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Simulation of single phase reactive transport on pore-scale images Zaki Al Nahari, Branko Bijeljic, Martin Blunt
Simulation of single phase reactive transport on pore-scale images
Zaki Al Nahari, Branko Bijeljic, Martin Blunt
Outline
• Motivation• Modelling reactive transport• Geometry & flow field• Transport• Reaction rate• Validation against analytical solutions• Results• Future work
Motivation
• Contaminant transport:• Industrial waste• Biodegradation of landfills…etc
• Carbon capture and storage:• Acidic brine.• Over time, potential dissolution
and/or mineral trapping.
• However….• Uncertainty in reaction rates
• The field <<in the lab.
• No fundamental basis to integrate flow, transport and reaction in porous media.
Physical description of reactive transport
Geometry • Pore-scale image
Flow• Pressure field• Velocity field
Transport• Advection• Diffusion
Reaction • Reaction rate
Geometry & Flow
• Micro-CT scanner uses X-rays to produce a sequence of cross-sectional tomography images of rocks in high resolution (µm)
• To obtain the pressure and velocity field at the pore-scale, the Navier-Stokes equations are fundamental approach for the flow simulation.• Momentum balance
• Mass balance
• For incompressible laminar flow, Stokes equations can be used:
g
uP- uu
tu 2
0u0uP- 2
g
0u
Pore space
Velocity field
Pressure field
Transport
Track the motion of particles for every time step by:
• Advection along streamlines using a novel formulation accounting for zero flow at solid boundaries. It is based on a semi-analytical approach: no further numerical errors once the flow is computed at cell faces.
• Diffusion using random walk. It is a series of spatial random displacements that define the particle transitions by diffusion.
sinsin
6
ttt
m
yy
tD
cossin ttt xx cos ttt zz
Reaction Rate
• Bimolecular reactionA + B → C
• The reaction occurs if two conditions are met:• Distance between reactant is less than or
equal the diffusive step ( )• If there is more than one possible reactant, the
reaction will be with nearest reactant..
• The probability of reaction (P) as a function of reaction rate constant (k):
d
Validation for bulk reaction
• Reaction in a bulk system against the analytical solution:• no porous medium• no flow
• Analytical solution for concentration in bulk with no flow.
• Number of Voxels:• Case 1: 10×10×10• Case 2: 20×20×20• Case 3: 50×50×50
• Number of particles:• A= 100,000 density= 0.8 Np/voxel• B= 50,000 density= 0.4 Np/voxel
• Parameters:• Dm= 7.02x10-11 m2/s• k= 2.3x109 M-1.s-1
• Time step sizes:• Δt= 10-3 s P= 3.335×10-3
• Δt= 10-4 s P= 1.055×10-2
• Δt= 10-5 s P= 3.335×10-2
Case 1: Number of Voxels= 10×10×10
Δt= 10-5 s
Δt= 10-4 sΔt= 10-3 s
Case 1: Number of Voxels= 10×10×10
Case 2: Number of Voxels= 20×20×20
Δt= 10-5 s
Δt= 10-4 sΔt= 10-3 s
Case 2: Number of Voxels= 20×20×20
Case 3: Number of Voxels= 50×50×50
Δt= 10-4 s
Δt= 10-3 s
Results for reactive transport
Case 1: Parallel injection• Both reactants (A and B) injected
at the top and bottom half of the inlet.
Case 2: Injection• Reactant, A, is resident in the
pore space, while reactant B is injected at the inlet face.
