A Semi-Analytic (p/z) Rate-Time Relation for the Analysis and Prediction of Gas Well Performance

Abstract

Abstract This paper presents a rigorous theoretical development of long term boundary-dominated flow solutions which involve direct coupling of the stabilized flow equation with the gas material balance equation. Due to the highly non-linear nature of the gas flow equation, pseudopressure and pseudotime functions have been used over the years for the analysis of production rate and cumulative production data. While the pseudopressure and pseudotime functions 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 relate the viscosity-compressibility product with the reservoir pressure (p/z) profile. These models provide approximate, but direct, solutions for modelling 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 rate prediction. We also provide verification of our new flowrate and pressure solutions using numerical simulation results and we demonstrate the application of these solutions using a field example. Introduction This paper focuses on the development and application of semianalytic solutions for modelling gas well performance - with particular emphasis on production rate analysis using decline type curves. Our emphasis on decline curve analysis arises both from its utility 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.) versus time 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. Several efforts 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, 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 these so-called "Arps equations" is primarily due to the explicit form of the relations, which makes them easy for practical applications.