Homogenization of ComplexFlows in Porous Mediaand …Homogenization of ComplexFlows in Porous Mediaand Applications Harsha Hutridurga Ramaiah CMAP, ´Ecole Polytechnique, Palaiseau

Homogenization of Complex Flows in

Porous Media and Applications

Harsha Hutridurga Ramaiah

CMAP, École Polytechnique, Palaiseau.

17 September 2013

Harsha Hutridurga (Polytechnique) Homogenization of Reactive Flows 17 September 2013 1 / 55

Objective

Study of Solute Transport in Porous Media:

• Interplay among Molecular Diffusion, Mean Flow Field,Chemistry.

• Derive Effective Diffusion (Dispersion) for solute transport.

• Derive Effective Equation starting with a Porescale Description.


Outline

1. Domain description and Homogenization techniques.

2. Homogenization of single component transport.

∗ Modelling.

∗ Little detour.

∗ Linear reaction model.

∗ Nonlinear reaction model.

∗ Numerical study related to both models.

3. Homogenization of multicomponent transport.


Motivations

Different transport models studied here, find applications in

• Groundwater contaminant transport.

• CO2 sequestration.

• Underground storage of nuclear wastes.

• Oil reservoir simulations.

• Extraction plants in chemical engineering.


Periodic Porous MediaLength Scales : L (Observation length); l (Microscopic length).

Seperation of scales : l ≪ L i.e., ε =l

L≪ 1.

Paving : For j ∈ Zd ; Fluid : Y jε := ε(Y 0 + j) ; Solid : Sjε := ε(∂Σ0 + j).

Ωε := ∪j∈ZdYjε . We assume connected.

∂Ωε := ∪j∈ZdSjε . No assumption of connectedness.

Y0

Ʃ0

Y∂Ʃ

0

Y = Y0∪ Ʃ

0

Periodic Porous media

ε} ℝ2

UnitCell

FluidPart Part

Solid


Homogenization Techniques

Replacing governing equations at heterogeneities length scale with ahomogeneous model posed in an equivalent macroscopic medium.

• Representative Elementary Volume (REV) method: Whitaker’99.

• Two-scale asymptotic expansion method (formal):Babuska’74; Bensoussan, Lions, Papanicolaou’78; Sanchez-Palencia’80;Bakhvalov, Panasenko’90.

• Rigorous justification:Γ-convergence of De Giorgi’75G-convergence of Spagnolo’76H-convergence of Tartar’77.Two-scale convergence of Nguetseng’89; Allaire’92.Periodic Unfolding method of Cioranescu, Damlamian, Griso’02

• We have used Two-scale asymptotic expansion with drift:Papanicolaou’95; Donato, Piatnitski’05; Marusic-Paloka, Piatnitski’05;Allaire, Raphael’07.


Single Solute TransportModelling

Little detour

Linear Reaction Model

Nonlinear Reaction Model


Transport & Reaction (porescale modelling) I

Fluid part Ωf := ∪j∈Zd(Y0 + j); Solid part Ωs := R

d \ Ωf .

Adsorption : Surface reaction of mass exchange.

Unknown solute concentration : u in Ωf ; v on ∂Ωs.

In chemical equilibrium (at fluid-pore interface) : v = f(u)

Adsorption isotherm : f(u) =

{K u Linear K > 0

αu/(1 + βu) Langmuir α, β > 0

Non-equilibrium regime :dv

dτ= κ(f(u)− v); Reaction rate κ > 0.

If surface convection and diffusion are considered, ODE becomes:

∂v

∂τ+ bs · ∇sv − divs(Ds∇sv) = κ(f(u)− v) on (0, ζ) × ∂Ωs.


Transport & Reaction (porescale modelling) II

Projection matrix : For y ∈ ∂Σ0, G(y) = Id− n(y)⊗ n(y).

n(y) : Exterior normal to Y 0 at y.

Surface gradient : ∇s := G(y)∇

Surface divergence : divsΨ := div(G(y)Ψ).

Coupled convection-diffusion-reaction equations for u(τ, y) and v(τ, y):

∂u∂τ

+ b(y) · ∇u− div(D∇u) = 0 in (0, ζ)× Ωf ,

−D∇u · n = κ(f(u)− v) on (0, ζ)× ∂Ωs,

∂v∂τ

+ bs · ∇sv − divs(Ds∇sv) = κ(f(u)− v) on (0, ζ)× ∂Ωs.

