Klaus Helbig

Orthorhombic media: Modeling elastic wave behavior in a vertically fractured earth

Vertical fractures and horizontal fine layering combine to form a long, wavelength equivalent orthorhombic medium. Such media constitute a subset of the set of all orthorhombic media. Orthorhombic elastic symmetry is the lowest symmetry for which the slowness surface (the solution of the Christoffel equation) is bicubic rather sextic. Various properties of orthorhombic media, such as the number and location of conical points and longitudinal directions, may be derived from the slowness surface or, because of its bicubic character, the squared slowness surface, which is a cubic surface. From the occurrence and angular orientation of some of these distinctive features, conclusions can be drawn with respect to the properties of the medium and to the parameters of the assumed underlying causes of the anisotropy. The estimation of these more subtle properties gains greater importance with the proliferation of multiazimuthal seismic surveys and the ability to drill along ever more complicated 3‐D well trajectories.

75-plus years of anisotropy in exploration and reservoir seismics: A historical review of concepts and methods

The idea that the propagation of elastic waves can be anisotropic, i.e., that the velocity may depend on the direction, is about 175 years old. The first steps are connected with the top scientists of that time, people such as Cauchy, Fresnel, Green, and Kelvin. For most of the 19th century, anisotropic wave propagation was studied mainly by mathematical physicists, and the only applications were in crystal optics and crystal elasticity. During these years, important steps in the formal description of the subject were made. At the turn of the 20th century, Rudzki stressed the significance of seismic anisotropy. He studied many of its aspects, but his ideas were not applied. Research in seismic anisotropy became stagnant after his death in 1916. Beginning about 1950, the significance of seismic anisotropy for exploration seismics was studied, mainly in connection with thinly layered media and the resulting transverse isotropy. Very soon it became clear that the effect of layer-induced anisotropy on data ac...

75-plus years of anisotropy in exploration and reservoir seismics: Ahistorical review of concepts and methods, GEOPHYSICS, 70, 9ND–23ND

On page 18ND, right column, the phrase “in the Valhall oil field in Norway” was incorrect. We apologize for any inconvenience caused by this error.

75th Anniversary Paper: 75-plus years of anisotropy in exploration and reservoir seismics: A historical review of concepts and methods

The idea that the propagation of elastic waves can be anisotropic, i.e., that the velocity may depend on the direction, is about 175 years old. The first steps are connected with the top scientists of that time, people such as Cauchy, Fresnel, Green, and Kelvin. For most of the 19th century, anisotropic wave propagation was studied mainly by mathematical physicists, and the only applications were in crystal optics and crystal elasticity. During these years, important steps in the formal description of the subject were made. At the turn of the 20th century, Rudzki stressed the significance of seismic anisotropy. He studied many of its aspects, but his ideas were not applied. Research in seismic anisotropy became stagnant after his death in 1916. Beginning about 1950, the significance of seismic anisotropy for exploration seismics was studied, mainly in connection with thinly layered media and the resulting transverse isotropy. Very soon it became clear that the effect of layer-induced anisotropy on data ac...

Elastic medium equivalent to Fresnel’s double-refraction crystal

In 1821, Fresnel obtained the wave surface of an optically biaxial crystal, assuming that light waves are vibrations of the ether in which longitudinal vibrations (P waves) do not propagate. An anisotropic elastic medium mathematically analogous to Fresnels crystal exists. The medium has four elastic constants: a P-wave modulus, associated with a spherical P wave surface, and three elastic constants, c(44), c(55), and c(66), associated with the shear waves, which are mathematically equivalent to the three dielectric permittivity constants epsilon(11), epsilon(22), and epsilon(33) as follows: mu(0)epsilon(11)<==>rho/c(44), mu(0)epsilon(22)<==>rho/c(55), mu(0)epsilon(33)<==>rho/c(66), where mu(0) is the magnetic permeability of vacuum and rho is the mass density. These relations also represent the equivalence between the elastic and electromagnetic wave velocities along the principal axes of the medium. A complete mathematical equivalence can be obtained by setting the P-wave modulus equal to zero, but this yields an unstable elastic medium (the hypothetical ether). To obtain stability the P-wave velocity has to be assumed infinite (incompressibility). Another equivalent Fresnels wave surface corresponds to a medium with anomalous polarization. This medium is physically unstable even for a nonzero P-wave modulus.

