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SPE 89952 Wavelets in Petroleum Industry: Past, Present and Future

Copyright 2004, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Houston, Texas, U.S.A., 26–29 September 2004. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

L. Guan, SPE, Texas A&M U., Y. Du, SPE, New Mexico Institute of Mining and Technology, and L. Li, U. of Houston step to the next. Therefore, the implementation of Harr’s wavelet in the approximation problem for continuous functions and differential functions was rather bad. So Harr’s method was forgotten for many years. Until the middle of 1980s, the development of wavelet was very slow and not many people knew the wavelet functions beyond individual specialized communities. But in 1984, the Harr’s method got rebirth. Jean Morlet, a French geophysicist employed by the oil company Elf Aquitaine, started a scientific revolution in seismic signal processing.3 He developed his own way of analyzing the seismic signals to overcome the deficiencies in the Fourier method. He created a new method to decompose seismic signals into what he called "wavelets of constant shape," shown in Fig.1. Later, the functions became "Morlet wavelets" and the word of "wavelet" is first appeared in Grossman and Morlet’s paper.4-5 A Morlet wavelet is a wave-like function with short extension: its graph oscillates around zero only over a short distance. The Morlet wavelets maintain the same shape whether they are compressed or dilated. Shortly after the publication of Grossman and Morlet’s work, Meyer and his doctorate student, Mallat6 made the wavelet transform as easy to work as Fourier transform and linked the theory of wavelets to the existing subband coding. Mallat discovered the relationship between quadrature mirror filters (a pair of high and low pass filter) and orthogonal wavelet bases. His multiresolution analysis method made the wavelets enter the mainstream of digital signal processing. Another great boost for the application of wavelet came from Daubechies7 at the later 1980s. Based on the work of Mallet, Daubechies constructed a set of basic orthogonal wavelet functions, which can be easily implemented using simple digital filter ideas, shown in Fig.1. Thanks to Daubechies’ work, wavelet transform has been widely used in many fields such as pattern recognition, image compression, mechanical fault diagnostic, signal de-noising, signal compression, earthquake diction, and other areas since 1990s. Introduction The recent developments of wavelet transforms have fascinated the scientific, engineering, and mathematic communities. With wavelet theory securely in place by the outstanding research of Grossman, Morlet, Mallet, and Daubechies, the application of wavelet has grown rapidly over the last decide. For example, a distribution list on wavelets

Abstract Wavelet is first introduced by Alfred Haar in 1910, but the development of wavelet was very slow until the middle of 1980s. In 1984, Jean Morlet made the wavelet rebirth by developing a new way of analyzing the seismic signals to overcome the deficiencies in the Fourier method. Since then, the new developments of wavelets have fascinated the scientific and engineering communities. A wavelet is a waveform of effectively limited duration that has an average value of zero and wavelets are a family of basis functions, which can separate a signal into distinct frequency packets that are localized in the time domain. Thus, wavelets are well suited to analyze nonstationary data. They can smooth the basic signals and keep the details of basic signals. Therefore, they provide a multiresolution framework for data representation. Wavelet analysis is a rapidly developing area in many disciplines of science and engineering and it is used in a wide variety of applications in the areas of medicine, biology, data compression, etc. In recent years, wavelet analysis has found its application in the petroleum industry. This paper reviews the recent application of wavelet analysis in the industry. Various application examples are discussed, especially the examples in the areas of reservoir characterization, geological model upscaling, and well testing. History of Wavelet Wavelet is first introduced by a Hungarian mathematician named Alfred Haar1 in 1910. The function he introduced is now called "Haar wavelet." The Haar wavelet function consists simply of a short positive pulse followed by a short negative pulse, shown in Fig.1. Although this function is easy to understand, its application was very limited since it was based on step-functions, and the step-functions jump from one

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that began with 40 names in 1990 is now an online newsletter with more than 17,000 subscribers.8 This paper begins with a brief discussion of the disadvantages of Fourier transform and advantages of wavelet transform. Then, the paper reviews all the examples found in the open literature search in the petroleum industry to demonstrate the application of the wavelet methods in the industry. The scope of this review is the wavelet application in the industry. The details discussion of numerical algorithms for implementation of wavelet theory is outside the scope of this paper, but it can be found in references. 3-7

information is lost. Therefore, when looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place.

