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The difficulty in achieving well-to-seismic ties due to errors arising from wrong time-to-depth conversions has been as a result of ignoring anisotropy in seismic processing. Anisotropy plays a vital role in the processing and interpretation of seismic data. In this work, an inversion method based on the elastic stiffness tensors was adopted to estimate and quantify anisotropy in two depobelts using petrophysical well logs in Niger delta (Central Swamp and Greater Ughelli). Results show that the estimated delta ( δ ), epsilon ( ε ), gamma ( γ ) and eta ( η ) exhibit a high degree of anisotropy in the shales than in the sands. The parameters were observed to be higher in the Central Swamp than the Greater Ughelli depobelt. This behavior could be associated with the alternating sequence of massive shale and sand beds geologically observed within this depobelt. This work was also able to derive empirical relations that could be used in estimating these parameters in the depobelts once appropriate information or data for any one parameter is available.

A medium measured at same point and has different values depending on the direction of measurement is said to be anisotropic. This is an inherent property of most rock mass due to inhomogeneities in structure and deposition. Anisotropy has been studied extensively for hydrocarbon exploration for the reason that it provides improvement in subsurface imaging. This has dramatically increased over the past two decades due to the inadequacy of isotropic velocity models in depth imaging and advances in parameter estimation, the transition from poststack to prestack depth imaging, the wider offset and azimuthal coverage of 3D surveys, and acquisition of high-quality multicomponent data.

Consequently, transverse isotropic (TI) models with a vertical (VTI) and tilted (TTI) axes of symmetry have become practically standard prestack imaging projects all over the world. Currently, many seismic processing and inversion methods operate with anisotropic models and therefore, it is important to treat anisotropy as an inherent part of the velocity fields for proper imaging of subsurface structures (Tsvankin, et. al., 2010) [

Incorporating anisotropy into velocity analysis requires estimation of several independent, spatially variable parameters from P-wave reflection travel times. Hence, the progress in P-wave processing can be largely attributed to breakthroughs in parameterization of transversely isotropic (TI) models. A proper design of model parameterization is one of the most critical issues in seismic data analysis for anisotropic media. Though the stiffness coefficients (c_{ij}) are convenient to use in forward-modeling algorithms, they are not well-suited for application in seismic processing and inversion. Thomsen (1986) [_{P0} and V_{S0}, respectively) with three dimensionless parameters (ε, δ, and γ), which characterize the magnitude of anisotropy and the P-wave time- processing parameter η (Alkhalifah and Tsvankin, 1995) [

The parameter ε is close to the fractional difference between the P-wave velocities in the directions perpendicular and parallel to the symmetry axis. It defines what is often simplistically called the “P-wave anisotropy.” Likewise, γ represents the same measure for SH-waves, while δ governs the P-wave velocity variation away from the symmetry axis and also influences the SV-wave velocity.

The study area (Central Swamp and Greater Ughelli Depobelts) in Niger Delta (

The non-hyperbolic moveout of events in the depobelts is mostly associated with this anisotropy, which is manifested as hockey sticks in seismic records (

The aim of the present study therefore, is to estimate ε, δ, γ and η parameters to characterize anisotropy in the depobelts, and finally, derive empirical relations for the estimation of these parameters in the two depobelts.

The stratigraphic sequence of the Niger Delta comprises three broad lithostratigraphic units, namely, a continental shallow massive sand sequence, the Benin Formation, a coastal marine sequence of alternating sands and shales, the Agbada Formation, and a basal marine shale unit, the Akata Formation. Outcrops of these units are exposed at various localities within the Niger delta (

The Benin Formation is characterized by high sand percentage (70% - 100%) and forms the top layer of the Niger Delta depositional sequence. The massive sands were deposited in continental environment comprising the fluvial realms (braided and meandering systems) of the upper delta plain. The Agbada Formation consists of alternating sands and shales representing sediments of the transitional environment comprising the lower delta plain (mangrove swamps, floodplain, and marsh) and the coastal barrier and fluviomarine realms. The sand percentage within the Agbada Formation varies from 30% to 70%, which results from the large number of depositional offlap cycles (Obaje, 2005) [

Petroleum in the Niger Delta is produced from the unconsolidated sands in the Agbada Formation and the reservoir properties are controlled by depositional environment and agents of deposition, structure and depth of burial.

