Jesper Spetzler

The Fresnel volume and transmitted waves

In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high-frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite-frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite-frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band-limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite-frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite-frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat’s theorem in the context of geometric ray theory is a special case of the finite-frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.

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.

Are we exceeding the limits of the great circle approximation in global surface wave tomography

The ray theoretical great circle approximation in global surface wave tomography is found to be limited to Earth models with am aximum degreel 30 for sur- face waves at 40 s and l 20 for surface waves at 150 s. This result holds for both phase velocity and group velocity maps. The highest resolution in present-day global surface wave tomography is close to these limits of ray theory. In ordertoobtainhigherdegreeresolutionmodelsoftheEarth in future surface wave tomography, it is necessary to take the scattering of surface waves into account. Increasing the datacoverageinseismological networkswillnotimprovethe details of tomographic images if ray theory is still applied. It is essential to include the nite-frequency eects as well.

Pore pressure and water saturation variations; Modification of Landrø's AVO ap- proach

Landro (2001) used in his approach the amplitude-versusoffset (AVO) techniques to distinguish between effects of pore pressure and water saturation, and to quantify the changes in fluid-solid media over time. Meadows (2001) presented two improvements to this method. In this paper, we suggest a new approach to fit the relation between effective stress variations (resulting from overburden stress and pore pressure) and changes in seismic impedance. We also improve Meadows’s (2001) second approach, where the changes in pore pressure and water saturation are inverted from impedances instead from intercept and gradient. In our approach we are able to quantify more accurately the changes in effective stress and water saturation.

Application of a linear finite-frequency theory to time-lapse crosswell tomography in ultrasonic and numerical experiments

Time-lapse seismic monitoring is the geophysical discipline whereby multiple data sets recorded at the same location but at different times are used to locate and quantify temporal changes in the elastic parameters of the subsurface. We validate a time-lapse monitoring method by crosswell tomography using two types of wavefield-modeling experiments: (1) a 3D real ultrasonic waveform experiment and (2) 2D synthetic finite-difference wavefield simulations. For both wavefield experiments, a time-lapse structure simulating a fluid sweep in a reservoir layer is applied. The time-lapse tomographic monitoring approach is based on the standard ray theory and a finite-frequency wavefield theory, where the latter takes into account the finite-frequency properties of recorded wavefields. The inverted time-lapse models compiled with either the ray theory or the finite-frequency wavefield theory locate and correctly quantify the flooding zone in the simulated fluid sweep model. Both wavefield theories provide an adequate result because the flooding zone is comparable in size to the Fresnel volume.

Time-Lapse Seismic Crosswell Monitoring of CO2 injected in an Onshore Sandstone Aquifer

We present a case study of time-lapse seismic monitoring of CO2 injection by time delay tomography. During 550 days, 10.400 tonnes of CO2 were injected into a porous reservoir sandstone at 1100 m depth. Crosswell data were acquired before and after the CO2 injection. The estimated tomographic velocity image shows a clear time-lapse velocity anomaly on the order of -10 % below the CO2 injection well head.

Time-Lapse Seismic Crosswell Monitoring of Steam Injection in Tar Sand

Time-lapse tomography of crosswell seismic data can be used to monitor hydrocarbon reservoirs for P-wave velocity changes due to production over time. A finitefrequency wave theory for phase and amplitude attributes is applied in the forward modelling part. In the inversion part, the common problem of poor illumination in crosswell tomography is taken into account. The developed tomographic imaging technique is used on real data in a crosswell experiment with two source wells and two receiver wells (i.e., in total there are four cross sections). Hot steam was injected via a horizontal pipeline going through a reservoir of tar sand during 72 days between the baseline and monitor survey. The time delay and relative amplitude variation observed in the time-lapse data were used separately and jointly to estimate the time-lapse velocity models for all four cross sections. In general, the four cross sections show a negative velocity anomaly close to the pipeline location. This observation is in agreement with rock physics modelling experiments that a temperature increase results in a velocity reduction of heavy oil reservoirs.

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.

Ultrasonic experiments for time‐lapse monitoring of CO2 sequestration

Potential of time-lapse croswell tomography in monitoring CO2 underground storage is tested in a case of specific geological setting – thin coalbed seams. In contrast to tick reservoirs of sandstone aquifers, thin coal seams are difficult to image. In first European pilot project of CO2 injection in thin coal seams (RECOPOL) seismic monitoring was not successful. In laboratory, using ultrasonic measurements we investigate possible reasons and main problems. Furthermore, we test two different theories (ray and finite-frequency wave theory) underlying imaging method and their performance in monitoring of CO2 injection.

Discrimination between phase and amplitude attributes in time-lapse seismic streamer dataTime-lapse seismic monitoring

Abnormally high pore pressures in the subsurface pose a hazard to drilling operations worldwide. The problem is not unusual on the Norwegian Continental Shelf. Knowledge of the pore pressure prior to drilling may reduce the risk related to drilling in high pressure zones. Pore pressure is also a vital paramter for producinig fields, and knowledge of how the pressure develops over time can be important for increased oil recovery. Seismic data contain information on the pore pressure and may contribute to increased understanding of subsurface pressure conditions. The thesis deals with methods for estimation of pressure from seismic velocity and amplitude data.