A Markov-Chain-Based Method to Characterize Anomalous Diffusion Phenomenon in Unconventional Reservoir

The recent success in developing unconventional reservoirs has aroused many new challenges to the theory of reservoir engineering. In this paper, we try to investigate the anomalous diffusion phenomenon caused by the heterogeneity due to the fracture network on the reservoir scale. Firstly, we revisit the physical background of the single-phase flow diffusivity equation by discussing the equivalent single particle diffusion. Combining the characteristics of single particle diffusion with complex fracture geometry, it is indicated that anomalous diffusion phenomenon will be dominant on the reservoir scale, even for single phase production behavior. Then a model based on Markov chain is presented to demonstrate the proposed anomalous diffusion by simulating the particle normal diffusion on a geometric graph and then calculating the relation of the mean square displacement vs. time in the embedding Euclidean space. Based on the simulation results, in consequence, we make discussions on the characteristic size of the heterogeneity due to the fracture network on the reservoir scale, summarize two types of pattern for the anomalous diffusion, and accordingly provide a supportive argument for using the fractional diffusivity equation, in place of the classical one, to model the flow and production behavior in highly fractured unconventional reservoirs.