Non-Graphical Solutions for Average Reservoir Pressure From Production or Buildup Data

Abstract

SPE Member Abstract Simple equations were developed for determination of the distance from wellbore at which the reservoir pressure is the same as average formation pressure. This volumetric averaging method was derived for both constant boundary pressure and no-flow across boundary type of reservoirs. The technique was then extended to transient flow equations to obtain mathematical relationships for determination of average reservoir pressure during either production or buildup periods without the need of a Horner plot or any other published graphs. These equations were then reduced to much simpler forms, by which approximate values of average reservoir pressure could be ascertain within 0.01%, provided that r /r ratio were greater than seven. Two example problems at the end of this paper compare the results of the proposed equations with the ones obtained by using Miller-Dyes-Hutchinson Matthews-Brons-Hazebroek, Dietz, and Ramey-Cobb methods. Furthermore, the examples show that the approximate equations are still providing very accurate results (within 0.1%) inspite of the rinv /re ratio being much less than seven. Introduction The analysis of bottom-hole flowing and shutin pressures for obtaining average reservoir pressure has been the subject of numerous papers. Miller-Byes-Hutchinson (MDH) presented a method of determining the average reservoir pressure by solving the differential equation for fluid flow in a cylindrical reservoir with one well located at the center. Their solution provided dimensionless curves for both constant-boundary pressure and bounded type of reservoirs. By using either of these MDH curves the average reservoir pressure can be determined. The MOH method requires a pseudo-steady state flow condition in the reservoir prior to shut in. Pitzer used the MDH method to obtain theoretical pressure buildup (PBU) curves for few other reservoir geometries. Matthews-Brons-Hazebroek (MBH) developed a method for calculating average pressure in a bounded reservoir. They presented graphs of dimensionless pressure (PDMBH) vs. dimensionless time (tDA) for different drainage-area geometry and well reservoir. The PD MBH and tDA are given as: .....(1) .....(2) p* is extrapolation of P on a Horner PBU plot to where the value of log () equals zero. Kh is determined from the slope of Horner's plot, and t is either the total production or the pseudo-production time (t) prior to shut-in. Based on MBH concept, Dietz developed an equation for determining the average reservoir pressure. His method uses geometrical shape factors (C.) for different well locations and drainage-area geometry. For bounded reservoirs, both Dietz and MBH methods provide identical results if stabilized-flow conditions pseudo-steady state) have prevailed prior to shut-in i.e. when MBH graphs become linear on semi-log plots). .....(3) If geometry of the reservoir is known, then Eqs. 1 through 3 and Horner PBU plot can be used to obtain P at stabilized flow conditions without the use of MBH dimensionless graphs. Eq. 4 developed by Dietz can be used to determine P directly off the Horner plot. This equation gives a t value which when substituted for At. the corresponding P was on the Homer plot equals P. ..........(4) P. 207^