Michael Schoenberg

Seismic anisotropy of fractured rock

A simple method for including the effects of geologically realistic fractures on the seismic propagation through fractured rocks can be obtained by writing the effective compliance tensor of the fractured rock as the sum of the compliance tensor of the unfractured background rock and the compliance tensors for each set of parallel fractures or aligned fractures. The compliance tensor of each fracture set is derivable from a second rank fracture compliance tensor. For a rotationally symmetric set of fractures, the fracture compliance tensor depends on only two fracture compliances, one controlling fracture compliance normal, the other, tangential, to the plane of the fractures. The stiffness tensor, which is more useful in the consideration of elastic wave propagation through rocks, can then be obtained by inversion. The components of the excess fracture compliance tensor represent the maximum amount of information that can be obtained from seismic data. If the background rock is isotropic and the normal a...

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.

Finite-difference modeling of faults and fractures

For the purposes of seismic propagation, a slip fault may be regarded as a surface across which the displacement caused by a seismic wave is discontinuous while the stress traction remains continuous. The simplest assumption is that this slip and the stress traction are linearly related. Such a linear slip interface condition is easily modeled when the fault is parallel to the finite-difference grid, but is more difficult to do for arbitrary nonplanar fault surfaces. To handle such situations we introduce equivalent medium theory to model material behavior in the cells of the finite-difference grid intersected by the fault. Virtually identical results were obtained from modeling the fault by (1) an explicit slip interface condition (fault parallel to the grid) and (2) using the equivalent medium theory when the finite-difference grid was rotated relative to the fault and receiver array. No additional computation time is needed except for the preprocessing required to find the relevant cells and their associated moduli. The formulation is sufficiently general to include faults in and between arbitrary anisotropic materials with slip properties that vary as a function of position.

ELASTIC WAVES THROUGH A SIMULATED FRACTURED MEDIUM

Ultrasonic velocities were measured on a block composed of lucite plates with roughened surfaces pressed together with a static normal stress to simulate a fractured medium. The measurements, normal, parallel, and oblique to the fractures, show that for wavelengths much larger than the thickness of an individual plate, the block can be modeled as a particular type of transversely isotropic (TI) medium that depends on four parameters. This TI medium behavior is the same as that of an isotropic solid in which are embedded a set of parallel linear slip interfaces, specified by (1) the excess compliance tangential to the interfaces and (2) the excess compliance normal to the interfaces. At all static stress levels, we inverted the data for the background isotropic medium parameters and the excess compliances. The background parameters obtained were basically independent of stress level and agreed well with the bulk properties of the lucite. The excess compliances decreased with increasing static closing stres...

Poroelastic modeling of seismic boundary conditions across a fracture.

Permeability of a fracture can affect how the fracture interacts with seismic waves. To examine this effect, a simple mathematical model that describes the poroelastic nature of wave-fracture interaction is useful. In this paper, a set of boundary conditions is presented which relate wave-induced particle velocity (or displacement) and stress including fluid pressure across a compliant, fluid-bearing fracture. These conditions are derived by modeling a fracture as a thin porous layer with increased compliance and finite permeability. Assuming a small layer thickness, the boundary conditions can be derived by integrating the governing equations of poroelastic wave propagation. A finite jump in the stress and velocity across a fracture is expressed as a function of the stress and velocity at the boundaries. Further simplification for a thin fracture yields a set of characteristic parameters that control the seismic response of single fractures with a wide range of mechanical and hydraulic properties. These boundary conditions have potential applications in simplifying numerical models such as finite-difference and finite-element methods to compute seismic wave scattering off nonplanar (e.g., curved and intersecting) fractures.

