Skip to main content
SpringerLink
Log in

A scalable multistage linear solver for reservoir models with multisegment wells

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

Wellbore flow and interactions between wells and the reservoir can be complex. Accurate modeling of these behaviors is especially important for multilateral and other advanced wells. This paper describes a new scalable linear solver for flow simulation of detailed reservoir models with advanced wells and well groups. A general purpose research simulator serves as the computational platform, in which a multisegment well (MsWell) model is used to describe wellbore flow. In the MsWell model, the wellbore is discretized into a number of segments. Hence, the MsWell model adds a large number of equations and unknowns, which are fully coupled to the reservoir model. Operating constraints on groups of wells add one more level of complexity to the system. The new linear solver is a generalized two-stage constrained pressure residual preconditioner. A global pressure system is obtained algebraically in the first stage. The system represents the pressure coupling between the reservoir and wells accurately. The well groups are disaggregated into individual multisegment wells, which then are further reduced to a standard well-like form. The two-stage scheme serves as the inner loop of a generalized minimum residual solver. Algebraic multigrid is used to compute the first-stage pressure solution; a special block-based incomplete lower–upper preconditioner is used for the second stage. We demonstrate the superior performance of this new solver compared with state-of-the-art methods using a variety of highly detailed reservoir models with complex wells and well groups.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Science Publishers, London (1979)

    Google Scholar 

  2. Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B., Zhang, H.: PETSc Users Manual, Revision 3.2. Mathematics and Computer Sciences Division, Argonne National Laboratory (2011)

  3. Behie, G.: Practical considerations for incomplete factorization methods in reservoir simulation. In: Proceedings of the 7th SPE Symposium on Reservoir Simulation, SPE 12263, San Francisco, CA (1983)

  4. Byer, T.: Preconditioned Newton methods for simulation of reservoir with surface facilities. Ph.D. thesis, Stanford University (2000)

  5. Byer, T., Edwards, M., Aziz, K.: Preconditioned Newton methods for fully coupled reservoir and surface facility models. In: Proceedings of the SPE annual technical conference and exhibition, SPE 49001, New Orleans, LA (1998)

  6. Cao, H.: Development of techniques for general purpose simulation. Ph.D. thesis, Stanford University (2002)

  7. Cao, H., Tchelepi, H., Wallis, J., Yardumian, H.: Parallel scalable CPR-type linear solver for reservoir simulation. In: Proceedings of the SPE annual technical conference and exhibition, SPE 96809, Dallas, TX (2005)

  8. Chien, M.C., Tchelepi, H.A., Yardumian, H.E., Chen, W.H.: A scalable parallel multi-purpose reservoir simulator. In: Proceedings of the 14th SPE Reservoir Simulation Symposium, SPE 37976, Dallas, TX (1997)

  9. Christie, M., Blunt, M.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng., SPE 72469 4(4), 308–317 (2001)

    Google Scholar 

  10. Coats, B., Fleming, G., Watts, J., Rame, M., Shiralkar, G.: A generalized wellbore and surface facility model, fully coupled to a reservoir simulator. SPE Reserv. Eval. Eng., SPE 87913 7(2), 132–142 (2004)

    Google Scholar 

  11. Coats, K.: A note on IMPES and some IMPES based simulation models. SPE J., SPE 49774 5(3), 245–251 (2000)

    Google Scholar 

  12. Deimbacher, F., Heinemann, Z.: Time-dependent incorporation of locally irregular grids in large reservoir simulation models. In: Proceedings of the 12th SPE Symposium on Reservoir Simulation, SPE 25260, New Orleans, LA (1993)

  13. Fan, Y.: Chemical reaction modeling in a subsurface flow simulator with application to in-situ upgrading and CO2 mineralization. Ph.D. thesis, Stanford University (2010)

  14. Fung, L.S., Dogru, A.H.: Parallel unstructured-solver methods for simulation of complex giant reservoirs. SPE J., SPE 106237 13(4), 440–446 (2008)

    Google Scholar 

  15. Gee, M.W., Siefert, C.M.: ML 5.0 Smoothed Aggregation User’s Guide. Computational Math & Algorithms, Sandia National Laboratories (2007)

  16. GeoQuest, S.: Eclipse Technical Description. Multi-Segment Wells (2005)

  17. Ghorayeb, K., Holmes, J., Torrens, R., Grewal, B.: A general purpose controller for coupling multiple reservoir simulations and surface facility networks. In: Proceedings of the 17th SPE reservoir simulation symposium, SPE 79702, Houston, TX (2003)

  18. Gong, B.: Effective models of fractured systems. Ph.D. thesis, Stanford University (2007)

  19. Holmes, J.A.: Enhancements to the strongly coupled, fully implicit well model: wellbore crossflow modeling and collective well control. In: Proceedings of the 7th SPE Reservoir Simulation Symposium, SPE 12259, San Francisco, CA (1983)

  20. Holmes, J., Barkve, T., Lund, O.: Application of a multisegment well model to simulate flow in advanced wells. In: Proceedings of the SPE European petroleum conference, SPE 50646, The Hague, Netherlands (1998)

  21. Jiang, Y.: Tracer flow modeling and efficient solvers for GPRS. Master’s thesis, Stanford University (2004)

  22. Jiang, Y.: Techniques for modeling complex reservoirs and advanced wells. Ph.D. thesis, Stanford University (2007)

  23. Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discrete fracture model applicable for general-purpose reservoir simulators. SPE J., SPE 79699 9(2), 227–236 (2004)

