A fractional decline curve analysis model for shale gas reservoirs

In the past several decades, in order to have quick and direct methods to perform production forecasting and reserves estimation in practice, petroleum engineers have designed various techniques to interpolate the production rate both analytically and numerically, among which many decline curve analysis models have been proposed and widely used because of their simplicity and efficiency. Although all decline curve analysis models could be employed in some cases under certain assumptions, each has its own limitations and is not applicable for all cases. With the increasing interest in shale gas reservoirs, engineers have found a common long-tail behavior for gas production profile of shale gas wells, which cannot be well described by the current decline curve models. In this paper, based on the anomalous diffusion phenomena that also have the long-tail behavior, we developed a new fractional decline curve (FDC) model with three fitting parameters using the general solution of the fractional diffusion equations, which is a special case of so-called Mittag-Leffler function. In addition, we proposed a four-step scheme according to the asymptotic properties of the Mittag-Leffler function to quantify the three parameters. We verified the new FDC model against a numerical reservoir model. In addition, we applied the FDC model to perform history matching and production forecasting for five actual shale-gas wells from the Fayetteville Shale. The results show that the new model is easy to use and provides a reliable estimated ultimate recovery (EUR), which can help the petroleum industry to perform data analysis rapidly and forecast production more accurately in shale gas reservoirs.