Tobias M. Müller

Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review

One major cause of elastic wave attenuation in heterogeneous porous media is wave-induced flow of the pore fluid between heterogeneities of various scales. It is believed that for frequencies below 1 kHz, the most important cause is the wave-induced flow between mesoscopic inhomogeneities, which are large compared with the typical individual pore size but small compared to the wavelength. Various laboratory experiments in some natural porous materials provide evidence for the presence of centimeter-scale mesoscopic heterogeneities. Laboratory and field measurements of seismic attenuation in fluid-saturated rocks provide indications of the role of the wave-induced flow. Signatures of wave-induced flow include the frequency and saturation dependence of P-wave attenuation and its associated velocity dispersion, frequency-dependent shear-wave splitting, and attenuation anisotropy. During the last four decades, numerous models for attenuation and velocity dispersion from wave-induced flow have been developed with varying degrees of rigor and complexity. These models can be categorized roughly into three groups according to their underlying theoretical framework. The first group of models is based on Biot’s theory of poroelasticity. The second group is based on elastodynamic theory where local fluid flow is incorporated through an additional hydrodynamic equation. Another group of models is derived using the theory of viscoelasticity. Though all models predict attenuation and velocity dispersion typical for a relaxation process, there exist differences that can be related to the type of disorder periodic, random, space dimension and to the way the local flow is incorporated. The differences manifest themselves in different asymptotic scaling laws for attenuation and in different expressions for characteristic frequencies. In recent years, some theoretical models of wave-induced fluid flow have been validated numerically, using finite-difference, finite-element, and reflectivity algorithms applied to Biot’s equations of poroelasticity. Application of theoretical models to real seismic data requires further studies using broadband laboratory and field measurements of attenuation and dispersion for different rocks as well as development of more robust methods for estimating dissipation attributes from field data.

Wave-induced fluid flow in random porous media: attenuation and dispersion of elastic waves.

A detailed analysis of the relationship between elastic waves in inhomogeneous, porous media and the effect of wave-induced fluid flow is presented. Based on the results of the poroelastic first-order statistical smoothing approximation applied to Biots equations of poroelasticity, a model for elastic wave attenuation and dispersion due to wave-induced fluid flow in 3-D randomly inhomogeneous poroelastic media is developed. Attenuation and dispersion depend on linear combinations of the spatial correlations of the fluctuating poroelastic parameters. The observed frequency dependence is typical for a relaxation phenomenon. Further, the analytic properties of attenuation and dispersion are analyzed. It is shown that the low-frequency asymptote of the attenuation coefficient of a plane compressional wave is proportional to the square of frequency. At high frequencies the attenuation coefficient becomes proportional to the square root of frequency. A comparison with the 1-D theory shows that attenuation is of the same order but slightly larger in 3-D random media. Several modeling choices of the approach including the effect of cross correlations between fluid and solid phase properties are demonstrated. The potential application of the results to real porous materials is discussed.

One‐dimensional random patchy saturation model for velocity and attenuation in porous rocks

Porous rocks encountered in hydrocarbon reservoirs are often saturated with a mixture of two or more fluids. Generation of synthetic seismograms as well as interpretation of in-situ attenuation measurements require a theoretical understanding of the relation between the heterogeneous distribution of fluid patches and the acoustic properties of rocks. Thus, the problem of calculating acoustic properties of rocks saturated with a mixture of two fluids has attracted considerable interest (White, 1975; Murphy, 1982; Gist, 1994; Mavko and Mukerji, 1998; Pride et al., 2004). At the same time, this problem is also interesting from the theoretical point of view because partially saturated rocks represent a particularly interesting situation when the effects of dynamic poroelasticity may be significant at seismic or sonic frequencies. Indeed, it is a radical departure from the situation with a porous material fully saturated with a single fluid. Such a fully saturated material exhibits frequency-dependent effects only at frequencies comparable with Biot9s characteristic frequency (Bourbie et al., 1987) ω c = ηϕ/κρ f , where ρ f is the fluid density, η is fluid viscosity, κ is permeability, and ϕ is porosity. For frequencies much lower than ω c , the dynamic effects can be ignored and Gassmann theory applies.

