Azimuthal Fourier Research Articles

Superresolution and analysis of three-dimensional velocity fields of underexpanded jets in different screech modes

Time-resolved (TR), three-dimensional (3D) velocity fields of screeching, underexpanded jets are estimated using non-time-resolved particle image velocimetry and simultaneous TR microphone measurements. Specifically, we aim to reconstruct TR 3D velocity fluctuation fields associated with the A2, B, and C modes of a screeching jet using a linear regression model and to analyze screech dynamics of these modes. The linear regression model is constructed on the basis of a linear relationship between the velocity and acoustic fields. Three nozzle pressure ratios (NPRs) of 2.30, 2.97, and 3.40 are employed. The dominant azimuthal modes for three cases are investigated using azimuthal Fourier coefficients of the acoustic data obtained by the azimuthal array of eight microphones placed near the nozzle exit. The dominant azimuthal modes at NPRs of 2.30, 2.97, and 3.40 are m=0, 1, and 1, respectively. The first two proper orthogonal decomposition (POD) modes in these azimuthal modes are dominant at all NPRs and are associated with screech. 3D velocity fluctuation fields associated with screech are reconstructed from these leading POD modes of the acoustic data. The reconstructed 3D velocity fluctuation fields at NPRs of 2.97 and 3.40 exhibit two helical structures with opposite rotation directions. The present results demonstrate that, in the B mode, the flapping structure exhibits random clockwise and counterclockwise rotations over an extended time domain, while maintaining a consistent direction within short time domains. In addition, in the C mode, two helical structures with opposite rotation directions, as well as the flapping structure, are observed. Published by the American Physical Society 2024

Causal analysis among azimuthal Fourier modes in linear plasma based on multivariate time series models

This study investigates the causal relationships among azimuthal Fourier modes in linear plasma turbulence using multivariate time series models. We elucidate the dynamics of mode interactions in magnetized plasmas by employing the vector autoregressive model and Granger causality analysis. Our analysis, based on data from the plasma assembly for nonlinear turbulence analysis, reveals significant variations in causality with changing pressure conditions. Modes form weakly coupled clusters at lower pressures, while higher pressures lead to stronger coupling and larger clusters. The impulse response function further provides insights into the temporal propagation and nature of influences between modes. These findings enhance the understanding of spatial pattern formation in magnetized plasmas and offer a quantitative framework for analyzing plasma turbulence dynamics.

Azimuthal Fourier decomposition for loss analysis of hollow-core tube lattice fibers, Part II: Tube thickness variation effects

The effect on confinement loss of thickness variations along the perimeter of the tubes composing the cladding of inhibited-coupling guiding Tube Lattice hollow-core fibers is investigated by using the Azimuthal Fourier Decomposition technique (developed in Part I) for the description of the cladding modal dynamics and their interaction with the fundamental core mode. The results show that the thickness inhomogeneity affects the confinement loss spectrum through confinement loss increase and frequency red- and blue-shift of the high-loss spectral regions. The magnitudes of the confinement loss increase and the high-loss region frequency shift strongly depend on the spatial distribution of the thickness inhomogeneity. The study provides insight into the loss mechanism of non-ideal tube lattice fibers, it allows to quantify the impact of such kind of structural deformations, identifying the route to make fibers more resilient to such fabrication imperfections, and highlighting once again the importance played in inhibited-coupling fibers by the interaction between core modes and the intricate set of cladding modes.

Horizontally polarized kink oscillations supported by solar coronal loops in an asymmetric environment

Context. Kink oscillations are ubiquitously observed in solar coronal loops, and understanding them is crucial in the contexts of coronal seismology and atmospheric heating. Aims. We studied kink modes supported by a straight coronal loop embedded in an asymmetric environment using 3D magnetohydrodynamic simulations. Methods. We implemented the asymmetric effect by setting different exterior densities below and above the loop interior and initiated the simulation using a kink-like velocity perturbation perpendicular to the loop plane, mimicking the frequently measured horizontally polarized kink modes. Results. We find that the external velocity fields show fan-blade structures propagating in the azimuthal direction as a result of the successive excitation of higher azimuthal Fourier modes. Resonant absorption and phase-mixing can still occur despite an asymmetric environment, leading to the development of small-scale structures at loop boundaries. These small-scale structures nonetheless develop asymmetrically at the upper and lower boundaries due to the different gradients of the Alfvén speed. Conclusions. These findings enrich our understanding of kink modes in coronal loops embedded within an asymmetric environment, providing insights that will be helpful for future high-resolution observations.

