# Optimal Well-Placement Using Probabilistic Learning

• Roger Ghanem
• Christian Soize
• Charanraj Thimmisetty
Original Article

## 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.

$${\mathcal {C}}_{\mathbf {w}}$$

$${\mathcal {C}}_{\mathbf {w}_{0}}$$

: 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)
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)
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)
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)
7. 7.
A. Bowman, A. Azzalini. Applied Smoothing Techniques for Data Analysis (Oxford University Press, Oxford, 1997)
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)
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)
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)
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)
14. 14.
R. Ghanem, Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resour. Res. 34(9), 2123–2136 (1998)
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)
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)
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)
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)
22. 22.
C. Soize, Polynomial chaos expansion of a multimodal random vector. SIAM/ASA Journal on Uncertainty Quantification. 3(1), 34–60 (2015).
23. 23.
C. Soize, R. Ghanem, Data-driven probability concentration and sampling on manifold. J. Comput. Phys. 321, 242–258 (2016).
24. 24.
J. Spall. Introduction to stochastic searh and optimization (Wiley-Interscience, New York, 2003)
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)
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 problem
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)
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)