Widths of Hydraulic Fractures
Authors: Perkins T.K. , Kern L.R. 
Published: December 2007

Abstract
A study of fluid mechanics, rupture of brittle materials and the theory of elastic deformation of rocks shows that, for a given formation, crack width is essentially controlled by fluid pressure drop in the fracture. Operating conditions which cause high pressure drop along the crack (such as high injection rate and viscous fluids) will result in relatively wide cracks. Conversely, operating conditions which cause low pressure drop (low injection rates and thin fluids) will result in relatively narrow cracks. Charts and equations have been derived which permit the estimation of fracture widths for a variety of flow conditions and for both horizontal and vertical fractures.

Introduction
There has been considerable speculation concerning the geometry of hydraulically created fractures in the earth's crust. One of the questions of practical importance is the width of fractures under dynamic conditions, i.e., while the fracture is being created and extended. Such width information could be used, for instance, to help estimate the area of a fracture generated under various conditions. Also, there has been a recent trend toward the use of large propping particles. Therefore is desirable to know what factors can be varied in order to assure entry of the large particles into the fracture. There has been some work on fracture widths reported in the literature. In particular, there have been several Russian publications dealing with this subject. These papers have dealt principally with the elastic theory and the application of this theory to hydraulic fractures. These studies have not led to an engineering method for estimating fracture widths under dynamic conditions. A recent papers has reviewed and summarized the Russian concepts. An earlier paper from our laboratories also discussed the application of the elastic theory to hydraulic fractures. This first approach, based largely on photoelastic studies, has proved to be too simplified to accurately describe the fracturing process. However, these early thoughts have served as a guide during the development of more exact concepts. We would like to present in this paper our current concepts regarding fracture widths and some estimates of hydraulic fracture widths for several conditions. We believe that it is now possible to predict with fair accuracy the factors influencing fracture widths. Furthermore, the method of prediction has been reduced to a simple and convenient graphical or numerical calculation.

CRACKS IN A BRITTLE, ELASTIC MATERIAL
Many investigators have shown that competent rocks behave elastically over some range of stresses. Of course, if the tensile stress imposed upon a rock exceeds some limiting value, then the rock will fail in tension. In similar manner, there are some limiting shear stresses that can be imposed upon rocks. Hubbert and Willis have discussed the shear conditions which will lead to failure. Under moderate stress conditions (such as those likely to be encountered when hydraulically fracturing) and when stresses are rapidly applied, relatively, most rocks will fail in a brittle manner. Hence, for this discussion of hydraulic fractures in the earth's crust, we assume the rocks behave as brittle, elastic materials. Let us develop the discussion in the following way. (The following thoughts are applicable only to brittle materials.)First we consider a brittle, elastic system. An energy balance will show the minimum pressure necessary to fracture rock, and from this pressure we calculate the minimum crack width resulting from extension of a hydraulic fracture.Then we will show that, under ordinary fracturing conditions, fracture widths are appreciably greater than the minimum widths of extending fractures. In fact, we will find that crack width is controlled by fluid pressure drop in the fracture.We will discuss pressure drops in fractures and resulting crack widths for various operating conditions and both vertical and horizontal fractures.Finally, we will discuss the significance of these concepts, their relationship to fracturing pressures. etc. First, consider minimum fracture extension pressures.
We can shed some light on this question by considering the theory proposed by Griffith to explain the rupture of brittle, elastic materials. Griffith recognized that solid materials exhibit a surface energy (similar to surface tension in a liquid).
JPT
P. 937^