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HYDRAULIC FRACTURING 17/255
m ¼ fluid viscosity, cp models is the requirement to specify the fracture height or
G ¼ E=2(1 þ n), shear modulus, psia to assume that a radial fracture will develop. It is not
h f ¼ fracture height, ft always obvious from data such as logs where, or whether,
the fracture will be contained. In addition, the fracture
17.3.3 The PKN model height will usually vary from the well to the tip of the
Perkins and Kern (1961) also derived a solution for a fixed- fracture, as the pressure varies.
height vertical fracture as illustrated in Fig. 17.6. Nordgren There are two major types of pseudo–three-dimensional
(1972) added leakoff and storage within the fracture (due to (P3D) models: lumped and cell based. In the lumped (or
increasing width) to the Perkins and Kern model, deriving elliptical) models, the fracture shape is assumed to consist
what is now known as the PKN model. The average width of of two half-ellipses joined at the center. The horizontal
the PKN fracture is expressed as length and wellbore vertical tip extensions are calculated
1=4 at each time-step, and the assumed shape is made to match
ð
q i m 1 nÞx f p
w ¼ 0:3 g , (17:9) these positions. Fluid flow is assumed to occur along
G 4
streamlines from the perforations to the edge of the ellipse,
where g 0:75. It is important to emphasize that even for with the shape of the streamlines derived from simple
contained fractures, the PKN solution is only valid when analytical solutions. In cell-based models, the fracture
the fracture length is at least three times the height. shape is not prescribed. The fracture is treated as a series
The three models discussed in this section all assume of connected cells, which are linked only via the fluid flow
that the fracture is planar, that is, fracture propagates in a from cell to cell. The height at any cross-section is calcu-
particular direction (perpendicular to the minimum stress), lated from the pressure in that cell, and fluid flow in the
fluid flow is one-dimensional along the length (or radius) vertical direction is generally approximated.
of the fracture, and leakoff behavior is governed by a Lumped models were first introduced by Cleary (1980),
simple expression derived from filtration theory. The and numerous papers have since been presented on their
rock in which the fracture propagates is assumed to be a use (e.g., Cleary et al., 1994). As stated in the 1980 paper,
continuous, homogeneous, isotropic linear elastic solid, ‘‘The heart of the formulae can be extracted very simply by
and the fracture is considered to be of fixed height (PKN a non-dimensionalization of the governing equations; the
and KGD) or completely confined in a given layer (radial). remainder just involves a good physics-mathematical
The KGD and PKN models assume respectively that the choice of the undetermined coefficients.’’ The lumped
fracture height is large or small relative to length, while the models implicitly require the assumption of a self-similar
radial model assumes a circular shape. Since these models fracture shape (i.e., one that is the same as time evolves,
were developed, numerous extensions have been made, except for length scale). The shape is generally assumed to
which have relaxed these assumptions. consist of two half-ellipses of equal lateral extent, but with
different vertical extent.
In cell-based P3D models, the fracture length is discre-
17.3.4 Three-Dimensional and Pseudo-3D Models tized into cells along the length of the fracture. Because
The planar 2D models discussed in the previous section only one direction is discretized and fluid flow is assumed
are deviated with significant simplifying assumptions. to be essentially horizontal along the length of the fracture,
Although their accuracies are limited, they are useful for the model can be solved much more easily than planar 3D
understanding the growth of hydraulic fractures. The models. Although these models allow the calculation of
power of modern computer allows routine treatment fracture height growth, the assumptions make them pri-
designs to be made with more complex models, which are marily suitable for reasonably contained fractures, with
solved numerically. The biggest limitation of the simple length much greater than height.

Figure 17.6 The PKN fracture geometry.