Simulation and solution of a proppant migration and sedimentation model for hydraulically fractured inhomogeneously wide fractures

The distribution of proppant during hydraulic fracturing may directly contribute to the flow conductivity of the proppant fracture, so research on the migration and sedimentation of proppant in the fracture and the final distribution pattern is of great relevance. The mass conservation equations of proppant solids and fracturing fluid were adopted to describe the distribution of proppant migration and sedimentation in the fracture, improving the additional gravity coefficient by utilizing the density difference between proppant and fracturing fluid and the concentration of proppant, together with the proppant sedimentation velocity at different Reynolds numbers in this study. The flow coefficients were obtained by discretizing the system of equations through the finite volume method combined with the harmonic mean method and the upstream weight method. The concentration additional pressure gradient term was computed by using the Superbee format innovatively to improve the solution convergence of the model. Numerical simulations with identical parameters were compared with indoor test results, which fully verified the correctness of the model and the accuracy of the discrete solution based on the finite volume method. The effects of flow rate of fracturing fluid, ratio of injected sand, viscosity of fracturing fluid, grain size of proppant and density of proppant on proppant migration and sedimentation based on three elliptical fracture morphologies: even-wide, top-wide and bottom-narrow as well as top-narrow and bottom-wide were investigated and analyzed through comparing the rate of proppant front movement, the level of sweeping range and the degree of inhomogeneity in the range under different conditions. Findings of this study suggest that: (1) top-wide and bottom-narrow fractures are more preferable for homogenous sanding in the early stage of proppant injection, and top-narrow and bottom-wide fractures are best for sanding in the later stage; (2) the viscosity of fracturing fluid is the most influential factor on proppant migration and sedimentation, which increases in the range of 100 mPa.s to enhance the sweeping range and homogeneity of proppant sanding and therefore achieve a better fracturing effect.