Abstract
Ensemble methods are remarkably powerful for quantifying geological uncertainty. However, the use of the ensemble of reservoir models for robust optimization (RO) can be computationally demanding. The straightforward computation of the expected net present value (NPV) requires many expensive simulations. To reduce the computational burden without sacrificing accuracy, we present a fast and effective approach that requires only simulation of the mean reservoir model with a bias correction factor. Information from distinct controls and model realizations can be used to estimate bias for different controls. The effectiveness of various bias-correction methods and a linear or quadratic approximation is illustrated by two applications: flow optimization in a one-dimensional model and the drilling-order problem in a synthetic field model. The results show that the approximation of the expected NPV from the mean model is significantly improved by estimating the bias correction factor, and that RO with mean model bias correction is superior to both RO performed using a Taylor series representation of uncertainty and deterministic optimization from a single realization. Use of the bias-corrected mean model to account for model uncertainty allows the application of fairly general optimization methods. In this paper, we apply a nonparametric online learning methodology (learned heuristic search) for efficiently computing an optimal or near-optimal robust drilling sequence on the REEK Field example. This methodology can be used either to optimize a complete drilling sequence or to optimize only the first few wells at a reduced cost by limiting the search depths.
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Abbreviations
- f ∗ :
-
Actual expected objective function value of optimal control
- α :
-
Bias correction factor
- \(\bar \alpha \) :
-
Average value of bias correction factor
- \(\bar m\) :
-
Expected value of model parameter
- \(\bar \mu _{n_{s}}\) :
-
Mean single-step ratio of variability in estimated value
- β :
-
Partial correction factor for individual realizations
- Δ t :
-
Time interval
- Δ x :
-
Length interval
- δ :
-
Distance between controls
- γ :
-
Single-step ratio of estimated value
- \(\hat \alpha _{\text {loc}}\) :
-
Local estimate
- \(\hat \alpha _{r}\) :
-
Regularized estimate
- \(\hat \gamma \) :
-
Forecast error of initial evaluation function value
- \(\hat f \) :
-
Learned evaluation function
- λ :
-
Regularization parameter/Lagrangian parameter (depending on context)
- b 0 :
-
Vector of β at a fixed control x0
- b :
-
Vector of β from random controls and realizations
- P :
-
Vector of drilling sequence
- w :
-
Weight vector of β
- μ :
-
Viscosity of fluid
- ω :
-
Weight
- ρ :
-
Distance-based weight
- \(\sigma ^{2}_{\alpha }\) :
-
Variance of bias correction factor α
- \(\sigma ^{2}_{\beta }\) :
-
Variance of partial correction factor β
- θ :
-
Permeability
- A :
-
Cross-sectional area
- b :
-
Discount rate
- C :
-
Covariance of model parameter
- d :
-
Number of remaining actions
- E :
-
Expected value of objective function
- f :
-
Objective function/evaluation function (depending on context)
- f m :
-
First derivatives of objective function with respect to model parameter
- f m m :
-
Second derivatives of objective function with respect to model parameter
- g :
-
Actual economic value from previous actions
- h :
-
Heuristic function/estimated maximum future value
- k :
-
Log-permeability
- L :
-
Taper length
- m :
-
Model parameter
- n eff :
-
Effective sample size
- n s :
-
Environment state at decision stage
- N e :
-
Total number of model realizations
- N s :
-
Number of selected wells
- N w :
-
Total number of wells
- N x :
-
Total number of random controls
- NPV :
-
Net present value
- P :
-
Pressure
- q :
-
Production/injection rate
- r :
-
Reward/cost
- T :
-
Total number of time steps
- t :
-
Time
- W :
-
Cost of drilling well
- x :
-
Control variable
- Δ t Φ :
-
Learning time period
- Φ :
-
Learning technique index
- f n :
-
Evaluation function at a specific state
- h n :
-
Heuristic function at a specific state
- i :
-
Control index/learning period index/cell index (depending on context)
- j :
-
Model realization index/time step index(depending on context)
- k :
-
Well index
- o :
-
Oil
- s :
-
Decision stage
- w :
-
Water/well (depending on context)
- wi :
-
Water injection
- \(\cos \limits \) :
-
Cosine distance
- L 1 :
-
Manhattan distance
- L 2 :
-
Euclidean distance
- eff:
-
Effective sample size
- \(\bar m\) :
-
Reservoir mean model
- Φ :
-
Learning
- i :
-
Injection
- p :
-
Production
- lin:
-
Linear approximation
- quad:
-
Quadratic approximation
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Funding
Open Access funding provided by NORCE Norwegian Research Centre AS. This research was supported through the DIGIRES project by the Research Council of Norway and industry partners Equinor, Petrobras, Aker BP, Neptune Energy, WintershallDEA, Lundin Norway, and Vår Energy. Access to Eclipse licenses was provided by Schlumberger.
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Appendices
Appendix 1
To derive Eq. 24, recall that:
Assuming stationarity, this simplifies to:
Similarly:
The unbiasedness condition gives that:
Appendix 2
The covariance function C is related to the variance and the semivariogram:
where σ2 is the variance, h is the distance between two observations, and γ(h) is the semivariogram at distance h.
Since the observation β is obtained from random controls of individual realizations, we need to model the covariance function of β with two terms, i.e., variability in β at a fixed model realization and variability in β at a fixed control variable:
where Cβ(hx) is the covariance function for β at fixed model realization; hx is distance between observations; Cβ(hm) is the covariance function for β at fixed control variable; hm is the distance of realizations corresponding to the observed controls, i.e., hm = 0 if observed values are from the same realization, hm = 1 if observed values are from different realizations.
Cβ(hx) can be obtained from the average variogram of β of all ensemble realizations:
where \(\sigma ^{2}_{\beta }(m_{j})\) and γ(hx,mj) are the variance and variogram of β at a fixed realization mj, respectively.
For fixed control variable, the values of βij and \(\beta _{ij^{\prime }}\) will be correlated for model realizations mj and \(m_{j^{\prime }}\) at a small distance. Here, we assume that the model realizations are far enough apart such that the observed values of β from different realizations are independent. Then the covariance function for β for fixed control variable can be described as:
where \(\sigma ^{2}_{\beta }(x_{k})\) is the variance of observed β from different realizations at a fixed control xk.
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Wang, L., Oliver, D.S. Fast robust optimization using bias correction applied to the mean model. Comput Geosci 25, 475–501 (2021). https://doi.org/10.1007/s10596-020-10017-y
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DOI: https://doi.org/10.1007/s10596-020-10017-y