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A Multiscale Fractal Transport Model with Multilayer Sorption and Effective Porosity Effects

  • J. G. WangEmail author
  • Bowen Hu
  • Di Wu
  • Fakai Dou
  • Xiaolin Wang
Article
  • 54 Downloads

Abstract

In order to study gas transport properties of fractured shale gas reservoirs for the accurate estimation of shale gas production, a new multiscale fractal transport model with an effective porosity model was proposed based on the fractal theory and the multilayer fractal Frenkel–Halsey–Hill (FHH) adsorption. In shale matrix, both fractal microstructures of pores (such as pore size distribution, flow path tortuosity, and pore surface roughness) and multiscale flow mechanisms (including slip flow and Knudsen diffusion) were coupled. In fracture network, fractal fracture length distribution, stress compaction, and gas pressure were introduced to formulate a new fracture permeability model. These permeability and effective porosity models were then incorporated into the governing equations of gas flow and the deformation equation of reservoirs to form a numerical model. This numerical model was solved within COMSOL Multiphysics for shale gas recovery. Both transport models in shale matrix and fracture network were validated by experimental data or compared with other models. Finally, sensitivity analysis was conducted to identify key parameters to gas recovery enhancement. It was found that the multilayer gas adsorption and fractal microstructures have great impacts on gas production in shale reservoirs. The cumulative gas production can be increased by 26% after 8000 days when the multilayer adsorbed gas is considered. Larger surface fractal dimension and larger tortuosity fractal dimension represent more roughness pore surface, higher flow resistance, and lower cumulative gas production. Bigger pore diameter fractal dimension means more pores, higher permeability, and higher cumulative gas production. Our model with fractal FHH adsorption was in better agreements with field data from Marcellus and Barnett shale reservoirs than other models.

Keywords

Fractal FHH model Effective porosity Multiscale fractal transport Surface fractal dimension Tortuosity fractal dimension 

Notes

Acknowledgements

The authors are grateful to the financial support from the Fundamental Research Funds for the Central Universities (Grant No. 2018ZZCX04).