Azimuthal Fourier decomposition for loss analysis of hollow-core tube lattice fibers part I: Ideal fibers

This is the first part of two papers where we propose and apply a methodology for confinement loss analysis in tube lattice fibers (TLFs). The methodology is based on azimuthal Fourier decomposition (AFD) of the fiber’s cladding and core modes along the perimeters of the cladding tubes composing. This technique, combined with coupled mode theory, constitutes an effective approach to gain insight in the inhibited coupling waveguiding mechanism and design, along with fiber non-idealities impact on confinement loss. In this part I, we describe the approach and apply it to loss analysis of ideal TLFs. The approach is then applied to the analysis of the effects of tube thickness variation in part II.

Unsteady large-scale wake structure behind levitated free-stream-aligned circular cylinder

The relationships between characteristic large-scale wake structures appearing behind a free-stream-aligned circular cylinder are investigated and discussed from the velocity field obtained by wind tunnel tests. The tests were conducted under a supportless condition using a magnetic suspension and balance system and stereo PIV measurements at a Reynolds number of $3.46\times 10^4$ . The velocity fields were analysed with a modal decomposition combining azimuthal Fourier decomposition and proper orthogonal decomposition. The wake behind the free-stream-aligned circular cylinder with three different fineness ratios of 1.0, 1.5 and 2.0 was investigated, and the wake structures in a non-reattaching flow formed by the cylinder at a fineness ratio of 1.0 are mainly discussed in the present study. Four characteristic large-scale wake structures of the recirculation bubble pumping, azimuthal shear mode, large-scale vortex shedding and streaks are identified and mainly focused on in the present study. The state of the vortex shedding is classified into three: anticlockwise/clockwise circular and flapping patterns. Each state has a relationship with the azimuthal shear mode and it tends to appear when the state is circular. Furthermore, from the analysis of the relationship between modes, the recirculation bubble pumping is found to be related to the vortex shedding position in the radial direction and the strength of the streaks. Particularly, analysis of causality shows that the recirculation bubble pumping is affected by them in the low-frequency range.

Torque wiggles – a robust feature of the global disc–planet interaction

ABSTRACT Gravitational coupling between planets and protoplanetary discs is responsible for many important phenomena such as planet migration and gap formation. The key quantitative characteristic of this coupling is the excitation torque density – the torque (per unit radius) imparted on the disc by planetary gravity. Recent global simulations and linear calculations found an intricate pattern of low-amplitude, quasi-periodic oscillations in the global radial distribution of torque density in the outer disc, which we call torque wiggles. Here, we show that torque wiggles are a robust outcome of global disc–planet interaction and exist despite the variation of disc parameters and thermodynamic assumptions (including β-cooling). They result from coupling of the planetary potential to the planet-driven density wave freely propagating in the disc. We developed analytical theory of this phenomenon based on approximate self-similarity of the planet-driven density waves in the outer disc. We used it, together with linear calculations and simulations, to show that (a) the radial periodicity of the wiggles is determined by the global shape of the planet-driven density wave (its wrapping in the disc) and (b) the sharp features in the torque density distribution result from constructive interference of different azimuthal (Fourier) torque contributions at radii where the planetary wake crosses the star–planet line. In the linear regime, the torque wiggles represent a weak effect, affecting the total (integrated) torque by only a few per cent. However, their significance increases in the non-linear regime, when a gap (or a cavity) forms around the perturber’s orbit.

Base pressure fluctuations on levitated freestream-aligned circular cylinder

Base pressure fluctuations associated with the large-scale wake structures behind a freestream-aligned circular cylinder and aerodynamic force fluctuations related to them are experimentally investigated in the wind tunnel tests. Measurements at ReD=6.97×104 and 1.04×105 were conducted using pressure-sensitive paint (PSP) and a magnetic suspension and balance system (MSBS) for creating a supportless condition. The obtained pressure fields were mainly analyzed by a modal decomposition combining azimuthal Fourier decomposition and proper orthogonal decomposition (POD). The pressure fluctuations caused by large-scale vortex shedding were observed from the results of frequency analysis for mode coefficients. The states of the fluctuations were classified into three patterns, which are anticlockwise/clockwise circular and flapping patterns. These patterns have been observed in the previous studies for velocity fluctuations in the wake of a freestream-aligned circular cylinder. The conditional sampling analysis revealed that the trend in the amplitude of the pressure fluctuations is different by the state, and the flapping pattern causes a large pressure difference across the cylinder axis. Furthermore, the relationship between the antisymmetric pressure fluctuations and lift fluctuations, which act in the lateral direction of the cylinder, is confirmed.

