Two stochastic versions of the Arps decline curve

• Analyse of stochastic curve declines. • Introduction of uncertainty in curve decline. • Oil and gas application. Based on the Arps equation, we propose two stochastic models for decline curve useful in oil engineering context. These stochastic models allow to take into account non-monotonic curve and to integrate randomness. Theoretical properties and simulations of these models are provided. The first passage time distribution of these stochastic models to a constant level is then studied. In conclusion, we introduce the oil production cumulative rate defined as the integral of the studied stochastic process over a given time interval. Statistical inference of the parameters could based on this integrated stochastic models. All these models provide a theoretical framework that could be applied to data in future work.