For global mass conservation, we take the reaction term as Neumanndata.


Notation : Subscript # indicates a space of Y -periodic functions.

Flow fields: b ∈ L∞# (Ωf ;Rd); bs ∈ L∞# (∂Ωs;R

d).

divyb = 0 in Ωf , b · n = 0 on ∂Ωs, divsyb

s = 0 on ∂Ωs.

We assume that the flow fields are given and are independent of time.

In physical reality, bs is the tangential trace of b on ∂Ωs.

Coercive Diffusion D ∈ L∞# (Ωf ;Rd×d); Ds ∈ L∞# (∂Ωs;R

d×d).

Previous works:

ODE for v:Allaire, Brizzi, Mikelic, Piatnitski’10, Chem.Eng.SciAllaire, Mikelic, Piatnitski’10, SIAM.J.Math.Anal.

We have generalized the model via surface convection and diffusion.Allaire, Hutridurga’12, IMA.J.Appl.Math.


Scaled model (Strong Convection regime)

Parabolic scaling : x = ε y (observation scale); t = ε2 τ (longer time)

Final time : T = ε2ζ. As ζ = O(ε−2), T = O(1).

Y -periodic coefficients become ε-periodic: bε(x) = b(x/ε), Dε(x) = D(x/ε).

∂uε∂t

+1

εbε · ∇uε − div(Dε∇uε) = 0 in (0, T ) × Ωε,

−1

εDε∇uε · n =

κ

ε2(f(uε)− vε) on (0, T )× ∂Ωε,

∂vε∂t

+1

εbsε · ∇

svε − divs(Dsε∇

svε) =κ

ε2(f(uε)− vε) on (0, T )× ∂Ωε,

uε(0, x) = uin(x) x ∈ Ωε,; vε(0, x) = v

in(x) x ∈ ∂Ωε.

ε−1 next to the velocity and ε−2 next to reaction term means, the convection

diffusion and reaction are of same order of magnitude at the porescale.


Oscillations enter the system in different ways :

∗ Underlying microstructure Ωε

∗ Periodic coefficients

∗ Singular scaling

∗ Initial data

Questions

⋆ Do the oscillations in the coefficients affect the solution ?

⋆ Does the singular convection term affect the solution ?

⋆ What is the effective behaviour of the ε-problem ?

⋆ Does there exist unique (u0, v0) such that limε→0

uε = u0; limε→0

vε = v0 ?

⋆ Does the effective diffusion depend on convection and reaction ?


A Little Detour

Two-scale asymptotic expansion with drift(Extension of two-scale expansions for convection dominated problems.)

Papanicolaou’95; Donato, Piatnitski’05; Allaire, Raphael’07.


Convection-Diffusion in (0, T )× Rd :

∂uε∂t

+1

εbε · ∇uε − div(Dε∇uε) = 0. (C-D)

Homogenization of (C-D) is classical when∫Y

b(y) dy = 0.

(see e.g. McLaughlin, Papanicolaou, Pironneau’85).

Two-scale expansion with drift: uε(t, x) =∞∑

i=0

εiui

(t, x−

b∗

εt,x

ε

)

where the constant b∗ ∈ Rd is the drift.The coefficients ui(t, x, y) are Y -periodic in y.

The drift b∗ is unknown. We compute b∗ during upscaling.

Plug the postulated ansatz in (C-D). At order ε−2: (with periodic b.c.)

b(y) · ∇yu0 − divy(D∇yu0) = 0 in Y. =⇒ u0(t, x, y) ≡ u0(t, x)


(At order ε−1:) (with periodic b.c.)

b(y) · ∇yu1 − divy(D(∇xu0 +∇yu1)) = (b∗ − b) · ∇xu0 in Y.

Fredholm Alternative (source terms in equilibrium): b∗ =∫Y

b(y) dy.

(Homogenization Result)

uε(t, x) ≈ u0

(t, x−

b∗t

ε

)+ ε

d∑

i=1

χi

(xε

) ∂u0∂xi

(t, x−

b∗t

ε

),

where u0 satisfies:∂u0∂t

− div (D∇u0) = 0 and χi satisfy a cell problem.

