Analytical and Numerical Analysis for Temporal Anomalous Diffusion Flow in Unconventional Fractured Reservoirs
Authors: Al-Rbeawi Salam , Owayed Jalal 
Published: March 2019

Abstract
Understanding the phenomenal anomalous diffusion flow mechanisms in unconventional fractured porous media is one of the objectives of this paper. It aims to convert this understanding to analytical and numerical models that could help in managing reservoir performance. The second objective is studying the impact of several reservoir parameters that collaborate with temporal anomalous diffusion flow mechanisms on pressure distribution, flow regimes, production rate decline, productivity index behavior during the entire production life of these reservoirs. The motivation is eliminating the possibilities of misunderstanding reservoir performance if reservoir fluid transportation in porous media is assumed undergoing classic (normal) diffusion flow mechanisms.
Several analytical and numerical models are presented in this study for pressure drop and decline rate as wells as cumulative decline rate and productivity index in ultralow permeability reservoirs that depleted by multiple hydraulic fractures. The reservoirs of interest are assumed consisting of stimulated part called stimulated reservoir volume (SRV) where hydraulic fractures are propagating and unstimulated part called unstimulated reservoir volume (USRV) where no hydraulic fractures. The analytical models are developed from trilinear dual-porosity flow models (TDP) presented in the literature with a consideration given to the anomalous diffusion flow. Two inner wellbore conditions are considered: The first is constant sandface flow rate and the second is constant wellbore pressure. While a linearized implicit finite difference method is used to simulate numerically pressure distribution. In this technique, the temporal domain is discretized to several time steps while the space between hydraulic fractures is divided into several block-centered grids. Early time solutions are developed for pressure behavior and decline rate wherein transient state is the dominant flow condition. Different reservoir configurations and different temporal anomalous diffusion flow exponents (α) are investigated.
The outcomes of this study are: 1) Understanding the impact of temporal anomalous diffusion flow mechanisms on unconventional reservoir performance. 2) Developing analytical and numerical models for pressure behavior, flow rate, cumulative flow rate, and productivity index considering diffusion flow mechanisms. 3) Developing analytical models for different flow regimes that could be seen during the entire production life of reservoirs. 4) Studying the impact of reservoir configurations and temporal anomalous diffusion flow conditions on transient and stabilized pseudo-steady state productivity index. The study has pointed out: 1) Anomalous diffusion flow mechanisms have significant impact on pressure drop, flow rate, and productivity index of unconventional reservoirs especially during transient state flow. 2) Wellbore pressure drop declines rapidly when temporal diffusion flow mechanism becomes dominant flow pattern in the porous media during early and intermediate production time, however, the trend is reversed at late production time. 3) Anomalous diffusion flow exhibits transient productivity index higher than normal diffusion flow while the index of normal diffusion is bigger than anomalous diffusion during pseudo-steady state flow. 4) Different slopes are observed for early linear flow regime corresponding to different anomalous diffusion flow mechanism. 5) The maximum ultimate production capacity is obtained by normal diffusion flow mechanism eventhough cumulative production given by anomalous diffusion flow at early and intermediate production time is bigger than normal diffusion. 6) Reservoirs with anomalous diffusion flow mechanism may have better performance than reservoir with normal diffusion flow mechanism.