Henk Keers

Estimation of reservoir properties using transient pressure data: An asymptotic approach

An asymptotic formulation of the inverse problem for flow reveals that the inversion may be partitioned into two complementary subproblems. In the first problem the arrival time associated with the peak slope of the transient curve is directly related to reservoir properties. The second inverse problem is similar to current methods for interpreting flow data; the transient head amplitudes are related to reservoir storage and conductivity. The first subproblem, the arrival time inversion, involves much less computation than does amplitude matching. Furthermore, it appears to be more robust with respect to the starting model. Therefore the solution to the arrival time inversion provides a starting model for amplitude matching. The methodology is particularly suited to the analysis of observations from well tests. We apply the approach to observations from two interference tests conducted at the Borehole Test Facility in Oklahoma. Using the transient pressure measurements, we image a shallow conductive fracture. The existence and location of the fracture has been verified by both geophysical and borehole data. In particular, core from a slant well contains an open, vertical fracture which coincides with our conductive feature.

Acoustic crosswell imaging using asymptotic waveforms

Seismic waveforms are inverted using an asymptotic method. The asymptotic method models amplitudes correctly at the caustics and takes nonstationary raypaths into account when computing the waveforms, and thus is an extension of geometrical ray theory. Using numerical differencing, partial derivatives of the data with respect to the model are computed. As expected, these partial derivatives (or sensitivity functions) are concentrated along, but not confined to, raypaths. The sensitivity functions enable the formulation of a waveform inversion algorithm, which is applied to a synthetic crosswell experiment and a laboratory crosswell experiment. The synthetic experiment shows the advantages of the waveform inversion method over conventional traveltime inversion methods. Boundaries of anomalies are better defined, and smearing is reduced. The waveform inversion produces a much lower misfit than the traveltime inversion. The goal of the laboratory experiment was the detection of a nonaqueous phase liquid (NAPL) in water saturated sand. The sand was imaged before and after injection of the NAPL. Using the waveform inversion method, low-velocity anomalies were imaged that correlate well with post-experiment determination of NAPL concentrations. The low-velocity anomaly defocuses the seismic energy. However, the amplitude reduction due to the low-velocity anomaly is not enough to explain the observed low amplitudes. We suggest that other mechanisms (such as multiple scattering, 3-D effects, or intrinsic attenuation) not included in the asymptotic waveform modeling play an important role in decreasing the amplitude.

Application of the Maslov Seismogram Method in Three Dimensions

Asymptotic methods provide an efficient way to compute seismograms in heterogeneous media. However, zeroth-order ray theory, the simplest of the asymptotic methods, often fails because of the presence of caustics. Maslov theory is an extension of zeroth-order ray theory, which gives a uniformly valid expression of the wavefield everywhere, including the caustics. This result is given in terms of an integral of ray data over one or two ray parameters. It is shown in this paper how geometrical arrivals are constructed in the one and two-parameter Maslov integrals.In practice Maslov seismograms have been computed using only one ray parameter. However, in three-dimensional media two parameters are needed to uniquely define a ray. In this paper we present an efficient algorithm to compute two-parameter Maslov integrals. The Maslov integral is evaluated by computing the frequency-to-time Fourier transform prior to integration over the ray parameters. The wavefield is then discretized by smoothing with a boxcar function. The resulting expression, which only requires the results of ordinary kinematic and dynamic ray tracing, cen be computed efficiently and robustly. A numerical example is given that illustrates the use of this algorithm.

Zeroth-order asymptotics: Waveform inversion of the lowest degree

Sensitivity computation is an integral part of many waveform inversion algorithms. An accurate and efficient technique for sensitivity computation follows from the zero‐order asymptotic solution to the elastodynamic equation of motion. Given the particular form of the asymptotic solution, we show that perturbations in high‐frequency waveforms are primarily sensitive to perturbations in phase. The resulting expression for waveform sensitivity is the time derivative of the synthetic seismogram multiplied by the phase sensitivity. All of the necessary elements for a step in the waveform inversion algorithm result from a single forward simulation. A comparison with sensitivities calculated using a purely numerical perturbation technique demonstrates that zero‐order sensitivities are accurate. Based upon the methodology, we match 330 waveforms from a crosswell experiment at a bacterial transport site near Oyster, Virginia. Each iteration of the waveform inversion takes approximately 18 minutes of CPU time on a...

