Summary
In this paper we present a rigorous theoretical development of solutions for boundary-dominated gas flow during reservoir depletion. These solutions were derived by directly coupling the stabilized flow equation with the gas material balance equation. Due to the highly nonlinear nature of the gas flow equation, pseudo pressure and pseudotime functions have been used over the years for the analysis of production rate and cumulative production data. While the pseudo pressure and pseudotime functions do provide a rigorous linearization of the gas flow equation, these transformations do not provide direct solutions. In addition, the pseudotime function requires the average reservoir pressure history, which in most cases is simply not available.
Our approach uses functional models to represent the viscosity-compressibility product as a function of the reservoir pressure/z-factor (p/z) profile. These models provide approximate, but direct, solutions for modeling gas flow during the boundary-dominated flow period. For convenience, the solutions are presented in terms of dimensionless variables and expressed as type curve plots. Other products of this work are explicit relations for p/z and Gp(t). These solutions can be easily adapted for field applications such as the prediction of rate or cumulative production.
We also provide verification of our new flow rate and pressure solutions using the results of numerical simulation and we demonstrate the application of these solutions using a field example.
Introduction
We focus here on the development and application of semi-analytic solutions for modeling gas well performance¾with particular emphasis on production rate analysis using decline type curves.
Our emphasis on decline curve analysis arises both from its usefulness in viewing the entire well history, as well as its familiarity in the industry as a straightforward and consistent analysis approach. More importantly, the approach does not specifically require reservoir pressure data (although pressure data are certainly useful).
Decline curve analysis typically involves a plot of production rate, qg and/or other rate functions (e.g., cumulative production, rate integral, rate integral derivative, etc.) vs. time (or a time-like function) on a log-log scale. This plot is matched against a theoretical model, either analytically as a functional form or graphically in the form of type curves. From this analysis formation properties are estimated. Production forecasts can then be made by extrapolation of the matched data trends.
The specific formation parameters that can be obtained from decline curve analysis are original gas in place (OGIP), permeability or flow capacity, and the type and strength of the reservoir drive mechanism.
In addition, we can establish the future performance of individual wells, and the estimated ultimate recovery (EUR).
Attempts to theoretically model the production rate performance of gas and oil wells date as far back as the early part of this century. In 1921, a detailed summary of the most important developments in this area was documented in the Manual for the Oil and Gas Industry.1
Several efforts2,3 were made over the years immediately thereafter, and probably the most significant contribution towards the development of the modern decline curve analysis concept is the classic paper by Arps,2 written in 1944. In this work Arps presented a set of exponential and hyperbolic equations for production rate analysis. Although the basis of Arps' development was statistical (and therefore empirical), these historic results have found widespread appeal in the oil and gas industry. The continuous use of the so-called "Arps equations" is primarily due to the explicit form of the relations, which makes these equations quite useful for practical applications.
The next major development in production decline analysis technology occurred in 1980, when Fetkovich4 presented a unified type curve which combined the Arps empirical equations with the analytical rate solutions for bounded reservoir systems.
Pre-Drilling Production Forecasting of Parent and Child Wells Using a 2-Segment Decline Curve Analysis (DCA) Method Based on an Analytical Flow-Cell Model Scaled by a Single Type Well