References

  1. Ahmad, A.L., Mustafa, N.N.: Pore surface fractal analysis of palladium-alumina ceramic membrane using Frenkel–Halsey–Hill (FHH) model. J. Colloid Interface Sci. 301(2), 575–584 (2006)Google Scholar
  2. Beskok, A., Karniadakis, G.: Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng. 3(1), 43–77 (1999)Google Scholar
  3. Brunauer, S., Emmett, P., Teller, E.: Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 60, 309–319 (1938)Google Scholar
  4. Cai, J., Lin, D., Singh, H., Wei, W., Zhou, S.: Shale gas transport model in 3D fractal porous media with variable pore sizes. Mar. Pet. Geol. 98, 437–447 (2018)Google Scholar
  5. Cai, J., Yu, B.: A discussion of the effect of tortuosity on the capillary imbibition in porous media. Transp. Porous Media 89(2), 251–263 (2011)Google Scholar
  6. Cao, R., Wang, Y., Cheng, L., Ma, Y.Z., Tian, X., An, N.: A new model for determining the effective permeability of tight formation. Transp. Porous Media 112(1), 21–37 (2016)Google Scholar
  7. Civan, F.: Effective correlation of apparent gas permeability in tight porous media. Transp. Porous Media 82(2), 375–384 (2010)Google Scholar
  8. Cui, X., Bustin, R.M.: Volumetric strain associated with methane desorption and its impact on coalbed gas production from deep coal seams. AAPG Bull. 89(89), 1181–1202 (2005)Google Scholar
  9. Darabi, H., Ettehad, A., Javadpour, F., Sepehrnoori, K.: Gas flow in ultra-tight shale strata. J. Fluid Mech. 710(12), 641–658 (2012)Google Scholar
  10. Eftekhari, B., Marder, M., Patzek, T.: Field data provide estimates of effective permeability, fracture spacing, well drainage area and incremental production in gas shales. J. Nat. Gas Sci. Eng. 56, 141–151 (2018)Google Scholar
  11. Fan, D., Ettehadtavakkol, A.: Semi-analytical modeling of shale gas flow through fractal induced fracture networks with microseismic data. Fuel 193, 444–459 (2017)Google Scholar
  12. Ghanbarian, B., Javadpour, F.: Upscaling pore pressure-dependent gas permeability in shales. J. Geophys. Res. Solid Earth 122(4), 2541–2552 (2017)Google Scholar
  13. Ghanbarian, B., Perfect, E., Liu, H.: A geometrical aperture-width relationship for rock fractures. Fractals. 27(1), 1940002 (2019)Google Scholar
  14. Hu, B., Wang, J.G., Wu, D., Wang, H.: Impacts of zone fractal properties on shale gas productivity of a multiple fractured horizontal well. Fractals. 27(2), 1950006 (2019).  https://doi.org/10.1142/S0218348X19500063 Google Scholar
  15. Hunt, A., Ghanbarian, B., Saville, K.: Unsaturated hydraulic conductivity modeling for porous media with two fractal regimes. Geoderma 207, 268–278 (2013)Google Scholar
  16. Javadpour, F.: Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Pet. Technol. 48(8), 16–21 (2009)Google Scholar
  17. Jia, B., Li, D., Tsau, J.S., Barati, R.: Gas permeability evolution during production in the Marcellus and Eagle Ford shales: coupling diffusion/slip-flow, geomechanics, and adsorption/desorption. In: SPE/AAPG/SEG Unconventional Resources Technology Conference, 24–26 July, Austin, Texas, USA (2017)Google Scholar
  18. Karniadakis, G.E., Beskok, A., Aluru, N.R.: MicroFlows and Nanoflows—Fundamentals and Simulation. Interdisciplinary Applied Mathematics Series. Springer, New York (2005)Google Scholar
  19. Klimczak, C., Schultz, R.A., Parashar, R., Reeves, D.M.: Cubic law with aperture-length correlation: implications for network scale fluid flow. Hydrogeol. J. 18(4), 851–862 (2010)Google Scholar
  20. Klinkenberg, L.J.: The permeability of porous media to liquids and gases. Socar Proc. 2(2), 200–213 (1941)Google Scholar
  21. Lee, S., Fischer, T.B., Stokes, M.R., Klingler, R.J., Ilavsky, J., Mccarty, D.K., Wigand, M.O., Derkowski, A., Winans, R.E.: Dehydration effect on the pore size, porosity, and fractal parameters of shale rocks: ultrasmall-angle x-ray scattering study. Energy Fuels 28(11), 6772–6779 (2014)Google Scholar
