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We develop a numerical algorithm for simulation of wave propagation in linear nonisothermal poroelastic media, based on Biot theory and a generalized Fourier law of heat transport in analogy with Maxwell model of viscoelasticity. A plane wave analysis indicates the presence of the classical P and S waves and two slow waves, namely, the Biot and the thermal slow modes of propagation, which present diffusive behavior under certain conditions, depending on viscosity, frequency, and the thermoelastic constants. The wavefield is computed with a direct meshing method using the Fourier differential operator to calculate the spatial derivatives. We propose two alternative time‐stepping algorithms, namely, a first‐order explicit Crank‐Nicolson method and a second‐order splitting method. The Fourier differential operator provides spectral accuracy in the calculation of the spatial derivatives. Modeling the thermal diffusive mode is relevant for high‐temperature high‐pressure fields and since it leads to mesoscopic attenuation by mode conversion of the fast waves to the thermal waves. Key Points: The differential equations of dynamic thermoporoelasticity are solved with a direct grid methodThe solver is based on the Fourier pseudospectral method to compute the spatial derivativesThe simulation reveals the behavior of four wave modes, namely, the fast P, slow P, S, and the thermal wave-diffusion field [ABSTRACT FROM AUTHOR]

We provide a new derivation of the Kramers-Kronig relations on the basis of the Sokhotski-Plemelj equation with detailed mathematical justifications. The relations hold for a causal function, whose Fourier transform is regular (holomorphic) and square-integrable. This implies analyticity in the lower complex plane and a Fourier transform that vanishes at the high-frequency limit. In viscoelasticity, we show that the complex and frequency-dependent modulus describing the stiffness does not satisfy the relation but the modulus minus its high-frequency value does it. This is due to the fact that despite its causality, the modulus is not square-integrable due to a non-null instantaneous response. The relations are obtained in addition for the wave velocity and attenuation factor. The Zener, Maxwell, and Kelvin-Voigt viscoelastic models illustrate these properties. We verify the Kramers-Kronig relations on experimental data of sound attenuation in seabottoms sediments. [ABSTRACT FROM AUTHOR]

We present a detailed approach to implement a moment-tensor point source to compute displacements, particle velocities and accelerations using direct grid methods. Here, the wave modeling algorithm is based on pseudospectral methods to compute the partial derivatives. A comparison to the analytical solution in the 3D acoustic case verifies the discrete implementation of the source in the mesh. Then, the more general 3D elastic case is illustrated and simulations, with and without free surface, are performed that can be used as a reference solution for other grid methods. [ABSTRACT FROM PUBLISHER]

In designing the optics of an imaging multi-etalon spectro polarimeter as a post-focus instrument for the Advanced Technology Solar Telescope (ATST), many constraints must be considered. Among these are the large entrance pupil diameter of the telescope (4 m), the demanded large field of view (≥90 arc sec), high spectral resolving power (≥200,000), and limited field-dependent blue-shift of the instrumental profile [≤3 full width at half maximum (FWHM)], which require Fabry-Perot interferometers of large diameter (≥200 mm), lighted by highly collimated beams. This implies large optical elements and long optical paths. Moreover, to use interference pre-filters with a relatively small diameter (≤70 mm) and placed between the interferometers to reduce the inter-reflections in axial-mount, a "pupil adapter" must be included with a further increase of the optical path length. Although a multi-etalon spectropolarimeter works in quasi-monochromatic light, the Fraunhofer lines of interest cover a wide range of wavelengths (850 to 1650 nm), which demands a good chromatic aberration control. A low instrumental polarization (≤0.5%) is also required to allow a high polarimetric precision. Finally, some secondary optical paths are required to perform the initial instrumental setup and to secure the best instrumental performances. A diffraction-limited optical solution for ATST is described that fulfills all the above requirements in a relative small volume. [ABSTRACT FROM AUTHOR]

Laboratory investigations suggest that a precise relationship exists between Poisson's ratio, pore pressure and fluid type. Values of Poisson's ratio for dry samples are significantly smaller than those for fluid-saturated samples. The values are anomalously high for high pore pressure, with the possibility of differentiating between gas-saturated, brine-saturated and oil-saturated porous rocks. The present study considers two overpressure models, based on oil/gas conversion and disequilibrium compaction, to obtain Poisson's ratio versus differential pressure (confining pressure minus pore pressure). The model results are in good agreement with experiments. Poisson's ratio is approximately constant at high differential pressures and increases (decreases) for saturated (dry) rocks at low differential pressures. Fluid type can be determined at all differential pressures from Poisson's ratio. The analysis is extended to the anisotropic case by computing the three Poisson's ratios of a transversely isotropic rock versus differential pressure. While one of them is practically independent of effective pressure, the others increase with increasing pore pressure. Experiments performed on cores under different pressure conditions, and calibration of the models with these data, provide a tool for inverting pore pressure from seismic data. [ABSTRACT FROM AUTHOR]

