Dispersion in periodic porous media -...

Dispersion in periodic porous mediaB. Amazian, A. Bourgeat, M. Jurak

Dispersion in periodic porous media – p.1/49

Dimensionless field scale equations

φ(x

ε)∂cε

∂t+ ~qε(x) · ∇cε =

ε

Pediv(D(

x

ε)∇cε) in Ω × (0, T )

~q ε = −K(x

ε)∇P ε, div~q ε = 0 in Ω

+ some initial and boundary conditions

ε� 1 (separation of scales);

φ, K, D are periodic;

Pe = O(1) (local Peclet number);

Time scale of global convection.


Assumptions

Domain Ω ⊂ R2 is bounded;

Porosity φ(y) > 0;

Permeability and diffusion tensors:

K =

[

k1,1(y) 0

0 k2,2(y)

]

, D =

[

d1,1(y) 0

0 d2,2(y)

]

,

are positive definite;

All functions φ, ki,j , di,j are Y = (0, 1)2-periodic.


Asymptotic expansion

J.-L. AURIAULT, L. LEWANDOWSKA (1996),Diffusion/adsorption/advection macro transport insolids, Eur. J. Mech., A/Solids, 15 (4) 681-704.

M.B. PANFILOV (1995), Macro-scale Models of FlowThrough Heterogeneous Porous Media. Theory andApplications of Transport in Porous Media, Vol. 16, Kluwer.

A. BOURGEAT, M. JURAK, A.L. PIATNITSKI (2003), Averaging atransport equation with small diffusion and oscillatingvelocity, Math. Meth. Appl. Sci., 26 95-117.Justification in Ω = Rd × (0, L) by use of Schauder’sestimates.


Effective equations

〈φ〉∂c1,ε

∂t+ (~q 0 + ε~q 1) · ∇c1,ε =

ε

Pediv(Dh(x)∇c1,ε)

in Ω × (0, T )

~q 0 = −Kh∇P 0, div~q 0 = 0 in Ω

+ the same boundary conditions for c1,ε and P 0


Effective equations

〈φ〉∂c1,ε

∂t+ (~q 0 + ε~q 1) · ∇c1,ε =

ε


in Ω × (0, T )

~q 0 = −Kh∇P 0, div~q 0 = 0 in Ω


〈·〉 is mean value with respect the fast variable y;


Effective equations

〈φ〉∂c1,ε

∂t+ (~q 0 + ε~q 1) · ∇c1,ε =

ε


in Ω × (0, T )

~q 0 = −Kh∇P 0, div~q 0 = 0 in Ω


Kh is effective permeability tensor;


Effective equations

〈φ〉∂c1,ε

∂t+ (~q 0 + ε~q 1) · ∇c1,ε =

ε


in Ω × (0, T )

~q 0 = −Kh∇P 0, div~q 0 = 0 in Ω


~q 1 is the first order correction to effective Darcy’svelocity ~q 0 (defined later);


Effective equations

〈φ〉∂c1,ε

∂t+ (~q 0 + ε~q 1) · ∇c1,ε =

ε


in Ω × (0, T )

~q 0 = −Kh∇P 0, div~q 0 = 0 in Ω


Dh(x) = Dh(Pe∇P 0(x)) is effective diffusion tensor (withdispersion effects);


Effective permeability

Permeability cell problem (i = 1, 2): χi ∈ H1(Y )

divy (K(y)(∇yχi + ~ei)) = 0 in Y

Matrix velocity Q(y):

Q(y)~e1 = −K(y)(∇χ1(y) + ~e1),

Q(y)~e2 = −K(y)(∇χ2(y) + ~e2),

Effective permeability and effective Darcy’s velocity:

Kh = −〈Q〉, ~q 0(x) = −Kh∇P 0(x).


Effective diffusion

Diffusion cell problem (i = 1, 2): ψi ∈ H1(Y )

−divy (D(y)(∇yψi + ~ei)) + Pe Q(y)~λ · ∇yψi

= Pe

[

φ(y)

〈φ〉〈Q〉~λ− Q(y)~λ

]

· ~ei

Dh(Pe~λ)~ei = 〈D(∇yψi + ~ei)〉 + Pe〈(φ

〈φ〉〈Q〉~λ− Q~λ)ψi〉,

Dh(x) = Dh(Pe∇P 0(x)).


