Skip to main content
SpringerLink
Log in

Application of diffuse source basis functions for improved near well upscaling

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

Abstract

Near well flow can have a significant impact on the accuracy of the upscaling of geologic models. A recent benchmark study has shown that these errors may dominate over other aspects of upscaling in commercial reservoir simulators. This same study showed the advantage of “Diffuse Source” (DS) upscaling over previous approaches. We now demonstrate the application of the DS basis functions to the calculation of the upscaled well index and the well cell intercell transmissibilities. DS upscaling is an extension of pseudo-steady-state (PSS) flow based upscaling that utilizes the diffusive time of flight to distinguish well-connected and weakly-connected sub-volumes. DS upscaling retains the localization advantage of a PSS calculation: unlike steady state flow, the local upscaling problem does not couple to adjacent regions, and local-global iterations are not required. DS upscaling has been developed and utilized for the calculation of the intercell transmissibility, but we now apply it to the calculation of the upscaled well index. Consistent with other researchers, we adjust the intercell transmissibilities in the near well region. We also consider the upscaling of the well index for a reservoir model in which the well trajectory and the high resolution geologic model are not simultaneously available. For many practitioners, this remains the most common reservoir modeling workflow. The result is an algebraic well index upscaling calculation, which also improves upon commercial applications. The industry standard for the well index follows Peaceman. We show that PSS/DS upscaling reduces to Peaceman’s well index on a coarse grid, and is consistent with Peaceman’s numerical convergence analysis. (In contrast, steady state upscaling for the well index reduces to the Dietz well index.) The current approach is a generalization of Peaceman’s well index, but now extended to represent near well reservoir heterogeneity and with the arbitrary placement of a well perforation within a simulation well cell. Consistent with steady state upscaling, we find an advantage in adjusting the intercell transmissibility in the near well region. However, we have found that it is only necessary to do so for the well cell itself, which may be a consequence of the improved localization of the current calculation. The new methodology performs very well. It is tested for two models, including the SPE10 reference model and the Amellago carbonate outcrop model. We compare the results to steady state flow based upscaling, the algebraic well index upscaling described above, and algorithms found in commercial applications.

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. Ding, D.: Near-well upscaling for reservoir simulations. Oil Gas Sci. Technol.-Revue De L Institut Francais Du Petrole. 59, 157–165 (2004). https://doi.org/10.2516/ogst:2004012

    Article  Google Scholar 

  2. Chawathe, A., Taggart, I.: Insights into upscaling using 3D streamlines. SPE Reserv. Eval. Eng. 7(04), 285–296 (2004). https://doi.org/10.2118/88846-PA

    Article  Google Scholar 

  3. Chen, Y., Durlofsky, L.J., Gerritsen, M., Wen, X.H.: A coupled local–global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26(10), 1041–1060 (2003). https://doi.org/10.1016/S0309-1708(03)00101-5

    Article  Google Scholar 

  4. Christie, M.A.: Upscaling for reservoir simulation. J. Pet. Technol. 48, 1004–1010 (1996). https://doi.org/10.2118/37324-JPT

    Article  Google Scholar 

  5. King, P.R.: The use of renormalization for calculating effective permeability. Transp. Porous Media. 4(1), 37–58 (1989). https://doi.org/10.1007/bf00134741

    Article  Google Scholar 

  6. Warren, J.E., Price, H.S.: Flow in heterogeneous porous media. Soc. Pet. Eng. J. 1(03), 153–169 (1961). https://doi.org/10.2118/1579-G

    Article  Google Scholar 

  7. Nunna, K., Liu, C.-H., King, M.J.: Application of diffuse source functions for improved flow upscaling. Comput. Geosci. 24, 493–507 (2019). https://doi.org/10.1007/s10596-019-09868-x

    Article  Google Scholar 

  8. Durlofsky, L.J., Milliken, W.J., Bernath, A.: Scaleup in the near-well region. SPE J. 5(01), 110–117 (2000). https://doi.org/10.2118/61855-PA

