Crystal dissolution and precipitation in porous media: formal - PowerPoint PPT Presentation

Crystal dissolution and precipitation in porous media: formal homogenization and numerical experiments T.L. van Noorden, I.S. Pop Centre for Analysis, Scientific Computing and Applications Department of Mathematics, Technische Universiteit Eindhoven Part of this research has been funded by the Dutch BSIK/BRICKS project Dubrovnik, 15-10-2008 .

Outline • Introduction: crystals in porous media • Model: free boundary problem y=1/2 l � w � (t) g y v flow q n • Thin strip x � (t) d(x,t) � � w y=-1/2 l x=0 x=L *+ %"#&$'() !"#$ • Perforated domain + !"#$%&'$ ("&)*$ • Open Problems / Future Directions

Flow through porous medium porous medium, fully saturated dissolved ions transported by the flow, e.g. sodium ( Na + ) and chlorine ( Cl − ) ions crystals attached to the grain surface (porous matrix), e.g. sodium chloride ( NaCl ) precipitation/dissolution reaction on the grain surface M 12 ⇆ n 1 M 1 + n 2 M 2

Model equations y=1/2 l � w � (t) g y v flow q n x � (t) d(x,t) � � w y=-1/2 l x=0 x=L Flow: q – fluid velocity (m/s) p – pressure inside fluid (Pa) µ – dynamic viscosity (kg/(ms))  µ ∆ q = ∇ p,  Stokes flow:  in Ω t and q = Kv n ν, on Γ t , = 0 . ∇ · q with K = ρ f − ( n 1 + n 2 ) ρ c . (Using the assumption c f + c 1 + c 2 ≡ ρ f ) ρ f

Model equations y=1/2 l � w � (t) g y v flow q n x � (t) d(x,t) � � w y=-1/2 l x=0 x=L Ion concentration: Precipitation, dissolution reaction: M 12 ⇆ n 1 M 1 + n 2 M 2 , Mass conservation for ion concentrations c i (mol/m 3 ) ( i = 1 , 2): in fluid ∂ t c i + ∇ · ( qc i − D ∇ c i ) = 0 for x ∈ Ω t ( n i ρ − c i ) v n = Dν · ∇ c i for x ∈ Γ t

Dissolution and precipitation rate Thickness of crystalline layer: normal velocity of interface between cristals and fluid v n = r p − r d , 1) Precipitation rate r p (mol/m 2 s): r p = k p r ( c 1 , c 2 ) = k p [ c 1 ] n 1 + [ c 2 ] n 2 + 2) Dissolution rate r d (mol/m 2 s) r d ∈ k d H ( d ( x, Γ w )) where H denotes the set-valued Heaviside graph { 0 } , if u < 0 ,   H ( u ) = [0 , 1] , if u = 0 , { 1 } , if u > 0 . 

2D Model: dimensionless equations Denote ǫ := l L , ... Assumptions: symmetry w.r.t. y -axis, c 1 = c 2 = c ref u ǫ t = ∇ · ( D ∇ u ǫ − q ǫ u ǫ ) ,  u ǫ   ǫ 2 µ ∆ q ǫ = ∇ p ǫ ,   in Ω ǫ ( t ) , ∇ · q ǫ = 0 ,   u ǫ , q ǫ and p ǫ symmetric around y = 0 ,    � x ) 2 , d ǫ t = k ( r ( u ǫ ) − w ) 1 + ( ǫd ǫ    w ∈ H ( d ǫ ) ,  on Γ ǫ ( t ) ν ǫ · ( D ∇ u ǫ − q ǫ u ǫ ) = − ǫk ( r ( u ǫ ) − w )( ρ − u ǫ ) ,   q ǫ = − ǫKk ( r ( u ǫ ) − w ) ν ǫ ,   where Ω ǫ ( t ) := { ( x, y ) | 0 ≤ x ≤ 1 , − ǫ (1 / 2 − d ǫ ( x, t )) ≤ y ≤ ǫ (1 / 2 − d ǫ ( x, t )) } , and where � ν ǫ = ( ǫ∂ x d ǫ , − 1) T / 1 + ( ǫ∂ x d ǫ ) 2 ,

