Numerical Investigation of the Perforation Friction Loss and Discharge Coefficient

Among the efforts to optimize hydraulic fracturing treatments in multi-stage horizontal wells, extreme limited entry (XLE) perforating has evolved as a technique to increase the stage length and reduce the total stage count with no compromise to the well productivity. The frictional losses in such a choked fluid diversion practice is tremendous, and it is known to change dynamically due to the perforation erosion process. The orifice-based pressure drop equation, commonly used to estimate the perforation friction loss, contains a dimensionless kinetic energy correction factor, the coefficient of discharge (C_d). C_d is used to correct the ideal rate of discharge (Q_i) obtained by solving Bernoulli's equation, to the actual rate (Q) due to the non-ideal effects of flow contraction and friction.

Utilizing Computational Fluid Dynamics (CFD), this work modeled the flow through perforations and developed a quantitative understanding of the C_d for sharp-edge drilled and semi-round jet perforations. The study also investigated the C_d sensitivity to several design parameters such as the perforation diameter, tunnel length, and fluid viscosity. In addition to the steady state simulations, a transient erosion model with dynamic mesh capability was developed to predict the time-dependent C_d behavior due to the erosional process.

The results identified a C_d value of 0.72 for a 0.35" sharp-edge drilled perforation. The discharge efficiency of actual jet and eroded perforations is higher. The model estimated a C_d ranging from 0.75 to 0.83 for a 0.35" jet perforation due to the semi-round entry and inlet burr effect. The erosion process, which dynamically changes the perforation size and shape, also improves the perforation's discharge efficiency. The transient erosion model predicted a 23% increase in the perforation size and a post-erosion C_d of 0.94. The sensitivity analysis indicated that the smaller the perforation size, the longer the tunnel, the higher the viscosity and proppant concentration, and the smaller the proppant size, the lower theC_d. This numerical work provides an insight on the C_d value and its dependency on the perforation parameters, which helps to understand the wide range of values reported in the literature and reduce the uncertainty in estimating the perforation friction.