  22. Lim, K.T., Aziz, K.: Matrix-fracture transfer shape factors for dual-porosity simulators. J. Pet. Sci. Eng. 13(3–4), 169–178 (1995)Google Scholar
  23. Liu, K., Ostadhassan, M.: Multi-scale fractal analysis of pores in shale rocks. J. Appl. Geophys. 140, 1–10 (2017)Google Scholar
  24. Liu, R., Li, B., Jiang, Y., Huang, N.: Review: mathematical expressions for estimating equivalent permeability of rock fracture networks. Hydrogeol. J. 24, 1623–1649 (2016)Google Scholar
  25. Mengal, S., Wattenbarger, R.: Accounting for adsorbed gas in shale gas reservoirs. Society of Petroleum Engineers. SPE-141085-MS (2011)  https://doi.org/10.2118/141085-ms
  26. Miao, T., Yu, B., Duan, Y., Fang, Q.: A fractal analysis of permeability for fractured rocks. Int. J. Heat Mass Transf. 81(81), 75–80 (2015)Google Scholar
  27. Michel, G., Sigal, R., Civan, F., Devegowda, D.: Parametric investigation of shale gas production considering nano-scale pore size distribution, formation factor, and non-Darcy flow mechanisms. In: SPE Technical Conference & Exhibition. Society of Petroleum Engineers (2011)  https://doi.org/10.2118/147438-ms
  28. Miller, A.A.: The Variance of Methane Adsorption and Its Relation to Thermal Maturity in the Marcellus Shale. Master’s thesis, University of Texas at Arlington (2015)Google Scholar
  29. Millán, H., Govea-Alcaide, E., García-Fornaris, I.: Truncated fractal modeling of H2O-vapor adsorption isotherms. Geoderma 206, 14–23 (2013)Google Scholar
  30. Neuzil, C.E., Tracy, J.V.: Flow through fractures. Water Resour. Res. 17(1), 191–199 (1981)Google Scholar
  31. Olson, J.E.: Sublinear scaling of fracture aperture versus length: an exception or the rule? J. Geophys. Res. Solid Earth. (2003).  https://doi.org/10.1029/2001jb000419 Google Scholar
  32. Pang, Y., Soliman, M.Y., Deng, H., Emadi, H.: Analysis of effective porosity and effective permeability in shale-gas reservoirs with consideration of gas adsorption and stress effects. SPE J. 26(2), 1739–1759 (2017)Google Scholar
  33. Patzek, T., Male, F., Marder, M.: A simple model of gas production from hydrofractured horizontal wells in shales. AAPG Bull. 98(12), 2507–2529 (2014)Google Scholar
  34. Patzek, T., Male, F., Marder, M.: Gas production in the Barnett shale obeys a simple scaling theory. Proc. Natl. A Sci. 110(49), 19731–19736 (2013)Google Scholar
  35. Pfeifer, P., Obert, M., Cole, M.W.: Fractal BET and FHH theories of adsorption: a comparative study. Proc. R. Soc. Lond. A 423, 169–188 (1989a).  https://doi.org/10.1098/rspa.1989.0049 Google Scholar
  36. Pfeifer, P., Wu, Y.J., Cole, M.W., Krim, J.: Multilayer adsorption on a fractally rough surface. Phys. Rev. Lett. 62(17), 1997–2000 (1989b).  https://doi.org/10.1103/PhysRevLett.62 Google Scholar
  37. Sheng, G., Javadpour, F., Su, Y.: Effect of microscale compressibility on apparent porosity and permeability in shale gas reservoirs. Int. J. Heat Mass Transf. 120, 56–65 (2018)Google Scholar
  38. Sun, H., Yao, J., Fan, D.Y., Wang, C.C., Sun, Z.: Gas transport mode criteria in ultra-tight porous media. Int. J. Heat Mass Transf. 83, 192–199 (2015)Google Scholar
  39. Tan, X.H., Kui, M.Q., Li, X.P., Mao, Z.L., Xiao, H.: Permeability and porosity models of bi-fractal porous media. Int. J. Mod. Phys. B 31(29), 1750219 (2017)Google Scholar
  40. Turcio, M., Reyes, J.M., Camacho, R., Lira-Galeana, C., Vargas, R.O., Manero, O.: Calculation of effective permeability for the BMP model in fractal porous media. J. Pet. Sci. Eng. 103(3), 51–60 (2013)Google Scholar
  41. Vajda, P., Felinger, A.: Multilayer adsorption on fractal surfaces. J. Chromatogr. A 1324, 121–127 (2014)Google Scholar
  42. Wang, F., Liu, Z., Jiao, L., Wang, C., Guo, H.: A fractal permeability model coupling boundary-layer effect for tight oil reservoirs. Fractals 25(3), 1750042 (2017)Google Scholar
