Upscaling of the Reaction-Advection-Diffusion Equation in Porous - PowerPoint PPT Presentation

Upscaling of the Reaction-Advection-Diffusion Equation in Porous Media with Monod-Like Kinetics Florin A. Radu Helmholtz Center for Environmental Research - UFZ, Permoserstr. 15, D-04318 Leipzig, Germany University of Jena, W¨ ollnitzerstr. 7, D-07749, Jena, Germany mailto:florin.radu@ufz.de Joint work with F. Hesse, S. Attinger and M. Thullner

Motivation • Macroscale simulations based on microscale parameters are normaly overestimating the degradation, which leads to a false prognoze • There is therefore a strong need for effective, macroscale degradation rates • For the zero- or first-order degradation, the derivation of effective parameters is well understood. Not the same can be said about Monod-like kinetics F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 2

OBJECTIVE • Starting with the 2D pore scale model to derive an 1D model by upscaling in the transversal direction ⇒ • To determinate effective rates for Monod-like degradation • To consider the effect of bioavailability on upscaling F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 3

Bioavailability Diffusion-limited regime: • Diffusion is low compared with the degradation rates • The contaminant is degraded very fast at the surface F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 4

Bioavailability Diffusion-limited regime: • Diffusion is low compared with the degradation rates • The contaminant is degraded very fast at the surface Reaction-limited regime: • Diffusion is fast compared with the degradation rates • The process is controlled by reaction F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 4

Bioavailability Diffusion-limited regime: • Diffusion is low compared with the degradation rates • The contaminant is degraded very fast at the surface Transition regime Reaction-limited regime: • Diffusion is fast compared with the degradation rates • The process is controlled by reaction • The effective degradation rates are influenced by diffusion (and convection)! F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 5

Mathematical Model (2D Pore Scale Model) ∂ ∂t c + v · ∇ c = D ∆ c in Ω p , D ∇ c · n = R ( c ) on Γ s , Γ i c = c 0 on f , Γ o ∇ c · n = 0 on f . where R ( c ) = − k max c K m + c or R ( c ) = − kc F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 6

Simplifications • The system is made dimensionless • We consider steady state • We neglect the longitudinal diffusion • The velocity has a component only in the flow direction F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 7

(simplified) Mathematical Model = D ∂ 2 Pe v(y) ∂ ∂ xc ∂y 2 c in Ω p , D ∇ c · n = R ( c ) on Γ s , Γ i c = 1 on f , Γ o ∇ c · n = 0 on f . where Φ 2 c or R ( c ) = − Φ 2 c R ( c ) = − c 1 + K m F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 8

AIM: an 1D upscaled Model ∂ 2 ∂ v eff ∂x � c � y = D eff ∂x 2 � c � y − R eff ( � c � y ) in V x , � �� � � �� � � �� � reaction advection diffusion � c � y = 1 on Inlet. • We need to determine the effective coefficients v eff , D eff and R eff . F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 9

Different scenarios ❳❳❳❳❳❳❳❳❳❳❳❳ Reaction First-order Monod Velocity ❳ uniform v = 1 v = 1 Φ 2 c R ( c ) = − Φ 2 c R ( c ) = − 1+ c/K m parabolic � 1 − y 2 � � 1 − y 2 � v = 1 . 5 v = 1 . 5 Φ 2 c R ( c ) = − Φ 2 c R ( c ) = − 1+ c/K m F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 10

First-order kinetics and uniform velocity profile ❳❳❳❳❳❳❳❳❳❳❳❳ Reaction First-order Monod Velocity ❳ uniform v = 1 v = 1 Φ 2 c R ( c ) = − Φ 2 c R ( c ) = − 1+ c/K m parabolic � 1 − y 2 � � 1 − y 2 � v = 1 . 5 v = 1 . 5 Φ 2 c R ( c ) = − Φ 2 c R ( c ) = − 1+ c/K m F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 11

