A mathematical model for calculating rod-shaped proppant conductivity under the combined effect of compaction and embedment

Hydraulic fracturing is a widely applied technique used to improve productivity in the oil and gas industry. Spherical proppants are commonly used in hydraulic fracturing, and massive studies have been conducted on proppants of this shape. Rod-shaped proppant has raised considerable interest in recent years, but few researches have been reported on the quantitative evaluation of the new shape proppant. In this paper, based on Hertz theory of elastic contacts, mathematical models were derived to calculate fracture conductivity, reduction in fracture aperture, proppant embedment, and deformation of rod-shaped proppants. The models for calculating the porosity of the fracture with proppants in loose and close packing modes were also established in this study. The results showed that, compared with spherical proppants, rod-shaped ones are theoretically more promising for hydraulic fracturing, especially under low elastic moduli and high closure pressures. From these derived models, rod-shaped proppant of big size might provide better fracture conductivity. It is concluded that the analytical models with correction coefficients could match the rod-shaped proppant embedment and fracture conductivity at different closure pressures. They are of great significance in selecting proppants and helpful to achieve high oil or gas production from hydraulic fracturing with proppants.