Abstract
In this study, we propose the use of a first-order gradient framework, the adaptive moment estimation (Adam), in conjunction with a stochastic gradient approximation, to well location and trajectory optimization problems. The Adam framework allows the incorporation of additional information from previous gradients to calculate variable-specific progression steps. As a result, this assists the search progression to be adjusted further for each variable and allows a convergence speed-up in problems where the gradients need to be approximated. We argue that under computational budget constraints, local optimization algorithms provide suitable solutions from a heuristic initial guess. Nonlinear constraints are taken into account to ensure the proposed solutions are not in violation of practical field considerations. The performance of the proposed algorithm is compared against steepest descent and generalized pattern search, using two case studies — the placement of four vertical wells and placement of 20 nonconventional (deviated, horizontal and/or slanted) wells. The results indicate that the proposed algorithm consistently outperforms the tested methods in terms computational efficiency and final optimum value. Additional discussions regarding nonconventional parameterization provide insights into simultaneous perturbation gradient approximations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahmad, A., El-Shafie, A., Razali, S.F.M., Mohamad, Z.S.: Reservoir optimization in water resources: a review. Water. Resour. Manag. 28(11), 3391–3405 (2014)
Cameron, D.A., Durlofsky, L.J.: Optimization of well placement, co2 injection rates, and brine cycling for geological carbon sequestration. Int. J. Greenhouse Gas Control 10, 100–112 (2012). https://doi.org/10.1016/j.ijggc.2012.06.003, https://www.sciencedirect.com/science/article/pii/S1750583612001296
Arouri, Y., Sayyafzadeh, M.: An accelerated gradient algorithm for well control optimization. J. Pet. Sci. Eng. 190, 106872 (2020). https://doi.org/10.1016/j.petrol.2019.106872, http://www.sciencedirect.com/science/article/pii/S0920410519312884
Alrashdi, Z., Sayyafzadeh, M.: (+) evolution strategy algorithm in well placement, trajectory, control and joint optimisation. J. Pet. Sci. Eng. 177, 1042–1058 (2019). https://doi.org/10.1016/j.petrol.2019.02.047, http://www.sciencedirect.com/science/article/pii/S0920410519301846
Sayyafzadeh, M.: Reducing the computation time of well placement optimisation problems using self-adaptive metamodelling. J. Pet. Sci. Eng. 151, 143–158 (2017)
Bittencourt, A.C., Horne, R.N.: Reservoir development and design optimization. In: SPE Annual Technical Conference and Exhibition. https://doi.org/10.2118/38895-MS. Society of Petroleum Engineers (1997)
Yeten, B., Durlofsky, L.J., Aziz, K.: Optimization of nonconventional well type, location, and trajectory. SPE J. 8(03), 200–210 (2003)
Onwunalu, J.E., Durlofsky, L.J.: Application of a particle swarm optimization algorithm for determining optimum well location and type. Comput. Geosci. 14(1), 183–198 (2010). https://doi.org/10.1007/s10596-009-9142-1
Sayyafzadeh, M., Alrashdi, Z.: Well controls and placement optimisation using response-fed and judgement-aided parameterisation: Olympus optimisation challenge. Comput. Geosci. https://doi.org/10.1007/s10596-019-09891-y(2019)
Bouzarkouna, Z., Ding, D.Y., Auger, A.: Well placement optimization with the covariance matrix adaptation evolution strategy and meta-models. Comput. Geosci. 16(1), 75–92 (2012)
Martí, R: Multi-start Methods, pp. 355–368. Springer (2003)
O’Donoghue, B., Candàs, E: Adaptive restart for accelerated gradient schemes. Found. Comput. Math. 15(3), 715–732 (2015). https://doi.org/10.1007/s10208-013-9150-3
Bellout, M.C., EcheverríaCiaurri, D, Durlofsky, L.J., Foss, B., Kleppe, J.: Joint optimization of oil well placement and controls. Comput. Geosci. 16(4), 1061–1079 (2012). https://doi.org/10.1007/s10596-012-9303-5
Isebor, O.J., EcheverríaCiaurri, D, Durlofsky, L.J.: Generalized field-development optimization with derivative-free procedures. SPE J. 891–908. https://doi.org/10.2118/163631-PA (2014)
Wang, H., Ciaurri, D.E., Durlofsky, L.J.: Use of retrospective optimization for placement of oil wells under uncertainty. In: Proceedings of the 2010 Winter Simulation Conference, pp 1750–1757 (2010)
Wang, H., Echeverría-Ciaurri, D, Durlofsky, L., Cominelli, A.: Optimal well placement under uncertainty using a retrospective optimization framework. SPE J. 17(01), 112–121 (2012). https://doi.org/10.2118/141950-PA
Jansen, J.D.: Adjoint-based optimization of multi-phase flow through porous media - a review. Comput. Fluids 46(1), 40–51 (2011). https://doi.org/10.1016/j.compfluid.2010.09.039, http://www.sciencedirect.com/science/article/pii/S0045793010002677
Sarma, P., Chen, W.H.: Efficient well placement optimization with gradient-based algorithms and adjoint models. In: Intelligent Energy Conference and Exhibition (2008)
