A variational method for numerically solving the inverse problem of the gas lift process of oil production

B. T. Zhumagulov, Nurlan Temirbekov, А.К. Тураров, Syrym E. Kasenov, A. M. Temirbekova

2025

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DOI 10.47533/2025.1606-146x.2-10
Data confidence 70 medium
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Abstract

In this paper, we propose a constructive method for numerically solving direct and inverse problems arising in the gas lift process of oil production, which is described by a hyperbolic system of differential equations. To solve the direct problem, a second-order difference scheme is used, which ensures the 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 target functional is carried out using the gradient method. The calculation of the gradient of the objective function is based on a conjugate problem constructed using the Lagrange identity and the duality principle, which guarantees the mathematical rigor of the approach. Numerical experiments have confirmed the effectiveness of the proposed method for solving the inverse problem and optimizing the input parameters of the gas lift process. The conjugate problem contains valuable information about the solution of the direct problem, so the gradients of the functional are equal to the solution of the conjugate problem and its first derivative in time at t = 0. Numerical calculations show that the values of the minimized functional decrease monotonously and are limited from below. This means that the iterative method used converges. The additional conditions set at T = 0 for the direct problem are used to formulate the condition of the conjugate problem. The developed algorithm contributes to the development of a numerical implementation of the conjugate optimization method of the inverse problem for a hyperbolic equation. The tasks of the type under study are of great practical importance and can be used to calculate the intensity of the gas lift process of oil production.