colloids transport in porous media: analysis and applications. · 09-04-2014 · carbon transport...

Colloids transport in porous media: analysis andapplications.

Oleh Kreheljoint work with Adrian Muntean and Peter Knabner

CASA, Department of Mathematics and Computer Science.Eindhoven University of Technology.

April 9, 2014

What are colloids?

Colloid aggregation is important in:

Cheese production

Colloid filtration

Carbon transport in seawater

Nanoparticles engineering

Water treatment

Our goal: see the effects of aggregation on colloidal transport in porousmedia.Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 2 / 27

Soret effect: the heat makes the particles move

Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 3 / 27

Dufour effect: the particles carry the heat


Model assumptions

Figure : View of a “real” (a) and a “model”(b) aggregate

u1 u2 u3 u4 u5

size class 1 size class 2 size class 3

Figure : Each aggregate is uniquely identified by the number of monomers it’scomposed of.


Population balance (Smoluchowski) equation

Assuming presently a well-mixed solution, we have:

dukdt

= R(u) :=1

2

∑i+j=k

αijβijuiuj − uk

N∑i=1

αijβkiui − bkuk +N∑i=k

dkibiui

uk – concentration of species composed of k monomers

ak – the effective diameter for uk

βij = βij(ai , aj) – aggregation rate

αij = αij(ai , aj) – collision efficiency

bk – breakage coefficient

dki – breakage distribution


Porous model geometry

ε

ε

Yij

Γ

ε

ε

Yij

Γ

Figure : Microstructure of Ωε (isotropic case on the left, anisotropic case on theright). Yij is the periodic cell.


Microscopic model equations

System for Pε:

∂tθε +∇ · (−κε∇θε)− τ ε

N∑i=1

∇δuεi · ∇θε = 0, in Ωε,

∂tuεi +∇ · (−dεi ∇uεi )− f εi ∇δθε · ∇uεi = Ri (u

ε), in Ωε,

∂tvεi = aεi u

εi − bεi v

εi , on (0,T )× Γε,

with boundary conditions:

− κε∇θε · n = 0, on (0,T )× Γε,

− dεi ∇uεi · n = ε(aεi uεi − bεi v

εi ), on (0,T )× Γε,

θε(x , 0) = θε(x , 1) x ∈ [0, 1],

θε(0, y) = θε(1, y) y ∈ [0, 1],

uεi (x , 0) = uεi (x , 1) x ∈ [0, 1], i ∈ 1, . . . ,Nuεi (0, y) = uεi (1, y) y ∈ [0, 1], i ∈ 1, . . . ,N


Model challenges

∂tθε +∇ · (−κε∇θε)− τ ε

N∑i=1

∇δuεi · ∇θε = 0, in Ωε,

∂tuεi +∇ · (−dεi ∇uεi )− f εi ∇δθε · ∇uεi = Ri (u

ε), in Ωε,

∂tvεi = aεi u

εi − bεi v

εi , on (0,T )× Γε,

with boundary conditions:

− κε∇θε · n = 0, on (0,T )× Γε,

− dεi ∇uεi · n = ε(aεi uεi − bεi v

εi ), on (0,T )× Γε.

Non-linear cross-diffusion terms with a gradient

Non-linear coupling via the aggregation reaction

PDE-ODE coupling via the deposition condition


Model well-posedness

For Pε, under the assumptions:

Positivity of initial data

dεi > dε,0i > 0, κε > κε,0 > 0

aεi > 0, bεi > 0

we show (for weak solutions):

Existence

Uniqueness

Positivity

Boundedness

Proof idea:

Galerkin method is used to prove existence of two auxiliary problems

Banach fixed point theorem is used to prove the existence anduniqueness of the full problem


Rigorous upscaling

Definition (Two-scale convergence)

Let (uε) ∈ L2(0,T ; L2(Ω)), Ω ⊂ Rn.(uε) two-scale converges to a unique functionu0(t, x , y) ∈ L2((0,T )× Ω× Y ) ⇐⇒ ∀φ ∈ C∞0 ((0,T )× Ω,C∞# (Y )):

limε→0

T∫0

∫Ω

uεφ(t, x ,x

ε)dxdt =

1

|Y |

T∫0

∫Ω

∫Y

u0(t, x , y)φ(t, x , y)dydxdt

Denote as uε2 u0.


