Pressure-Buildup Test Analysis- Variable-Rate Case: A New Approach

Abstract

Summary This paper presents a new method of estimating a constant-rate analog for a variable-rate flow followed by a buildup test. The time and rate m the analog are used to generate a conventional Homer plot for the buildup test. The approximation is derived rigorously from the Horner buildup and variable-rate superposition equations. The major advantage of this method over other proposals is that it not only forces the production volumes to be equal (material balance constraint) but also forces the pressures at a particular time to be equal. Our approximate method has been verified for several variable-rate schemes with a finite-difference numerical simulator. In effect of wellbore stomp was studied at both surface and sandface rates. In effects of a short shut-in time and a shut-in time equal to the duration of the production period were also investigated. The results of the new method seldom varied more than 5% from the results with rigorous superposition. The new is not intended to replace superposition, but rather to give the best possible constant-rate analog. This method enhances our ability to use the conventional (constant-rate) Horner theory to generate accurate results. Also, our new method requires a single superposition calculation rather than a superposition calculation for each data point, as in the case of the superposition plot. Introduction The purpose of this paper is to present a new method to the Horner method to analyze buildup mm following variable-rate production before shut-m. Previous studies focused on ‘intuitive" results or less accurate approximations that require the buildup test to exceed the duration of the flow period. The new method is derived to match exactly the pressure and material balance at a single time. This means that error will be introduced when a Horner plot is made and a slope and intercept are constructed through this point and its nearest neighbors. We will demonstrate, however, that this method almost always gives more accurate results than the other two approximate methods, results that approach the accuracy obtained with superpostion. Without this new method, we would be forced to use superposition to obtain consistently accurate results. Because the new method was derived from the superposition solution for a buildup test following variable-rate production, the superpostion plotting function must be calculated to use this method However, our new method requires only a single superposition calculation. This means that both superpostion and Horner plots can be made from the available data. One could argue that the Horner plot is unnecessary, but the ability to model the variable-rate system in terms of a constantly analog is useful. Several authors State equations to calculate the superposition plotting function for a buildup test. In the section Development of the Analysis Technique, we present a root equation (Eq. 1) that results from equating the Horner constant-rate equation and the superposition plotting funcion, Xb. Three methods of solving the root equation are given m the Appendices. Appendix A outlines the development of the root equation and its solution by Newton iteration as described by Hornbeck. Appendix B outlines the solution of the root equation by interpolation and correlation equations. Any of these solution techniques yields sufficiently accurate results. In the study described in Verification of the New Method, we used a finite-difference numerical model to generate the buildup test data following variable-rate production. The effects of wellbore storage and shut-in duration were studied. This study verified that the new is the most accurate constant-rate analog available. This study also documented some positive and negative aspects of the Horner and Odeh and Selig constant-rate analog methods. We demonstrate the application of the new method with an example in Application of the New Method. This example, which uses hand calculations, is intended to familiarize the reader with the method, but it is important to realize that, for more than a few changes in rate, the calculations should be performed on a computer. Development of the Analysis Technique Results of pressure-buildup test analysis have long been used to characterize reservoirs in the petroleum industry. A major obstacle in well testing is the attempt to maintain a constant production rate during the test. The constant production rate is required for basic drawdown and buildup test analysis methods, although these methods can be extended to account for variable-rate flow. The extension requires that the production rate be known as a function of time or that it can at least be estimated. Superposition is an exact formulation that leads to correct estimates of formation properties. There may be a need, however, f or an accurate constant-rate analog for a buildup test following variable-rate flow. To fill this need, we derived our new method from the Horner constant-rate equation and the superposition plotting function Xb. The result of equating the Horner constant-rate equation and the superposition plotting function, Xb, is given in Appendix A as y=(1/x) 1n (1+x)...........................(1) ..........................................(2) ..........................................(3) where tL=last shut-in time, qm=lat production rate, Q=cumu lative production, XbL=buildup superposition plotting function at last shut-in time tpn=pseudoproducing time for new method, and qn=pseudoproducing rate for new method. This method can be applied to any point (time) on the buildup test graph. WE ordinarily choose the last data point because it is the least likely to be distorted by wellbore storage. However, if a boundary is encountered, the last data point before the boundary was encountered should be used. Also, if the data are completely distorted by wellbore storage, neither the new method nor superposition is likely to provide correct results. The new solution assumes transient radial flow into the well over the net pay thickness, homogeneous and isotropic porous medium, uniform net pay thickness, porosity and permeability constant (independent of pressure), fluid of small and constant compressibility, constant fluid viscosity, negligible gravity forces, single-phase flow, and any rate schedule. The Horner equation, modified for our pseudoproducing time, tpn, and rate, qn is ..........................................(4) where.....................................(5) Eq. 4 has exactly the same form as the constant-rate Horner equation and can be interpreted with exactly the same relations for formation permeability, k, skin factor, s, and extrapolated pressure, pi (or p* in a finite system). Verification of the new Method We compared the new method to the Horner, odeh-Selig, and superposition methods.