Embedment of proppants with non-uniform particle sizes into rock fracture surface

Proppants are often used in hydraulic fracturing to place in fractures to prevent fracture closure. Due to the stress concentration at the contact area between proppants and rock and their difference in strength and stiffness, proppants become embedded into the rock fracture surface. An elastic-plastic model has been proposed to study the embedment of a single proppant, but there are few theoretical studies on the embedment of randomly dispersed and non-uniform size proppants. This paper proposed and applied a theoretical model to calculate the embedment of both monolayer and multi-layer proppants to rock fracture surface. The theoretical model of proppant embedment depth considering the effective influence height is developed based on the embedment characteristics of single proppant following the Hertz contact theory, and the surface deformation of multiple proppants is established based on the superposition principle. The theoretical model was realized in MATLAB and validated by four experimental studies of the embedment of mono- and multi-layer proppants under different closure stresses. The verified calculation model is subsequently applied to parameter analysis, and the influence of proppant heterogeneity is discussed. The results show that, under the same closure stress, lower proppant placement concentration, larger proppant particle sizes, and smaller Young s modulus contribute to greater proppant embedment depth. The heterogeneity of proppant has great influence on the embedment depth of single-layer proppants but a relatively smaller influence on multi-layer proppants. The proposed model provides a theoretical basis for considering proppant embedment in the optimization and application of proppants in the field.