New Mathematical Models for Calculating the Proppant Embedment and Conductivity

Proppant embedment plays a significant role in decreasing fracture aperture and conductivity, especially for weakly consolidated sandstones, shale (oil and gas) rock, and coal beds. Empirical and semi-empirical models were usually used to calculate the embedment of proppants. However the accuracy of matching or predicting the proppant embedment using these existing models may not be satisfactory in some cases. On the other hand, it is difficult to determine the coefficients of these models. In this study, analytical models were derived to compute the proppant embedment and fracture conductivity. These new models can be used to calculate the proppant embedment, proppant deformation, the change in fracture aperture and fracture conductivity in the ideal or experimental situations of either single-layer or multi-layer patterns in the fractures under closure pressures. The new models showed that the proppant embedment and fracture conductivity are affected by the factors of closure pressure, fracture aperture, the elastic modulus of proppant and coal bed, the size of proppant, the concentration of proppant-paving, etc. Experimental data of proppant embedment in fractures and fracture conductivity of different proppants at different closure pressures were used to test the models derived in this study. The results from matching the experimental data using the new and the existing models were compared. The results showed that the new models especially the revised new models could match the experimental data in all of the cases studied. The new models for calculating the proppant embedment and fracture conductivity with a better accuracy are of great significance in selecting proppants, which is helpful to achieve high fracture conductivity and then high oil or gas productions of conventional, especially unconventional resources such as shale oil, shale gas, and coal bed methane.