DG approximation of two-component miscible liquid-gas porous media - PowerPoint PPT Presentation

Setting Numerical method Results DG approximation of two-component miscible liquid-gas porous media flows Alexandre Ern and Igor Mozolevski Universit´ e Paris-Est, CERMICS, Ecole des Ponts, France RICAM Workshop, Linz, October 2011 Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Introduction ◮ Multicomponent multiphase porous media flows are encountered in several applications ◮ petroleum engineering (oil-water) ◮ agricultural engineering, groundwater remediation (air-water) ◮ Such flows have received enhanced attention recently ◮ gas sequestration ◮ underground repositories of radioactive waste Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results The issue of radioactive waste I ◮ Production in France estimated at 2 kg/year/citizen ◮ various sources: electro-nuclear plants, health treatments, industrial processes, research, army ◮ Total conditioned radioactive waste amounts to 1.2 Mm 3 ◮ 0.2% of this volume contains 99% of radioactivity ◮ HAVL waste High Activity, Long Life 1 Myears Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results The issue of radioactive waste II ◮ HAVL waste could be stored in underground repository ◮ ANDRA (French Agency for Radioactive Waste Management) ◮ preliminary study established feasibility in 2005 ◮ clay host rock located at ≈ 500 m depth ◮ decision to be taken in 2014 , operating could start in 2025 Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results The issue of radioactive waste III ◮ Underground research facility currently operating ◮ Various academic research programs have been launched ◮ GNR MOMAS Mathematical modeling and numerical simulation www.gdrmomas.org Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results The issue of radioactive waste IV ◮ Multiple barriers to contain radionuclides (RN) ◮ conditioning of waste, steel containers ◮ manufactured barriers (bentonite, steel) ◮ host rock (quite favorable properties) ◮ Main time scales ◮ 10 2 years operating and observing the facility, reversibility ◮ 10 4 years degradation of manufactured barriers ◮ 10 6 years migration of RN through geosphere up to biosphere Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Hydrogen production and migration ◮ Corrosion of metallic components (and marginally water radiolysis) Fe + 2H 2 0 → Fe(OH) 2 + H 2 ◮ Understand hydrogen migration through host rock ◮ Two-phase (liquid, gas), two-component (water, hydrogen) flow, gas phase is compressible ◮ Gas phase (dis)appearance gas saturation at inflow position of gas phase front 2 200 1.8 180 1.6 160 Saturation front position Gas saturation at x=0 1.4 140 1.2 120 1 100 0.8 80 0.6 60 0.4 40 0.2 20 0 0 0 2 4 6 8 10 0 2 4 6 8 10 t (year) 5 t (year) 5 x 10 x 10 Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Outline ◮ Setting ◮ Numerical method ◮ Results Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Setting ◮ Governing equations ◮ Choice of main unknowns ◮ Mathematical model Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Governing equations I ◮ Basic notation ◮ subscript α ∈ { l , g } for phase ◮ superscript β ∈ { w , h } for component ◮ ̺ β α density of component β in phase α ◮ s α saturation of phase α , s l + s g = 1 ◮ p α pressure of phase α ◮ Mass conservation equation for each component � � ∂ t ( s α ̺ β ∇ · ( ̺ β α q α + j β α ) = F β Φ α ) + α ∈{ l , g } α ∈{ l , g } with Φ porosity, q α volumetric flow rate of phase α , j β α mass diffusion flux of component β in phase α ◮ [Bear ’78, Chavent & Jaffr´ e ’78, Helmig ’97] Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Governing equations II ◮ Darcy–Muskat law for volumetric flow rates (neglecting gravity) q α = − K λ α ( s α ) ∇ p α ∀ α ∈ { l , g } with K absolute permeability, λ α mobility of phase α ( λ α = 0 if phase α is absent) ◮ Capillary pressure π : [0 , 1) → [0 , + ∞ ) p g = p l + π ( s g ) ◮ Assume incompressibility in liquid phase and neglect water vaporization ̺ w l = ̺ std ̺ w g = 0 l Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Governing equations III ◮ Ideal gas law in gas phase and hydrogen phase changes in thermodynamic equilibrium (Henry’s law) ̺ h ̺ h g = C g p g l = C h p g ◮ Fick’s law for dissolved hydrogen diffusion flux (dilute approximation) j h l = − Φ s l D h l ∇ ̺ h j w l = − j h l l ◮ Governing equations Φ ̺ std ∂ t s l + ∇ · ( ̺ std q l − j h l ) = F w l l Φ ∂ t ( ̺ h l s l + C g p g s g ) + ∇ · ( ̺ h l q l + C g p g q g + j h l ) = F h Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Choice of main unknowns I ◮ Possible absence of gas phase in a priori unknown parts of domain ◮ Choosing one of the saturations as one of the main unknowns is inappropriate if gas phase disappears ◮ s g is identically 0 ◮ s l is identically 1 ◮ Unified formulation of governing equations highly desirable to avoid intricate numerical solvers ◮ Artificially enforcing s g ≥ ǫ > 0 is inappropriate (can lead to both dissolved hydrogen and gas pressure overestimation) Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Choice of main unknowns II ◮ Following [Bourgeat, Jurak & Sma¨ ı ’09], we choose as main unknowns y 2 := ̺ h y = ( y 1 , y 2 ) , y 1 := p l , l allowing for a unified formulation including gas phase disappearance p l ̺ l s g = 0, no gas h = C h p l s g = ǫ s g > 0, gas present ̺ h l ◮ See also ◮ [Jaffr´ e & Sboui ’10] for a reformulation based on complementary constraints ◮ [Abadpour & Panfilov ’09] for method with negative saturations Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Choice of main unknowns III ◮ Gas saturation recovered from capillarity and thermodynamic equilibrium � y 2 � s g ( p l , ̺ h l ) = s g ( y ) = π − 1 − y 1 C h π − 1 : R → [0 , 1) inverse capillary pressure function extended by zero 1 0 ◮ s g is a continuous function of y ◮ continuously differentiable for van Genuchten capillary pressure model ◮ not differentiable at entry pressure for Brooks–Corey model Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Mathematical model I ◮ Nondimensional form ◮ reference pressure p 0 (1 MPa), reference density C h p 0 (15g/m 3 ) ◮ mass conservation equations scaled by ̺ std and C h p 0 l ◮ Governing equations ∂ t b 1 ( y ) − ∇ · ( A 11 ( y ) ∇ y 1 + A 12 ( y ) ∇ y 2 ) = F 1 ∂ t b 2 ( y ) − ∇ · ( A 21 ( y ) ∇ y 1 + A 22 ( y ) ∇ y 2 ) = F 2 with b i ( y ) nondimensional and A ij ( y ) in m 2 /s ◮ IC on y , BC either Dirichlet on y or Neumann on total fluxes � σ i ( y ) = A ij ( y ) ∇ y j j ∈{ 1 , 2 } Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Mathematical model II ◮ System coefficients b 1 ( y ) = − Φ s g ( y ) b 2 ( y ) = Φ a ( s g ( y )) y 2 A 11 ( y ) = p 0 K λ l (1 − s g ( y )) A 12 ( y ) = − ( C h p 0 /̺ std )Φ(1 − s g ( y )) D h l l A 21 ( y ) = y 2 p 0 K λ l (1 − s g ( y )) A 22 ( y ) = y 2 p 0 K λ g ( s g ( y )) ω + Φ(1 − s g ( y )) D h l with a ( s ) = 1 + ( ω − 1) s and ω = C g / C h ( ≈ 50) ◮ First equation is parabolic in y 1 , degenerating into elliptic if gas phase disappears ◮ Second equation is parabolic in y 2 , degenerating into elliptic if dissolved hydrogen disappears Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows

Setting Numerical method Results Mathematical model III ◮ Structure of space differential operator ◮ Ellipticity iff ( A 12 + A 21 ) 2 < 4 A 11 A 22 ◮ Typical orders of magnitude (in µ m 2 /s) A 11 ≈ 50 , A 12 ≈ 0 . 7 , A 21 ≈ 50 , A 22 ≈ 450 ◮ Under assumption A 12 ≈ 0, ellipticity iff p 0 K λ l (1) < 4Φ D h l ◮ Alternative assumption for ellipticity is smallness condition on hydrogen [Mikeli´ c ’09] Alexandre Ern Universit´ e Paris-Est, CERMICS Two-component two-phase flows