First-order kinetics and uniform velocity profile • The effective averaged equation reads: Pe ∂ ∂ x � c � y = − Φ 2 eff � c � y , • Effective degradation rate Φ 2 eff = η Φ 2 with η = c | y =1 � c � y • c | y =1 is the bioavailable concentration, whereas � c � y the y -averaged one. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 12

First-order kinetics and uniform velocity profile • For small Φ 2 the global and local behavior is coupled ( reaction-limited regime ). • For large Φ 2 the global reaction rate Φ 2 eff saturates ( diffusion-limited regime ). F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 13

First-order kinetics and uniform velocity profile Diffusion limited regime • In the reaction-limited regime quantitatively differences of both curves. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 14

First-order kinetics and uniform velocity profile Diffusion-limited regime • In the diffusion-limited regime both curves are far apart. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 15

Uniform velocity profile and Monod kinetics ❳❳❳❳❳❳❳❳❳❳❳❳ Reaction First-order Monod Velocity ❳ uniform v = 1 v = 1 Φ 2 c R ( c ) = − Φ 2 c R ( c ) = − 1+ c/K m parabolic � 1 − y 2 � � 1 − y 2 � v = 1 . 5 v = 1 . 5 Φ 2 c R ( c ) = − Φ 2 c R ( c ) = − 1+ c/K m F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 16

Uniform velocity profile and Monod kinetics • The effective averaged equation reads: Φ 2 eff � c � y Pe ∂ ∂ x � c � y = − , 1 + � c � y / K m • Effective degradation rate eff = η Φ 2 und K m, eff = K m /η Φ 2 with η = c | y =1 � c � y • c | y =1 is the bioavailable concentration, whereas � c � y the y -averaged one. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 17

Uniform velocity profile and Monod kinetics K m = 1 • For K m > c behavior of η is similar to first-order reaction. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 18

Uniform velocity profile and Monod kinetics K m = 0 . 1 For K m � c nonlinearities increase. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 19

Uniform velocity profile and Monod kinetics K m = 0 . 01 • Constant approximations for effective parameters through fitting. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 20

Uniform velocity profile and Monod kinetics One parameter fit • Fitting for η is not able to reproduce behavior in the transition regime. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 21

Two parameter fit • With two parameters the characteristic is well preserved. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 22

Parabolic velocity profile ❳❳❳❳❳❳❳❳❳❳❳❳ Reaction First-order Monod Velocity ❳ uniform v = 1 v = 1 Φ 2 c R ( c ) = − Φ 2 c R ( c ) = − 1+ c/K m parabolic � 1 − y 2 � � 1 − y 2 � v = 1 . 5 v = 1 . 5 Φ 2 c R ( c ) = − Φ 2 c R ( c ) = − 1+ c/K m F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 23

Parabolic velocity profile • Idea: variable separation � c ( x, y ) = c i ( x )Ψ i ( y ) i ≥ 1 • Effective equation for the first mode c 1 ∂ 2 ∂ v eff ∂xc 1 = D eff ∂x 2 c 1 − R eff ( c 1 ) in V x , � �� � � �� � � �� � reaction advection diffusion � c � y = 1 on Inlet. • The averaged concentration is approximated by c 1 � Ψ 1 � y . F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 24

Parabolic velocity profile Effective velocity • Higher reaction rate emphasize the effect of velocity profile. • The effective velocity v eff saturates for high values of Φ 2 F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 25

Parabolic velocity profile Effective diffusion • Weak dependecy of the dipersivity from reaction rate. • The effective dispersivity D eff saturates for high values of Φ 2 F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 26

Parabolic velocity profile Monod kinetics • Same fitting procedure as with uniform velocity can be applied. • Slightly different behavior is observed. F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 27

Summary • We considered a 2D pore scale model for advective-diffusive-reactive transport of a contaminant • A dimension reducing upscaling model is presented • The special case of degradation following Monod kinetics is treated • Constant effective parameters are determinated through numerical fitting F . A. Radu Upscaling 2008, Dubrovnik 13 - 16.10.2008 28