HOMOGENIZATION OF REACTIVE TRANSPORT IN POROUS MEDIA Gr egoire - PowerPoint PPT Presentation

Homogenization of reactive transport 1 G. Allaire HOMOGENIZATION OF REACTIVE TRANSPORT IN POROUS MEDIA Gr´ egoire ALLAIRE, CMAP, Ecole Polytechnique Andro MIKELIC, ICJ, Universit´ e Lyon 1 Andrey PIATNITSKI, Narvik University. Work partially supported by the GDR MOMAS CNRS-2439 Dedicated to Alain Bourgeat 1. Introduction 2. Main result 3. Two-scale asymptotic expansions with drift 4. Rigorous proof

Homogenization of reactive transport 2 G. Allaire -I- INTRODUCTION We consider a periodic porous medium: fluid part Ω f , solid part R n \ Ω f . convection diffusion of a single solute: ∂c ∗ ∂t ∗ + b ∗ · ∇ x ∗ c ∗ − div x ∗ ( D ∗ ∇ x ∗ c ∗ ) = 0 in Ω f × (0 , T ∗ ) , with a linear adsorption process on the pore boundaries: c ∗ c ∗ − D ∗ ∇ x ∗ c ∗ · n = ∂ ˆ ∂t ∗ = k ∗ ( c ∗ − ˆ on ∂ Ω f × (0 , T ∗ ) , K ∗ ) The incompressible fluid velocity b ∗ ( x ∗ , t ∗ ) is assumed to be known. The unknowns are the concentrations c ∗ in the fluid and ˆ c ∗ on the solid boundary.

Homogenization of reactive transport 3 G. Allaire ✞ ☎ Scaling ✝ ✆ We adimensionalize the equations as follows: ✗ Characteristic lengthscale L R and timescale T R . ℓ ✗ Period ℓ << L R : we introduce a small parameter ǫ = L R . ✗ Characteristic velocity b R . ✗ Characteristic concentrations c R and ˆ c R . ✗ Characteristic diffusivity D R . ✗ Characteristic adsorption rate k R and adsorption equilibrium constant K R . New adimensionalized variables and constants: x = x ∗ , t = t ∗ , b ǫ ( x, t ) = b ∗ ( x ∗ , t ∗ ) , D = D ∗ , k = k ∗ , K = K ∗ L R T R b R D R k R K R

Homogenization of reactive transport 4 G. Allaire ✞ ☎ Scaling (continued) ✝ ✆ New unknowns: u ǫ = c ∗ , v ǫ = ˆ c c R ˆ c R and dimensionless equations ∂u ǫ ∂t + V R T R b ǫ · ∇ x u ǫ − D R T R div x ( D ∇ x u ǫ ) = 0 in Ω ǫ × (0 , T ) L 2 L R R and − DD R c R ∇ x u ǫ · n = ˆ c R ∂v ǫ ∂t = k R k ( c R u ǫ − ˆ c R v ǫ ) on ∂ Ω ǫ × (0 , T ) . L R T R KK R T diff eclet number: Pe = L R b R P´ = D R T advec = T diff Damkohler number: Da = L R k R D R T react We choose a diffusion timescale, i.e., we assume T R = L 2 D R = T diff . R

Homogenization of reactive transport 5 G. Allaire ✞ ☎ Scaling (continued) ✝ ✆ ∂u ǫ ∂t + Pe b ǫ · ∇ x u ǫ − div x ( D ∇ x u ǫ ) = 0 in Ω ǫ × (0 , T ) and ˆ c R ∂v ǫ c R v ǫ ˆ − D ∇ x u ǫ · n = ∂t = Da k ( u ǫ − ) on ∂ Ω ǫ × (0 , T ) . c R L R c R KK R We assume ˆ c R = T adsorp c R ˆ = T adsorp Pe = ǫ − 1 , Da = ǫ − 1 , = ǫ, = 1 c R L R T react c R K R T desorp

