J. Germán Rubino

Equivalent viscoelastic solids for heterogeneous fluid-saturated porous rocks

Differenttheoreticalandlaboratorystudiesonthepropagation ofelasticwavesinrealrockshaveshownthatthepresenceofheterogeneities larger than the pore size but smaller than the predominant wavelengths mesoscopic-scale heterogeneities may producesignificantattenuationandvelocitydispersioneffectson seismic waves. Such phenomena are known as “mesoscopic effects” and are caused by equilibration of wave-induced fluid pressure gradients.We propose a numerical upscaling procedure to obtain equivalent viscoelastic solids for heterogeneous fluidsaturated rocks. It consists in simulating oscillatory compressibility and shear tests in the space-frequency domain, which enable us to obtain the equivalent complex undrained plane wave and shear moduli of the rock sample. We assume that the behavior of the porous media obeys Biot’s equations and use a finiteelementproceduretoapproximatethesolutionsoftheassociated boundary value problems. Also, because at mesoscopic scales rock parameter distributions are generally uncertain and of stochastic nature, we propose applying the compressibility and sheartestsinaMonteCarlofashion.Thisfacilitatesthedefinition of average equivalent viscoelastic media by computing the moments of the equivalent phase velocities and inverse quality factors over a set of realizations of stochastic rock parameters described by a given spectral density distribution.We analyzed the sensitivity of the mesoscopic effects to different kinds of heterogeneities in the rock and fluid properties using numerical examples.Also,theapplicationoftheMonteCarloprocedureallowed us to determine the statistical properties of phase velocities and inverse quality factors for the particular case of quasi-fractal heterogeneities.

Seismoacoustic signatures of fracture connectivity

Wave-induced fluid flow (WIFF) between fractures and the embedding matrix as well as within connected fractures tends to produce significant seismic attenuation and velocity dispersion. While WIFF between fractures and matrix is well understood, the corresponding effects related to fracture connectivity and the characteristics of the energy dissipation due to flow within fractures are largely unexplored. In this work, we use oscillatory relaxation simulations based on the quasi-static poroelastic equations to study these phenomena. We first consider synthetic rock samples containing connected and unconnected fractures and compute the corresponding attenuation and phase velocity. We also determine the relative fluid displacement and pressure fields in order to gain insight into the physical processes involved in the two manifestations of WIFF in fractured media. To quantify the contributions of the two WIFF mechanisms to the total seismic attenuation, we compute the spatial distribution of the local energy dissipation. Finally, we perform an exhaustive sensitivity analysis to study the role played by different characteristics of fracture networks on the seismic signatures. We show that in the presence of connected fractures both P wave attenuation and phase velocity are sensitive to some key characteristics of the probed medium, notably to the lengths, permeabilities, and intersection angles of the fractures as well as to the overall degree of connectivity of the fracture network. This, in turn, indicates that a deeper understanding of these two manifestations of WIFF in fractured media may eventually allow for the extraction of some of these properties from seismic data.

P-wave seismic attenuation by slow-wave diffusion: Numerical experiments in partially saturated rocks

P-waveattenuationbyslow-wavediffusionisasignificant loss mechanism at seismic frequencies. This effect is known as mesoscopic loss, because it is a consequence of fluid flow at mesoscopic-scale inhomogeneities. These are larger than the pore size but smaller than the wavelength, typically tens of centimeters, and are due to local variations in lithological properties or to patches of immiscible fluids. Basically, a P-wave traveling in a porous medium induces a fluid-pressuregradientinregionsofdifferentproperties,suchaspatches saturated with different fluids, generating slow P-waves, which diffuse away from the interfaces separating the fluids. This mechanism can be explained by the combined effect of mesoscopic-scale inhomogeneities and mode conversion at interfaces. We consider a periodically stratified medium and perform numerical experiments to determine the P-wave quality factor in partially saturated rocks. The modeling method is an iterative domain-decomposition 2D finite-elementalgorithmforsolvingBiotequationsofmotioninaparallel computer, which is a requirement to run the numerical experiments at seismic frequencies. The simulated pulses show evidence of the mesoscopic-loss mechanism, and the quality factors estimated with the spectral-ratio and frequency-shift methods are in good agreement with the theoretical valuespredictedbytheWhitetheory.Errorsintheestimation of the quality factor are less than 5% spectral ratio and 3% frequencyshift.

