**4. Conclusion**

A numerical model is developed for contact analysis of proppant embedment in the formation based on experimental investigation. Initially, the model was developed based on the experimental design of proppant embedment in the laboratory where the load is applied uni-axially from the top. Then, the study is extended by applying load from top as well as from the bottom side of the proppant in the fractured surface to simulate the actual reservoir condition. The amount of proppant embedment has been computed on both sides of the proppant in the fracture surface. Also, the deformation and normal stress profile have been plotted along with the formation and proppant. The total penetration of the surfaces has been recorded 141 μm on each side as the equal loads have been applied on both sides of formations around the proppant. This shows that actual proppant embedment is very high if stresses are present on both sides of the proppant in the fracture. The computational contact mechanics analyses have been able to capture the actual conductivity of fracture showing that the finite element method can be used to estimate embedment depth and has comparable results with experimental measurements. Long-term production of hydrocarbon from shale reservoirs is directly related to fracture conductivity in the hydraulically stimulated reservoir volume. This study shows that the uncertainty and reduction in hydrocarbon production profile with time can be mimicked by exact estimation of proppant embedment and fracture closure with finite element method since it relates to fracture conductivity. The presented method can serve as a valuable criterion to effectively reduce the loss of hydraulic fracture conductivity in shale reservoirs with time. Based on this numerical model, the required fracture conductivity can be achieved by keeping the extra width of fracture in the design criteria to reduce the conductivity loss in the formation.