Modeling acoustic waves generated by a localized source is always vexed by the nagging problem of spurious reflections and wraparound arising when the wavefront reaches the boundary of the numerical mesh. This difficulty may be circumvented by using a very large computational domain, which is very inefficient, or can be tackled by using some kind of absorbing boundary technique, which has not yet found a universally satisfactory solution. In this work, the wave equation is modified by introducing a term that is nonzero only in a narrow strip near the boundary. Then, a splitting technique permits to compute part of the solution analytically (hence, at no computational cost), while an application of Weyl's formula for the exponential of a matrix leads to a second-order accurate scheme that completes the algorithm. An application to SH seismic wave modeling shows that the performance of the present method is competitive with standard ones. Moreover, there is evidence for a potential application to the modeling of wave propagation in porous media, where stiff differential equations arise. [ABSTRACT FROM AUTHOR]

The applicability of the super-slip boundary condition in wind-driven quasigeostrophic ocean circulation models is reexamined. The energy and enstrophy characteristics of the super-slip condition are discussed for the equilibrium state. A model is constructed with super-slip on the western boundary and free slip on the other boundaries. Both linear and nonlinear solutions are presented. Compared to the case with all free-slip boundaries, this new model gives a more energetic and narrower western boundary current, but otherwise the differences are not very great. [ABSTRACT FROM AUTHOR]

Describes a computer method to overcome the difficulties arising in the application of the Marquardt algorithm in fitting a logistic curve to a set measured data. Use of the software Mathematica; Logistic differential equation.

Investigates the wave propagation properties of nonuniform plane waves in an anisotropic conducting material. Dispersion relation for nonuniform TM waves; Complex Umov-poynting theorem; Energy velocity; Time-average electric and magnetic energy densities; Polar representations of the energy velocity for a nonuniform plane wave.

We establish a generalization of the thermoelasticity wave equation to the porous case, including the Lord–Shulman (LS) and Green–Lindsay (GL) theories that involve a set of relaxation times ( τ i , i = 1 , ... , 4 ). The dynamical equations predict four propagation modes, namely, a fast P wave, a Biot slow wave, a thermal wave, and a shear wave. The plane-wave analysis shows that the GL theory predicts a higher attenuation of the fast P wave, and consequently a higher velocity dispersion than the LS theory if τ 1 = τ 2 > τ 3 , whereas both models predict the same anelasticity for τ 1 = τ 2 = τ 3 . We also propose a generalization of the LS theory by applying two different Maxwell–Vernotte–Cattaneo relaxation times related to the temperature increment ( τ 3 ) and solid/fluid strain components ( τ 4 ), respectively. The generalization predicts positive quality factors when τ 4 ≥ τ 3 , and increasing τ 4 further enhances the attenuation. The wavefields are computed with a direct meshing algorithm using the Fourier pseudospectral method to calculate the spatial derivatives and a first-order explicit Crank–Nicolson time-stepping method. The propagation illustrated with snapshots and waveforms at low and high frequencies is in agreement with the dispersion analysis. The study can be useful for a comprehensive understanding of wave propagation in high-temperature high-pressure fields. [ABSTRACT FROM AUTHOR]

Abstract: We obtain the full-wave solution for the wave propagation at the surface of anisotropic media using two spectral numerical modeling algorithms. The simulations focus on media of cubic and hexagonal symmetries, for which the physics has been reviewed and clarified in a companion paper. Even in the case of homogeneous media, the solution requires the use of numerical methods because the analytical Green’s function cannot be obtained in the whole space. The algorithms proposed here allow for a general material variability and the description of arbitrary crystal symmetry at each grid point of the numerical mesh. They are based on high-order spectral approximations of the wave field for computing the spatial derivatives. We test the algorithms by comparison to the analytical solution and obtain the wave field at different faces (stress-free surfaces) of apatite, zinc and copper. Finally, we perform simulations in heterogeneous media, where no analytical solution exists in general, showing that the modeling algorithms can handle large impedance variations at the interface. [Copyright &y& Elsevier]

Abstract: The acoustic behavior of biologic media can be described more realistically using a stress-strain relation based on fractional time derivatives of the strain, since the fractional exponent is an additional fitting parameter. We consider a generalization of the Kelvin-Voigt rheology to the case of rational orders of differentiation, the so-called Kelvin-Voigt fractional-derivative (KVFD) constitutive equation, and introduce a novel modeling method to solve the wave equation by means of the Grünwald-Letnikov approximation and the staggered Fourier pseudospectral method to compute the spatial derivatives. The algorithm can handle complex geometries and general material-property variability. We verify the results by comparison with the analytical solution obtained for wave propagation in homogeneous media. Moreover, we illustrate the use of the algorithm by simulation of wave propagation in normal and cancerous breast tissue. [Copyright &y& Elsevier]