A viscoelastic representation of wave attenuation in porous media

The theories developed by White and co-workers describe the complex moduli of a medium partially saturated with spherical gas pockets and those of stratified layers composed of two heterogeneous porous media. A generalization to gas patches of arbitrary shape has been given by Johnson. These models represent the mesoscopic-loss mechanism, which is one of the most significant causes of attenuation of seismic waves in reservoir rocks. Comparison of Whites and Johnsons models show that, as the patch shape complexity increases, the patch geometry affects much more the relaxation frequency than it affects the maximum loss. The simulation of synthetic seismograms requires solving Biots differential equations with very small grid spacings, because the loss mechanism involves the conversion of fast P-wave energy to diffusion energy in the form of the Biot slow wave. Because the wavelength of this wave can be very small, the poroelastic solution requires a very large amount of storage and computer time. An efficient approach is to approximate Whites moduli by the Zener model and then solve the single-phase viscoelastic differential equations.

Reflection and transmission of waves in composite porous media: A quantification of energy conversions involving slow waves

This paper studies the reflection and transmission of plane elastic waves at interfaces in fluid-saturated poroviscoelastic media in which the solid matrix is composed of two weakly coupled solids. The analysis of this problem, not formally performed before, is based on a theory recently developed by some of the authors, which allows us to derive expressions for the reflection and transmission coefficients at a plane interface within this kind of media and their relationship with the energy flux (Umov-Poynting) vector. The results of the present derivation were applied to study the energy splitting that takes place when a plane fast compressional wave strikes obliquely an interface defined by a change in ice content within a sample of water saturated partially frozen sandstone. The numerical results show wave mode conversions from fast to slow compressional and shear waves, with maximum energy conversion on the order of 20% from fast to slow wave modes near the critical angle. This phenomenon was observed...

Seismic characterization of thin beds containing patchy carbon dioxide-brine distributions: A study based on numerical simulations

We studied the seismic attenuation and velocity dispersion effects produced by wave-induced fluid flow in weakly consolidated sandstones containing patchy carbon dioxide CO2 ðÞ -brine distributions. The analysis also focuses on the velocity pushdown because of the presence of this gas, as well as on the role of the wave-induced fluid flow (mesoscopic) effects on the amplitude variation with angle (AVA) seismic response of thin layers containing CO2, such as those found at the Utsira Sand, Sleipner field, offshore Norway. We found that this loss mechanism may play a key role on conventional surface seismic data, suggesting that further data analysis may provide useful information on the characteristics of the fluid distributions in these environments. Numerical experiments let us observe that although mesoscopic effects can be very significant in this kind of media, the seismic response of a given isolated thin layer computed considering such effects is very similar to that of a homogeneous elastic thin layer with a compressional velocity equal to that of the original porous rock averaged in the effective data bandwidth. This suggests that the thin-bed prestack spectral inversion method published by the authors could be used to estimate representative compressional velocities and layer thicknesses in these environments. In effect, results using realistic synthetic prestack seismic data show that isolated CO2bearing thin beds can be characterized in terms of their thicknesses and representative compressional velocities. This information can be qualitatively related to CO2 saturations and volumes; thus, the prestack spectral inversion method could find application in the monitoring of the evolution of CO2 plumes at injection sites similar to that at the Sleipner field.

Seismic wave attenuation and dispersion due to wave-induced fluid flow in rocks with strong permeability fluctuations

Oscillatory fluid movements in heterogeneous porous rocks induced by seismic waves cause dissipation of wave field energy. The resulting seismic signature depends not only on the rock compressibility distribution, but also on a statistically averaged permeability. This so-called equivalent seismic permeability does not, however, coincide with the respective equivalent flow permeability. While this issue has been analyzed for one-dimensional (1D) media, the corresponding two-dimensional (2D) and three-dimensional (3D) cases remain unexplored. In this work, this topic is analyzed for 2D random medium realizations having strong permeability fluctuations. With this objective, oscillatory compressibility simulations based on the quasi-static poroelasticity equations are performed. Numerical analysis shows that strong permeability fluctuations diminish the magnitude of attenuation and velocity dispersion due to fluid flow, while the frequency range where these effects are significant gets broader. By comparing the acoustic responses obtained using different permeability averages, it is also shown that at very low frequencies the equivalent seismic permeability is similar to the equivalent flow permeability, while for very high frequencies this parameter approaches the arithmetic average of the permeability field. These seemingly generic findings have potentially important implications with regard to the estimation of equivalent flow permeability from seismic data.