    Article  Google Scholar 

  9. King, M.J.: Upgridding and upscaling: current trends and future directions. SPE Distinguished Lecture (2007). https://doi.org/10.2118/112810-DL

  10. Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation (includes associated paper 6988). SPE J. 18, 183–194 (1978). https://doi.org/10.2118/6893-PA

    Article  Google Scholar 

  11. Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. SPE J. 23, 531–543 (1983). https://doi.org/10.2118/10528-PA

    Article  Google Scholar 

  12. Peaceman, D.W.: Interpretation of wellblock pressures in numerical reservoir simulation: part 3 -- off-center and multiple wells within a wellblock. SPE Reserv. Eng. 5(02), 227–232 (1990). https://doi.org/10.2118/16976-pa

    Article  Google Scholar 

  13. Peaceman, D.W.: Representation of a horizontal well in numerical reservoir simulation. SPE Adv. Technol. Ser. 1(01), 7–16 (1993). https://doi.org/10.2118/21217-PA

    Article  Google Scholar 

  14. Karimi-Fard, M., Durlofsky, L.: Accurate resolution of near-well effects in upscaled models using flow-based unstructured local grid refinement. SPE J. 17(04), 1084–1095 (2012). https://doi.org/10.2118/141675-PA

    Article  Google Scholar 

  15. Fu, J., Axness, C.L., Gomez-Hernandez, J.J.: Upscaling transmissivity in the near-well region for numerical simulation: a comparison on uncertainty propagation. Eng. Appl. Comput. Fluid Mech. 5(1), 49–66 (2011). https://doi.org/10.1080/19942060.2011.11015352

    Article  Google Scholar 

  16. Liu, C.-H., Nunna, K., Syed, I., King, M.J.: Evaluation of upscaling approaches for the Amellago carbonate outcrop model. Paper presented at the SPE Europec featured at 81st EAGE Conference and Exhibition. (2019). https://doi.org/10.2118/195560-MS

  17. Desbarats, A.J.: Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media. Math. Geol. 24(3), 249–267 (1992). https://doi.org/10.1007/bf00893749

    Article  Google Scholar 

  18. Ding, Y.: Scaling-up in the vicinity of wells in heterogeneous field. Paper presented at the SPE Reservoir Simulation Symposium. (1995). https://doi.org/10.2118/29137-MS

  19. Muggeridge, A.H., Cuypers, M., Bacquet, C., Barker, J.W.: Scale-up of well performance for reservoir flow simulation. Pet. Geosci. 8(2), 133–139 (2002). https://doi.org/10.1144/petgeo.8.2.133

    Article  Google Scholar 

  20. Holden, L., Nielsen, B.F.: Global upscaling of permeability in heterogeneous reservoirs: the output least squares (OLS) method. Transp. Porous Media. 40(2), 115–143 (2000). https://doi.org/10.1023/a:1006657515753

    Article  Google Scholar 

  21. Chen, Y., Durlofsky, L.J.: Adaptive local–global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Media. 62(2), 157–185 (2006). https://doi.org/10.1007/s11242-005-0619-7

    Article  Google Scholar 

  22. Renard, P., de Marsily, G.: Calculating equivalent permeability: a review. Adv. Water Resour. 20(5), 253–278 (1997). https://doi.org/10.1016/S0309-1708(96)00050-4

    Article  Google Scholar 

  23. Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4, 308–317 (2001). https://doi.org/10.2118/72469-PA

    Article  Google Scholar 

  24. Geiger, S.: Amellago Carbonate Reservoir Model, ExxonMobil (FC)2 Alliance. In. International Centre for Carbonate Reservoirs, Heriot-Watt University, (2016)

  25. Shekhar, R., Sahni, I., Benson, G., Agar, S., Amour, F., Tomás, S., Christ, N., Alway, R., Mutti, M., Immenhauser, A., Karcz, Z., Kabiri, L.: Modelling and simulation of a Jurassic carbonate ramp outcrop, Amellago, high Atlas Mountains, Morocco. Pet. Geosci. 20(1), 109–123 (2014). https://doi.org/10.1144/petgeo2013-010

