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changes across the receiver array, standard methods for estimating slowness, such as STC processing, which relies on


wave-shape


similarity, must be adapted to handle dispersive waves. Dispersive STC processing identifies the slowness of individual frequency components.1 1 A plot of flexural-wave slowness versus frequency is called a dispersion curve (below). Dispersion-curve analysis compares modeled acoustic dispersion curves for homogeneous isotropic formations with curves measured by borehole sonic tools.1 2


The radial depth of investigation of flexural waves is approximately one wavelength. Low- frequency flexural waves probe deep into the formation, and high-frequency flexural waves have shallower depths of investigation. Analysis of flexural-mode slowness as a function of frequency can therefore provide detailed information about the formation near and far from the borehole. At zero frequency, flexural-wave slowness is the true formation shear slowness. Plotting flexural-wave slowness versus frequency and identifying the zero-frequency limit of the curve allow estimation of formation shear slowness. In this way, analysis of flexural-wave dispersion allows estimation of shear slowness in fast or slow formations.1 3


Up to now, this article has concentrated on the simplest case of a homogeneous isotropic formation and monopole and dipole sources. Such a formation has one P-wave slowness, one Stoneley-wave slowness and one S-wave slowness. Most of the applications for using sonic-logging results to infer formation porosity, permeability, fluid type, elastic moduli, lithology or mineralogy have been developed for homogeneous isotropic formations. Additional complexities arise in inhomogeneous or anisotropic formations. The rest of this article addresses anisotropy first, then looks at inhomogeneous formations.


Anisotropy


The spatial alignment of mineral grains, layers, fractures or stress causes wave velocity to vary with direction, a property called anisotropy.1 4


In


seismic surveys, the anisotropy of the overburden shales is known to cause imaging difficulties that need to be corrected to place reservoir targets at the correct location. Information about aniso- tropy is also needed whenever an understanding of rock mechanics is required. Directional drilling, drilling in tectonically active areas, designing oriented-perforating jobs, planning hydraulic-fracturing operations and developing pressure-supported recovery plans all benefit from knowledge of elastic anisotropy. The natural processes that cause anisotropy


4 0 0 3 0 0 Stoneley


also cause it to have one of two main orientations: horizontal or vertical. To a first approximation, horizontal layers create an anisotropic medium that may be considered isotropic in


2 0 0


Dipole flex ural


1 0 0 Shear 0 0 2 4 6 8 F requency, kH z


> Dispersion curves characterizing slow ness at different freq uencies in an isotropic form ation. Shear w aves are not dispersive; all their freq uency com ponents travel at the sam e slow ness. Stoneley w aves are only slightly dispersive. Flex ural m odes ex cited b y a dipole source ex hib it large dispersion in this form ation. At the zero-freq uency lim it,  ex ural-w ave slow ness tends to the shear-w ave slow ness ( dotted line) .


anisotropic vertically. Such a medium is known as transversely isotropic with a vertical axis of symmetry (TIV) (above right). Similarly, vertical fractures create a simplified anisotropic medium that may be considered isotropic in any direction aligned with fracture planes, and anisotropic in the direction orthogonal to fracture planes. This medium is known as transversely isotropic with a horizontal axis of symmetry (TIH). Sonic waves are sensitive to these directional differences in material properties. Waves travel fastest when the direction of particle motion, called polarization, is parallel to the direction of greatest


stiffness. Compressional waves have particle motion in the direction of propagation, so P-waves travel fastest in directions parallel to layering and fractures, and travel more slowly when perpendicular to layering and fractures.


y x


H orizontal axis of symmetry


z


T I V y


V ertical axis of symmetry


x z


T I H


> Sim pli ed geom etries in elastic anisotropy . In horizontal lay ers ( top) , elastic properties m ay b e uniform horizontally , b ut vary vertically . Such a


m edium m ay b e approx im ated as transversely isotropic w ith a vertical ax is of sy m m etry ( TIV) ,


m eaning that the form ation m ay b e rotated ab out the ax is to produce a m edium w ith the sam e properties. In form ations w ith vertical fractures


( b ottom ) , elastic properties m ay b e uniform in vertical planes parallel to the fractures, b ut m ay vary in the perpendicular direction. This m edium m ay b e approx im ated as transversely isotropic w ith a horizontal ax is of sy m m etry ( TIH) .


all horizontal directions, but


10. K im b all CV and Marzetta TL: “ Sem b lance Processing of Borehole Acoustic Array Data, ” Geophy sics4 9 , no. 3 ( March 19 84 ) : 27 4 – 281.


11. K im b all CV: “ Shear Slow ness Measurem ent b y Dispersive Processing of the Borehole Flex ural Mode, ” Geophy sics63 , no. 2 ( March– April 19 9 8) : 3 3 7 – 3 4 4 .


12. Murray D, Plona T and Valero H-P: “ Case Study of Borehole Sonic Dispersion Curve Analy sis, ” Transactions of the SPWLA 4 5 th Annual Logging Sy m posium , J une 6– 9 , 2004 , Noordw ij k , The Netherlands, paper BB.


The k ey param eters req uired for dispersion-curve m odeling are form ation slow ness, form ation density , b orehole- uid velocity , b orehole- uid density and b orehole diam eter.


13 . Sinha BK and Z eroug S: “ Geophy sical Prospecting Using Sonics and Ultrasonics, ” in Web ster J G ( ed) : Wiley Ency clopedia of Electrical and Electronic Engineers Vol. 8. New Y ork City : J ohn Wiley and Sons, Inc. ( 19 9 9 ) : 3 4 0– 3 65 .


14 . This holds for alignm ents on scales that are sm aller than the w avelength of the w aves in q uestion.


Arm strong P, Ireson D, Chm ela B, Dodds K , Esm ersoy C, Miller D, Hornb y B, Say ers C, Schoenb erg M, Leaney S and Ly nn H: “ The Prom ise of Elastic Anisotropy , ” Oil eld Review 6, no. 4 ( Octob er 19 9 4 ) : 3 6– 4 7 .


Spring 2006


39


Slowness, µs/ ft


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