Relative permeability model for shale reservoirs considering water distribution and imbibition with two fractal regions

Shale gas is a key transitional low-carbon energy source, offering a cleaner-burning alternative to coal, reducing greenhouse gas emissions, and supporting energy security during the transition to renewable energy systems. However, its efficient production is challenged by issues such as fracturing fluid retention, which adversely affects gas flow. During well shut-in and flowback following hydraulic fracturing, fracturing fluid imbibes into the formation, redistributing within its complex pore structure and altering gas-water flow dynamics. This study develops a comprehensive gas-water relative permeability model for shale reservoirs, using fractal theory and porous media mechanics. The model incorporates key factors, including pore size distribution, fractal geometry, tortuosity, pore connectivity, and fluid-phase interactions within distinct fractal regions of the shale. The analysis shows that increases in fractal tortuosity, critical water saturation, fractal dimension in the large-pore fractal region, displacement probability in the small-pore fractal region, and imbibition time reduce water relative permeability. In contrast, increasing displacement probabilities in both fractal regions, pore coordination number, and imbibition time enhance gas relative permeability. These findings highlight the importance of mitigating aqueous phase trapping and optimizing fracturing fluid flowback to minimize formation damage and improve production rates. By advancing our understanding of fluid behavior under complex reservoir conditions, this study provides a theoretical framework for designing operational strategies that enhance shale gas recovery, supporting its role in meeting energy demands while reducing greenhouse gas emissions.