Moving coordinates : Observer sees only diffusion.

To come back to fixed frame of reference:

ũε(t, x) := u0

(t, x−

b∗

εt

);∂ũε∂t

+1

εb∗ · ∇ũε − div (D∇ũε) = 0.

where ε−1b∗ is the effective velocity and D is the effective diffusion.Harsha Hutridurga (Polytechnique) Homogenization of Reactive Flows 17 September 2013 15 / 55

Two-scale convergence withdrift

Marusic-Paloka, Piatnitski’05

This generalizes Two-scale convergence (Nguetseng’89, Allaire’92).

Two-scale convergence was adapted to periodic surfaces in:Allaire, Damlamian, Hornung’95; Neuss-Radu’96.

We have adapted Two-scale convergence with drift to periodic surfaces.


Definition (Two-scale convergence with drift)

Let b∗ ∈ Rd be a constant drift. A sequence uε(t, x) ∈ L2((0, T )×Rd) is

said to two-scale converge with drift b∗ to a limit

u0(t, x, y) ∈ L2((0, T ) × Rd;L2#(Y ))

if, for any function φ(t, x, y) ∈ C∞c ((0, T ) × Rd;C∞# (Y )), we have:

limε→0

∫ T

0

∫

Rd

uε(t, x)φ(t, x−

b∗

εt,x

ε

)dx dt =

∫ T

0

∫

Rd

∫

Y

u0(t, x, y)φ(t, x, y) dy dx dt.

We denote this convergence by uε2−drift−−−−⇀ u0.

We apply this notion to homogenize:

∂uε∂t

+1

εbε · ∇uε − div(Dε∇uε) = 0.


Theorem (Compactness)

Let b∗ ∈ Rd be a constant. For any sequence uε(t, x) satisfying‖uε‖L2((0,T );H1(Rd)) ≤ C, there exists u0 ∈ L

2((0, T );H1(Rd)),

u1 ∈ L2((0, T ) × Rd;H1#(Y )) and a subsequence (still denoted by ε) s.t.

uε2−drift−−−−⇀ u0,

∇uε2−drift−−−−⇀ ∇xu0 +∇yu1.

The a priori estimates for the simplified problem:

Lemma (a priori estimates)

Let uε be the solution of the simplified model. Then, there exists aconstant C, independent of ε, s.t.

‖uε‖L∞((0,T );L2(Rd)) + ‖∇uε‖L2((0,T )×Rd)d ≤ C.

Above a priori estimates help us extract converging subsequences andtwo-scale with drift limits.


Linear Isotherm


Theorem (Homogenization Result) (Part I)

The solution (uε, vε) of the linear model satisfy:

uε(t, x) ≈ u0

(t, x−

b∗t

ε

)+ ε

d∑

i=1

χi

(xε

) ∂u0∂xi

(t, x−

b∗t

ε

),

vε(t, x) ≈ K u0

(t, x−

b∗t

ε

)+ ε

d∑

i=1

ωi

(xε

) ∂u0∂xi

(t, x−

b∗t

ε

),

where the drift b∗ is given by

b∗ = (|Y 0|+K|∂Σ0|)−1

∫

Y 0

b(y) dy +K

∫

∂Σ0

bs(y) dσ(y)

.

Drift is a weighted average of both velocity fields (with K as weight).

Transport and Chemistry are coupled!

Even when bs ≡ 0, surface effects contribute to the drift.


Theorem (Homogenization Result) (Part II)

The zero order approximation u0 satisfies:

Kd∂u0∂t

− divx (D∇xu0) = 0 in (0, T ) × Rd,

Kd u0(0, x) = |Y0|uin(x) + |∂Σ0|vin(x) x ∈ Rd,

where Kd = |Y0|+K|∂Σ0| and the dispersion D is given by:

Dij =

∫

Y 0

D(y) (∇yχi + ei) · (∇yχj + ej) dy

+κ

∫

∂Σ0

(χi −

ωiK

)(χj −

ωjK

)dσ(y)

+K−1∫

∂Σ0

Ds(y)(Kei +∇

syωi

)·(Kej +∇

syωj

)dσ(y).

Effective diffusion D is symmetric, positive definite and constant.


