Fast upscaling of the hydraulic conductivity of three-dimensional fractured porous rock for reservoir modeling. (English) Zbl 1428.86007
Summary: A fast upscaling procedure for determining the equivalent hydraulic conductivity of a three-dimensional fractured rock is presented in this paper. A modified semi-analytical superposition method is developed to take into account, at the same time, the hydraulic conductivity of the porous matrix \((K_M)\) and the fractures \((K_F)\). The connectivity of the conductive fracture network is also taken into account. The upscaling approach has been validated by comparison with the hydraulic conductivity of synthetic samples calculated with full numerical procedures (flow simulations and averaging). The extended superposition approach is in good agreement with numerical results for infinite size fractures. For finite size fractures, an improved model that takes into account the connectivity of the fracture network through multiplicative connectivity indexes determined empirically is proposed. This improved model is also in good agreement with the numerical results obtained for different configurations
of fracture networks.
MSC:
| 86A05 | Hydrology, hydrography, oceanography |
| 74L10 | Soil and rock mechanics |
| 74R10 | Brittle fracture |
Keywords:
fractured porous medium; three-dimensional upscaling; equivalent hydraulic conductivity; numerical simulations; Darcy; random fracture sets; network connectivitySoftware:
BIGFLOWReferences:
| [1] | Ababou R (1990) Identification of effective conductivity tensor in randomly heterogeneous and stratified aquifers. In: Bachu S (ed) Proceedings Fifth Canadian-American conference on hydrogeology: parameter identification and estimation for aquifers and reservoirs, Calgary, Alberta, Canada, Sep. 18-20, 1990. Nat. Water Well Assoc., Water Well Journal Publish. Co., Dublin, Ohio, 1990, pp 155-157 |
| [2] | Ababou R, Bagtzoglou AC(1993) BIGFLOW: a numerical code for simulating flow in variably saturated, heterogeneous geologic media (theory & user’s manual, Version 1.1). Report NUREG/CR-6028, U.S. Nuclear Regul. Commission, Gov. Printing Office, Washington, DC |
| [3] | Ababou R., Renard P (2011) Equivalent permeability tensor in fractured media: an algebraic approach. In: Amaziane B, Barrera D, Mraoui H, Rodriguez ML, Sbibih D (eds) Proceedings MAMERN11: 4th internat. conf. on approx. methods and numer. model. in envir. and natur. resour. (Saïdia, Morocco, May 23-26, 2011). Univ. Granada (2011), ISBN: 078-84-338-5230-4, 2011 |
| [4] | Ababou, R.; McLaughlin, D.; Gelhar, LW; Tompson, AF, Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media, Transp Porous Media, 4, 549-565 (1989) · doi:10.1007/BF00223627 |
| [5] | Ababou, R.; Millard, A.; Treille, E.; Durin, M.; Plas, F.; Peters, XA (ed.); etal., Continuum modeling of coupled thermo-hydro-mechanical processes in fractured rock, 651-658 (1994), Netherlands · doi:10.1007/978-94-010-9204-3_79 |
| [6] | Ababou, Rachid; Cañamón Valera, Israel; Poutrel, Adrien, Macro-permeability distribution and anisotropy in a 3D fissured and fractured clay rock: ‘Excavation Damaged Zone’ around a cylindrical drift in Callovo-Oxfordian Argilite (Bure), Physics and Chemistry of the Earth, Parts A/B/C, 36, 1932-1948 (2011) · doi:10.1016/j.pce.2011.07.032 |
| [7] | Adler PM, Thovert J-F (1999) Fractures and fracture networks. Kluwer Academic Publishers, Dordrecht · doi:10.1007/978-94-017-1599-7 |
| [8] | Adler PM, Thovert J-F, Mourzenko VV (2012) Fractured porous media. Oxford University Press, Oxford · doi:10.1093/acprof:oso/9780199666515.001.0001 |
| [9] | Balberg, I.; Anderson, CH; Alexander, S.; Wagner, N., Excluded volume and its relation to the onset of percolation, Phys Rev B, 30, 3933 (1984) · doi:10.1103/PhysRevB.30.3933 |
