Tensor Analysis of the Relative Permeability in Naturally Fractured Reservoirs
- Mohammad H. Sedaghat (University of Queensland) | Siroos Azizmohammadi (Montanuniversität Leoben) | Stephan K. Matthäi (University of Melbourne)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- February 2020
- Document Type
- Journal Paper
- 162 - 184
- 2020.Society of Petroleum Engineers
- relative permeability upscaling, naturally fractured reservoirs, relative permeability tensor
- 3 in the last 30 days
- 147 since 2007
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Fluid evidence shows that prediction of water breakthrough and oil recovery from fractured reservoirs cannot be performed accurately without upscaled relative permeability functions. Relative permeability is commonly assumed to be a scalar quantity, although the justification of that—specifically for naturally fractured reservoirs (NFRs)—is rarely attempted. In this study, we investigate the validity of this scalar-quantity assumption and how it affects fracture/matrix equivalent relative permeabilities, kri(Sw), achieved by a numerical simulation of unsteady-state waterflooding of discrete-fracture/matrix models (DFMs).
Numerical determination of relative permeability requires a realistic-model, a spatially adaptive simulation approach, and a sophisticated analysis procedure. To fulfil these requirements, we apply the discrete-fracture/matrix modeling to well-characterized outcrop analogs at the hectometer to kilometer scale. These models are parameterized with aperture and capillary entry pressure data, taking into account variations from fracture segment to segment, trying to emulate in-situ conditions. The finite-element-centered finite-volume method is used to simulate two-phase flow in the fractured rock, while also considering a range of wettability conditions from water-wet to oil-wet.
Our results indicate that the fracture/matrix equivalent relative permeability is a weakly anisotropic property. The tensors are not necessarily symmetric, and the absolute-permeability tensor is the most influential factor, determining the level of anisotropy of kri. The anisotropy ratio (AR) changes with saturation, is influenced by the fracture/matrix-interface wetted area (Awf), and differs for each phase. In addition, the diagonal terms of the equivalent relative permeability tensor (krii), determined using our novel approach, can be different from those obtained using the assumption that kri is scalar. The magnitude of the difference is controlled by the absolute permeability, wettability, flow rate, and orientation of the fractures in the model. It is worth mentioning that the type and direction of imbibition can be determined by off-diagonal terms of the kri tensor. Furthermore, krii largely depends on the direction of the waterflood along the i-axis.
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Ababou, R. 1988. Three-Dimensional Flow in Random Porous Media. PhD dissertation, Massachusetts Institute of Technology, Cambridge, Massachusetts (January 1988).
Akin, S. and Bagci, S. 2001. A Laboratory Study of Single-Well Steam-Assisted Gravity Drainage Process. J Pet Sci Eng 32 (1): 23–33. https://doi.org/10.1016/S0920-4105(01)00145-0.
Azizmohammadi, S. and Matthäi, S. K. 2017. Is the Permeability of Naturally Fractured Rocks Scale Dependent? Water Resour Res 53 (9): 8041–8063. https://doi.org/10.1002/2016WR019764.
Barenblatt, G. and Gil’man, A. 1987. Nonequilibrium Counterflow Capillary Impregnation. J Eng Phys Thermophys 52 (3): 335–339. https://doi.org/10.1007/BF00872519.
Barton, N., Bandis, S., and Bakhtar, K. 1985. Strength, Deformation and Conductivity Coupling of Rock Joints. Int J Rock Mech Min Sci 22 (3): 121–140. https://doi.org/10.1016/0148-9062(85)93227-9.
Batycky, J. P., McCaffery, F. G., Hodgins, P. K. et al. 1981. Interpreting Relative Permeability and Wettability From Unsteady-State Displacement Measurements. SPE J. 21 (3): 296–308. SPE-9403-PA. https://doi.org/10.2118/9403-PA.
Bear, J. 1972. Dynamics of Fluids in Porous Media. New York City: American Elsevier Publishing Company.
Bedrikovetsky, P. 1993. Mathematical Theory of Oil and Gas Recovery: With Applications to Ex-USSR Oil and Gas Fields. Springer Science and Business Media.
Belayneh, M., Geiger, S., and Matthäi, S. K. 2006. Numerical Simulation of Water Injection Into Layered Fractured Carbonate Reservoir Analogs. AAPG Bull 90 (10): 1473–1493. https://doi.org/10.1306/05090605153.
