This paper describes two techniques for hyperbolic decline parameter identification. The first technique uses a genetic algorithm in the optimization procedure. The genetic algorithms are potentially useful in solving optimization problems when the objective function contains irregularities. The second technique uses linear regression for fitting a decline curve to data. The method weights equally the production rates during curve fitting, resulting in a stable solution. Consequently, the results are reproducible for a wide range of applications. Both methods were tested against field and literature data, demonstrating rapid, stable convergence and reproducible curves.
Decline curve analysis is on of the oldest and one of the most practical tools used by petroleum engineers to estimate reserves and to predict wells and reservoir performance. The method is deterministic, the estimation of the remaining reserves being based on known historical data.
The conventional decline analysis method with its extrapolation procedure is empirical, with no fundamental theoretical foundation. However, the method is very popular, partially due to the success of the predicted forecast, and partially, because of the non-adequate predictions by other methods. This is especially true in gas reservoir evaluation where material balance methods do not work well, and the problem is exacerbated in tight and layered reservoirs where petrophysical information is scarce. Moreover, advanced decline curve analysis and computer simulation methods require knowledge of fracture, reservoir geometry as well as detailed history of flowing rates and pressure, which is most often unavailable in practical cases.
The purpose of this paper is to consider alternative solutions to an old problem, when only summary reservoir and production information is available. We consider two algorithms for forecasting the production of gas wells. One is an improved conventional method, which optimizes simultaneously the three parameters of the hyperbolic decline curve. The other method uses a genetic algorithm to fit functions to field data. The genetic algorithm offers a high probability of locating the global optimum resulting in a robust solution. Finally, both algorithms allow optimization of square of regression coefficients using nonlinear equation solvers.
Mathematical and graphical approaches to extrapolate oil and gas production started almost one-century ago1,2. However, Arps3 was the first to summarize the previous developments and to classify the decline curves using a loss-ration method.
He considered three basic decline equations:
constant- percentage decline (exponential) with a decline coefficient b=0,
hyperbolic decline with a decline coefficient between 0 and 1, and
harmonic decline with a decline coefficient of 1.
Arps' equations are listed in Table 1. The equations have no solid fundamental basis and their extrapolation is strictly empirical. Arps cited that reliable oil and gas reserves are needed during the early stages when only a minimum amount of information is available. Consequently, with a limited production history, a wide range of interpretations is possible at latter times.
Gentry4 studied the effect of reservoir and fluid properties on decline curve behavior and found that the behavior of the decline curves is affected mainly by relative permeability, rock heterogeneity, drive mechanism, fluid characteristics and operation considerations. He also concluded that reservoir heterogeneity would cause the decline coefficient to increase to values above 1.0.