Wang, C., Li, G., Reynolds, A.C.: Optimal well placement for production optimization. In: Eastern Regional Meeting. https://doi.org/10.2118/111154-MS, p 5. Society of Petroleum Engineers (2007)
Zandvliet, M., Handels, M., van Essen, G., Brouwer, R., Jansen, J-D: Adjoint-based well-placement optimization under production constraints. J. Pet. Sci. Eng. https://doi.org/10.2118/105797-PA (2008)
Bangerth, W., Klie, H., Wheeler, M.F., Stoffa, P.L., Sen, M.K.: On optimization algorithms for the reservoir oil well placement problem. Comput. Geosci. 10(3), 303–319 (2006). https://doi.org/10.1007/s10596-006-9025-7
Jesmani, M., Jafarpour, B., Bellout, M.C., Hanea, R.G., Foss, B.: Application of simultaneous perturbation stochastic approximation to well placement optimization under uncertainty. In: ECMOR XV-15th European Conference on the Mathematics of Oil Recovery, pp cp–494–00133. European Association of Geoscientists & Engineers (2016)
Vlemmix, S., Joosten, GerardJP, Brouwer, R., Jansen, J-D: Adjoint-based well trajectory optimization. In: EUROPEC/EAGE Conference and Exhibition. Society of Petroleum Engineers (2009)
Volkov, O., Bellout, M.C.: Gradient-based constrained well placement optimization. J. Pet. Sci. Eng. 171, 1052–1066 (2018). https://doi.org/10.1016/j.petrol.2018.08.033, http://www.sciencedirect.com/science/article/pii/S0920410518306995
Leeuwenburgh, O., Egberts, P.J., P.Abbink, O.A.: Ensemble methods for reservoir life-cycle optimization and well placement. In: SPE/DGS Saudi Arabia Section Technical Symposium and Exhibition. https://doi.org/10.2118/136916-MS, p 8. Society of Petroleum Engineers (2010)
Li, L., Jafarpour, B., Mohammad-Khaninezhad, M.R.: A simultaneous perturbation stochastic approximation algorithm for coupled well placement and control optimization under geologic uncertainty. Comput. Geosci. 17(1), 167–188 (2013). https://doi.org/10.1007/s10596-012-9323-1
Guyaguler, B., Horne, R.N.: Uncertainty assessment of well placement optimization. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2001)
Ding, Y.: Optimization of well placement using evolutionary methods. In: Europec/EAGE Conference and Exhibition. Society of Petroleum Engineers (2008)
Forouzanfar, F., Reynolds, A.C.: Well-placement optimization using a derivative-free method. J. Pet. Sci. Eng. 109, 96–116 (2013). https://doi.org/10.1016/j.petrol.2013.07.009, http://www.sciencedirect.com/science/article/pii/S0920410513001952
Emerick, A.A., Silva, B., Almeida, L.F., Szwarcman, D., Pacheco, M.A.C., Vellasco, M.M.B.R.: Well placement optimization using a genetic algorithm with nonlinear constraints. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2009)
Wu, Y., Schuster, M., Chen, Z., Le, Q.V., Norouzi, M., Macherey, W., Krikun, M., Cao, Y., Gao, Q., Macherey, K.: Google’s neural machine translation system: Bridging the gap between human and machine translation. arXiv:1609.08144 (2016)
Gregor, K., Danihelka, I., Graves, A., Rezende, D.J., Wierstra, D.: Draw: A recurrent neural network for image generation. arXiv:1502.04623 (2015)
Ledig, C., Theis, L., Huszár, F, Caballero, J., Cunningham, A., Acosta, A., Aitken, A., Tejani, A., Totz, J., Wang, Z.: Photo-realistic single image super-resolution using a generative adversarial network. In: Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp 4681–4690 (2017)
Spall, J.C.: Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control 37(3), 332–341 (1992)
Sadegh, P., Spall, J.C.: Optimal random perturbations for stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control. 43(10), 1480–1484 (1998)
Spall, J.C.: Implementation of the simultaneous perturbation algorithm for stochastic optimization. IEEE Trans. Aerospace. Electron. Syst. 34(3), 817–823 (1998)
Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv:1412.6980(2014)
Lefebvre, M.: Applied Probaility and Statistics, 1st edn. Springer, New York (2006)
MathWorks. Matlab (2018)
Floris, F.J.T., Bush, M.D., Cuypers, M., Roggero, F., Syversveen, A-R: Methods for quantifying the uncertainty of production forecasts: a comparative study. Pet. Geosci. 7(S), S87–S96 (2001). https://doi.org/10.1144/petgeo.7.S.S87, https://www.earthdoc.org/content/journals/10.1144/petgeo.7.S.S87
Acknowledgements
The authors would like to thank Steve Begg for his support and fruitful discussion. The authors would also like to thank David Echeverria Ciaurri for his help with the code relating to the inter-well constraint. The authors are also grateful to Rock Flow Dynamics for providing licenses for tNavigator. This work was supported with supercomputing resources provided by the Phoenix HPC service at the University of Adelaide.
Funding
Open Access funding enabled and organized by CAUL and its Member Institutions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Arouri, Y., Sayyafzadeh, M. An adaptive moment estimation framework for well placement optimization. Comput Geosci 26, 957–973 (2022). https://doi.org/10.1007/s10596-022-10135-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-022-10135-9