    Google Scholar 

  24. Li, D., Beckner, B., Kumar, A.: A new efficient averaging technique for scaleup of multimillion-cell geologic models. SPE Reserv. Eval. Eng., SPE 56554 4(4), 297–307 (2001)

    Google Scholar 

  25. Litvak, M., Darlow, B.: Surface network and well tubinghead pressure constraints in compositional simulation. In: Proceedings of the 13th SPE Reservoir Simulation Symposium, SPE 29125, San Antonio, TX (1995)

  26. Nacul, E., Lepretre, C., Pedrosa Jr., O., Girard, P., Aziz, K.: Efficient use of domain decomposition and local grid refinement in reservoir simulation. In: Proceedings of the SPE Annual Technical Conference and Exhibition, SPE 20740, New Orleans, LA (1990)

  27. Nolen, J.: Treatment of wells in reservoir simulation. Tech. Rep. (1990)

  28. Pedrosa Jr., O.A., Aziz, K.: Use of a hybrid grid in reservoir simulation. SPE Reservoir Engineering, SPE 13507 1(6), 611–621 (1986)

    Google Scholar 

  29. Saad, Y.: ILUT: a dual threshold incomplete LU factorization. Numer. Linear Algebra Appl. 1(4), 387–402 (1994)

    Article  Google Scholar 

  30. Saad, Y.: Iterative Methods for Sparse Linear Systems, 1st edn. PWS Publishing, Boston (1996)

    Google Scholar 

  31. Saad, Y., Schultz, M.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  Google Scholar 

  32. Saad, Y., Suchomel, B.: ARMS: an algebraic recursive multilevel solver for general sparse linear systems. Numer. Linear Algebra Appl. 9, 359–378 (2002)

    Article  Google Scholar 

  33. Schiozer, D.: Simultaneous simulation of reservoir and surface facilities. Ph.D. thesis, Stanford University (1994)

  34. Schiozer, D., Aziz, K.: Use of domain decomposition of simultaneous simulation of reservoir an surface facilities. In: Proceedings of the SPE western regional meeting, SPE 27876, Long Beach, CA (1994)

  35. Shi, H., Holmes, J., Diaz, L., Durlofsky, L.J., Aziz, K.: Drift-flux parameters for three-phase steady-state flow in wellbores. SPE J., SPE 89836 10(2), 130–137 (2005)

    Google Scholar 

  36. Shi, H., Holmes, J., Durlofsky, L.J., Aziz, K., Diaz, L., Alkaya, B., Oddie, G.: Drift-flux modeling of two-phase flow in wellbores. SPE J., SPE 84228 10(1), 24–33 (2005)

    Google Scholar 

  37. Shiralkar, G., Watts, J.: An efficient formulation for simultaneous solution of the surface network equations. In: Proceedings of the 18th SPE Reservoir Simulation Symposium, SPE 93073, The Woodlands, TX (2005)

  38. Smith, B., Bjorstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  39. Stueben, K.: Algebraic multigrid (AMG): experiences and comparisons. In: Proceedings of the International Multigrid Conference (1983)

  40. Stueben, K.: An introduction to algebraic multigrid. Appendix in book ‘Multigrid’ pp. 413–532 (2001)

  41. Stueben, K., Clees, T.: SAMG User’s Manual. Fraunhofer Institute SCAI (2003)

  42. Voskov, D., Tchelepi, H.: Compositional space parameterization: theory and application for immiscible displacements. SPE J., SPE 106029 14(3), 431–440 (2009)

    Google Scholar 

  43. Voskov, D., Younis, R., Tchelepi, H.: General nonlinear solution strategies for multiphase multicomponent eos based simulation. In: Proceedings of the 20th SPE Reservoir Simulation Symposium, SPE 118996, The Woodlands, TX (2009)

  44. Wallis, J.: Incomplete Gaussian elimination as a preconditioning for generalized conjugate gradient acceleration. In: Proceedings of the 7th SPE symposium on reservoir simulation, SPE 12265, San Francisco, CA (1983)

  45. Wallis, J., Kendall, R., Little, T., Nolen, J.: Constrained residual acceleration of conjugate residual methods. In: Proceedings of the 8th SPE symposium on reservoir simulation, SPE 13536, Dallas, TX (1985)

  46. Younis, R.: Modern advances in software and solution algorithms for reservoir simulation. Ph.D. thesis, Stanford University (2012)

  47. Zhou, Y.: Multistage preconditioner for well groups and automatic differentiation for next generation GPRS. Master’s thesis, Stanford University (2009)

  48. Zhou, Y., Tchelepi, H.A., Mallison, B.T.: Automatic differentiation framework for compositional simulation on unstructured grids with multi-point discretization schemes. In: Proceedings of the 21st SPE Reservoir Simulation Symposium, SPE 141592, The Woodlands, TX (2011)

Download references

Author information

Author notes

  1. Yuanlin Jiang

    Present address: Quantum Reservoir Impact, LLC, 909 Fannin St., Houston, TX, 77010, USA

Authors and Affiliations

  1. Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA

    Yifan Zhou, Yuanlin Jiang & Hamdi A. Tchelepi

Authors
  1. Yifan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuanlin Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hamdi A. Tchelepi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yifan Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, Y., Jiang, Y. & Tchelepi, H.A. A scalable multistage linear solver for reservoir models with multisegment wells. Comput Geosci 17, 197–216 (2013). https://doi.org/10.1007/s10596-012-9324-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-012-9324-0

Keywords

Search

Navigation