Pressure Analysis on Hydraulic Fracturing
Authors: Rosolen Marcos A. 
Published: December 2007

Abstract
The interpretation of the pressure behavior observed in hydraulic fracturing treatments was introduced by Nolte by means of a well-known technique that carries his name, and it has been the best available tool to calibrate and characterize the necessary data for fracture design and evaluation. To enhance the methodology, this paper proposes the virtual fracture concept to equate the fluid loss across its walls during the treatment, simplifying the Carter's solution for the filtration and allowing the determination of the fracture volume in a simple and explicit way. It includes, also, the formulation for the pressure drop due to fluid flow inside the fracture into the system of equations, establishing the influence of the fracturing fluid rheology in the process and improving the fracture dimensions calculation. The introduction of these concepts yields the main parameters that control this process in an independent way, reducing the number of free variables.

Introduction
The hydraulic fracture modeling has been formalized since 1957 and its evolution can be followed in specialized papers. These models have arisen from the coupling of some developed theories of the Fluid Mechanics, Porous Media Flow and Rock Mechanics areas. Two classical models for fracture design are established in the literature: the PKN method, formulated by Perkins & Kern and complemented by Nordgren, based on the formulation proposed by Sneddon for the fracture propagation and on the Carter's model for the filtration; and the KGD method, developed by Geertsma & de Klerk and revised by Daneshy, based on the theory proposed by Muskhelishvili, England & Green and Khristianovich & Zheltov for fracture propagation, on the Barenblatt criterion for its stability and, also, on the Carter's model. Both models consider Fluid Mechanics principles in the formulation of the fracturing fluid flow.
Since 1979, a sequence of Nolte's papers has introduced a new approach to the process. It is equated as a function of the variables recorded during the treatment (pressure, pumped volume, fluid properties and others), allowing its real-time analysis. This procedure has been enhanced by several contributions.
This paper introduces the virtual fracture concept to simplify the Carter's solution for the filtration, allowing the direct determination of the fracture volume. It also considers in this type of analysis the influence of the rheology in the fracturing fluid flow from its tensorial formulation, estimating the fracture width and determining the fracture geometry.
The following hypotheses are assumed:–The fracture is vertical, confined in its height and propagates symmetrically to the wellbore showing a rectangular transversal section. The height is h, the length is 2L and the width at the wellbore is w0;–The fluid is incompressible, pumped with a constant rate q at the surface. The fluid moves into the fracture under a steady shear flow, and the fluid viscosity is described by the Power-Law model. Its equation of state is given by the generalized Newtonian law–After the pumping, the fracture closes preserving the final length and height;–By convention, the compression signal is positive.

Volumetric Analysis
The volume and the dimensions of hydraulic fractures are controlled by the fluid filtration across the fracture walls which is quantified by means of an empirical curve obtained in laboratory.