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- uut

u 2

0u

0uP- 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 reaction

A + 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 Blunt

Emirates Foundation for funding this project

Series of Images

Image 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

x

uuu

1112 vyy

y

vvv

1112 wzz

z

www

zyx ,,min

1121

2

12

lnxxuuxu

xu

uu

x

px

1121

2

12

lnyyvvxv

yv

vv

v

py

1121

2

12

lnzzwwzw

zw

ww

z

pz

z

ww

pe

y

vuv

pe

x

uu

pe

ezzww

zw

ww

zwzz

eyyvu

yvv

vv

yvyy

exxuu

xu

uu

xuxx

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 algorithms

3 algorithms

Mostaghimi et al. (2010)

Transport

• Particles Motion:• Advection• Diffusion.

• To measure the spreading of particles in porous media

• Peclet number

2

2

2

1

titi

L

XX

dt

dD

S

m

av

A

VL

D

LuPe

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

IBa

IAz

mmEmID

T (°C) A B Ions

0 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

z

RTD

DD

Ψ

DD

Ψ

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)

349.6

50.1

73.5

106

119

199.1

76.35

78.1

44.5

138.6

025, Ci

HNa

K2Mg

2CaOH

-ClBr

-3HCO-2

3CO

Ions

All 0.02

0.0139

0.018

1C

HOH