The optimal hydraulic fracture geometry under non-Darcy flow effects

The presence of non-Darcy flow in a hydraulic fracture significantly reduces the effective conductivity of the fracture, and hence adversely affects the productivity of a hydraulically fractured gas well. It has been reported that under non-Darcy flow a shorter and wider fracture geometry provides better productivity than a longer and narrower fracture; conversely, for Darcy flow a longer and narrower fracture yields higher productivity.The objective of this work is to optimize hydraulic fracture geometry under non-Darcy flow effects in hydraulically fractured well in order to obtain maximum well productivity and economic benefit. A rigorous mathematical model is established to describe transient non-Darcy flow in hydraulic fractures and coupled with Darcy fluid flow in the reservoir. A semi-analytical method is further presented to solve this model. Then, on the basis of the simulation results, the fracture geometry is optimized for two goals. The first goal is to determine the maximum well productivity for a given fracture volume. On the basis of theoretical analysis and hundreds of simulation results, a semi-experiential correlation is developed to calculate the optimal fracture length for the given fracture volume. The equation generated suggests that the optimal fracture length is proportional to (1/β)1/3, which means that under a given fracture volume, if non-Darcy flow is increased 8 times, the optimal fracture length will decrease by half. The optimal fracture length is also a function of the fracture volume, formation permeability, fluid viscosity and density, net pay thickness and flow rate. The second goal is to achieve the maximum economic profit through optimizing the fracture volume, since larger fracture volume generally brings larger productivity, but also leads to higher hydraulic fracturing cost. Analyses show that the more severe the non-Darcy flow, the smaller the optimal fracture volume required. Therefore, fracture geometry optimization should involve two stages: fracture volume optimization and fracture length optimization.The proposed optimal fracture length equation has been applied to analyze several cases from the literature. The results show that the design criterion to eliminate the non-Darcy flow effects will cause such a short fracture length that the well productivity would be significantly reduced. Therefore, it is suggested that well productivity and economic benefit should be systematically considered in the design criterion for hydraulic fracturing stimulation.