The Eleventh International Workshop on Seismic Anisotropy (11IWSA)

The 11th International Workshop on Seismic Anisotropy (11IWSA), hosted by Memorial University of Newfoundland and chaired by Michael Slawinski, was held July 25–30, 2004, at St. Johns, Newfoundland, Canada. Attendees presented more than 40 papers covering recent developments of the theory of seismic wave propagation in anisotropic media and its various applications to seismic exploration.

Kelvin's eigensystems in anisotropic poroelasticity

Correct interpretation and processing of seismic data must integrate a correct description of the mechanical behavior of rocks, taking into account facts such as the presence of anisotropy and porosity with or without a saturating fluid. This work discusses elasticity of porous media of arbitrary anisotropy type, with emphasis on the study of deformation states and the associated elastic constants. The stress-strain law is represented in seven dimensions. Dynamic parameters (i.e., the six stress components and fluid pressure) are linked with kinematical parameters (i.e., the six strain components and the local increase of fluid content) by a 7D poroelastic tensor. The model is based on the following mechanical interpretation: each eigenvector (eigenstrain) of the poroelastic tensor defines a fundamental deformation state of the medium and the seven eigenvalues (eigenstiffnesses) representthe genuine poroelastic parameters. The set of seven eigenstrains and corresponding eigenstiffnesses constitute the eig...

Fifty years of amplitude control

The amplitudes of seismic waves have always been a foremost concern of the seismologist to which considerable ingenuity was devoted. In the 1920s the problem was to magnify the ground motion sufficiently for detection. This was done at first by simple levers that moved mechanical pens. But at the start of exploration seismology, this had already been superseded by optical levers, photographic recording, and (soon after) electromechanical transduction followed by amplification. From the 1930s to about the early ’60s, devices of increasing complexity were introduced to compress the large amplitude difference between the first arrivals and the weakest reflections of interest to the limited dynamic range of the recording medium: first the paper record, then magnetic storage media, and finally the digital magnetic tape. This period can be identified with techniques known as automatic gain control (AGC). Soon after the introduction of digital recording techniques, the emphasis shifted: with intermediate digital...

From reflection elements to structure — A look at the history of data interpretation

Traditionally, input acquired in the field consisted of the original paper records; output submitted to the client consisted of structural sections and depth-contour maps of selected interfaces. Before the introduction of magnetic recording, it was common practice to do the conversion in the field office. Tools for this conversion ranged from slide rules and desk calculators to wavefront charts. These tools were based on the geometry of rays in media where velocity is a function of depth only. The detailed algorithms underlying the conversion were often developed in the exploration companies and — originally — were carefully guarded. But at least the underlying principles were exchanged throughout the industry through books, journal articles, and presentations at meetings, such as noted in nearly 300 references in C. H. Dix’s Seismic Prospecting for Oil (1952) . The techniques of data acquisition and data interpretation have changed considerably, but the underlying principles of ray geometry are the same....

Anisotropy and dispersion — Two sides of a coin

Anisotropy is the dependence of the velocity on direction; dispersion is the dependence of the velocity on frequency f (or wavelength λ ). These apparently disjoint phenomena can be dealt with together if one uses as the common independent variable the wave vector k= ki , a vector in the direction of the wave and normal with length proportional to the wavenumber ω∕V=2πf∕V=2π∕λ . The phase velocity is V=ω∕k and the group velocity gi = gradk (ω) , the gradient of ω in k -space. The most convenient display of anisotropy and dispersion is by surfaces of equal ω (or equal f=ω∕2π ), to be determined with the help of a dispersion equation. In isotropic conditions, theseiso-omega surfaces are spheres. If there is no dispersion, the spheres are equidistant. For wave-number regions with dispersion, the spheres have variable spacing. For anisotropic conditions, the iso-omega surfaces are nonspherical but similar and equispaced if the medi-um is nondispersive. If dispersion occurswith anisotropy, the iso-omega surfac...