Marco Milani

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.

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.

Numerical upscaling of frequency-dependent P- and S-wave moduli in fractured porous media: Upscaling of P- and S-wave moduli

Relating seismic attributes to the characteristics of mesoscopic fractures is inherently challenging, yet these heterogeneities tend to dominate the mechanical and hydraulic properties of the medium. Analytical approaches linking the effects of material properties on seismic attributes, such as attenuation and velocity dispersion, tend to be limited to simple geometries, low fracture densities, and/or non-interacting fractures. Furthermore, the influence of fluid flow within interconnected fractures on P-wave and S-wave attenuation is difficult to accommodate in analytical models. One way to overcome these limitations is through numerical upscaling. In this paper, we apply a numerical upscaling approach based on the theory of quasi-static poroelasticity to fluid-saturated porous media containing randomly distributed horizontal and vertical fractures. The inferred frequency-dependent elastic moduli represent the effective behaviour of the underlying fractured medium if the considered sub-volume has at least the size of a representative elementary volume. We adapt a combined statistical and numerical approach originally proposed for elastic composites to explore wether the overall statistical properties of simple fracture networks can be captured by computationally feasible representative-elementary-volume sizes. Our results indicate that, for the considered scenarios, this is indeed possible and thus represent an important first step towards the estimation of frequency-dependent effective moduli of realistic fracture networks.

Fluid pressure diffusion effects on the seismic reflectivity of a single fracture

When seismic waves travel through a fluid-saturated porous medium containing a fracture, fluid pressure gradients are induced between the compliant fracture and the stiffer embedding background. The resulting equilibration through fluid pressure diffusion (FPD) produces a frequency dependence of the stiffening effect of the fluid saturating the fracture. As the reflectivity of a fracture is mainly controlled by the stiffness contrast with respect to the background, these frequency-dependent effects are expected to affect the fracture reflectivity. The present work explores the P- and S-wave reflectivity of a fracture modeled as a thin porous layer separating two half-spaces. Assuming planar wave propagation and P-wave incidence, this article analyzes the FPD effects on the reflection coefficients through comparisons with a low-frequency approximation of the underlying poroelastic model and an elastic model based on Gassmanns equations. The results indicate that, while the impact of global flow on fracture reflectivity is rather small, FPD effects can be significant, especially for P-waves and low incidence angles. These effects get particularly strong for very thin and compliant, liquid-saturated fractures and embedded in a high-permeability background. In particular, this study suggests that in common environments and typical seismic experiments FPD effects can significantly increase the seismic visibility of fractures.

A generalized effective anisotropic poroelastic model for periodically layered media accounting for both Biot's global and interlayer flows: Effective anisotropic poroelastic model

We present a generalized effective poroelastic model for periodically layered media in the mesoscopic scale range, which accounts for both Biot’s global and interlayer wave-induced fluid flow, as well as for the anisotropy associated with the layering. Correspondingly, it correctly predicts the existence of the fast and slow P-waves as well as quasi and pure S-waves. The proposed analytical model is validated through comparisons of the P-wave and S-wave phase velocity dispersion and attenuation characteristics with those inferred from a one-dimensional numerical solution of Biot’s poroelastic equations of motion. We also compare our model with the classical mesoscopic model of White for a range of scenarios. The results demonstrate that accounting for both wave-induced fluid flow mechanisms is essential when Biot’s global flow prevails at frequencies that are comparable or smallerwith respect to those governing interlayer flow. This is likely to be the case in media of high permeability, such as, for example, unconsolidated sediments, clean sandstones, karstic carbonates, or fractured rocks. Conversely, when interlayer flow occurs at smaller frequencies with respect to Biot’s global flow, the predictions of this model are in agreement with White’s model, which is based on quasi-static poroelasticity.

Seismic attenuation and velocity dispersion in fractured rocks: The role played by fracture contact areas

The presence of fractures in fluid-saturated porous rocks is usually associated with strong seismic P-wave attenuation and velocity dispersion. This energy dissipation can be caused by oscillatory wave-induced fluid pressure diffusion between the fractures and the host rock, an intrinsic attenuation mechanism generally referred to as wave-induced fluid flow. Geological observations suggest that fracture surfaces are highly irregular at the millimetre and sub-millimetre scale, which finds its expression in geometrical and mechanical complexities of the contact area between the fracture faces. It is well known that contact areas strongly affect the overall mechanical fracture properties. However, existing models for seismic attenuation and velocity dispersion in fractured rocks neglect this complexity. In this work, we explore the effects of fracture contact areas on seismic P-wave attenuation and velocity dispersion using oscillatory relaxation simulations based on quasi-static poroelastic equations. We verify that the geometrical and mechanical details of fracture contact areas have a strong impact on seismic signatures. In addition, our numerical approach allows us to quantify the vertical solid displacement jump across fractures, the key quantity in the linear slip theory. We find that the displacement jump is strongly affected by the geometrical details of the fracture contact area and, due to the oscillatory fluid pressure diffusion process, is complex-valued and frequency-dependent. By using laboratory measurements of stress-induced changes in the fracture contact area, we relate seismic attenuation and dispersion to the effective stress. The corresponding results do indeed indicate that seismic attenuation and phase velocity may constitute useful attributes to constrain the effective stress. Alternatively, knowledge of the effective stress may help to identify the regions in which wave induced fluid flow is expected to be the dominant attenuation mechanism.