Using Finite-System Buildup Analysis To Investigate Fractured, Vugular, Stimulated, and Horizontal Wells (includes associated papers 22059 and 22060)
Authors: Mead Homer N. 
Published: December 2007

Summary.
The rectangular hyperbolic method (RHM)is a steady-state, finite-system concept that has proved to be amazingly accurate in determining final static reservoir pressure at shut-in. The method also calculates slope, skin, percentage of open reservoir thickness contributing to flow, effective drainage radius, and permeability. It apparently also gives reasonable answers in horizontal-well buildups. It has been found that all wellbore reservoir pressure buildups after all wellbore and storage effects have died out follow rectangular hyperbolic curves. The analysis is based solely on Darcy's steady-state radial-flow equation, the basic transient-pressure-buildup equation, and the determination of effective drainage radius from in an empirical equation based on field data obtained from the literature. The RHM is not related to any method based on an infinite-system concept.

Introduction
A method is presented here for transient-pressure behavior analysis that is based on a finite-system concept. As a matter of fact, it is simply another form for Darcy's steady-state radial-flow equation. It can be used for buildups and falloffs in any type of reservoir. It is a procedure that is based on examination of that portion of the reservoir from the wellbore out to the effective radius of drainage, rd. When a well is put on production at constant rate, more reservoir volume is produced into the well than can be produced by the reservoir close to the well; consequently. the drainage radius must increase.
When the well begins to produce, an effective circular band begins to move out from the wellbore. The inside effective circular radius, rf, is the farthest radius from the wellbore through which the same rate is flowing as from the wellbore. The outside effective circular radius, re, is the closest effective radius to the wellbore through which there is no flow. We have assumed that rd &gt;rf and that rd less than re. We assume that In rd = 1n r, -1/4. This is based on the fact that ln rf = ln r, -1/2. When re reaches the boundary of the reservoir system, it stops increasing in size. At the same time, rd and rf also stabilize when the well is producing continuously at constant rate.
The above situation has long been observed in the field. It is especially noticeable in stabilized wells. If a stabilized pumping well is shut in for a period of time, the effective radius of drainage decreases. If the well is shut in long enough, rd will eventually decrease to rwe. When the well is again placed on production, it will produce initially at a substantially higher rate before dropping constantly to the rate at which it was producing before it was shut in. Pseudo-steady-state conditions will have been reached again. The preceding condition will be exhibited in some manner in any producing well.
RHM was initially presented in 1981. Since then, two other papers on RHM have been written. In both these papers, however, the approach was based on an infinite-system concept. Several factors have been developed in the past 7 years that were not considered in the 1981 paper: R, tte, he, ke, and a way to make p* = ps on the Homer plot.
RHM can be used in any buildup or falloff as long as pseudo-steady-state conditions are reached. The purpose of this paper is to show that it can be used on buildups or falloffs in wells that are completed in naturally fractured reservoirs or vugular-type reservoirs, as well as the normal sandstone or limestone reservoirs. It can be used in normally or artificially stimulated wells, it can also be considered for horizontal wells. "Stimulated" is defined in this regard as the point where the effective wellbore radius is greater than wellbore radius. This definition does not include wells in which the wellbore diameter is enlarged for gravel packing.
This method is one of matching measured and calculated data points on buildups and falloffs. It is based on the fact that when all wellbore and storage effects have died out, the linear plot of shut-in time vs. bottomhole pressure (BHP) is a rectangular hyperbola specific for that particular buildup. The analysis is made with three data points on any portion of the linear plot. Three constants are determined, B, C, and D. With these three constants, calculated points from zero-time shut-in to final measured point are obtained and compared with the measured data points for that particular portion of the actual linear plot of measured pressures. The methods for calculating B, C, and D and determining data points from zerotime shut-in to infinite-time shut-in are described fully in Ref. 1. Care must be taken that all anomalous-pressure-behavior(APB) effects in later buildup are accounted for.
Derivations will be shown for the basic equations needed for REM analysis. The new concepts of R, tm, tte, he, ke, and the reason that using tte for tt in a Homer plot makes p* = ps are presented without proof. Then analyses for the five examples chosen are presented. Finally, overall conclusions are drawn.

Basic Derivations and Allied Data
The basic equations are developed as follows. Darcy's radial-flow equation is
(1) The basic transient-pressure equation is
(2) Equating Eq. I to Eq. 2 yields (3)
where is the delta pressure in the reservoir between rd and rw. Therefore, (4)
where ps =effective static reservoir pressure and pwf is bottomhole wellbore flowing pressure at the instant of shut-in, = (ps -Pwf). So, =, - . It has been found empirically that effective drainage radius, rd, is directly related to on a log-log plot of rd to according to Eq. 5:
(5)
JPT
P. 1361^