A Simple and Accurate Description of Nonlinear Fluid Leakoff in High-Permeability Fracturing
- P.J. van den Hoek (Shell Intl. E&P B.V.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2002
- Document Type
- Journal Paper
- 14 - 23
- 2002. Society of Petroleum Engineers
- 5.4.1 Waterflooding, 2.4.3 Sand/Solids Control, 6.5.2 Water use, produced water discharge and disposal, 5.2 Reservoir Fluid Dynamics, 2.4.6 Frac and Pack, 2.5.2 Fracturing Materials (Fluids, Proppant), 2.4.5 Gravel pack design & evaluation, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 3 Production and Well Operations, 2.5.1 Fracture design and containment
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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 leakoff 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 minifracture radii and lower total leakoff coefficients. This will result in more aggressive pump schedules.
Fluid leakoff from hydraulic fractures is normally described by a 1D (Carter) fluid-flow model. In its simplest form, the leakoff rate within this model is, for a propagating fracture of constant height h, given by the equation
where Qt = the leakoff rate at time t; h and L=fracture height and length, respectively; CT = the total leakoff coefficient; and t(x) = 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 leakoff 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 magnitude. 1,2 In this case, Eq. 1 needs to be replaced by a proper description of the reservoir fluid flow around the fracture.1-4 Also, for hydraulic fracture stimulation (frac-packing)5-11 and cuttings reinjection (CRI)12-14 in high-permeability reservoirs, leakoff 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 in which the reservoir flow contribution to total leakoff is the controlling factor, as can be the case for frac-packing operations.5,9,10
A number of efforts have been undertaken to improve upon Eq. 1 by incorporating 2D (nonlinear) reservoir flow effects. Gringarten et al.15 derived an approximate formula for fully transient elliptical flow around a static fracture. Settari2 presented an approximation for 2D leakoff from a propagating fracture in a homogeneous reservoir for constant injection rate. Koning1 showed that by substitution of the appropriate dimensionless parameters, the Gringarten formula15 can also be used for propagating fractures under constant injection rate. This point will be further addressed in the description of elliptical leakoff below. More recently, Valko et al.10 developed a radial leakoff model to capture the 2D reservoir flow associated with high-permeability fracturing.
To date, however, none of the efforts addressing nonlinear leakoff fluid flow around a hydraulic fracture have resulted in a model that can be used for a fracture propagating at arbitrary, not necessarily constant, velocity, i.e., that can be used to describe the growth of a fracture that propagates through a multilayer reservoir, with stress contrasts [leading to (temporary) retardation/ acceleration of fracture growth] and rock mechanical property contrasts, and that can also be used to describe the fracture closure after shut-in. There is no model, moreover, that in its simplicity is comparable to the Carter formula (Eq. 1), and as such is easy to implement into any hydraulic fracturing code. It is the aim of the current paper to resolve that deficiency.
An exact numerical solution has been derived of the fully transient elliptical (leakoff) fluid flow equation around a hydraulically induced fracture propagating with any, not necessarily constant, velocity and based on an arbitrary fracture growth history. Note that this solution (i.e., the leakoff rate profile around the fracture) at any timestep will affect the fracture volume balance and pressure, and, therefore, the fracture propagation rate in the next timestep. The numerical solution is presented in the description of the numerical leakoff model to come.
A simple analytical formula for elliptical leakoff rate is presented in the description of elliptical leakoff. It is shown that this formula approximates the numerical results within a few percent, 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 leakoff rates (i.e., from low-permeability fracture stimulation on one hand to high-permeability waterflood fracturing on the other). Typical application areas are not only high-permeability fracture stimulation and CRI, but also waterflood fracturing in low- to medium-permeability reservoirs and/or reservoirs with medium-to-high-viscosity oil, in which the late-time transient ("pseudoradial") solution that is often used1,3,4 tends to underestimate fracture size.
In the section describing the limits of linear leakoff, the above result is applied to a variety of hydraulic (propped) fracturing and CRI field examples to explore the limits of the linear (Carter) leakoff assumption, both in pressure-decline analysis of minifractures and subsequent computation of optimized pump schedules and in the computation of fracture dimensions in CRI.
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