Ensemble Methods for Reservoir Life-Cycle Optimization and Well Placement

Leeuwenburgh, O.; Egberts, P.J.P.; Abbink, O.A.

doi:10.2118/136916-ms

Cited by 20 publications

(10 citation statements)

References 7 publications

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“…is no longer valid. However, one can still obtain an approximation of C U ∇ J E ( u ℓ ) by first using a simplex gradient to approximate ∇ J E ( u ℓ ) and then multiplying this gradient by C U . For information on the simplex gradient formulation in general see, for example, .…”

Section: Comments On Stochastic Simplex Approximate Gradientmentioning

confidence: 99%

“…and the definitions of Eqs. and , it easily follows that for each k , k = 1,2,⋯ N e ,

\begin{matrix} \sum_{j = 1}^{N_{p}} ({\hat{u}}_{ℓ, j, k} - u_{ℓ}) (J (m_{k}, {\hat{u}}_{ℓ, j, k}) - J (m_{k}, u_{ℓ})) = Δ {\hat{U}}_{ℓ, k} (Δ j_{ℓ, k}) \\ \approx (Δ {\hat{U}}_{ℓ, k} {(Δ {\hat{U}}_{ℓ, k})}^{T}) \nabla_{u} J (m_{k}, u_{ℓ}) . \end{matrix}

Thus, as in , an approximate stochastic gradient, denoted by ∇ u ,sto J ( m k , u ℓ ), is given by

\nabla_{boldu, sto} J ({boldm}_{k}, {boldu}_{ℓ}) = {((), normalΔ {\overset{boldU}{true}}_{ℓ, k} {(normalΔ {\overset{boldU}{true}}_{ℓ, k})}^{T})}^{+} normalΔ {\overset{boldU}{true}}_{ℓ, k} (normalΔ {boldj}_{ℓ, k}) \approx \nabla_{boldu} J ({boldm}_{k}, boldu

…”

Section: Comments On Stochastic Simplex Approximate Gradientmentioning

confidence: 99%

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A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

Fonseca

Chen

Jansen

et al. 2016

Numerical Meth Engineering

157

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SUMMARYWe consider a technique to estimate an approximate gradient using an ensemble of randomly chosen control vectors, known as Ensemble Optimization (EnOpt) in the oil and gas reservoir simulation community. In particular, we address how to obtain accurate approximate gradients when the underlying numerical models contain uncertain parameters because of geological uncertainties. In that case, 'robust optimization' is performed by optimizing the expected value of the objective function over an ensemble of geological models. In earlier publications, based on the pioneering work of Chen et al. (2009), it has been suggested that a straightforward one-to-one combination of random control vectors and random geological models is capable of generating sufficiently accurate approximate gradients. However, this form of EnOpt does not always yield satisfactory results. In a recent article, formulate a modified EnOpt algorithm, referred to here as a Stochastic Simplex Approximate Gradient (StoSAG; in earlier publications referred to as 'modified robust EnOpt') and show, via computational experiments, that StoSAG generally yields significantly better gradient approximations than the standard EnOpt algorithm. Here, we provide theoretical arguments to show why StoSAG is superior to EnOpt.

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Section: Comments On Stochastic Simplex Approximate Gradientmentioning

confidence: 99%

“…and the definitions of Eqs. and , it easily follows that for each k , k = 1,2,⋯ N e ,

\begin{matrix} \sum_{j = 1}^{N_{p}} ({\hat{u}}_{ℓ, j, k} - u_{ℓ}) (J (m_{k}, {\hat{u}}_{ℓ, j, k}) - J (m_{k}, u_{ℓ})) = Δ {\hat{U}}_{ℓ, k} (Δ j_{ℓ, k}) \\ \approx (Δ {\hat{U}}_{ℓ, k} {(Δ {\hat{U}}_{ℓ, k})}^{T}) \nabla_{u} J (m_{k}, u_{ℓ}) . \end{matrix}

Thus, as in , an approximate stochastic gradient, denoted by ∇ u ,sto J ( m k , u ℓ ), is given by

