Solution of the direct and inverse problem of gas lift oil production process by the optimal control method

The article presents a numerical method for solving the direct and inverse problems of the gas-lift process of oil production described by one-dimensional Navier-Stokes equations for compressible gas. To solve the direct problem, a family of difference schemes was developed and an analysis of the correctness of the difference problem was carried out depending on the parameter. The solution of the inverse problem is reduced to an optimal control problem, where the target functional is formed using an additional condition. To minimize the target functional, the gradient method is used, and its gradient is calculated by solving the conjugate problem. The conjugate problem contains important information about the solution of the direct problem and is based on the Lagrange identity and the condition of equality to zero of the integral terms. A feature of the conjugate problem is its retrospective nature, since the additional condition on the volumetric gas flow rate and pressure is specified at a certain point in time. The iteration method determines the initial conditions for the volumetric gas flow rate and pressure through the solution of the conjugate retrospective problem. The conducted computational experiment confirmed that the proposed algorithm can be used to determine the initial values of the volumetric gas flow rate and pressure with high accuracy under a given additional condition. The developed method also allows plotting the performance curve of the gas lift process. Keywords: gas-lift oil production process; Navier-Stokes equations; conjugate equation; inverse problem; optimal control; gradient method; finite-difference method.