Black Hole Polarimetry I. A Signature of Electromagnetic Energy Extraction

In 1977, Blandford and Znajek showed that the electromagnetic field surrounding a rotating black hole can harvest its spin energy and use it to power a collimated astrophysical jet, such as the one launched from the center of the elliptical galaxy M87. Today, interferometric observations with the Event Horizon Telescope (EHT) are delivering high-resolution, event-horizon-scale, polarimetric images of the supermassive black hole M87* at the jet launching point. These polarimetric images offer an unprecedented window into the electromagnetic field structure around a black hole. In this paper, we show that a simple polarimetric observable—the phase ∠β 2 of the second azimuthal Fourier mode of the linear polarization in a near-horizon image—depends on the sign of the electromagnetic energy flux and therefore provides a direct probe of black hole energy extraction. In Boyer–Lindquist coordinates, the Poynting flux for axisymmetric electromagnetic fields is proportional to the product B ϕ B r . The phase ∠β 2 likewise depends on the ratio B ϕ /B r , thereby enabling an observer to determine the direction of electromagnetic energy flow in the near-horizon environment experimentally. Data from the 2017 EHT observations of M87* are consistent with electromagnetic energy outflow. Currently envisioned multifrequency observations of M87* will achieve higher dynamic range and angular resolution, and hence deliver measurements of ∠β 2 closer to the event horizon as well as better constraints on Faraday rotation. Such observations will enable a definitive test for energy extraction from the black hole M87*.

Super-resolution of time-resolved three-dimensional density fields of the B mode in an underexpanded screeching jet

The estimation of time-resolved three-dimensional (3D) density fields of an underexpanded jet at the nozzle pressure ratio of 2.42, a so-called “spatiotemporal super-resolution” was conducted using non-time-resolved three-dimensional background-oriented schlieren (3D-BOS) and time-resolved microphone measurements. This approach aims to reconstruct three-dimensional density fields associated with the intermittent and switching behavior of the B mode of a screeching jet from the microphone data by constructing a linear regression model. An azimuthal Fourier decomposition is applied to the 3D-BOS and microphone data, and the proper orthogonal decomposition (POD) is performed for each of their azimuthal Fourier modes. The m=1 azimuthal Fourier mode is dominant in both cases, and the leading two POD modes in the m=1 azimuthal mode of the microphone data are associated with the B mode. The linear regression model is constructed from the POD modes of the m=1 azimuthal 3D-BOS data and the first two microphone POD modes of the m=1 azimuthal mode of the microphone data. The three-dimensional density fields reconstructed from each POD mode of the m=1 azimuthal mode of the microphone data have helical structures with opposite rotation directions. The amplitudes of those POD modes change with time, and the azimuthal structure associated with the B mode is determined depending on those amplitudes. The present result showed that intermittency in the flapping to helical structures and their strength can be interpreted by the temporal changes in the strengths of two rotating helical structures with opposite rotation directions.

Alfvénic Motions Arising from Asymmetric Acoustic Wave Drivers in Solar Magnetic Structures

Alfvénic motions are ubiquitous in the solar atmosphere and their observed properties are closely linked to those of photospheric p-modes. However, it is still unclear how a predominantly acoustic wave driver can produce these transverse oscillations in the magnetically dominated solar corona. In this study we conduct a 3D ideal MHD numerical simulation to model a straight, expanding coronal loop in a gravitationally stratified solar atmosphere which includes a transition region and chromosphere. We implement a driver locally at one foot-point corresponding to an acoustic–gravity wave which is inclined by θ = 15° with respect to the vertical axis of the magnetic structure and is similar to a vertical driver incident on an inclined loop. We show that transverse motions are produced in the magnetic loop, which displace the axis of the waveguide due to the breaking of azimuthal symmetry, and study the resulting modes in the theoretical framework of a magnetic cylinder model. By conducting an azimuthal Fourier analysis of the perturbed velocity signals, the contribution from different cylindrical modes is obtained. Furthermore, the perturbed vorticity is computed to demonstrate how the transverse motions manifest themselves throughout the whole non-uniform space. Finally we present some physical properties of the Alfvénic perturbations and present transverse motions with velocity amplitudes in the range 0.2–0.75 km s−1 which exhibit two distinct oscillation regimes corresponding to 42 and 364 s, where the latter value is close to the period of the p-mode driver in the simulation.