Correction to Darcy’s velocity

The first order correction to Darcy’s velocity has the form:

~q 1(x) = −2

∑

i,j=1

~N2i,j∂2P 0(x)

∂xi∂xj− Kh∇P 1(x),

where vectors ~N2i,j are given by

~N2i,j · ~el = 〈K(∇χl + ~el)χi − K(∇χi + ~ei)χl〉 · ~ej .

(χi = solutions to permeability cell problems.)


Second pressure P 1(x)

div(

Kh∇P 1)

= 0 in Ω

completed by some boundary conditions that depend onexact calculation of all boundary layers.

In general, we do not know P 1(x)!


Example of P 1(x)

Let Ω = R × (0, L). Then

div(

Kh∇P 1)

= 0 in Ω

P 1(x) =2

∑

i=1

ν+i ∂iP0(x) on x2 = 0

P 1(x) =2

∑

i=1

ν−i ∂iP0(x) on x2 = L.

Constants ν±i are introduced by boundary layer functions.


ν+i and ν−i

Boundary layer problem:χ+i (i = 1, 2) is bounded solution of the problem

−divy(

K∇yχ+i

)

= 0 in (0, 1) × (0,∞)

χ+i (y1, 0) = −χi(y1, 0) for y1 ∈ (0, 1)

y1 7→ χ+i (y1, y2) is 1 − periodic,

ν+i = limy2→∞χ+i (y1, y2) (exponentially).

Similarly for ν−i .


Moving convection to the RHS

~q 1(x) = −2

∑

i,j=1

~N2i,j∂2P 0(x)

∂xi∂xj− Kh∇P 1(x),

can be written as

ε~q 1 · ∇c1,ε = −εdiv(

(~N · ∇P 0)J∇c1,ε)

− εKh∇P 1(x) · ∇c1,ε

where

~N = 〈K(∇χ2 + ~e2)χ1 − K(∇χ1 + ~e1)χ2〉, J =

[

0 1

−1 0

]

.


Moving convection to the RHS

ε~q 1 · ∇c1,ε = −ε[

2∑

i,j=1

~N2i,j∂2P 0(x)

∂xi∂xj+ Kh∇P 1(x)

]

· ∇c1,ε

can be written as

ε~q 1 · ∇c1,ε = −εdiv(

(~N · ∇P 0)J∇c1,ε)

− εKh∇P 1(x) · ∇c1,ε

where

~N = 〈K(∇χ2 + ~e2)χ1 − K(∇χ1 + ~e1)χ2〉, J =

[

0 1

−1 0

]

.


Effective equations

〈φ〉∂c1,ε

∂t+ (~q 0 + ε~q 1) · ∇c1,ε =

ε

Pediv(D

h(∇P 0)∇c1,ε)

~q 0 = −Kh∇P 0, ~q 1 = −Kh∇P 1,

div~q 0 = 0, div~q 1 = 0

+boundary and initial conditions

Dh(∇P 0) = Dh(Pe∇P 0) + Pe(~N · ∇P 0)J.


Material symmetries in Darcy’s law

Consider complete symmetry: ∀(y1, y2)

K(y1,−y2) = K(y1, y2) and K(−y1, y2) = K(y1, y2).

Then:

Kh is diagonal;

~N = 0;

In rectangular domain all constant ν±i are equal to zero;Therefore P 1 = 0.

This result suggests that in any domain P 1 = 0 if thereis enough symmetry.


Material symmetries (continued)

Consider the group of orthogonal matrices:

N = {C ∈ Md×d : C−1 = Cτ , CY = k + Y for k ∈ Zd}.

Assume C ∈ N and ∀z ∈ Rd,

CτK(Cz)C = K(z).

Then it holds

Q(Cz) = CQ(z)Cτ , Kh = CKhCτ ,

for all z ∈ Rd.


Symmetry of effective diffusion

Effective diffusion tensor is generally not symmetric.Symmetric and skew-symmetric parts are given by

Dh(~λ) = Dh(~λ)symm + Dh(~λ)skew.

where

Dh(~λ)symmi,j = 〈D(∇yψj + ~ej) · (∇yψi + ~ei)〉

Dh(~λ)skewi,j = Pe〈Q~λ · ∇yψiψj〉 + 〈D∇yψj · ~ei〉 − 〈D∇yψi · ~ej〉.

