Transport in Porous Media

, Volume 111, Issue 1, pp 97–121 | Cite as

Flow Consistency Between Non-Darcy Flow in Fracture Network and Nonlinear Diffusion in Matrix to Gas Production Rate in Fractured Shale Gas Reservoirs

  • Jia Liu
  • J. G. Wang
  • Feng Gao
  • Yang Ju
  • Xiangxiang Zhang
  • Lai-Chang Zhang


Due to complex pore structures and ultra-low permeability in unconventional gas reservoirs, the flow consistency between the macro-flow in fracture network and the micro-flow or gas diffusion in matrix may significantly impact the production rate of fractured gas reservoirs. This study investigated the impact of this flow consistency on the production rate through the development of a numerical simulation model and its application to a shale gas reservoir. In this model, a fractured gas reservoir consists of fracture network and matrix. In the fracture network, gas flow was assumed to follow the non-Darcy law. In the matrix, a nonlinear diffusion model was proposed for the gas micro-flow through a non-empirical apparent permeability. This nonlinear diffusion model considered the advection and diffusion of the free-phase gas in nanoporous channels as well as the gas desorption in matrix. Further, the mass exchange rate between fracture network and matrix was calculated via a diffusion time which comprehensively considers both the diffusion capability and the size of matrix block. This numerical model was verified through history matching of the production data from two shale wells and then applied to a typical production well to investigate the effects of pore size in matrix, fracture spacing, and initial fracture permeability on production curve (i.e., production rate versus time). It is found that the production curve is significantly affected by this flow consistency. Pore size and initial fracture permeability play the key roles in this flow consistency. Fracture spacing and fracture permeability can alter the production curve. In this sense, the production curve can be designable through this flow consistency. Production efficiency can be improved through appropriate control of the fracturing degree of shale reservoir. Meanwhile, accurate measurement of shale pore size distribution provides an important parameter to the design of this flow consistency.


Fracture-matrix system Flow consistency Non-empirical apparent permeability Apparent diffusion coefficient Sorption–desorption 



The authors are grateful to the financial support from Creative Research and Development Group Program of Jiangsu Province (2014), National Natural Science Fund for Distinguished Young Scholars of China (Grant No. 51125017), and National Natural Science Foundation of China (Grant No. 51374213, 51404250).