Fig.2 - The Fourier transform2

Fig.3- Decomposition of a Fourier signal2 Fig.1 Three different types of wavelets: (A) Haar wavelet, (B) Morlet wavelet, and (C) Daubechies wavelet.

Why Goes Beyond Fourier Perhaps the most powerful mathematical tool for analyzing and processing signals and images is the Fourier method, invented in the 1920s by Joseph Fourier, a French mathematician and physicist. A signal is considered as a superposition of sine and cosine waves with different frequencies in Fourier transform. Fig.2 shows the Fourier Transform and Fig.3 shows the decomposition of a Fourier signal. The success of the Fourier method is due to the fact that in practice a limited amount of the sine or cosine functions suffice to discover the main characteristics of a signal. For many signals, Fourier transform is extremely useful because the signal’s frequency content is of great importance, especially for the “stationary signal” - that is the signal properties do not change much over time. But the ‘Fourier coefficient’ for a certain frequency gives the average strength of that frequency in the full signal. As a sine or cosine function keeps undulating to infinity, the Fourier coefficients don’t provide direct information about the local behavior of a signal; they only give a kind of average information on the signal as a whole. The Fourier transform uses a sum of sine and cosine functions. However, the sine and cosine functions are periodic functions that are non-local; that is they go on to plus and minus infinity on both ends of the real line. Therefore, any change at a particular point of the time domain has an effect that is left over the entire real line. So in transforming the signal from time domain to frequency domain, time

In an effort to correct this deficiency of Fourier transform, Dennis Gabor9 adapted the Fourier transform to analyze only a small section of the signal at a time. This adaptation is called Short-Time Fourier Transform (STFT) which maps a signal into a two-dimensional function of time and frequency. Fig.4 shows the theory of this transform.

Fig.4- The short-time Fourier transform a signal2

The STFT represents a sort of compromise between the time and frequency based view of signal. It provides some information about both when and at what frequencies a signal event happens. However, the precision of the STFT is determined by the size of the window. Therefore, the deficiency of the STFT is that once you choose a particular size for the time window, that window is the same for all frequencies. However, many signals require a more flexible transform which can vary the window size to determine much more accurately about both when and at what frequencies a signal event happens either time of frequency. In summary, although Fourier transform is a power mathematical tool, it is not very good at detecting rapid

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changes in signals even for the revised Fourier transform. The major disadvantage of Fourier transform lies in its lack of localization and it considers phenomena in an infinite interval which is very far from our everyday point of view. Furthermore, many signals such as occurring in seismic data and well test data in petroleum industry, display structure at many different scales. The Fourier method misses most of this multi-scale structure. Fig.5 clearly illustrates this point. Therefore, we either “can’ t see the forest for the trees” or “can’ t see the trees for the forest” in the Fourier transform.

Fig. 5 - The bottom left figure is the Fourier transform of the upper sinusoidal signal with a small discontinuity. The bottom right is the wavelet transform result of the same signal. It is obvious that the wavelet transform clearly shows the exact tiny discontinuity location in time domain.2

duration and a wavelet basis consists of a father wavelet that represents the smooth baseline trend and a mother wavelet that is dilated and shifted to construct different level of details. At high scales, the wavelet has a small time support, enabling it to zoom in details and short period duration events such as spikes. At low scales, wavelets capture long period duration events. Fig. 7 shows the wavelet transform of a signal. To understand wavelet more easily, let’ s first take a look at the way of how our eye look at the forest in the real world. In the real world, when we look at the forest from the window of a flying helicopter, the forest appeared to be a blanket of green. But from the window of an automobile on the ground, the blanket of green resolves into individual trees. If we get out of the car and walk into the forest, you will begin to see the branches and leaves. When we come closer to a tree and look at some leaves of the tree, we might find a butterfly standing on a leaf. So when we came closer to the forest we find much more details of the forest that we cannot see from distance. But when we take a picture of the forest and see what we find. We can find the individual tree and leaves too. A forest is still a forest no matter the extent we enlarge the forest picture. Although our eyes can see the forest at many scales of resolution, but the camera can only show one scale at a time. To explain this in a simple way, the Fourier transform is like the way we look at the forest from the camera which does not with zoom in function, while wavelet transform is like the way we see the forest with our eyes.

