Pressure History Inversion for Interpretation of Fracture Treatments
Authors: Gulrajani Sunil N. , Mack Mark G. , Elbel Jack 
Published: December 2007

Abstract
Conventional analysis of hydraulic fracturing treatments with analytical solutions is based on idealized assumptions, which limits its usefulness in many applications. Manual (trial and error) pressure matching techniques using a fracture simulator are laborious and consider only one parameter at a time. This paper presents an alternative technique for a consistent analysis of pressure records during both injection and decline, including changes due to rate and fluid viscosity variations. This allows it to be used to analyze calibration as well as propped treatment pressures, giving it broad field application. The technique includes a formal inversion procedure coupled to a hydraulic fracture simulator which includes both 2-D fracture models and a pseudo three-dimensional height growth model. The algorithm has also been extended to identify the occurrence of non-uniqueness, an inherent limitation in any fracturing pressure interpretation.
Comprehensive case studies from fracture treatments in different formations in North America, the North Sea and Africa are presented. Calibration treatments are first analyzed using this technique following which propped treatment pressures are predicted and then confirmed with field measurements. The analysis is applied to both normal and tip-screenout treatments, some with pressure records showing near wellbore pressure losses. Good agreement between the predicted and measured pressures in these cases shows the validity of the algorithm and confirms its use for field application.

Introduction
The seminal work by Nolte presented a theoretical basis for the evaluation of fracture pressure decline for calculation of fracture geometry parameters, fluid leakoff coefficient and treatment efficiency. This technique, when coupled with an analysis of pressures during injection, provides a systematic framework for minifracturing test evaluation and fracturing diagnosis.
Although this analytical approach is attractive, assumptions regarding height growth during injection and fracture geometry changes following shut-in hinder its application in a majority of field cases. Variants, both analytical and numerical, have been proposed to account for fracture tip or toughness pressures, fracture recession and height growth, fracture extension during shut-in, fluid compressibility and formation heterogeneties. These, however, are either based on numerical experimentation or may require additional input (fluid leakoff exponent, for instance) that may not be readily available. They also fail to rigorously account for additional deviations from ideal conditions such as rheology, multiple fluids and near-wellbore pressure drops that potentially could have significant effects on the results.
Manual pressure matching has also been employed to evaluate calibration treatments. This technique requires repeated execution of a fracture simulator with varying parameter inputs to visually match simulated pressures with field measurements. Pressures for any formation parameterization and treatment schedule can be evaluated using this procedure. Additionally, no assumptions other than those incorporated in the fracture simulator have to be made for its application. However, given the limited number of parameters of a formation which can be accurately characterized using wireline logs or laboratory tests, the procedure could require significant effort when several parameters are not known.
Interpretation of pressure records can alternatively be formulated as an "Inverse Problem".
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