Estimating Average Reservoir Pressure Using The Hyperbola Approach: New Algorithm And Field Examples
Authors: Kabir C.S. , Hasan A.R. 
Published: December 2007

Estimating Average Reservoir Pressure Using The Hyperbola Approach: New Algorithm And Field Examples.

Abstract
Estimation of average reservoir pressure (p) from pressure buildup tests by using the rectangular hyperbola approach was introduced in the early 1980s. The method was based on the approximation of the Horner equation and was therefore conceptually restricted to treating the infinite-acting data alone. This work extends the theoretical envelope of the previous studies and proposes a new algorithm for computing p.
In the proposed algorithm, we used Cartesian-derivative of pressure to aid interpretation. We handle data noise implicitly in this procedure. This work also provides some theoretical justification for including the, boundary-dominated data for the hyperbola analysis. In addition, pressure falloff examples are included to extend its application.
Initially, we used a set of known, synthetic examples to show the applicability of this method. Field examples constitute the paper's backbone, however. Five types of problems are examined here. They include buildup tests preceded by short- and long-producing times, reservoir with unknown heterogeneity, and involving noisy late-time data. Finally, we discuss a falloff test dataset, acquired with a mechanical gauge, to demonstrate the need for data smoothing, which is an important part of the new algorithm.

Introduction
Establishing the average reservoir pressure either from a pressure buildup or a falloff test is a very important task of transient test interpretation. This piece of information is an essential ingredient for managing a reservoir. Computing p is rather straightforward in a new well or when the transients have not had sufficient time to encounter reservoir heterogeneities. However, problems surface when the late-time data encounter outer boundaries of unknown or ill-defined shape, such as those in developed fields or in long-duration tests of an exploratory well.
The problem stems from our inability to identity the correct well/reservoir configuration for which mathematical solutions exist. Geological and geophysical data can potentially aid in identifying the reservoir boundaries, However, converging (buildup) or diverging (falloff) flow streamlines that make up the drainage geometry are dependent upon local heterogeneities, which are rather difficult to discern in field cases.
When an analyst can define the well/reservoir drainage configuration, then several methods become available for p computation. These methods include MBH-Horner, Ramey-Cobb, Odeh-Al-Hussainy, and Dietz, to name a few, for the no-flow outer-boundary condition. For the constant-pressure outer boundary, Kumar and Ramey and Ramey et al. presented appropriate methods. Total voidage replacement is presupposed for the latter case, however.
Methods that do not require the drainage shape definition for establishing p are available. For instance, the Muskat method, presented as early as in 1937, falls into this category. However, Ramey and Cobb's subsequent work showed that the Muskat method is applicable only to very late-time data and that it works well in water-drive systems. The other method, developed in the early 1980s, is based on the rectangular hyperbola approach. This method has been used with success in many producing provinces of the world, although some reservations were expressed initially. A theoretical basis for treating the late-time data, beyond the infinite-acting posed, was not given in the original work, however. Equally important, the method's ability to deal with noisy data remained unexplored, especially those encountered at late times.
This investigation extends earlier work and provides a theoretical framework for fitting hyperbola through a data segment, leading to p computation objectively. Field examples comprising both buildup and falloff tests were used to verify the new approach.

Theoretical Considerations
New Formulation. As shown in the Appendix, the buildup equation can be approximated by a rectangular hyperbola regardless of the well/reservoir configuration and outer-boundary conditions. P. 888