This research is motivated by the well productivity index (PI) concept of the reservoir engineering. It was observed by petroleum engineers that the PI stabilizes to a constant value. The goal of this paper is to develop a mathematically rigorous framework to prove the existence of a certain time invariant for the transient dynamic processes associated with a non-linear flow. Calculus of variations is applied to estimate this invariant, which is, in fact, the PI of the well. As a result of the developed framework, in order to compute the PI, it is sufficient to solve an auxiliary steady state problem. At the same time, the solution of each BVP minimizes the energy functional in the corresponding Sobolev space. This link between the PI and calculus of variations allows us to estimate dependency of the PI on the geometry of the reservoir/well system and to identify the optimal configurations of the well without solving an auxiliary BVP.
|Journal||Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., Advances in Dynamical Systems, suppl. S2|
|State||Published - 2007|