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Pore to continuum upscaling of permeability in heterogeneous porous media using mortars

Published Online:pp 249-266https://doi.org/10.1504/IJOGCT.2012.046323

Pore-scale modelling has become an accepted method for estimating macroscopic properties (such as permeability) that describe flow and transport in porous media. In many cases extracted macroscopic properties compare favourably to experimental measurements. However, computational and imaging restrictions generally limit the network size to the order of 1.0 mm3 and these models often ignore effects of surrounding flow behaviour.

In this work permeability is upscaled in large (~106 pores), heterogeneous pore-scale network models using an efficient domain decomposition method. The large pore network is decomposed into 100 smaller networks (sub-domains) and then coupled with the surrounding models to determine accurate boundary conditions. Finite element mortars are used as a mathematical tool to ensure interfacial pressures and fluxes are matched at the network boundaries. The results compare favourably to the more computationally intensive (and impractical) approach of upscaling the medium as a single model. Additionally, the results are more accurate than straightforward hierarchical upscaling methods.

Keywords

pore-scale modelling, continuum scale, mortar coupling, upscaling, porous media, heterogeneous, multiscale modelling, network modelling

References

  • 1. Adler, P.M. , Jacquin, C.G. , Thovert, J. (1992). ‘The formation factor of reconstructed porous media’. Water Res.. 28, 6, 1571-1576 Google Scholar
  • 2. Al-Raoush, R. , Thompson, K.E. , Willson, C.S. (2003). ‘Comparison of network generation techniques for unconsolidated porous media’. Soil Sci. Soc. Am. J.. 67, 6, 1687-1700 Google Scholar
  • 3. Arbogast, T. , Cowsar, L.C. , Wheeler, M.F. , Yotov, I. (2000). ‘Mixed finite element methods on non-matching multi-block grids’. SIAM J. Numerical Analysis. 37, 4, 1295-1315 Google Scholar
  • 4. Arbogast, T. , Pencheva, G. , Wheeler, M.F. , Yotov, I. (2007). ‘A multiscale mortar mixed finite element method’. Multiscale Model. Simul.. 6, 1, 319-346 Google Scholar
  • 5. Bakke, S. , Oren, P.E. (1997). ‘3-D pore-scale modelling of sandstones and flow simulations in the pore networks’. SPE Journal. 2, 2, 136-149 Google Scholar
  • 6. Balhoff, M.T. , Thompson, K.E. (2004). ‘Modeling the steady flow of yield stress fluids in packed beds’. AICHE Journal. 50, 12, 3034-3048 Google Scholar
  • 7. Balhoff, M.T. , Wheeler, M.F. (2009). ‘A predictive pore scale model of non-darcy flow in porous media’. SPE J.. 14, 4, 579-587 Google Scholar
  • 8. Balhoff, M.T. , Thomas, S.G. , Wheeler, M.F. (2008). ‘Mortar coupling and upscaling of pore-scale models’. Computational Geosciences. 12, 1, 15-27 Google Scholar
  • 9. Balhoff, M.T. , Thompson, K.E. , Hjortso, M. (2007). ‘Coupling pore networks to continuum models’. Computers and Geosciences. 33, 3, 393-410 Google Scholar
  • 10. Bernardi, C. , Maday, Y. , Patera, A.T. (1994). ‘A new nonconforming approach to domain decomposition: the mortar element method. Non-linear partial differential equations and their applications’. 299, CollTege de France Seminar, Pitman Research Notes Mathematics Series, Harlow:Longman Science Technology , 13-51 Google Scholar
  • 11. Bhagmane, J. (2009). ‘Using mortars to upscale permeability in heterogeneous porous media from the pore to continuum scale’. University of Texas at Austin, MS thesis Google Scholar
