Interpretation And Analysis of Hydraulic Fracture Pressures
Authors: Guo F. , Morgenstern N.R. , Scott J.D. 
Published: March 2010

Abstract
Fracture pressure analysis is used to estimate the geometry and type of hydraulic fractures. Many fracture pressure analysis models have been put forward since the 1970's. However, few comparisons between the models' predictions and experimental data are found in the literature. This paper applies fracture pressure analysis to pressure data produced in the laboratory. The analysis shows that fracture pressure generally declines after breakdown and that fractures seem to propagate in a manner similar to the KGD or radial models. However, neither model can predict a slow decline in fracture pressure due to the conventional leak-off assumption, and a rapid decline due to the continuity equation.

Introduction
One main factor that complicates the process of hydraulic fracturing is that it cannot be directly observed. To overcome this deficiency, fracture pressure analysis was developed. Fracture pressure analysis can be used to estimate fracture geometry through the analysis of pressures during and after injection. Models which predict fracture geometry and bottomhole pressure were first developed. The models and the parameters in the models can be adjusted so that the change in bottomhole pressure reasonably matches that observed. Consequently, the corresponding fracture geometry becomes the possible fracture geometry. Fracture pressure analysis is routinely performed in hydraulic fracturing. The intent of this paper is to understand the limitations of the current methods.

Principles of Fracture Pressure Analysis
Fracture pressure analysis was developed by Nolte(1–5). The basic principles are analogous to those for pressure analysis of transient fluid flow in the reservoir. Both provide a means to interpret complex phenomena occurring underground by analyzing the pressure response resulting from fluid movement in rock formations. The difference is that fracture pressure analysis involves the process of fracture propagation. Therefore, fracture mechanics fluid mechanics, and flow in porous media must be incorporated. In order to obtain a closed form solution, this complex solid/fluid mechanics interaction problem, like two-dimensional (2D) hydraulic fracturing models, must be simplified by introduction of some assumptions. The simplification procedure is as follows.
Fracture shape and the direction of fracture propagation are specified, not controlled by strength theory or principles of fracture mechanics. Like 2D hydraulic fracturing models, fracture pressure analysis accepts three kinds of fracture geometry: PKN, KGD, and radial model. The three models are illustrated in Figure 1. Both PKN and KGD models have a rectangular extension mode. The difference between them is that PKN model uses an elliptical cross section, while KGD model has a rectangular cross section. The radial model has a circular shape and propagates in the radial direction.
FIGURE 1: Schematic of fracture models. (Available in full paper)
Fracture width can be expressed as an explicit function of fracture pressure and fracture length. This relationship is called fracture compliance. Fracture compliance can be expressed as
Equation (1) (Available in full paper)
Equation (1a) (Available in full paper)
for PKN model,
Equation (1b) (Available in full paper)
for KGD model, and
Equation (1c) (Available in full paper)
for radial model.
The previous relation (Equation 1) for compliance is for the case in which the pressure within a fracture is constant across the area.