Two-Phase Flow Numerical Modeling : Application to a Geological Nuclear Waste Disposal Sylvie Granet Sylvie Granet Sylvie Granet Sylvie Granet – – – – Clément Clément Chavant Clément Clément Chavant Chavant Chavant EDF R&D EDF R&D EDF R&D EDF R&D Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 1 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
OUTLINE � Industrial context : Nuclear waste storage � A classical two-phase flow model � An example of application : modeling of a disposal cell for an intermediate level long-lived waste (Benchmark Couplex Gaz 1 submitted by Andra) � Presentation of the problem � Numerical methods : • Classical FE Scheme : Discretization and results • Hybrid Finite Volume method : overview and first results � A first 3D study : modeling of a modulus of High level long-lived waste (Benchmark Couplex Gaz 2 submitted by Andra) � Results � Perspectives � Conclusions Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 2 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
Industrial context : Underground waste storage � A fully coupled THMC problem on a complex geometry ! � Main reasons of two-phase flow modelling : � Presence of initially unsaturated media (plugs, sealing …) � Ventilation of the galleries � Thermal drying � Hydrogen production due to corrosion of steel components (containers, casing) � Hydraulic specificities : � High level of capillary pressure (50 Scheme for an underground mined repository Mpa) and high level of gas pressure (due to corrosion) � Saturation closed to one in the geological media Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 3 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
A classical two-phase flow model � 2 components (H 2 and H 2 O) in 2 phases (liquid and gas) = = − P f S P P � Capillary relation ( ) c l g l � Mass conservation for water and hydrogen : ∂ ( ) ( ) + c + c = = m Div F F Q c H H O 2 , c l g c ∂ t 2 k k l S � Transport equations : . ( ) ( ) = − ∇ − ρ g F r l P l l l µ • Darcy’s law for each phase l g k k S ( ) . ( ) = − r g ∇ − ρ g F P g g g µ g • Diffusion law linking component velocities in each mixture (Fick’s Law) H O H F F H O H F F 2 2 2 2 g g + = − ∇ + = − ∇ F C F C l l l l g g ρ H O ρ H ρ H O ρ H 2 2 2 2 l l g g H P ρ H 2 2 g ≤ l • Dissolution (Henry’s Law) ol M K H H 2 • Vaporization (equilibrium equation) Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 4 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
Main modelisation difficulties � Injection of gas in a saturated porous media • Geological media initially saturated : what is the initial concentration of hydrogen or air in the liquid ? • High level of gas pressure � Presence of multiple barriers • Very high level of heterogeneities of the different materials • Initial level of saturation very dependant of the material � Huge non linearities • Capillary and Relative permeabilities functions (influencing type of equations and front shape) Ex. for relative permeabilities ( ) ( ) m 2 m l = − − k S S 1 / 1 1 (Van Genuchten) ( ) rel l l ∂ φρ l S k k S . ( ) ( ) + ∇ = l l Div r l P Q + λ ( ) A B l = k S S 1 / (Brooks&Corey) l ∂ µ t rel l l l l = k S 3 (cubic) rel l Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 5 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
COUPLEX-GAZ I (Andra 2007): Exercice definition (1/2) Modeling of a disposal cell for an intermediate level long-lived waste : P l = 4.2 Mpa Q=0 Q=0 100 m P l = 5.5 Mpa � Anisotropic problem (in the clay K H ≠ K v ) � Total hydrogen Flux for each primary package : Q H2 6.25 mol/year 0.5 mol/year 500 years 10 000 years t(year) Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 6 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