Fracture detection, fluid identification and brittleness evaluation in the gas-bearing reservoir with tilted transverse isotropy via azimuthal Fourier coefficients

Seismic characterization of the sweet spot plays a crucial role in the exploration and exploitation of oil and gas reservoirs. Therefore, we focused on fluid discrimination, natural fracture detection and brittleness evaluation in the reservoir permeated by an inclined set of parallel fractures using pre-stacked seismic inversion method. To this end, we proved by numerical experiments that the suggested fluid indicator is available for identifying the gas-bearing reservoir due to its high sensitivity to the gaseous hydrocarbon and extremely weak dependences on the porosity and the fracture density of the gas-bearing formation. To reduce the cumulative errors induced by indirect calculations, we derived the reflection coefficient of the TTI (tilted transverse isotropy) rock in terms of Young's modulus, Poisson's ratio, the fluid indicator and the fracture parameters and then rewrote it in the form of the Fourier series. By comparing with the published results, the derived reflectivity equation was proven rational and of satisfactory accuracy. Furthermore, we proposed a hierarchical inversion method incorporating the discrete Fourier transformation (DFT) and the Bayesian inversion strategy, which consists of three steps. Firstly, the DFT method is utilized to decouple anisotropic seismic components from the isotropic seismic component. In the last two steps, the Bayesian inversion approach is taken to acquire the stable and reliable estimations of the fluid and brittleness indicators and the fracture parameters from the second-order and zeroth-order seismic components, respectively. The synthetic example proved that we can realize rational and reliable predictions of the unknown parameters using the hierarchical method even for the case with moderate noise. The field case demonstrated that the proposed method is feasible and stable, and the suggested sweet spot indicators worked well for gaseous hydrocarbon discrimination, natural fracture detection and brittleness evaluation.

Characteristics of collective modes in anisotropic plasmas

We use a quasi-particle picture and the hard-loop approximation to study collective modes in anisotropic plasmas. We study a general class of anisotropic distribution functions and derive and solve the dispersion equations. We focus on imaginary modes because of their potential influence on plasma dynamics, and we study their dependence on the parameters that characterize the anisotropy of the system. We show that the strength of the imaginary modes is related to the oblateness of the distribution, and the size of the azimuthal Fourier coefficients.

Estimation of in situ stresses from PP-wave azimuthal seismic data in fracture-induced anisotropic media

Horizontally transverse isotropy (HTI) induced by vertical or subvertical aligned fractures is common for unconventional fractured porous shale oil or gas reservoirs. Compared with the unfractured rocks, the seismic response characteristics of PP-wave azimuthal amplitudes are usually disturbed by the fractures and the in situ stresses. Knowledge of fracture properties, as well as in situ stresses, is required to optimize horizontal well planning and hydraulic fracturing during production, and the seismic inversion for in situ stresses from the PP-wave azimuthal amplitude data in fracture-induced anisotropic media is an essential step. Using the linear-slip theory and the effective stress law, we derive the fluid-saturated effective elastic stiffness tensors parameterized by background elastic moduli, the effective stress coefficient of isotropic host rocks, fluid modulus, porosity, and fracture parameters based on the anisotropic Gassmann’s fluid substitution equation. Combining the perturbations in saturated stiffness tensors and scattering theory, we formulate the reflection coefficient equation of PP-wave data as a function of background porosity-related stress parameter and two (i.e., normal and shear) fracture weakness parameters. Following Bayes’ rule, we estimate the porosity-related stress parameter and fracture weaknesses using the inversion method of azimuthal Fourier coefficients. Finally, we compute the effective horizontal and vertical in situ stresses using the estimated elastic moduli, porosity-related stress parameter, and fracture weaknesses. Synthetic and real data sets demonstrate that our proposed inversion approach based on the derived reflection equation provides us another way to obtain reasonable estimates of in situ stresses in a complex-fractured porous shale reservoir.

