Embedment behavior of variable-radius indenters in shale: A combined experimental and theoretical approach

During the shale gas extraction, a larger fracture closure stress causes proppants to embed in the shale surface, greatly reducing fracture conductivity. The heterogeneity of shale makes its mechanical properties exhibit obvious scale effects, and the embedment behavior of proppants of different sizes is thus affected by multi-scale mechanical responses. To investigate the effect of particle size on the embedment mechanism, this work conducted micro-indentation tests using indenters of varying radii to model the embedment process of a single proppant. The test results show that when the embedment depth is 1 μm, with the radius increasing from 25 μm to 400 μm, the corresponding load does not show a linear growth trend, while the Young's modulus increases by about 76%. When the embedment depth is 3 μm, 5 μm, and 10 μm, respectively, the load gradually increases with the increase in radius. It is worth noting that when the embedment depth reaches 10 μm, the Young's modulus corresponding to each particle size tends to be stable, about 25 GPa. Based on the Drucker-Prager yield criterion, an analysis model of proppant embedment depth considering the particle size effect was established. Compared with the traditional Hertz and Thornton models, the proposed model can more accurately reflect the plastic behavior and strengthening mechanism of shale, and is applicable for predicting and evaluating multi-scale embedment responses.