Skip to main content
Log in

Multiscale model reduction for shale gas transport in fractured media

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

In this paper, we develop a multiscale model reduction technique that describes shale gas transport in fractured media. Due to the pore-scale heterogeneities and processes, we use upscaled models to describe the matrix. We follow our previous work (Akkutlu et al. Transp. Porous Media 107(1), 235–260, 2015), where we derived an upscaled model in the form of generalized nonlinear diffusion model to describe the effects of kerogen. To model the interaction between the matrix and the fractures, we use Generalized Multiscale Finite Element Method (Efendiev et al. J. Comput. Phys. 251, 116–135, 2013, 2015). In this approach, the matrix and the fracture interaction is modeled via local multiscale basis functions. In Efendiev et al. (2015), we developed the GMsFEM and applied for linear flows with horizontal or vertical fracture orientations aligned with a Cartesian fine grid. The approach in Efendiev et al. (2015) does not allow handling arbitrary fracture distributions. In this paper, we (1) consider arbitrary fracture distributions on an unstructured grid; (2) develop GMsFEM for nonlinear flows; and (3) develop online basis function strategies to adaptively improve the convergence. The number of multiscale basis functions in each coarse region represents the degrees of freedom needed to achieve a certain error threshold. Our approach is adaptive in a sense that the multiscale basis functions can be added in the regions of interest. Numerical results for two-dimensional problem are presented to demonstrate the efficiency of proposed approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akkutlu, I.Y., Efendiev, Y., Savatorova, V.: Multi-scale asymptotic analysis of gas transport in shale matrix. Transp. Porous Media 107(1), 235–260 (2015)

    Article  Google Scholar 

  2. Akkutlu, I.Y., Fathi, E.: Multiscale gas transport in shales with local kerogen heterogeneities. SPE J. 17(04), 1–002 (2012)

    Article  Google Scholar 

  3. Ambrose, R.J., Hartman, R.C., Diaz-Campos, M., Akkutlu, I.Y., Sondergeld, C.H.: Shale gas-in-place calculations part i: new pore-scale considerations. SPE J. 17(01), 219–229 (2012)

    Article  Google Scholar 

  4. Baca, R., Arnett, R., Langford, D.: Modeling fluid flow in fractured porous rock masses by finite element techniques. Int. J. Numer. 4, 337–348 (1984)

    Article  Google Scholar 

  5. Calo, V., Efendiev, Y., Galvis, J., Li, G.: Randomized oversampling for generalized multiscale finite element methods. arXiv:1409.7114.pdf

  6. Calo, V.M., Efendiev, Y., Galvis, J., Ghommem, M.: Multiscale empirical interpolation for solving nonlinear pdes. J. Comput. Phys. 278, 204–220 (2014)

    Article  Google Scholar 

  7. Chaturantabut, S., Sorensen, D.C.: Application of pod and deim to dimension reduction of nonlinear miscible viscous fingering in porous media. Math. Comput. Model. Dyn. Syst. 17, 337–353 (2011)

    Article  Google Scholar 

  8. Chen, Z., Yue, X.: Numerical homogenization of well singularities in the flow transport through heterogeneous porous media. Multiscale Model. Simul. 1(2), 260–303 (2003)

    Article  Google Scholar 

  9. Chung, E.T., Efendiev, Y., Leung, W.T.: Residual-driven online generalized multiscale finite element methods. arXiv preprint arXiv:1501.04565 (2015)

  10. Chung, E.T., Efendiev, Y., Li, G.: An adaptive GMsFEM for high-contrast flow problems. J. Comput. Phys. 273, 54–76 (2014)

    Article  Google Scholar 

  11. Dietrich, P., Helmig, R., Sauter, M., Hötzl, H., Köngeter, J., Teutsch, G.: Flow and transport in fractured porous media. Springer Science & Business Media (2005)

  12. Durlofsky, L.J.: Numerical calculation of equivalent grid bock permeability tensors for heterogeneous porous media. Water Resour. Res. 27(5), 699–708 (1991)

    Article  Google Scholar 

  13. Efendiev, Y., Galvis, J., Gildin, E.: Local-global multiscale model reduction for flows in highly heterogeneous media. J. Comput. Phys. 231(24), 8100–8113 (2012)

    Article  Google Scholar 

  14. Efendiev, Y., Galvis, J., Hou, T.: Generalized multiscale finite element methods. J. Comput. Phys. 251, 116–135 (2013)

    Article  Google Scholar 

  15. Efendiev, Y., Galvis, J., Li, G., Presho, M.: Generalized multiscale finite element methods. oversampling strategies. Int. J. Multiscale Comput. Eng., accepted (2013)

  16. Efendiev, Y., Galvis, J., Wu, X.H.: Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230, 937–955 (2011)

    Article  Google Scholar 

  17. Efendiev, Y., Hou, T., Ginting, V.: Multiscale finite element methods for nonlinear problems and their applications. Commun. Math. Sci. 2, 553–589 (2004)

    Article  Google Scholar 

  18. Efendiev, Y., Lee, S., Li, G., Yao, J., Zhang, N.: Hierarchical multiscale modeling for flows in fractured media using generalized multiscale finite element method. arXiv preprint arXiv:1502.03828. to appear in Int. J. Geomath. doi:10.1007/s13137-015-0075-7 (2015)

  19. Gangi, A.F.: Variation of whole and fractured porous rock permeability with confining pressure. In: International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol. 15, pp. 249–257. Elsevier (1978)

  20. Gong, B., Karimi-Fard, M., Durlofsky, L.J.: Upscaling discrete fracture characterizations to dual-porosity, dual-permeability models for efficient simulation of flow with strong gravitational effects. SPE J. 13(1), 58–67 (2008)

