Analytical solution of the boundary value problem of mathematical modeling for non-stationary oil flows through the trunk pipeline in the presence of internal pressure sources

Abstract

The main oil pipeline, due to its spatial extent, can be considered as a control object with distributed parameters (ODP). The dependences on time and coordinates of the flow velocity and pressure in the pipeline are considered as controlled output values of the ODP. The boundary value problem of mathematical modeling of the process of pipeline transportation of oil in the standard form is presented in the form of a linear partial differential equation of the second order. The paper presents a solution to the boundary value problem of mathematical modeling of unsteady oil flow through a trunk pipeline in the presence of internal concentrated pressure sources in the form of functions describing the dependences on time and spatial coordinates of pressures and average cross-section pipeline oil flow rates. To represent the solution of the boundary value problem in the form of convolution integrals, Green's functions and standardizing functions are obtained, which makes it possible to use non-smooth (discontinuous) dependencies to describe programs for changing the values of internal concentrated pressure sources over time. The solutions obtained make it possible to use the methods of the theory of optimal control of systems with distributed parameters to solve the problems of optimal control of the process of pipeline transportation of oil.