Beatriz Quintal

Biot's theory of poroelasticity is widely recognized as a fundamental framework for describing wave‐induced fluid flow in mesoscopic heterogeneous porous media. However, dynamic numerical modeling of velocity dispersion and attenuation in heterogeneous porous media remains challenging due to (a) the stiffness of Biot's equations and (b) the wavelength ‐dependent behavior relative to heterogeneity size . The former, arising from the contrasting discretization requirements for fast and slow P waves, has largely been mitigated by semi‐implicit schemes. The latter persists because accurate simulations require periodically repeated representative elementary volumes: when , the domain becomes prohibitively large; when , scattering masks mesoscale fluid‐flow effects. These constraints have led to a reluctant yet often questioned reliance on quasi‐static simulations neglecting inertial effect. We introduce a dimensionless framework enabling dynamic simulations in the same domain by directly controlling the diffusive process, eliminating frequency sweeps and reproducing quasi‐static analytical and numerical solutions while correctly including inertial effects. Our results demonstrate that quasi‐static approximations remain physically valid for modeling mesoscale fluid flow when the total contribution of global flow is very small. These results enable a more flexible and informed selection of modeling and measurement approaches. Meanwhile, our numerical framework can be used to model wave propagation in 3D heterogeneous porous media, accounting for both mesoscopic and global flow effects, which has been impossible until now.

To bridge seismic P- and S-wave velocities and attenuation across the scales, two experimental techniques were found to capture rocks’ dissipative character and complex moduli: (i) the resonant-bar and (ii) the stress–strain forced-oscillation methods. Although none of these methods could directly measure P- and S-wave velocities, their inference had not been questioned: If the fluid-saturated rock was isotropic, any solicitation mode would yield properties that are still isotropic, and only two independent constants are required. In practice, however, reported laboratory data in undrained isotropic sandstones — when three independent complex elastic properties were measured under the same stress states — revealed inconsistencies that challenged this view. Undrained bulk moduli measured from hydrostatic oscillations Kudiso were significantly lower than those inferred from axial oscillations Kudax, although the dry rock showed Kdax∼Kdiso. This discrepancy existed in microcracked sandstones only at lowest effective pressures Peff and was related to anomalously high Poisson’s ratio νud when compared with well-validated inclusion theories. Most reported νud worldwide for microcracked rocks at low Peff highlighted the same inconsistency. Although the well-known concept of “static” stress-induced anisotropy was invalidated by Kdax∼Kdiso, what if a directional measurement induces a “dynamic” stress-induced anisotropy? A fully coupled 3D finite-element model incorporating compliant cracks, with either hydrostatic or axial loading and a simple geometry, was designed to check this question. Although it yielded Kdax∼Kdiso under dry conditions, Kudax was 25% larger than Kudiso under liquid saturation. The undrained mechanical response of an isotropic microcracked rock might have been that of an anisotropic rock at the time of the measurement if the loading method itself was anisotropic.

Seismic wave propagation in fluid-saturated rocks exhibits complex attenuation and dispersion characteristics across a wide frequency range, which are influenced by various mechanisms of energy dissipation. An integrated poroviscoelastic model that combines three primary attenuation processes was introduced: Biot global flow, squirt flow, and the intrinsic viscoelasticity of the solid skeleton. An integrated time-domain Biot-crack approach was developed by incorporating a 1D squirt flow representation within Biot’s framework. The squirt flow mechanism was described using a generalized Zener model, which approximates its core term through a rational function, with coefficients optimized via a genetic algorithm. Furthermore, the model was extended to the poroviscoelastic medium by also incorporating the standard linear solid (Zener) viscoelastic model to account for intrinsic frame losses. This unified methodology effectively identified three distinct attenuation peaks across the frequency spectrum, each corresponding to a specific mechanism: intrinsic frame viscoelasticity at low frequencies, squirt flow at intermediate frequencies, and Biot’s global flow at higher frequencies. The simulations in heterogeneous media demonstrated significant wave attenuation and velocity dispersion effects. This integrated framework enhanced the characterization of wave behavior in fluid-saturated rocks, with important implications for reservoir characterization and seismic interpretation.

Under stress, fluid-saturated porous rocks progressively undergo crack closure, stress accumulation, and crack growth leading to mechanical failure, significantly changing crack density, permeability, and fluid pressurization. The understanding of the stress-dependence of elastic wave propagation in such rocks has great potential applications to infer stress states as well as predicting physical properties such as permeability at depth. It however involves understanding physically the complex coupling processes at play for the observed nonlinear stress-dependence of elastic properties in poroelastic rocks, and finding convenient comprehensive modeling strategies to capture it. In parallel to modeling cracks nonlinear closure or growth, various piecewise modeling approaches such as nonlinear elasticity, hyperelasticity or inelasticity, have been proposed to encompass that complex behavior. Because the nonlinear Padé acoustoporoelasticity theory provides such an unified tensor-based framework to capture these coupling processes, it is of interest to investigate the physical meaning of involved elastic constants for stressed fluid-saturated cracked rocks. This knowledge gap is addressed by experimental validation using low- and high porosity Fontainebleau sandstone samples, saturated with water and glycerin. The results show that (1) the acoustoelastic 3oeCs account for hyperelastic strains due to stress accumulation in the background (comprising grains and stiff pores) and should be estimated from the dry-rock experimental data in the high-pressure regime, (2) Padé coefficients account for strong nonlinear strains due to crack closure, showing a clear power-law correlation with the normalized crack indicator expressed as a function of crack density and aspect ratio, (3) Fitting with the physics underlying Eshelby-type inclusion modeling approaches, Padé coefficients show no dependence to the saturating fluids, and (4) the solid–fluid coupling described by Biot’s poroelasticity is significantly affected by acoustoelastic effects.