Theorem (Homogenization Result) (Part III)

(χi, ωi)1≤i≤d satisfy the following cell problem:

b(y) · ∇yχi − divy(D(∇yχi + ei)) = (b∗ − b) · ei in Y

0,

bs(y) · ∇syωi − divsy(D

s(∇syωi +Kei))

= K(b∗ − bs) · ei + κ(χi −K

−1ωi)

on ∂Σ0,

−D(∇yχi + ei) · n = κ(χi −K

−1ωi)

on ∂Σ0,

y → (χi, ωi) Y − periodic.

Solvability given by Fredholm Alternative (source terms in equilibrium):

∫

Y 0

(b∗ − b) · ei dy +

∫

∂Σ0

K(b∗ − bs) · ei dσ(y) = 0.

This gives the expression for the drift b∗.


Variational Formulation

a(uε, vε;φε, ψε) =

T∫

0

∫

Ωε

[∂uε∂t

φε+1

εbε · ∇uεφε +Dε∇uε · ∇φε

]dx dt

+ε

T∫

0

∫

∂Ωε

1

K

[∂vε∂t

ψε+1

εbsε · ∇

svεψε +Dsε∇

svε · ∇sψε

]dσε(x) dt

+εκ

ε2

T∫

0

∫

∂Ωε

(uε −

vεK

)(φε −

ψεK

)dσε(x) dt = 0.


Energy Equality :

1

2

d

dt‖uε‖

2L2(Ωε)

+ε

2K

d

dt‖vε‖

2L2(∂Ωε)

+

∫

Ωε

Dε∇uε · ∇uε dx

+ε

K

∫

∂Ωε

Dsε∇svε · ∇

svε dσ(x) +κε

ε2

∫

∂Ωε

(uε −

vεK

)2dσ(x) = 0

A priori estimates can be derived via the Energy estimates.

We then use the compactness results from two-scale convergence withdrift to derive homogenized equation.


Another scaling for the reaction rate:

Take b∗1 =1

|Y 0|

∫Y 0b(y) dy and b∗2 =

1|∂Σ0|

∫∂Σ0

bs(y) dσ(y)

Let us rescale the reaction rate ε−2κ→ κ in the linear model.If b∗1 = b

∗2, we have the following coupled homogenized system:

|Y 0|∂u0∂t

+ κ|∂Σ0|(u0 −

v0K

)= div(A∗∇xu0) in (0, T )× R

d,

|∂Σ0|∂v0∂t

− κ|∂Σ0|(u0 −

v0K

)= div(B∗∇xv0) in (0, T )× R

d.

If b∗1 6= b∗2, we have the following decoupled homogenized system:

|Y 0|∂u0∂t

+ κ|∂Σ0|u0 = div(A∗∇xu0) in (0, T ) × R

d,

|∂Σ0|∂v0∂t

+κ

K|∂Σ0|v0 = div(B

∗∇xv0) in (0, T )× Rd.


Numerical Study - Linear Model

Using Freefem++ package.


Lagrange P1 finite elements. Number of vertices 33586.

Obstacle : circular disk of radius 0.2

velocity field b(y) in Y 0

Surface velocity bs ≡ 0. Mean velocity∫Y 0b(y) dy = (0.0385,−2.67× 10−5)

Drift velocity b∗ = (0.0180,−1.25 ∗ 10−5)

For a given problem, ε is fixed. Effective velocity = b∗/ε.


Bulk diffusion D = 1., Surface diffusion Ds = 1.Reaction rate κ = 1., Equilibrium constant K = 1.Effective diffusion D depends on b(y) via cell solutions (χi, ωi).Behaviour of D11 w.r.t increasing velocity field for different values of D

s.

0

100

200

300

400

500

600

700

800

0 100 200 300 400 500 600 700 800 900 1000

Lon

gitu

dina

l Dis

pers

ion

Peclet Number

DS=0DS=100DS=200DS=300DS=400DS=500

Effective diffusion increases with the increasing velocity.Harsha Hutridurga (Polytechnique) Homogenization of Reactive Flows 17 September 2013 28 / 55

Bulk diffusion D = 1.Reaction rate κ = 1., Equilibrium constant K = 1.In previous experiment, D increased with surface diffusion Ds.Behaviour of D w.r.t increasing Ds with other parameters fixed.

