Abstract
In this paper, we analyze the time scales associated with instable fingering induced by density contrasts in miscible displacement porous media flow. We perform numerical simulations of a two-dimensional domain with boundaries that are closed to flow and identify the three regimes of the dynamics, namely the development of a stable diffusive boundary layer, the onset and growth of instabilities, and the fully nonlinear dynamics. Special focus is given to the onset of the fully nonlinear regime. The results are generic in the sense that there are no parameters in the non-dimensional model problem. Large ensembles are studied and an error estimate is given based on the combined effect of numerical errors and sampling errors. The nonlinear time scales show a dependence on the size of initial perturbations. We estimate this size for three formations used for CO2 storage and find that the onset of enhanced convective mixing is considerably delayed compared with the linear onset time.
Similar content being viewed by others
References
Alexander, R.: Implicit Runge–Kutta methods for stiff O.D.E.’s. SIAM J. Numer. Anal. 14(6), 1006–1021 (1977)
Baklid, A., Korbøl, R., Owren, G.: Sleipner Vest CO2 disposal, CO2 injection into a shallow underground aquifer. SPE 36600, 269–277 (1996)
Batzle, M., Wang, Z.: Seismic properties of pore fluids. Geophysics 11, 1396–1408 (1992)
Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic, Norwell (1991)
Chadwick, R.A., Zweigel, P., Gregersen, U., Kirby, G.A., Holloway, S., Johannessen, P.N.: Geological reservoir characterization of a CO2 storage site: the Utsira Sand, Sleipner, northern North Sea. Energy 29, 1371–1381 (2004)
Diersch, H.-J.G., Kolditz, O.: Variable-density flow and transport in porous media: approaches and challenges. Adv. Water Resour. 25, 899–944 (2002)
Duan, Z., Sun, R.: An improved model calculating CO2 solubility in pure water and aqueous NaCl solution from 273 to 533 K and from 0 to 2000 bar. Chem. Geol. 193, 257–271 (2003)
Elder, J.W.: Steady free convection in a porous medium heated from below. J. Fluid Mech. 27, 29–48 (1967)
Elder, J.W.: Transient convection in a porous medium. J. Fluid Mech. 27(3), 609–623 (1967)
Elenius, M.T., Nordbotten, J.M., Kalisch, H.: Effects of a capillary transition zone on the stability of a diffusive boundary layer. IMA J. Appl. Math. (2012, accepted for publication)
Ennis-King, J.P., Paterson, L.: Role of convective mixing in the long-term storage of carbon dioxide in deep saline formations. SPE J. 10(3), 349–356 (2005)
Ennis-King, J.P., Preston, I., Paterson, L.: Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. Phys. Fluids 17(8), 084107 (2005)
Fein, E.: d 3 f-Ein Programmpaket zur Modellierung von Dichteströmungen. GRS, Braunschweig, GRS-139 (1998)
Frolkovic, P., De Schepper, H.: Numerical modelling of convection dominated transport coupled with density driven flow in porous media. Adv. Water Resour. 24, 63–72 (2001)
Garcia, J.E.: Density of aqueous solutions of CO2. Report of Lawrence Berkley National Laboratory (2001)
Green, C., Ennis-King, J., Pruess, K.: Effect of vertical heterogeneity on long-term migration of CO2 in saline formations. Energy Proc. 1, 1823–1830 (2009)
Hidalgo, J.J., Carrera, J.: Effect of dispersion on the onset of convection during CO2 sequestration. J. Fluid Mech. 640, 441–452 (2009)
Iding, M., Ringrose, P.: Evaluating the impact of fractures on the performance of the In Salah CO2 storage site. Int. J. Greenhouse Gas Control 4, 242–248 (2010)
