10/12/2008 Upscaling “dissolution” “dissolution” Upscaling mechanisms in porous media mechanisms in porous media M. Quintard Institut de�Mécanique des�Fluides de�Toulouse ������������ F.�Golfier,�Y.�Aspa,�C.�Cohen,�G.�Vignoles,�R.Lenormand,�B.�Bazin,�D.�Ding,�S.� Véran,�F.�Fichot,�J.�Belloni,�B.�Goyeau,�D.�Gobin,�C.�Pierre,�F.�Plouraboué Financial�support:�IFP,�CNRS/INSU/PNRH,�SNECMA,�DGA 1 M. Quintard dissolution
10/12/2008 Outline Outline Outline � Background � Background � Pore-scale model and effective surface � Pore-scale model and effective surface � Darcy-scale models � Darcy-scale models � Stability � Stability � Large-scale models? � Large-scale models? � Conclusions � Conclusions 2 dissolution M. Quintard
10/12/2008 Introduction Daccord�et�al.,�1993 Introduction Introduction � Dissolution: � Dissolution: geochemistry, karsts, salt geochemistry, karsts, salt mines, NAPL, petrol. mines, NAPL, petrol. Engng, aerospace Engng, aerospace Ha�Long�Bay industry… industry… � Problems: � Problems: Igue de� – Multiple-scale analysis – Multiple-scale analysis Planagrèze – History effects – History effects – Instabilities – Instabilities 3 dissolution M. Quintard
10/12/2008 Example 1: acidizing treatment Example 1: acidizing acidizing treatment treatment Example 1: 4 dissolution M. Quintard
10/12/2008 Example 2: ablation of composite Example 2: ablation of composite Example 2: ablation of composite structures structures structures 5 dissolution M. Quintard
10/12/2008 Upscaling Surface Heterogeneities: Upscaling Surface Surface Heterogeneities Heterogeneities: : Upscaling The concept of Effective Surface The concept concept of of Effective Surface Effective Surface The 6 dissolution M. Quintard
10/12/2008 Non-ablative case: various approaches Non- -ablative case: various approaches ablative case: various approaches Non Direct�Simulation micro + ɶ c c � Effective�surface,� � � � effective�BC�(jump conditions) �� ����������������� ����������� micro meso !���"��#����� ������ Meso�scale modelling,�GTE� � � � � Effective�surface,�effective�BC �� ������������ �������������� ������ micro meso Domain decomposition ������ ������������� �������������� micro meso 7 dissolution M. Quintard
10/12/2008 Extension of Wood et al. (2000) Extension of Wood et al. (2000) Extension of Wood et al. (2000) Flow:�Blasius u . c . D c ���in�the�fluid�domain ∇ = ∇ ∇ n . D c k ( ) ���at��� x c − ∇ = Σ γκ c y ( h ) C ��at�top�of�B.L. = = 0 c x ( x ) c x ( x l ) = = = + 0 0 8 dissolution M. Quintard
10/12/2008 Extension of Wood et al. (2000) Extension of Wood et al. (2000) Extension of Wood et al. (2000) ɶ c c c = + with u . c . D c ���in�the�fluid�domain ∇ = ∇ ∇ n . D c k ( ) x c ���at��� − ∇ = Σ γκ Σ and ɶ ɶ u . c . D c ���in�the�fluid�domain ∇ = ∇ ∇ ɶ ɶ ɶ ɶ n . D c k c k c k c ���at��� − ∇ = + − Σ γκ Σ 9 dissolution M. Quintard
10/12/2008 Extension of Wood et al. (2000) Extension of Wood et al. (2000) Extension of Wood et al. (2000) ɶ c s ( ) x c ... = + with�� � u . s . D s ���in�the�fluid�domain ∇ = ∇ ∇ ɶ ɶ n . D s k s k k s ���at��� − ∇ = + − Σ γκ Σ + ɶ k k k s = eff Σ Σ 10 dissolution M. Quintard
10/12/2008 Results for circular patches (far from Results for circular patches (far from Results for circular patches (far from entrance region) entrance region) entrance region) ɶ K k / k 3� = = m f Sc / D 1 = ν = Da kl D / = 11 dissolution M. Quintard
10/12/2008 Results Results Results � two limit cases � two limit cases – Da <<1, c ( x ) = C 0 � k eff = 〈 k 〉 – Da <<1, c ( x ) = C 0 � k eff = 〈 k 〉 – Da >>1, k eff = k * (harmonic mean of the – Da >>1, k eff = k * (harmonic mean of the reactivities) reactivities) � general case � general case – influence of geometry – influence of geometry – slight influence of Re – slight influence of Re (for low Re… ) (for low Re… ) 12 dissolution M. Quintard