0.828

0.83

0.832

0.834

0.836

0.838

0.84

0.842

0.844

0.846

0 1 2 3 4 5 6 7 8 9 10

Lon

gitu

dina

l Dis

pers

ion

Molecular Diffusion

0.826

0.828

0.83

0.832

0.834

0.836

0.838

0.84

0.842

0.844

0.846

0 1 2 3 4 5 6 7 8 9 10

Tra

nsve

rse

Dis

pers

ion

Molecular Diffusion

Observe that the transverse diffusion D22 is slightly lower than thelongitudinal diffusion D11.

Observe also that the effective diffusion becomes asymptotically constant for

relatively lower values of Ds.


Bulk diffusion D = 1., Surface diffusion Ds = 1.Equilibrium constant K = 1.Reaction rate κ appears in the expression for effective diffusion D.Behaviour of D w.r.t increasing κ with other parameters fixed.

1.55

1.6

1.65

1.7

1.75

1.8

1.85

0 50 100 150 200 250 300

Lon

gitu

dina

l Dis

pers

ion

Reaction Rate 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 50 100 150 200 250 300T

rans

vers

e D

ispe

rsio

nReaction Rate

Observe that D11 > D22.

Observe also that the effective diffusion becomes asymptotically constant for

very high values of κ.


Bulk diffusion D = 1., Surface diffusion Ds = 1.Reaction rate κ0 = 1., Equilibrium constant K = 1.Behaviour of cell solution χ1 w.r.t increasing Reaction rate κ.

IsoValue-0.118263-0.101224-0.0898652-0.0785062-0.0671471-0.0557881-0.0444291-0.03307-0.021711-0.0103520.001007060.01236610.02372510.03508420.04644320.05780220.06916120.08052030.09187930.120277

IsoValue-0.0166337-0.0141745-0.0125351-0.0108956-0.00925613-0.00761666-0.0059772-0.00433773-0.00269826-0.001058790.0005806720.002220140.003859610.005499070.007138540.008778010.01041750.01205690.01369640.0177951

IsoValue-0.0181629-0.0155051-0.0137331-0.0119612-0.0101893-0.0084174-0.00664549-0.00487357-0.00310166-0.001329750.0004421670.002214080.003985990.005757910.007529820.009301730.01107360.01284560.01461750.0190473

κ = κ0 κ = 5 κ0 κ = 8 κ0

This inversion phenomenon occurs with the cell solution χ2 too.

As χi are just the correctors, it would be interesting to test how increasing

reaction rate affects reconstructed solution.


Bulk diffusion D = 1., Surface diffusion Ds = 1.Reaction rate κ0 = 1., Equilibrium constant K = 1.Behaviour of reconstructed solution χ1 + y1 w.r.t increasing Reaction rate κ.

κ = κ0 κ = 6 κ0 κ = 19 κ0

As κ increases, the presence of surface diffusion results in the transition of theobstacles from being repulsive to attractive.

χ1 + y1 corresponds to local linearization of first two terms of ansatz for uε.


Langmuir Isotherm


Theorem (Homogenization Result) (Part I)

Assume b∗ =1

|Y 0|

∫

Y 0

b(y) dy =1

|∂Σ0|

∫

∂Σ0

bs(y) dσ(y).

The solution (uε, vε) of the nonlinear model satisfy:

uε(t, x) ≈ u0

(t, x−

b∗t

ε

)+ ε

d∑

i=1

χi

(u0,

x

ε

) ∂u0∂xi

(t, x−

b∗t

ε

),

vε(t, x) ≈ f(u0)

(t, x−

b∗t

ε

)+ εf ′(u0)

d∑

i=1

ωi

(u0,

x

ε

) ∂u0∂xi

(t, x−

b∗t

ε

),

where f(u) = αu/(1 + β u) is the Langmuir isotherm.

Remark that the cell solutions χi, ωi depend on u0.


