Unsteady Flow of Fluids in Oil Reservoirs

Abstract

This paper is an analysis of the unsteady flow of a compressible fluid flowing radially to a well in a sand formation. The phenomenon of unsteady flow occurs as a result of fluid expansion. When the pressure in the formation is lowered, the fluids therein expand and the increase in volume imparts motion to the fluid which flows towards the region of lowest pressure in the formation. This process is continuous in the reservoir and extends further away from the well with increased production. By the ``equation of continuity,'' the solution for two specific cases of unsteady flow are derived. The first of these is that in which the fluid from a sand reservoir of limited size flows to a well in which the pressure at the level of the producing sand always remains constant. The variation of the pressure gradient, the rate of production, and the cumulative production with respect to time are given in Eqs. (11), (24), (29) and (30). The second case is that in which the flow of fluid to a well is such that the rate of production at the well is always constant. This case is derived by the assumption that the cylindrical body of sand, which is influenced by the well, is subject to a steady depletion of fluid, and in order that the rate remain constant at the well, fluid must flow into the reservoir from some extraneous source in increasing amounts. Eventually, however, the rate of flow of fluid into the reservoir becomes equal to the rate at which fluid is withdrawn from the well and steady flow is established in the sand. The equations for the pressure gradient and the decline of pressure at the well with respect to time are (39), (52) and (56).