On the Design of Vertical Hydraulic Fractures
Authors: Daneshy Abbas Ali 
Published: December 2007

Some of the assumptions involved in designing vertical hydraulic fractures should be critically examined as to their validity. This is done here on the basis of a new width equation and a numerical design procedure. It is found that some of the assumptions related to the fluid mechanics of the problem greatly affect the computed results and therefore deserve special problem greatly affect the computed results and therefore deserve special attention.

Introduction
The problem of computing the geometry of a hydraulic fracture is closely related to two separate engineering disciplines: fluid mechanics and fracture mechanics. The role of fluid mechanics is to establish the pressure distribution inside the fracture due to the flow of fluid, and fracture mechanics determines what shape the resulting hydraulic fracture will have. The computation of the geometry of hydraulic fractures essentially consists of the mathematical matching of the fluid and fracture mechanics of the process. process. Over the years, the procedure used for computing the geometry of hydraulic fractures has undergone several major developments. The list of those who have contributed is long, and among them, Howard and Fast, Perkins and Kern, Christianovich and Zheltov, Geertsma and de Klerk, Williams, and Nordgren deserve special attention. At the present state of the art one can determine the geometry of a vertical or a horizontal hydraulic fracture from several equations or equivalent charts. These equations are usually derived on the basis of certain assumptions that link the rigorous mathematical solutions of the various aspects of hydraulic fracturing. A good number of these assumptions stem from a mathematical necessity and are unavoidable. Some of them are based on experimental evidence and therefore are considered legitimate. Still there are some assumptions whose presence is not essential for the solution of the problem and whose validity has not yet been tested. These assumptions will receive special attention here. As will be shown, some of them can be eliminated without greatly changing the computed results, whereas eliminating others will change the outcome considerably.
The procedure used for the design and the study of hydraulic fractures in this paper considers the treatment fluid to behave like a power-law model, which is more realistic than assuming it to be Newtonian. The computation of the fluid leak-off does not depend on any specific mode of fracture propagation. Finally, the width equation is derived propagation. Finally, the width equation is derived on the basis of a nonconstant pressure distribution inside the fracture.
In designing hydraulic fractures one generally considers three events that occur simultaneously. These are (1) the opening of the fracture, (2) the pressurizing of the treatment fluid, and (3) the pressurizing of the treatment fluid, and (3) the leaking off of the fluid. For the sake of completeness, a brief discussion of the mechanism governing these events is given below.

Equilibrium Fractures
The concept of equilibrium fracture, developed by Barenblatt, states that the stress concentration at the tip of a fracture should be finite. For hydraulic fractures this means that the fracture should have a geometry similar to Fig. 1. It also means that the treatment fluid can never penetrate the entire length of the fracture.
JPT
P. 83