Why Wavelet Transform Works The Fourier Transform only works well under the assumption that the original time domain signal is periodic in nature. Then the time domain signal can be translated into frequency domain with a combination of sine and cosine waves with infinite duration. The nature of Fourier transformation makes it difficult to deal with functions that have transient components, especially the function with sharp transitions. Further more, most signals we meet in our daily life are non-stationary and of finite duration. But another new developed mathematical transform can make us “seeing the forest and the trees” at the same time. It is the wavelet transform. The wavelet theory has its root in Fourier analysis, but there are important differences. Unlike the Fourier transform, the wavelets functions do not have infinite duration and they allow users to divide a complicated signal into several components and process the components individually. This important property along with the high efficiency of its algorithm makes wavelet analysis a very attractive tool for the signal processing. A wavelet is a wave-like function with short extension and its graph damps very fast with the mean value of zero over the whole domain. Fig.6 shows the wavelet decomposition of a signal. The wavelet can be non-zero only for a short range of

Fig.6 – Decomposition of a wavelet signal2

Fig.7 - The wavelet transform2

The new development of wavelets in the latest twenty years overcame the problem of Fourier Transform. Currently, the wavelet method provided with a sound mathematical basis, has become a powerful tool in the field of data compression, noise suppression and images/signals process.

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Applications of Wavelet in Petroleum Industry Although wavelets are best known for image compression, it was originally applied in geophysics in the early 1980s for the analysis of seismic signals by Morlet. A seismic signal is characterized by its travel time, frequency, and phase information, as well as noise. But a seismic signal is nonstationary data and usually has rapid change information in it, which limits the powerfulness of Fourier transform. Since the pioneering work of Morlet in seismic, wavelet transform has already been widely researched and used in many areas of geophysics. The theory and application in geophysics are well documented.10-14 The development of the wavelet theory in the 1990s has enabled a suite of applications in various fields of data analysis. But application of wavelets in the petroleum industry is relatively rare. Among several thousand papers on the research and application of wavelets written with the last 15 year,2 less than a dozen papers are on the data analysis from the petroleum industry. At the same time, most of the applications in the industry are on geological model upscaling and well test data analysis. In the following sections we will review all the applications we found in the open literature. Hydrocarbon Detection Wavelet transforms can provide the high-resolution visualization of subtle stratigraphic features and illustrate the effects of hydrocarbon reservoirs on surface seismic data. 10-14 Several such effects have been seen to detect hydrocarbons: (1) High attenuation of seismic waves traveling through thicker gas reservoirs; (2) Low frequency shadow zones related with thin-bed reservoirs; (3) “ Thin-bed” tuning associated with potential reservoirs.