10/12/2008 Ablative case: transient � � DNS � � � � DNS � � Ablative case: transient � � � � DNS Ablative case: transient � Ablation leads to non-differentiable surfaces � Ablation leads to non-differentiable surfaces � Limits of ALE and phase field methods � � � Limits of ALE and phase field methods � � � � � � adapted VOF method (Aspa, 2006) adapted VOF method (Aspa, 2006) 13 dissolution M. Quintard
10/12/2008 Example 1: steady-state surface Example 1: steady- -state surface state surface Example 1: steady 14 dissolution M. Quintard
10/12/2008 Example 2: porous composites Example 2: porous composites Example 2: porous composites Weaved Layered Da ≈ 3 Da ≈ 6�10 82 Note:� T changes�the�“diffusion”�coefficient 15 dissolution M. Quintard
10/12/2008 K eff ? case Da<<1 ? case Da<<1 K eff K eff ? case Da<<1 � Projected Areas: A m,p and � Projected Areas: A m,p and A f,p A f,p � Steady-state ablation: � Steady-state ablation: uniform velocity implies uniform velocity implies ⇒ k cos( ) k cos( ) ξ θ = ξ θ f f m m A A A ɶ f ⇒ f k k m k = = f m A A A f p , m p , f p , k k k k = ≈ ≠ eff m t 0 = 16 dissolution M. Quintard
10/12/2008 K eff ? ? K eff K eff ? � Limit Cases: simple � Limit Cases: simple models models – Da<<1, max( k ) – Da<<1, max( k ) – Da>>1, k -harmonic mean – Da>>1, k -harmonic mean � Intermediate Da � � � � � Intermediate Da � � � � complex simulations complex simulations Simplified model 17 dissolution M. Quintard
10/12/2008 Dissolution: Darcy-scale (core-scale) Dissolution: Darcy- -scale (core scale (core- -scale) scale) Dissolution: Darcy models? models? models? pore8scale to Darcy8scale σ � Pore-scale: non-local � Pore-scale: non-local β effects (space and effects (space and V ����������� time) time) wormhole ����������� ω • Direct�Simulation� :�Bekri et�al.� porous�medium η ������������ V ∞ (1995),�Mercet (2000),�Zhang�et� Smith�(2001),�… ���������� • -etwork�models (Fredd,�Hoefner� Darcy8scale to et�al.) core8scale 18 dissolution M. Quintard
10/12/2008 Target 1: Dissolution Instabilities Target 1: Dissolution Instabilities Target 1: Dissolution Instabilities (Wormholing) (Wormholing Wormholing) ) ( FLOW�RATE HCl�calcite� system (Fredd et�al.,�SPE) water�"aCl system (Zarcone et�al.) 19 dissolution M. Quintard
10/12/2008 Target 2: “Optimum flow rate” Target 2: “Optimum flow rate” Target 2: “Optimum flow rate” Optimum�injection�rate: minimum�injected�acid�volume�to�breakthrough Q opt =�f�(length�core,�C NaCl ...) 1.00E+00 10 injected volume to breakthrough dimensionless length (L/Lm) 8.00E-01 Q= 5 cm3/heure 6.00E-01 Q = 10 cm3/heure (V/VP) Q = 30 cm3/heure Q= 50 cm3/heure 4.00E-01 Q= 100 cm3/heure Q= 150 cm3/heure OPTIMUM� 2.00E-01 Q= 200 cm3/heure FLOW�RATE salt concentration of 150g/l Q= 250 cm3/heure 0.00E+00 1 0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 1 10 100 1000 injected volume (V/VP) injection flux rate (cm3/hour) "aCL Concentration�of�150�g/l 20 dissolution M. Quintard
10/12/2008 Simplified Pore-scale problem Simplified Pore- -scale problem scale problem Simplified Pore - . S . − + c ∂ + ∇ ( ) ( ) . v c . D c ����in�the�fluid�domain = ∇ ∇ t ∂ c C ��at� A βσ = eq "ote�(binary�case) : n . D c kc �� ��� c C 0��at� A ��if� Da 1 − ∇ = ⇔ = ≈ >> eq βσ 21 dissolution M. Quintard
10/12/2008 Darcy-scale: Local Non- Darcy- -scale: Local Non scale: Local Non- - Darcy Equilibrium Models Equilibrium Models Equilibrium Models β C c C = = � Local equilibrium dissolution: � Local equilibrium dissolution: produces produces A A eq β β sharp fronts sharp fronts � LNE: Heuristic model classically used in Chemical Engineering � LNE: Heuristic model classically used in Chemical Engineering (discussion in Quintard & Whitaker, 1994, 1999) (discussion in Quintard & Whitaker, 1994, 1999) 1 β ∫ C c c ( x y ) dV = = + A A A β β β V x y β V β β β 8phase C ∂ ( ) * V . C . D . C C ε + ∇ = ∇ ∇ − α n βσ t ∂ r β x additional�terms + V σ 8phase +�other�macro8scale�equations 22 dissolution M. Quintard
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