A mixed-integer non-linear problem formulation for miscible WAG injection

Abstract

The paper presents a formulation to evaluate optimal injection scenarios or production strategies in an oil reservoir. The decisions variables involved in such strategy formulations are a mix of both continuous and integer nature variables. The time-line of the operation or the recovery strategy addressed in this paper is divided into three action periods: cyclic injection (a single or multiple/alternating cycles), continuous injection, and depletion. Two reservoir models description used in this study: a pattern sector model with a single producer and single injectors based on the SPE 5th comparative project miscible WAG (Killough and Kossack, 1987), and a multi-well case with 5 producers and 8 injectors with a more complex reservoir geology that is an extension of the SPE 5th comparative project. The study addresses two independent production modes, each with a different set of production constraints, artificial-lift from the start to the end of the timeline and natural-flow from the start to the end of the study period. For the two simplest injection strategies: continuous gas or continuous water injection, the only control variable is tubing-head pressure in the injection wells. For more complex injection strategies more decision variables are required including tubing-head pressures, injection volume, and time to change the injection strategy. The 8 studied strategies with their relevant decision variables are stated in a case matrix table in the paper. Each of the 8 strategies is studies with either natural flow mode or artificial lift mode. The objective function is the maximum of a Net Present Value (NPV) formula using revenue with a time escalation sales price, and a simple Operating Expenditures (OPEX) and Capital Expenditure (CAPEX). The optimization function is simple but captures the important trends in comparing oil recovery strategies. The optimization program is run for each strategy stated in the case matrix table, searching for the optimum by varying the continuous decision variables of that particular strategy. To ensure a proper location of the global optimum of each strategy, the optimization search employs several starting values of the continuous variables, with the number of starting values increasing with increased strategy complexity. The starting values are randomly generated. The overall optimum operation strategy is the one with the highest calculated NPV. The entire optimization study is conducted in a semi-automated manner. A proprietary program runs and manages the integration of the reservoir simulator, the NPV model and the optimizer. This is done automatically for each investigated strategy. The proposed methodology is applicable to any oil reservoir where both surface water and gas injection is available. This work contributes to the literature by establishing a general mixed-integer problem formulation for water and gas injection and providing an efficient heuristic method for solving the problem.