Multi-stage hydraulic fracturing is a key technique used for improving well productivity in shale gas/oil reservoirs. However, the outcome of this technique is inconsistent from area to area. This is believed to be due to the rough design of hydraulic fracturing because of lack of knowledge of how fracture branches are initiated and populated. The objective of this study was to develop an analytical method to predict the population and the average length of fracture branches for maximizing Stimulated Reservoir Volume (SRV).

This study used calculus of variations to derive an analytical model for predicting the time-dependent population and dimension of fracture branches clusters of perforations. The rate of propagation of created fracture branches is simulated along the path of the minimum energy in the system. Lagrangian of the system is determined by the energy transfer from kinetic energy to pressure energy, elastic-potential energy, surface energy and heat loss due to friction. The condition for Hamilton's integral to be stationary is that the Lagrangian of the system satisfies the Euler equation.

The SRV was formulated as a function of dimension of fracture branches, number of perforation clusters and the orientation of the horizontal wellbore relative to the maximum horizontal stress. A case study of Woodford Shale indicates that the concept of FCI does reflect the complexity of fractures in the real world. The mathematical model reveals that after rapid growth at the very beginning, the number of fracture branches increases slowly. The growth rate of fracture branch declines with time. A sensitivity analysis with the analytical model indicates that the population of fracture branches at any given time increases with fluid injection rate. It is concluded that the growth rate of SRV in the reservoir declines with pumping time. The efficiency of creating SRV should be maximized by quickly pumping dense and viscous hydraulic fluid over a short time period in each stage of fracturing. The distance between perforation clusters should be optimized based on the average length of fracture branches given by the optimal stage-pumping time.

This study first time presents an analytical model to predict population and the average length of fracture branches. It provides petroleum engineers a rigorous method for maximizing the SRV in volume stimulation of shale gas/oil reservoirs.

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