A Novel Mittag-Leffler Function Decline Model for Production Forecasting in Multi-Layered Unconventional Oil Reservoirs

Abstract

Over the past several decades, Arps decline curve analysis (DCA) has proved to be effective and efficient for production forecasts and EUR estimates due to its simplicity and applicability. However, as multi-stage hydraulically-fractured horizontal wells have unlocked the economic potential of unconventional reservoirs, forecasting future production accurately using Arps decline models becomes more challenging because of the complicated fluid flow mechanisms characterizing stimulated multi-layered ultra-low permeability porous media. Many field studies indicate unreliable forecasts and limitations in multi-layered field applications in particular. This paper presents a Mittag-Leffler (ML) function decline model which enhances the reliability of forecasts for multi-layered unconventional oil reservoirs by honoring anomalous diffusion physics for each layer. Many traditional decline curve models fail to honor the sub- or super-diffusion phenomenon under the paradigm of anomalous diffusion. The general form of our proposed two-factor ML function consolidates anomalous diffusion and classical diffusion into a single model, specifically including Arps hyperbolic, harmonic, exponential decline models and the stretched exponential decline model (SEPD) as special cases. Comparisons show that the ML model falls between the predictions of Arps and SEPD models in which the estimates are consistently either "overly optimistic" or "too conservative." For a multi-fractured horizontal well, the fracture height partially penetrating different layers leads to a layer-wise flow pattern which is reflected and captured in the production profile by curve-fitting the corresponding ML function parameters. We provide a workflow to guarantee consistency when applying the approach to each layer in field cases. We applied the workflow to one synthetic case using embedded discrete fracture modeling (EDFM) and to two field cases. We used hindcasting to demonstrate efficacy of the model by matching early-to-middle time production histories, forecasting future production, and comparing forecasted performance to hidden histories as well as to the corresponding EURs. The comparisons demonstrate the validity and reliability of the proposed ML function decline curve model for multi-layered unconventional oil reservoirs. Overall, this study shows that the novel ML-function DCA model is a robust alternative to forecast production and EUR in multi-layered unconventional oil reservoirs. The workflow presented was validated using one synthetic case and two actual field cases. This method further provides unique insight into multi-fractured horizontal well production profile characterization and facilitates well-spacing optimization, thereby improving reservoir development in layered unconventional reservoirs.