The Hubbert diffusion process: Estimation via simulated annealing and variable neighborhood search procedures—application to forecasting peak oil production

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 stochastic model 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.