We obtain the wave velocities of clay-bearing sandstones as a function of clay content, porosity and frequency. Unlike previous theories, based simply on slowness and/or moduli averaging or two-phase models, we use a Biot-type three-phase theory that considers the existence of two solids (sand grains and clay particles) and a fluid. The theory, which is consistent with the critical porosity concept, uses three free parameters that determine the dependence of the dry-rock moduli of the sand and clay matrices as a function of porosity and clay content. Testing of the model with laboratory data shows good agreement between predictions and measurements. In addition to a rock physics model that can be useful for petrophysical interpretation of wave velocities obtained from well logs and surface seismic data, the model provides the differential equation for computing synthetic seismograms in inhomogeneous media, from the seismic to the ultrasonic frequency bands. [ABSTRACT FROM AUTHOR]

P-wave conversion to slow diffusion (Biot) modes at mesoscopic (small-scale) inhomogeneities in porous media is believed to be the most important attenuation mechanisms at seismic frequencies. This study considers three periodic thin layers saturated with gas, oil, and water, respectively, a realistic scenario in hydrocarbon reservoirs, and perform finite-element numerical simulations to obtain the wave velocities and quality factors along the direction perpendicular to layering. The results are validated by comparison to the Norris-Cavallini analytical solution, constituting a cross-check for both theory and numerical simulations. The approach is not restricted to partial saturation but also applies to relevant properties in reservoir geophysics, such as porosity and permeability heterogeneities. This paper considers two cases, namely, the same rock skeleton and different fluids, and the same fluid and different dry-rock properties. Unlike the two-layer case (two fluids), the results show two relaxation peaks and the agreement between numerical and analytical solutions is excellent. [ABSTRACT FROM AUTHOR]

Abstract: We present a review of wave propagation at the surface of anisotropic media (crystal symmetries). The physics for media of cubic and hexagonal symmetries has been extensively studied based on analytical and semi-analytical methods. However, some controversies regarding surfaces waves and the use of different notations for the same modes require a review of the research done and a clarification of the terminology. In a companion paper we obtain the full-wave solution for the wave propagation at the surface of media with arbitrary symmetry (including cubic and hexagonal symmetries) using two spectral numerical modeling algorithms. [Copyright &y& Elsevier]

Abstract: 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 White''s and Johnson''s 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 Biot''s 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 White''s moduli by the Zener model and then solve the single-phase viscoelastic differential equations. [Copyright &y& Elsevier]

Hydrocarbon depletion and fluid injection cause compaction and stretching of the reservoir and overburden layers. 4D prestack seismic data can be used to detect these changes because compaction/stretching causes changes in traveltimes and seismic velocities. We show that, by using two different petro-elastic models at varying effective pressures, a good approximation is to assume that the fractional changes in layer thickness, Δ L/ L, and seismic velocity, Δ v/ v, are related by a linear function of Δ L/ L. The slope of this function (the dilation factor, ) is negative and its absolute value generally decreases (shale, low porosity) or increases (sandstone, high porosity) with increasing layer thickness and decreasing effective pressure. The analysis is mainly performed for isotropic deformations. The dilation factor for uniaxial deformations is smaller in absolute value. The dilation factor, which can be calculated from time-lapse data, can be used to predict reservoir compaction/stretching as a function of depth and surface subsidence. [ABSTRACT FROM AUTHOR]

We present an iterative algorithm formulated in the space-frequency domain to simulate the propagation of waves in a bounded poro-viscoelastic rock saturated by a two-phase fluid. The Biot-type model takes into account capillary forces and viscous and mass coupling coefficients between the fluid phases under variable saturation and pore fluid pressure conditions. The model predicts the existence of three compressional waves or Type-I, Type-II and Type-III waves and one shear or S-wave. The Type-III mode is a new mode not present in the classical Biot theory for single-phase fluids. Our differential and numerical models are stated in the space-frequency domain instead of the classical integrodifferential formulation in the space-time domain. For each temporal frequency, this formulation leads to a Helmholtz-type boundary value problem which is then solved independently of the other frequency problems, and the time-domain solution is obtained by an approximate inverse Fourier transform. The numerical procedure, which is first-order correct in the spatial discretization, is an iterative nonoverlapping domain decomposition method that employs an absorbing boundary condition in order to minimize spurious reflections from the artificial boundaries. The numerical experiments showing the propagation of waves in a sample of Nivelsteiner sandstone indicate that under certain conditions the Type-III wave can be observed at ultrasonic frequencies. [ABSTRACT FROM AUTHOR]