Burnet and Castagna 14 investigated the attenuation of seismic waves through clean gas sands reservoirs using northeastern Mexico data. Fig. 8 illustrates the reflectivity of gas sands higher than the surrounding facies. This amplitude anomaly may be explained that the time-thickness of this sand is close to tuning at higher frequency. It seems that the variations of amplitude with frequency can be used to help detect hydrocarbons. Explorationists or engineers have become to realize that the wavelet transforms (or, spectrally decomposed) based analysis, rather than using the broad-band seismic sections, is a valuable aid in the search for hydrocarbon potentials today. And it is still on the way of getting more successes among seismic interpretation and processing. Source Rock Characterization Well logging data are commonly used in the study of subsurface sedimentary rocks and gamma-ray logs are particularly useful for distinguishing clay-rich from clay-poor sediments. Some stratigraphic discontinuity, such as sequence boundaries, may be detectable in well logs. However, small faults and slumps in deep marine sediments do not show obvious sedimentary facies changes and thus are not directly detectable from well logs. Prokoph et al.15-17 used two computer models to simulate possible variations in the sedimentary rocks, such as gradual and abrupt changes of sedimentation rate, short-time events, and abrupt facies changes. They found that the wavelet analysis is very useful in evaluating the quality of oil-source rock data and high resolution stratigraphy from well log data. The oil source rock from offshore eastern Canada (Egret Member) indicates positive correlation between stratigraphic completeness and percentage of non-source rock, as well as total mass of organic material. By using wavelet analysis, Prokoph et al. successfully recognized the sedimentary features from oil source rocks based on the gammy-ray log data. The results of this study demonstrate that wavelet analysis can be used in the evaluation of source rock distribution. Reservoir Property Estimation The ultimate goal of reservoir characterization is to estimate the spatial distribution of reservoir properties such as porosity, permeability, and oil saturation. Accurate estimation of reservoir properties is crucial to the oil or gas field management. Panda et al.18 applied wavelet transforms to onedimensional and two-dimensional permeability data to determine the locations of layer boundaries and other discontinuities. They analyzed the permeability structures of arbitrary size by binning in the time-frequency plane with wavelet transform. Wavelet transforms were also applied to scaling up spatially correlated heterogeneous permeability fields. In this study, the authors applied the wavelet transforms to permeability data to demonstrate (1) scaling of permeability, (2) removal of white noise from data, and (3) application of

Fig. 8-Gas-charged reservoir shows different reflectivity. The reservoir is not anomalous at 20hz but clearly stands out at 40hz. 14

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the edge-detection algorithm. There are two example applications presented here: the first one contains onedimensional permeability measured over a vertical distance of 40 m on a Page Sandstone outcrop core in northern Arizona, and the second presents a two-dimensional permeability field. Fig.9 shows the denoising results of the Page Sandstone permeability data.

is to take a fine scale geological description and to produce effective properties appropriate for coarse scale model that preserve the key fine scale features. To make an upscaling work, the coarse scale model has to capture the important details of the fine scaled model. This means that relatively homogeneous areas can be greatly upscaled while areas with large heterogeneity must be less upscaled to preserve the important details that control the dynamics of the flow. Wavelet analysis is a multiresolution framework and, thus, it is well suited for upscaling rock and flow properties in a multiscale heterogeneous reservoir. The large family of wavelets provides a flexible way to control the smoothness of the resulted upscaling properties. Panda et al.,18 Jansen and Kelkar,21 have used wavelet analysis to upscale fine geological models. Fig.10 shows the fine-scale permeability field. Fig.11 and Fig.12 shows the upscaling results based on pressure-solver method and wavelet-based upscaling method, respectively.

Fig.9-Denoising Page Sandstone Permeability data using wavelet transform18

Panda et al. demonstrate the use of orthogonal wavelets to analyze permeability data to identify local discontinuities. At the same time, it also demonstrates the use of the wavelet transforms to remove white noise and to scale-up permeability data. This is particularly useful for numerical flow simulation where reduction of permeability data often becomes necessary. The most remarkable beauty of this method, which is lacking in other traditional data analysis method, is that this method can decompose and reconstruct nonstationary data efficiently and inexpensively. Lu and Horne,19 Datta-Gupta et al.20 also demonstrate that the properties of wavelet analysis are very desirable in reservoir characterization. The wavelet analysis not only can overcome the shortcoming of conventional reservoir parameter estimation methods, but also can make the estimation algorithm very efficient and stable. Reservoir Model Upscaling The recent development of computer software and hardware technology benefits reservoir engineers as well as geologist. The highly detailed geological model built by diligent geologist with the help of geostatistical software continues to exceed the capacities of conventional fluid flow simulators by a significant margin. This resolution gap has driven the development of upscaling techniques. The objective of the scale up is to reduce the number of cells from fine geological models without losing the important flow behavior of the fine models. The philosophy of scale up

Fig.10 – Original fine-scale permeability data used for wavelet analysis and upscaling: grid size = 256×256.18

Fig.11 – Scaleup 2D permeability field using pressure-solver method.18 Scaledup grid = 64×64.

Fig.12 - Scaleup 2D permeability field using wavelet-based method.18 Scaled up grid = 64×64.