Homogenization of reactive transport 6 G. Allaire ✞ ☎ Scaling (continued) ✝ ✆  ∂u ǫ ∂t + 1 ǫ b ǫ · ∇ x u ǫ − div x ( D ǫ ∇ x u ǫ ) = 0 in Ω ǫ × (0 , T )        u ǫ ( x, 0) = u 0 ( x ) , x ∈ Ω ǫ ,   − D ǫ ∇ x u ǫ · n = ǫ∂v ǫ ∂t = k ǫ ( u ǫ − v ǫ   K ) on ∂ Ω ǫ × (0 , T )        v ǫ ( x, 0) = v 0 ( x ) , x ∈ ∂ Ω ǫ Assumptions: ✗ Unit cell Y = (0 , 1) n = Y ∗ ∪ O with fluid part Y ∗ � x � with div y b = 0 in Y ∗ ✗ Stationary incompressible periodic flow b ǫ ( x ) = b ǫ and b · n = 0 on ∂ O (not a necessary assumption, see Allaire-Raphael 2007) � x � ✗ Periodic symmetric coercive diffusion D ǫ ( x ) = D ǫ

Homogenization of reactive transport 7 G. Allaire ✞ ☎ Goal of homogenization ✝ ✆ Find the effective diffusion tensor. This is the so-called problem of Taylor dispersion (1953). Many previous works, including Adler, Auriault, van Duijn, Knabner, Mauri, Mikelic, Quintard, Rosier, Rubinstein, etc.

Homogenization of reactive transport 8 G. Allaire -II- MAIN RESULT Theorem. The solution ( u ǫ , v ǫ ) satisfies � � � � t, x − b ∗ t, x − b ∗ u ǫ ( t, x ) ≈ u 0 ǫ t and v ǫ ( t, x ) ≈ Ku 0 ǫ t with the effective drift � b ∗ = ( | Y ∗ | + | ∂ O| n − 1 K ) − 1 Y ∗ b ( y ) dy and u 0 the solution of the homogenized problem  ∂u 0 in R n × (0 , T ) ∂t − div ( A ∗ ∇ u 0 ) = 0    u 0 ( t = 0 , x ) = | Y ∗ | u 0 ( x ) + | ∂ O| n − 1 v 0 ( x )  in R n   | Y ∗ | + K | ∂ O| n − 1

Homogenization of reactive transport 9 G. Allaire ✞ ☎ Precise convergence ✝ ✆ � � � � t, x − b ∗ t, x − b ∗ + r u + r v u ǫ ( t, x ) = u 0 ǫ t ǫ ( t, x ) and v ǫ ( t, x ) = Ku 0 ǫ t ǫ ( t, x ) with � T � ( t, x ) | 2 dt dx = 0 , R n | r u,v lim ǫ ǫ → 0 0

Homogenization of reactive transport 10 G. Allaire ✞ ☎ Homogenized coefficients ✝ ✆ The homogenized diffusion tensor is A ∗ = ( | Y ∗ | + K | ∂ O| n − 1 ) − 1 � � A ∗ 1 + A ∗ 2 � 1 = K 2 k | ∂ O| n − 1 b ∗ ⊗ b ∗ and A ∗ Y ∗ D ( I + ∇ y w ( y ))( I + ∇ y w ( y )) T dy with A ∗ 2 = where the components w i ( y ), 1 ≤ i ≤ n , of w ( y ) are solutions of the cell problem  b ( y ) · ∇ y w i − div y ( D ( y ) ( ∇ y w i + e i )) = ( b ∗ − b ( y )) · e i in Y ∗     D ( y ) ( ∇ y w i + e i ) · n = Kb ∗ · e i on ∂ O     y → w i ( y ) Y -periodic Remark that the value of b ∗ is exactly the compatibility condition for the existence of w i .

Homogenization of reactive transport 11 G. Allaire ✞ ☎ Equivalent homogenized equation ✝ ✆ � � t, x − b ∗ Define ˜ u ǫ ( t, x ) = u 0 ǫ t . Then, it is solution of  ∂ ˜ ∂t + 1 u ǫ ǫ b ∗ · ∇ ˜ in R n × (0 , T ) u ǫ − div ( A ∗ ∇ ˜  u ǫ ) = 0   u ǫ ( t = 0 , x ) = | Y ∗ | u 0 ( x ) + | ∂ O| n − 1 v 0 ( x )  in R n ˜   | Y ∗ | + K | ∂ O| n − 1

