Abstract
Upscaling is a major issue regarding mechanical and transport properties of rocks. This paper examines three issues relative to upscaling. The first one is a brief overview of Effective Medium Theory (EMT), which is a key tool to predict average rock properties at a macroscopic scale in the case of a statistically homogeneous medium. EMT is of particular interest in the calculation of elastic properties. As discussed in this paper, EMT can thus provide a possible way to perform upscaling, although it is by no means the only one, and in particular it is irrelevant if the medium does not adhere to statistical homogeneity. This last circumstance is examined in part two of the paper. We focus on the example of constructing a hydrocarbon reservoir model. Such a construction is a required step in the process of making reasonable predictions for oil production. Taking into account rock permeability, lithological units and various structural discontinuities at different scales is part of this construction. The result is that stochastic reservoir models are built that rely on various numerical upscaling methods. These methods are reviewed. They provide techniques which make it possible to deal with upscaling on a general basis. Finally, a last case in which upscaling is trivial is considered in the third part of the paper. This is the fractal case. Fractal models have become popular precisely because they are free of the assumption of statistical homogeneity and yet do not involve numerical methods. It is suggested that using a physical criterion as a means to discriminate whether fractality is a dream or reality would be more satisfactory than relying on a limited data set alone.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ababou R. (1988) Three-dimensional flow in random porous media. Ph.D. dissertation, M.I.T., Cambridge, MA, USA.
Boe, O. (1994), Analysis of an upscaling method based on conservation of dissipation, Transport in Porous Media 17, 77–86.
Cardwell, W., and Parsons, R.L. (1945), Average permeabilities of heterogeneous oil sands, Trans. Am. Inst. Mining. Met. Pet. Eng. 34–42.
Chen, Y., Durlofsky, L.J., Gerritsen, M., and Wen, X.H. (2003), A coupled local-global approach for simulating flow in highly heterogeneous formations, Advances Water Resources 26, 1041–1060.
Durlofsky, L.J. (1991), Numerical calculation of equivalent grid block permeability tensors for heterogeneous media, Water Resources Res. 27, 699–708.
Durlofsky, L.J. (2003), Upscaling of geocellular models for reservoir flow simulation: A review of recent progress, Proc. 7th International Forum on Reservoir Simulation, Bühl/Baden-Baden, Germany.
Gallouët, T., and Guérillot, D. (1994), Averaged heterogeneous porous media by minimisation of error on the flow rate, 4th Europ. Conf. Math. Oil Rec.
Gelhar, L.W., and Axness, C.L. (1983), Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resources Research, 19, 161–180.
Gomez-Hernández, J.J., and Journel, A.G. (1994), Stochastic characterization of grid block permeability, SPE Form. Eval. 9(2), 93–99.
Gubernatis, J.E., and Krumhansl, J.A. (1975), Macroscopic engineering properties of polycristalline materials: Elastic properties, J. Appl. Phys. 46(5), 1875–1883.
Guéguen, Y., Chelidze, T., and Le Ravalec, M. (1997), Microstructures, percolation thresholds and rock physical properties, Tectonophysics 2798, 23–35.
Guéguen, Y., David, C., and Gavrilenko, P. (1991), Percolation and fluid transport in the crust, Geophys. Res. Lett. 18,5, 931–934.
Guéguen, Y., and Palciauskas, V., Introduction to the Physics of rocks (Princeton Univ. Press, 1994) 294 pp.
Hashin, Z., and Shtrikman, S. (1962), A variational approach to the theory of the elastic behaviour of polycristals, J. Mech. Phys. Sol. 10, 343–352.
Holden, L., and Nielsen, B.F. (2000), Global upscaling of permeability in heterogeneous reservoirs: The Output Lest Squares (OLS) methods, Transp. Porous Media 40, 115–143.
Holden, L. and Lia, O. (1992), A tensor estimator for the homogeneization of absolute permeability, Transp. Porous Media 8(1), 37–46.
Hou, T.Y., and Wu, X.H. (1997), A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comp. Phys. 134(1), 169–189.