Theorem (Homogenization Result) (Part II)

The zero order approximation u0 satisfies:

[|Y 0|+ |∂Σ0|f ′(u0)

] ∂u0∂t

− divx(D(u0)∇xu0) = 0 in (0, T )× Rd,

[|Y 0|u0 + |∂Σ

0|f(u0)](0, x) = |Y 0|uin(x) + |∂Σ0|vin(x) in Rd,

where D(u0) is the effective diffusion. The cell problem is:

b(y) · ∇yχi − divy(D∇yχi) = (b∗ − b(y)) · ei + divy(Dei) in Y

0,

−D(y) (∇yχi + ei) · n = κf′(u0) (χi − ωi) on ∂Σ

0,

bS · ∇syωi − divsy(D

s∇syωi)− (b∗ − bs) · ei − div

sy(D

sei)

= κ (χi − ωi) on ∂Σ0.

Fredholm Alternative :∫Y 0

(b∗ − b(y)) dy + f ′(u0)∫

∂Σ0(b∗ − bs) dσ(y) = 0.

As b∗ is a constant, we impose the matching drift assumption.


Dispersion (non-constant, non-symmetric, positive definite):

Dij(u0) =

∫

Y 0

D(y) (∇yχi + ei) · (∇yχj + ej) dy

+f ′(u0)

∫

∂Σ0

[χi − ωi] [χj − ωj] dσ(y)

+f ′(u0)

∫

∂Σ0

Ds(y)(∇syωi + ei

)·(∇syωj + ej

)dσ(y)

+

∫

Y 0

D(y)(∇yχj · ei −∇yχi · ej

)dy

+f ′(u0)

∫

∂Σ0

Ds(y)(∇syωj · ei −∇

syωi · ej

)dσ(y)

+

∫

Y 0

(b(y) · ∇yχi

)χj dy + f

′(u0)

∫

∂Σ0

(bs(y) · ∇syωi

)ωj dσ(y)

Remark the convective effects in effective diffusion.Harsha Hutridurga (Polytechnique) Homogenization of Reactive Flows 17 September 2013 36 / 55

Some notions on rigorous proofRemark that f(u) = αu/(1 + βu) blows up at u = −1/β.

Maximum Principles : 0 ≤ uin, vin ≤M implies 0 ≤ uε, vε ≤ M.

Maximum Principles and Energy Estimates lead to a priori estimates.

A priori estimates help us obtain limits (two-scale with drift).

Pass to limit, as ε→ 0, in the variational formulation.

As f(uε) is nonlinear, we need strong compactness of uε.

Difficulty: The presence of large drift ε−1b∗.

We prove compactness in moving coordinates i.e., of ûε defined as

ûε(t, x) = uε

(t, x+

b∗t

ε

).

Some ideas of the proof borrowed from:

Homogenization in a fixed domain (0, T )× Rd:Marusic-Paloka, Piatnitski’05, J. London. Math. Soc.


Difficulty: Ωε is unbounded, thus Rellich theorem doesn’t apply.

Lemma (localization in moving coordinates)

Let uε be the bulk concentration of the nonlinear model. Then, for anyδ > 0, there exists R(δ) > 0 such that, for any t ∈ [0, T ],

∥∥∥ûε(t, x)∥∥∥L2(Ω̂ε(t)∩Qc

R(δ))≤ δ,

where QcR(δ) = R

d \QR(δ) with QR(δ) =]−R(δ),+R(δ)[d.

As uε is defined in Ωε, we translate the domain: Ω̂ε(t) ={x+

b∗t

ε: x ∈ Ωε

}.

Difficulty: Localization gives compactness in space, but not in time.

We prove Equicontinuity in time (Arzelà-Ascoli type).Harsha Hutridurga (Polytechnique) Homogenization of Reactive Flows 17 September 2013 38 / 55

Numerical Study - Nonlinear Model

Using Freefem++ package.


Drift velocity : b∗ =1

|Y 0|

∫

Y 0

b(y) dy =1

|∂Σ0|

∫

∂Σ0

bs(y) dσ(y).

In 2d, connected fluid part =⇒ solid part is not.

Incompressibility : divsbs(y) = 0 on ∂Σ0 =⇒

∫

∂Σ0

bs(y) dσ(y) = 0.

In 2d, for the nonlinear model, we need to have∫Y 0b(y) dy = 0.

Vec Value00.1511960.3023930.4535890.6047860.7559820.9071781.058371.209571.360771.511961.663161.814361.965552.116752.267952.419142.570342.721542.87273

Vec Value00.2461870.4923730.738560.9847471.230931.477121.723311.969492.215682.461872.708052.954243.200433.446613.69283.938994.185174.431364.67755

Symmetric velocity field Non-symmetric velocity field


Bulk diffusion D = 1., Surface diffusion Ds = 1.Reaction rate κ = 1.Effective diffusion D depends on the homogenized solution u0.Behaviour of D when the magnitude of u0 increases when the velocity fieldb(y) is symmetric.