• Berea Sandstone• Number of Voxels: 300×300×300• Number of particles:
• A= 400,000 density= 1.481×10-2 Np/voxel• B= 200,000 density= 7.407×10-3 Np/voxel
• Pe= 200
Results; Case 1 - Parallel injection
2-D
3-D
x (μm)
x (μm)
y (μ
m)
z (μm)
y (μ
m)
Results; Case 1 - Parallel injection after 1 sec
2-D
3-D
x (μm)
x (μm)
y (μ
m)
z (μm)
y (μ
m)
C= 1087
Results; Case 2 - Front injection
2-D
3-D
x (μm)
x (μm)
y (μ
m)
z (μm)
y (μ
m)
Results; Case 2 – Front injection after 1 sec
2-D
3-D
x (μm)
x (μm)
y (μ
m)
z (μm)
y (μ
m)
C= 713
Future Work
1. Fluid-Fluid interactions• Predict experimental data;
Gramling et al. (2002)
2. Fluid-solid interactions• Dissolution and/or precipitation• Change the pore space
geometry and hence the flow field over time
Gramling et al. (2002)
23
2323
32
323
32
3
COMCOHMCO
2HCOMCOHMCO
HCOMHMCO
THANK YOU
Acknowledgements: Dr. Branko Bijeljic and Prof. Martin BluntEmirates Foundation for funding this project
Series of ImagesImage Mirror
(0, 0, 0) (x, 0, 0)
(0, y, 0) (x, y, 0)
(0, 0, z)
(0, y, z) (x, y, z)
(x, 0, z)
(0, 0, 0)(x, 0, 0)
(0, y, 0)(x, y, 0)
(0, 0, z)
(0, y, z)(x, y, z)
(x, 0, z)
(0, 0, 0)
(0, 0, z)
(0, y, 0)
(0, y, z) (x, y, z)
(2x, 0, z)
(2x, 0, 0)
(2x, y, 0)
(0, 0, 0)
(0, 0, z)
(0, y, 0)
(0, y, z) (x, y, z)
(2x, 0, z)
(2x, 0, 0)
(2x, y, 0)
(0, 0, 0)
(0, 0, z)
(0, y, 0)
(0, y, z) (x, y, z)
(2x, 0, z)
(2x, 0, 0)
(2x, y, 0)
Image 1 Image 2NumberImages
Image + Mirror
Model
Geometry• Obtaining micro-CT images
Flow
• Obtaining the pressure and velocity fields in the rock image
Transport• Track particles motion through the pore space
Reaction• Geochemical reactions
Couple transport with reactions
Advection
• General Pollock’s algorithm with no solid boundaries:1. To obtain the velocity at position inside a voxel
2. To estimate the minimum time for a particle to exit a voxel:
3. To determine the exit position of a particle in the neighbouring voxel
1112 uxx
xuuu
1112 vyy
yvvv
1112 wzz
zwww
zyx ,,min
1121
2
12
lnxxuuxu
xuuux
px
1121
2
12
lnyyvvxv
yvvvv
py
1121
2
12
lnzzwwzw
zwwwz
pz
zww
pe
yvuv
pe
xuu
pe
ezzwwzw
wwzwzz
eyyvuyvv
vvyvyy
exxuuxu
uuxuxx
12
12
12
112
1
12
11
112
1
12
11
112
1
12
11
Mostaghimi et al. (2010)
Advection
6 algorithms 3 algorithms
12 algorithms 8 algorithms
12 algorithms 12 algorithms3 algorithms
Mostaghimi et al. (2010)
Transport
• Particles Motion:• Advection• Diffusion.
• To measure the spreading of particles in porous media
• Peclet number 2
2
21
titi
L
XX
dtdD
S
m
av
AVL
DLuPe
Bijeljic and Blunt (2006)
Heterogeneous reactions
• Assumption:• Temperature is constant• CO2 is dissolved in brine.• No vaporisation process.• No biogeological reactions
• Carbonate dissolution and precipitation kinetic constant rate are taken from Chou et al. (1989).
iii
b
f
ma
aakaakaakr
kakakr
23
23
222
32
HCOM62
HCOM5HCOM4
3COH2H1
23
2323
32
323
32
3
COMCOHMCO
2HCOMCOHMCO
HCOMHMCO
Heterogeneous reactions
• Activity Coefficients are estimated using Harvie-Moller-Weare (HMV) methods (Bethke, 1996).
2
2
5.0
1log
lnln
ii
oi
idhi
j j kkjijkjij
dhii
zMI
IBaIAz
mmEmID
T (°C) A B Ions0 0.4883 0.3241 35 0.4921 0.3249 3.510 0.4960 0.3258 4-4.515 0.5 0.3262 4.520 0.5042 0.3272 525 0.5085 0.3281 630 0.5130 0.3290 835 0.5175 0.3297 940 0.5221 0.330550 0.5319 0.332160 0.5425 0.3338
m 10 10oia
Br,Cl ,K -
OH-3HCO ,Na
-23CO2Ba
22 Fe ,Ca2Mg
H
Heterogeneous reactions
• Nigrini (1970) approach are used to estimate diffusion coefficient
CT
zRTD
DD
Ψ
DDΨ
DD
Cii
i
ii
251
F
1
1
11ln
1
1
025,
0
2
00
02
01
23
2
2
31
3
02
01
010
12
Ions (10-4 S.m2/s)
349.650.173.5106119
199.176.3578.144.5
138.6
025, Ci
HNa
K2Mg
2CaOH
-ClBr
-3HCO-2
3CO
IonsAll 0.02
0.01390.018
1C
HOH
