# Optimal Well-Placement Using Probabilistic Learning

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## 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

- ᅟ
A lower case letter

*x*is a real variable.- ᅟ
A boldface lower case letter

**x**is a real vector.- ᅟ
An upper case letter

*X*is a real random variable.- ᅟ
A boldface upper case letter

**X**is a real random vector.- ᅟ
A lower case letter between brackets [

*x*] is a real matrix.- ᅟ
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

*N*_{0}initial design variables- \({\mathcal {C}}_{\mathbf {w}_{g}}\)
: set of

*N*_{g}design variables on search grid- ᅟ
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

**w**^{opt}: 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

**w***r*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**- ᅟ
\({\mathcal {Q}}\) at a given

**w***x*_{I},*y*_{I}: coordinates of injection well

*x*_{P},*y*_{P}: coordinates of production well

**W**: random well locations

**Q**: random production levels for given

**W****X**: (

**W**,**Q**)**w**_{g}: optimization variable on search grid

**w**_{0}: optimization variable on initial set of points

*N*_{0}: number of initially available solutions

*N*_{g}: number of points for grid search by optimization algorithm

*𝜃*: element of sample space

*Θ**n*_{rep}: number of repetitions for each design variable

*n*_{MC}: 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).

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