  43. Wang, J., Hu, B., Liu, H., Han, Y., Liu, J.: Effects of ‘soft-hard’ compaction and multiscale flow on the shale gas production from a multistage hydraulic fractured horizontal well. J. Pet. Sci. Eng. 170, 873–887 (2018)Google Scholar
  44. Warpinksi, N., Teufel, L.: Influence of geologic discontinuities on hydraulic fracture propagation. J. Pet. Technol. 39(2), 209–220 (1987).  https://doi.org/10.2118/13224-PA Google Scholar
  45. Wu, J., Yu, B.: A fractal resistance model for flow through porous media. Int. J. Heat Mass Transf. 71(3), 331–343 (2008)Google Scholar
  46. Wu, K., Li, X., Guo, C., Chen, Z.: Adsorbed gas surface diffusion and bulk gas transport in nanopores of shale reservoirs with real gas effect-adsorption-mechanical coupling. SPE Reservoir Simulation Symposium 23–25 February, Houston, Texas, USA. SPE-173201-MS (2015)  https://doi.org/10.2118/173201-ms
  47. Xia, Y., Cai, J., Wei, W., Hu, X., Wang, X., Ge, X.: A new method for calculating fractal dimensions of porous media based on pore size distribution. Fractals 26(3), 1850006 (2018)Google Scholar
  48. Xu, P., 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), 74–81 (2008)Google Scholar
  49. Yang, C., Zhang, J., Wang, X., Tang, X., Chen, Y., Jiang, L., Gong, X.: Nanoscale pore structure and fractal characteristics of marine-continental transitional shale: a case study from the lower permian Shanxi shale in the southeastern ordos basin, China. Mar. Pet. Geol. 88, 54–68 (2017)Google Scholar
  50. Yeager, B., Meyer, B.: Injection/fall-off testing in the Marcellus shale: using reservoir knowledge to improve operational efficiency. In: SPE 139067, Presented at SPE Eastern Regional Meeting, October 12–14, Morgantown, WV (2010).  https://doi.org/10.2118/139067-ms
  51. Yu, B., Cheng, P.: A fractal permeability model for bi-dispersed porous media. Int J. Heat Mass Transf. 45(14), 2983–2993 (2002)Google Scholar
  52. Yu, B., Li, J.: Some fractal characters of porous media. Fractals 9(03), 365–372 (2001)Google Scholar
  53. Yu, W., Sepehrnoori, K.: Simulation of gas desorption and geomechanics effects for unconventional gas reservoirs. Fuel 116(1), 455–464 (2014)Google Scholar
  54. Yu, W., Sepehrnoori, K., Patzek, T.: Modeling gas adsorption in marcellus shale with Langmuir and BET Isotherms. SPE J. 21(2), 589–600 (2016)Google Scholar
  55. Yu, W., Zhang, T., Song, D., Sepehrnoori, K.: Numerical study of the effect of uneven proppant distribution between multiple fractures on shale gas well performance. Fuel 142, 189–198 (2015)Google Scholar
  56. Zamirian, M., Aminian, K., Ameri, S., Fathi, E.: New steady-state technique for measuring shale core plug permeability. Society of Petroleum Engineers, SPE-171613-MS (2014).  https://doi.org/10.2118/171613-ms
  57. Zhang, J., Li, X., Wei, Q., Sun, K., Zhang, G., Wang, F.: Characterization of full-sized pore structure and fractal characteristics of marine–continental transitional Longtan formation shale of Sichuan basin, South China. Energy Fuels 30(10), 10490–10504 (2017)Google Scholar
  58. Zhang, L., Li, J., Tang, H., Guo, J.: Fractal pore structure model and multilayer fractal adsorption in shale. Fractals 22(03), 1440010 (2014)Google Scholar
  59. Zheng, Q., Fan, J., Li, X., Wang, S.: Fractal model of gas diffusion in fractured porous media. Fractals 26(3), 1850035 (2018)Google Scholar
  60. Zhou, S., Xue, H., Ning, Y., Guo, W., Zhang, Q.: Experimental study of supercritical methane adsorption in Longmaxi shale: insights into the density of adsorbed methane. Fuel 211, 140–148 (2018)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mechanics and Civil EngineeringChina University of Mining and TechnologyXuzhouChina
  2. 2.State Key Laboratory for Geomechanics and Deep Underground EngineeringChina University of Mining and TechnologyXuzhouChina
  3. 3.School of EngineeringUniversity of TasmaniaHobartAustralia

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