Estimation of two-phase relative permeabilities based on treelike fractal model and pressure drop around gas bubbles in porous media. (English) Zbl 1482.76125
Summary: In order to estimate relative permeabilities of brine-gas, a semi-analytical model based on fractal theory is proposed. The model simplifies porous media as treelike nanotubes, the single nanotube’s flow has been branched out into whole pore network of core sample through treelike fractal theory. Two-phase flow description in sole nanotube, based on a setting of gas bubbles pushing continuous water phase ahead, solves pressure drop around brine and gas bubbles by Stokes’ viscous force, Laplace equation and Fairbrother-Stubbs bubble film thickness formula, Bretherton bubble pressure drop. By substituting the other three classic formulas into Bretherton expression, the iteration equation of pressure drop around a single gas bubble is acquired, the number of gas bubbles is multiplied and the gas phase pressure drop is calculated. Through merging small bubbles into a large bubble, the relative permeabilities of two phases can be solved by introducing Poiseuille’s law and Darcy’s law. After getting the fundamental data, pore size distribution data and viscosity of two phases, relative permeability curves can be drawn, revealing comprehensive elements including wettability, fluid viscosity, saturation, morphological characteristics. The calculated curves cross each other on the right half of the \(x\)-axis, meet the features of real production, flowback difficulties of hydraulic liquid and well production dropping quickly, also show identical trend with experimental data. Through a series of sensitivity analysis, to some extent, the influence on relative permeabilities is compared of different fractal models, different film thickness, bubble length, water sorption layer thickness, elastic deformation of pores.
MSC:
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
| 76S05 | Flows in porous media; filtration; seepage |
| 28A80 | Fractals |
Keywords:
fractal model; brine-gas phase; relative permeability; pressure drop; Stokes viscous force; Bretherton bubble pressure modelReferences:
| [1] | Kaviany, M., Principles of Heat Transfer in Porous Media (Springer-Verlag, 1995). · Zbl 0889.76002 |
| [2] | Chen, D., Pan, Z. and Ye, Z., Dependence of gas shale fracture permeability on effective stress and reservoir pressure: Model match and insights, Fuel139 (2015) 383-392. |
| [3] | Xu, P. and Yu, B., Developing a new form of permeability and Kozeny-Carman constant for homogeneous porous media by means of fractal geometry, Adv. Water Resour.31(1) (2008) 74-81. |
| [4] | Wei, W.et al., Kozeny-Carman constant of porous media: Insights from fractal-capillary imbibition theory, Fuel234 (2018) 1373-1379. |
| [5] | Guan, C.et al., Prediction of oil-water relative permeability with a fractal method in ultra-high water cut stage, Int. J. Heat Mass Transf.130 (2019) 1045-1052. |
| [6] | Liu, R., Jing, H., Li, X., Yin, Q., Xu, Z. and He, M., An experimental study on fractal pore size distribution and hydro-mechanical properties of granites after high temperature treatment, Fractals29 (2021) 2150083. |
| [7] | Dong, S.et al., A novel fractal model for estimating permeability in low-permeable sandstone reservoirs, Fractals28 (2020) 2040005. |
| [8] | Wu, T. and Wang, S., A fractal permeability model for real gas in shale reservoirs coupled with Knudsen diffusion and surface diffusion effects, Fractals28(1) (2020) 2050017. |
| [9] | Hu, B.et al., Evolution of fractal dimensions and gas transport models during the gas recovery process from a fractured shale reservoir, Fractals27(8) (2019) 1950129. |
| [10] | Liang, M.et al., A fractal study for the effective electrolyte diffusion through charged porous media, Int. J. Heat Mass Transf.137 (2019) 365-371. |
| [11] | Ye, W.et al., A fractal model for threshold pressure gradient of tight oil reservoirs, J. Pet. Sci. Eng.179 (2019) 427-431. |
| [12] | Wu, C. Q., Xu, H. J. and Zhao, C. Y., A new fractal model on fluid flow/heat/mass transport in complex porous structures, Int. J. Heat Mass Transf.162 (2020) 120292. |
| [13] | Zhang, T.et al., An apparent liquid permeability model of dual-wettability nanoporous media: A case study of shale, Chem. Eng. Sci.187 (2018) 280-291. |
| [14] | He, G., Zhao, Z. and Ming, P., A fractal model for predicting permeability and liquid water relative permeability in the gas diffusion layer (GDL) of PEMFCs, J. Power Sources163 (2007) 846-852. |
| [15] | Xiao, B., Fan, J. and Ding, F., A fractal analytical model for the permeabilities of fibrous gas diffusion layer in proton exchange membrane fuel cells, Electrochim. Acta134 (2014) 222-231. |
| [16] | Shi, Y., Cheng, S. and Quan, S., Fractal-based theoretical model on saturation and relative permeability in the gas diffusion layer of polymer electrolyte membrane fuel cells, J. Power Sources209 (2012) 130-140. |
