A simultaneous perturbation stochastic approximation algorithm for coupled well placement and control optimization under geologic uncertainty

Abstract

Development of subsurface energy and environmental resources can be improved by tuning important decision variables such as well locations and operating rates to optimize a desired performance metric. Optimal well locations in a discretized reservoir model are typically identified by solving an integer programming problem while identification of optimal well settings (controls) is formulated as a continuous optimization problem. In general, however, the decision variables in field development optimization can include many design parameters such as the number, type, location, short-term and long-term operational settings (controls), and drilling schedule of the wells. In addition to the large number of decision variables, field optimization problems are further complicated by the existing technical and physical constraints as well as the uncertainty in describing heterogeneous properties of geologic formations. In this paper, we consider simultaneous optimization of well locations and dynamic rate allocations under geologic uncertainty using a variant of the simultaneous perturbation and stochastic approximation (SPSA). In addition, by taking advantage of the robustness of SPSA against errors in calculating the cost function, we develop an efficient field development optimization under geologic uncertainty, where an ensemble of models are used to describe important flow and transport reservoir properties (e.g., permeability and porosity). We use several numerical experiments, including a channel layer of the SPE10 model and the three-dimensional PUNQ-S3 reservoir, to illustrate the performance improvement that can be achieved by solving a combined well placement and control optimization using the SPSA algorithm under known and uncertain reservoir model assumptions.

Robust Ensemble-based Multi-objective Optimization

“…In this study we used a reactive control based performance to rank and choose the set of models. Li et al (2012), using the SPSA technique to approximate a gradient, suggested to randomly select a subset of models at every iteration of the optimization, instead of using an a-priori chosen subset of models. With this randomly chosen set, similar to the selected models approach, individual gradients are estimated to estimate a single robust gradient.…”

“…Additionally, Raniolo et al (2013) and Li et al (2012), have investigated the applicability of approximate gradient techniques for life-cycle robust water flooding optimization while Yang et al (2011) applied the robust optimization principle to a Steam-Assisted Gravity Drainage (SAGD) application. Yasari et al 2013, Pajonk et al 2011 and Awotunde and Sibaweihi (2011) have investigated the applicability of robust multi-objective optimization using evolutionary algorithms for well control and well placement optimization with objectives varying from economic criteria to voidage replacement ratio and cumulative production volumes.…”

Recent Developments in Closed-Loop Approaches for Real-Time Mining and Petroleum Extraction

“…For the use in robust optimization, see van Essen et al (2009); for a general overview, see Jansen (2011). Alternative, less codeintrusive, robust methods use approximate gradient and/or stochastic methods (Chen et al 2009;Chen and Oliver, 2010;Li et al 2013;Fonseca et al 2015Fonseca et al , 2016 or 'non-classical' methods such as, e.g., streamline methods (Alhutali et al 2008), evolutionary strategies (Pajonk et al 2011), or polynomial chaos expansions in combination with response surfaces (Babaei et al 2015), with further references given in Echeverrıa Ciaurri et al (2011).…”

Optimal Well-Placement Using Probabilistic Learning

“…For instance, particle swarm optimization (PSO) [17] and the simultaneous perturbation stochastic approximation (SPSA) [4,24] were introduced to accelerate the convergence of the iterative process towards an optimal solution. The SPSA was also applied to an optimal well-placement problem under geologic uncertainty [18], where the optimal solution was the maximum over an average of a statistical ensemble of size 100 reflecting uncertainty in permeability. It was observed that accounting for geologic uncertainty in this manner had a noticeable effect on the optimal wellplacement.…”

“…In this context, both robust optimization [13,28], involving a combination of mean and variance, risk-based optimization involving probabilities of exceedance [2], conditional value-at-risk (CVaRO) [8] dealing with risks associated with low returns, and utility functions [16] have been considered. Several recent studies dealing with optimal well-placement have stressed the value of joint location/control optimization whereby both the location of the wells and production strategy over some time horizon are simultaneously optimized [5,6,18,28,29]. Clearly, in such cases, the computational aspects of the optimization problem are significantly more challenging, specially when accounting for uncertainties in geology, investment, and pricing.…”