1. Department of Earth Science and Engineering Imperial College Consortium on Pore-scale Modelling Ali Raeini, Branko Bijeljic and Martin Blunt

2. Micro-CT image Capillary pressure Relative permeability Macroscopic description of the flow Microscopic description of flow

3. Capillary pressure Relative permeability Pore network model Pore network modelling Micro-CT image

4. Capillary pressure Relative permeability Numerical Simulation Micro-CT image Finite volume mesh

5. Capillary pressure equation Mass balance Momentum balance Advection of indicator function

6. OpenFOAM: open source CFD toolbox interFOAM code : Discretisation of PDEs:Finite Volume Method Capturing interface position:Volume of Fluid method Surface tension forces:Continuous Surface Force (CSF) model (Brackbill et al., 1992) www.openfoam.com + our modifications......

7. Initial condition Equilibrium solution

8. (m/s)  Presence of spurious velocities in CSF formulation makes the simulations impossible at low capillary numbers

9. Sharp Surface Force (SSF) formulation (m/s) (Pa)

10. Capillary waves Spurious currents

11. Jagged solid walls Smooth solid walls solid wall (m/s)  Non-physical velocities are generated close to the jagged solid walls in both CSF and SSF formulations

12. Filtering capillary forces parallel to the interface (m/s) (Pa)

13. CSF formulation SSF formulation

14. Filtering capillary fluxes  Filtering is necessary to obtain accurate results for the velocity field

15. Filtering capillary forces parallel to the interface Filtering capillary fluxes Sharp Surface Force formulation.....

17. High Capillary Numbers Low Capillary Numbers Single-phase flow (>10 -7 )

18. (Pa) PdPd PdPd  Obtaining Capillary pressure and relative permeability curves from transient experiments  Improving the efficiency of the numerical model for low capillary numbers PcPc PcPc

19.  Steady two - phase flow experiments, using cyclic boundary conditions  Steady two - phase flow experiments, using cyclic boundary conditions  Multiphase micro - CT images Transient experiments

20. Thanks to: o Stephen Neethling, Peyman Mostaghimi, Christopher Pentland and many other colleagues, for useful discussions o Sponsors of the project: DTI, EPSRC, ENI, Saudi Aramco, BG, BHP, JOGMEC, Schlumberger, Shell, Statoil and Total and

21. References Brackbill, J. U., Kothe, D. B., and Zemach, C. A continuum method for modeling surface tension. J. Comput. Phys., 100(2):335–354, 1992

22. Spare slides

23. Nawaw cwcw θC pc C fc,filt|| SKWSKC φ c,filt NcNc PcPc P c rel. error |u| avg |u| rel. deviaton |u| max. error Effect of: 1000300.20.0521 0.008E-0672220.053.5E-4-0.560.23 Filtering coefficient for capillary fluxes ( C φ c,filt ) 0.0058E-671650.047.6E-4-0.040.05 0.0108E-671220.047.9E-4-0.010.07 1000300.2 0.00 210.005 8E-672090.055.8E-4-0.280.33 Filtering coefficient for capillary forces parallel to the interface ( C fc,filt|| ) 0.108E-671170.047.5E-4-0.060.15 1000300.20.05 0 10.0058E-6 72370.058.0E-40.000.04 α smoothing kernel (SK) 171960.057.8E-4-0.030.03 1000300.20.052 0 0.0058E-6 85030.249.3E-40.170.90 n w smoothing kernel (WSK) 371510.047.4E-4-0.070.03 100030 0.1 0.05210.0058E-6 71840.058.4E-40.050.04 Capillary pressure sharpness coefficient ( C pc ) 0.371690.047.6E-4-0.040.11 1000 0 0.20.05210.0058E-6 82080.048.4E-40.060.21 Different values of contact angle ( θ ) 902-5.3E-4-0.33-0.01 15084630.075.8E-4-0.270.08 10 10 300.20.052 0 0.0058E-6 88500.326.1E-4-0.2729.63 The wall smoothing kernel (WSK) for more jagged solid walls 166930.008.4E-40.011.25 367000.008.3E-40.001.06 10.5 091560.333.8E-4-0.5236.96 16643-0.045.9E-4-0.254.83 36628-0.046.5E-4-0.184.43 5 00300.20.05210.005 9E-66179-0.064.2E-4-0.530.03 Mesh refinement (N) 208E-672550.057.6E-4-0.040.29