A simple and accurate description of nonlinear fluid leakoff in high-permeability fracturing

Fluid leakoff in hydraulic fracturing is conventionally described using the well-known linear (Carter) model. Although this works well in low-permeability formations, the linear leakoff assumption may lead to sometimes significant overestimation of fracture dimensions in medium- to high-permeability formations. At present, no methodology is available that allows an easy estimation of the impact of nonlinear fluid leakoff on fracture dimensions and on pressure decline (minifracture) analysis. This paper aims to resolve that deficiency. An exact numerical solution is presented to the fully transient elliptical fluid-flow equation around a propagating hydraulic fracture for arbitrary pump rate(s). In addition, a simple analytical formula for elliptical leakoff rate is presented that is shown to yield an excellent approximation of the numerical results, both during fracture growth and after shut-in. This formula can be easily incorporated into any existing hydraulic fracture model, and it is applicable over the entire range of fluid leak-off rates (i.e., from low-permeability fracture stimulation on one hand to high-permeability waterflood fracturing on the other). The above result is applied to a variety of hydraulic fracturing field examples to explore the Limits of the linear (Carter) leakoff assumption, both in pressure-decline analysis of minifractures and in fracture design. It is shown that in frac-packing and high-perm cuttings reinjection (CRI), the linear leakoff assumption may lead up to a tenfold overestimation of fracture dimensions. Conversely. incorporating nonlinear leakoff in the minifracture interpretation of high-permeability fractures will yield larger mini-fracture radii and lower total leakoff coefficients. This will result in more aggressive pump schedules.