Bogdanov, I., Mourzenko, V., Thovert, J.-F. et al. 2003a. Two-Phase Flow Through Fractured Porous Media. Phys. Rev. E 68: 026703. https://doi.org/10.1103/PhysRevE.68.026703.
Bogdanov, I., Mourzenko, V., Thovert, J. F. et al. 2003b. Effective Permeability of Fractured Porous Media in Steady State Flow. Water Resour Res 39 (1): 1023. https://doi.org/10.1029/2001WR000756.
Bogdanov, I., Mourzenko, V., Thovert, J.-F. et al. 2007. Effective Permeability of Fractured Porous Media With Power-Law Distribution of Fracture Sizes. Phys. Rev. E 76: 036309. https://doi.org/10.1103/PhysRevE.76.036309.
Borisenko, A. I. and Tarapov, I. E. 1979. Vector and Tensor Analysis With Applications. Mineola, New York: Dover Publications.
Bosworth, W., Khalil, S., Clare, A. et al. 2014. Integration of Outcrop and Subsurface Data During the Development of a Naturally Fractured Eocene Carbonate Reservoir at the East Ras Budran Concession, Gulf of Suez, Egypt. Geol. Soc. London Spec. Publ. 374: 333–360. https://doi.org/10.1144/SP374.3.
Braun, E. M. and Blackwell, R. J. 1981. A Steady-State Technique for Measuring Oil-Water Relative Permeability Curves at Reservoir Conditions. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 4–7 October. SPE-10155-MS. https://doi.org/10.2118/10155-MS.
Brooks, R. H. and Corey, A. T. 1964. Hydraulic Properties of Porous Media. Hydrology Papers, No. 3, Colorado State University, Fort Collins, Colorado, USA.
Corey, A. T., Rathjens, C. H., Henderson, J. H. et al. 1956. Three-Phase Relative Permeability. J Pet Technol 8 (11): 63–65. SPE-737-G. https://doi.org/10.2118/737-G.
Dershowitz, W. and Miller, I. 1995. Dual Porosity Fracture Flow and Transport. Geophys Res Lett 22 (11): 1441–1444. https://doi.org/10.1029/95GL01099.
Durlofsky, L. 1991. Numerical Calculation of Equivalent Grid Block Permeability Tensors of Heterogeneous Porous Media. Water Resour Res 27 (5): 699–708. https://doi.org/10.1029/91WR00107.
Durlofsky, L., Milliken, W., Dehghani, K. et al. 1994. Application of a New Scale Up Methodology to the Simulation of Displacement Processes in Heterogeneous Reservoirs. Presented at the International Petroleum Conference and Exhibition of Mexico, Veracruz, Mexico, 10–13 October. SPE-28704-MS. https://doi.org/10.2118/28704-MS.
Durlofsky, L. J. 2005. Upscaling and Gridding of Fine Scale Geological Models for Flow Simulation. Oral presentation given at the 8th International Forum on Reservoir Simulation, Iles Borromees, Stresa, Italy, 20–24 June.
Eichel, H., Helmig, R., Neuweiler, I. et al. 2005. Upscaling of Two-Phase Flow Processes in Porous Media. In Upscaling Multiphase Flow in Porous Media: From Pore to Core and Beyond, ed. D. B. Das and S. M. Hassanizadeh, 237–257. Dordrecht, The Netherlands: Springer.
Elfeel, M. A. and Geiger, S. 2012. Static and Dynamic Assessment of DFN Permeability. Paper presented at the SPE Europec/EAGE Annual Conference, Copenhagen, Denmark, 4–7 June. SPE-154369-MS. https://doi.org/10.2118/154369-MS.
Firoozabadi, A. and Aziz, K. 1991. Relative Permeabilities From Centrifuge Data. J Can Pet Technol 30 (5): 33–42. PETSOC-91-05-02. https://doi.org/10.2118/91-05-02.
Fourar, M. and Lenormand, R. 2000. Inertial Effects in Two-Phase Flow Through Fractures. Oil Gas Sci Technol Rev. IFP 55 (3): 259–268. https://doi.org/10.2516/ogst:2000018.
Gallouët, T. and Guérillot, D. 1991. An Optimal Method for Averaging the Absolute Permeability. Oral presentation given at the 3rd International Reservoir Characterization Technical Conference, Tulsa, 3–5 November.
Gelhar, L. W. and Axness, C. L. 1983. Three-Dimensional Stochastic Analysis of Macrodispersion in Aquifers. Water Resour Res 19 (1): 161–180. https://doi.org/10.1029/WR019i001p00161.