Fluid and solid motion in the neighborhood of a fluid-filled borehole due to the passage of a low-frequency elastic plane wave

The exact problem of the interaction of an arbitrary plane wave in a homogeneous elastic medium with a fluid‐filled circularly cylindrical borehole is formulated to analyze three‐axis, full‐waveform borehole seismic data. The plane elastic wave (which may be incident upon the borehole from any direction) is partially scattered around the hole and an acoustic field is induced in the borehole fluid. Since seismic wavelengths tend to be much larger than diameters of boreholes, the exact solution may be expanded in powers of Ω (a small number), the incident wavenumber times the borehole radius. In the limit as Ω→0, the scattered field in the elastic medium is of O(Ω); in this limit the borehole crosssection is undistorted and moves with the velocity of the incident wave as if no borehole were present. Neglecting terms proportional to Ω2 or higher leaves three kinds of θ‐dependent terms, where θ is the azimuthal angle about the borehole axis measured with respect to the wavenumber vector of the incident wave. ...

CROSS-BOREHOLE TOMOGRAPHY IN ANISOTROPIC MEDIA

Anisotropy has significant effect on traveltime cross‐borehole tomography. Even relatively weak anisotropy cannot be ignored if accurate velocity estimates are desired, since isotropic traveltime tomography treats anisotropy as inhomogeneity. Traveltime data in our examples were synthetically generated by a ray‐tracing code for anisotropic media, and the computed quasi‐P‐wave traveltimes were subsequently inverted using the “dual tomography” technique (Carrion, 1991). The results of the tomographic inversion show typical artifacts due to the anisotropy, and that accurate imaging is impossible without taking the anisotropy into account.

Fractured reservoirs: An analysis of coupled elastodynamic and permeability changes from pore-pressure variation

Equivalent-medium theories can describe the elastic compliance and fluid-permeability tensors of a layer containing closely spaced parallel fractures embedded in an isotropic background. We propose a relationship between effective stress (background or lithostatic stress minus pore pressure) and both permeability and elastic constants. This relationship uses an exponential-decay function that captures the expected asymptotic behavior, i.e., low effective stress gives high elastic compliance and high fluid permeability, while high effective stress gives low elastic compliance and low fluid permeability. The exponential-decay constants are estimated for physically realistic conditions. With relationships coupling pore pressure to permeability and elastic constants, we are able to couple hydromechanical and elastodynamic modeling codes. A specific coupled simulation is demonstrated where fluid injection in a fractured reservoir causes spatially and temporally varying changes in pore pressure, permeability, and elastic constants. These elastic constants are used in a 3D finite-difference code to demonstrate time-lapse seismic monitoring with different acquisition geometries. Changes in amplitude and traveltime are seen in surface seismic P-to-S reflections as a function of offset and azimuth, as well as in vertical seismic profile P-to-S reflections and in crosswell converted S-waves. These observed changes in the seismic response demonstrate seismic monitoring of fluid injection in the fractured reservoir.

Approximate dispersion relations for qP-qSV-waves in transversely isotropic media

To decouple qP and qSV sheets of the slowness surface of a transversely isotropic (TI) medium, a sequence of rational approximations to the solution of the dispersion relation of a TI medium is introduced. Originally conceived to allow isotropic P-wave processing schemes to be generalized to encompass the case of qP-waves in transverse isotropy, the sequence of approximations was found to be applicable to qSV-wave processing as well, although a higher order of approximation is necessary for qSV-waves than for qP-waves to yield the same accuracy. The zeroth‐order approximation, about which all other approximations are taken, is that of elliptical TI, which contains the correct values of slowness and its derivative along and perpendicular to the medium’s axis of symmetry. Successive orders of approximation yield the correct values of successive orders of derivatives in these directions, thereby forcing the approximation into increasingly better fit at the intervening oblique angles. Practically, the first‐o...

Elastic Properties of Rocks With Multiple Sets of Fractures

Recent work on equivalent elastic medium theory is modified and extended beyond fine-layered anisotropic rocks to include multiple sets of fractures, each lying in any plane. The group-theoretic structure is still basic, but group elements are now in terms of compliances rather than stiffnesses, and this much simplifies the algebra, and makes the commutative property clear rocks can be fractured in any order. This then leads to a method for modeling the effect of fracture distributions. This new model also provides a unifying framework to compare the several specialized fracture models in the literature. Several simple but physically reasonable examples are given.