Application of Wavelet Transform to the Analysis of Pressure-Transient Data
- M.Y. Soliman (Halliburton Energy Services) | J. Ansah (Halliburton Energy Services) | S. Stephenson (Halliburton Energy Services) | B. Mandal (Halliburton Energy Services)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- April 2003
- Document Type
- Journal Paper
- 89 - 99
- 2003. Society of Petroleum Engineers
- 4.3.4 Scale, 5.1.2 Faults and Fracture Characterisation, 5.6.3 Pressure Transient Testing, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 5.5 Reservoir Simulation, 3 Production and Well Operations, 5.2 Reservoir Fluid Dynamics, 5.6.4 Drillstem/Well Testing
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Although the mathematical concept of "wavelet transform" was developed in the early part of the 20th century, it did not receive practical application until the 1980s. Wavelet transform is now used in a wide variety of applications in the areas of medicine, biology, and data compression, among others. A significant benefit provided by wavelet transform is its capability to provide two functions:
Smoothing of the basic signal.
Retention or even enhancement of the details.
In data analysis, these characteristics allow the user to recognize hidden signals quickly. In well testing, these capabilities can enhance signals that can be masked by other events or by the data frequency itself. Because the derivative techniques currently used in well testing tend to smooth data and conceal certain events, the use of the wavelet transform to analyze the raw data could prove to be very valuable as a major step in enhancing the techniques of modern data analysis.
This paper briefly reviews the application of the wavelet transform methodology to data analysis in general and shows why its application to well-testing data in particular is important. It also provides guidelines for the application of this technology to well testing.
Several field examples are provided to demonstrate the application of the wavelet transform to well testing and to show how the qualitative and quantitative uses of the method determine wellbore anomalies, boundary effects, and other well-testing phenomena.
A significant amount of information can be carried within one signal. By mathematically manipulating the raw signal, the hidden or masked information can be revealed. This mathematical manipulation is called transformation. The transform of a signal (i.e., a vector) is a new representation of that signal. A signal that has been "transformed" by any of the available mathematical transformations is the processed signal.
There are many mathematical transformations that may be applied to a raw signal, the most popular of which are the Fourier transforms. In their raw format, most signals are time-domain signals, meaning that the signal is a function of time. In many cases, distinguishable pieces of information are hidden in the frequency content of the signal when viewing a signal in time domain (e.g., amplitude vs. time).
The frequency spectrum of a signal includes the frequency components of that signal. A mathematical transform of the signal changes the signal into the frequency domain and, consequently, reveals hidden information carried by the signal.
A mathematical transform is applied routinely to a signal to reveal the hidden information discussed above. There are numerous applications in which this transformation is used, not only in technical applications but also in daily life. Examples include analysis of electric light, car repair, and electrocardiography (ECG). An ECG signal gives the electric signature of the heart. Anomalies in the signal are hard to detect in the time domain; however, after transformation into the frequency domain, the anomalies become relatively easy to determine, thus leading to a correct diagnosis.
Application of Signal Analysis to Well-Test Analysis
Conventional well-test analysis of data uses a log-log plot of the pressure change vs. time. Logarithmic plotting is a very powerful technique because it maintains the relative shape of the major physical parameters affecting the performance of a well. Thus, an analyst may quickly identify the reservoir type and characteristics by using this technique. Maintenance of the shape of the data also allows an analyst to visually match the observed data to a theoretically developed model.1 Logarithmic plotting, however, smoothes data and may hide subtle effects of some other physical parameters.
To overcome this problem, the derivative method was developed and used extensively.2 Theoretically, the derivative technique enhances subtle changes between data points and subsequently overcomes some of the problems associated with the log-log plotting. Three features, however, compromise its effectiveness.
First, the logarithmic plotting technique is also used in the derivative technique, which may have some negative effect.
Second, because the derivative between two neighboring points will cause a very noisy graph and may be counterproductive in practice, the derivative is taken over a window of data. One has to strike the right balance during this process. Too small a window will produce a noisy derivative plot, while too large a window will overly smooth the data and reduce the effectiveness of the plot.
Third, as shown in Eq. 1, for an infinite-acting reservoir, the derivative is actually the product of time and the pressure derivative with respect to time. Although the use of the product of the derivative and time has some excellent justifications that we will not discuss here, it has the effect of stretching the plot and thus obscuring the effect of some of the well and reservoir effects.
The application of signal-analysis techniques to raw, unprocessed data to filter out noise and/or reach a better understanding of either the source or the path of a signal has been used in many fields. Similarly, the application of these techniques to raw, unprocessed pressure signals would help in the identification of some of the wellbore and reservoir anomalies that may be misinterpreted or even missed in conventional analyses. Surface or downhole valve shut-in, segregation of gas, liquid phases inside the wellbore, change of the fluid phase around the gauge, presence of boundaries, and change of flow regime may cause anomalies in a pressure- transient signal.
|File Size||5 MB||Number of Pages||11|