\nabla_{boldu, sto} J ({boldm}_{k}, {boldu}_{ℓ}) = {((), normalΔ {\overset{boldU}{true}}_{ℓ, k} {(normalΔ {\overset{boldU}{true}}_{ℓ, k})}^{T})}^{+} normalΔ {\overset{boldU}{true}}_{ℓ, k} (normalΔ {boldj}_{ℓ, k}) \approx \nabla_{boldu} J ({boldm}_{k}, boldu

…”

Section: Comments On Stochastic Simplex Approximate Gradientmentioning

confidence: 99%

A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

Fonseca

Chen

Jansen

et al. 2016

Numerical Meth Engineering

157

View full text Add to dashboard Cite

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“…In an ideal case we would like to additionally meet such short-term economic objectives whilst still maintaining the life cycle objectives. Thus we choose a secondary 10 Undiscounted NPV ($)…”

Section: Objective Functionsmentioning

confidence: 99%

Ensemble-based hierarchical multi-objective production optimization of smart wells

2014

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In an earlier study two hierarchical multi-objective methods were suggested to include short-term targets in life-cycle production optimization. However this earlier study has two limitations: 1) the adjoint formulation is used to obtain gradient information, requiring simulator source code access and an extensive implementation effort, and 2) one of the two proposed methods relies on the Hessian matrix which is obtained by a computationally expensive method. In order to overcome the first of these limitations, we used ensemble-based optimization (EnOpt).EnOpt does not require source code access and is relatively easy to implement. To address the second limitation, we used the Broyden-Flecther-Goldfarb-Shanno (BFGS) algorithm to obtain an approximation of the Hessian matrix. We performed experiments in which a water flood was optimized in a geologically realistic multi-layer sector model. The controls were inflow control valve settings at pre-defined time intervals. Undiscounted Net Present Value (NPV) and highly discounted NPV were the long-term and short-term objective functions used. We obtained an increase of approximately 14% in the secondary objective for a decrease of only 0.2-0.5% in the primary objective. The study demonstrates that ensemble-based hierarchical multi-objective optimization can achieve results of practical value in a computationally efficient manner.

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“…The control samples are drawn from a multivariate Gaussian distribution with a user-defined (constant) covariance matrix and a known mean. Several publications (Chen 2008;Chen and Oliver 2012;Leeuwenburgh et al 2010;Su and Oliver 2010) have shown that EnOpt can achieve good results of practical value on a variety of different reservoir models and recovery techniques. A major drawback, however, is the significantly lower computational efficiency and accuracy compared to the adjoint method.…”

Section: Introductionmentioning

confidence: 99%

A Theoretical look at Ensemble-Based Optimization in Reservoir Management

2015

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Ensemble-based optimization has recently received great attention as a potentially powerful technique for life-cycle production optimization, which is a crucial element of reservoir management. Recent publications have increased both the number of applications and the theoretical understanding of the algorithm. However, there is still ample room for further development since most of the theory is based on strong assumptions. Here, the mathematics (or statistics) of Ensemble Optimization is studied, and it is shown that the algorithm is a special case of an already well-defined natural evolution strategy known as Gaussian Mutation. A natural description of uncertainty in reservoir management arises from the use of an ensemble of history-matched geological realizations. A logical step is therefore to incorporate this uncertainty description in robust life-cycle production optimization through the expected objective function value. The expected value is approximated with the mean over all geological realizations. It is shown that the frequently advocated strategy of applying a different control sample to each reservoir realization delivers an unbiased estimate of the gradient of the expected objective function. However, this procedure is more variance prone than the deterministic strategy of applying the entire ensemble of perturbed control samples to each reservoir model realization. In order to reduce the variance of the gradient estimate, an importance sampling algorithm is proposed and tested on a toy problem with increasing dimensionality.

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Ensemble Methods for Reservoir Life-Cycle Optimization and Well Placement

Cited by 20 publications

References 7 publications

A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

A Stochastic Simplex Approximate Gradient (StoSAG) for optimization under uncertainty

Ensemble-based hierarchical multi-objective production optimization of smart wells

A Theoretical look at Ensemble-Based Optimization in Reservoir Management

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