| [10] | Bamberger A (1977) Approximation des coefficients d’opérateurs elliptiques, stables pour la G-convergence. Rapport du Centre de mathématiques appliquées, École polytechnique, n° MAP/15 |
| [11] | Berkowitz, B.; Adler, PM, Stereological analysis of fracture network structure in geological formations, J Geophys Res [Solid Earth], 103, 15339-15360 (1998) · doi:10.1029/98JB01072 |
| [12] | Bouwer, H., Planning and interpreting soil permeability measurements, J Irrig Drain Div, ASCE, 95, 391-402 (1969) |
| [13] | Brown, SR, Fluid flow through rock joints: the effect of surface roughness, J Geophys Res, 92, 1337-1347 (1987) · doi:10.1029/JB092iB02p01337 |
| [14] | Budiansky, B., On the elastic moduli of some heterogeneous materials, J Mech Phys Solids, 13, 223 (1965) · doi:10.1016/0022-5096(65)90011-6 |
| [15] | Cañamón I (2006) Analysis and modeling of coupled thermo-hydro-mechanical phenomena in three-dimensional fractured media. PhD thesis, Institut National Polytech. de Toulouse & Univ. Politécnica de Madrid |
| [16] | Cardwell, WT; Parsons, RL, Average permeabilities of heterogeneous oil sands, Trans Am Inst Mining Metall Pet Eng, 160, 34-42 (1945) |
| [17] | Charlaix, E.; Guyon, E.; Rivier, N., A criterion for percolation threshold in a random array of plates, Solid State Commun, 50, 999-1002 (1984) · doi:10.1016/0038-1098(84)90274-6 |
| [18] | Dagan, G., Models of groundwater flow in statistically heterogeneous porous formations, Water Resour Res, 15, 47-63 (1979) · doi:10.1029/WR015i001p00047 |
| [19] | Desbarats, AJ, Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media, Math Geol, 24, 249-267 (1992) · doi:10.1007/BF00893749 |
| [20] | Deutsch, C., Calculating effective absolute permeability in sandstone/shale sequences, SPE Form Eval, 4, 343-348 (1989) · doi:10.2118/17264-PA |
| [21] | Dimitrakopoulos, R.; Desbarats, AJ, Geostatistical modelling of grid block permeabilities for 3D reservoir simulators, SPE Reservoir Eng, 8, 13-18 (1997) · doi:10.2118/21520-PA |
| [22] | Farmer, CL, Upscaling: a review, Int J Numer Meth Fluids, 40, 63-78 (2002) · Zbl 1058.76574 · doi:10.1002/fld.267 |
| [23] | Hashin, Z.; Shtrikman, S., A variational approach to the theory of elastic behaviour of multiphase materials, J Mech Phys Solid, 11, 127-140 (1963) · Zbl 0108.36902 · doi:10.1016/0022-5096(63)90060-7 |
| [24] | Journel AG, Deutsch, C, Debarats AJ (1986) Power averaging for block effective permeability: SPE 15128, presented at the 56th California Regeonal Meeting of the SPE, Oakland, California, April 2-4, 1986 |
| [25] | Kfoury, M.; Ababou, R.; Noetinger, B.; Quintard, M., Upscaling fractured heterogeneous media: permeability and mass exchange coefficient, J Appl Mech (JAM), Trans ASME, 73, 41-46 (2006) · Zbl 1111.74480 · doi:10.1115/1.1991864 |
| [26] | Kiraly, L., Anisotropie et hétérogénéité de la perméabilité dans les calcaires fissurés, Eclogae Geol Helv, 62, 613-619 (1969) |
| [27] | Lang, PS; Paluszny, A.; Zimmerman, RW, Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions, J Geophys Res Solid Earth, 119, 6288-6307 (2014) · doi:10.1002/2014JB011027 |
| [28] | Le Loc’h, G (1988) An efficient strategy for combining the permeabilities: practical application on a simulated reservoir. In: Proc. of the 3rd internat. Geostatistics congress, Avignon, Sept 5-9 |
| [29] | Li, L.; Zhou, H.; Gómez-Hernández, JJ, A comparative study of three dimensional hydraulic conductivity upscaling at the macro-dispersion experiment (MADE) site, Columbus Air Force Base, Mississippi (USA), J Hydrol, 404, 278-293 (2011) · doi:10.1016/j.jhydrol.2011.05.001 |
