Effect of Wellbore Storage on Pulse-Test Pressure Response

Abstract

This paper presents the first results on the effect of wellbore storage on the response of a pulse test. In a pulse test between two wells a distance L apart in a homogeneous and uniform formation containing a single fluid of constant compressibility, the pressure response at the responding well is given by t dp1(t-t′)P(t;L) = Pp(t;L) + q(t′)dt′........(1)o dt′ The first term on the right side of Eq. 1 is the pressure response that would be observed if there was no flow into the responding well; the second term is the contribution caused by the flow into the responding well as a result of the pressure response itself. Here, P1 is the unit function per unit production rate of van Everdingen and Hurst. per unit production rate of van Everdingen and Hurst. The flow rate into the responding well, q, is related to the time rate of change of the observed pressure response. In terms of a constant wellbore storage, C, d p(t;L)q = r w2 h cC ...................(2)dt The results obtained by solving Eqs. 1 and 2 for an instantaneous injection of a fluid volume Vi at t = 0 (an instantaneous pulse) are shown in Figs. 1 and 2. (A brief outline of the solution method is given in the Appendix.) Fig. 1 shows that, for a given distance between the pulsed and the responding well (L/rw), the wellbore storage causes delay in the time at which the maximum pressure is observed, t (Pmax)/to. Here, to is the time at which the maximum pressure would be observed if there were no wellbore storage. Thus, when C = 0, t (Pmax) = to. Fig. 1 also shows that the value of t (Pmax)/to approaches unity as the interwell distance increases and as the wellbore storage parameter decreases. Once the value of the time at which the maximum pressure response occurs is determined, the value of the pressure response occurs is determined, the value of the maximum pressure response can be determined from Fig. 2. Fig, 2 shows that wellbore storage can also reduce the maximum observed pressure response appreciably. Since t (Pmax) = to when there is no wellbore storage, Fig. 2 shows there is no reduction in the maximum pressure response under those conditions. But the reduction in the maximum p assure response may be pronounced for large values of C. These effects can be best illustrated using examples. We take rw = 1/4 ft, L = 250 ft, kh/ = 6.228 x 10(3) md-ft/cp, ch = 1.051 × 10(-3) ft/psi, and Vi = 100 bbl for both examples, and use C = 10(3) for Example 1 (representative of a liquid-filled well) and C = 5 × 10(4) for Example 2 (representative of a well containing lowpressure gas). Results are listed in Table 1. There are two ways that wellbore storage effects should be used. One is in the design of pulse tests, in preparing the equipment and procedures to increase the preparing the equipment and procedures to increase the likelihood of being able to measure the anticipated signal over the indicated period of time. For instance, it could be very important to be aware that the high wellbore storage in Example 2 gave a time to reach maximum response that was 90 percent longer than would have been otherwise expected. The second use is in interpreting results. In Example 2, the calculated reservoir storage capacity (ch) would be too high by 33 percent if wellbore storage effects were ignored. Also, the calculated reservoir hydraulic diffusivity (k/ c) would be too low by 47 percent if wellbore storage effects were ignored. percent if wellbore storage effects were ignored.It should be pointed out that uncertainties in reservoir parameters of a factor of 2 are sometimes tolerable. parameters of a factor of 2 are sometimes tolerable. JPT P. 707