Indelman, P., and Dagan, G. (1993), Upscaling of permeability of anisotropic heterogeneous formations, 1-The general framework, Water Resour. Res. 29(4), 917–923.
Kachanov, M. (1992), Effective elastic properties of cracked solids: critical review of some basic concepts, Appl. Mech. Rev. 45, 304–335.
Kachanov, M. (1993), Elastic solids with many cracks and related problems, Advan. Appl. Mechan. 30, 259–445.
Kardar, M., Parisi, G., and Yi-Chen Zhang (1986), Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56, 889.
Katz, A.J., and Thompson, A.H. (1985), Fractal sandstone pores: Implications for conductivity and pore formation, Phys. Rev. Lett. 54, 1325.
King, M.J., Application and analysis of tensor permeability to crossbedded reservoirs, SPE Paper 26118, Western Regional Meeting (Anchorage, Alaska, USA, 1993).
Le Ravalec, M., and Guéguen, Y. (1996), High and low frequency elastic moduli for saturated porous cracked rock, differential self consistent and poroelastic theories, Geophys. 61, 1080–1094.
Madden, T.R. (1983), Microcrack connectivity in rocks: a renormalization group approach to the critical phenomena of conduction and failure incrystalline rocks, J. Geophys. Res. 88, 585–592.
Matheron, G., Eléments pour une théorie des milieux poreux (Masson, eds, Paris, 1967).
Pickup, G.E., Jensen, J.L., Ringrose, P.S., and Sorbie, K.S. (1992), A method for calculating permeability tensors using perturbed boundary conditions, 3rd Europ. Conf. Math. Oil Rec. 225–238.
Pickup, G.E., Ringrose, P.S., and Sorbie, K.S. (1994), Permeability tensors for sedimentary structures, Math. Geol. 26(2), 227–250.
Renard, P., and De Marsily, G. (1997), Calculating equivalent permeability: A review, Advan. Water Resources 20(5–6), 253–278.
Ricard, L., Le Ravalec-Dupin, M., and Guéguen, Y. (2005), submitted.
Rubin, Y. and Gomez-Hernandez, J.J. (1990), A stochastic approach to the problem of upscaling of conductivity in disordered media: Theory and unconditional numerical simulations, Water Resources Res. 26(4), 691–701.
Schubnel, A., and Guéguen, Y. (2003), Dispersion and anisotropy of elastic waves in cracked rocks, J. Geophys. Res. 108, 2101–2116.
Stauffer, D., and Aharony, A., Introduction to Percolation Theory (Taylor and Francis, London, 1992) 181 pp.
Warren, J., and Price H. (1961), Flow in heteregeneous porous media, Soc. Petrol. Engin. J. 1, 153–169.
Wen, X.H., Durlofsky, L.J., and Edwards, M.G. (2003), Use of border regions for improved permeability upscaling, Math. Geology 35, 521–547.
Wen, X.H., and Gomez-Hernandez, J.J. (1996), Upscaling hydraulic conductivities in heterogeneous media: An overview, J. Hydrol. 183(1–2), ix–xxxii.
White, C.D., and Horne, R.N., Computing absolute transmissibility in the presence of fine scale heterogeneity, SPE 16011, SPE Symp. Reservoir Simulation, San Antonio, TX, 1987).
Wilson, K.G. (1971), Renormalization group and critical phenomena, Phys. Rev. B4, 3174–3205.
Wu, X.H., Efendiev, Y.R., and Hou, T.Y. (2002), Analysis of upscaling absolute permeability, Discrete Contin. Dyn. Syst., Ser. B 2(2), 185–204.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag, Basel
About this chapter
Cite this chapter
Guéguen, Y., Le Ravalec, M., Ricard, L. (2006). Upscaling: Effective Medium Theory, Numerical Methods and the Fractal Dream. In: Dresen, G., Zang, A., Stephansson, O. (eds) Rock Damage and Fluid Transport, Part I. Pageoph Topical Volumes. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7712-7_15
Download citation
DOI: https://doi.org/10.1007/3-7643-7712-7_15
Received:
Revised:
Accepted:
Published:
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7711-3
Online ISBN: 978-3-7643-7712-0
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)