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Abstract
In this work, attenuation and phase shift information acquired from pressure pulse tests at multiple frequencies was used to estimate the permeability distribution of reservoirs. By preprocessing the time series pressure data at the active well and the observation well, not only are the sizes of both the time and frequency conditioning data reduced tremendously, but also there is no need to know the flow rate. However, to put this method into practice and to know its limits, an investigation was conducted under various reservoir conditions to resolve the issues of detrending (removing transient upward trends in the pressure data) and to develop the Fourier transform procedures that are necessarily involved with this approach.

In addition to estimate horizontal heterogeneous permeability from a radial multicomposite reservoir, the method utilizing multiple frequency attenuation and phase shift information was applied to a partially penetrating well in a multilayered reservoir with crossflow to reveal the heterogeneous vertical permeability distribution.
The performance of the frequency method was investigated in comparison to straightforward pressure history matching. Quasi-Newton line-search optimization with gradient information was used for both methods. In addition, we investigated the sensitivity of the permeability estimation to perturbation in both pressure data and attenuation and phase shift information.

A heuristic method for detrending was devised which helps in obtaining accurate attenuation and phase shift information. Cases with different values of storage, skin, and boundary conditions were considered. The impact of varying the number of periods and the sampling rate were checked to determine the sensitivity of Fourier transformation.

Introduction
Most reservoirs are naturally heterogeneous. Studies have been conducted to extract the heterogeneous permeability from well test pressure data (for example, Feitosa et al. 1994; Oliver 1992). When a periodic pumping test is conducted, the measured time series pressure data can be transformed to the frequency domain. Rosa (1991) demonstrated an analytical relationship between the effective radii of cyclic influence according to frequencies in pulse testing. This work suggested that a longer periodicity of pulsing should be sourced to reach further distances.

Multiple frequency information has the ability to reveal the radial permeability distribution between wells as demonstrated in the radial ring model (Ahn et al. 2010). Another applicable model to inspect the use of multiple frequency information is to reveal the vertical permeability distribution in a multilayered reservoir.
Estimating vertical permeability distribution is important due to the impact of the layer permeability values on the primary and secondary recovery processes. 
Kaneda et al. (1991) obtained permeability for two layers from the pressure time series data. Ayan et al. (1995) showed that both horizontal and vertical permeability can be obtained using a wireline formation tester. They formulated convolution with respect to the sourcing flow rate and extracted the effective average permeability. To accomplish this, the vertical probe is displaced a short distance from the sink probe. Layer pulse testing was described by Saeedi et al. (1987) with two Repeat Formation Tester (RFT) surveys, and the horizontal and vertical permeabilities were obtained using a numerical simulator in a real data case.

A partially penetrating well with crossflow induces vertical flow components in the vicinity of the well. The sourcing pressure is measured at the perforated layer and the observed pressure is recorded at some depth where the pressure magnitude is reduced and delayed from the sourced pressure in the perforated layer. This attenuation and phase shift information in the pressure signal is used here for characterization of petrophysical parameters. When this information is recorded at multiple frequencies in square pulsing, it is of high interest to see if it reveals the heterogeneity of the well as it has done in the radial ring model (Ahn et al. 2010).