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When we compare the two upscaling results from pressure-solver method and wavelet method, we may easily conclude that the wavelet-based method more accurately preserves local discontinuities of the permeability data. A Further study shows as the degree of upscaling increases the difference between the pressure solver and wavelet-scaled results increase sharply. The wavelet-based scaling preserves the layer boundaries unlike the pressure solver method, shown in Fig.13-14.

basis for the new standard for compression of images for JPEG2000.

Fig.15 – The image on the left is the original; the one on the right is reconstructed from a 26:1 compression.22

Fig.13 – Scaleup 2D permeability field using pressure-solver method.18 Scaledup grid = 16×16.

Fig.14 – Scaleup 2D permeability field using wavelet-based method.18 Scaledup grid = 16×16.

Both of these two upscaling studies show that the multiresolution wavelet transform improve the computational efficiency and accuracy of upscaling for generating equivalent rock and rock-fluid properties under various geological and flow conditions. The advantage of the wavelet transform in geological model upscaling is that since the equivalent properties at different length scales are computed recursively, the interdependent influences of the heterogeneities on the scales are included effectively. Downhole Data Compression Data compression is the most widely used area for wavelet transform. Currently, one of the most popular successes of the wavelets is the compression of the FBI fingerprints. Fig. 15 shows the comparison of a compression fingerprint and original one. By applying wavelets the FBI managed to compression its file of 200 million fingerprints to a few percent of its original magnitude without any loss of the prints’ identity in the 1990s!22 Moreover, it is recently used as

However, only one data compression application has been found in the petroleum industry. It seems that petroleum engineers have not realized the powerfulness of wavelet in data compression except Bernasconi and his three coworkers from Holland. Bernasconi, et al.23 presented a data compression algorithm based on the wavelet transform. They studied the possibility of compressing the downhole data using a waveletbased compression algorithm that was able to maintain high quality in the stored signals. The algorithm is very simple but very effective and it can be easily embedded in the processing software of the current generation downhole equipment without major modifications. Data decompression can be accomplished on a standard personal computer after data retrieval and the users can decide the quality of the reconstructed signals. Extensive simulations on real data show that compression ratio up to 15:1 may be achieved for most signals without significant data degradation. A compression ratio up to 15:1 means an important increase of acquisition time for down-hole measurement while drilling (MWD) equipment and a noticeable speed increase of transmitted information through mud-pulse telemetry. Their study results indicated that wavelet transform is suitable for downhole implementation and might be successfully applied both online and offline solutions to the data compression for MWD. Well Test Analysis Gringarten’ s24 log-log plot of the pressure change versus time is widely used in the conventional well test analysis. But the logarithmic plotting smoothes data and may hide subtle effects of some other physical parameters. In 1983, Bourder et al.25 developed a new method which took the derivative of the Gringarten drawdown type curves in well testing to overcome the logarithmic plotting problem. The derivative technique does enhance subtle changes between data points and subsequently overcomes some of the drawbacks associated with the log-log plotting. Soliman et al.26 applied the Daubechies family of wavelets for analyzing pressure signals. They took a three-step method in their study. First step is to reconstruct the pressure data on an evenly spaced time scale. Second step is to consider the

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wavelet transform of the signal and graph the results. The final step is to pick wellbore and reservoir events using the best wavelet type or a combination of wavelet types for the particular data. Soliman et al. demonstrated the application of wavelet transform in well testing by applying it to 4 field examples. Fig.16 shows the results from Example 2 in their study. Their results show wavelet transform is an effective tool for analyzing pressure transient data. During wavelet transformation, wellbore and toll events are revealed as abrupt and sharp discontinuities in the pressure transient signal. These events may not be clearly detected on all sampling levels since they are characterized by short duration of occurrence.

data. They tested the wavelet transform with several sets of simulated data and actual field data. Later, Athichanagorn, et al.29 developed a multi-step method for processing and interpretation of long-term pressure data based on wavelet transform and they found it to be an effective approach for the analysis of long-term pressure data. The multi-step method can be summarized in a seven-step procedure: x x x x x x x Outlier removal De-noising Transient identification Data reduction Flow-history reconstruction Behavioral filtering Moving window analysis

