72. June 8_Colloid-facilitated transort model (progress and future work)-all media_Reimus

1. Used Fuel Disposition Campaign
Next-Gen PA Model: Approach for
Colloid-Facilitated Transport
Paul Reimus and Mavrik Zavarin
Model Integration Session
UFD Working Group Meeting
June 8, 2016

2. Used
Fuel
Disposition
2
0.00001
0.0001
0.001
0.01
0.1
1
0 50 100 150 200
Concentration(C/Co)
Pore Volumes
Conservative RN
Retarded RN with No Colloids
CFT Kinetic Control
CFT Equilibrium Control
6/8/2016 CFT Lightning
Function of ([exp(-kdesτ) x exp(-kfiltτ)])
d
B
K1R
φ
ρ
+=
CK1
K1
CK1
CKK1
R
colcold,
d
B
colcold,
colcold,d
B
eff
+
+
≈
+
++
=
φ
ρ
φ
ρ
Two Transport Regimes – Easy to
Model, Difficult to Mechanistically
Parameterize

3. Used
Fuel
Disposition
Colloid Transport Equations
 Colloid Transport (General):
𝝏𝝏𝑪𝑪𝒄𝒄
𝝏𝝏𝒕𝒕
+ 𝒅𝒅𝒅𝒅 𝒅𝒅 𝒗𝒗𝒄𝒄 𝑪𝑪𝒄𝒄 + 𝑫𝑫𝒄𝒄 𝛁𝛁𝑪𝑪𝒄𝒄 =
−𝒌𝒌𝒇𝒇𝒇𝒇
𝑺𝑺𝑺𝑺
𝑽𝑽
𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑺𝑺𝒄𝒄
𝑺𝑺𝒄𝒄,𝒎𝒎𝒎𝒎𝒎𝒎
+ 𝒌𝒌𝒓𝒓𝒓𝒓
𝑺𝑺𝑺𝑺
𝑽𝑽
𝑺𝑺𝒄𝒄 − 𝒌𝒌𝒇𝒇𝒇𝒇,𝒊𝒊𝒊𝒊
𝑺𝑺𝑺𝑺
𝑽𝑽
𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊,𝒎𝒎𝒎𝒎𝒎𝒎
+
𝑺𝑺𝑺𝑺
𝑽𝑽
𝑷𝑷𝒄𝒄
𝝏𝝏𝑺𝑺𝒄𝒄
𝝏𝝏𝒕𝒕
= 𝒌𝒌𝒇𝒇𝒇𝒇 𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑺𝑺𝒄𝒄
𝑺𝑺𝒄𝒄,𝒎𝒎𝒎𝒎𝒎𝒎
− 𝒌𝒌𝒓𝒓𝒓𝒓 𝑺𝑺𝒄𝒄
𝝏𝝏𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝝏𝝏𝒕𝒕
= 𝒌𝒌𝒇𝒇𝒇𝒇,𝒊𝒊𝒊𝒊 𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊,𝒎𝒎𝒎𝒎𝒎𝒎
 At a true steady state,
𝝏𝝏𝑺𝑺𝒄𝒄
𝝏𝝏𝒕𝒕
= 𝟎𝟎, 𝐚𝐚𝐚𝐚𝐚𝐚 𝑷𝑷𝒄𝒄 ≅ 𝒌𝒌𝒇𝒇𝒇𝒇,𝒊𝒊𝒊𝒊 𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊,𝒎𝒎𝒎𝒎𝒎𝒎
3
Reversible Filtration Irreversible Filtration Production
Mobile Colloids
Reversibly Immobilized Colloids
Irreversibly Immobilized Colloids
6/8/2016 CFT Lightning
Assume colloids limited to flowing domain