| [30] | Long, JCS; Remer, JS; Wilson, CR; Witherspoon, PA, Porous media equivalents for networks of discontinuous fractures, Water Resour Res, 18, 645-658 (1982) · doi:10.1029/WR018i003p00645 |
| [31] | Marchant.J (1977) Sur la résistance équivalente d’un réseau aléatoire de structure irrégulière. CR Acad Sci Paris, t.284, Série B-85:88 |
| [32] | Matheron G (1967) Eléments pour une Théorie des Milieux Poreux. Masson et Cie, Paris, p 166 |
| [33] | Mourzenko, VV; Thovert, JF; Adler, PM, Percolation of three-dimensional fracture networks with power-law size distribution, Phys Rev E, 72, 036103 (2005) · doi:10.1103/PhysRevE.72.036103 |
| [34] | Mourzenko V, Thovert J-F, Adler PM (2009) Proceedings of the international conference on rock joints and jointed rock masses, Tucson, Arizona |
| [35] | Oda, M., Permeability tensor for discontinuous rock masses, Géotechnique, 35, 483-495 (1985) · doi:10.1680/geot.1985.35.4.483 |
| [36] | Oda, M., An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock masses, Water Resour Res, 22, 1845-1856 (1986) · doi:10.1029/WR022i013p01845 |
| [37] | Pouya, A.; Fouché, O., Permeability of 3D discontinuity networks: new tensors from boundary-conditioned homogenization, Adv Water Resour, 32, 303-314 (2009) · doi:10.1016/j.advwatres.2008.08.004 |
| [38] | Pozdniakov, S.; Tsang, C-F, A self-consistent approach for calculating the effective hydraulic conductivity of a binary, heterogeneous medium, Water Resour Res, 40, w05105 (2004) · doi:10.1029/2003WR002617 |
| [39] | Renard, P.; Marsily, G., Calculating equivalent permeability: A review, Adv Water Resour, 20, 253-278 (1997) · doi:10.1016/S0309-1708(96)00050-4 |
| [40] | Renard P, Ababou R (2009) Relation between the definition and properties of the equivalent permeability tensor in heterogeneous and fractured porous media. In: Amaziane B et al (eds) Proceedings MAMERN 09: 3rd international conference on approximation methods and numerical modeling in environment and natural resources (Pau, France, 8-11 June 2009), Editorial Univ. de Granada, ISBN: 978-84-338-5006-5 |
| [41] | Renard, Ph; Le Loc’h, G.; Ledoux, E.; de Marsily, G.; Mackay, R., A fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous porous media, Water Resour Res, 36, 3567-3580 (2000) · doi:10.1029/2000WR900203 |
| [42] | Sævik, PN; Berre, I.; Jakobsen, M.; Lien, M., A 3D computational study of effective medium methods applied to fractured media, Transp Porous Media, 100, 115-142 (2013) · doi:10.1007/s11242-013-0208-0 |
| [43] | Sahimi M (1995) Flow and transport in porous media and fractured rock. VCH, New York · Zbl 0849.76001 |
| [44] | Snow, DT, Anisotropic permeability of fractured media, Water Resour Res, 5, 1273-1289 (1969) · doi:10.1029/WR005i006p01273 |
| [45] | Tsang, YW, The effect of tortuosity on fluid flow through a single fracture, Water Resour Res, 20, 1209-1215 (1984) · doi:10.1029/WR020i009p01209 |
| [46] | Vanmarcke E (1983) Random fields (analysis and synthesis). The MIT Press, Cambridge · Zbl 1206.60051 |
| [47] | Warren, JE; Price, HS, Flow in heterogeneous porous media, Soc Pet Eng J, 1, 153-169 (1961) · doi:10.2118/1579-G |
| [48] | Wen, XH; Gomez-Hernandez, JJ, Upscaling hydraulic conductivities in heterogeneous media: an overview, J Hydrol, 183, ix-xxxii (1996) · doi:10.1016/S0022-1694(96)80030-8 |
| [49] | Wiener, O., Abh. Math.-Phys. Klasse Königlich Sächsischen Des Wiss, Leipzig, 32, 509-604 (1912) |
| [50] | Zinn, B.; Harvey, CF, When good statistical models of aquifer heterogeneity go bad: a comparison of flow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields, Water Resour Res, 39, 1051 (2003) · doi:10.1029/2001WR001146 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