Gerke, K. M., Karsanina, M. V., and Mallants, D. 2015. Universal Stochastic Multiscale Image Fusion: An Example Application for Shale Rock. Sci Rep 5, Article Number 15880. https://doi.org/10.1038/srep15880.
Guerillot, D., Rudkiewicz, J., Ravenne, C. et al. 1990. An Integrated Model for Computer Aided Reservoir Description: From Outcrop Study to Fluid Flow Simulations. Rev. Inst. Fr. Pét. 45 (1): 71–77. https://doi.org/10.2516/ogst:1990005.
Hagoort, J. 1980. Oil Recovery by Gravity Drainage. SPE J. 20 (3): 139–150. SPE-7424-PA. https://doi.org/10.2118/7424-PA.
Hassanpour, R., Manchuk, J., Leuangthong, O. et al. 2010. Calculation of Permeability Tensors for Unstructured Gridblocks. J Can Pet Technol 49 (10): 65–74. SPE-141305-PA. https://doi.org/10.2118/141305-PA.
He, C. and Durlofsky, L. 2006. Structured Flow-Based Gridding and Upscaling for Modeling Subsurface Flow. Adv Water Resour 29 (12): 1876–1892. https://doi.org/10.1016/j.advwatres.2005.12.012.
Heinemann, Z. and Mittermeir, G. 2014. Naturally Fractured Reservoir Engineering, Vol. 5. Leoben, Austria: Textbook Series, PHDG.
Honarpour, M. M., Koederitz, F., and Herbert, A. 1986. Relative Permeability of Petroleum Reservoirs. Boca Raton, Florida: CRC Press.
Hu, X. and Huang, S. 2017. Physical Properties of Reservoir Rocks. In Physics of Petroleum Reservoirs, ed. X. Hu, S. Hu, F. Jin, et al. 7–164. Berlin: Springer Minerology Series, Springer.
Huang, D. D. and Honarpour, M. M. 1998. Capillary End Effects in Coreflood Calculations. J Pet Sci Eng 19 (1–2): 103–117. https://doi.org/10.1016/S0920-4105(97)00040-5.
Huo, D. and Benson, S. M. 2016. Experimental Investigation of Stress-Dependency of Relative Permeability in Rock Fractures. Transp Porous Media 113 (3): 567–590. https://doi.org/10.1007/s11242-016-0713-z.
Johnson, E. F., Bossler, D. P., and Bossler, V. O. N. 1959. Calculation of Relative Permeability From Displacement Experiments. In Petroleum Transactions, AIME, Vol. 216, 370–372, SPE-1023-G. Richardson, Texas: Society of Petroleum Engineers.
Jones, S. C. and Roszelle, W. O. 1978. Graphical Techniques for Determining Relative Permeability From Displacement Experiments. J Pet Technol 30 (5): 807–817. SPE-6045-PA. https://doi.org/10.2118/6045-PA.
Jonoud, S. and Jackson, M. D. 2008. New Criteria for the Validity of Steady-State Upscaling. Transp Porous Media 71 (1): 53–73. https://doi.org/10.1007/s11242-007-9111-x.
Kamat, B. R., Brown, L. F., Manseau, E. J. et al. 1995. Expression of Vascular Permeability Factor/Vascular Endothelial Growth Factor by Human Granulosa and Theca Lutein Cells: Role in Corpus Luteum Development. Am J Pathol 146 (1): 157–165.
Karimi-Fard, M. 2004. Growing Region Technique Applied to Grid Generation of Complex Fractured Porous Media. Oral presentation given at ECMOR IX–9th European Conference on the Mathematics of Oil Recovery, Cannes, France, 30 August–2 September.
Kasap, E. and Lake, L. W. 1990. Calculating the Effective Permeability Tensor of a Gridblock. SPE Form Eval 5 (2): 192–200. SPE-18434-PA. https://doi.org/10.2118/18434-PA.
Keilegavlen, E., Nordbotten, J. M., and Stephansen, A. F. 2012. Tensor Relative Permeabilities: Origin, Modeling and Numerical Discretization. Int J Numer Anal Model 9 (3): 701–724.
Kim, J. G. and Deo, M. D. 2000. Finite Element, Discrete-Fracture Model for Multiphase Flow in Porous Media. AIChE J. 46 (6): 1120–1130. https://doi.org/10.1002/aic.690460604.
