Adaptive control design for a nonlinear parabolic PDE: Application to water coning

Abstract

AbstractWater coning is usually responsible for the production of undesirable water from oil wells. This phenomenon may cause a decrease in oil production rate, increase in water cut production, and costs, which subsequently leads to early shutdown of the well. Although the boundary control of the production rate was suggested for managing the problem, due to the uncertainty associated with the physical nature of petroleum reservoirs, it failed to be implemented in practice. To overcome this issue, the paper employs the adaptive control approach for the distributed parameter system, which is modelled using a nonlinear partial differential equation (PDE). For this purpose, an adaptive control law and an update law for estimating the uncertain parameter are developed using the direct Lyapunov method. Next, the global stability of the closed‐loop system with the abovementioned laws is proven. Finally, the effectiveness and performance of the proposed idea is demonstrated by numerical simulations. The results show that the thickness of an oil column tends to zero as time tends to infinity for the whole spatial domain. In other words, as time elapses, the whole oil column will be depleted before the cone breakthrough. The numerical simulation demonstrates that though water cone breakthrough is inevitable in the conventional way of production, the adaptive control approach successfully controls the cone growth up, even with no knowledge of reservoir permeability. The results of this study can be applied to any type of reservoir subjected to water coning.