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Optimal Well-Placement Using Probabilistic Learning

  • Roger GhanemEmail author
  • Christian Soize
  • Charanraj Thimmisetty
Original Article
  • 547 Downloads

Abstract

A new method based on manifold sampling is presented for formulating and solving the optimal well-placement problem in an uncertain reservoir. The method addresses the compounded computational challenge associated with statistical sampling at each iteration of the optimization process. An estimation of the joint probability density function between well locations and production levels is achieved using a small number of expensive function calls to a reservoir simulator. Additional realizations of production levels, conditioned on well locations and required for evaluating the probabilistic objective function, are then obtained by sampling this jpdf without recourse to the reservoir simulator.

Keywords

Sampling on manifolds Probabilistic learning Diffusion on manifold MCMC Optimal well placement Petroleum engineering 

Notations

A lower case letter x is a real variable.

A boldface lower case letter x is a real vector.

An upper case letter X is a real random variable.

A boldface upper case letter X is a real random vector.

A lower case letter between brackets [x] is a real matrix.

A boldface upper case letter between brackets [X] is a realrandom matrix.

\(\mathbb {N} = \{0,1,2,\ldots \}\)

: set of all the integers.

\(\mathbb {R}\)

: set of all the real numbers.

\(\mathbb {R}^{n}\)

: Euclidean vector space on \({\mathbb {R}}\) of dimension n.

x

: usual Euclidean norm in \({\mathbb {R}}^{n}\).

\(\mathbb {M}_{n,N}\)

: set of all the (n × N) real matrices.

\({\mathcal {C}}_{\mathbf {w}}\)

: admissible set for optimization

\({\mathcal {C}}_{\mathbf {w}_{0}}\)

: set of N0 initial design variables

\({\mathcal {C}}_{\mathbf {w}_{g}}\)

: set of Ng design variables on search grid

Open image in new window is the indicator function of set Open image in new window if \(a\in {\mathcal {A}}\) and = 0 if \(a\notin {\mathcal {A}}\).

E

: Mathematical expectation.

α

: production confidence level for optimal solution

wopt

: optimal solution

\({\mathbf {w}}_{r}^{\text {opt}}\)

: reference optimal solution

\({\mathbf {w}}_{d}^{\text {opt}}\)

: optimal solution using statistical surrogate

\({\mathbf {w}}_{\text {ar}},{\mathbf {x}}_{\text {ar}}\)

: additional realizations of W and X

\(q_{r}^{\text {opt}}\)

: production level at wr opt

\(q_{d}^{\text {opt}}\)

: production level at \({\mathbf {w}}_{d}^{\text {opt}}\)

\({\mathcal {Q}}({\mathbf {w}})\)

: random cumulative production after 2000 days

\(F_{\mathcal {Q} (\mathbf {w})} (q)\)

: probability distribution function for h(q,w):\(1-F_{\mathcal {Q} (\mathbf {w})}(q)\)

\(p_{\mathcal {Q} (\mathbf {w})} (q; \mathbf {w})\)

: probability density function for \({\mathcal {Q}}\) at given w

\({\mathcal {Q}}\) at a given w

xI,yI

: coordinates of injection well

xP,yP

: coordinates of production well

W

: random well locations

Q

: random production levels for given W

X

: (W, Q)

wg

: optimization variable on search grid

w0

: optimization variable on initial set of points

N0

: number of initially available solutions

Ng

: number of points for grid search by optimization algorithm

𝜃

: element of sample space Θ

nrep

: number of repetitions for each design variable

nMC

: number of samples drawn from statistical surrogate

Notes

Acknowledgments

This research was supported by the US department of energy under the Scidac Institute for Uncertainty Quantification under Extremes (Quest).

