The Effect of Poisson's Ratio on Fracture Height
Authors: Labudovic Vladimir 
Published: December 2007

Summary
In past fracturing programs, the height of a formed fracture generally was assumed. If the layer was homogeneous with clearly defined boundaries of shale, then the height of the formed fracture most frequently coincided with the layer thickness. Reservoirs having several intercalations and reservoirs of secondary porosity made it very difficult to presume fracture height. For these reasons, a method to determine hydraulic fracture height by use of the Poisson ratio of reservoir rocks was designed.

Introduction
The success of hydraulic fracturing also depends on whether the assumed fracture height is the same as the height formed by hydraulic fracturing. The importance of precise fracture height determination increases with massive hydraulic fracturing (MHF).
Many authors have dealt with hydraulic fracture size determination. Daneshy showed how to determine the length and the width of a fracture provided that the fracture height is assumed constant along the fracture formed. Perkins and Kern made an important contribution to fracture-width determination and particularly to the penny-shape crack theory. Their work on forming and shaping of a fracture is used in this paper. They, also assume the fracture height to be constant along the formed fracture.
Dobkins developed a practical method for fracture height determination in a well considered for fracturing treatment on the basis of data obtained from a well already fractured in the same reservoir area.
Others such as Rosepilier, Simonson et al., and Prats have dealt with fracture extension in relation to the physics-mechanical properties of reservoir rocks. Simonson et al. analyze factors affecting, the extension of hydraulic fracture in height-i.e., fracture propagation into overlying and underlying barrier formations. Essential factors for fracture propagation are (1) material properties in the pay zone and the barrier zone, (2) insitu stress in the pay zone and in the barrier zone, and (3) the effect of hydrostatic pressure gradients.
Considerations of fracture extension here were developed on a similar basis but, to simplify the calculation procedure of fracture height, only the Poisson's ratio is used as an essential element.
Elements of Hydraulic Fracture-Height Determination Method by, Use of Poisson's Ratio of Reservoir Rocks. A vertical fracture formed b), hydraulic fracturing extends in length, height, and width. The fracture tends to extend circularly according to the penny-shape crack theory but, in the case where barrier layers have larger horizontal stress and stiffness, the pay-zone fracture then must adjust to the shape of barrier layers.
Most frequently, shales are barriers possessing higher horizontal stress than sandstones so that they limit the fracture extension upward. In reservoirs with several shale and sandstone intercalations without sharp boundaries, it is difficult to anticipate fracture extension-particularly in a new, unexploited reservoir. Also, it is very difficult to anticipate fracture height in hydraulic fracturing of naturally fractured reservoir rocks.
For this reason, a new, practical method of fractureheight determination by use of Poisson's ratio was devised. Laboratory measurements determined that shales have a higher Poisson's ratio and a lower Young,'s modulus than sandstones from the same reservoir.
Theoretical basis for application of this method is provided by equations for calculating fracture gradient from Matthews and Kelly and Eaton, where the functional dependence between formation fracture gradient and Poisson's ratio of reservoir rocks is given.
In the laboratory of the Inst. for Mining and Rock Mechanics of Zagreb U., Poisson's ratio of rock from oil and gas reservoirs was determined by use of static and dynamic methods and was compared with results of Eaton's calculations. Results of measurements and calculations are given in Table 1.
Table I shows that there are considerable deviations in measured values from Eaton's. For this reason the method of calculation of Poisson's ratio from velocities of longitudinal and transversal waves given by Pirson and Rzhevsky and Novik was used.
Hydraulic fracture height is determined by calculating, Poisson's ratios of the rock in question for the fracturing area and entering them into tables and diagrams. The area of lower values of Poisson's ratio is the area of fracture height extension, and increased values represent barrier of vertical fracture extension. The obtained magnitudes for calculated fracture height should be compared with a log diagram for resistivity (R) and spontaneous potential (SP) from the same interval. Calculated height is also corrected with results of fracture-height measurements after fracturing and in this way reliability of this method is considerably increased.
Example 1. Oil Well W-1 was fractured with a hydrocarbon-base gel. The reservoir is composed of sandstone with thick deposits of shale on the bottom and shaly sandstone on the top.
JPT
P. 287^