Numerical investigation of 1D anomalous diffusion in fractured nanoporous reservoirs

Abstract

This dissertation presents a numerical anomalous-diffusion model for well performance analysis in unconventional wells and investigates its applicability to actual field data. Motivation for this research arises from the limitations of current modeling approaches to properly capture the production characteristics observed in complex nanoporous formations. First, a linear (1D) anomalous-diffusion model for single-phase slightly compressible flow is developed. To do so, Darcy’s Law is replaced by a more general fractional flux law incorporating long-range spatial and temporal pressure dependencies. Cases of constant-terminal-rate and constant-terminal-pressure production are considered and validated with available analytical solutions for both transient- and boundary-dominated flow periods. It is shown that the severity of sub- or super-diffusion can be determined from common diagnostic log-log plots of pressure drop or rate decline vs. time. Based on these insights, a novel approach for production data analysis in unconventional wells is proposed and applied to two Barnett shale-gas wells, one displaying characteristics of super-diffusion, the other of sub-diffusion. The numerical, slightly compressible flow model is converted to gas flow and conventional PVT correlations are used. Although the example analyses are based on limited production, reservoir, and completion data, the model shows potential in capturing the production characteristics of both wells. Finally, the anomalous-diffusion model is extended to two-phase, immiscible oil-water flow using an Implicit Pressure Explicit Saturation (IMPES) formulation, and the effects of sub- and super-diffusion on phase flow-rates and saturation distributions are studied. While simplifying assumptions are made with respect to phase behavior and fluid-rock interactions in nanoporous formations, the model is meant as a proof of concept for the future study of multi-phase flow under anomalous diffusion.