A Simplified Analytical Elastoplastic Embedment Depth Model of Proppant Based on the Drucker–Prager Yield Criterion

In deep shale gas reservoir exploration, elastoplastic embedment plays an increasingly important role in analyses of the conductivity of fracture networks because it is affected by high temperature, high in-situ stress. To obtain a full understanding of proppant embedding behavior, an analytical model of the embedment depth of a single spherical proppant was developed based on the Drucker–Prager (D–P) yield criterion and shale elastoplastic mechanical behavior. Shale indentation tests with rigid balls with diameters of 0.8 mm and 1.0 mm were conducted to validate the reliability of the theoretical model. Compared with the Hertz, Thornton and non-linear models, the new model developed in this paper was in better agreement with embedment experimental data. Each parameter of the new model has clear physical meaning and can be obtained by a series of triaxial compression tests. An Analysis of Variance (ANOVA) method was applied to discuss the sensitivity of the model to the parameters, and the results showed that the strength parameters α and Y contributed the most to, and had nonlinear effects on, the proppant embedment depth.