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A solution for the transition zone isosats in two-phase primary drainage in the presence of gravity

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Abstract

Primary drainage in a water-wet saturated medium in the absence of capillarity is typically a combination of shock (discontinuous) and rarefaction (continuous) waves. Using nonlinear relative permeability functions for the host fluid and the invading fluid leads to the existence of a shock wave front, and the degree of nonlinearity of the relative permeability functions has an inverse relationship with the size of the shock wave (i.e., difference of saturation between upstream and downstream of the shock wave), whereas for linear relative permeability functions, the shock wave size approaches 0. Injection of a lower-viscosity immiscible phase such as gas or solvent into a water-wet porous medium in the presence of large capillary pressure leads to development of an extended and growing saturation transition zone that follows the discontinuous shock wave front. In this article, a semianalytical solution for the position of equisaturation contours (isosats) in the transition zone in the presence of gravity is obtained for a set of linearized relative permeability functions. The capillary (diffusive) and buoyancy terms are neglected, and the generalized convective equation for mass conservation is obtained. The set of equations is then reduced to a one-dimensional steady-state differential equation through forcing the isosat formulation to obey mass conservation. This scheme allows the isosat distribution to be solved, and the case of injection into an axisymmetric geometry for a confined planar configuration is solved and presented. A finite element model was developed to demonstrate the reasonable agreement between analytical and numerical solutions.

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References

  1. Jenkinson, D.S., Adams, D.E., Wild, A.: Model estimates of CO2 emissions from soil in response to global warming. Nature. 351, 304–306 (1991)

    Article  Google Scholar 

  2. Stevens, S.H., Kuuskraa, V.A., Gale, J., Beecy, D.: CO2 injection and sequestration in depleted oil and gas fields and deep coal seams: worldwide potential and costs. Environ. Geosci. 8(3), 200–209 (2008)

    Google Scholar 

  3. Whitaker, S.: Flow in porous media II: the governing equations for immiscible, two-phase flow. Transp. Porous Media. 1(2), 105–125 (1986)

    Article  Google Scholar 

  4. Chen, Z.: Some invariant solutions to two-phase fluid displacement problems including capillary effect. SPE Reserv. Eng. 3(2), 691–700 (1988)

    Google Scholar 

  5. Doughty, C., Karsten, P.: A similarity solution for two-phase water, air, and heat flow near a linear heat source in a porous medium. J. Geophys. Res. 97(B2), 1821–1838 (1992)

    Article  Google Scholar 

  6. Pascal, H.: Similarity solutions to some two-phase flows through porous media. Int. J. Non-Lin. Mech. 27(4), 565–573 (1992)

    Article  Google Scholar 

  7. Nordbotten, J.M., Celia, M.A.: Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech. 561, 307–327 (2006)

    Article  Google Scholar 

  8. Barenblatt, I.G., Entov, V.M., Ryzhik, V.M.: Theory of fluid flow in natural rock. Springer, Boston (1990)

    Book  Google Scholar 

  9. Holden, L.: On the strict hyperbolicity of the Buckley–Leverett equation for three phase flow in porous media. SIAM J. Appl. Math. 50(3), 667–682 (1990)

    Article  Google Scholar 

  10. Juanes, R., Patzek, T.: Analytical solution to the Riemann problem of three-phase flow in porous media. Transp. Prous Media. 55(4), 47–70 (2004)

    Article  Google Scholar 

  11. Smoller, J.: Shock waves and reaction–diffusion equations. Springer-Verlag, New York (1983)

    Book  Google Scholar 

  12. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Springer, Berlin (2000)

    Book  Google Scholar 

  13. Plug, W.J., Bruining, J.: Capillary pressure for the sand–CO2–water system under various pressure conditions. Application to CO2 sequestration. Adv. Water Resour. 30(11), 2339–2353 (2007)

    Article  Google Scholar 

  14. Yortsos, Y.C.: A theoretical analysis of vertical flow equilibrium. Transp. Porous Media. 18(2), 107–129 (1995)

    Article  Google Scholar 

  15. Gasda, E.S., Nordbotten, J.M., Celia, M.A.: Vertical equilibrium with sub-scale analytical methods for geological CO2 sequestration. Comput. Geosci. 13(4), 469–481 (2009)