King, M. J. 1993. Application and Analysis of Tensor Permeability to Crossbedded Reservoirs. Presented at the SPE Western Regional Meeting, Anchorage, 26–28 May. SPE-26118-MS. https://doi.org/10.2118/26118-MS.
Király, L. and Morel, G. 1976. Remarques sur L’Hydrogramme des Sources Karstiques Simulépar Modèles Mathématiques. Bull. du Centre d’Hydrogéologie 1: 37–60.
Kranzz, R., Frankel, A., Engelder, T. et al. 1979. The Permeability of Whole and Jointed Barre Granite. Int J Rock Mech Min Sci Geomech 16 (4): 225–234. https://doi.org/10.1016/0148-9062(79)91197-5.
Lang, P., Paluszny, A., and Zimmerman, R. 2014. Permeability Tensor of Three-Dimensional Fractured Porous Rock and a Comparison to Trace Map Predictions. J Geophys Res Solid Earth 119 (8): 6288–6307. https://doi.org/10.1002/2014JB011027.
Levine, J. S. 1954. Displacement Experiments in a Consolidated Porous System. In Petroleum Transactions, AIME, Vol. 201, 57–66, SPE-308-G. Richardson, Texas: Society of Petroleum Engineers.
Lian, P. and Cheng, L. 2012. The Characteristics of Relative Permeability Curves in Naturally Fractured Carbonate Reservoirs. J Can Pet Technol 51 (2): 137–142. SPE-154814-PA. https://doi.org/10.2118/154814-PA.
Lohne, A., Virnovsky, G. A., and Durlofsky, L. J. 2006. Two-Stage Upscaling of Two-Phase Flow: From Core to Simulation Scale. SPE J. 11 (3): 304–316. SPE-89422-PA. https://doi.org/10.2118/89422-PA.
Long, J., Remer, J., Wilson, C. et al. 1982. Porous Media Equivalents for Networks of Discontinuous Fractures. Water Resour Res 18 (3): 645–658. https://doi.org/10.1029/WR018i003p00645.
Mattha¨i, S., Bazrafkan, S., Lang, P. et al. 2012. Numerical Prediction of Relative Permeability in Water-Wet Naturally Fractured Reservoir Rocks. Proc., ECMOR XIII–13th European Conference on the Mathematics of Oil Recovery, Biarritz, France, 10–13 September. https://doi.org/10.3997/2214-4609.20143167.
Matthäi, S., Geiger, S., Roberts, S. et al. 2007. Numerical Simulation of Multi-Phase Fluid Flow in Structurally Complex Reservoirs. Geol. Soc. London Spec. Publ. 292: 405–429. https://doi.org/10.1144/SP292.22.
Matthäi, S., Mezentsev, A., and Belayneh, M. 2005. Control-Volume Finite-Element Two-Phase Flow Experiments With Fractured Rock Represented by Unstructured 3D Hybrid Meshes. Presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 31 January–2 February. SPE-93341-MS. https://doi.org/10.2118/93341-MS.
Matthäi, S. K. and Nick, H. M. 2009. Upscaling Two-Phase Flow in Naturally Fractured Reservoirs. AAPG Bull 93 (11): 1621–1632. https://doi.org/10.1306/08030909085.
Min, K.-B., Jing, L., and Stephansson, O. 2004. Determining the Equivalent Permeability Tensor for Fractured Rock Masses Using a Stochastic REV Approach: Method and Application to the Field Data from Sellafield, UK. Hydrogeology Journal 12 (5): 497–510. https://doi.org/10.1007/s10040-004-0331-7.
Mohanty, K. and Miller, A. 1991. Factors Influencing Unsteady Relative Permeability of a Mixed-Wet Reservoir Rock. SPE Form Eval 6 (3): 349–358. SPE-18292-PA. https://doi.org/10.2118/18292-PA.
Muskat, M. and Wyckoff, R. D. 1937. The Flow of Homogeneous Fluids Through Porous Media. Ann Arbor, Michigan: J. W. Edwards.
Oda, M. 1985. Permeability Tensor for Discontinuous Rock Masses. Géotechnique 35 (4): 483–495. https://doi.org/10.1680/geot.19184.108.40.2063.
Odling, N., Gillespie, P., Bourgine, B. et al. 1999. Variations in Fracture System Geometry and Their Implications for Fluid Flow in Fractured Hydrocarbon Reservoirs. Pet Geosci 5 (4): 373–384. https://doi.org/10.1144/petgeo.5.4.373.