References

  1. 1.
    S. Aanonsen, A. Eide, L. Holden, J. Aasen, in Optimizing reservoir performance under uncertainty with application to well location. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, (1995), pp. 67–76Google Scholar
  2. 2.
    V. Artus, L. Durlofsky, J. Onwunala, K. Aziz, Optimization of nonconventional wells under uncertainty using statistical proxies. Comput. Geosci. 10, 389–404 (2006)CrossRefGoogle Scholar
  3. 3.
    M. Babaei, A. Alkhatib, I. Pan, Robust optimization of subsurface flow using polynomial chaos and response surface surrogates. Comput. Geosci. 19, 979–998 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    W. Bangerth, H. Klie, M. Wheeler, P. Stoffa, M. Sen, On optimization algorithms for the reservoir oil well placement problem. Comput. Geosci. 10, 303–319 (2006)CrossRefGoogle Scholar
  5. 5.
    B. Beckner, X. Song, in Field development planning using simulated annealing-optimal economic well scheduling and placement. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, (1995)Google Scholar
  6. 6.
    M. Bellout, D. Echeverria-Ciaurri, L. Durlofsky, B. Foss, J. Kleppe, Joint optimization of oil well placement and controls. Comput. Geosci. 16, 1061–1079 (2012)CrossRefGoogle Scholar
  7. 7.
    A. Bowman, A. Azzalini. Applied Smoothing Techniques for Data Analysis (Oxford University Press, Oxford, 1997)zbMATHGoogle Scholar
  8. 8.
    L. Christiansen, A. Capolei, J. Jørgensen, Time-explicit methods for joint economical and geological risk mitigation in production optimization. J. Pet. Sci. Eng. 146, 158–169 (2016)CrossRefGoogle Scholar
  9. 9.
    M. Christie, M. Blunt, Tenth spe comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4, 308–317 (2001)Google Scholar
  10. 10.
    R. Coifman, S. Lafon, Diffusion maps, applied and computational harmonic analysis. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner, S. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. PNAS. 102(21), 7426–7431 (2005)CrossRefGoogle Scholar
  12. 12.
    ECLIPSE: Reference manual. Schlumberger, Houston, Texas (2009)Google Scholar
  13. 13.
    G. van Essen, M. Zandvilet, P. V. den Hof, O. Bosgra, J. Jansen, Ribust waterflooding optimization of multiple geological scenarios. SPE J. 14(1), 202–210 (2009)CrossRefGoogle Scholar
  14. 14.
    R. Ghanem, Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resour. Res. 34(9), 2123–2136 (1998)CrossRefGoogle Scholar
  15. 15.
    R. Ghanem, C. Soize, Probabilistic non-convex constrained optimization with fixed number of function evaluations. Int. J. Numer. Methods Eng. to appear (2017)Google Scholar
  16. 16.
    B. Guyaguler, R. Horne, Uncertainty assessment of well placement optimization. SPE Reserv. Eval. Eng. 7 (1), 23–32 (2004)Google Scholar
  17. 17.
    M. Jesmani, M. Bellout, R. Hanea, B. Foss, Well placement optimization subject to realistic field development constraints. Comput. Geosci. 20, 1185–1209 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    L. Li, B. Jafarpour, M. Mohammad-Khaninezhad, A simultaneous perturbation stochastic approximation algorithm for coupled well placement and control optimization under geologic uncertainty. Comput. Geosci. 17, 167–188 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    K. Rashid, W. Bailey, B. Couet, D. Wilkinson, An efficient procedure for expensive reservoir-simulation optimization under uncertainty. SPE Economics & Management. 5(4), 21–33 (2013)CrossRefGoogle Scholar
  20. 20.
    D. Rian, A. Hage, in Automatic optimization of well locations in a north sea fractured chalk reservoir using a front tracking reservoir simulator. International Petroleum Conference and Exhibition of Mexico. Society of Petroleum Engineers, (1994)Google Scholar
  21. 21.
    D. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization, 2nd edn. (Wiley, New York, 2015)zbMATHGoogle Scholar
  22. 22.
    C. Soize, Polynomial chaos expansion of a multimodal random vector. SIAM/ASA Journal on Uncertainty Quantification. 3(1), 34–60 (2015).  https://doi.org/10.1137/140968495 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    C. Soize, R. Ghanem, Data-driven probability concentration and sampling on manifold. J. Comput. Phys. 321, 242–258 (2016).  https://doi.org/10.1016/j.jcp.2016.05.044 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    J. Spall. Introduction to stochastic searh and optimization (Wiley-Interscience, New York, 2003)CrossRefGoogle Scholar
  25. 25.
    W. Sun, L. Durlofsky, A new data-space inversion procedure for efficient uncertainty quantification in subsurface flow problems. Math. Geosci. 49, 679–715 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    C. Thimmisetty, P. Tsilifis, R. Ghanem, Paper petroleum. Artificial Intelligence for Engineering Design, Analysis and Manufacturing. 31(3), 265–276 (2017). Homogeneous chaos basis adaptation for design optimization under uncertainty: Application to the oil well placement problemCrossRefGoogle Scholar
  27. 27.
    H. Wang, D. Echeverria-Ciaurri, L. Durlofsky, A. Cominelli, Optimal well placement under uncertainty using a retrospective optimization framework. SPE J. 17(1), 112–121 (2012)CrossRefGoogle Scholar
  28. 28.
    B. Yeten, L. Durlofsky, K. Aziz, in Optimization of nonconventional well type, location and trajectory. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, (2002)Google Scholar
  29. 29.
    Y. Zhang, R. Lu, F. Forouzanfar, A. Reynolds, Well placement and control optimization for wag/sag processes using ensemble-based method. Comput. Chem. Eng. 101, 193–209 (2017)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Southern CaliforniaLos AngelesUSA
  2. 2.Laboratoire Modélisation et Simulation Multi-EchelleUniversité Paris-EstMarne-la-ValléeFrance
  3. 3.Lawrence Livermore National LaboratoryCenter for Applied Scientific ComputingLivermoreUSA

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