Jeroen Jocker

Validation of first-order diffraction theory for the traveltimes and amplitudes of propagating waves

Ultrasonic measurements of acoustic wavefields scattered by single spheres placed in a homogenous background medium (water) are presented. The dimensions of the spheres are comparable to the wavelength and the wavelength and represent both positive (rubber) and negative (teflon) velocity anomalies with respect to the background medium. The sensitivity of the recorded wavefield to scattering in terms of traveltime delay and amplitude variation is investigated. The results validate a linear (first-order) diffraction theory for wavefields propagating in heterogeneous media with anomaly contrasts on the order of ±15%. The diffraction theory is compared further with the exact results known from literature for scattering from an elastic sphere, formulated in terms of Legendre polynomials. To investigate the 2D case, the first-order scattering theory is tested against 2D elastic finite-difference calculations. As the presented theory involves a volume integral, it is applicable to any geometric shape, and the scattering object does not need to be spherical or any other specific symmetrical shape. Furthermore, it can be implemented easily in seismic data inversion schemes, which is illustrated with examples from seismic crosswell tomography and a reflection experiment.

Fracture characterization at multiple scales using borehole images, sonic logs, and walkaround vertical seismic profile

We present a quantitative forward-modeling methodology to link and interpret several measurements relevant to mechanical properties of fractures such as borehole images, sonic anisotropy logs, and borehole seismic anisotropy. The analysis is applied to a case study from a north African tight gas field using data from a vertical well. Two studies are conducted independently using the same geological fracture data to model fracture-induced anisotropy. In the first study, we use the orientation of the natural and drilling-induced fractures interpreted on the image log to model the azimuthal fracture-induced anisotropy at the sonic scale. The mechanical effects of natural and drilling-induced fractures are treated using different compliance parameters for each fracture type. We show that modeled sonic fast shear azimuths could be biased by the presence of noncompliant fractures in each fracture type, and we propose an empirical selection criterion to reject noncompliant fractures prior to compliance estimation. Then, we estimate the fracture compliances and confirm that natural open fractures have larger compliances than drilling-induced fractures. In the second study, we apply interpreted borehole images toward modeling of the azimuthal vertical seismic profile (VSP) attributes as a function of source azimuthal position. Natural fractures inside a window of height, h, and located at depth, d, are included, and several volume sizes and positions (i.e., h and d) are considered. We find a good agreement between modeled and observed transverse-over-radial displacement trends using natural fractures within windows located at the depth of the VSP receiver, and having window heights on the order of one to two VSP shear wavelengths.

Incremental linear-elastic response of rocks containing multiple rough fractures: Similarities and differences with traction-free cracks

Fractures in the subsurface serve as conduits for fluids and gas, connecting remote hydrocarbon reservoir sections to production wells. Seismic and sonic data are popular sources for information on fracture properties. The most commonly used model to extract fracture information from such data is based on the paradigm of the displacement discontinuity interface, without a direct link to relevant characteristics such as the surface roughness properties of a fracture. Indeed, fractures can be modeled as displacement discontinuity surfaces, and in this sense they resemble traction-free cracks. In literature, cracks and fractures are not always properly distinguished, perhaps because the terms are often perceived as synonyms. However, microstructural parameters that control magnitudes of the discontinuities — and thus the effective stiffnesses — are entirely different: statistics of contacts for fractures versus crack density for traction-free cracks. We explore the effective elasticity of rocks containing multiple fractures using a model of a fracture as two rough surfaces with isolated contacts. This is done in the context of the incremental, linear elastic response to small stress changes, typical in wave-propagation problems. Fractures are dry or may have diverse orientations, and contacts may or may not be Hertzian. A link exists between contact characteristics and effective stiffness of single and multiple fractures. Our work examines and accounts for the strong effect of interactions between individual contacts by means of a double sum over mutual positions as well as outlines the differences and similarities between theories for cracks and fractures.

Ultrasonic measurements on poroelastic slabs : determination of reflection and transmission coefficients and processing for Biot input parameters

Ultrasonic reflection and transmission measurements on saturated, plane-parallel, poroelastic slabs are presented. A data processing technique is proposed to obtain poroelastic parameters from transmission measurements. A special experimental data acquisition and processing technique is applied to minimize the finite beam effects of the transducers. This technique yields results which can be compared with poroelastic plane-wave theory. Results of normal- and oblique-incidence measurements are presented, and transmission data are processed to yield wave speeds, sample thickness, angle of incidence, tortuosity, and permeability. The results show good agreement with independent measurements, and they are subsequently used as input for a forward modeling of the complete transmitted and reflected waveforms utilizing Biot theory. The agreement between recorded and modeled signals is good, both in time and frequency domain.