Generally it holds

Dh(−~λ) = Dh(~λ)τ .


Material symmetries (continued)

Let C ∈ N and assume that ∀z ∈ Rd,

CτK(Cz)C = K(z), CτD(Cz)C = D(z), φ(Cz) = φ(z).

ThenD(~λ) = CD(Cτ~λ)Cτ

Corollary:

If we can take C = −I (complete symmetry) then D(~λ) is

symmetric.


Pe 7→ Dh(Pe~λ) as Pe → ∞

R.N. BHATTACHARYA, V.K. GUPTA, H.F. WALKER (1989),Asymptotics of Solute Dispersion in Periodic Porous Media,SIAM J. Appl. Ma th. 49(1) 86-98.A. AVELLANEDA, A. J. MAJDA (1991): An IntegralRepresentation and Bounds of the Effective Diffusivity inPassive Advection by Laminar and Turbulent Flows,Commun. Math. Phys., 138, 339-391.

|Dh(Pe~λ)i,j | ≤ C(Pe)2.

A.J. MAJDA, P.R. KRAMER (1999), Simplified models for tur-

bulent diffusion: Theory, numerical modeling, and physical

phenomena, Physics Reports 314, 237-574.



R.N. BHATTACHARYA, V.K. GUPTA, H.F. WALKER (1989),Asymptotics of Solute Dispersion in Periodic Porous Media,SIAM J. Appl. Ma th. 49(1) 86-98.

A. AVELLANEDA, A. J. MAJDA (1991): An IntegralRepresentation and Bounds of the Effective Diffusivity inPassive Advection by Laminar and Turbulent Flows,Commun. Math. Phys., 138, 339-391.







R.N. BHATTACHARYA, V.K. GUPTA, H.F. WALKER (1989),Asymptotics of Solute Dispersion in Periodic Porous Media,SIAM J. Appl. Ma th. 49(1) 86-98.A. AVELLANEDA, A. J. MAJDA (1991): An IntegralRepresentation and Bounds of the Effective Diffusivity inPassive Advection by Laminar and Turbulent Flows,Commun. Math. Phys., 138, 339-391.






Limit behavior

For simplicity we consider only diagonal terms.

Minimal enhanced diffusion: Dh(Pe~λ)i,i = O(1);

Maximal enhanced diffusion: Dh(Pe~λ)i,i = O(Pe2);


Minimal/maximal enhancement

Minimal enhanced diffusion is the case where there exists aperiodic function hi ∈ H1(Y ) such that

Q(y)~λ · ∇hi =

[

φ(y)

〈φ〉〈Q〉~λ− Q(y)~λ

]

· ~ei

in Y .For maximal enhanced diffusion there must exist a nontrivial periodic function h ∈ H1(Y ) such that

Q(y)~λ · ∇h = 0,

in Y .




Q(y)~λ · ∇hi =

[

φ(y)

〈φ〉〈Q〉~λ− Q(y)~λ

]

· ~ei

in Y .

For maximal enhanced diffusion there must exist a nontrivial periodic function h ∈ H1(Y ) such that

Q(y)~λ · ∇h = 0,

in Y .




Q(y)~λ · ∇hi =

[

φ(y)

〈φ〉〈Q〉~λ− Q(y)~λ

]

· ~ei

in Y .For maximal enhanced diffusion there must exist a nontrivial periodic function h ∈ H1(Y ) such that

Q(y)~λ · ∇h = 0,

in Y .


Numerical simulations

J. SALLES, J.-F. THOVERT, R. DELANNAY, L. PREVORS, J.-L. AURIAULT,M. ADLER: Taylor dispersion in porous media. Determinationof the dispersion tensor, Phys. Fluids A 5(10), October1993.M. QUINTARD, M. KAVIANY, S. WHITAKER: Two-medium treatmentof heat transfer in porous media: numerical results foreffective properties, Advances in Water Resources, Vol. 20,No 2-3, pp. 77-94, 1997.