  1. Beskok, A., Karniadakis, G.E.: A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng. 3(1), 43–77 (1999)CrossRefGoogle Scholar
  2. Bustin, R.M., Bustin, A.M.M.: Importance of rock properties on the producibility of gas shales. Int. J. Coal Geol. 103, 132–147 (2012)CrossRefGoogle Scholar
  3. Chen, D., Pan, Z., Ye, Z.: Dependence of gas shale fracture permeability on effective stress and reservoir pressure: model match and insights. Fuel 139, 383–392 (2015)CrossRefGoogle Scholar
  4. Civan, F., Rai, C., Sondergeld, C.: Shale-gas permeability and diffusivity inferred by improved formulation of relevant retention and transport mechanisms. Transp. Porous Media 86(3), 925–44 (2011)CrossRefGoogle Scholar
  5. Civan, F.: Effective correlation of apparent gas permeability in tight porous media. Transp. Porous Media 82(2), 375–384 (2010)CrossRefGoogle Scholar
  6. Cooke Jr, C.E.: Conductivity of fracture proppants in multiple layers. J. Petrol. Technol. 25(9), 1101–1110 (1973). (SPE-4117-PA)CrossRefGoogle Scholar
  7. Cooper, S.M., Cruden, B.A., Meyyappan, M., Raju, R., Roy, S.: Gas transport characteristics through a carbon nanotubule. Nano Lett. 4(2), 377–381 (2004)CrossRefGoogle Scholar
  8. Cui, X., Bustin, A.M., Bustin, R.: Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids 9, 208–223 (2009)CrossRefGoogle Scholar
  9. Darabi, H., Ettehad, A., Javadpour, F., Sepehrnoori, K.: Gas flow in ultra-tight shale strata. J. Fluid Mech. 710, 641–658 (2012)CrossRefGoogle Scholar
  10. Deng, J., Zhu, W., Ma, Q.: A new seepage model for shale gas reservoir and productivity analysis of fractured well. Fuel 124, 232–240 (2014)CrossRefGoogle Scholar
  11. Dongari, N., Sharma, A., Durst, F.: Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes. Microfluid. Nanofluid. 6(5), 679–692 (2009)CrossRefGoogle Scholar
  12. Friedel, T., Voigt, H.D.: Investigation of non-Darcy flow in tight-gas reservoirs with fractured wells. J. Petrol. Sci. Eng. 54(3–4), 112–128 (2006)CrossRefGoogle Scholar
  13. Fuentes-Cruz, G., Valko, Peter P.: Revisiting the dual-porosity/dual-permeability modeling of unconventional reservoirs: the induced-interporosity flow field. SPE J. 20(1), 125–141 (2015)CrossRefGoogle Scholar
  14. Guo, C., Xu, J., Wu, K., Wei, M., Liu, S.: Study on gas flow through nano pores of shale gas reservoirs. Fuel 143, 107–117 (2015)CrossRefGoogle Scholar
  15. Javadpour, F., Fisher, D., Unsworth, M.: Nanoscale gas flow in shale sediments. J. Can. Pet. Technol. 46(10), 55–61 (2007)CrossRefGoogle 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)CrossRefGoogle Scholar
  17. Klinkenberg, L.J.: The permeability of porous media to liquids and gases. In API Drilling and Production Practice, pp. 200–213. (1941)Google Scholar
  18. Lim, K., Aziz, K.: Matrix-fracture transfer shape factors for dual-porosity simulators. J. Petrol. Sci. Eng. 13, 169–178 (1995)CrossRefGoogle Scholar
  19. Ma, J., Sanchez, J., Wu, K., Couples, G., Jiang, Z.: A pore network model for simulating non-ideal gas flow in micro- and nano-porous material. Fuel 116, 498–508 (2014)CrossRefGoogle Scholar
  20. Mi, L., Jiang, H., Li, J.: The impact of diffusion type on multiscale discrete fracture model numerical simulation for shale gas. J. Nat. Gas Sci. Eng. 20, 74–81 (2014)CrossRefGoogle Scholar
  21. Naraghi, M.E., Javadpour, F.: A stochastic permeability model for the shale-gas systems. Int. J. Coal Geol. 140, 111–124 (2015)CrossRefGoogle Scholar
  22. Qanbari, F., Clarkson, C.R.: A new method for production data analysis of tight and shale gas reservoirs during transient linear flow period. J. Nat. Gas Sci. Eng. 14, 55–65 (2013)CrossRefGoogle Scholar
  23. Qanbari, F., Clarkson, C.R.: Analysis of transient linear flow in stress-sensitive formations. SPE Reserv. Eval. Eng. 17, 98–104 (2014)CrossRefGoogle Scholar
  24. Rahmanian, M., Solano, N., Aguilera, R.: Storage and Output Flow From Shale and Tight Gas Reservoirs. SPE Western Regional Meeting, Anaheim, 7–29 May 2010, SPE-133611-MS (2010)Google Scholar
  25. Singh, H., Javadpour, F.: A new non-empirical approach to model transport of fluids in shale gas reservoirs. In Unconventional Resources Technology Conference (In SPE, AAPG and SEG), Denver. doi: 10.1190/urtec2013-127 (2013)
  26. Song, H., Yu, M., Zhu, W., Wu, P., Lou, Y., Wang, Y., Killough, J.: Numerical investigation of gas flow rate in shale gas reservoirs with nanoporous media. Int. J. Heat Mass Transf. 80, 626–635 (2015)CrossRefGoogle Scholar
  27. Veltzke, T., Thöming, J.: An analytically predictive model for moderately rarefied gas flow. J. Fluid Mech. 698, 406–422 (2012)CrossRefGoogle Scholar
  28. Wang, H.T.: Performance of multiple fractured horizontal wells in shale gas reservoirs with consideration of multiple mechanisms. J. Hydrol. 510, 299–312 (2014)CrossRefGoogle Scholar
  29. Wang, J.G., Kabir, A., Liu, J., Chen, Z.: Effects of non-Darcy flow on the performance of coal seam gas wells. Int. J. Coal Geol. 93, 62–74 (2012)CrossRefGoogle Scholar
  30. Wang, J.G., Peng, Y.: Numerical modeling for the combined effects of two-phase flow, deformation, gas diffusion and \({\rm CO}_{2}\) sorption on caprock sealing efficiency. J. Geochem. Explor. 144, 154–167 (2014)CrossRefGoogle Scholar
  31. Wang, J.G., Ju, Y., Gao, F., Peng, Y., Gao, Y.: Effect of \({\rm CO}_{2}\) anisotropic sorption and swelling on caprock sealing efficiency. J. Clean. Prod. 103, 685–695 (2015)CrossRefGoogle Scholar
  32. Warren, J., Root, P.: The behavior of naturally fractured reservoirs. Soc. Petrol. Eng. J. 3(3), 245–255 (1963)CrossRefGoogle Scholar
  33. Wu, K., Li, X., Wang, C., Yu, W., Guo, C., Ji, D., Ren, G., Chen, Z.: Apparent permeability for gas flow in shale reservoirs coupling effects of gas diffusion and desorption. SPE-2014-1921039-MS, presented at the SPE/AAPG/SEG Unconventional Resources Technology Conference, 25–27 August 2014, Denver, Colorado, USA (2014)Google Scholar
  34. Xiong, X., Devegowda, D., Michel, G., Sigal, R., Civan, F.: A fully-coupled free and adsorptive phase transport model for shale gas reservoirs including non-Darcy flow effects. SPE 159758, presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 8–10 October, 2012 (2012)Google Scholar
  35. Ye, Z., Chen, D., Wang, J.G.: Evaluation of the non-Darcy effect in coalbed methane production. Fuel 121, 1–10 (2014)CrossRefGoogle Scholar
  36. Yu, W., Sepehmoori, K.: Simulation of gas desorption and geomechanics effects for unconventional gas reservoirs. Fuel 116, 455–464 (2014)CrossRefGoogle Scholar
  37. Zhang, H.B., Liu, J., Elsworth, D.: How sorption-induced matrix deformation affects gas flow in coal seam: a new FE model. Int. J. Rock Mech. Min. Sci. 4(8), 1226–1833 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Jia Liu
    • 1
  • J. G. Wang
    • 1
    • 2
  • Feng Gao
    • 1
    • 2
  • Yang Ju
    • 1
    • 3
  • Xiangxiang Zhang
    • 2
  • Lai-Chang Zhang
    • 4
  1. 1.State Key Laboratory for Geomechanics and Deep Underground EngineeringChina University of Mining and TechnologyXuzhouChina
  2. 2.School of Mechanics and Civil EngineeringChina University of Mining and TechnologyXuzhouChina
  3. 3.State Key Laboratory of Coal Resources and Safe MiningChina University of Mining and Technology at BeijingBeijingChina
  4. 4.School of EngineeringEdith Cowan UniversityJoondalup, PerthAustralia

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