Analytical analysis of fracture conductivity for sparse distribution of proppant packs

Conductivity optimization is important for hydraulic fracturing due to its key roles in determining fractured well productivity. Proppant embedment is an important mechanism that could cause a remarkable reduction in fracture width and, thus, damage the fracture conductivity. In this work a new analytical model, based on contact mechanics and the Carman–Kozeny model, is developed to calculate the embedment and conductivity for the sparse distribution of proppant packs. Features and controlling factors of embedment, residual width and conductivity are analyzed. The results indicate an optimum distance between proppant packs that has the potential to maintain the maximum conductivity after proppant embedment under a sparse distribution condition. A change in the optimum distance is primarily controlled by closure pressure, the rock elastic modulus and the proppant elastic modulus. The proppant concentrations and the poroelastic effect do not influence this optimum distance.