  • 12. Bryant, S.L. , Mellor, D.W. , Cade, C.A. (1993). ‘Physically representative network models of transport in porous media’. AIChE Journal. 39, 3, 387-396 Google Scholar
  • 13. Chen, J. , Wilkinson, D. (1985). ‘Pore-scale viscous fingering in porous media’. Phys. Rev. Lett.. 55, 18, 1892-1895 Google Scholar
  • 14. Chen, S. , Doolen, G. (1998). ‘Lattice Boltzman method for fluid flows’. Annual Review of Fluid Mechanics. 30, 329-364 Google Scholar
  • 15. Christie, M. (1996). ‘Upscaling for reservoir simulation’. Journal of Petroleum Technology. 48, 11, 1004-1010 Google Scholar
  • 16. Durlofsky, L. (1998). ‘Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques’. Computational Geosciences. 2, 2, 73-92 Google Scholar
  • 17. Fatt, I. (1956). ‘The network model of porous media, I, Capillary pressure characteristics’. Pet. Trans. AIME. 207, 144-159 Google Scholar
  • 18. Koplik, J. , Lasseter, T.J. (1985). ‘Two-phase flow in random network models of porous media’. SPE Journal. 25, 1, 89-100 Google Scholar
  • 19. Liang, Z. , Ioannidis, A. , Chatzis, I. (2000). ‘Permeability and electrical conductivity of porous media from 3D replicas of the microstructure’. Chem. Eng. Sci.. 55, 22, 5247-5262 Google Scholar
  • 20. Lindquist, W.B. , Lee, S. , Coker, D.A. , Jones, K.W. , Spanne, P. (1996). ‘Medial axis analysis of void structure in three-dimensional tomographic images of porous media’. J. Geophys. Res.. 101, B4, 8297-8310 Google Scholar
  • 21. Lindquist, W.B. , Venkatarangan, A. , Dunsmuir, J. , Wong, T. (2000). ‘Pore and throat size distributions measured from synchrotron X-ray tomographic images of Fontainebleau sandstones’. J. Geophys. Res.. 105, B9, 21509-21528 Google Scholar
  • 22. Lopez, X. , Valvatne, P. , Blunt, M. (2003). ‘Predictive network modeling of single-phase non-Newtonian flow in porous media’. Journal of Colloid and Interface Science. 264, 1, 256-265 Google Scholar
  • 23. Payatakes, A.C. , Ng, K.M. , Flumerfelt, R.W. (1980). ‘Oil ganglion dynamics during immiscible displacement: model formulation’. AIChE Journal. 26, 3, 430-443 Google Scholar
  • 24. Peszynska, M. , Wheeler, M. , Yotov, I. (2002). ‘Mortar upscaling for multiphase flow in porous media’. Computational Geosciences. 6, 1, 73-100 Google Scholar
  • 25. Prodanovic, M. , Bryant, S.L. (2006). ‘A level set method for determining critical curvatures for drainage and imbibitions’. Journal of Colloid and Interface Science. 304, 2, 442-458 Google Scholar
  • 26. Sok, R.M. , Knackstedt, M.A. , Sheppard, A.P. , Pinczewski, W.V. , Lindquist, W.B. , Venkatarangan, A. , Paterson, L. (2002). ‘Direct and stochastic generation of network models from tomographic images. Effect of topology on residual saturations’. TIPM. 46, 2–3, 345-371 Google Scholar
  • 27. Tartakovsky, A. , Meakin, P. (2005). ‘Simulation of unsaturated flow in complex fractures using smoothed particle hydrodynamics’. Vadose Zone Journal. 4, 3, 848-855 Google Scholar
  • 28. Valvatne, P. , Blunt, M. (2004). ‘Predictive pore-scale modeling of two-phase flow in mixed wet media’. Water Resources Research. 40, 1-21 Google Scholar
  • 29. Venkatarangan, A.B. (2000). ‘Geometric and statistical analysis of porous media’. State University of New-York at Stony Brook, Thesis Google Scholar
  • 30. Wen, X.H. , Durlofsky, L.J. , Edwards, M.G. (2003). ‘Use of border regions for improved permeability upscaling’. Mathematical Geology. 35, 5, 521-547 Google Scholar
  • 31. Willson, C.S. , Al-Raoush, R.I. (2005). ‘Extraction of physically realistic pore network properties from three-dimensional synchrotron Xray microtomography images of unconsolidated porous media systems’. J. Hydrol.. 300, 1, 44-64 Google Scholar