F. CHERBLANC, A. AHMADI, M. QUINTARD: Two-medium descrip-

tion of dispersion in heterogeneous porous media: Calcula-

tion of macroscopic properties, Water Resources Res. Vol

39, No 6, 2003.


Numerical simulations

Heterogeneous simulation;

Homogeneous simulation;

Homogeneous simulation without dispersion.


Numerical methods

Darcy’s equation: Mixed-hybrid finite element method(RT0 elements on non structured triangulation);

Concentration equation: Vertex-centered finite volumemethod with semi-implicit time stepping;

Unit cell problems: Finite volume method with finitedifferences on uniform mesh with up-winding;

Solvers: CG and GMRES; preconditioners: ILU, SSOR;


Heterogeneous simulation

Ω = (0, 10) × (0, 10), ε =1

10

∂cε

∂t+ ~qε(x) · ∇cε = div(D(

x

ε)∇cε)

~q ε = −K(x

ε)∇P ε, div~q ε = 0

D(x

ε)∇cε · ~n = 0 on ∂Ω

P ε = Pbdr on ∂Ω

+ initial condition for cε


Heterogeneous simulation: data

Ki, Di

Km, Dm

Ki = 1.0, Km= 8.0; Kh=4.416;

Di = 0.01, Dm= 0.02; Dh=0.01569

We have complete symmetry.


Heterogeneous simulation: pressurePressure

−6

−4

−2

0

2

4

6

8

10

12

XY

Z

Z−

Axi

s

−6.009

0

10

12.01

Y−Axis

0

10 X−Axis

0

10


Heterogeneous simulation: velocity

5 6 7 7.2834.831

5

6

7

7.331

Darcy’s velocity

X−Axis

Y−

Axi

s

5 6 7 7.2834.831

5

6

7

7.331


Heterogeneous simulation: cinit

0 100

10

Concentration: step = 0, t=0.000000

X−Axis

Y−

Axi

s

1.2

1.5

1.8

2.1

2.4

2.7

3

3.3

3.6

3.9

0 100

10


Heterogeneous simulation: c, t = 1.0

0 100

10


X−Axis

Y−

Axi

s

1.04

1.12

1.2

1.28

1.36

1.44

1.52

1.6

1.68

1.76

0 100

10


Heterogeneous simulation: Pe

Local Peclet number:

Pe =|~q|l

|D∇c|.

Pe ≈ 25 −−250 for t = 0

Pe ≈ 80 −−800 for t = 1


Homogeneous simulation

We have complete symmetry.

∂ch

∂t+ ~q 0 · ∇ch = div(Dh∇ch)

~q 0 = −Kh∇P 0, div~q 0 = 0,

Dh∇ch · ~n = 0 on ∂Ω

P 0 = Pbdr on ∂Ω

+ initial condition for ch

Dh = Dh(l∇P 0) (l = the cell length).


Homogeneous simulation

Procedure:

Solve permeability cell problems and calculate Kh;

Run Darcy’s flow simulation and calculate ∇P 0;

For each triangle T ⊂ Ω solve diffusion cell problem with~λ = l∇P 0|T and calculate Dh|T = D

h(~λ).

Since there are too many triangles perform first someclustering to reduce the amount of the calculations.


n×m clustering

Divide [0, 2π] into n equal subintervals Ii, i = 1, 2, . . . ,n;

Divide [min |~q|,max |~q|] into m equal subintervals Jj ,j = 1, 2, . . . ,m;

Replace all vectors in Ii × Jj by their mean value.

Example: 10 × 10 clustering.Number of vectors before clustering: 54452Number of vectors after clustering: 22L2–error introduced by clustering: absolute = 6.113882,

relative = 10.1492 %.