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 1 2 3 4 5 6 7 8 9 10

Dis

pers

ion

Magnitude homogenized solution

Horizontal disp. symmetric vfVertical disp. symmetric vf

Observe that D11 and D22 are almost the same and they are asymptotically

constant for greater values of u0.Harsha Hutridurga (Polytechnique) Homogenization of Reactive Flows 17 September 2013 41 / 55

Bulk diffusion D = 1., Surface diffusion Ds = 1.Reaction rate κ = 1.Effective diffusion D depends on the homogenized solution u0.Behaviour of D when the magnitude of u0 increases when the velocity fieldb(y) is non-symmetric.

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 1 2 3 4 5 6 7 8 9 10

Dis

pers

ion

Magnitude homogenized solution

Horizontal disp. asymmetric vfVertical disp. asymmetric vf

Observe that D22 > D11 and they are asymptotically constant for greater

values of u0.Harsha Hutridurga (Polytechnique) Homogenization of Reactive Flows 17 September 2013 42 / 55

Bulk diffusion D = 1., Surface diffusion Ds = 1.Reaction rate κ = 1. Homogenized solution u0 = 2.5Behaviour of D11 when the magnitude of D

s increases.

0.778

0.779

0.78

0.781

0.782

0.783

0.784

0.785

0 1 2 3 4 5 6 7 8 9 10


Bulk diffusion D = 1., Surface diffusion Ds = 1.Reaction rate κ = 1. Homogenized solution u0 = 2.5Behaviour of D11 when the magnitude of κ increases.

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0 100 200 300 400 500 600 700 800


Bulk diffusion D = 1., Surface diffusion Ds = 1.Reaction rate κ = 1.Compare exact solution with homogenized and reconstructed solution.ũε(t, x) = u0(t, x) + ε(χ1(u0, x/ε)∂x1u0 + χ2(u0, x/ε)∂x2u0)

Domain =]0, 1[×]0, 1[ with circular obstacles of radius 0.05arranged with period ε = 0.25.

IsoValue00.511.522.533.544.555.566.577.588.5

Initial Data

Initial data with compact support in ]0, 1[×]0, 1[

Exact solution uε is computed with Characteristic-Galerkin method for theconvection term.


Time t = 0.001

Exact soln. Homogenized son. Reconstructed soln.

Observe the oscillations in the reconstructed solution.


Time t = 0.005



Time t = 0.01


Homogenized and Reconstructed solution seem to diffuse quicker thanthe Exact solution.We believe it is because of the period ε = 0.25, which may not be toosmall to see the effect of homogenization.


MulticomponentTransport

Weakly coupled parabolic system of equations.


Consider N solutes dissolved in a fluid.

Unknown solute concentration : uα for each 1 ≤ α ≤ N .

(Governing Equations: For each 1 ≤ α ≤ N)

ρα∂uα∂τ

+ bα · ∇uα − div(Dα∇uα) +N∑

β=1

Παβuβ = 0 in (0, ζ)× Ωf ,

Dα∇uα · n = 0 on (0, ζ) × ∂Ωs.

Convective fields : bα ∈ L∞# (Ωf ;R

d). (not divergence free)

Porosity coefficients : ρα ∈ L∞# (Ωf ;R>0).

Coercive diffusion matrices : Dα ∈ L∞# (Ωf ;R

d).

Previous works:Neutron diffusion model:

Capdeboscq’02, Proc. Roy. Soc. Edinburgh


Constant Coupling Matrix Π:• Cooperative : Παβ ≤ 0 for α 6= β.

• Irreducible

Assumptions borrowed from (for wellposedness): Sweers’92, Math.Z.

Spectral problem and its Adjoint with Y -periodic b.c.:

bα(y) · ∇yϕα − divy

(Dα∇yϕα

)+

N∑

β=1

Παβϕβ = λραϕα in Y0,

Dα∇yϕα · n = 0 on ∂Σ0.

