Application of Wavelets to Production Data in Describing Inter-Well Relationships
Authors: Jansen F.E. , Kelkar M.G. 
Published: December 2007

Abstract
Production data is one of the most abundant sources of data available about the reservoir and the well behavior. Several authors have shown how the cross correlation between pairs of wells can provide information about the flow path in the reservoir. The calculation of the cross correlation itself has often proved to be problematic. This is due to the non-linear and non-stationary nature of the inter-well relationship. The inter-well relationship is a function of the boundary conditions imposed by the wells themselves and the reservoir properties. The Wavelet transformation is a new tool, which unlike the Fourier transform, allows for a non-stationary treatment of the data. This opens new possibilities with respect to a robust cross correlation between wells and the use of these data to more consistently determine the cause of the well behavior and its influence on surrounding wells. This paper presents a brief introduction into the use of the Wavelet transformation to decompose the production data into a combination of their frequency (details) and smoothed (scaled) components. The frequency components are then used for the analysis of the inter-well relationship. The time localizing ability of the transform is exploited to generate a robust and time dependent cross correlation between pairs of wells, using traditional cross correlation routines such as the Spearman Rank correlation. This cross correlation can then be used to estimate the degree of well interference, preferential flow paths and the existence of flow barriers. The proposed method is validated using data from the North Robertson Field in West Texas. This field has been under waterflood since 1987. For additional information regarding the waterflood, please see Reference 19.

Introduction
Production data represent a source of information about the dynamic boundary conditions as well as the static reservoir properties. The in-situ flow process depends upon both the reservoir properties and the boundary conditions in the form of mass and pressure transfer. In a water flood the injection wells provide a dynamic boundary condition. Thus, if we view the reservoir in a simplistic manner (Fig. 1), we can assume that the production rates and pressures are a function of the combination of both the injection rates and the reservoir properties. This allows us to make certain assumptions with respect to the information present in the production data.
Production data are by nature dynamic and represents a composite of many different events, such as well control, reservoir decline and near wellbore damage, as well as the influence from nearby wells. By examining the production data, we find that it is often difficult to establish a clear and consistent patterns between pairs of wells. Figure 2 shows an example of the production rate as a function of time plotted along with four surrounding well injection rates. The data are taken from the North Robertson Field in West Texas. For reference the well locations are shown in Figure 3. It is not easy to calculate a cross correlation between a pair of wells. The cross correlation between a pair of wells tends to be nonlinear, thus, non-parametric cross correlation schemes such as the Spearman rank correlation has seen some success. The response between the wells can contain a time lag, in addition the cross correlation itself is also a function of time. The production data therefore represents a non-stationary signal, which require special treatment. Due to the pressure superposition among the wells, a direct and constant relationship between the wells will therefore not be easily detectable. A relatively strong change in the injection rate is required to generate enough of a change in the nearby producers to detect a significant correlation. This nonstationarity causes a time dependency in the cross correlation between the wells, thus, regular cross correlation techniques tend to fail.
The Wavelet transform is one method to break down the data into its frequency spectra and still retain the time dependency. P. 323^