Ouyang and Kikani30 also applied the wavelet transform to the analysis of long-term downhole pressure data and confirmed the usefulness of the wavelet transform in longterm pressure data analysis. Interwell Relationship Several authors31-33 have reported the applicability of the cross correlation of the production data to identify structural trends and preferred flow paths, thus, identifying fracture pattern, stress directions, barriers, location of water cycling, poor sweeped area and continuity regions. A strong correlation is an indication of continuity between the wells and can give an indication of the preferred flow path. Examining the cross correlation between all of the wells in a field will provide us with more than just a direct inter-well relationship between single pairs of wells, it can also provide us with the general trends of the field. But Jansen and Kelkar34 indicated that calculation of the cross correlation itself might be problematic. This is due to the non-linear and non-stationary nature of the inter-well relationship. Their study shows the inter-well relationship is a function of the boundary conditions imposed by the wells themselves and the reservoir properties. It is therefore necessary to capture this transient nature of the cross correlation, or else an otherwise strong correlation might be lost in the averaging process. A strong increase in the injection rate might therefore be required to generate a significant response in the producer. Jansen and Kelkar demonstrate the usefulness of the wavelet transform as a method to both conditions the data and the cross correlation by applying the method to the North Robertson Filed in West Texas. They concluded that the success of any cross correlation scheme depended upon the proper treatment of the data and their transient nature, capturing the important features controlling the well behavior. The wavelet transformation allows for a non-stationary treatment of the data and it opens new possibilities with respect to a robust cross correlation between wells and the used of these data more consistently determine the cause of the well behavior and its influence to surrounding wells.

Fig.16 – Wavelet coefficients clearly detect the two boundaries in the Well-2 data.26

The authors concluded that the windowing ability of the wavelet techniques provides an excellent opportunity to detect events much more quickly, and in near real-time speed and they believe that the wavelet transform could complement conventional pressure transient analysis techniques for comprehensive interpretation of well test data. But they also note that no single wavelet is capable of detecting all possible discontinuities present in a pressure signal. Gonzalex-Tamez et al.27 applied wavelet transform to denoise the well test data. Their studied showed that the frequency spectrum of noise in the pressure transient test data is in the same frequency band with the reservoir signals. By using wavelet transform, the signal-noise ratio of pressure tests with truncation noises is improved. Permanent Downhole Gauge Data Analysis The long-term data from permanent downhole gauge is tremendous and it is impossible to include the entire data set in the interpretation because of the limited computer resources. Therefore, petroleum engineers need to reduce the amount of permanent downhole gauge data to a manageable size. Kikani et al.28 first applied the wavelet transform to pressure transient data to process the long-term downhole

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Solution of Two-phase Flow In addition to the variety areas mentioned above, the wavelet transform has also been used in the numerical solution of partial differential equations. Moridis et al.35 presented the use of wavelets to solve the non-linear Partial Differential Equation (PDE) of two-phase flow in one dimension. They tested the use of wavelets in the solution of the one-dimensional Buckley-Leverett problem against analytical solutions and solutions obtained from standard numerical models. Two classes of wavelet bases (Daubechies and Chui-Wang) and two methods (Galerkin and collocation) were investigated. They found the Chui-Wang wavelets and a collocation method provided the optimum wavelet solution for this type of problem. Fig.17 shows the analytical and numerical solution for t=1500 days using wavelet method. The study results of this research indicated that wavelet transforms could be an effective and accurate method, which does not suffer from oscillations or numerical smearing in the presence of steep fronts. But they also noted that the application of wavelet transform in the solution of partial differential equations is a new and unexplored area, about which the existing information base is very limited. They would like to approach their results only as a first indication of the performance of wavelets in the solution of PDEs. Unfortunately, no further study found in this area in the petroleum industry.