Matrix propagator method for layered porous media: Analytical expressions and stability criteria

We fully expand the Thomson-Haskell propagator matrix method to obtain closed-form analytical expressions for the reflectivity and transmittivity of a horizontally stratified poroelastic Biot-type medium that is bounded from above and below by fluid half-spaces. Stable and unstable regions of the propagator matrix method are determined by means of comparison with the inherently stable reflectivity method. It was found that the stability is mainly determined by the imaginary part of the vertical wavenumber of the Biot slow wave.

Rough contacting surfaces: similarities and differences with traction-free cracks

SUMMARY The effective elasticity of rocks containing multiple rough fractures is discussed. This is done in the context of the incremental, linear-elastic response to small stress changes typical in wave propagation problems. Fractures may have diverse orientations, and contacts may or may not be Hertzian. Fractures can be modeled as displacement discontinuity surfaces, and in this sense they resemble traction-free cracks. However, contacts drastically reduce the fracture compliance, and thus its sonic ”footprint”, making the fracture equivalent to a tractionfree crack of a much smaller size. Hence, microstructural parameters that control magnitudes of the discontinuities - and thus the effective stiffnesses - are different: statistics of contacts for rough fractures versus a sum of crack-sizes cubed for traction-free cracks. Understanding and quantifying these differences is essential since one is otherwise forced to treat crack density as a fitting parameter, thus losing the link between the effective elasticity and fracture surface characteristics.

Validation of first‐order wave diffraction theory by an ultrasonic experiment

Ray theory is inadequate to explain the behavior of finitefrequency wave propagation in media with structures smaller in size than wavelength and the Fresnel zone. In such complex structures, wave diffraction effects are important. By performing an ultrasonic wave experiment, a newly developed theory for finite-frequency wave propagation is successfully validated. The presented wave theory has a large potential in high-resolution seismic crosswell and VSP tomographic experiments.

Analysis of walkaround VSP azimuthal response using borehole images

We applied interpreted fracture data from borehole images towards modeling of azimuthal walkaround VSP attributes. The methodology is tested with a case study on data coming from a tight-gas field. Natural fractures inside a window of height h and located at depth d are included in an excess compliance model to obtain the formation anisotropic stiffness tensor. Five tensors are generated to model fast shear wave velocity as a function of VSP source azimuth. In addition, best approximative transverse-isotropy tensors are derived and applied in combination with a simple relation describing general walkaround attribute behavior. It is shown that with this approach, borehole images can be applied successfully to predict the walkaround VSP TR-ratio dependency on source azimuth. Good agreement between model and observation is achieved for a volume corresponding to the last two VSP shear wavelengths.

Numerical investigation of alternative fracture stiffness measures and their respective scaling behaviours

ABSTRACT We study the mechanical deformation of fractures under normal stress, via tangent and specific fracture stiffnesses, for different length scales using numerical simulations and analytical insights. First, we revisit an equivalent elastic layer model that leads to two expressions: the tangent stiffness is the sum of an “intrinsic” stiffness and the normal stress, and the specific stiffness is the tangent stiffness divided by the fracture aperture at current stress. Second, we simulate the deformation of rough fractures using a boundary element method where fracture surfaces represented by elastic asperities on an elastic half‐space follow a self‐affine distribution. A large number of statistically identical “parent” fractures are generated, from which sub‐fractures of smaller dimensions are extracted. The self‐affine distribution implies that the stress‐free fracture aperture increases with fracture length with a power law in agreement with the chosen Hurst exponent. All simulated fractures exhibit an increase in the specific stiffness with stress and an average decrease with increase in length consistent with field observations. The simulated specific and tangent stiffnesses are well described by the equivalent layer model provided the “intrinsic” stiffness slightly decreases with fracture length following a power law. By combining numerical simulations and the analytical model, the effect of scale and stress on fracture stiffness measures can be easily separated using the concept of “intrinsic” stiffness. We learn that the primary reason for the variability in specific stiffness with length comes from the fact that the typical aperture of the self‐affine fractures itself scales with the length of the fractures.

The energy balance at linear slip interfaces

SUMMARY We present an analysis of the energy balance at interfaces defined by linear slip boundary conditions. The media above and below the interface are purely elastic, whereas the interface itself behaves either elastic or viscoelastic. We obtain an expression for the (time-averaged) amount of elastic energy converted to heat, as a function of the interface properties. All terms in the energy balance are identified, and the balance itself is demonstrated in an example where - at sufficiently high frequencies - up to 8% of the incident P, and up to 30% of the incident SH-wave energy is lost to heat.