Fluid leak-off in hydraulic fracturing is conventionally described using the well-known linear (Carter) model. Although this works well in low-permeability formations, the linear leak-off 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 non-linear fluid leak-off on fracture dimensions and on pressure decline (mini-fracture) analysis. The current paper aims at resolving 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 leak-off 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 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 hand.

The above result is applied to a variety of hydraulic fracturing field examples to explore the limits of the linear (Carter) leak-off assumption, both in pressure-decline analysis of mini-fractures and in fracture design. It is, amongst others, shown that in frac-packing and high-perm CRI, the linear leak-off assumption may lead up to a tenfold overestimation of fracture dimensions. This result helps to explain the common field observation that CRI in unconsolidated sandstones results in fractures that appear to be significantly smaller than predicted by conventional hydraulic fracture models.


Fluid leak-off from hydraulic fractures is normally described by a one-dimensional (Carter) fluid flow model. In its simplest form, the leak-off rate within this model is, for a propagating fracture of constant height h, given by the equation

Equation 1

where Ql is the leakoff rate at time t, h and L are fracture height and length, respectively, CT is the total leakoff coefficient, and t(x) is the first time of exposure of x to injection fluid.

It is well-known that Eq. (1) only works properly if the fracture propagation rate is large compared to the leak-off diffusion rate. If this is not the case, the use of Eq. (1) can lead to overestimation of fracture length. For example, in waterflooding under fracturing conditions, this overestimation may be up to two orders of magnitude1,2. In this case, Eq. (1) needs to be replaced by a proper description of the reservoir fluid flow around the fracture1–4. Also, for hydraulic fracture stimulation (frac-packing)5–11 and cuttings re-injection (CRI)12–14 in high-permeability reservoirs, leak-off rate may be high enough compared to fracture propagation rate to the extent that using the 1D Carter model Eq. (1) is not justified anymore. This is especially true for those cases where the reservoir flow contribution to total leak-off is the controlling factor, as can be the case for fracpacking operations5,9,10.

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