a powerful technique for processing non-stationary and of finite duration signals. Moreover, many signals in petroleum industry, such as seismic data, logging data, well test data are non-stationary and of finite duration data. In this paper we have reviewed the successful applications of wavelets for the analysis of reservoir signals in the petroleum industry. Wavelets already open a large, unexplored territory to petroleum engineers. We believe this is the only start of the application for wavelets in the petroleum industry. Once more petroleum engineers start to learn the powerfulness of the wavelet transforms, they will find more applications in the industry. Nomenclature

MWD PDE STFT =Measurement while drilling =Partial differential equation =Short-time Fourier transform

Fig.17 – Wavelet solution for the Buckley-Leverett problem34

Conclusion Fourier method is a very powerful tool to analyze signals. But when we looking at Fourier transform of a signal, it is impossible to tell when a particular event took place because the time information is lost in the Fourier transforms. However, the wavelet transforms allow users to divide a complicated signal into several components and process the components individually. This important feature along with the high efficiency of its algorithm makes wavelet analysis a very attractive tool for the signal processing. The wavelet transform can localize information in both time and frequency domains simultaneously. It thus provides

15. Prokoph, A. and Agterberg, F.P.: “ Wavelet Analysis of welllogging data from oil source rock, Egret Member, offshore eastern Canada,” AAPG Bull., 2000, 84 No. 10, 1617-1632.

13. Castagna, J., Sun, S. and Siegfried, R., “ Instaneous spectral analysis: Dection of low-frequency shadows associated with hydrocarbon” , The Leading Edge, Februray, 2003,120-127. 14. Burnett, M., and Castagna, J., “ Application of spectral decomposition to gas basins in Mexico” , The Leading Edge, November, 2003, 1130-1134.

Haar, A.: Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69(1910), 331-371. 2. Wavelets: A New Tool for Signal Analysis (http://www.mathworks.com/access/helpdesk/help/toolbox/wave let/ch01_int.shtml), MATLAB tutorial. 3. Morlet, J., Arens, G., Fourgeau, E., and Giard, D.: “ Wave Propagation and Sampling Theory - part I: Complex Signal and Scattering in Multilayered Media,” Geophysics, 47(2), 203-221, 1982. 4. Grossmann, A., and Morlet, J.: “ Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape,” SIAM J. Math. Anal., 15(4), 723-736, 1984. 5. Grossman, A., and Morlet, J.: Mathematics and Physics, 2, L. Streit, Ed., World Scientific Publishing, Singapore. 6. Mallat, S.: “ A theory for Multiresolution Signal Decomposition: The Wavelet Representation,” IEEE Transaction: Pattern Analysis and Machine Intelligence, 1989, 31, 679-693. 7. Daubechies, I.: “ Orthonormal Bases of Compactly Supported Wavelets,” Communications on Pure Applied Mathematics, XLI, 901-996, 1988. 8. Mackenzie, D.: Wavelets: Seeing the Forest and the Trees, National Academy of Sciences, December 2001. 9. Gabor, D.: Theory of Communication, J. IEE, November 1946, Vol.93, Part III, No. 26, 426-457. 10. Behrens, R.A., Mackleod, M.K., Tran, T.T., and Alimi, A.O.: “ Incorporating Seismic Attribute Maps in 3D Reservoir Models,” SPERE, April 1998, 122-126. 11. Chakraborty, A., and Okaya, D.: “ Frequency-time decomposition of seismic data using wavelet-based methods,” Geophysics, 1995, 60, 1906-1916. 12. Miao, X.G., and Moon, W.M.: “ Application of Wavelet Transform in Reflection Seismic Data Analysis,” Geosciences Journal, Vol 3, No.3, p171-179, September 1999.