Two-scale compactness

Theorem (Two-scale compactness on domains)

(i) From each bounded sequence (uε) in L2(0,T ; L2(Ω)), a subsequencemay be extracted which two-scale converges tou0(t, x , y) ∈ L2((0,T )× Ωε × Y ).

(ii) Let (uε) be a bounded sequence in H1(0,T ,H1(Ω)), then thereexists u ∈ L2((0,T )× H1

#(Y )) such that up to a subsequence (uε)

two-scale converges to u0 ∈ L2(0,T ; L2(Ω)) and

∇uε 2 ∇xu0 +∇y u.


Two-scale convergence for the thermal colloids problem

Lemma

With positive initial data and coercive and bounded diffusion and heatconduction coefficients, the following holds:

(i) ui ui and θε θ in L2(0,T ;H1(Ω)),

(ii) ui∗ ui and θε

∗ θ in L∞((0,T )× Ω),

(iii) ∂tui ∂tui and ∂tθε ∂tθ in L2(0,T ;H1(Ω)),

(iv) ui → ui and θε → θ strongly in L2(0,T ;Hβ(Ωε)) for 12 < β < 1 and√

ε‖ui − ui‖L2((0,T )×Γε) → 0 as ε→ 0,

(v) ui2 ui , ∇ui

2 ∇xui +∇yu

1i where u1

i ∈ L2((0,T )× Ωε;H1#(Y )),

(vi) θε2 θ, ∇θε 2

∇xθ +∇yθ1 where θ1 ∈ L2((0,T )× Ωε;H1

#(Y )),

(vii) vi2 vi ∈ L∞((0,T )× Ωε × (0,T )× Γε) and

∂tvi2 ∂tvi ∈ L2((0,T )× Ωε × (0,T )× Γε).


A model for colloids transport with deposition

∂tui +∇ · (−di∇ui ) = Ri (u) in Ω, (1)

∂tvi = aiui − bivi on Γ, (2)

with the boundary conditions

− di∇ui · n = aiui − bivi on Γ, (3)

− di∇ui · n = 0 on ΓN , (4)

ui (t, x) = uD(t, x) on ΓD , (5)

and the initial conditions

ui (0, x) = u0i (x) in Ω, (6)

vi (0, x) = v0i (x) on Γ. (7)


Nondimensionalized system

After rescaling, we obtain this system:

∂tui +∇ · (−di∇ui ) = ΛRi (u) in Ω, (8)

∂tvi = Bi(aiui − bivi ) on Γ, (9)


− di∇ui · n = ε(aiui − bivi ) on Γ, (10)

− di∇ui · n = 0 on ΓN , (11)

ui (t, x) =uD(t, x)

u0on ΓD , (12)

and the initial conditions

ui (0, x) =u0i (x)

u0in Ω, (13)

vi (0, x) =v0i (x)

v0on Γ. (14)


Rigorous homogenization result

The final upscaled system:

∂tui −∇ · (Di∇ui ) + Aiui − Bivi = ΛRi (u) in Ω, (15)

∂tvi = Aiui − Bivi in Ω (16)

with effective parameters:

Dijk =

∫Ydεi (y)(δjk +∇ywi )dy i ∈ 1, . . . ,N (17)

Ai := Bi

∫Γε

aεi (y) dσ(y) i ∈ 1, . . . ,N (18)

Bi := Bi

∫Γε

bεi (y) dσ(y) i ∈ 1, . . . ,N (19)

where wj(y) solve the cell problem:

∇y · (dεi (y)∇wj) = −(∇dεi (y))j j ∈ 1, . . . , d, y ∈ Y (20)


Cell problem solution: isotropic case

Figure : Solutions to the cell problems that correspond to isotropic periodicgeometry.

