Mathematical model of fracture complexity indicator in multistage hydraulic fracturing

The multistage hydraulic fracturing has been considered as the major stimulation method used for enhancing well productivity in unconventional low-permeability reservoirs. A definite positive relationship has been showed between fracture complexity and productivity. However, the fracture complexity is currently unknown due to the fact that it cannot be obtained through any existing diagnostic techniques or numerical simulation. In this paper, the pseudo number of fractures is defined as the fracture complexity indicator to describe the complexity of fracture network during multistage hydraulic fracturing. A model for predicting the fracture complexity indicator is established based on the principle of energy balance and minimum energy. Sensitivity analysis of this analytical model demonstrate that the fracture complexity indicator increases with injection time, injection rate and elastic modulus, while decreases with the height of fractures. Case study is performed to check the validibility of this analytical model in four different wells. It indicates that the SRV obtained from micro-seismic date cannot be used as a general criterion for evaluating the effectiveness of fracturing, since larger SRV does not mean higher complexity of fracture network or higher production. However, fracture complexity indicator described in this paper can illustrate the fracture complexity, which is a parameter to evaluate the effectiveness of multistage hydraulic fracturing. This study provides a general method to optimize the fracturing design so as to increase fracture complexity and evaluate the effectiveness of multistage hydraulic fracturing.