The Solution Of Problems Associated With Constant Well Pressure
Authors: Cox Dave O. 
Published: December 2007

Abstract
A method is presented that allows any solution for the pressure effects associated with production at constant rate to be converted into a solution for producing rate when well pressure is constant. The producing rate when well pressure is constant. The method, which is suitable for computer application, involves expansion of the pressure solution for constant production rate into a pseudosteady-state form plus a transient series of negative exponentials. plus a transient series of negative exponentials. The producing rate for constant well pressure then is obtained directly from the coefficients of the pressure expansion. pressure expansion. Examples are presented for the producing rate from a well in an infinite reservoir, the pressure distribution around a well in an infinite reservoir, and the producing rate under pseudosteady-state conditions from a bounded reservoir. An exponential decline in producing rate results from pseudosteady state flow with constant well pressure. pseudosteady state flow with constant well pressure. By increasing the number of terms used in the expansion of a constant rate solution, any desired accuracy for the corresponding constant pressure may be obtained. For rapid estimation, a zero-order approximation that dimensionless rate is the reciprocal of dimensionless pressure usually will suffice. A first-order (exponential decline) approximation is normally accurate enough for engineering purposes, while a second-order approximation may be used for all but the most precise calculations or very early times.

Introduction
In the past, it was a common practice to test oil and gas wells at a constant producing rate; because of production allowables, in some instances a well would be produced at an essentially constant rate throughout its life. Today, as tighter reservoirs are being exploited and more wells are being produced at capacity rates, a constant well pressure produced at capacity rates, a constant well pressure is often more characteristic of production than a constant rate.
Hundreds of solutions of the diffusivity equation for constant producing rate exist in the literature, but only a few for constant well pressure. Most notable among the latter are those for pressure. Most notable among the latter are those for an infinite reservoir, a circular reservoir with a central well, and an infinite reservoir with an induced fracture. With the large number of constant rate solutions available, it would be advantageous to have a method to convert these solutions into constant pressure solutions. One method to do this is through superposition of incremental production rates and pressure functions to obtain production rates and pressure functions to obtain succeeding rates at later times. This method requires extensive computations and considerable care in formulating a suitable solution algorithm. Furthermore, because of the lesser number of increments at early time, poor accuracy is obtained at early time, while because of the larger number of previous increments, at later times the number of calculations required per time step increases and limits the effective range of this method. A more direct method for converting constant rate solutions into constant pressure solutions thus is desired. An excellent paper by Clegg provides such a method. Clegg demonstrated that approximate inversion methods for Laplace transforms could be used to determine constant pressure solutions. Several difficulties arise with this approach, however. First, the Laplace transform of the constant rate solution is necessary; this often requires numerical integration since analytic forms do not exist for all problems. Second, two different functions may be identical for a time range of interest, but have greatly different Laplace transforms. Third, it is difficult to evaluate the errors incurred with this approach. Finally, while the methods presented by Clegg work well for infinite reservoirs, they are not applicable to finite reservoirs.
For converting constant rate solutions into constant pressure solutions, it is the purpose of this paper to present a method that does not require the use of the Laplace transform, is applicable to bounded reservoirs, and allows an evaluation of errors incurred.