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Simulation of single phase reactive transport on pore-scale images Zaki Al Nahari, Branko Bijeljic, Martin Blunt

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Outline Motivation Modelling reactive transport Geometry & flow field Transport Reaction rate Validation against analytical solutions Results Future work

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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 <

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Physical description of reactive transport Geometry Pore-scale image Flow Pressure field Velocity field Transport Advection Diffusion Reaction Reaction rate

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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

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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.

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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):

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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: D m = 7.02x10 -11 m 2 /s k= 2.3x10 9 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

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Case 1: Number of Voxels= 10×10×10 Δt= 10 -5 s Δt= 10 -4 sΔt= 10 -3 s

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Case 1: Number of Voxels= 10×10×10

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Case 2: Number of Voxels= 20×20×20 Δt= 10 -5 s Δt= 10 -4 sΔt= 10 -3 s

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Case 2: Number of Voxels= 20×20×20

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Case 3: Number of Voxels= 50×50×50 Δt= 10 -4 s Δt= 10 -3 s

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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

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Results; Case 1 - Parallel injection 2-D 3-D x (μm) y (μm) z (μm) y (μm)

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Results; Case 1 - Parallel injection after 1 sec 2-D 3-D x (μm) y (μm) z (μm) y (μm) C= 1087

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Results; Case 2 - Front injection 2-D 3-D x (μm) y (μm) z (μm) y (μm)

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Results; Case 2 – Front injection after 1 sec 2-D 3-D x (μm) y (μm) z (μm) y (μm) C= 713

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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)

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THANK YOU Acknowledgements: Dr. Branko Bijeljic and Prof. Martin Blunt Emirates Foundation for funding this project

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Series of Images ImageMirror (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 1Image 2 Number Images Image + Mirror

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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

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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 Mostaghimi et al. (2010)

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Advection 6 algorithms3 algorithms 12 algorithms 8 algorithms 12 algorithms 3 algorithms Mostaghimi et al. (2010)

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Transport Particles Motion: Advection Diffusion. To measure the spreading of particles in porous media Peclet number Bijeljic and Blunt (2006)

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Heterogeneous reactions Assumption: Temperature is constant CO 2 is dissolved in brine. No vaporisation process. No biogeological reactions Carbonate dissolution and precipitation kinetic constant rate are taken from Chou et al. (1989).

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Heterogeneous reactions Activity Coefficients are estimated using Harvie- Moller-Weare (HMV) methods (Bethke, 1996). T (°C)ABIons 00.48830.32413 50.49210.32493.5 100.49600.32584-4.5 150.50.32624.5 200.50420.32725 250.50850.32816 300.51300.32908 350.51750.32979 400.52210.3305 500.53190.3321 600.54250.3338

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Heterogeneous reactions Nigrini (1970) approach are used to estimate diffusion coefficient Ions (10 -4 S.m 2 /s) 349.6 50.1 73.5 106 119 199.1 76.35 78.1 44.5 138.6 Ions All0.02 0.0139 0.018

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