Fractal fractional dual porosity diffusion model

Abstract

Unconventional reservoirs are characterized by having a wide pore-size distribution, ranging from nanopores to hydraulic fractures. Given these heterogeneities, non-uniform flow behavior is likely to be developed, and new models considering these complexities should be sought. This research presents a fractal-fractional model to consider the anisotropy, heterogeneity, and anomalous diffusive flow, that may occur inside unconventional reservoirs. This is done by incorporating a more general flux law and power-law relationships. Subdiffusive flow is considered in the fracture network inside the SRV. Time-fractional derivatives in the flux law model this anomalous diffusion behavior (fractional approach). Additionally, it is assumed that the stimulation induces certain fractal characteristics in between the propped hydraulic fractures. Inside this region, petrophysical properties and matrix block size are assigned through power-law relationships (fractal approach). The model is solved numerically through a finite difference scheme, the results show good agreement with existing numerical and analytical works. The generated responses cannot be obtained with existing models when anisotropy-heterogeneity and anomalous diffusion are present. Thus, the typical slopes are not recovered when the fractal dimension (dmf), connectivity index (θ), and anomalous diffusion exponent (α) take other than normal diffusion values (dmf = 1, θ = 0, α = 1). According to this study, the anomalous diffusion approach represents an alternative for analyzing well responses from complex systems. It is also shown that, when designing stimulation treatments, attention should be paid to increasing unpropped fracture density-connectivity. To the best of the authors' knowledge, this is the first time that the effect anisotropy and heterogeneity in unconventional reservoirs, is analyzed through a 3D combination of fractal-fractional diffusion.