Paluszny, A., Matthäi, S., and Hohmeyer, M. 2007. Hybrid Finite Element–Finite Volume Discretization of Complex Geologic Structures and a New Simulation Workflow Demonstrated on Fractured Rocks. Geofluids 7 (2): 186–208. https://doi.org/10.1111/j.1468-8123.2007.00180.x.
Pickup, G. E. and Sorbie, K. S. 1996. The Scaleup of Two-Phase Flow in Porous Media Using Phase Permeability Tensors. SPE J. 1 (4): 369–382. SPE-28586-PA. https://doi.org/10.2118/28586-PA.
Pirker, B. and Heinemann, Z. E. 2008. Method to Preliminary Estimation of the Reserves and Production Forecast for Dual Porosity Fractured Reservoirs. Presented at the Europec/EAGE Conference and Exhibition, Rome, Italy, 9–12 June. SPE-113378-MS. https://doi.org/10.2118/113378-MS.
Prévost, M., Lepage, F., Durlofsky, L. J. et al. 2005. Unstructured 3D Gridding and Upscaling for Coarse Modelling of Geometrically Complex Reservoirs. Pet Geosci 11 (4): 339–345. https://doi.org/10.1144/1354-079304-657.
Qadeer, I. 1988. Health Services System in India: An Expression of Socio-Economic Inequalities. Gt Concern March (1): 3–8.
Qi, L. and Luo, Z. 2017. Tensor Analysis: Spectral Theory and Special Tensors. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics.
Renard, P., Genty, A., and Stauffer, F. 2001. Laboratory Determination of the Full Permeability Tensor. J Geophys Res Solid Earth 106 (B11): 26443–26452. https://doi.org/10.1029/2001JB000243.
Ringrose, P., Sorbie, K., Corbett, P. et al. 1993. Immiscible Flow Behaviour in Laminated and Cross-Bedded Sandstones. J Pet Sci Eng 9 (2): 103–124. https://doi.org/10.1016/0920-4105(93)90071-L.
Rodriguez, A. A., Klie, H., Sun, S. et al. 2006. Porous Media Upscaling of Hydraulic Properties: Full Permeability Tensor and Continuum Scale Simulations. Presented at the SPE/DOE Symposium on Improved Oil Recovery, Tulsa, 22–26 April. SPE-100057-MS. https://doi.org/10.2118/100057-MS.
Roehl, P. O. and Choquette, P. W. ed. 1985. Carbonate Petroleum Reservoirs. New York City: Casebooks in Earth Sciences Series, Springer-Verlag.
Romm, E. 1966. Fluid Flow in Fractured Rocks, trans. W. R. Blake. Moscow: Nedra Publishers (1972).
Rose, L. R. F. 1987. Toughening Due to Crack-Front Interaction With a Second-Phase Dispersion. Mech Mater 6 (1): 11–15. https://doi.org/10.1016/0167-6636(87)90018-4.
Rustad, A. B., Theting, T. G., and Held, R. J. 2008. Pore Space Estimation, Upscaling and Uncertainty Modelling for Multiphase Properties. Presented at the SPE Symposium on Improved Oil Recovery, Tulsa, 20–23 April. SPE-113005-MS. https://doi.org/10.2118/113005-MS.
Saidi, A. M. 1983. Simulation of Naturally Fractured Reservoirs. Presented at the SPE Reservoir Simulation Symposium, San Francisco, 15–18 November. SPE-12270-MS. https://doi.org/10.2118/12270-MS.
Sedaghat, M., Azizmohammadi, S., and Matthäi, S. K. 2016a. How Fracture Capillary Pressure Affects Ensemble Relative Permeability of Naturally Fractured Reservoirs. Proc., ECMOR XV–15th European Conference on the Mathematics of Oil Recovery, Amsterdam, The Netherlands, 29 August–1 September. https://doi.org/10.3997/2214-4609.201601807.
Sedaghat, M., Gerke, K., Azizmohammadi, S. et al. 2016b. Numerical-Simulation-Based Determination of Relative Permeability in Laminated Rocks. Proc., 78th EAGE Conference and Exhibition, Vienna, Austria, 30 May–2 June. https://doi.org/10.3997/2214-4609.201600789.
Sedaghat, M. H., Gerke, K., Azizmohammadi, S. et al. 2016c. Simulation-Based Determination of Relative Permeability in Laminated Rocks. Energy Procedia 97 (November): 433–439. https://doi.org/10.1016/j.egypro.2016.10.041.
