Abstract
Relative permeability curves of two-phase flow in a fracture have been a subject of study in recent years. The importance of these curves have been widely observed in multidisciplines, such as water subsurface resources, geothermal energy and underground hydrocarbon resources, especially fractured oil and gas reservoirs. Extensive experimental studies have been cited alongside the numerical studies in this area. However, simple analytical and practical solutions are still attractive. In the current study, wettability effects and phase interference explicitly were tried to be implemented in a simple analytical formula. The wettability effects are represented by residual saturations which resulted in direct calculation of relative permeability end points. In addition, the phase interference part affected the shape of the curves that allowed to quantify the degree of phase interference from no phase interference, assigned as zero, to ultimate phase interference, assigned as infinity. The results were compared and validated with the available experimental data in the literature. The proposed formulation is applicable for both smooth and rough fracture assemblies.
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Alturki, A., Maini, B., Gates, I.: The effect of wall roughness on two-phase flow in a rough-walled Hele-Shaw cell. J. Pet. Explor. Prod. Technol. 4(4), 397–426 (2014). doi:10.1007/s13202-013-0090-x
Babadagli, T., Raza, S., Ren, X., Develi, K.: Effect of surface roughness and lithology on the water-gas and water-oil relative permeability ratios of oil-wet single fractures. Int. J. Multiph. Flow 75, 68–81 (2015a). doi:10.1016/j.ijmultiphaseflow.2015.05.005
Babadagli, T., Ren, X., Develi, K.: Effects of fractal surface roughness and lithology on single and multiphase flow in a single fracture: an experimental investigation. Int. J. Multiph. Flow 68, 40–58 (2015b). doi:10.1016/j.ijmultiphaseflow.2014.10.004
Bertels, S.P., DiCarlo, D.A., Blunt, M.J.: Measurement of aperture distribution, capillary pressure, relative permeability, and in situ saturation in a rock fracture using computed tomography scanning. Water Resour. Res. 37(3), 649–662 (2001). doi:10.1029/2000WR900316
Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn. Wiley, New York (2002)
Brown, S.R.: Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. Solid Earth 92(B2), 1337–1347 (1987). doi:10.1029/JB092iB02p01337
Chen, C.Y., Horne, R.N.: Two-phase flow in rough-walled fractures: experiments and a flow structure model. Water Resour. Res. 42(3), W03–430 (2006). doi:10.1029/2004WR003837
Chen, C.Y., Horne, R.N., Fourar, M.: Experimental study of liquid–gas flow structure effects on relative permeabilities in a fracture. Water Resour. Res. 40(8), W08–301 (2004). doi:10.1029/2004WR003026
Durlofsky, L., Brady, J.F.: Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30(11), 3329–3334 (1987). doi:10.1063/1.866465
Fourar, M., Lenormand, R.: A viscous coupling model for relative permeabilities in fractures. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (1998). doi:10.2118/49006-MS
Gross, S., Reusken, A.: Numerical Methods for Two-Phase Incompressible Flows, vol. 40. Springer, Berlin (2011)
Hanks, R.W.: The laminar-turbulent transition for flow in pipes, concentric annuli, and parallel plates. AIChE J. 9(1), 45–48 (1963). doi:10.1002/aic.690090110
Honarpour, M.M., Koederitz, F., Herbert, A.: Relative Permeability of Petroleum Reservoirs. CRC Press Inc, Boca Raton (1986)
Huo, D., Benson, S.M.: Experimental investigation of stress-dependency of relative permeability in rock fractures. Transp. Porous Med. 113(3), 567–590 (2016). doi:10.1007/s11242-016-0713-z
Lian, P., Cheng, L., Ma, C.Y.: The characteristics of relative permeability curves in naturally fractured carbonate reservoirs. J. Can. Pet. Technol. 51(02), 137–142 (2012). doi:10.2118/154814-PA
