Variational method of numerical solution of the inverse problem of gas lift oil production process

This paper proposes a constructive method for numerically solving direct and inverse problems arising in the gas-lift oil production process, which is described by a hyperbolic system of differential equations. To solve the direct problem, a second-order difference scheme is used, which ensures stability and accuracy of calculations in the space-time domain. The inverse problem is formulated as an optimal control problem, where the minimization of the objective functional is carried out using the gradient method. The calculation of the gradient of the objective function is based on the constructed adjoint problem using the Lagrange identity and the duality principle, which guarantees the mathematical rigor of the approach. Numerical experiments confirmed the efficiency of the proposed method for solving the inverse problem and optimizing the input parameters of the gas lift process. The adjoint problem contains valuable information about the solution of the direct problem, so the gradients of the functional are equal to the solution of the adjoint problem and its first derivative with respect to time at t = 0. Numerical calculations show that the values of the minimized functional decrease monotonically and remain bounded below. This means that the used iterative method converges. Additional conditions set at T =0 for the direct problem are used to formulate the condition of the adjoint problem. The developed algorithm contributes to the development of the numerical implementation of the adjoint optimization method of the inverse problem for a hyperbolic equation. The problem of the type under study is of great practical importance and can be used to calculate the intensity of the gas lift process of oil production.