Johannsen, K.: On the validity of the Boussinesq approximation for the Elder problem. Comput. Geosci. 7(3), 169–182 (2003)
Johannsen, K.: Numerische Aspekte dichtegetriebener Strömung in porösen Medien. Professorial dissertation (habilitation), Heidelberg, Germany (2004)
Johannsen, K., Oswald, S., Held, R., Kinzelbach, W.: Numerical simulation of three-dimensional saltwater-freshwater fingering instabilities observed in a porous medium. Adv. Water Resour. 29, 1690–1704 (2006)
Johnson, R.A.: Miller and Freund’s Probability and Statistics for Engineers. Prentice-Hall, Englewood Cliffs (1994)
Kneafsey, T.J., Pruess, K.: Laboratory flow experiments for visualizing carbon dioxide-induced, density-driven brine convection. Transp. Porous Med. 82, 123–139 (2010)
Maldal, T., Tappel, I.M.: CO2 underground storage for Shøhvit gas field development. Energy 29, 1403–1411 (2004)
Mathieson, A., Wright, I., Roberts, D., Ringrose, P.: Satellite imaging to monitor CO2 movement at Krechba, Algeria. Energy Proc. 1, 2201–2209 (2009)
Pamukcu, Y., Hurter, S., Jammes, L., Vu-Hoang, D., Pekot, L.: Characterizing and predicting short term performance for the In Salah Krechba field CCS joint industry project. Energy Proc. 4, 3371–3378 (2011)
Pau, G.S.H., Bell, J.B., Pruess, K., Almgren, A.S., Lijewski, M.J., Zhang, K.: High-resolution simulation and characterization of density-driven flow in CO2 storage in saline aquifers. Adv. Water Resour. 33, 443–455 (2010)
Pham, T.H.V., Maast, T.E., Hellevang, H., Aagard, P.I.: Numerical modeling including hysteresis properties for CO2 storage in Tubåen formation, Snøhvit field, Barents Sea. Energy Proc. 4, 3746–3753 (2011)
Riaz, A., Hesse, M., Tchelepi, H.A., Orr, F.M.: Onset of convection in a gravitationally unstable diffusive boundary layer in porous media. J. Fluid Mech. 548, 87–111 (2006)
Rochelle, C.A., Moore, Y.A.: The solubility of supercritical CO2 into pure water and synthetic Utsira porewater. British Geological Survey, CR/02/052 (2006)
Slim, A.C., Ramakrishnan, T.S.: Onset and cessation of time-dependent, dissolution-driven convection in porous media. Phys. Fluids 22, 124103 (2010)
Tewes, F., Boury, F.: Formation and rheological properties of the supercritical CO2–water pure interface. J. Phys. Chem. 548, 892–898 (2005)
van Duijn, C.J., Pieters, G.J.M., van der Ploeg, A., Wooding, R.A.: Stability criteria for the vertical boundary layer formed by throughflow near the surface of a porous medium. In: Raats, P.A.C., Smiles, D., Warrick, A.W. (eds.) Environmental Mechanics; Water, Mass and Energy Transfer in the Biosphere—The Philip Volume, vol. 129, pp. 155–169 (2002)
Wooding, R.A.: Steady state free thermal convection of liquid in a saturated permeable medium. J. Fluid Mech. 2, 273–285 (1957)
Wooding, R.A., Tylor, S.W., White, I.: Convection in groundwater below an evaporating salt lake: 1. Onset of instability. Water Resour. Res. 33(6), 1199–1217 (1997)
Wooding, R.A., Tylor, S.W., White, I., Anderson, P.A.: Convection in groundwater below an evaporating salt lake: 2. Evolution of fingers and plumes. Water Resour. Res. 33(6), 1219–1228 (1997)
Xu, X., Chen, S., Zhang, D.: Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers. Adv. Water Resour. 29, 397–407 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Elenius, M.T., Johannsen, K. On the time scales of nonlinear instability in miscible displacement porous media flow. Comput Geosci 16, 901–911 (2012). https://doi.org/10.1007/s10596-012-9294-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-012-9294-2