1. Finite-Element-Based Characterisation of Pore- scale Geometry and its Impact on Fluid Flow Lateef Akanji Supervisors Prof. Martin Blunt Prof. Stephan Matthai

2. 2 Outline 1.Research Objectives 2.Development of Single-phase Pore-scale Formulation and Numerical Model 3.Workflow and Model Verification 4.Validation: Application to Porous Media

3. 3 Research Objectives  To characterize pore-scale geometries and derive the constitutive relationship governing single and multiphase flow through them  To contribute to a better understanding of the physics of fluid flow in porous media based on first principle numerical approach  To investigate the dependency of fluid flow on the pore geometry which is usually neglected on the continuum scale  To develop a constitutive relationship which allows a more rigorous assessment of fluid flow behavior with implications for the larger scale

4. 4 Outline 1.Research Objectives 2.Development of Single-phase Pore-scale Formulation and Numerical Model 3.Workflow and Model Verification 4.Validation: Application to Porous Media

5. 5 Development of Single-phase Pore-scale Formulation and Numerical Model The general p.d.e. governing fluid flow at pore scale is given by the Navier – Stokes equations as: For an incompressible fluid conservation of mass takes the form For a steady-state system, the substantial time derivative goes to zero i.e. For slow laminar viscous flow with small Reynold’s number, the advective acceleration term drops out and we have the linear Stokes equations: (1/2)

6. FEM discretisation and solution sequence Define a function that obeys: Step 1: We solve Poisson’s equation for with homogeneous b.c. Step 2: We compute the pressure field using – this ensures that Since we define the velocity by: 6 Development of Single-phase Pore-scale Formulation and Numerical Model (2/2) μuμu fluid pressure, P Dependent variables are placed at the nodes. tetrahedron

7. 7 Outline 1.Research Objectives 2.Development of Single-phase Pore-scale Formulation and Numerical Model 3.Workflow and Model Verification 4.Validation: Application to Porous Media

8. 8 Workflow and Model Verification (1/7)

9. 9 Model Verification, Step1: Porosity Porosity Pore Volume / (Grain Volume + Pore Volume) (2/7)

10. 10 Model Verification, Step2: Pore Radius Computation Pore radii Derivative of f(x,y) Pore Radius (μm) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 3.35 µm GRAIN PORES (3/7)

11. 11 Model Verification, Step3: Pore Velocity Placement of 7 FEM Placement of 14 FEM Placement of 21 FEM (4/7)

12. 12 Model Verification, Step3: Pore Velocity Error analysis Caseabc Pressure gradient (Pa-m -1 )9860 Channel length (µm)30 Number of Elements71421 Channel velocity mismatch b/w analytical and numerical (%) 22.622.540.92 Volume flux mismatch b/w analytical and numerical (%) 22.813.642.0 (5/7)

13. 13 Model Verification, Step3: Pore Velocity Velocity (µms -1 ) (6/7)

14. 14 Model Verification, Step4: Effective Permeability (7/7)

15. 15 Outline 1.Research Objectives 2.Development of Single-phase Pore-scale Formulation and Numerical Model 3.Workflow and Model Verification 4.Validation: Application to Porous Media (Results)

16. 16 (Validation) Porous Media with Cylindrical Posts (1/10)

17. 17 (Talabi et al., SPE 2008) (2/10) Application to Porous Media Sample I: Ottawa sandstone Micro-CT scan CAD Hybrid mesh Velocity profile Velocity (x 10 -5 ms -1 ) 0 2 4 6 8 10 12 14 simulation thresholding meshing 4.5mm Velocity (x 10 -5 ms -1 ) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

18. 18 Pore Radius (μm) 0 10 20 30 40 50 60 70 80 Ottawa Sandstone Application to Porous Media Pore radius distribution (3/10) Pore Radius (μm) 0 10 20 30 40 50 60 70 80 LV60 Sandstone Sombrero beach carbonate

19. Application to Porous Media 19 3D Lab Expt 2D Num. Simulation Ottawa sand Dimension (mm) 4.5 x 4.5 x 4.5 4.5 x 4.5 Porosity (%) 35 39 Permeability (D) 45 31 LV60 sand Dimension (mm) 4.1 x 4.1 x 4.1 4.1 x 4.1 Porosity (%) 37 40 Permeability (D) 40 29 Sombrero beach carbonate sand Dimension (mm) - 4.5 x 4.5 Porosity (%) - 36 Permeability (D) - 28 Computed versus Measured Permeability (4/10)

20. 20 Application 3D Granular Packs (5/10)

21. 21 3D Granular Packs (6/10) Xavier Garcia

22. 22 3D Granular Packs Fluid Pressure (7/10) CAD geometry

23. 23 Sample 1 2.4 mm Φ= 33.52 Φ= 37.02 Φ= 38.43 Φ= 32.3 Φ= 35.80 (8/10)

24. 24 Sample 2 2.4 mm Φ= 32.43 Φ= 33.52 Φ= 36.81 Φ= 35.57 Φ= 37.63 Does the detail really matter? (9/10)

25. Permeability versus Porosity 25 X 10 -5 (10/10)

26. 26 Single-phase Advection in Porous Media (1/2) Ottawa

27. 27 Single-phase Advection in Porous Media (2/2) LT-M

28. 28 Conclusions  I have presented a Finite-Element-Based numerical simulation work flow showing pore scale geometry description and flow dynamics based on first principle  This is achieved by carrying out several numerical simulation on micro-CT scan, photomicrograph and synthetic granular pack of pore scale model samples  In order to accurately model fluid flow in porous media, the φ, r, pc, k distribution must be adequately captured (1/1)

29. 29 Future work  Two-phase flow with interface tracking testing for snap-off and phase trapping using level set method (Masa Prodanovic – University of Texas @ Austin)  Investigate dispersion in porous media (Branko Bijeljic) drainage imbibition Courtesy: (Masa Prodanovic – University of Texas @ Austin) Capturing snap-off during imbibition Courtesy: (Masa Prodanovic – University of Texas @ Austin) (1/1)

30. 30 Acknowledgements PTDF Nigeria CSMP++ Group

31. THANK YOU! 31