The Effect of Water Influx on p/z-Cumulative Gas Production Curves
Authors: Bruns J.R. , Fetkovich M.J. , Meitzen V.C. 
Published: December 2007

Abstract
The relationship between p/z and cumulative gas production for typical gas reservoirs was studied by calculating pressure response to various modes of gas production and water encroachment. Water encroachment methods considered were Schilthuis, Hurst simplified and van Everdingen-Hurst. In the method, the assumptions normally made in water encroachment calculations were accepted. Normally, pressures are measured and the gas reserves and water encroachment found implicitly. Conversely, in this work various encroachment factors, reserves and reservoir-aquiler geometry were assumed and the pressures solved implicitly. The results show the spectrum of p/z shapes that can be expected for real reservoirs. With normal encroachment rates for closed aquifers the p/z chart exhibits the typical inflection at early times. This has sometimes been interpreted as all measurement error. These studies have shown that a new look should be taken at interpretation. It is rather dangerous to extrapolate "straight-line" p/z charts if encroachment from an aquifer is suspected.

Introduction
A common method of predicting gas reserves is the graphical solution to the gas material balance equation. A special case of the material balance equation is linear in p/z with cumulative gas production (Gp) which predicts the initial in-place gas when p/z is extrapolated to zero. Derivation of this form is based on the equation of state, corrected for compressibility (pV = znRT), and, particularly, on the reservoir being closed (no water encroachment). A straight line on the p/z chart results when these conditions hold. However, an apparent straight line on the chart does not assure that the reservoir is closed. Many of the curves show a rapid decline in the early stages of production after which they flatten out. Confusion arises as to whether these characteristics are caused totally by pressure measurements. To answer this question in part, a series of controlled mathematical experiments was performed in which a typical gas field was produced subject to various forms of water encroachment. These runs were specifically designed to eliminate measurement errors by calculating pressures at the inner boundary of the aquifer. The resultant p/z charts were thus made available for study and direction in predicting reserves and to indicate the curvature that can be expected in addition to that caused by normal measurement error.

Solution of the Basic Equation for p/z
The basic equation solved for p and p/z is derived in Appendix A. It is
(1)
Ga and Gr are the apparent and real values of original gas in place and are derived by assuming a closed reservoir for Ga, and one open to an aquifer for Gr. The function S(p, t) is defined by three methods - Schilthuis, Hurst simplified or van Everdingen-Hurst. The definitions of these functions are given in Appendix B. Eq. 1 is the linear function that is commonly plotted (Ga vs S(p, t)/B-Bi) with the intercept predicting the original gas in place and slope predicting the water encroachment factor. This is a graphical solution of Eq. 1 when histories on pressure and cumulative productions are known. In some cases the equation has been rearranged so a plot can be made such that the encroachment factor is predicted by the intercept and the reserve by the slope. In the calculations presented in this paper, in-place values, water encroachment factors, rock fluid properties, and cumulative production were set. Eq. 1 was solved implicitly for p/z.
JPT
P. 287ˆ