Temporal and Spatial Anomalous Diffusion Flow Mechanisms in Structurally Complex Porous Media: The Impact on Pressure behavior, Flow regimes, and Productivity Index
Authors: Al-Rbeawi Salam 
Published: November 2019

Abstract
The main objective of this paper is understanding the phenomenal anomalous diffusion flow mechanisms in unconventional fractured porous media. This understanding is crucial for estimating the impact of these flow mechanisms on pressure behavior, flow regimes, and transient and pseudo-steady state productivity index of the two cases of inner wellbore conditions: constant sandface flow rate and constant wellbore pressure. The targets are hydraulically fractured unconventional reservoirs characterized by porous media with complex structures. These media are consisted of a matrix and naturally induces fractures embedded in the matrix as well as hydraulic fractures.
Several analytical models for pressure drop and decline rate as wells productivity index in ultralow permeability reservoirs are presented in this study for the two inner wellbore conditions. A numerical solution is also presented in this study for pressure behavior using a linearized implicit finite difference method. The analytical models are developed from trilinear flow models presented in the literature with a consideration given to the temporal and spatial fractional pressure derivative for the ano malous diffusion flow that could be the dominant flow mechanism in the stimulated reservoir volume between hydraulic fractures. Mittag-Leffler functions are used for solving fractional derivatives of pressure and flow rate considering that temporal and spatial fractional exponents are less than one. Two solutions are developed in this study for the two inner wellbore conditions. The first represents the transient state condition that controls fluid flow in unconventional reservoirs for very long produc tion time. The second is the solution of pseudo-steady state condition that might be observed after transient state flow. The second solution is used for estimating stabilized pseudo-steady state productivity index considering different reservoir conditions. In the numerical solution, the temporal and spatial domains are discretized into several time steps and block-centered grids respectively. The results of the analytical models are compared with numerical solutions.
The outcomes of this study are: 1) Understanding the impact of temporal and spatial diffusion flow mechanisms on pressure behavior, flow rate declining pattern, and productivity index scheme during early and late production time. 2) Developing analytical and numerical models for fractional derivatives of pressure and flow rate considering diffusion flow mechanisms 3) Developing analytical models for different flow regimes that could be developed during the entire production life of reservoirs. 4) Studying the impact of reservoir configuration and wellbore type as well as different temporal and spatial diffusion flow conditions on stabilized pseudo-steady state productivity index. The study has pointed out: 1) Temporal and spatial diffusion flow have a significant impact on pressure drop, flow rate, and productivity index. 2) Wellbore pressure drop for constant Sandface flow rate declines rapidly as the temporal diffusion flow mechanism is the dominant flow pattern in the porous media. 3) Wellbore pressure drop for constant Sandface flow rate slightly increases during transient state flow as the spatial diffusion flow mechanisms increase and rapidly increases during pseudo-steady state flow. 4) Productivity index of diffusion flow is higher than the index of normal diffusion flow during transient and pseudo-steady state conditions. 5) The linear flow regime is most affected by anomalous diffusing flow and can be used to characterize the type of diffusion flow.