The fracture models developed by Khristianovich and Zheltov, Perkins and Kern, and Nordgren are compared to field fracturing-treatment data. The Khristianovich- Zheltov model agrees with the field data as adequately as the Perkins-Kern-Nordgren model does, but both models need to account for changing instantaneous shut-in pressure (ISIP). A method for on-the-job fluid-loss pressure (ISIP). A method for on-the-job fluid-loss coefficient measurement based on measurements of increasing ISIP is developed. Furthermore, examples illustrating the fluid-loss coefficient measurement method are discussed. Finally, examples that show excellent agreement between field data and production estimates based on Khristianovich-Zheltov fracture parameters are presented. presented. Introduction
The ultimate goal of fracturing research is to understand fracture growth so that optimal treatments can be designed and applied successfully to wells. An integral part of this effort has been the development of mathematical models of the complex fracturing process. Two models have evolved from a different set of boundary conditions applied to the rock mechanics theory. One model developed by Khristianovich and Zheltov (hereafter referred to as KZ) involves the assumption of a rectangular shape for the vertical cross section of the frac~ reotAnr oelemo dl elovedoptrPby Perkins and Kern that was modified by Nordgren (hereafter referred to as PKN) involves the assumption that the vertical cross-sectional shape is elliptical. The models have been compared and contrasted in various ways by several authors; Geertsma and Haafkens and Daneshy provide good bibliographies. Recent publications contend that only the PKN model fits field-observed fracture responses. This paper, however, shows field data that are consistent with the KZ model as well as with the PKN model.
Fig. 1 illustrates the difference between the KZ and PKN models in vertical cross section. Scale drawings of PKN models in vertical cross section. Scale drawings of the two shapes are shown for equal area at three different aspect ratios. Visual differences are apparent only up to an aspect ratio of about 100. A typical fracture would have an aspect ratio greater than 1,000. For example, a fracture 0.5 in. [1.27 cm] wide that is 250 ft [76.2 m] high would have an aspect ratio of 6,000. Therefore, the visual shape of the fracture is not the most significant difference between the two models. The important differences are in fracture widths and fracture friction pressure losses.
The difference in vertical cross sections (Fig. 1) can be related to the presence or absence of slip at boundary interfaces. According to this interpretation, the KZ model permits slip, while the PKN model does not. Another permits slip, while the PKN model does not. Another interpretation of either shape is that it represents an approximation to a more complex shape, such as the S-shaped fractures referred to by Veatch in his Fig. 12. The shape of an actual fracture that is 200 to 300 ft [61 to 91 m] high and about 0.5 in. [1.27 cm] wide is not represented exactly by either model. As rock lithology and stress levels vary over vertical distance, fracture width may respond locally with a complicated, curved cross section.
The variation in fracture shape between the KZ and PKN models leads to significantly different fracture widths PKN models leads to significantly different fracture widths and lengths because the frictional pressure loss along the two fractures is different. The KZ fracture experiences decreasing pressure loss as the fracture becomes wider and longer. It widens at a faster rate than it lengthens, so the increased friction loss to be expected by increased length is more than offset by increased width. The PKN fracture behaves in the opposite way; the width does not increase fast enough to overcome the effects of increased length. The PKN fracture experiences increased friction loss as it becomes longer. The level of friction loss (or equivalently, the overpressure level in the fracture) influences the distribution of fracture volume into width and length. The PKN fracture would be roughly twice as long and half as wide as an equivalent fracture generated under the KZ assumptions.
Even with the differences resulting from fracture shape, the two models are similar in the following ways: (1) the rock is assumed to behave in a linear elastic manner, and the fluid does not fill the leading edge; (2) the horizontal cross section is elliptical with major pressure losses occurring at the fracture tip; and (3) the vertical fracture height is fixed with fracture growth occurring only in width and horizontal distance away from the wellbore.
On the basis of field pressure observations, a difference in fracture friction loss is not enough to discard either model. When increasing ISIP behavior is taken into account, both models predict increasing bottomhole treating pressure independent of the fracture friction trend. pressure independent of the fracture friction trend. Furthermore, the increasing ISIP trend can be used as an indicator of fluid-loss behavior during a treatment so that treatment parameters can be altered during a job to maximize effectiveness. Finally, actual field data over a wide range of conditions are modeled closely by KZ-shaped fractures, giving credence to this model as a tool for fracture design and surveillance.