Seismic wave propagation in fluid-saturated porous rocks exhibits significant dispersion and attenuation due to pore-scale fluid flow, known as squirt flow. Squirt flow occurs at the microscopic scale and takes place in compliant pores (e.g., grain contacts or microcracks) connected to stiffer pores. Recently, an analytical model for squirt flow in wedge-shaped cracks connected to a stiff pore was developed to investigate wave dispersion and attenuation in isotropic porous rocks. To validate this model, 3D numerical simulations of squirt flow were conducted using the finite element method. The numerical results revealed a significant discrepancy between the published analytical and the present numerical solutions. After modifying the published analytical model, the updated analytical solution showed a good agreement with the numerical solution. The study also analyzed how wedge-shaped crack aperture and crack geometry affect the attenuation and dispersion responses. The results demonstrate that, using an equivalent crack aperture, the penny-shaped crack geometry serves as a physically consistent substitute for the wedge-shaped geometry. Furthermore, in numerical simulations of squirt flow, using penny-shaped geometries presents great advantages compared to using wedge-shaped geometries due to the heavier discretization requirements of the latter.

The seismoelectric effect, characterized by the coupling of seismic and electromagnetic waves in fluid-saturated porous media, offers a promising avenue for subsurface exploration and earthquake seismology. However, the complexity of the Earth’s subsurface, particularly the presence of anisotropic velocity and attenuation, poses challenges for accurate modeling and interpretation. Here, we introduce a novel approach to model seismoelectric wave propagation in viscoelastic anisotropic porous media by incorporating the nearly constant Q model into the seismoelectric wave equations. Using a series of 2D and 3D models, we analyze the influence of both velocity and attenuation anisotropy on the propagation of seismoelectric waves. Our results show that both velocity and attenuation anisotropies contribute to the attenuation of the seismoelectric wavefield. Spatial changes in the elastic and electrical parameters of the subsurface media generate interface response of electromagnetic waves, although these are weaker compared to coseismic electric fields. Our extended seismoelectric models provide a robust description of wave propagation in poro-viscoelastic anisotropic media, which can allow for enhancing the current understanding of the Earth’s interior.

Poroelastic coupling between fractures and the surrounding rock is important to numerous applications in geosciences. We measure the in‐situ fluid pressure and local strain response of a fractured carbonate sample to hydrostatic pressure oscillations. A linear poroelastic model that represents the rock sample is parameterized using X‐ray imaging and ultrasonic wave transmission measurements. The numerical solution, based on Biot's quasistatic equations, is consistent with the measured frequency dependent dispersion of the apparent bulk modulus of the background matrix and the in‐situ pore pressure response, which is caused by fluid pressure diffusion from the compliant fractures into the stiffer matrix. The observed fluid pressure diffusion is causally related to the numerically quantified intrinsic attenuation at seismic frequencies, which is a major contributor to the dissipation of seismic waves. Our analysis supports the use of a simple approximation of fractures as compliant and planar inclusions in numerical simulations based on linear poroelasticity.

Seismic waves propagating in fluid-saturated porous rocks exhibit attenuation and velocity dispersion in a broad range of frequencies. At sonic and ultrasonic frequencies, the attenuation is predominantly caused by fluid flow in cracks and grain contacts, so-called squirt flow. This physical mechanism for attenuation also may be relevant at seismic frequencies. We develop a simple and accurate analytical model for attenuation and dispersion caused by squirt flow in isotropic porous rocks. The input material properties for a specific rock model can be directly measured in a laboratory or calculated using analytical and numerical approaches. The results from our squirt flow model are compared with inherently accurate 3D numerical solutions for the same pore geometries. The analytical and numerical results are in good agreement. Furthermore, we observe that our analytical model is more accurate than the currently available analytical solution for squirt flow in isotropic porous rocks. MATLAB routines to reproduce the presented results are made available.

SUMMARY: Seismic inversion of amplitude versus offset (AVO) data in viscoelastic media can potentially provide high-resolution subsurface models of seismic velocities and attenuation from offset/angle seismic gathers. P- and S-wave quality factors (Q), whose inverse represent a measure of attenuation, depend on reservoir rock and pore fluid properties, in particular, saturation, permeability, porosity, fluid viscosity and lithology; however, these quality factors are rarely taken into account in seismic AVO inversion. For this reason, in this work, we aim to integrate quality factors derived from physically based models in AVO inversion by proposing a gradient descent optimization-based inversion technique to predict the unknown model properties (P- and S-wave velocities, the related quality factors and density). The proposed inversion minimizes the non-linear least-squares misfit with the observed data. The optimal solution is iteratively obtained by optimizing the data misfit using a second-order limited-memory quasi-Newton technique. The forward model is performed in the frequency–frequency-angle domain based on a convolution of broad-band signals and a linearized viscoelastic frequency-dependent AVO (FAVO) equation. The optimization includes the adjoint-state-based gradients with the Lagrangian formulation to improve the efficiency of the non-linear seismic FAVO inversion process. The inversion is tested on synthetic seismic data, in 1-D and 2-D, with and without noise. The sensitivity for seismic quality factors is evaluated using various rock physics models for seismic attenuation and quality factors. The results demonstrate that the proposed inversion method reliably retrieves the unknown elastic and an-elastic properties with good convergence and accuracy. The stability of the inverse solution especially seismic quality factors estimation relies on the noise level of the seismic data. We further investigate the uncertainty of the solution as a function of the variability of the initial models.