    Article  Google Scholar 

  26. Abramowitz, M., Stegun, I., Romer, R.: Exponential integral and related functions. In: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, American Journal of Physics 56, 958-958 (1988). https://doi.org/10.1119/1.15378

  27. Schlumberger: ECLIPSE Version 2017.1 Reference Manual. (2017)

  28. King, M.J., MacDonald, D.G., Todd, S.P., Leung, H.: Application of novel upscaling approaches to the Magnus and Andrew reservoirs. Paper presented at the SPE European Petroleum Conference. (1998). https://doi.org/10.2118/50643-MS

  29. Li, H., Durlofsky, L.J.: Upscaling for compositional reservoir simulation. SPE J. 21(03), 873–887 (2016). https://doi.org/10.2118/173212-PA

    Article  Google Scholar 

  30. Zhou, Y., King, M.J.: Improved upscaling for flow simulation of tight gas reservoir models. Paper presented at the SPE Annual Technical Conference and Exhibition. (2011). https://doi.org/10.2118/147355-MS

  31. Ding, D.Y.: Modeling formation damage for flow simulations at reservoir scale. SPE J. 15(03), 737–750 (2010). https://doi.org/10.2118/121805-PA

    Article  Google Scholar 

  32. Li, H., Chen, Y., Rojas, D., Kumar, M.: Development and application of near-well multiphase upscaling for forecasting of heavy oil primary production. J. Pet. Sci. Eng. 113, 81–96 (2014). https://doi.org/10.1016/j.petrol.2014.01.002

    Article  Google Scholar 

  33. Mascarenhas, O., Durlofsky, L.J.: Coarse scale simulation of horizontal wells in heterogeneous reservoirs. J. Pet. Sci. Eng. 25(3), 135–147 (2000). https://doi.org/10.1016/S0920-4105(00)00009-7

    Article  Google Scholar 

  34. Nakashima, T., Li, H., Durlofsky, L.J.: Near-well upscaling for three-phase flows. Comput. Geosci. 16(1), 55–73 (2012). https://doi.org/10.1007/s10596-011-9252-4

    Article  Google Scholar 

  35. Soeriawinata, T., Kasap, E., Kelkar, M.: Permeability upscaling for near-wellbore heterogeneities. SPE Form. Eval. 12(04), 255–262 (1997). https://doi.org/10.2118/36518-PA

    Article  Google Scholar 

  36. Wen, X.-H., Durlofsky, L.J., Chen, Y.: Efficient 3D implementation of local-global upscaling for reservoir simulation. SPE J. 11(04), 443–453 (2006). https://doi.org/10.2118/92965-PA

    Article  Google Scholar 

  37. Fincham, A.E., Christensen, J.R., Barker, J.W., Samier, P.: Up-gridding from geological model to simulation model: review, applications and limitations. Paper presented at the SPE Annual Technical Conference and Exhibition. (2004). https://doi.org/10.2118/90921-ms

  38. Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Methods: Theory and Applications, vol. 4. Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York, New York (2009). https://doi.org/10.1007/978-0-387-09496-0

  39. Stern, D.: Practical aspects of Scaleup of simulation models. J. Pet. Technol. 57(09), 74–81 (2005). https://doi.org/10.2118/89032-jpt

    Article  Google Scholar 

  40. Wu, X.-H., Stone, M., Parashkevov, R., Stern, D., Lyons, S.L.: Reservoir modeling with global scale-up. Paper presented at the SPE Middle East Oil and Gas Show and Conference. (2007). https://doi.org/10.2118/105237-ms

  41. Cardwell Jr., W.T., Parsons, R.L.: Average permeabilities of heterogeneous oil sands. Trans. AIME. 160(01), 34–42 (1945). https://doi.org/10.2118/945034-G

