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

Abstract

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.

Robust Ensemble-based Multi-objective Optimization

“…The gradient is used in a simple steepest ascent scheme, where we determine the step size with the aid of an inexact line search and the Arjimo conditions (Nocedal and Wright 2006). We use the switching approach for hierarchical multi-objective optimization with implementation details as described in Fonseca et al (2014 …”

“…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. Furthermore, the robust EnOpt implementation as introduced by Chen et al (2008), which uses a 1:1 ensemble ratio, is computationally competitive to the adjoint method, because one adjoint simulation is required for each model realization.…”

Determination of lower and upper bounds of predicted production from history-matched models

“…However, it is, in theory, also possible to implement our method using approximate gradients obtained with, e.g. the simultaneous perturbation stochastic approximation (SPSA) technique (see [19]) or ensemble optimization (EnOpt) (see [3] for the basics of the method and [8] for an implementation in hierarchical optimization). The latter (EnOpt) approach also allows for the inclusion of uncertainty in the reservoir models; see [9].…”

“…Moreover, we note that other weak-constrained optimization methods could be applied to solve the hierarchical optimization problem. The current implementation has shown to be robust in various applications using both adjoint-based and ensemble-based techniques [4,8,23].…”

“…However, the multiple objectives are not of the same importance; priority lies with obtaining a good history match, while determining a lower or upper bound on predicted economic performance serves as a secondary objective. To that end, the multiobjective optimization problem may be cast into a hierarchical optimization problem, as presented in [10] and more recently specifically for oil production optimization in [4,8,23]. In this structure, optimization of a (secondary) economic cost function J is constrained by the requirement that the (primary) quantitative history-matching cost function V must remain close to its minimal value V min .…”