Comparison of the simulations

Notation: cε=heterogeneous solution; ch=homogeneoussolution; ch,0=homogeneous solution without dispersion;

time‖cε−ch‖L2(Ω)‖cε‖L2(Ω)

‖cε−ch,0‖L2(Ω)‖cε‖L2(Ω)

0.111111 9.49 % 10.91 %0.222222 8.50 % 11.30 %0.333333 6.15 % 10.19 %0.444444 5.00 % 10.01 %0.555556 4.51 % 10.13 %0.666667 4.03 % 9.92 %0.777778 3.62 % 9.71 %0.888889 3.34 % 9.55 %



(heterogeneous and homogeneous with dispersion)

0 100

10


X−Axis

Y−

Axi

s

1.04

1.12

1.2

1.28

1.36

1.44

1.52

1.6

1.68

1.76

0 100

10

0 100

10


X−Axis

Y−

Axi

s

1.08

1.17

1.26

1.35

1.44

1.53

1.62

1.71

1.8

1.89

0 100

10



(heterogeneous and homogeneous without dispersion)

0 100

10


X−Axis

Y−

Axi

s

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 100

10

0 100

10


X−Axis

Y−

Axi

s

1.08

1.17

1.26

1.35

1.44

1.53

1.62

1.71

1.8

1.89

0 100

10


Dh(Pe~λ)11, ~λ = (1, 0)

K=1

D=0.01

K=8, D=0.02

0 10 20 30 400

1000

1547

PLOT

X−Axis

Y−

Axi

s

0 10 20 30 400

1000

1547


Dh(Pe~λ)22, ~λ = (1, 0)

K=1

D=0.01

K=8, D=0.02

0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

0.0528

PLOT

X−Axis

Y−

Axi

s

0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

0.0528


Dh(Pe~λ)11, ~λ = (1, 1)

K=1

D=0.01

K=8, D=0.02

0 10 20 30 400

10

20

30

40

50

60

70

80

lambda=(1,1)

Pe

Dxx

0 10 20 30 400

10

20

30

40

50

60

70

80


ψ1(y) for ~λ = (1, 1), Pe = 20

0 10

1

Cell Diffusion soluton (direction=x)

X−Axis

Y−

Axi

s

−1.2

−1 −0.8

−0.

8

−0.6

−0.6

−0.

6

−0.4

−0.

4

−0.4

−0.4

−0.2

−0.

2

−0.2

−0.

2

−0.2

1.94

e−16

1.94

e−16

1.94e−1

6

1.94

e−16

1.94e−16

1.94

e−16

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.4

0.4

0.40.

4

0.4

0.4

0.4

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.8

1

1

1

1

1

1

1

1

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.4

1.4

1.4

1.4

1.4

1.4

1.4

1.4

1.6

1.6

1.6

1.6

1.6

1.6

1.6

1.6

1.8

1.8

1.8

1.8

1.8

0 10

1


Non symmetric cell

K=1

D=0.01

K=8, D=0.02

K =

[

5.76 −0.185

−0.185 5.76

]

, D =

[

0.0176 −7.7 · 10−5

−7.7 · 10−5 0.0176

]

The same initial and boundary conditions.



Notation: cε=heterogeneous solution; ch=homogeneoussolution; ch,0=homogeneous solution without dispersion;

time‖cε−ch‖L2(Ω)‖cε‖L2(Ω)

‖cε−ch,0‖L2(Ω)‖cε‖L2(Ω)

0.111111 7.34 % 8.14 %0.222222 4.77 % 6.23 %0.333333 3.56 % 5.54 %0.444444 3.45 % 5.84 %0.555556 3.01 % 5.67 %0.666667 2.74 % 5.57 %0.777778 2.54 % 5.45 %0.888889 2.34 % 5.34 %



(heterogeneous and homogeneous with dispersion)

0 100

10


X−Axis

Y−

Axi

s

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

0 100

10

0 100

10


X−Axis

Y−

Axi

s

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 100

10


Conclusion

The results suggest that acceptable effective diffusioncan be obtained with rather coarse clustering of Darcy’svelocity;

Influence of ~q 1 is not (yet) demonstrated.


Taylor expansion with respect to Pe

Solutions of diffusion cell problems: ψi = ψi(y; Pe~λ).

Taylor expansion

ψi = ψ0i + Peψ

1i + Pe

2 ψ2i + Pe3 ψ3i + . . .

ψ0i = ψ0i (y), ψ

1i (y,

~λ) =d

∑

k=1

ψ1i;k(y)λk,

ψ2i (y,~λ) =

d∑

k,l=1

ψ1i;k,l(y)λkλl, . . .