1D model Assumptions: • no flow: q = 0 • 1D wall salt wall  Fluid ∂ t v = ∂ 2 for x ∈ (0 , h ( t )) , x v,  v(x,t)   ∂ x v = 0 , for x = 0 ,    x=h(t) ∂ x v = ( ρ − v ) h ′ ( t ) , for x = h ( t ) , x=0 h ′ ( t ) = D a ( w ( t ) − r ( v )) , x=1  for x = h ( t ) ,     w ( t ) ∈ H (1 − h ( t )) .  Theorem: There exists a unique, positive and bounded solution. (Pop, v.N. IMA J. Appl. Math. 2008) , 2D/3D: existence and uniqueness are open

2D Simulation: dissolution in strip (Movie)

Thin strip: upscaling Formal assymptotics for ǫ → 0 Assume u 0 ( x, y ǫ, t ) + ǫu 1 ( x, y u ǫ ( x, y, t ) ǫ, t ) + ǫ 2 ( ... ) , = q 0 ( x, y ǫ, t ) + ǫq 1 ( x, y q ǫ ( x, y, t ) ǫ, t ) + ǫ 2 ( ... ) , = p 0 ( x, y ǫ, t ) + ǫp 1 ( x, y p ǫ ( x, y, t ) ǫ, t ) + ǫ 2 ( ... ) , = d ǫ ( x, t ) d 0 ( x, t ) + ǫd 1 ( x, t ) + ǫ 2 ( ... ) . = The vertical coordinate of the variables u i ( x, z, t ), q i ( x, z, t ) and p ǫ ( x, z, t ) are rescaled. They are defined on Ω( t ) := { ( x, z ) | 0 ≤ x ≤ 1 , − 1 / 2 + d ǫ ≤ z ≤ 1 / 2 − d ǫ } .

Formal asymptotics Substituting the asymptotic expansions, integrating along the z -coordinate, and retaining only terms in- dependent of ǫ , yields  ∂ t ((1 − 2 d 0 ) u 0 + 2 ρd 0 ) = ∂ x ( D (1 − 2 d 0 ) ∂ x u 0 − ¯ qu 0 ) ,    ∂ x ¯ q − 2 K∂ t d 0 = 0 ,  ∂ t d 0 ∈ k ( r ( u 0 ) − H ( d 0 )) ,   where � 1 / 2 − d 0 ( x,t ) − 1 / 2+ d 0 ( x,t ) q (1) ¯ q ( x, t ) = ( x, z, t ) dz. 0

Thin strip: upscaled vs. original equations u d x x Profiles of both 2-D and effective model, for t = 20 and t = 40. Thin line: solution of the effective model Dashed line: 2-D model with ǫ = 0 . 1 Dots: 2-D model with ǫ = 0 . 01

Thin strip: traveling wave Non-negative traveling wave solutions: u = u ( η ), d = d ( η ) and q = q ( η ) with η = x − at , and d < 1 / 2, satisfying − a ((1 − 2 d ) u + 2 ρd ) ′ − ((1 − 2 d ) Du ′ − qu ) ′ = 0 ,   − ad ′ ∈ k ( r ( u ) − H ( d )) ,  in R . q ′ + 2 aKd ′ = 0 ,   and boundary conditions u ( −∞ ) = u ∗ , u ( ∞ ) = u ∗ , d ( −∞ ) = d ∗ , d ( ∞ ) = d ∗ , q ( −∞ ) = q ∗ , where 0 ≤ u ∗ , u ∗ , q ∗ and 0 ≤ d ∗ , d ∗ < 1 / 2.

Thin strip: traveling wave (2) d ∗ = 0 � d ∗ > 0 , I (dissolution wave) 0 ≤ u ∗ < u s u ∗ = u s , d ∗ > 0 , � d ∗ = 0 II (precipitation wave) u ∗ = u s , 0 ≤ u ∗ < u s Theorem. No traveling wave exists with boundary conditions from class II. Theorem. For any set of boundary conditions from class I, there exists a traveling wave (unique up to a shift). (v.N. EJAM 2008) (Compare to results in Knabner, Van Duijn, EJAM 1997: crystal layer has infinitesimal thickness, can be obtained as for- mal limit ρ → ∞ )

Perforated Domain *+ %"#&$'() !"#$ + !"#$%&'$ ("&)*$ Level set function S such that Γ = { S = 0 } . Evolution of Γ given by S t + |∇ S | v n = S t − 1 ( k p r ( c 1 , c 2 ) − k d w ( x )) |∇ S | = 0 ρ c Expand S ǫ

