Abstract
This article presents a mathematical model describing flow of two fluid phases in a heterogeneous porous medium. The medium contains disconnected inclusions embedded in the background material. The background material is characterized by higher value of the non-wetting-phase entry pressure than the inclusions, which causes non-standard behavior of the medium at the macroscopic scale. During the displacement of the non-wetting fluid by the wetting one, some portions of the non-wetting fluid become trapped in the inclusions. On the other hand, if the medium is initially saturated with the wetting phase, it starts to drain only after the capillary pressure exceeds the entry pressure of the background material. These effects cannot be represented by standard upscaling approaches based on the assumption of local equilibrium of the capillary pressure. We propose a relevant modification of the upscaled model obtained by asymptotic homogenization. The modification concerns the form of flow equations and the calculation of the effective hydraulic functions. This approach is illustrated with two numerical examples concerning oil–water and CO2–brine flow, respectively.
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Amaziane B., Bourgeat A., Koebbe J.: Numerical simulation and homogenization of two-phase flow in heterogeneous porous media. Transport Porous Med. 6(5–6), 519–547 (1991)
Barker J., Thibeau S.: A critical review of the use of pseudo relative permeabilities for upscaling. SPE Reservoir Eng. 12(2), 138–143 (1997)
Braun C., Helmig R., Manthey S.: Macro-scale effective constitutive relationships for two phase flow processes in heterogeneous porous media with emphasis on the relative permeability-saturation relationship. J. Contam. Hydrol. 76(1–2), 47–85 (2005)
Brooks, R., Corey, A.: Hydraulic Properties of Porous Media. Hydrology Paper 3. Colorado State University, Fort Collins, CO
Burdine N.: Relative permeability calculations from pore size distribution data. Trans. Am. Inst. Min. Metall. Petrol. Eng. 198, 71–77 (1953)
Eichel H., Helmig R., Neuweiler I., Cirpka O.: Upscaling of two-phase flow processes in porous media. In: Das, D., Hassanizadeh, S. (eds) Upscaling Multiphase Flow in Porous Media, pp. 237–257. Springer, Dordrecht (2005)
Ekrann S., Aasen J.: Steady-state upscaling. Transport Porous Med. 41(3), 245–262 (2000)
Flemisch, B., Fritz, J., Helmig, R., Niessner, J., Wohlmuth, B.: DUMUX: a multiscale multi-physics toolbox for flow and transport processes in porous media. In: Ibrahimbegovic, A., Dias, F., Matthies, H., Wriggers, P. (eds.) ECCOMAS Thematic Conference on Multi-scale Computational Methods for Solids and Fluids. Cachan, France, November 28–30 (2007)
Helmig R.: Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of the Hydrosystems. Springer, Berlin (1997)
Jonoud S., Jackson M.: New criteria for the validity of steady-state upscaling. Transport Porous Med. 71(1), 53–73 (2008)
Kopp A., Class H., Helmig R.: Investigations on CO2 storage capacity in saline aquifers—Part 1: dimensional analysis of flow processes and reservoir characteristics. Int. J. Greenhouse Gas Control 3(3), 263–276 (2009a). doi:10.1016/j.ijggc.2008.10.002
Kopp A., Class H., Helmig R.: Investigations on CO2 storage capacity in saline aquifers—Part 1: estimation of storage capacity coefficients. Int. J. Greenhouse Gas Control 3(3), 277–287 (2009b). doi:10.1016/j.ijggc.2008.10.001
Lewandowska J., Laurent J.P.: Homogenization modeling and parametric study of moisture transfer in an unsaturated heterogeneous porous medium. Transport Porous Med. 45(3), 321–345 (2001)
Mualem Y.: A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12(3), 513–522 (1976). doi:10.1029/WR012i003p00513
Quintard M., Whitaker S.: Two-phase flow in heterogeneous porous media: the method of large scale averaging. Transport Porous Med. 3(4), 357–413 (1988)
Saadatpoor E., Bryant S., Sepehrnoori K.: Effect of capillary heterogeneity on buoyant plumes: a new local trapping mechanism. Energy Procedia 1, 3299–3306 (2009a). doi:10.1016/j.egypro.2009.02.116
Saadatpoor, E., Bryant, S., Sepehrnoori, K.: New trapping mechanism in carbon sequestration. Transport Porous Med. (2009b). doi:10.1007/s11242-009-9446-6
Saez A., Otero C., Rusinek I.: The effective homogeneous behavior of heterogeneous porous media. Transport Porous Med. 4(3), 213–238 (1989)
Span R., Wagner W.: A new equation of state for carbon dioxide covering the fluid region from the triple point temperature to 1100 K at pressures up to 800 MPa. J. Phys. Chem. Ref. Data 25, 1509–1596 (1996)
Stephen K., Pickup G., Sorbie K.: The local analysis of changing force balances in immiscible incompressible twophase flow. Transport Porous Med. 45(1), 63–88 (2001)
Szymkiewicz A.: Calculating effective conductivity of heterogeneous soils by homogenization. Arch. Hydro-eng. Environ. Mech. 52(2), 111–130 (2005)
van Duijn C., Mikelic A., Pop I.: Effective equations for two phase flow with trapping on the micro scale. SIAM J. Appl. Math. 62, 531–1568 (2002)
van Duijn C., Eichel H., Helmig R., Pop I.: Effective equations for two-phase flow in porous media: the effect of trapping at the micro-scale. Transport Porous Med. 69(3), 411–428 (2007). doi:10.1007/s11242-006-9089-9
van Genuchten M.: A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980)
Vasin M., Lehmann P., Kaestner A., Hassanein R., Nowak W., Helmig R., Neuweiler I.: Drainage in heterogeneous sand columns with different geometric structure. Adv. Water Res. 31(9), 1205–1220 (2008)
Vereecken H., Kasteel R., Vanderborght J., Harter T.: Upscaling hydraulic properties and soil water flow processes in heterogeneous soils: a review. Vadose Zone J. 6(1), 1–28 (2007)
Virnovsky G., Friis H., Lohne A.: A steady-state upscaling approach for immiscible two-phase flow. Transport Porous Med. 54(2), 167–192 (2004)
Acknowledgements
The author A.S. would like to thank the German Research Foundation (DFG) for providing the financial support for the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. In addition, support from the International Research Training Group “Nonlinearities and Upscaling in Porous Media” (NUPUS) for the same author is kindly acknowledged here. The authors would like to thank Andreas Lauser for help in numerical implementation of the model, and two anonymous reviewers for their helpful suggestions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Szymkiewicz, A., Helmig, R. & Kuhnke, H. Two-Phase Flow in Heterogeneous Porous Media with Non-Wetting Phase Trapping. Transp Porous Med 86, 27–47 (2011). https://doi.org/10.1007/s11242-010-9604-x
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DOI: https://doi.org/10.1007/s11242-010-9604-x