Pressure Drawdown Analysis, Variable-Rate Case

Abstract

Abstract A theoretical development is presented which provides a straightforward method of handling the drawdown analysis for both oil and gas wells flowing at variable rates. In the past our inability to analyze variable-rate drawdowns has been a major obstacle in using them for formation evaluation. This method should permit wider use of these tests in the future. Introduction Pressure build-up analyses are widely used to obtain reservoir data such as the effective flow capacity of the formation. On the other hand, although pressure draw- downs have been run for a long time, they have not been used extensively to evaluate reservoirs because they are difficult to analyze. one of the difficulties in analyzing pressure drawdowns has been that the theory developed until now has required the test to be run at a constant rate. For both oil and gas wells this is a difficult requirement to satisfy, especially during the early period of a well's life when drawdown tests are usually run. in spite of this limitation, drawdown tests are often run with build-ups as a check. In this paper a theory, which handles the drawdown analysis for variable rates, is presented for both oil and gas wells. Thus, one of the biggest deterrents to running drawdowns is removed, and they should become a much more useful engineering tool in the future. THEORETICAL CONSIDERATIONS OIL RESERVOIRS The general equation used for describing the unsteady- state radial flow of slightly compressible fluids in homogeneous porous media can be written in cylindrical coordinates as (1) where r is the radial distance in cm, t is the time in seconds, p is the pressure at r and t in atmospheres, phi is the fractional porosity, mu is the viscosity in cp, c is the compressibility in vol/vol/atm, and k, is the permeability in darcies. There are a number of solutions of Eq. I for various boundary conditions. One of these is the so-called "point- source" solution, which approximates the case of an infinite oil reservoir with a well located at r = 0 and produced at a constant rate. The point-source solution of Eq. 1 is (2) and h is the net pay thickness in cm, q is the production rate in cc/sec, p is the original pressure at t = 0, p is the pressure at any t and r, and r, is the well radius. Eq. 2 describes the pressure drawdown of a well where the formation around it is neither damaged nor improved. If a condition of permeability damage or improvement exists, the equation must be corrected for these effects. van Everdingen introduced an additional term: where S is the total flowing time for n constant-rate flow periods, t - t, t - t, ...t - t with rates q, q, ...q . If we use Eq. 4 in its present form to construct a straight-line pressure drawdown plot, k, r, phi, mu, c and S must be known. This is so because for every t a variable factor is left at the right side of the equation. This has been a major difficulty in analyzing a variable-rate pressure draw- down. JPT P. 960ˆ