The resulting effective diffusion tensor:

D =

[0.75 0.171476

0.171476 0.75

]Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 17 / 27

Cell problem solution: anisotropic case

Figure : Solutions to the cell problems that correspond to anisotropic periodicgeometry.

The resulting effective diffusion tensor:

D =

[0.817467 0.07863380.214942 0.817467

]Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 18 / 27

A model with a blocking function for the deposition

∂tn = −vp · ∇n + Dh∆n − f

πa2p

∂tθ, (21)

∂tθ = πa2pknB(θ), (22)


n(t, 0) =

n0 t ∈ [0, t0]

0 t > t0

, (23)

∂n

∂ν(t, L) = 0, (24)

and initial conditions

n(0, x) = 0, (25)

θ(0, x) = 0. (26)


Simulation of the aggregation effects on deposition - 1

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

ma

ss in

th

e s

yste

m

u1

u2

u (together)

u (single species)

Figure : Simulation comparison for a single species system versus an aggregatingsystem. The straight line is the breakthrough curve for the colloidal mass for theproblem without aggregation. The dashed line is the breakthrough curve for thecolloidal mass for the problem with aggregation. It is obtained by summingmass-wise the breakthrough curves for the monomers u1 and dimers u2.


Simulation for the Langmuirian blocking function

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time

mass in

th

e s

yste

m

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time

mass in

th

e s

yste

m

u1

u2

v1

v2

Figure : The effect of the Langmuirian dynamic blocking function B(θ) = 1− βθon the deposition (right) versus no blocking function (left). u1 and u2 are thebreakthrough curves, while v1 and v2 are the concentrations of the depositedspecies.


Simulation for the RSA blocking function

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time

mass in

th

e s

yste

m

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time

mass in

th

e s

yste

m

u1

u2

v1

v2

Figure : The effect of the RSA dynamic blocking functionB(θ) = 1− 4θ∞βθ + 3.308(θ∞βθ)2 + 1.4069(θ∞βθ)3 on the deposition (right)versus no blocking function (left). u1 and u2 are the breakthrough curves, whilev1 and v2 are the concentrations of the deposited species.


Simulation of the aggregation effects on deposition - 2

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time

ma

ss in

th

e s

yste

m

u1(s)

u2(s)

v1(s)

v2(s)

u1(d)

u2(d)

v1(d)

v2(d)

Figure : The effect of aggregation rates on the breakthrough curves. On the left,the default rate of aggregation is used, on the right - it’s doubled. A change ofaggregation rate can be achieved by varying the concentration of salt in thesuspension, according to DLVO theory.


Implementation details

cell problem solver (FEM):deal.II - a numeric library for parallel computationshttp://www.dealii.org/

spatial discretization (FEM):DUNE - a numeric library for parallel computationshttp://www.dune-project.org/

time discretization:CVODE library from the SUNDIALS packagehttps://computation.llnl.gov/casc/sundials/main.html

Fully coupled or operator splitting

Code is 2D/3D


http://www.dealii.org/

http://www.dune-project.org/

https://computation.llnl.gov/casc/sundials/main.html

Multiscale FEM

General idea:

Lεuε = f in Ω

uε = 0 on ∂Ω

Approximate uε with uh ∈ V h = spanφiK : K ∈ Kh s.t.:

Lεφi = 0 in K ∈ Kh

a(uh, v) = f (v) ∀v ∈ V h

Error estimate:

‖uε − uh‖1,Ω ≤ C1h‖f ‖0,Ω + C2

√ε

h


Outlook

Multiscale Finite Element method implementation for the system

Pore clogging due to deposition

Corrector estimates for the cross-diffusion problem

MSFEM estimates and implementation for the cross-diffusion problem


Last slide

Thank you for your attention.


colloids transport in porous media: analysis and applications. · 09-04-2014 · carbon transport...

Documents