    Article  Google Scholar 

  21. Hajibeygi, H., Karvounis, D., Jenny, P.: A loosely coupled hierarchical fracture model for the iterative multiscale finite volume method. Soc. Petrol. Eng. 230(24), 8729–8743 (2011). doi:10.2118/141991-MS

    Google Scholar 

  22. Hajibeygi, H., Bonfigli, G., Hesse, M.A., Jenny, P.: Iterative multiscale finite-volume method. J. Comput. Phys. 227(19), 8604–8621 (2008)

    Article  Google Scholar 

  23. Hajibeygi, H., Jenny, P.: Adaptive iterative multiscale finite volume method. J. Comput. Phys. 230(3), 628–643 (2011)

    Article  Google Scholar 

  24. Hajibeygi, H., Karvounis, D., Jenny, P., et al.: A loosely coupled hierarchical fracture model for the iterative multiscale finite volume method. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2011)

  25. Jenny, P., Lunati, I.: Modeling complex wells with the multi-scale finite-volume method. J. Comput. Phys. 228(3), 687–702 (2009)

    Article  Google Scholar 

  26. Lee, S.H., Zhou, H., Tchelepi, H.A.: Adaptive multiscale finite-volume method for nonlinear multiphase transport in heterogeneous formations. J. Comput. Phys. 228(24), 9036–9058 (2009)

    Article  Google Scholar 

  27. Lee, S.H., Jensen, C., Lunati, I.: Modeling and simulation of shale gas production in multi-staged hydraulic-fractured formations. In: ECMOR XIII-13th European Conference on the Mathematics of Oil Recovery (2012)

  28. Lee, S.H., Lough, M.F., Jensen, C.L.: Hierarchical modeling of flow in naturally fractured formations with multiple length scales. Water Resour. Res. 37(3), 443–455 (2001)

    Article  Google Scholar 

  29. Li, J., Wang, C., Ding, D., Wu, Y.-S., Di, Y.: A generalized framework model for simulation of gas production in unconventional gas reservoirs. In: Paper SPE 163609 Presented at SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, pp. 18–20 (2013)

  30. Li, L., Lee, S.H.: Efficient field-scale simulation of black oil in naturally fractured reservoir through discrete fracture networks and homogenized media. SPE Reserv. Eval. Eng.

  31. Loucks, R.G., Reed, R.M., Ruppel, S.C., Jarvie, D.M.: Morphology, genesis, and distribution of nanometer-scale pores in siliceous mudstones of the mississippian barnett shale. J. Sediment. Res. 79(12), 848–861 (2009)

    Article  Google Scholar 

  32. Lough, M.F., Lee, S.H., Kamath, J.: A new method to calculate effective permeability of gridblocks used in the simulation of naturally fractured reservoirs. SPE. doi:10.2118/36730-PA

  33. Lunati, I., Tyagi, M., Lee, S.H.: An iterative multiscale finite volume algorithm converging to the exact solution. J. Comput. Phys. 230(5), 1849–1864 (2011)

    Article  Google Scholar 

  34. Karimi-Fard, M.M., Firoozabadi, A.: Numerical simulation of water injection in 2d fractured media using discrete-fracture model. SPE REE J. 4, 117–126 (2003)

    Article  Google Scholar 

  35. Noorishad, J., Mehran, M.: An upstream finite element method for solution of transient transport equation in fractured porous media. Water Resour. Res. 18(3), 588–596 (1982)

    Article  Google Scholar 

  36. Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R: A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv. Water Resour. 29(7), 1020–1036 (2006)

    Article  Google Scholar 

  37. Sondergeld, C.H., Ambrose, R.J., Rai, C.S., Moncrieff, J., et al.: Micro-structural studies of gas shales. In: SPE Unconventional Gas Conference. Society of Petroleum Engineers (2010)

  38. Ţene, M., Al Kobaisi, M.S., Hajibeygi, H., et al.: Algebraic multiscale solver for flow in heterogeneous fractured porous media. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2015)

  39. Wasaki, A., Akkutlu, I.Y, et al.: Permeability of organic-rich shale. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2014)

  40. Wolfsteiner, C., Lee, S.H., Tchelepi, H.A.: Well modeling in the multiscale finite volume method for subsurface flow simulation. Multiscale Model. Simul. 5(3), 900–917 (2006)

    Article  Google Scholar 

  41. Wu, Y.-S., Di, Y., Kang, Z., Fakcharoenphol, P.: A multiple-continuum model for simulating single-phase and multiphase flow in naturally fractured vuggy reservoirs. J. Pet. Sci. Eng. 78(1), 13–22 (2011)

    Article  Google Scholar 

  42. Wu, Y.-S., Qin, G., Ewing, R.E., Efendiev, Y., Kang, Z., Ren, Y.: A multiple-continuum approach for modeling multiphase flow in naturally fractured vuggy petroleum reservoirs. In: SPE-104173, Presented at the 2006 SPE International Oil & Gas Conference and Exhibition in China held in Beijing, China

  43. Yao, J., Sun, H., Fan, D.-Y., Wang, C.-C., Sun, Z.-X.: Numerical simulation of gas transport mechanisms in tight shale gas reservoirs. Pet. Sci. 10(4), 528–537 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yalchin Efendiev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akkutlu, I.Y., Efendiev, Y. & Vasilyeva, M. Multiscale model reduction for shale gas transport in fractured media. Comput Geosci 20, 953–973 (2016). https://doi.org/10.1007/s10596-016-9571-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-016-9571-6

Keywords

Navigation