Hydraulic fracturing is a common stimulation technique in unconventional reservoirs to create fractures systems and allow hydrocarbon production. Proppant (granular material) is normally injected during hydraulic fracturing to keep open the fracture network and enhance hydrocarbon production performance. Proppant has a strong influence on fracture compliance and therefore will affect the characteristics of the generated seismic wavefield. To account for the effect of proppant in fracture compliance, we have developed new analytical formulations to obtain normal and tangential compliance for the case of dry and fluid-saturated fractures. We derive these expressions based on Hertz-Mindlin contact theory. Results from the compliance sensitivity analyses provide insights into the effects of proppant distribution and mechanical properties on fracture compliance. We also applied the innovative generalized multiscale finite-element method (GMsFEM) to simulate wave propagation through discrete hydraulic fractures filled with proppant. The GMsFEM approach represents individual fractures on a finely discretized mesh; this fine mesh is used to capture fracture properties by generating quantities (basis functions) that are used for modeling wave propagation on a much coarser grid. This methodology reduces the size of the computational problem, allowing faster results. Simulation results indicate the changes of the scattered wavefield as the proppant placement varies in different parts of the fractures and as the number of fracture stages increases.

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