Sedaghat, M., Azizmohammadi, S., and Matthäi, S. K. 2017. Numerical Investigation of Fracture-Rock Matrix Ensemble Saturation Functions and Their Dependence on Wettability. J Pet Sci Eng 159 (November): 869–888. https://doi.org/10.1016/j.petrol.2017.10.013.
Seers, T. and Hodgetts, D. 2013. Closed Form Unsupervised Registration of Multi-Temporal Structure From Motion-Multiview Stereo Data Using Non-Linearly Weighted Image Features. Oral presentation given at the American Geophysical Union Fall Meeting 2013, San Francisco, 9 –13 December.
Sheng, Q. and Thompson, K. 2013. Dynamic Coupling of Pore-Scale and Reservoir-Scale Models for Multiphase Flow. Water Resour Res 49 (9): 5973–5988. https://doi.org/10.1002/wrcr.20430.
Skjaeveland, S., Siqveland, L., Kjosavik, A. et al. 1998. Capillary Pressure Correlation for Mixed-Wet Reservoirs. Presented at the SPE India Oil and Gas Conference and Exhibition, New Delhi, India, 17–19 February. SPE-39497-MS. https://doi.org/10.2118/39497-MS.
Slightam, C. 2012. Characterizing Seismic-Scale Faults Pre- and Post-Drilling; Lewisian Basement, West of Shetlands, UK. In Advances in the Study of Fractured Reservoirs, Geological Society Special Publication No. 374, ed. G. H. Space, J. Redfern, R. Aguilera, et al. Chap. 15. London: Geological Society of London.
Sonntag, C. and Von Gunten, U. 2012. Chemistry of Ozone in Water and Wastewater Treatment: From Basic Principles to Applications, Vol. 11. London: IWA Publishing.
Sudicky, E. and McLaren, R. 1992. The Laplace Transform Galerkin Technique for Large-Scale Simulation of Mass Transport in Discretely Fractured Porous Formations. Water Resour Res 28 (2): 499–514. https://doi.org/10.1029/91WR02560.
Tran, N. H., Chen, Z., and Rahman, S. S. 2007. Practical Application of Hybrid Modelling to Naturally Fractured Reservoirs. Pet Sci Technol 25 (10): 1263–1277. https://doi.org/10.1080/10916460500423445.
Van Spronsen, E. 1982. Three-Phase Relative Permeability Measurements Using the Centrifuge Method. Presented at the SPE Enhanced Oil Recovery Symposium, Tulsa, 4–7 April. SPE-10688-MS. https://doi.org/10.2118/10688-MS.
Wang, J. and Buckley, J. 1999. Wettability and Rate Effects on End-Point Relative Permeability to Water. Presented at the International Symposium of the Society of Core Analysts, Golden, Colorado, 1–4 August. SCA-9937.
Welge, H. J. 1952. A Simplified Method for Computing Oil Recovery by Gas or Water Drive. J Pet Technol 4 (4): 91–98. SPE-124-G. https://doi.org/10.2118/124-G.
Wen, X., Durlofsky, L., and Edwards, M. 2003. Use of Border Regions for Improved Permeability Upscaling. Math Geol 35 (5): 521–547. https://doi.org/10.1023/A:1026230617943.
White, C. and Horne, R. 1987. Computing Absolute Transmissibility in the Presence of Fine-Scale Heterogeneity. Presented at the SPE Symposium on Reservoir Simulation, San Antonio, Texas, 1–4 February. SPE-16011-MS. https://doi.org/10.2118/16011-MS.
Wu, H. and Pollard, D. D. 2002. Imaging 3-D Fracture Networks Around Boreholes. AAPG Bull 86 (4): 593–604. https://doi.org/10.1306/61EEDB52-173E-11D7-8645000102C1865D.
Wu, K., Nunan, N., Crawford, J. W. et al. 2004. An Efficient Markov Chain Model for the Simulation of Heterogeneous Soil Structure. Soil Sci Soc Am J 68 (2): 346–351. https://doi.org/10.2136/sssaj2004.3460.
Yeh, T. C. J., Gelhar, L. W., and Gutjahr, A. L. 1985. Stochastic Analysis of Unsaturated Flow in Heterogeneous Soils: 3. Observations and Applications. Water Resour Res 21 (4): 465–471. https://doi.org/10.1029/WR021i004p00465.
Zienkiewicz, O. C. 1977. The Finite Element Method, third edition. London: McGraw-Hill.