Mathematics of non-Darcy CO2 injection into saline aquifers Ana Mijic Imperial College London Department of Civil and Environmental Engineering PMPM Research Network 2015 Annual Meeting, Edinbourgh Thanks to Tara LaForce and Ann Muggeridge Simon Mathias Sebastian Geiger Grantham Institute for Climate Change John Archer Fund ORSAS Fund Problem background Context • • • • CO2 injection is limited by the formation fracture pressure Injected CO2 can induce the resident saline water’s evaporation resulting in salt deposition CO2 is a compressible gas whose properties vary with the change of the reservoir pressure At high injection rates inertial effects may become significant Analytical solutions to CO2-brine displacement Inaccurate predictions at high injection rates (Lu et al., 2009) Nordbotten et al. (2005) Nordbotten and Celia (2006) Segregated flow approach Vilarrasa et al. (2010) Compositional displacement Phase Compressibility Mathias et al. (2009) Non-Darcy flow Burton et al. (2008) Zeidouni et al. (2009) Diffuse flow approach Analytical solutions to CO2-brine displacement Analysis of critical processes in the near-well region Nordbotten et al. (2005) Nordbotten and Celia (2006) Segregated flow approach Vilarrasa et al. (2010) Compositional displacement Phase Compressibility Mathias et al. (2009) Non-Darcy flow Burton et al. (2008) Zeidouni et al. (2009) Diffuse flow approach Mathematics Buckley-Leverett model • • • Linear oil-water system Momentum equation by Darcy’s law Shock front by Method of Characteristics (MOC) Modelling approach • • • 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 Radial convection equation for gas phase Gas = phase 1 Liquid = phase 2 radial distance r Fractional flow function Two-phase extension of the Forchheimer equation Two-phase extension of the Darcy’s law Fractional flow function Two-phase extension of the Forchheimer equation Darcy fraction of the gas velocity, q10 Two-phase extension of the Darcy’s law Non-Darcy scaling factor, b ≥ 1.0 Non-Darcy fractional flow function Non-Darcy gas phase velocity Darcy fraction of the gas velocity Forchheimer flow parameter Quadratic representation of gas phase velocity Forchheimer factor Non-Darcy flow calculation algorithm Solution for saturation by modified MOC Equal area rule 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. Miscible displacement F1 Da r cyf lo 1 F1( C1,r2) F1( C1,r1) For ch h ei merf lo s1c 1 C1 Solutions for saturation and pressure as extension of the immiscible model! Salt precipitation • Salt saturation • Permeability reduction Model from Zeidouni et al. (2009) Correction for gas compressibility (1) 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. Fractional flow module Computational scheme Saturation profile module Pressure profile module Compressibility correction module Results Relative permeability 1 CO modelled 2 0.9 Brine modelled CO2 experimental • Corey (1954) and van Genuchten (1980) models as a function of the end-point relative permeability Relative permeability (−) 0.8 Brine experimental 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 CO2 saturation (−) 0.6 0.7 0.8 Exp. data from Krevor et al. (2012) Forchheimer coefficient for the gas phase • 14 −1 Forchheimer parameter for the gas phase (m ) 10 Wong (1970) Evans and Evans (1988) Janicek and Katz (1955) Geertsma (1974) Kutasov (1993) Frederick and Graves (1994) 12 10 10 • 10 8 Carbonates • 10 6 10 4 10 Ottawa sand 2 10 −14 10 −13 10 10 −12 10 −11 −10 10 2 Effective permeability (m ) 10 −9 Limited experimental data Modified Janicek and Katz (1955) formulation Range of W between 3.2 10-9 and 3.2 10-7 m1.5 Non-Darcy fractional flow curves 1 CO2 0.9 0.2 m 1m 10 m 0.8 100 m 2 CO fractional flow (−) Darcy 0.7 r 0.6 0.5 • 0.4 0.3 • 0.2 m 1m 10 m 100 m Darcy 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 CO saturation (−) 2 0.6 0.7 0.8 rd 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 Non-Darcy saturation profiles 0.8 0.8 Q =30 kg s Darcy flow, W=0 −8 1.5 Non−Darcy flow, W=3.2 10 m −7 1.5 Non−Darcy flow, W=3.2 10 m 0.7 Q =100 kg s−1 0.7 m Q =120 kg s 0.6 CO saturation (−) 0.5 0.4 −1 m 0.6 0.5 0.4 2 2 CO saturation (−) −1 m 0.3 0.3 0.2 0.2 0.1 0.1 0 0 5 10 15 20 Radial distance (m) 25 30 Effects of non-Darcy flow 35 0 0 5 10 15 20 Radial distance (m) 25 Effects of injection rate 30 Non-Darcy pressure distribution a) H=50 m and k=10 −13 m 2 b) H=200 m and k=10 100 24 80 70 60 50 40 30 22 Pressure (MPa) Darcy flow, t=12 h Non−Darcy flow, t=12 h Darcy flow, t=5 d Non−Darcy flow, t=5 d Darcy flow, t=50 d Non−Darcy flow, t=50 d 90 Pressure (MPa) −13 20 18 16 14 20 0 10 2 10 Radial distance (m) 12 0 10 2 10 Radial distance (m) Effects of non-Darcy flow and formation properties m 2 Effects of partial miscibility Overall fractional flow curves in single- and two-phase regions Permeability reduction Effects of the Forchheimer coefficient variability in incompressible displacement on permeability reduction Incompressible displacement Compressible displacement 1 1 Darcy non−Darcy const. b 1 1 2 0.6 0.4 0.2 0 1 0.6 0.4 0 10 20 30 Radial distance (m) 40 0 50 21 21 20 20 19 19 18 17 16 15 15 13 2 10 Radial distance (m) 20 30 Radial distance (m) 16 13 0 10 17 14 10 0 18 14 12 non−Darcy var. b 0.2 d) Pressure (MPa) c) Pressure (MPa) Incompressible and compressible saturation and pressure profiles 1 0.8 non−Darcy var. b b) CO saturation (−) 2 a) CO saturation (−) 0.8 Effects of compressibility Darcy non−Darcy const. b 12 0 10 2 10 Radial distance (m) 40 50 Comparison with ECLIPSE 300 1 20 Simulator Incompressible Analytical Compressible Analytical 18 b) Pressure (MPa) 2 a) CO saturation (−) 0.8 19 0.6 0.4 17 16 15 14 0.2 13 0 0 50 100 Radial distance (m) 150 12 0 10 2 10 Radial distance (m) Excellent agreement between analytical and simulation results. However, the comparison is with constant values of the Forchheimer coefficient! 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 Implications of non-Darcy effects (2) In non-Darcy displacement CO2 injectivity is limited by the pressure increase at high rates and highly sensitive to formation permeability Conclusions The application example showed significant influence of non-Darcy effects in low permeability formations when CO2 is injected at high rates: • • • • CO2 displacement efficiency improvement Additional pressure increase at the well Salt precipitation reduction Alteration of injectivity function Where this can be taken • • • • • Estimation of Forchheimer coefficient for the gas phase in formations suitable for CO2 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 saturationdependent Forchheimer flow Where this can be taken • • • • • Estimation of Forchheimer coefficient for the gas phase in formations suitable for CO2 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 saturationdependent Forchheimer flow Ana Mijic ana.mijic@imperial.ac.uk @leiastarspear