4. Used
Fuel
Disposition
Solute Transport Equations
 Mobile Solute Transport Equation:
𝝏𝝏𝑪𝑪𝒂𝒂
𝝏𝝏𝒕𝒕
+ 𝒅𝒅𝒅𝒅 𝒅𝒅 𝒗𝒗𝒂𝒂 𝑪𝑪𝒂𝒂 + 𝑫𝑫𝒂𝒂 𝛁𝛁𝑪𝑪𝒂𝒂 =
� −𝒌𝒌𝒂𝒂𝒂𝒂,𝒊𝒊
𝝆𝝆𝑩𝑩
∅
𝑪𝑪𝒂𝒂 𝟏𝟏 −
𝑺𝑺𝒂𝒂,𝒊𝒊
𝑺𝑺𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊
+ 𝒌𝒌𝒔𝒔𝒔𝒔,𝒊𝒊
𝝆𝝆𝑩𝑩
∅
𝑺𝑺𝒂𝒂,𝒊𝒊
𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏
𝒊𝒊= 𝟏𝟏
+
� −𝒌𝒌𝒂𝒂𝒄𝒄,𝒊𝒊 𝑪𝑪𝒂𝒂 𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑪𝑪𝒂𝒂𝒄𝒄,𝒊𝒊 𝑪𝑪𝒄𝒄
𝑪𝑪𝒂𝒂𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊 𝑪𝑪𝒄𝒄
+ 𝒌𝒌𝒄𝒄𝒂𝒂,𝒊𝒊 𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒄𝒄
𝒏𝒏𝒄𝒄𝒄𝒄𝒄𝒄𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔𝒔
𝒊𝒊= 𝟏𝟏
+
�
𝑺𝑺𝑺𝑺
𝑽𝑽
−𝒌𝒌𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒂𝒂 𝑺𝑺𝒄𝒄 𝟏𝟏 −
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄
𝑪𝑪𝒂𝒂𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊 𝑺𝑺𝒄𝒄
+ 𝒌𝒌𝒄𝒄𝒄𝒄,𝒊𝒊 𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄 − 𝒌𝒌𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒂𝒂 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊 𝟏𝟏 −
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝑪𝑪𝒂𝒂𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏
𝒊𝒊= 𝟏𝟏
+ terms accounting for matrix diffusion or diffusion into secondary porosity
 Solute Sorbed onto Immobile Surfaces (nsites equations):
𝝏𝝏𝑺𝑺𝒂𝒂,𝒊𝒊
𝝏𝝏𝒕𝒕
= 𝒌𝒌𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒂𝒂 𝟏𝟏 −
𝑺𝑺𝒂𝒂,𝒊𝒊
𝑺𝑺𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊
− 𝒌𝒌𝒔𝒔𝒔𝒔,𝒊𝒊 𝑺𝑺𝒂𝒂,𝒊𝒊
4
(Flowing domain only; more terms
needed if secondary porosity)
6/8/2016 CFT Lightning

5. Used
Fuel
Disposition
Solute Associated with Colloids
Equations
 Solute Sorbed to Mobile Colloids (ncolsites equations):
𝝏𝝏𝑪𝑪𝒂𝒂𝒄𝒄,𝒊𝒊 𝑪𝑪𝒄𝒄
𝝏𝝏𝒕𝒕
+ 𝒅𝒅𝒅𝒅 𝒅𝒅 𝒗𝒗𝒂𝒂 𝑪𝑪𝒂𝒂𝒄𝒄,𝒊𝒊 𝑪𝑪𝒄𝒄 + 𝑫𝑫𝒄𝒄 𝛁𝛁𝑪𝑪𝒂𝒂𝒄𝒄,𝒊𝒊 𝑪𝑪𝒄𝒄 = 𝒌𝒌𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒂𝒂𝒄𝒄,𝒊𝒊 𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒄𝒄
𝑪𝑪𝒂𝒂𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊 𝑪𝑪𝒄𝒄
− 𝒌𝒌𝒄𝒄𝒄𝒄,𝒊𝒊 𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒄𝒄
−𝒌𝒌𝒇𝒇𝒇𝒇
𝑺𝑺𝑺𝑺
𝑽𝑽
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑺𝑺𝒄𝒄
𝑺𝑺𝒄𝒄,𝒎𝒎𝒎𝒎𝒎𝒎
+ 𝒌𝒌𝒓𝒓𝒓𝒓
𝑺𝑺𝑺𝑺
𝑽𝑽
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄 − 𝒌𝒌𝒇𝒇𝒇𝒇,𝒊𝒊𝒊𝒊
𝑺𝑺𝑺𝑺
𝑽𝑽
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊,𝒎𝒎𝒎𝒎𝒎𝒎
 Solute Sorbed to Immobile Colloids (ncolsites equations):
𝝏𝝏𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄
𝝏𝝏𝒕𝒕
= 𝒌𝒌𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒂𝒂 𝑺𝑺𝒄𝒄 𝟏𝟏 −
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄
𝑪𝑪𝒂𝒂𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊 𝑺𝑺𝒄𝒄
− 𝒌𝒌𝒄𝒄𝒄𝒄,𝒊𝒊 𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄 +
𝒌𝒌𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒂𝒂 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊 𝟏𝟏 −
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝑪𝑪𝒂𝒂𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
− 𝒌𝒌𝒄𝒄𝒄𝒄,𝒊𝒊 𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
−𝒌𝒌𝒇𝒇𝒇𝒇 𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑪𝑪𝒄𝒄 𝟏𝟏 −
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄
𝑪𝑪𝒂𝒂𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊 𝑺𝑺𝒄𝒄
+ 𝒌𝒌𝒓𝒓𝒓𝒓 𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄 − 𝒌𝒌𝒇𝒇𝒇𝒇,𝒊𝒊𝒊𝒊 𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄 𝟏𝟏 −
𝑪𝑪𝒂𝒂𝒂𝒂,𝒊𝒊 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
𝑪𝑪𝒂𝒂𝒂𝒂,𝒎𝒎𝒎𝒎𝒎𝒎,𝒊𝒊 𝑺𝑺𝒄𝒄,𝒊𝒊𝒊𝒊
56/8/2016 CFT Lightning

