A study on equivalent permeability models for tree-shaped fractures with different geometric parameters

Reasonably characterizing hydraulic fractures and evaluating their permeability after fracturing are vital for the development of low-permeability reservoirs. Field micro-seismic monitoring, indoor fracturing experiments, and numerical simulations have determined that artificial fractures will bifurcate, thus a tree-shaped (TS) network can be employed to characterize the complex fractures. Considering that the fracture sections in classical fracturing models are generally assumed to be elliptical or rectangular, equations for calculating the pressure drop in TS fractures were derived in this study. Then, the equivalent permeability models for TS fractures between a point and a line as well as disc-shaped networks were presented and validated through numerical simulations. Based on the equivalent permeability models, the effects of geometric parameters on the equivalent permeability were discussed, and the permeabilities with elliptical and rectangular cross-sections were compared. The results showed that (1) for an elliptical cross-section, the effects of the fracture width ratio and height ratio on the equivalent permeability were influenced by their relationship and the primary fracture aspect ratio, while the effect of the fracture width ratio was more significant than that of the height ratio for a rectangular cross-section. (2) For an elliptical cross-section, the equivalent permeability increased with the rising of the primary fracture aspect ratio when the width ratio was greater than the height ratio, while the trend was opposite when the fracture width ratio was smaller than the height ratio. (3) The equivalent permeability of the elliptical crosssection was smaller than that of the rectangular cross-section, and their difference increased with an increase in the primary fracture aspect ratio, width ratio, and bifurcation series, while it decreased with the fracture height ratio. This study may identify the direction of fracture morphology required for reservoir reconstruction and can provide a reference for permeability evaluations after fracturing.