    Article  Google Scholar 

  16. Nordabotten, J.M., Celia, M.A., Bachu, A.: Injection and storage of CO2 in deep saline aquifers. Transp. Porous Media. 58, 339–360 (2005)

    Article  Google Scholar 

  17. Juanes, R., MacMinn, C., Szulczewski, M.: The footprint of the CO2 plume during carbon dioxide storage in saline aquifers: storage efficiency for capillary trapping at the basin scale. Trans. Porous Media. 82(1), 19–30 (2009)

    Article  Google Scholar 

  18. Nordbotten, J.M.D.H.K.: Impact of the capillary fringe in vertically integrated models for CO2 storage. Water Resour. Res. 47(W02537), 11 (2009)

    Google Scholar 

  19. Golding, M.J., Neufeld, J.A., Hesse, M.A.: Two-phase gravity currents in porous media. J. Fluid Mech. 678, 248–270 (2011)

    Article  Google Scholar 

  20. Buckley, S.E., Leverett, M.C.: Mechanism of fluid displacement in sands. Trans. AIME, 1942

  21. Blunt, M., King, P.: Relative permeabilities from two- and three-dimensional pore-scale network modeling. Transp. Porous Media. 6(4), 407–433 (2004)

    Google Scholar 

  22. Taggart, I.: Extraction of dissolved methane in brine by CO2 injection: implication for CO2 sequestration. SPE J. 791–804 (2010)

  23. Hosseini, S.A., Simon A.M., Farzam J.: Analytical model for CO2 injection into brine aquifers-containing residual CH4. Transp. Porous Media. 94(3), 795–815 (2012)

    Article  Google Scholar 

  24. Malekzadeh, F.A., Pak, A.: A discretized analytical solution for fully coupled non-linear simulation of heat and mass transfer in poroelastic unsaturated media. Int. J. Anal. Num. Methods Geomech. 33(18), 1589–1611 (2009)

    Article  Google Scholar 

  25. Brooks, A.N., Hughes, J.R.T.: Streamline upwindin/Petrov–Galerkin formulations for convection dominated flows with particular emohasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982)

    Article  Google Scholar 

  26. Yin, S., Dusseault, M.B., Rothenburg, L.: Thermal reservoir modeling in petroleum geomechanics. Int. J. Anal. Num. Methods Geomech. 33(4), 449–485 (2009)

    Article  Google Scholar 

  27. Settari, T., Aziz, K.: Petroleum reservoir simulation. Applied Science, London (1979)

    Google Scholar 

  28. Chen, Z., Haun, G.: Computational methods for multiphase flows in porous media. SIAM, London (2006)

    Book  Google Scholar 

  29. Brooks, R.H., Corey, A.T.: Properties of porous media affecting fluid flow. J. Irrig. Drain. Div. 92(IR2), 61–88 (1966)

    Google Scholar 

  30. Huppert, H.E., Woods, A.W.: Gravity-driven flows in porous layers. J. Fluid Mech. 292, 55–69 (1995)

    Article  Google Scholar 

  31. Lake, L.W.: Recovery, enhanced oil. Prentice Hall Inc., New Jersey (1989)

    Google Scholar 

  32. Hosseini, S.A., Nicot, J.-P.: Numerical modeling of a multiphase water–oil–CO2 system using a water–CO2 system: application to the far field of a US Gulf Coast reservoir. Int. J. Greenh. Gas Control. 10, 88–99 (2012)

    Article  Google Scholar 

  33. Mathias, S.A., Gluyas, J.G., González Martínez de Miguel, G.J., Hosseini, S.A.: Role of partial miscibility on pressure buildup due to constant rate injection of CO2 into closed and open brine aquifers. Water Resour. Res. 47(12) (2011)

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Canada

    Farshad A. Malekzadeh & Maurice B. Dusseault

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  1. Farshad A. Malekzadeh
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  2. Maurice B. Dusseault
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Correspondence to Farshad A. Malekzadeh.

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Malekzadeh, F.A., Dusseault, M.B. A solution for the transition zone isosats in two-phase primary drainage in the presence of gravity. Comput Geosci 17, 757–771 (2013). https://doi.org/10.1007/s10596-013-9354-2

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