A transversely anisotropic medium with a vertical symmetry are characterized by three anisotropic parameters: ε, γ, and δ (Sayers, 2005 [

where C_{11} is the in-plane compressional modulus, C_{13} is an important constant that controls the shape of the wave surfaces, C_{33} is the out-of-plane compressional modulus, C_{44} is the out-of-plane shear modulus and C_{66} is the in-plane shear modulus. These five independent elastic constants C_{11}, C_{13}, C_{33}, C_{44} and C_{66} are for weak and transversely isotropic medium with vertical symmetry.

The P-wave time-processing parameter η (Alkhalifah and Tsvankin, 1995) [

Two wells A and B from each depobelt were used in this study (_{P}) and shear (V_{S}) wave velocity logs. The initial task in the anisotropy modeling is to solve for the five independent elastic constants using well data from which the anisotropy parameters were evaluated using appropriate equations.

The estimated anisotropy parameters in sands and shales for the two wells were subsequently cross-plotted in shales and sands for the two depobelts to characterize anisotropy.

The P-wave and S-wave anisotropy were cross-plotted for the Central Swamp and Greater Ughelli depobelts (

γ = 1.6172ε + 0.0026 with R^{2} = 0.5177 (5)

and

γ = 1.1721ε + 0.0036 with R^{2} = 0.6552 (6)

for Greater Ughelli, where ε and γ are the P-wave (ε) and S-wave (γ) anisotropy parameters respectively and R^{2} is the correlation coefficient.

This empirical relations implies that P-wave (ε) anisotropy can be estimated from S-wave (γ) anisotropy, and vice-versa, once sufficient information about one is known.

Swamp and Greater Ughelli Depobelts. The result show that η is higher than δ in both sand and shale formations. This behavior could be associated with the alternating sequence of massive shale and sand beds observed within this depobelt. An inverse relationship is observed between the two parameters and the best-fit regression line to data were estimated.

Result show that the best-fit regression line to data for the Central Swamp and Greater Ughelli respectively are:

δ = −0.6069η + 0.0021 with R^{2} = 0.5732 (7)

and

δ = −0.8586η + 0.0021 with R^{2} = 0.5165 (8)

Results from γ-ε cross-plots show that approximately 7% - 29% P-wave anisotropy and 10% - 40% S-wave anisotropy were observed in the shales; while 2.5% - 3% P-wave anisotropy and 2% - 10% S-wave anisotropy were observed in sands. Likewise, in

The estimated anisotropy parameters in the Central Swamp is considered to be high (with ε = 29%, γ = 40%, δ = 14%, and ŋ = 28%) as compared to that in Greater Ughelli depobelt (with ε = 7%, γ = 10%, δ = 8%, and δ = 10%).

It is also worthy to note that when a sand contains large amounts of authigenic clays, it may become intrinsically anisotropic (Wang, 2002) [

Based on the cross-plots of the anisotropy parameters and analyses of the best-fit regression line of the plots in shale, a relation was derived for ε-γ and δ-η parameters in the two depobelts. These relations are quite good considering their high correlation coefficients. The relations could be used for estimating any of the parameters in the two depobelts once there is appropriate information or data for any one parameter. Such estimation is independent of pressure, pore fluids and lithology.

It is very useful when S-wave (γ) is available, but P-wave (ε) anisotropy is not. This is done to avoid having to undergo the tedious algebraic derivation of the P-wave anisotropy using stiffness tensor. This also applies to δ-η in the study.

Our results in the Central Swamp and Greater Ughelli depobelts shows a higher degree of anisotropy in shales than sands, indicating that the Niger Delta shales are intrinsically anisotropic. The predominance of shales in the depobelts accounts for this high degree of anisotropy. The estimated anisotopy is higher in the Central Swamp than the Greater Ughelli depobelt. A relation has also been derived for the P-wave (ε) and S-wave (γ) anisotropies, and also for the Delta δ- and Eta ŋ-anisotropies in the Niger Delta.

The relevance of these parameters in seismic processing and interpretation makes it imperative to have these empirical relations which in effect could be applied in new areas within the Niger Delta to estimate anisotropy parameters for seismic processing and imaging.

We are grateful to Shell Petroleum Development Company, Port Harcourt, for the privilege to use their data in this study.

Iheanyichukwu P. C. Okorie,Joseph O. Ebeniro,Chukwuemeka N. Ehirim, (2016) Anisotropy and Empirical Relations for the Estimation of Anisotropy Parameters in Niger Delta Depobelts. International Journal of Geosciences,07,345-352. doi: 10.4236/ijg.2016.73027