Peanut Harmonic Expansion for a Fundamental Solution of Laplace’s Equation in Flat-Ring Coordinates

We derive an expansion for the fundamental solution of Laplace’s equation in flat-ring cyclide coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of coordinate surfaces which are peanut shaped and orthogonal to surfaces which are flat-rings. These internal and external peanut harmonic functions are expressed in terms of Lamé—Wangerin functions. Using the expansion for the fundamental solution, we derive an addition theorem for the azimuthal Fourier component in terms of the odd-half-integer degree Legendre function of the second kind as an infinite series in Lamé—Wangerin functions. We also derive integral identities over the Legendre function of the second kind for a product of three Lamé—Wangerin functions. In a limiting case we obtain the expansion of the fundamental solution in spherical coordinates.

On the efficient evaluation of the azimuthal Fourier components of the Green's function for Helmholtz's equation in cylindrical coordinates

In this paper, we develop an efficient algorithm to evaluate the azimuthal Fourier components of the Green's function for the Helmholtz equation in cylindrical coordinates. A computationally efficient algorithm for this modal Green's function is essential for solvers for electromagnetic scattering from bodies of revolution (e.g., radar cross sections, antennas). Current algorithms to evaluate this modal Green's function become computationally intractable when the source and target are close or when the wavenumber is large or complex. Furthermore, most state-of-the-art methods cannot be easily parallelized. We present an algorithm for evaluating the modal Green's function that has performance independent of both source-to-target proximity and wavenumber, and whose cost grows as O(m), where m is the Fourier mode. Our algorithm's performance is independent of whether the wavenumber is real or complex. Furthermore, our algorithm is embarrassingly parallelizable.

Integrating a ponderomotive guiding center algorithm into a quasi-static particle-in-cell code based on azimuthal mode decomposition

High fidelity modeling of plasma based acceleration (PBA) requires the use of three dimensional, fully nonlinear, and kinetic descriptions based on the particle-in-cell (PIC) method. In PBA an intense particle beam or laser (driver) propagates through a tenuous plasma whereby it excites a plasma wave wake. Three-dimensional PIC algorithms based on the quasi-static approximation (QSA) have been successfully applied to efficiently model the interaction between relativistic charged particle beams and plasma. In a QSA PIC algorithm, the plasma response to a charged particle beam or laser driver is calculated based on forces from the driver and self-consistent forces from the QSA form of Maxwell's equations. These fields are then used to advance the charged particle beam or laser forward by a large time step. Since the time step is not limited by the regular Courant-Friedrichs-Lewy (CFL) condition that constrains a standard 3D fully electromagnetic PIC code, a 3D QSA PIC code can achieve orders of magnitude speedup in performance. Recently, a new hybrid QSA PIC algorithm that combines another speedup technique known as an azimuthal Fourier decomposition has been proposed and implemented. This hybrid algorithm decomposes the electromagnetic fields, charge and current density into azimuthal harmonics and only the Fourier coefficients need to be updated, which can reduce the algorithmic complexity of a 3D code to that of a 2D code. Modeling the laser-plasma interaction in a full 3D electromagnetic PIC algorithm is very computationally expensive due the enormous disparity of physical scales to be resolved. In the QSA the laser is modeled using the ponderomotive guiding center (PGC) approach. We describe how to implement a PGC algorithm compatible for the QSA PIC algorithms based on the azimuthal mode expansion. This algorithm permits time steps orders of magnitude larger than the cell size and it can be asynchronously parallelized. Details on how this is implemented into the QSA PIC code that utilizes an azimuthal mode expansion, QPAD, are also described. Benchmarks and comparisons between a fully 3D explicit PIC code (OSIRIS), as well as a few examples related to laser wakefield acceleration, are presented.