    Article  Google Scholar 

  42. Nunna, K., King, M.J.: Dynamic downscaling and upscaling in high contrast systems. Paper presented at the SPE Reservoir Simulation Conference. (2017). https://doi.org/10.2118/182689-MS

  43. Nunna, K., King, M.J.: Dynamic diffuse-source upscaling in high-contrast systems. SPE J. 25(01), 347–368 (2020). https://doi.org/10.2118/182689-PA

    Article  Google Scholar 

  44. Nunna, K., Zhou, P., King, M.J.: Novel diffuse source pressure transient upscaling. Paper presented at the SPE Reservoir Simulation Symposium. (2015). https://doi.org/10.2118/173293-MS

  45. Peaceman, D.W.: Recalculation of Dietz shape factor for rectangles. SPE Reserv. Eng. 10 (1990). https://doi.org/10.2118/21256-MS

  46. Krogstad, S., Lie, K.-A., Møyner, O., Nilsen, H.M., Raynaud, X., Skaflestad, B.: MRST-AD – an open-source framework for rapid prototyping and evaluation of reservoir simulation problems. Paper presented at the SPE Reservoir Simulation Symposium. (2015). https://doi.org/10.2118/173317-MS

  47. Lie, K.-A.: An Introduction to Reservoir Simulation using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST). Cambridge University Press, Cambridge (2019). https://doi.org/10.1017/9781108591416

  48. Jayasinghe, S., Darmofal, D.L., Dow, E., Galbraith, M.C., Allmaras, S.R.: A discretization-independent distributed well model. SPE J. 24(06), 2946–2967 (2019). https://doi.org/10.2118/198898-PA

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to gratefully acknowledge the support of Energi Simulation through the Texas A&M chair in Robust Reduced Complexity Modeling and the support of the members of the MCERI joint industry project at Texas A&M University. We also acknowledge the support provided by the ExxonMobil (FC)2 Alliance together with Sebastian Geiger and the International Centre for Carbonate Reservoirs at Heriot-Watt University for access to the Amellago model, and for Schlumberger for the use of their reservoir modeling applications.

Abbreviations

Property

Description

Unit of Measure

Conversion to SI

(*) Exact Conversion

Latin

   

(i, j, k)

Fine cell indices

[−]

[−]

k

Permeability

md

9.869233 × 10−16m2

p

Pressure

psi

6.89476 × 103Pa

\( {\overline{p}}_f \)

Averaged pressure on surface

psi

6.89476 × 103Pa

p wf

Bottomhole flowing pressure

psi

6.89476 × 103Pa

q w

Well flux

ft3/day

((0.3048)3/86400)m3/sec

q f

Total face flux

ft3/day

((0.3048)3/86400)m3/sec

t

Time

day

86400sec (*)

T f

Intercell transmissibility

md ⋅ ft

(9.869233)(0.3048) × 10−16m3

V p

Pore volume

ft 3

(0.3048)3m3 (*)

WI

Well index

md ⋅ ft

(9.869233)(0.3048) × 10−16m3

Greek

   

α

Hydraulic diffusivity

ft2/day

((0.3048)2/86400)m2/sec (*)

ϕ

Porosity

[1]

1 (*)

μ

Fluid viscosity

cp

10−3Pa · sec (*)

τ

Diffusive time of flight

\( \sqrt{day} \)

\( \sqrt{86400\sec } \) (*)

Author information

Authors and Affiliations

  1. Texas A&M University, College Station, TX, 77840, USA

    Ching-Hsien Liu, Krishna Nunna & Michael J. King

  2. Oxy, 5 Greenway Plaza, Houston, TX, USA

    Krishna Nunna

Authors
  1. Ching-Hsien Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Krishna Nunna
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael J. King
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ching-Hsien Liu.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, CH., Nunna, K. & King, M.J. Application of diffuse source basis functions for improved near well upscaling. Comput Geosci 26, 823–846 (2022). https://doi.org/10.1007/s10596-021-10117-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-021-10117-3

Keywords

Search

Navigation