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16. Prokoph, A., and Agterberg, F.P.: “ Detection of Sedimentary Cyclicity and Stratigraphic Completeness by Wavelet Analysis: An Application to Late Albian Cyclostratigraphic of the Western Canada Sedimentary Basin. 17. Prokoph, A. and Barthelmes, F.: “ Detection of Nonstationarities in Geological Time Series: Wavelet Transform of Chaotic and Cyclic Sequences,” Comp. & Geo. (1996) 22, 1097-1108. 18. Panda, M.N., Mosher, C.C., and Chopra, A.K.: “ Application of Wavelet Transforms to Reservoir-Data Analysis and Scaling,” paper SPE 60845, SPE Journal, March 2000, 92-101. 19. Lu, P., and Horne, H.N.: “ A Multiresolution Approach to Reservoir Parameter Estimation Using Wavelet Analysis,” paper SPE 62985 presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, 1-4 October. 20. Datta-Gupta A, Xue, G, and Lee, S.H.: “ Nonparametric Transformations for Data Correlation and Integration: From Theory to Practice,” 1999 in R. Schatzinger and J. Jordan, eds., Reservoir Characterization-Recent Advances, AAPG Memoir 71, 381-396. 21. Jansen, F.E., and Kelkar, M.G.: “ Upscaling of Reservoir Properties Using Wavelets,” paper SPE 39495 presented at 1998 India Oil and Gas Conference and Exhibition, New Delhi, India, 17-19 February. 22. Graps, A.: An Introduction to Wavelet, 1995 (http://www.amara.com/IEEEwave/IW_fbi.html). 23. Bernasconi, G., Rampa, V., Abramo, F., and Bertelli, L.: “ Compression of Downhole Data,” , paper SPE/IADC 52806 presented at the 1999 SPE/IADC Drilling Conference, Amsterdam, Holland, 9-11 March. 24. Gringarten, A.C.: “ Interpretation of Tests in Fissured and Multilayered Reservoirs with Double Porosity behavior: Theory and Practice,” JPT, April 1984, 549. 25. Bourdet, D., Whittle, T.M., Douglas, A.A., and Pirard, Y.M.: “ A New Set of Type Curves Simplifies Well Test Analysis,” World Oil, May 1983, 95-106. 26. Soliman, M.Y., Ansah, J., Stephenson, S., and Manda, B.: “ Application of Wavelet Transform to Analysis of Pressure Transient Data,” paper SPE 83670, SPEREE, April 2003, 89-99. 27. Gonzalez-Tamez, F., Camacho-Velazquez, R., and EscalanteRamirez, B.: “ Truncation De-Noising in Transient Pressure Tests,” paper SPE 56422 presented at 1999 SPE Annual Technical Conference and Exhibition, Houston, 3-6 October. 28. Kikani, J., and He, M.: “ Multi-resolution Analysis of LongTerm Pressure Transient Data Using Wavelet Methods,” paper SPE 48966 presented at 1998 SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September 1998. 29. Athichanagorn, S., Horne, R.N., and Kikani, J.: “ Processing and Interpretation of Long - Term Data Acquired From Permanent Pressure Gauges,” SPEREE, October 2002, 384-390. 30. Ouyang, L., and Kikani, J.: “ Improving Permanent Downhole Gauge (PDG) Data Processing via Wavelet Analysis,” paper SPE 78290 presented at the SPE 13th European Petroleum Conference, Aberdeen, U.K., 29-31 October, 2002. 31. Jansen, F., and Kelkar M.: “ Exploration Data Analysis of Production Data,” paper SPE 35184 presented at the 1996 Permian Basin Oil and Gas Recovery Conference, Midland, TX, 27-29 March, 1996. 32. Jansen, F., and Kelkar M.: “ Non-Stationary Estimation of Reservoir Properties using Production Data,” paper SPE 38729 presented at SPE 1997 Annual Technical Conference & Exhibition, San Antonio, Texas, 5-8 October, 1997.

33. Heffer, K.J., Fox, R.J., and McGill, C.A.: “ Novel Techniques Show Links between Reservoir Flow Directionality, Earth Stress, Fault Structure and Geomechanical Changes in Mature Woodlands,” paper SPE 30711 presented at 1995 SPE Annual Technical Conference & Exhibition, Dallas, Texas, 22-25 October, 1995. 34. Jansen, F.E., and Kelkar, M.G.: “ Application of Wavelets to Production Data in Describing Inter-Well Relationships,” paper SPE 38876 presented at 1997 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5-8 October, 1997. 35. Moridis, G.J., Nikolaou, M., and You, Y.: “ The use of Wavelet Transforms in the Solution of Two-Phase Flow Problems,” paper SPE 29144, SPE Journal, June 1996, 169-177.

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