Ambient noise levels and detection threshold in Norway

Ambient seismic noise is caused by a number of sources in specific frequency bands. The quantification of ambient noise makes it possible to evaluate station and network performance. We evaluate noise levels in Norway from the 2013 data set of the Norwegian National Seismic Network as well as two temporary deployments. Apart from the station performance, we studied the geographical and temporal variations, and developed a local noise model for Norway. The microseism peaks related to the ocean are significant in Norway. We, therefore, investigated the relationship between oceanic weather conditions and noise levels. We find a correlation of low-frequency noise (0.125–0.25 Hz) with wave heights up to 900 km offshore. High (2–10 Hz) and intermediate (0.5–5 Hz) frequency noise correlates only up to 450 km offshore with wave heights. From a geographic perspective, stations in southern Norway show lower noise levels for low frequencies due to a larger distance to the dominant noise sources in the North Atlantic. Finally, we studied the influence of high-frequency noise levels on earthquake detectability and found that a noise level increase of 10 dB decreases the detectability by 0.5 magnitude units. This method provides a practical way to consider noise variations in detection maps.

Acoustic Waveform Inversion for Ocean Turbulence

AbstractThe seismic oceanography method is based on extracting and stacking the low-frequency acoustic energy scattered by the ocean heterogeneity. However, a good understanding on how this acoustic wavefield is affected by physical processes in the ocean is still lacking. In this work an acoustic waveform modeling and inversion method is developed and applied to both synthetic and real data. In the synthetic example, the temperature field is simulated as a homogeneous Gaussian isotropic random field with the Kolmogorov–Obukhov spectrum superimposed on a background stratified ocean structure. The presented full waveform inversion method is based on the ray-Born approximation. The synthetic seismograms computed using the ray-Born scattering method closely match the seismograms produced with a more computationally expensive finite-difference method. The efficient solution to the inverse problem is provided by the multiscale nonlinear inversion approach that is specifically stable with respect to noise. Full...

Hybrid Ray-born and Finite-difference Full Waveform Inversion

In this contribution we present a hybrid method for full seismic waveform inversion incorporating the ray-Born and finite-difference methods. We also compute synthetic seismograms using both Methods for several random seismic velocity models and perform a comparison of the pressure waveforms. The main motivation in this contribution is to show that the ray-Born modeling method, which is less computationally expensive, is accurate enough to provide an efficient alternative to purely numerical methods.

QLg wave tomography beneath Norway

The propagation of seismic waves is influenced by changes in crustal structure as for example the transition from continental to oceanic crust along the Norwegian margin. We analyzed Lg wave propagation to map lateral crustal changes in Norway and adjacent areas. We used 1369 observations from 279 earthquakes recorded mostly by the Norwegian National Seismic Network between 1990 and 2017. First, we classified Lg wave propagation in terms of efficiency through Lg/Pn ratios and found significant changes between ray paths crossing offshore and onshore areas. Then we derived an average QLg(f) = 529 f0.42 model for Norway, which is in the expected range for a stable tectonic environment. This was used as starting model for a tomographic inversion. We present tomographic models of Lg wave attenuation at frequencies 2 Hz, 4 Hz, and 6 Hz, respectively. We observed the most significant variation between offshore and onshore regions. This can be explained by changes in crustal structure and the occurrence of unconsolidated sediments in the offshore areas.

Travel time computations using a compact eikonal equation for vertical transverse isotropic media: New VTI eikonal equation

Eikonal solvers often have stability problems if the velocity model is mildly heterogeneous. We derive a stable and compact form of the eikonal equation for P-wave propagation in vertical transverse isotropic media. The obtained formulation is more compact than other formulations and therefore computationally attractive. We implemented ray shooting for this new equation through a Hamiltonian formalism. Ray tracing based on this new equation is tested on both simple as well as more realistic mildly heterogeneous velocity models.We show through examples that the new equation gives travel times that coincide with the travel time picks from wave equation modelling for anisotropic wave propagation.

Resources for Computational Geophysics Courses

An important skill that students in solid Earth physics need to acquire is the ability to write computer programs that can be used for the processing, analysis, and modeling of geophysical data and phenomena. Therefore, this skill (which we call “computational geophysics”) is a core part of any undergraduate geophysics curriculum. In this Forum, we share our personal experience in teaching such a course.