Integration of Dynamic Data for Reservoir Characterization in the Frequency Domain
Authors: Huang Xuri , Kelkar Mohan 
Published: December 2007

Abstract
In this paper, a new approach is proposed to integrate static data with dynamic information in the frequency domain. The spatial relationship-variogram- is represented by power spectra and self-correlation spectra in the frequency domain. It is assumed that the large scale information can be obtained by static data such as geological and seismic information. In the frequency domain, this will be represented by the low frequency. The dynamic data-rate versus time at early time- is assumed to be influenced by near well bore permeability values. In the frequency domain, these are represented by high frequencies. By selectively perturbing high frequency values, the dynamic data can be matched without affecting the large scale features represented by the low frequency information. The final reservoir description can both honor the static data and dynamic information. A case study is presented to validate the proposed method.

Introduction
The goal of reservoir characterization is to estimate the interwell distribution of the reservoir properties such as porosity and permeability distributions by integration of different data. For a better understanding of the future reservoir performance, it is critical to improve the modeling of the permeability by integrating the dynamic data with the other static information into reservoir characterization process. The static information includes seismic surveys, geological analysis, core and log data. By assigning appropriate weights to each type of static information, an integrated description of the reservoir properties can be constructed. In addition, if the dynamic data are integrated in a reservoir description process, it will further enhance the reservoir description, especially the dynamic performance of the reservoir.
Recently, several approaches, including the inverse theory, have been applied to integrate the dynamic data with the constraint of the spatial relationship. Several investigators have applied different techniques to incorporate the static and dynamic data. In 1992, Deutsch used simulated annealing to integrate the engineering data with spatial relationship given by variogram for geostatistical modeling. In 1994, Sagar applied simulated annealing to incorporate the well test data and spatial relationship for modeling the permeability field with a more efficient forward modeling. In 1994, Oliver applied the inversion methods to incorporate spatial relationship with well-test data to generate multiple realizations of the permeability field. Chu et al. implemented the sensitivity coefficient in a more efficient way to perform the inversion. In 1995, Reynolds et al. proposed a more efficient method by using reparameterization. All these methods use the spatial relationship and dynamic data as constraints for the inversion of the permeability field. It works well if the spatial relationship is available and the correlation range is long. However, if the spatial relationship is not clear or easy to obtain, the methods may not work well. Also, honoring the spatial relationship does not necessarily mean that other data such as geological description or seismic modeling results are honored. Recently, Huang and Kelkar proposed a method to integrate the dynamic data with the static data especially the seismic data to avoid the use of the spatial relationship defined by the variogram. In this work, we propose an alternate approach to integrate the static and the dynamic data. With the help of Fourier transform, we manipulate the integration in the frequency domain to match the two types of data. Fourier transform is a popular tool to analyze signal by transferring the signal from the time or space domain into the frequency domain. It states that a stationary signal can be represented by a summation of series of cosine or sine functions with different amplitudes and angle shifts. In turn, the cyclic value corresponds to the frequency, while the angle shift will be the phase. The signal can be easily decomposed into different frequency components which are also representations of different scales. For example, the basic patterns can be represented by the low frequency components, while the noise or short scale variability can be represented by the high frequency components.
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