−divy(bαϕ∗

α)− divy

(Dα∇yϕ

∗

α

)+

N∑

β=1

Π∗αβϕ∗

β = λραϕ∗

α in Y0,

Dα∇yϕ∗

α · n+ bα(y) · nϕ∗

α = 0 on ∂Σ0.

Sweers’92 gives existence of first eigenvalue λ > 0, eigenfunctions ϕα, ϕ∗

α > 0.


Scaled modelParabolic scaling : x = ε y (observation scale); t = ε2 τ (longer time)

ρεα∂uεα∂t

+1

εbεα · ∇u

εα − div(D

εα∇u

εα) +

1

ε2

N∑

β=1

Παβuεβ = 0 in(0, T ) × Ωε,

Dεα∇uεα · n = 0 on(0, T )× ∂Ωε,

uεα(0, x) = uinα (x) x ∈ R

d.

Not possible to derive a priori estimates.

Use Factorization principle :• Renormalization in time.• Factoring out the oscillations.Vanninathan’81, Proc.Indian.Acd.Sci.Kozlov’84, Transc. Moscow Math. Soc.This approach is followed in:Donato, Piatnitski’05; Allaire, Raphael’07; Allaire, Capdeboscq’00.


Change of unknown: vεα(t, x) = exp (λt/ε2)uεα(t, x)

ϕα

(xε

)

Governing equations for vεα in (0, T )× Ωε:

ϕαϕ∗αρ

εα

∂vεα∂t

+1

εb̃εα ·∇v

εα−div

(D̃εα∇v

εα

)+

1

ε2

N∑

β=1

Παβϕ∗αϕβ(v

εβ−v

εα) = 0.

D̃εα∇vεα · n = 0, on (0, T )× ∂Ωε, v

εα(0, x) =

uinα (x)

ϕα

(xε

) .

with b̃α(y) = ϕαϕ∗

αbα + ϕαDα∇yϕ∗

α − ϕ∗

αDα∇yϕα for every 1 ≤ α ≤ N

D̃α(y) = ϕαϕ∗

αDα for every 1 ≤ α ≤ N.

Note that ε−2 in the reaction term forces vα to be close to each other.

Remark that b̃α are such that∑N

α=1 divy(b̃α) = 0.


Theorem (Homogenization Result)

The solution uεα(t, x) of the multicomponent model is approximated as:

[exp (−λt/ε2)ϕα

(xε

)][v(t, x−

b∗

εt)+ ε

d∑

i=1

∂v

∂xi

(t, x−

b∗

εt)ωi,α

(xε

)]

where the drift velocity b∗ is given by b∗ = 1ρ∗

∑Nα=1

∫Y 0b̃α(y) dy,

effective porosity, ρ∗, is given by ρ∗ =∑N

α=1

∫Y 0ϕαϕ

∗

αρα(y) dy

v satisfies a scalar diffusion equation:

ρ∗∂v

∂t− div(D∇v) = 0. in (0, T )× Rd,

Every uεα(t, x) behaves as v(t, x) up to multiplication by oscillations.

Irreducible Π ensures microscopic equilibrium among solutes resulting in asingle homogenized concentration v(t, x)

i.e., if Π ≡ 0, we get N different homogenized concentrations.


comments and perspectives

All comments are in the context of strong convection regime.

comment 1: Because of ε−1b∗, only unbounded domains.

comment 2: The drift b∗ is a constant.⋆ Forces us to assume that the drifts match in the nonlinear case.⋆ Cannot consider locally periodic velocity fields i.e., bε(x) = b(x, x/ε).⋆ Cannot homogenize coupled fluid-transport equations simultaneously.

Perspective 1: Locally periodic diffusion i.e., Dε(x) = D(x, x/ε).⋆ To be verified if our results hold true in this case.

Perspective 2: Linear multicomponent model is quite restrictive.⋆ We have been working with V. Giovangigli to incorporate someaspects of thermodynamics and arrive at nonlinear multicomponentreactive transport models.

Perspective 3: Two-scale convergence already extended when b∗(t).Big Challenge: Extend the notion of two-scale convergence to b∗(x).


Documents

Homogenization of ComplexFlows in Porous Mediaand …Homogenization of ComplexFlows in Porous Mediaand Applications Harsha Hutridurga Ramaiah CMAP, ´Ecole Polytechnique, Palaiseau