Bi-Objective Optimization of Well Placement and Controls Using StoSAG

Abstract

Abstract When geological uncertainty is considered in production optimization problem, robust optimization on a set of plausible reservoir models is essential. We consider biobjective optimization where the two objectives are to find the optimal trajectories and/or control settings of injectors and producers that maximize the expected value of the life-cycle net-present-value (NPV) and minimize the risk where the risk is defined as the minimum NPV achieved for any realization. We consider cases where the vector of design (optimization) variables includes either well trajectories or well controls or both sets of variables. The trajectory of each well is parametrized in terms of the (x,y, z) coordinates of the two endpoints of the well segment whereas well controls are represented by the pressure or rates for all control steps. In this work, the optimization is performed using a slightly modified version of a new ensemble-based optimization algorithm known as StoSAG (stochastic simplex approximate gradient). We analyzed the inferiority of the simultaneous optimization using the StoSAG method compared to the sequential optimization scheme and proposed an iterative simultaneous procedure where one iteration of trajectory update is followed by one iteration of control update. In this work, the iterative simultaneous optimization procedure outperforms the sequential procedure (a complete well trajectory optimization followed by a complete control optimization), based on the numerical results of a channelized water flooding reservoir. We also focus on developing efficient implementations of the lexicographic optimization method to find a reasonable trade-off between the expected NPV and the risk. The biobjective optimization is firstly implemented to the well placement optimization problem to illustrate the efficiency of using the proposed splitted StoSAG search direction for the augmented Lagrangian function compared to the basic StoSAG search direction. Besides the channelized reservoir, an inclined gas flooding reservoir loosely modelled after the Oseberg field, is also tested. Finally, the biobjective optimization to both well controls and trajectories are carried out combining the lexicographic method with both the iterative simultaneous procedure and the sequential procedure. Furthermore, in order to avoid less attractive well trajectory solutions where wells are too close together, the constraints that minimum distance between any two well trajectories is greater than or equal to a prescribed distance are also enforced in this paper. To the best of our knowledge, this paper presents the first illustration of the joint optimization of well placement and control with the StoSAG method, considering minimum well spacing constraints. Also the implementation of a computationally efficient decoupled StoSAG method to the biobjective optimization problem is discussed.