New Analytical and Hybrid Model Considers the Impacts of Non-Darcy Flow Permeability, Stimulated Matrix Permeability, and Anomalous Diffusion in Fractal Porous Media
Authors: Al-Rbeawi Salam 
Published: May 2017

Abstract
A hybrid model for unconventional gas reservoirs couples three different parameters is presented in this paper. The first parameter is the anomalous diffusion in a fractal porous media depleted by multiple hydraulic fractures. The second is the stimulated or induced matrix permeability resulted in the stimulated reservoir volume (SRV) due to fracking process. The third is non-Darcy flow permeability in hydraulic fractures where Darcy law may not be useful to describe high potential fluid flow. This model is generated from multi-linear flow model for fractal reservoirs controlled by diffusive flow mechanism with adjustment for fluid flux through hydraulic fracture face considering minimum fracture relative permeability (kmr) and non-Darcy Number (FND). The formation of interest is considered to be consisting of three porous media: hydraulic fractures (HF), stimulated reservoir volume (close to the vicinity of the wellbore where hydraulic fractures propagate transversely with respect to the wellbore), and un-stimulated reservoir volume (USRV) in the outer drainage area close to reservoir boundaries where no hydraulic fractures exist.
Pressure distributions, flow regimes, and reservoir performances have been investigated for three types of unconventional gas reservoirs. The first is the formation, homogenous matrix permeability, where petrophysical properties of both stimulated and un-stimulated reservoir volume are the same i.e. fracking process does not change matrix permeability. The second is the fractal reservoirs having different petrophysical properties in the two volumes without considering normal or classic diffusion mechanism. The third is the fractal reservoirs where anomalous diffusive flow mechanism dominates fluid flow rather than random flow described by Darcy law. A set of comparisons has been generated between the three types based on different characteristics for hydraulic fractures and reservoir configurations as well as different phenomenological coefficients for anomalous diffusion. These comparisons help understanding clearly the impact of both non-Darcy flow permeability and stimulated matrix permeability on reservoir performance under normal and anomalous diffusion flow mechanisms.
The outcomes of this study can be summarized as: 1) Generating a new analytical model that describes pressure distribution in fractal unconventional reservoirs and couples non-Darcy flow permeability of hydraulic fractures, stimulated or induced matrix permeability with the anomalous diffusion in porous media. 2) Understanding the impact of these different parameters on reservoir performance. 3) Developing different models for all types of flow regimes that are expected to be observed during the entire production life of reservoirs. 4) Comparing the productivity index for reservoirs: having homogeneous matrix permeability, fractal with normal diffusion, and fractal with anomalous diffusion. The most interesting points in this study are: 1) Minimum fracture relative permeability (kmr) significantly enhances the productivity index by eliminating the impact of non-Darcy flow while non-Darcy flow number (FND) works conversely 2) Increasing stimulated matrix permeability enhances reservoir performance 3) The impact of non-Darcy flow permeability of hydraulic fractures is seen at early and intermediate production time where transient flow is dominant. 4) Two different trends are recognized for the impact of anomalous diffusion flow: the first is positive impact in the transient flow period and the second is negative in the pseudo-steady state flow.