Homogenization of reactive transport 12 G. Allaire -III- TWO-SCALE ANSATZ WITH DRIFT To motivate our result, let us start with a formal process. Standard two-scale asymptotic expansions should be modified to introduce an unknown large drift b ∗ ∈ R n + ∞ � � t, x − b ∗ t ǫ , x � ǫ i u i u ǫ ( t, x ) = , ǫ i =0 with u i ( t, x, y ) a function of the macroscopic variable x and of the periodic microscopic variable y ∈ Y = (0 , 1) n . Similarly + ∞ � � t, x − b ∗ t ǫ , x � ǫ i v i v ǫ ( t, x ) = ǫ i =0

Homogenization of reactive transport 13 G. Allaire We plug these ansatz in the system of equations and use the usual chain rule derivation � � �� � � � t, x − b ∗ t t, x − b ∗ t ǫ , x ǫ , x � ǫ − 1 ∇ y u i + ∇ x u i ∇ u i = , ǫ ǫ plus a new contribution   � � �� t, x − b ∗ t ∂ ǫ , x  ∂u i � t, x, x � ∂t − ǫ − 1 b ∗ · ∇ x u i u i =  ∂t ǫ ǫ � �� � new term

Homogenization of reactive transport 14 G. Allaire  ∂u ǫ ∂t + 1 ǫ b ǫ · ∇ x u ǫ − div x ( D ǫ ∇ x u ǫ ) = 0 in Ω ǫ × (0 , T )        u ǫ ( x, 0) = u 0 ( x ) , x ∈ Ω ǫ ,   − 1 ǫ D ǫ ∇ x u ǫ · n = ∂v ǫ ∂t = k ǫ 2 ( u ǫ − v ǫ   K ) on ∂ Ω ǫ × (0 , T )        v ǫ ( x, 0) = v 0 ( x ) , x ∈ ∂ Ω ǫ

Homogenization of reactive transport 15 G. Allaire ✞ ☎ Fredholm alternative in the unit cell ✝ ✆ Lemma. The boundary value problem  b ( y ) · ∇ y v ( y ) − div y ( D ( y ) ∇ y v ( y )) = g ( y ) in Y ∗    D ( y ) ∇ y v ( y ) · n = h ( y ) on ∂ O    y → v ( y ) Y -periodic admits a unique solution in H 1 ( Y ∗ ), up to an additive constant, if and only if � � Y ∗ g ( y ) dy + h ( y ) ds = 0 . ∂ O Recall that Y = Y ∗ ∪ O with Y ∗ = fluid part and O = solid obstacle.

Homogenization of reactive transport 16 G. Allaire ✞ ☎ Cascade of equations ✝ ✆ Equation of order ǫ − 2 :  b ( y ) · ∇ y u 0 − div y ( D ( y ) ∇ y u 0 ) = 0 in Y ∗    � � u 0 − v 0 D ( y ) ∇ y u 0 · n = 0 = k on ∂ O K    y → u 0 , v 0 ( t, x, y ) Y -p´ eriodique We deduce u 0 ( t, x, y ) ≡ u 0 ( t, x ) and v 0 ( t, x, y ) ≡ Ku 0 ( t, x )

Homogenization of reactive transport 17 G. Allaire Equation of order ǫ − 1 :  − b ∗ · ∇ x u 0 + b ( y ) · ( ∇ x u 0 + ∇ y u 1 ) − div y ( D ( y ) ( ∇ x u 0 + ∇ y u 1 )) = 0 in Y ∗     � � − D ( y ) ( ∇ x u 0 + ∇ y u 1 ) · n = − b ∗ · ∇ x v 0 · n = k u 1 − v 1 on ∂ O K     y → u 1 , v 1 ( t, x, y ) Y -periodic We deduce n v 1 = Ku 1 + K 2 ∂u 0 � k b ∗ · ∇ x u 0 u 1 ( t, x, y ) = ( t, x ) w i ( y ) and ∂x i i =1

Homogenization of reactive transport 18 G. Allaire ✞ ☎ Cell problem ✝ ✆  b ( y ) · ∇ y w i − div y ( D ( y ) ( ∇ y w i + e i )) = ( b ∗ − b ( y )) · e i in Y ∗     D ( y ) ( ∇ y w i + e i ) · n = Kb ∗ · e i on ∂ O     y → w i ( y ) Y -periodic The compatibility condition (Fredholm alternative) for the existence of w i is � b ∗ = ( | Y ∗ | + | ∂ O| n − 1 K ) − 1 Y ∗ b ( y ) dy.