Three-dimensional density field of a screeching under-expanded jet in helical mode using multi-view digital holographic interferometry

A synchronised multi-axis digital holographic interferometry set-up is presented for the study of 3-D flow fields with large density gradients. This optical configuration provides instantaneous interferograms with fine spatial resolution in six directions of projection. A regularised tomographic approach taking into account the presence of possible shock waves is furthermore considered to reconstruct 3-D density fields. Applied to a screeching under-expanded supersonic jet with helical dynamics, this set-up is used to provide dense optical phase measurements in the initial region of the jet. The jet mean density field is shown to be satisfactorily estimated with sharply resolved density gradients. In addition, an approach based on azimuthal Fourier transform and snapshot proper orthogonal decomposition (POD) applied to the instantaneous flow observations is proposed to study the main coherent dynamics of the jet. Relying on a cluster analysis of the azimuthal POD mode coefficients, a reduced dynamical model in the POD mode phase space is used as an approximation of the two observed limit cycles. A clear 3-D representation of the density field of a helical instability associated with screech mode C is then evidenced, with two equally probable directions of rotation. Switching between the two directions is reported, highlighting intermittency in the feedback loop. This helical structure is particularly seen to extend to the jet core, driving its internal dynamics and inducing out-of-phase density fluctuations between the outer and inner shear layers. These out-of-phase motions are related to the non-uniform radial distribution of fluctuation phase associated with the outer-layer Kelvin–Helmholtz instability wave.

Gegenbauer expansions and addition theorems for a binomial and logarithmic fundamental solution of the even-dimensional Euclidean polyharmonic equation

On even-dimensional Euclidean space for integer powers of the positive Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. This limit is accomplished through parameter differentiation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we are able to obtain addition theorems for the azimuthal Fourier coefficients. These logarithmic and binomial addition theorems are expressed in Vilenkin polyspherical geodesic polar coordinate systems and as well in generalized Hopf coordinates on hyperspheres in arbitrary even dimensions.

Characterization of seismic anisotropy using azimuthal AVO analysis (AVAz) - An application case study in the deep and tight carbonate reservoirs from Potwar Basin onshore Pakistan

Reliable characterization of subsurface fractures information within the carbonate's resources plays a key role in reservoir description in terms of the storage and movement of fluid. Commonly, anisotropy information classifies heterogeneity in the rock fabric due to fractures, pores, and local stress variations. Amplitude Variation with Azimuth investigation examines azimuthal variations in the seismic amplitude information to provide useful information about fractures and anisotropy of a horizontally transverse isotropic (HTI).This paper presents modeling, analysis, and interpretation of Azimuthal amplitude variation with angles (AVAz) from 3D seismic and calibrate results with the wells data to extract fractures information in terms of fractures magnitude and orientations. We employed two most used data-driven alternate approaches 1) near-offset Rüger's equation and 2) azimuthal Fourier Coefficient (FC) for analyzing the AVAz responses in seismic data to estimate the fractures density and their orientations, and results are calibrated with the bore hole image data (FMI). These approaches proved practically convenient to quantify the azimuthal amplitude variations with two sets of AVAZ attributes. The results of AVAz modeling and analysis required good quality of input data. Before AVAZ modeling and attributes analysis, seismic data is pre-conditioned, and the data quality of the 3D seismic dataset is improved. Several qualitative and quantitative QC measures (e.g., comparison of angle stacks, AVAz analysis, cross-plots of correlation coefficients, and time shifts) are employed to ascertain the improvement of the health of the seismic data. Azimuthal AVA modeling helped to understand the azimuthal amplitude response due to anisotropy caused by fractures/stress. The predicted seismic response from the models is used to assess the sensitivity of AVAZ with the parameters of the anisotropic model. After the AVAZ modeling, the outcomes of the AVAZ attributes are generated using to 3D seismic and calibrated with the well data set to identify the anomaly of anisotropic gradient (Bani) and second Fourier coefficients (R2) by relating to fractures density or stress magnitude. These attributes helped to identify the amplitude variations with offset and azimuth in the deep inhomogeneous tight carbonate reservoir interval (Patala) of DS hydrocarbon field discovered in the north Potwar Basin, onshore Pakistan. High score of Bani or R2 observed at well location at the tight Paleocene carbonate (Patala formation) reservoirs exhibited the presence of fractures which is confirmed from the image logs (e.g., FMI).The results demonstrated that the studied formations exhibit significant heterogeneity due to the faults/fractures and stresses associated with the complex geological structure of the basin as well as variations in the rock fabric. Hence, accurate analysis of the fractures information from seismic data is quite useful not only in understanding the reservoir heterogeneity to optimize the production, but also in predicting the completion and development strategies that may be most effective.