Patrick N. Okoye

Inversion technique for recovering the elastic constants of transversely isotropic materials

A least‐squares iterative inversion technique has been developed for the determination of the elastic parameter δ* of any transversely isotropic modeling material in the laboratory. For most applications in petroleum geophysics, the elastic parameter δ* is very important and is the crucial anisotropic parameter for near‐vertical P‐wave propagation. Despite the potential importance of δ* in seismic exploration and for resolution in an anisotropic medium, the conventional procedures adopted in estimating its value unfortunately are faced with many ambiguities and the reliability of its measurement is doubtful prior to the development of this technique. The anisotropic inverse modeling technique finds the best fitting solution. To optimize the accuracy of the results presented in this paper, analytical rather than numerical differentiations were implemented and the modeling procedures allow for controlled iterative adjustments in resolving the parameter δ*. Inversion of the first‐arrival traveltimes obtained...

Fresnel zones and spatial resolution for P- and SH-waves in transversely isotropic media

In an elastically anisotropic medium where the seismic wave velocity is a function of direction, the wavefront shape is nonspherical and, in most cases, assumes a nonelliptical shape. Numerical modelling techniques have been used to calculate the Fresnel‐zone diameter for compressional (P) and shear (SH) waves in transversely isotropic and isotropic media, respectively. The size of the Fresnel zone is found to be predominantly dependent on the curvatures and wavelength of the wavefront as well as the dip angle of the reflector. In addition, the anisotropic elastic parameters δ* (critical near‐vertical anisotropy), e (the P-wave anisotropy), and γ (the SH-wave anisotropy) are found to significantly affect the size of the Fresnel zone. Numerical modeling results show considerable differences between the Fresnel zones for anisotropic and isotropic velocity functions at various reflector dips. In addition, the Fresnel‐zone dimensions for anisotropic media exhibit asymmetry and considerable change with dip. By...

A study of the effects of transducer size on physical modeling experiments for recovering anisotropic elastic parameters

The effect of finite size transducers on elastic parameters recovered from laboratory measurements has been studied in this paper. Ultrasonic transmission experiments in the physical modeling laboratory were used to acquire the first arrival traveltime data at different ray angles, to simulate walkaway VSP surveys in the field. The elastic parameters of the materials were then recovered by our developed inversion techniques, using velocities at different directions inverted from the first arrival traveltimes. In our laboratory experiments, “point” transducers and “large” transducers were used to carry out the transmission experiments. The finite size of the transducers introduced systematic errors in the first break picks along the survey line, and hence the determined velocity field was inaccurate. These systematic errors cannot be corrected by the inversion technique. Numerical simulations were also conducted to study the differences between experiments using large and point transducers. From our experimental measurements, accurate elastic parameter values are obtained when the ratio of the sample-thickness to transducer-diameter is greater than 36. Decreasing the size of the transducers is highly recommended when conducting laboratory experiments to recover elastic parameters. A method of offset correction is also implemented in an attempt to decrease the measurement errors.

An experimental investigation of the small-offset P-wave NMO equation in transversely isotropic media.

Summary Numerical and physical modelling studies have been used to investigate the accuracy and limitations of the small-offset P-wave NMO velocity estimation in transversely isotropic media. An expression for the P-wave small-offset NMO velocity for a single horizontal reflector in a vertically transversely isotropic layer has been previously obtained: V nmo =+ αδ 0 12 where αο and δ respectively represent the vertical P-wave velocity and the P-wave critical anisotropy at oblique angles. The above equation is presented as being exact and valid for any degree of anisotropy and has been used in estimating the small-off set NMO velocity in the offshore Zaire seismic data acquired by Unocal in West Africa. The accuracy of this expression is subject t o practical testing in this paper. A one-layer horizontal model was used to carry out the numerical modelling studies. A computer program which calculates Pwave NMO velocities in transversely isotropic media was used. This program uses the exact equations for the phase and ray velocities in generating velocity functions to be used in ray tracing; it is known to give accurate results. Using the publishe d elastic parameters for anisotropic rocks, the small-offset NMO velocities are computed and results are compared with values obtained using the above expression. The degree of anisotropy varied from very weak to strong and contrasting velocity function s (positive and negative δ) were used in numerical modelling simulations. Phenolite materials with contrasting velocity functions were used to simulate transversely isotropic media with vertical axis of symmetry in physical modelling tests. Experimental reflection data were collected and velocity analysis was conducted for small-offset tr aces, and the NMO velocity determined by the maximum stack response. The elastic parameters αο and δ of the Phenolite materials were recovered by an anisotropic inversion technique. These parameters were used to estimate the small-offset P-wave NMO velocity in Phenolite with the above expression and also with our computer program. The NMO velocities obtained from these two methods were compared to that obtained from velocity analysis of the experimental seismic data. These comparisons enabled the validity of the above expressio n to yield accurate P-wave NMO velocity, in the limit of small offset, to be tested. Numerical and physical modelling results indicate that the accuracy of the small-offset P-wave NMO equation depends on the nature and degree of the anisotropy encountered in a given area, and is not valid for all degrees of P-wave anisotropy. The sma lloffset NMO (stacking) velocity from horizontal reflectors is known to influence the effects of vertical transverse isotropy on normal moveout (NMO), dip moveout (DMO) corrections and time migration. Determination of the correct small-offset P-wave NMO velocity is essential for obtaining accurate seismic images, and subsequently correct interpretations.

The dependence of lateral resolution on the orientation of symmetry axis and elastic parameter in transversely isotropic media

Similar models having positive and negative values and hence, contrasting wavefront curvatures were also used in conducting the experiments. Cylindrical holes of various sizes were drilled into the bottoms of the Phenolite models. The hole sizes ranged from 0.83 to 1.834 and from 0.86 to 2.054 times the expected Fresnel-zone diameters in the horizontally layered and vertically fractured Phenolites respectively.