Perforated Domain: homogenization Formal assymptotics for ǫ → 0 Assume u 0 ( x, x ǫ , t ) + ǫu 1 ( x, x u ǫ ( x, t ) ǫ , t ) + ǫ 2 ( ... ) , = q 0 ( x, x ǫ , t ) + ǫq 1 ( x, x q ǫ ( x, t ) ǫ , t ) + ǫ 2 ( ... ) , = p 0 ( x, x ǫ , t ) + ǫp 1 ( x, x p ǫ ( x, t ) ǫ , t ) + ǫ 2 ( ... ) , = S 0 ( x, x ǫ , t ) + ǫS 1 ( x, x S ǫ ( x, t ) ǫ , t ) + ǫ 2 ( ... ) . = Where u k ( · , y, · ), q k ( · , y, · ), p k ( · , y, · ) and S k ( · , y, · ) are 1-periodic in y .

Upscaled equations  y ∈ [0 , 1] 2 ∂ t S 0 ( x, y, t ) − f ( u 0 ( x, t ) , y ) |∇ y S 0 ( x, y, t ) | = 0    ∂ t ( | Y 0 ( x, t ) | u 0 ) = ∇ x · ( D A ( x, t ) ∇ x u 0 − ¯ qu 0 ) + | Γ 0 ( x, t ) | f ( u 0 ) ρ x ∈ Ω  q = − 1 ¯ µ K ( x, t ) ∇ x p 0 x ∈ Ω    ∇ x · ¯ q = | Γ 0 ( x, t ) | Kf ( u 0 ) x ∈ Ω  where f ( u 0 ( x, t ) , y ) = k ( u 2 0 − H δ ( dist ( y, Γ))) Y 0 ( x, t ) = { S 0 < 0 } Γ 0 = { S 0 = 0 } (Hard step: interchange ∇ x and integration � | Y 0 ( x, t ) | ∂ t u 0 = ∇ y · ( ∇ y u 2 + ∇ x u 1 − q 1 u 0 − q 0 u 1 ) dy Y 0 ( x,t ) � + ∇ x · ( ∇ y u 1 + ∇ x u 0 − q 0 u 0 ) dy Y 0 ( x,t ) (v.N. MSS 2008))

where the tensors A = ( a ij ) i,j and K = ( k ij ) i,j are given by � a ij = δ ij + ∂ y i v j dy, Y 0 ( x,t ) where v j solves the cell-problem  ∆ y v j = 0 y ∈ Y 0 ( x, t )   ν 0 ∇ y v j = − e j y ∈ Γ 0 ( x, t )  periodicity in y,  and � k ij = w ji dy, Y 0 ( x,t ) where the vector w j with components w ji solves the cell-problem  ∆ y w j = ∇ y π j + e j y ∈ Y 0 ( x, t )    ∇ y · w j = 0 y ∈ Y 0 ( x, t )  w j = 0 y ∈ Γ 0 ( x, t )    periodicity in y,  with π j the corresponding pressure.

Simplification: circular grains  ∂ t R ( x, t ) = f ( u 0 , R ( x, t )) := k ( u 2 0 − H δ ( R − R min )) x ∈ Ω   ∂ t ((1 − πR 2 ) u 0 ) = ∇ x · ( D A ( R ) ∇ x u 0 − ¯  qu 0 ) + 2 πRf ( u 0 , R ) ρ x ∈ Ω  q = − 1 ¯ µ K ( R ) ∇ x p 0 x ∈ Ω    ∇ x · ¯ q = 2 πRKf ( u 0 ) x ∈ Ω  Periodicity in x 2 direction (Similar simplification is possible for ellipses, not for squares!) .(,/0* ! % #$&$" ! % #$ ! " #$ ! " #" '()*+,-*

Perforated domain upscaled vs. original equations Profiles of both 2-D and effective model, for t = 10 and t = 40. Dots: 2-D model with ǫ = 0 . 01 Line: effective model

Open problems / Future directions * Existence, uniqueness, estimates for microscale free * boundary model in 2D/3D? * Rigorous upscaling (phase-field formulation, with * Ch. Eck) * Blocking of strip ( d = 1 / 2)? * Application to biofilm growth models (with R. Helmig)