Solutions of diffusion cell problems: ψi = ψi(y; Pe~λ).Taylor expansion

ψi = ψ0i + Peψ

1i + Pe

2 ψ2i + Pe3 ψ3i + . . .

ψ0i = ψ0i (y), ψ

1i (y,

~λ) =d

∑

k=1

ψ1i;k(y)λk,

ψ2i (y,~λ) =

d∑

k,l=1




Solutions of diffusion cell problems: ψi = ψi(y; Pe~λ).Taylor expansion

ψi = ψ0i + Peψ

1i + Pe

2 ψ2i + Pe3 ψ3i + . . .

ψ0i = ψ0i (y), ψ

1i (y,

~λ) =

d∑

k=1

ψ1i;k(y)λk,

ψ2i (y,~λ) =

d∑

k,l=1



Sequence of equations

Functions ψki are periodic solutions of the followingproblems on the unit cell Y :

−div(D(∇ψ0i + ~ei)) = 0

−div(D∇ψ1i ) + Q(y)~λ · ∇ψ0i =

[

φ(y)

〈φ〉〈Q〉~λ− Q(y)~λ

]

· ~ei

For k = 2, 3, . . .

−div(D∇ψki ) + Q(y)~λ · ∇ψk−1i = 0


Expansion for effective diffusion

Dh(Pe~λ) = Dh,0 +[

Pe Nh,1(~λ) + Pe2 Dh,2(~λ)]

+ · · ·

+ (−1)n−1[

Pe2n−1 Nh,2n−1(~λ) + Pe2n Dh,2n(~λ)]

+ · · ·

Dh,0i,j = 〈D(∇ψ

0j + ~ej) · (∇ψ

0i + ~ei)〉,

Nh,2n−1i,j = 〈Q

~λ · ∇ψnj ψni 〉 n = 1, 2, . . . skew-symmetric

Dh,2ni,j = 〈D∇ψ

nj · ∇ψ

ni 〉 n = 1, 2, . . . symmetric


Small Peclet number approximation

Dh(Pe~λ) ≈ Dh,0 + Pe Nh,1(~λ) + Pe2 Dh,2(~λ)(+ · · ·?)

The first terms are very small so that higher number ofterms should be taken;

Radius of convergence can be small;

Accuracy can be gained by the use of Padéapproximations.


Standard diffusivity tensor

Dstd(~λ) = Dh,0 + |~λ|(

αl~λ⊗ ~λ

|~λ|2+ αt(I −

~λ⊗ ~λ

|~λ|2))

For large Peclet numbers most often diffusivity growslinearly with Peclet number, while for small Peclet numbersthe growth can be quadratic.


Dimensionless field scale equationsAssumptionsAsymptotic expansionEffective equationsEffective permeabilityEffective diffusionCorrection to Darcy's velocitySecond pressure $P^1(x)$Example of $P^1(x)$$u ^{+}_{i}$ and $u ^{-}_{i}$Moving convection to the RHSEffective equationsMaterial symmetries in Darcy's lawMaterial symmetries (continued)Symmetry of effective diffusionMaterial symmetries (continued)$Pe mapsto {cal D}^h(Pe lamb )$as $Pe o infty $Limit behaviorMinimal/maximal enhancementNumerical simulations Numerical simulationsNumerical methodsHeterogeneous simulationHeterogeneous simulation: dataHeterogeneous simulation: pressureHeterogeneous simulation: velocityHeterogeneous simulation: $c_{m init}$Heterogeneous simulation: $c$, $t=1.0$Heterogeneous simulation: PeHomogeneous simulationHomogeneous simulationn$imes $m clusteringComparison of the simulationsComparison of the simulationsComparison of the simulations${cal D}^h(Pe lamb )_{11}$,$;lamb =(1,0)$${cal D}^h(Pe lamb )_{22}$,$;lamb =(1,0)$${cal D}^h(Pe lamb )_{11}$,$;lamb =(1,1)$$psi _1(y)$for $lamb =(1,1)$, $Pe =20$ Non symmetric cell Comparison of the simulationsComparison of the simulationsConclusionTaylor expansion with respect to $Pe $Sequence of equationsExpansion for effective diffusionSmall Peclet number approximationStandard diffusivity tensor

Dispersion in periodic porous media -...

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