| [17] | Li, K.et al., Microstructural characterisation of organic matter pores in coal-measure shale, Adv. Geo-Energy Res.4(4) (2020) 372-391. |
| [18] | Yan, H., Youjia, W. and Niansheng, W., A new method for measuring the relative permeability quantitatively, Pet. Explor. Dev.27(5) (2000) 66-68. |
| [19] | Purcell, W. R., Capillary pressures — Their measurement using mercury and the calculation of permeability therefrom, Pet. Trans. AIME186(2) (1949) 39-48. |
| [20] | Burdine, N. T., Relative permeability calculations from pore size distribution data, Pet. Trans. AIME198 (1953) 71-78. |
| [21] | Brooks, R. H. and Corey, A. T., Properties of porous media affecting fluid flow, J. Irrig. Drain. Div. Proc. Am. Soc. Civil Eng.92(2) (1964) 61-88. |
| [22] | Yu, B. and Li, J., Some fractal characters of porous media, Fractals3(9) (2001) 365-372. |
| [23] | Li, R.et al., A fractal model for gas-water relative permeability curve in shale rocks, J. Nat. Gas Sci. Eng.81 (2020) 103417. |
| [24] | Xu, P.et al., Analysis of permeability for the fractal-like tree network by parallel and series models, Physica A, Stat. Mech. Appl.369(2) (2006) 884-894. |
| [25] | Zhang, X., Wu, C. and Liu, S., Characteristic analysis and fractal model of the gas-water relative permeability of coal under different confining pressures, J. Pet. Sci. Eng.159 (2017) 488-496. |
| [26] | Xu, P.et al., Prediction of relative permeability in unsaturated porous media with a fractal approach, Int. J. Heat Mass Transf.64 (2013) 829-837. |
| [27] | Li, K., More general capillary pressure and relative permeability models from fractal geometry, J. Contaminant Hydrol.111(1-4) (2010) 13-24. |
| [28] | Su, H.et al., A fractal-Monte Carlo approach to model oil and water two-phase seepage in low-permeability reservoirs with rough surfaces, Fractals29(1) (2021) 2150003. · Zbl 1500.76088 |
| [29] | Miao, T.et al., Analysis of permeabilities for slug flow in fractal porous media, Int. Commun. Heat Mass Transf.88 (2017) 194-202. |
| [30] | Wang, D.et al., Gas-water two-phase transport properties in shale microfractures, Chin. Sci. Bull.64(31) (2019) 3232-3243. |
| [31] | Yu, B. and Cheng, P., A fractal permeability model for bi-dispersed porous media, Int. J. Heat Mass Transf.45(14) (2002) 2983-2993. · Zbl 1101.76408 |
| [32] | Katz, A. J. and Thompson, A., Fractal sandstone pores: Implications for conductivity and pore formation, Phys. Rev. Lett.54(12) (1985) 1325. |
| [33] | Chengzu, H. and Mingqi, H., Fractal geometry description of reservoir pore structure, Oil Gas Geol.19(1) (1998) 15-23. |
| [34] | Denn, M. M., Process Fluid Mechanics (Prentice-Hall, Englewood Cliffs, NJ, 1980). |
| [35] | Bretherton, F. P., The motion of long bubbles in tubes, Fluid Mech.11(6) (1960) 797-812. · Zbl 0649.76060 |
| [36] | Fairbrother, F. and Stubbs, A. E., 119. Studies in electro-endosmosis. Part VI. The “bubble-tube” method of measurement, J. Chem. Soc.6 (1935) 527-529. |
| [37] | Irandoust, S. and Andersson, B., Liquid film in Taylor flow through a capillary, Ind. Eng. Chem. Res.28(11) (1989) 1684-1688. |
| [38] | Angeli, P. and Gavriilidis, A., Hydrodynamics of Taylor flow in small channels: A review, Proc. Inst. Mech. Eng. \(C,\) J. Mech. Eng. Sci.222(5) (2008) 737-751. |
| [39] | Taylor, G. I., Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech.10(2) (1961) 161-165. · Zbl 0091.42203 |
| [40] | Marchessault, R. N. and Mason, S. G., Flow of entrapped bubbles through a capillary, Ind. Eng. Chem.52(1) (1960) 79-84. |
| [41] | Aussillous, P. and Quéré, D., Quick deposition of a fluid on the wall of a tube, Phys. Fluids12(10) (2000) 2367. · Zbl 1184.76038 |
| [42] | Chen, J., Measuring the film thickness surrounding a bubble inside a capillary, J. Colloid Interface Sci.109(2) (1986) 341-349. |
| [43] | Bear, J., Dynamics of Fluids in Porous Media (American Elsevier Pub. Co., 1972). · Zbl 1191.76001 |
| [44] | Li, K. and Horne, R. N., Experimental verification of methods to calculate relative permeability using capillary pressure data, in SPE Western Regional/AAPG Pacific Section Joint Meeting, Anchorage, Alaska, , 2002, https://doi.org/10.2118/7675. |
| [45] | J. I. Gates and W. T. Lietz, Relative permeabilities of California cores by the capillary-pressure method, Paper presented at the Drilling and Production Practice, New York, NY, 1950. |
| [46] | Li, K. and Horne, R. N., Experimental study of gas slippage in two-phase flow, SPE Reservoir Eval. Eng.7 (2004) 409-415. |
| [47] | Sandberg, C. R., Gournay, L. S. and Sippel, R. F., The effect of fluid-flow rate and viscosity on laboratory determinations of oil-water relative permeabilities, Trans. AIME213(1) (1958) 36-43. |
| [48] | Olbricht, K. L., Pore-scale prototypes of multiphase flow in porous media, Annu. Rev. Fluid. Mech.28 (1996) 187-213. |
| [49] | Li, C.et al., Application of PeakForce tapping mode of atomic force microscope to characterize nanomechanical properties of organic matter of the Bakken Shale, Fuel233 (2018) 894-910. |
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