A Workflow for Efficient Initialization of Local-Search Iterative Methods for Waterflooding Optimization
Authors: Asadollahi Masoud , Naevdal Geir , Markovinovic Renato , Shafieirad Ali 
Published: March 2015

Abstract
Due to large dimensionality of typical field optimization problems (in terms of either the number of the optimization variables or the number of equations that needs to be solved at every simulation time-step), most approaches for field optimization are local iterative methods. Due to their local nature, achievable optima by these methods are directly impacted by the choice of the initial guess for the sought optimal profiles. Here we propose a simple, low-cost framework to efficiently initialize localsearch optimization algorithms, with the focus on waterflooding optimization under a standard net-present value (NPV) objective.
The procedure relies on accepted reservoir engineering concepts and consists of two sequential steps. First, a common economical 'shut-in' water-cut (WCT) is determined by rewriting the (assumed) positive NPV contribution of the individual completion in terms of WCT. If the WCT reached during the production from a given completion is lower than the economical 'shut-in' WCT the objective function is insensitive to that particular control variable. An adjusted shut-in WCT is therefore defined for each completion as the minimum of the economic WCT and the 'reached' WCT. Then, using these shut-in WCT values in a (rather limited) number of reservoir simulation runs, coefficients in a group production guide rate formula for the preferred phase are determined yielding a maximized NPV. Such a guide rate formula is normally implemented in all state of the art reservoir simulators. The guide rate coefficients and the economic and adjusted WCT values are used to initialize the WCT optimization problem. The initialization strategy was applied to history matched models of the Brugge field [1]. The optimization algorithms were Hooke-Jeeves (HJ), Nelder-Mead and Sequential Quadratic Programming. With any of these methods, the optimized shut-in WCTs in terms of the achieved NPV clearly outperformed the best NPV solution obtained so far on these history matched models (HJ being the best choice). The combined economic and adjusted WCT gave additional improvement in NPV. The injecting strategy was chosen to be the unity voidage replacement ratio, which decreased the number of variables considerably.

Introduction
Finding an optimal depletion strategy for hydrocarbon production has naturally always been a key subject in reservoir management, the underlying problem to be solved generally being maximization of a key quantity as oil production, net present value (NPV), etc. In the past, optimal settings of the optimization parameters were almost exclusively determined in a manual fashion, which generally is a quite time consuming procedure with a high likelihood of obtaining suboptimal results. While manual approaches are still predominant strategies in the reservoir management practice, due to maturity of most existing major oilfields and a gradual decrease in large oil discoveries, research for more systematic optimization approaches has been initiated.
Not surprisingly, most 'advances' have so far been made in systematic optimization, at least in theory, of waterflooding recovery processes, see e.g. Yeten et al. (2003); Brouwer and Jansen (2004); Sarma et al. (2005); Lorentzen et al. (2006); Wang et al. (2007); Chen et al. (2008); Sarma et al. (2008), Sarma and Chen (2008) [2–9]. The spectrum of the proposed (waterflooding) optimization techniques is already quite broad. Most of the proposed methods are local-search iterative methods, ranging from basic steepest descent iterative updates of the control variables involving either deterministic or stochastic objectivefunction gradient/sensitivity approximations, e.g. Wang et al. (2007); Chen et al. (2008) [6–7] to 'direct search' methods, as e.g. Nelder-Mead or Hooke-Jeeves [10–11], not requiring any derivative information.