colloids transport in porous media: analysis and applications. · 09-04-2014 · carbon transport...
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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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 4 / 27
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.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 5 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 6 / 27
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.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 7 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 8 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 9 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 9 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 9 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 10 / 27
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.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 11 / 27
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.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 12 / 27
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).
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 13 / 27
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)
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 14 / 27
Nondimensionalized system
After rescaling, we obtain this system:
∂tui +∇ · (−di∇ui ) = ΛRi (u) in Ω, (8)
∂tvi = Bi(aiui − bivi ) on Γ, (9)
with the boundary conditions
− 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)
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 15 / 27
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)
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 16 / 27
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)
with the boundary conditions
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)
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 19 / 27
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.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 20 / 27
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.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 21 / 27
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.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 22 / 27
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.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 23 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 24 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 25 / 27
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
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 26 / 27
Last slide
Thank you for your attention.
Oleh Krehel, A.M. and P.K. (TU/e) PBE for colloids April 9, 2014 27 / 27
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