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Simulation of single phase reactive transport on pore-scale images

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### 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.

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:

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.

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):

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

Results for reactive transport

- 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

- 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.

Future Work

- Fluid-Fluid interactions
- Predict experimental data; Gramling et al. (2002)
- Fluid-solid interactions
- Dissolution and/or precipitation
- Change the pore space geometry and hence the flow field over time

Gramling et al. (2002)

Acknowledgements:

Dr. Branko Bijeljic and Prof. Martin Blunt

Emirates Foundation for funding this project

Thank youSeries of Images

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Model

- Couple transport with reactions

Advection

- General Pollock’s algorithm with no solid boundaries:
- To obtain the velocity at position inside a voxel
- To estimate the minimum time for a particle to exit a voxel:
- To determine the exit position of a particle in the neighbouring voxel

Mostaghimi et al. (2010)

Advection

6 algorithms

3 algorithms

8 algorithms

12 algorithms

3 algorithms

12 algorithms

12 algorithms

Mostaghimi et al. (2010)

Transport

- Particles Motion:
- Advection
- Diffusion.
- To measure the spreading of particles in porous media
- Peclet number

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).

Heterogeneous reactions

- Activity Coefficients are estimated using Harvie-Moller-Weare (HMV) methods (Bethke, 1996).

Heterogeneous reactions

- Nigrini (1970) approach are used to estimate diffusion coefficient

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