Steady Flow Equations for Wells in Partial Water-Drive Reservoirs

This paper presents steady-state radial flow and pressure distribution equations for wells under partial water drive or fluid injection. A parameter f is introduced to characterize the strength of water drive or fluid injection and to show its influence on well productivity. Introduction No convenient method exists in petroleum literature to establish the well producing rate and pressure-distribution behavior for wells in partial water drive reservoirs and in fluid injection projects under steady conditions. Such behavior is understood for wells in closed or depletiontype circular reservoirs and in reservoirs with constant pressure at the outer boundary. The latter situation is encountered in an active water drive reservoir or in fully developed five-spot injection patterns. For most wells, a realistic condition at the reservoir drainage boundary is one of partial flow because of limited aquifer or fluid injection in the aquifer or both. Brownscombe and Collins presented a producing rate equation for a well at steady flow in a closed reservoir. The corresponding flow equation for constant pressure is well known. The derivation of those equations is presented by Craft and Hawkins. Dietz suggested a method presented by Craft and Hawkins. Dietz suggested a method of incorporating the effect of partial water drive on drainage area mean pressure of a well by changing shape factor values. Steady radial flow equations for a well in a circular partial water drive reservoir are developed in the partial water drive reservoir are developed in the following discussion. Theory Consider a well producing at a constant rate from a circular oil reservoir. The formation is assumed to be uniform in thickness, porosity, and permeability. The fluid has a constant viscosity and compressibility at reservoir temperature. After the initial transients have died down, a well reaches steady flow conditions, where the time rate of change of pressure, dp/dt, is constant with time. It is normally termed pseudosteady or quasisteady state for nonzero values of this constant and "true" steady state when the constant is zero. Under true steady state, pressure at a point acquires a constant value with respect to pressure at a point acquires a constant value with respect to time. Under pseudosteady conditions the absolute value of pressure at a point is changing with time, but the time rate of change of pressure is constant. Let us consider the flow rate across a radius r under the influence of the following conditions at the outer reservoir boundary at r : 1. No-Flow at r : In a closed or depletion-type reservoir, there is no flow across the outer reservoir boundary, implying that the pressure gradient (dp/dr) at r is zero. The fluid volume at the well (q) is produced solely by expansion of fluid in a cylindrical shell between the radii r and r . In other words: (1) (2) as rw less than less than re. The term [pi r h phi c (dp/dt)] represents the expansibility of the fluid contained between the radii re and rw . JPT P. 1654