Non-Uniqueness in Well Test Analysis: Case Studies
Authors: Kelly R.T. 
Published: December 2007

R.T. Kelly, SPE, Ranger Oil (U.K.) Limited

Abstract
The analysis of well test pressure data can often result in a non-unique solution, particularly as a result of the mathematical complexity of matching late-time derivative data and the limitations of well test analysis software. Other frequent faux-pas are the inappropriate use of the radial composite model and p*, these issues are discussed. A series of case studies are presented and discussed with a view to establishing that non-uniqueness is sometimes a reality. Furthermore, it is demonstrated (through examples) that starting with one mathematical match to pressure data and deriving a geological model is the reverse approach and may be fraught with danger.

Introduction
Few health warnings have been published concerning the abuse of the plethora of assumptions underlying well test analysis theory, the over-ambitious use of well test analysis software, and in particular the application of a single, non-unique 'model' to the evaluation of a field. Save for the efforts of Ershaghi and Woodbury, few forthright statements have been made to keep the unwary within the realms of reasonableness. By way of analogy and contrast, simulation literature is full of warnings by an august body of authors, led perhaps by Coats's offering of realism. The advent of type curve matching and use of the pressure derivative analysis have been both an enhancement and a source of frustration. Enhancement because the extraction of more reservoir information has been made possible, and source of frustration because it is frequently impossible to obtain unique solutions and the wealth of underlying simplifying assumptions are too often forgotten. These last points are exacerbated by the fact that not only do we no longer require graph and tracing paper to perform our analyses, but the software in use is only as safe as the underlying theory, which remains heavily assumption-laden.
Furthermore, the reservoir 'models' offered in modem software packages are simplified idealisations of pressure response, none of which can be as complex as the underground systems one is usually dealing with.
The case studies presented herein will demonstrate that several of these models may mathematically match the pressure data, and that selecting just one of them can be very dangerous, especially if this non-unique model is then used to predict long-term reservoir performance (for example if it is used as the basis of a reservoir simulator). Much emphasis is placed on collating the physical evidence which the well test analysis must match (not the converse), namely logs, core, the depositional model, and don't forget adjacent or analogous fields. These sentiments are expressed by Streltsova as she sets the scene to her treatment of complex problems, stating that "…models of the depositional environment derived from geological… studies of logs and core data can assist in the quality of well test interpretations". The word must should perhaps replace can in this latter statement, the caveat being that well test interpretation requires an appreciation of the dynamics of the reservoir which static geology alone cannot provide.

Validity Of The Assumptions
If any one or more of the assumptions (underlying the theory in use) is wholly or partially invalid, then the derived solution suffers accordingly. Well test theory is adequately covered in the industry standard works, in related areas of science, and in the appropriate sections of standard, general textbooks. Accordingly, the theory itself will not be discussed here save for a review of the plethora of underlying assumptions.
Solutions to the basic radial flow equation form the building blocks of well test theory, but the basis of this equation is purely horizontal flow of a monophase fluid to a well centred in a circular volume.
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