An efficient boundary control for porous media equation: Motivated by water coning problem

Abstract

The porous media equation is a nonlinear partially differential equation (PDE) that is extensively applied in industries. One such case is regarding the water coning problem in oil reservoirs. This phenomenon may result in decreasing oil production rates and increasing water cut production as well as costs that may lead to the early closure of an oil well. Conventionally, well shutting-in is the only treatment approach when a well faces this problem; however, this does not lead to full reservoir depletion. Recently, the boundary control approach has been proposed to solve the water coning problem. The most important and challenging question is the following: which controllers guarantee that this issue will be resolved?To deal with the water coning issue, first, several types of controllers were considered and applied on the nonlinear PDE. Next, to investigate the stability behaviour of each controller, the direct Lyapunov approach was employed. Finally, to support the results of the analytical part of this paper and to compare the performance of those controllers, numerical simulations were performed. The results show that only for those control laws that guarantee the asymptotic and exponential stability of the nonlinear PDE, the thickness of the oil column tends to zero as time tends to infinity for the whole spatial domain. In addition, a novel controller, which is introduced in the present study, fulfills the exponential stability and is more efficient than other presented controllers. Using the controller, the whole oil column is drained out about 16 % faster without water breakthrough.Krsti c [5] derived a set of nonlinear boundary control laws that achieve global asymptotic stability for both viscous and the inviscid Burger's equation using both Neumann and Dirichlet boundary control. Moreover, the performance of the nonlinear Neumann boundary control was compared with that of an uncontrolled scenario. [6] Smaoui [7,8] analyzed the boundary stabilization of a more generalized form of the Burger's equation by the nonlinear boundary control for a mixed boundary condition and proved the exponential stability of the system. The exponential stability of the delayed KdVB equation was also proven for the case that the delay parameter was sufficiently small. KS EquationElsewhere, Liu and Krsti c [9] addressed the problem of the global boundary control of the Kuramoto-Sivashinsky (KS) equation (a nonlinear PDE equation) under both Dirichlet and Neumann boundary conditions. This control law guarantees L2-global exponential stability, H2-global asymptotic stability, and H2-semiglobal exponential stability of the nonlinear PDE equation in a certain parameter range, while the uncontrolled case with Neumann boundary conditions is not asymptotically stable.