1. Mathematics of non-Darcy CO 2 injection into saline aquifers Ana Mijic Imperial College London Department of Civil and Environmental Engineering PMPM Research Network 2015 Annual Meeting, Edinbourgh

2. Thanks to Tara LaForce and Ann Muggeridge Simon Mathias Sebastian Geiger Grantham Institute for Climate Change John Archer Fund ORSAS Fund

3. Problem background

4. Context CO 2 injection is limited by the formation fracture pressure Injected CO 2 can induce the resident saline water’s evaporation resulting in salt deposition CO 2 is a compressible gas whose properties vary with the change of the reservoir pressure At high injection rates inertial effects may become significant

5. Analytical solutions to CO 2 -brine displacement Segregated flow approach Nordbotten et al. (2005) Nordbotten and Celia (2006) Mathias et al. (2009) Vilarrasa et al. (2010) Burton et al. (2008) Zeidouni et al. (2009) Compositional displacement Phase Compressibility Non-Darcy flow Diffuse flow approach Inaccurate predictions at high injection rates (Lu et al., 2009)

6. Analytical solutions to CO 2 -brine displacement Segregated flow approach Analysis of critical processes in the near-well region Compositional displacement Phase Compressibility Non-Darcy flow Diffuse flow approach Nordbotten et al. (2005) Nordbotten and Celia (2006) Vilarrasa et al. (2010) Mathias et al. (2009) Burton et al. (2008) Zeidouni et al. (2009)

7. Mathematics

8. Buckley-Leverett model Linear oil-water system Momentum equation by Darcy’s law Shock front by Method of Characteristics (MOC)

9. Radial convection equation for a gas-liquid system Gas flow governed by the Forchheimer equation Fractional flow function depending both on saturation and radial distance from the well Solution for saturation by the modified MOC Solution for pressure by numerical integration Modelling approach

10. Radial convection equation for gas phase radial distance r Liquid = phase 2 Gas = phase 1

11. Fractional flow function Two-phase extension of the Forchheimer equation Two-phase extension of the Darcy’s law

12. Fractional flow function Two-phase extension of the Forchheimer equation Two-phase extension of the Darcy’s law Darcy fraction of the gas velocity, q 1 0 Non-Darcy scaling factor,  ≥ 1.0 Non-Darcy fractional flow function

13. Non-Darcy gas phase velocity Forchheimer flow parameter Darcy fraction of the gas velocity Quadratic representation of gas phase velocity Forchheimer factor

14. Non-Darcy flow calculation algorithm

15. Solution for saturation by modified MOC Equal area rule

16. Solving for pressure Transient solution (Theis, 1935) Impact of the boundary condition: Open aquifer (Thiem, 1906) Closed aquifer (Dake, 1983) More details can be found in: Mijic, A., and T. C. LaForce (2012), Water Resour. Res., 48, W09503, doi:10.1029/2011WR010961.

17. Miscible displacement Solutions for saturation and pressure as extension of the immiscible model!

18. Salt precipitation Salt saturation Permeability reduction Model from Zeidouni et al. (2009)

19. Correction for gas compressibility (1)

20. Correction for gas compressibility (2) Solved iteratively using using the mean flux as a convergence criterion More details can be found in: Mijic, A., T. C. LaForce, and A. H. Muggeridge (2014), Water Resour. Res., 50, 4163–4185, doi:10.1002/2013WR014893.

21. Fractional flow module Saturation profile module Pressure profile module Compressibility correction module Computational scheme

22. Results

23. Relative permeability Corey (1954) and van Genuchten (1980) models as a function of the end-point relative permeability Exp. data from Krevor et al. (2012)

24. Forchheimer coefficient for the gas phase Limited experimental data Modified Janicek and Katz (1955) formulation Range of W between 3.2 10 -9 and 3.2 10 -7 m 1.5 Carbonates Ottawa sand

25. Non-Darcy fractional flow curves The most significant influence of the non-Darcy effect is near the well For a given saturation value the inertial losses significantly slow down the gas phase flow

26. Non-Darcy saturation profiles Effects of non-Darcy flowEffects of injection rate

27. Non-Darcy pressure distribution Effects of non-Darcy flow and formation properties

28. Effects of partial miscibility Overall fractional flow curves in single- and two-phase regions

29. Permeability reduction Effects of the Forchheimer coefficient variability in incompressible displacement on permeability reduction

30. Effects of compressibility Incompressible and compressible saturation and pressure profiles

31. Comparison with ECLIPSE 300 Excellent agreement between analytical and simulation results. However, the comparison is with constant values of the Forchheimer coefficient!

32. Implications of non-Darcy effects (1) The error associated with neglecting the non-Darcy flow increases significantly with the injection rate and decreased formation permeability

33. In non-Darcy displacement CO 2 injectivity is limited by the pressure increase at high rates and highly sensitive to formation permeability Implications of non-Darcy effects (2)

34. Conclusions The application example showed significant influence of non-Darcy effects in low permeability formations when CO 2 is injected at high rates: CO 2 displacement efficiency improvement Additional pressure increase at the well Salt precipitation reduction Alteration of injectivity function

35. Where this can be taken Estimation of Forchheimer coefficient for the gas phase in formations suitable for CO 2 injection Correction of pressure estimation due to salt precipitation in near-well region Analytical modelling of effects of rock dissolution Multiple well analysis in two-phase displacement Reservoir simulator with spatially varying saturation- dependent Forchheimer flow

36. Where this can be taken Estimation of Forchheimer coefficient for the gas phase in formations suitable for CO 2 injection Correction of pressure estimation due to salt precipitation in near-well region Analytical modelling of effects of rock dissolution Multiple well analysis in two-phase displacement Reservoir simulator with spatially varying saturation- dependent Forchheimer flow

37. Ana Mijic ana.mijic@imperial.ac.uk @leiastarspear