6 p l,out P a Conclusions 5 10 5 T simul years http://sources.univ-lyon1.fr/cas test.html 18/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Numerical test : Implementation 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 ◮ Fully implicit time discretization of the space/time p.d.e. system 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
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Numerical test : Implementation 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 ◮ Fully implicit time discretization of the space/time p.d.e. system Physical assumptions Unsaturated flow ◮ Newton iteration to solve nonlinearities of the space pde equations system 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
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Numerical test : Implementation 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 ◮ Fully implicit time discretization of the space/time p.d.e. system Physical assumptions Unsaturated flow ◮ Newton iteration to solve nonlinearities of the space pde equations system Saturated flow equations ◮ Spatial discretization of the pde with a standard linear F.E. Construction of a saturated/unsaturated from the C++ LIBMESH Library model Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 19/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Numerical test : Implementation 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 ◮ Fully implicit time discretization of the space/time p.d.e. system Physical assumptions Unsaturated flow ◮ Newton iteration to solve nonlinearities of the space pde equations system Saturated flow equations ◮ Spatial discretization of the pde with a standard linear F.E. Construction of a saturated/unsaturated from the C++ LIBMESH Library model ◮ GMRES/LU methods (PETSC) Three Numerical Tests Setting Simulations Implementation in the IRSN code: DIPHPOM Numerical Test I Numerical Test II Numerical Test III Conclusions 19/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 1: Gas injection in a fully water saturated host rock Alain Bourgeat, First, only solved Hydrogen density increases; second, the gas phase Universit´ e Claude ( S g )appears ⇒ ∇ p g , ∇ p c ( S g ) and ∇ p l > 0 ; finally higher Bernard Lyon 1 Institut Camille ∇ p c ( S g ) ⇒ ∇ p l < 0 ; and no water injection slow down the p l Jordan-UMR 5208 Contributors: F.Sma¨ ı, IRSN 20 Fontenay aux Roses total H 2 density (mol/m3) 1.4 and ICJ-UCBLyon1; gas saturation (%) M.Jurak, Univ. Zagreb 16 1.2 1 Physical assumptions 12 0.8 Unsaturated flow 8 0.6 equations 0.4 Saturated flow 4 equations 0.2 Construction of a 0 0 saturated/unsaturated 0 40 80 120 160 200 0 40 80 120 160 200 model abscissa (m) abscissa (m) 1.12 Three Numerical Tests liquid pressure (MPa) Setting 1.1 Simulations 1.08 Implementation in the IRSN code: DIPHPOM 1.06 Numerical Test I Numerical Test II 1.04 Numerical Test III 1.02 Conclusions 1 0 40 80 120 160 200 abscissa (m) 20/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
Multiphase Flow and Analysis and simulation Transport in saturated-unsaturated porous media Test 2: Gas injection in a unsaturated host rock Injection of Hydrogen increases ρ h Alain Bourgeat, l , p g ; and ( p c ( S g ) small Universit´ e Claude Bernard Lyon 1 enough) ⇒ p l increases up to ”meet” S g = 0 : a plug of liquid Institut Camille Jordan-UMR 5208 appears and is pushed by the injected gas to the R.H.S. ( with Contributors: F.Sma¨ ı, IRSN unsaturated Dirichlet cond.); bringing back to Test 1. Fontenay aux Roses and ICJ-UCBLyon1; M.Jurak, Univ. Zagreb total H 2 density (mol/m3) 2.5 40 gas saturation (%) Physical assumptions 2 30 Unsaturated flow 1.5 equations 20 Saturated flow 1 equations 10 0.5 Construction of a saturated/unsaturated 0 0 model 0 40 80 120 160 200 0 40 80 120 160 200 abscissa (m) abscissa (m) Three Numerical Tests Setting 2.5 liquid pressure (MPa) Simulations Implementation in the IRSN code: DIPHPOM 2 Numerical Test I Numerical Test II 1.5 Numerical Test III Conclusions 1 0 40 80 120 160 200 abscissa (m) 21/26
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