Research note: Seismic attenuation due to wave-induced fluid flow at microscopic and mesoscopic scales

ABSTRACT Wave‐induced fluid flow at microscopic and mesoscopic scales arguably constitutes the major cause of intrinsic seismic attenuation throughout the exploration seismic and sonic frequency ranges. The quantitative analysis of these phenomena is, however, complicated by the fact that the governing physical processes may be dependent. The reason for this is that the presence of microscopic heterogeneities, such as micro‐cracks or broken grain contacts, causes the stiffness of the so‐called modified dry frame to be complex‐valued and frequency‐dependent, which in turn may affect the viscoelastic behaviour in response to fluid flow at mesoscopic scales. In this work, we propose a simple but effective procedure to estimate the seismic attenuation and velocity dispersion behaviour associated with wave‐induced fluid flow due to both microscopic and mesoscopic heterogeneities and discuss the results obtained for a range of pertinent scenarios.

Thin-bed prestack spectral inversion

Prestack seismic data has been used in a new method to fully determine thin-bed properties, including the estimation of its thickness, P- and S-wave velocities, and density. The approach requires neither phase information nor normal-moveout (NMO) corrections, and assumes that the prestack seismic response of the thin layer can be isolated using an offset-dependent time window. We obtained the amplitude-versus-angle (AVA) response of the thin bed considering converted P-waves, S-waves, and all the associated multiples. We carried out the estimation of the thin-bed parameters in the frequency (amplitude spectrum) domain using simulated annealing. In contrast to using zero-offset data, the use of AVA data contributes to increase the robustness of this inverse problem under noisy conditions, as well as to significantly reduce its inherent nonuniqueness. To further reduce the nonuniqueness, and as a means to incorporate a priori geologic or geophysical information (e.g., well-log data), we imposed appropriate bounding constraints to the parameters of the media lying above and below the thin bed, which need not be known accurately. We tested the method by inverting noisy synthetic gathers corresponding to simple wedge models. In addition, we stochastically estimated the uncertainty of the solutions by inverting different data sets that share the same model parameters but are contaminated with different noise realizations. The results suggest that thin beds can be characterized fully with a moderate to high degree of confidence below tuning, even when using an approximate wavelet spectrum.

Numerical upscaling in 2-D heterogeneous poroelastic rocks: Anisotropic attenuation and dispersion of seismic waves

The presence of stiffness contrasts at scales larger than the typical pore sizes but smaller than the predominant seismic wavelengths, can produce seismic attenuation and velocity dispersion in fluid-saturated porous rocks. This energy dissipation mechanism is caused by wave-induced fluid-pressure diffusion among the different components of the probed geological formations. In many cases, heterogeneities have elongated shapes and preferential orientations, which implies that the overall response of the medium is anisotropic. In this work, we propose a numerical upscaling procedure that permits to quantify seismic attenuation and phase velocity considering fluid-pressure diffusion effects as well as generic anisotropy at the samples scale. The methodology is based on a set of three relaxation tests performed on a 2D synthetic rock sample representative of the medium of interest. It provides a complex-valued frequency-dependent equivalent stiffness matrix through a least-squares procedure. We also derive an approach for computing various poroelastic fields associated with the considered sample in response to the propagation of a seismic wave with arbitrary incidence angle. Using this approach, we provide an energy-based estimation of seismic attenuation. A comprehensive numerical analysis indicates that the methodology is suitable for handling complex media and different levels of overall anisotropy. Comparisons with the energy-based estimations demonstrate that the dynamic-equivalent viscoelastic medium assumption made by the numerical upscaling procedure is reasonable even in presence of high levels of overall anisotropy. This work also highlights the usefulness of poroelastic fields for the physical interpretation of seismic wave phenomena in strongly heterogeneous and complex media.