Using the complex plane radial flow potential for modeling the working process of a downhole jet pump

Abstract

Based on the use of the mathematical apparatus of the theory of functions of a complex variable, a method is proposed for modeling the propagation of the working flow in the flow path of an ejection well system in the form of a point source placed on the same axis as the mixing chamber of a jet pump. The point source of the working flow is considered as a function of the current reduced to a complex form. The working fluid radially leaves the source symmetrically in all directions. The streamlines of the working flow along the radii are directed from the center of the source. The radial velocity of the outflow decreases with distance from the center in inverse proportion to the first power of the distance to it. Using the equation of the velocity potential and the stream function, a relation is obtained for the complex potential of the plane-radial and spatial working flow. The set of mutually perpendicular streamlines and lines of equal potentials forms a hydrodynamic grid of a plane-radial working flow, which determines the kinematic picture of the working medium motion. The velocity of the liquid remains unchanged along the equipotential lines and changes when moving from one line to another. The flow rate of the liquid is constant along the streamlines and changes when passing to an adjacent line. The spatial hydrodynamic grid is formed by equipotential surfaces and flow surfaces of stream functions and has the form of orthogonally placed coaxial spheres and radial meridional planes. The complex potential obtained in the course of research is a function of the operating flow rate in the flow path of the jet pump and can be used in modeling the process of its symmetrical rotation in the well.