Skip to main content
SpringerLink
Log in

Partitioned Time Stepping Method for a Dual-Porosity-Stokes Model

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this report, we study a partitioned time stepping algorithm for a dual-porosity-Stokes model, which consists of dual-porosity media and macrofractures/conduits in the coupled system. More specifically, the dual-porosity-Stokes model uses two pressures, the matrix pressure and the fracture pressure, to couple with the Stokes equations. There are four physically valid interface conditions to couple the two models on the interface, including a no-exchange condition, a mass balance condition, a force balance condition, and the Beavers–Joseph condition. To decouple the complex model into three simple sub-problems and solve them separately, we propose a partitioned time stepping method. It solves one, uncoupled matrix pressure equation, microfracture pressure equation and Stokes equation per time step. Under a modest time step restriction of the form \(\triangle t\le C\)(depending on physical parameters) , we prove the stability of the method. We also derive its optimal error estimates. Numerical tests verify the theoretical results.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Beavers, G., Joseph, D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)

    Article  Google Scholar 

  2. Connors, J.M.: Partitioned time discretization for atmosphere–ocean interaction. University of Pittsburgh, Pittsburgh (2010)

    Google Scholar 

  3. Cao, Y., Gunzburger, M., He, X., Wang, X.: Parallel, non-iterative, multi-physics, domain decomposition methods for the time-dependent Stokes–Darcy model. Math. Comput. 83(288), 1617–1644 (2014)

    Article  MATH  Google Scholar 

  4. Cao, Y., Gunzburger, M., Hu, X., Hua, F., Wang, X., Zhao, W.: Finite element approxiations for Stokes–Darcy model with Beavers–Joseph interface boundary condition. SIAM J. Numer. Anal. 6, 4239–4256 (2010)

    Article  MATH  Google Scholar 

  5. Cao, Y., Gunzburger, M., Hua, F., Wang, X.: Coupled Stokes–Darcy model with Beavers–Joseph interface boundary condition. Commun. Math. Sci. 8, 1–25 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao, Y., Gunzburger, M., He, X.M., Wang, X.: Robin–Robin domain decomposition methods for the steady Stokes–Darcy model with Beaver–Joseph interface condition. Numer. Math. 117(4), 601–629 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Connors, J.M., Howell, J.S., Layton, W.J.: Decoupled time stepping methods for fluid–fluid interaction. SIAM J. Numer. Anal. 50(3), 1297–1319 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Connors, J.M., Howell, J.S., Layton, W.J.: Partitioned time stepping for a parabolic two domain problem. SIAM J. Numer. Anal. 47(5), 3526–3549 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, W., Gunzburger, M., Sun, D., Wang, X.: Efficient and long-time accurate second-order methods for the Stokes–Darcy system. SIAM J. Numer. Anal 51(5), 2563–2584 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, W., Gunzburger, M., Sun, D., Wang, X.: An efficient and long-time accurate third-order algorithm for the Stokes–Darcy system. Numer. Math. 134, 857–879 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, C., Wei, M., Chen, H., He, X., Bai, B.: Improved numerical simulation for shale gas reservoirs, OTC-24913. In: Offshore Technology Conference Asia, Kuala Lumpur, Malaysia (2014). https://doi.org/10.4043/24913-MS

  13. Hou, J.Y., Qiu, M., He, X.M., Guo, C.H., Wei, M.Z., Bai, B.J.: A dual-porosity-Stokes model and finite element method for coupling dual-porosity flow and free flow. SIAM. J. Sci. Comput. 38–5, B710–B739 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. He, X.M., Li, J., Lin, Y., Ming, J.: A domain decomposition method for the steady-state Navier–Stokes–Darcy model with Beavers–Joseph interface condition. SIAM J. Sci. Comput. 37(5), S264–S290 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jaeger, W., Mikelic, A.: On the interface boundary conditions of Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kubacki, M.: Uncoupling evolutionary groundwater-surface water flows using the Crank–Nicolson LeapFrog method. Numer. Methods Partial Differ. Equ. 29, 248–272 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lions, J.-L., Magenes, E.: Probl\(\acute{e}\)mes aux limites nonhomog\(\grave{e}\)nes at applications, vol. 1. Dunod, Paris (1968)

    Google Scholar 

  18. Layton, W.J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40, 2195–2218 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Layton, W.J., Trenchea, C.: Stability of the IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations. Appl. Numer. Math. 62(2), 112–120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Layton, W.J., Tran, H., Trenchea, C.: Analysis of long time stability and errors of two partitioned method for uncoupling evolutionary groundwater-surface water flows. SIAM J. Numer. Anal. 51(1), 248–272 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Layton, W.J., Tran, H., Xiong, X.: Long time stability of four methods for splitting the evolutionary Stokes–Darcy problem into Stokes and Darcy subproblems. J. Comput. Appl. Math. 236, 3198–3217 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mu, M., Zhu, X.H.: Decoupled schemes for a non-stationary mixed Stokes–Darcy model. Math. Comput. 79, 707–731 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sun, D.: High order long-time accurate methods for the Stokes–Darcy system and uncertainty quantification of contaminant transport. Florida State University, dissertation (2015)

  24. Shan, L., Zheng, H.: Partitioned time stepping method for fully evolutionary Stokes–Darcy flow with Beavers–Joseph interface conditions. SIAM J. Numer. Anal 51(2), 813–839 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shan, L., Zheng, H., Layton, W.J.: A decoupling method with different subdomain time steps for the nonstationary Stokes–Darcy model. Numer Methods Partial Differ. Equ. 29, 549–583 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoies. Soc. Petrol. Eng. J. 3, 245–255 (1963). https://doi.org/10.2118/426-PA

    Article  Google Scholar 

  27. Zhang, Y., Hou, Y., Shan, L.: Stability and convergence analysis of a decoupled algorithm for a fluid–fluid interaction problem. SIAM J. Numer. Anal. 54(5), 2833–2867 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science, Liaoning Technical University, Fu Xin, 123000, People’s Republic of China

    Li Shan

  2. School of Mathematics, Northwest University, Xi’an, 710127, Shaanxi, People’s Republic of China

    Jiangyong Hou

  3. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Wenjing Yan & Jie Chen

Authors
  1. Li Shan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiangyong Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenjing Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jie Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiangyong Hou.

Additional information

This work is supported by NSFC (Grant Nos. 11401284, 11701451 and 91730306), NSFC of Liaoning Province (20180551138), Scientific Research Program Funded by Shaanxi Provincial Education Department (17JK0787), and Natural Science Foundation of Shaanxi Province (2018JQ1077), China Postdoctoral Science Foundation (2015T81012).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shan, L., Hou, J., Yan, W. et al. Partitioned Time Stepping Method for a Dual-Porosity-Stokes Model. J Sci Comput 79, 389–413 (2019). https://doi.org/10.1007/s10915-018-0879-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0879-3

Keywords

Search

Navigation