COUPLEX-GAZ I : caracteristic curves – Van Genuchten Mualem model S(Pc) relative permeabilities krelw (pack.) 1 krelw (pack. concrete) 0,9 1 krelw (gap) package 0,8 0,9 krelw (fil. concrete) pack. concrete 0,8 0,7 Krelw (FZ) clerance krelw (DZ) 0,6 0,7 filler concrete krelw (Cox) 0,6 0,5 S krelgz(pack.) FZ 0,5 0,4 krelgz(pack.concrete) DZ 0,4 0,3 krelgz(gap) 0,3 0,2 Cox krelgz(fil. concrete) 0,2 0,1 krelgz(FZ) 0,1 0 krelgz(DZ) 0 0,00E+00 2,00E+08 4,00E+08 6,00E+08 8,00E+08 1,00E+09 krelgz(Cox) 0 0,2 0,4 0,6 0,8 1 Pc (Pa) ( ) − ) ( ) S ( ) 1 ( = + m S wres S m 2 2 m g = − − m k S S 1 / l = − − k S S 1 / 1 1 l wres m 1 1 n rel we we rel we we P + c 1 P r 30000 Singularities for S = 1 : f’(Smax) = P’(Smax) 25000 20000 ∂ ∂ w S Pc ∂ g k S k S ( ) ( ) ( ) = ∞ = r w f(S) = ∞ r w 15000 0 ∂ ∂ S Pc ∂ S 10000 w w 5000 2 nd order polynomial C1 0 Interpolation for S > Smax 0,980 0,985 0,990 0,995 1,000 S (ex. Smax = 0,99) Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 7 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
COUPLEX-GAZ I : Exercice definition (2/2) � Couplex’s initial conditions : high contrast of saturation and capillary pressure S init Pc init K 10 -18 m 2 Filler concrete 0,7 3 Mpa 10 -19 m 2 Concrete of package 0,6 4,4 Mpa 10 -12 m 2 clearance 0,1 6 Mpa 10 -15 m 2 Primary package 0,2 0,8 Mpa • In the clay (healthy, disturbed or fractured) : S = 1 Hydrostatic liquid pressure � Actually, we consider a small gas pressure : Pg= 1atm (corresponding to a initial concentration of hydrogen in liquid) Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 8 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
COUPLEX-GAZ I : Numerical method (FE) � Couplex Gaz modelisation’s tool (www.code-aster.org) � Choice of the unknowns : Pc and Pg H P ρ H 2 • In our formulation, we write 2 g = l ol M K H H 2 = • In saturated area : S 1 l K = ρ H ⇒ = − ≤ P ˆ H Pc P ˆ P Variable transformation : 0 2 g l g l ol M H 2 � A Classical Finite Element method : • Finite Elements with Q1 elements • Lumping of the mass matrix : Non diagonal mass matrix => maximum principle not verified => Oscillations Integration points at the vertex of the elements • Time discretization = Implicite Euler • Newton method for non linear resolution Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 9 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
COUPLEX-GAZ I : Gas saturation and capillary pressure profiles – X = 103 m Time (year) Saturation X = 103m 1 0 0,9 Cox DZ FZ 240 0,8 0,7 500 0,6 5000 0,5 S Conc. Conc. 0,4 10000 0,3 50000 0,2 500000 0,1 gap � Vertical cross section package 0 50 55 60 65 70 75 Y (m) Capillary pressure X = 103m Time (years) 8,00E+06 0 6,00E+06 � 3 steps : 240 4,00E+06 1- capillary equilibrium (t<200 years) 500 Pc (Pa) 2,00E+06 5000 2- Small desaturation by gas 0,00E+00 production (t< 10 000 years) 10000 0 20 40 60 80 100 120 -2,00E+06 50000 3- The gas disappears graduately -4,00E+06 500000 -6,00E+06 � Complete saturation at 60 000 years ! Y (m) Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 10 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
COUPLEX-GAZ I : Gas pressure – X = 103 m Times (years) Gas pressure X = 103m 8,00E+06 0 240 7,00E+06 500 6,00E+06 1000 5,00E+06 5000 Pg (Pa) � Vertical cross section 4,00E+06 7000 10000 3,00E+06 18000 2,00E+06 38000 1,00E+06 50000 0,00E+00 500000 0 50 100 Y (m) Maximal Gas Pressure of 6,75 Mpa at 10 000 years– The pressure remains constant in the engineered area Scaling Up and Modeling for Transport and Flow in Porous Media - October 13-16, Dubrovnik 2008 11 Two-phase flow numerical modeling : Application to a Geological Nuclear waste disposal
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