Liu, H.H., Wei, M.Y., Rutqvist, J.: Normal-stress dependence of fracture hydraulic properties including two-phase flow properties. Hydrogeol. J. 21(2), 371–382 (2013). doi:10.1007/s10040-012-0915-6
Lomize, G.: Flow in fractured rocks. Gosenergoizdat Mosc. 127, 197 (1951)
Pan, X.: Immiscible two-phase flow in a fracture. Ph.D. thesis, University of Calgary, Canada (1999)
Pan, X., Wong, R., Maini, B., et al.: Steady state immiscible oil and water flow in a smooth-walled fracture. J. Can. Pet. Technol. 37(05), 52–59 (1998). doi:10.2118/98-05-04
Patir, N., Cheng, H.: An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. J. lubr. Technol. 100(1), 12–17 (1978). doi:10.1115/1.3453103
Persoff, P., Pruess, K.: Two-phase flow visualization and relative permeability measurement in natural rough-walled rock fractures. Water Resour. Res. 31(5), 1175–1186 (1995). doi:10.1029/95WR00171
Pruess, K., Tsang, Y.: On two-phase relative permeability and capillary pressure of rough-walled rock fractures. Water Resour. Res. 26(9), 1915–1926 (1990). doi:10.1029/WR026i009p01915
Pyrak-Nolte, L.J., Cook, N.G., Nolte, D.D.: Fluid percolation through single fractures. Geophys. Res. Lett. 15(11), 1247–1250 (1988). doi:10.1029/GL015i011p01247
Rangel-German, E., Akin, S., Castanier, L.: Multiphase-flow properties of fractured porous media. J. Pet. Sci. Eng. 51(3), 197–213 (2006). doi:10.1016/j.petrol.2005.12.010
Raza, S., Hejazi, S.H., Gates, I.D.: Two phase flow of liquids in a narrow gap: phase interference and hysteresis. Phys. Fluids 28(7), 074–102 (2016). doi:10.1063/1.4953238
Renshaw, C.E.: On the relationship between mechanical and hydraulic apertures in rough-walled fractures. J. Geophys. Res. Solid Earth 100(B12), 24629–24636 (1995). doi:10.1029/95JB02159
Romm, E.: Flow Characteristics of Fractured Rocks. Nedra, Moscow (1966)
Saboorian-Jooybari, H.: Analytical estimation of water-oil relative permeabilities through fractures. Oil Gas Sci. Technol. Rev. dIFP Energ. Nouv. 71(3), 31 (2016). doi:10.2516/ogst/2014054
Saltelli, A., Chan, K., Scott, E.M., et al.: Sensitivity Analysis, vol. 1. Wiley, New York (2000)
Shad, S., Gates, I.D.: Multiphase flow in fractures: co-current and counter-current flow in a fracture. J. Can. Pet. Technol. 49(02), 48–55 (2010). doi:10.2118/133205-PA
Sisavath, S., Al-Yaarubi, A., Pain. C.C., Zimmerman, R.W.: A simple model for deviations from the cubic law for a fracture undergoing dilation or closure. In: Thermo-Hydro-Mechanical Coupling in Fractured Rock, pp. 1009–1022. Springer, Berlin (2003). doi:10.1007/978-3-0348-8083-1_14
Watanabe, N., Sakurai, K., Ishibashi, T., Ohsaki, Y., Tamagawa, T., Yagi, M., Tsuchiya, N.: New \(\nu \)-type relative permeability curves for two-phase flows through subsurface fractures. Water Resour. Res. 51(4), 2807–2824 (2015). doi:10.1002/2014WR016515
Ye, Z., Liu, H.H., Jiang, Q., Liu, Y., Cheng, A.: Two-phase flow properties in aperture-based fractures under normal deformation conditions: Analytical approach and numerical simulation. J. Hydrol. 545, 72–87 (2017). doi:10.1016/j.jhydrol.2016.12.017
Yu, C.: A simple statistical model for transmissivity characteristics curve for fluid flow through rough-walled fractures. Transp. Porous Med. 108(3), 649–657 (2015). doi:10.1007/s11242-015-0493-x
Zimmerman, R.W., Kumar, S., Bodvarsson, G.: Lubrication theory analysis of the permeability of rough-walled fractures. Int. J. Rock Mech. Min. 28(4), 325–331 (1991). doi:10.1016/0148-9062(91)90597-F
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Ranjbaran, M., Shad, S., Taghikhani, V. et al. A Heuristic Insight on End-Point Calculation and a New Phase Interference Parameter in Two-Phase Relative Permeability Curves for Horizontal Fracture Flow. Transp Porous Med 119, 499–519 (2017). https://doi.org/10.1007/s11242-017-0895-z
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DOI: https://doi.org/10.1007/s11242-017-0895-z