A Theoretical look at Ensemble-Based Optimization in Reservoir Management

Abstract

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.

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

“…In EnOpt, u is considered to be a random vector with a Gaussian distribution centered at u ℓ and covariance matrix , that is, at iteration ℓ , . This notation indicates that it is possible to generate a new covariance matrix, for example, by covariance matrix adaptation . For our purposes, however, it suffices to consider the covariance matrix as fixed so we assume throughout that at iteration ℓ .…”

“…Stordal et al . recently provided a theoretical analysis of the single and multiple reservoir formulations of standard EnOpt in which they demonstrate that both can be interpreted as special versions of a machine learning algorithm called ‘Gaussian mutation optimization’ . Stordal et al .…”

“…Stordal et al . recently demonstrated that deterministic EnOpt can also be interpreted as a special case of a machine learning algorithm known as Gaussian mutation (Amari et al . ).…”

Robust Ensemble-based Multi-objective Optimization

“…Hereafter this will be referred to as the "original formulation". In Stordal et al (2014) it is shown how this implementation converges to the exact gradient in a natural evolutionary framework. Fonseca et al (2014) investigated the applicability of EnOpt for multi-objective optimization based on a single model realization and showed results which suggest that EnOpt is a viable alternative when the adjoint is unavailable.…”

Review of Field Development Optimization of Waterflooding, EOR, and Well Placement Focusing on History Matching and Optimization Algorithms

“…Stochastic methods vary uncertain input parameters from a probability distribution function to create an ensemble of model predictions or optimization results [45,46]. The predictions that most closely align with the measurements are selected to find an optimal solution to the history matching problem.…”