6. Used
Fuel
Disposition
Additional Solute Detail on Colloid
or Immobile Media Surfaces
 Transformations on Surfaces (from Mavrik, with extensions/generalizations
from Paul):
6
Surface Mediated-Reduction
With ‘Aging’ of Reduced
Surface Species
Ca
Sa,1
Sa,2
Aging Process Only
Ca
Sa,1 Sa,2
Ca
Two Sorption Sites
with Surface
Transformation
Surfaces could be Colloids or Immobile Media
6/8/2016 CFT Lightning

7. Used
Fuel
Disposition
Total Conservative Mass Fraction –
A Simplification
 ‘Convolution’ of fraction of nonfiltering colloids with fraction of
nondesorbing solutes
 That is, ‘fraction of solute not desorbing from fraction of colloids not
filtering over transport time τ ’, given by:
= � 𝑬𝑬𝑬𝑬𝑬𝑬 −
𝑺𝑺𝑺𝑺
𝑽𝑽
𝒌𝒌𝒇𝒇𝒇𝒇 𝝉𝝉 � 𝑬𝑬𝑬𝑬𝑬𝑬 −𝒌𝒌𝒄𝒄𝒄𝒄 𝝉𝝉 𝑷𝑷 𝒌𝒌𝒄𝒄𝒄𝒄
𝒌𝒌𝒄𝒄𝒄𝒄,𝒎𝒎𝒎𝒎𝒎𝒎
𝟎𝟎
𝒅𝒅𝒌𝒌𝒄𝒄𝒄𝒄
𝒌𝒌𝒇𝒇𝒇𝒇
𝑷𝑷 𝒌𝒌𝒇𝒇𝒇𝒇
𝒌𝒌𝒇𝒇𝒇𝒇,𝒎𝒎𝒎𝒎𝒎𝒎
𝟎𝟎
𝒅𝒅𝒌𝒌𝒇𝒇𝒇𝒇
76/8/2016 CFT Lightning

8. Used
Fuel
Disposition
Other Simplifications
 If the time scale of transport is long relative to both forward and reverse rates of
any given reaction, that reaction can be assumed to be at equilibrium
– Good rule of thumb is if (SA/V)kfτ > 5-10 and krτ > 5-10, then equilibrium assumption is good
– Also, if sorption/filtration site density (Smax) is very large, then linear Kd-type approach works
 Every linear equilibrium assumption reduces number of equations by one
(sorbed or filtered species equation is eliminated)
 Special case of all reactions being at equilibrium and having a steady-state
colloid concentration, Ccol :
8
0=+ 2
2
x
C
CKv
t
C
CK
t
C
K+
x
C
D
x
C
v
t
C
colcfcolcd
B
cf
∂
∂
+
∂
∂
+
∂
∂
∂
∂
−
∂
∂
∂
∂
φ
ρ
( ) 0=1+1 2
2
x
C
D
x
C
vCK
t
C
CKK cfcolccolcd
B
∂
∂
−
∂
∂
+
∂
∂






++
φ
ρ
colc
colcd
B
C
CK
CKK
R
+
++
=
1
1
φ
ρ
Solute Transport Eq.:
Rearranged:
Effective Solute Retardation Factor:
Note Kd = kas/ksa, Kc = kac/kca
6/8/2016 CFT Lightning