Accurately charting the progress of oil production is a problem of great current interest. Oil production is widely known to be cyclical: in any given system, after it reaches its peak, a decline will begin. With this in mind, Marion King Hubbert developed his peak theory in 1956 based on the bell-shaped curve that bears his name. In the present work, we consider a stochasticmodel based on the theory of diffusion processes and associated with the Hubbert curve. The problem of the maximum likelihood estimation of the parameters for this process is also considered. Since a complex system of equations appears, with a solution that cannot be guaranteed by classical numerical procedures, we suggest the use of metaheuristic optimization algorithms such as simulated annealing and variable neighborhood search. Some strategies are suggested for bounding the space of solutions, and a description is provided for the application of the algorithms selected. In the case of the variable neighborhood search algorithm, a hybrid method is proposed in which it is combined with simulated annealing. In order to validate the theory developed here, we also carry out some studies based on simulated data and consider 2 real crude oil production scenarios from Norway and Kazakhstan.

翻译：准确描绘石油产量的变化趋势是当前极具关注的问题。众所周知，石油产量具有周期性：在任何特定系统中，产量达到峰值后便会开始下降。基于这一认识，马里昂·金·哈伯特于1956年提出了以其命名的钟形曲线理论——峰值理论。本文研究了一种基于扩散过程理论且与Hubbert曲线相关联的随机模型，同时探讨了该模型参数的最大似然估计问题。由于该模型涉及复杂的方程组，经典数值方法无法保证其解的存在性，我们建议采用模拟退火和变邻域搜索等元启发式优化算法。针对解空间的界定问题提出了若干策略，并详细阐述了所选算法的应用方法。在变邻域搜索算法方面，提出了一种将其与模拟退火相结合的混合方法。为验证本文理论，我们基于模拟数据开展了相关研究，并分别以挪威和哈萨克斯坦两个真实原油生产场景为例进行分析。