Dynamic poroelasticity of thinly layered structures

Abstract Compressional seismic P -waves, propagating in poroelastic, fluid saturated, laminated sediments are strongly affected by the medium heterogeneity. Here, simple analytical expressions for the P -wave phase velocity and attenuation coefficient are derived. Both are functions of frequency and statistical medium parameters such as correlation lengths and variances. The theoretical results are compared with results from numerical simulations and show good agreement. In heterogeneous media, impedance fluctuations lead to poroelastic scattering ; variations of the layer compressibilities cause inter-layer flow (a 1-D macroscopic local flow) . From the seismic frequency range (10–100 Hz) up to ultrasonic frequencies, attenuation due to heterogeneity is strongly enhanced compared to homogeneous Biot models. The new theory automatically includes different asymptotic approximations, such as poroelastic Backus averaging in the quasi-static and the no-flow limit, geometrical optics, and intermediate frequency ranges.

Direct laboratory observation of patchy saturation and its effects on ultrasonic velocities

Maximizing the recovery of known hydrocarbon reserves is one of the biggest challenges facing the petroleum industry today. Optimal production strategies require accurate monitoring of production-induced changes of reservoir saturation and pressure over the life of the field. Time-lapse seismic technology is increasingly used to map these changes in space and time. However, until now, interpretation of time-lapse seismic data has been mostly qualitative. In order to allow accurate estimation of the saturation, it is necessary to know the quantitative relationship between fluid saturation and seismic characteristics (elastic moduli, velocity dispersion, and attenuation). The problem of calculating acoustic properties of rocks saturated with a mixture of two fluids has attracted considerable interest (Gist, 1994; Mavko and Nolen-Hoeksema, 1994; Knight et al., 1998. For a comprehensive review of theoretical and experimental studies of the patchy saturation problem see Toms et al., 2006).

Seismic signatures of permeability in heterogeneous porous media

In homogeneous poroelastic systems, the permeability tensor practically does not influence propagating seismic waves in the low frequency range (0–1000 Hz; see, e.g., Schmitt, 1989; Gelinsky and Shapiro, 1996). In this paper, we show that this situation changes in heterogeneous systems such as, layered or fractured sediments. Due to the heterogeneities of poroelastic structures, the attenuation of P-waves is influenced by the permeability in an enhanced way. We show, however, that such a “seismic permeability” can differ very strongly from the hydraulic permeability.

Finite-difference modeling of wave propagation and diffusion in poroelastic media

Numerical modeling of seismic waves in heterogeneous, porous reservoir rocks is an important tool for interpreting seismic surveys in reservoir engineering. Various theoretical studies derive effective elastic moduli and seismic attributes from complex rock properties, involving patchy saturation and fractured media. To confirm and further develop rock-physics theories for reservoir rocks, accurate numerical modeling tools are required. Our 2D velocity-stress, finite-difference scheme simulates waves within poroelastic media as described by Biot’s theory. The scheme is second order in time, contains high-order spatial derivative operators, and is parallelized using the domain-decomposition technique. A series of numerical experiments that are compared to exact analytical solutions allow us to assess the stability conditions and dispersion relations of the explicit poroelastic finite-differ-ence method. The focus of the experiments is to model wave-induced flow accurately in the vicinity of mesoscopic hete...

A first-order statistical smoothing approximation for the coherent wave field in random porous media

An important dissipation mechanism for waves in randomly inhomogeneous poroelastic media is the effect of wave-induced fluid flow. In the framework of Biot’s theory of poroelasticity, this mechanism can be understood as scattering from fast into slow compressional waves. To describe this conversion scattering effect in poroelastic random media, the dynamic characteristics of the coherent wavefield using the theory of statistical wave propagation are analyzed. In particular, the method of statistical smoothing is applied to Biot’s equations of poroelasticity. Within the accuracy of the first-order statistical smoothing an effective wave number of the coherent field, which accounts for the effect of wave-induced flow, is derived. This wave number is complex and involves an integral over the correlation function of the medium’s fluctuations. It is shown that the known one-dimensional (1-D) result can be obtained as a special case of the present 3-D theory. The expression for the effective wave number allows ...

Anisotropic dispersion and attenuation due to wave‐induced fluid flow: Quasi‐static finite element modeling in poroelastic solids

[1] Heterogeneous porous media such as hydrocarbon reservoir rocks are effectively described as anisotropic viscoelastic solids. They show characteristic velocity dispersion and attenuation of seismic waves within a broad frequency band, and an explanation for this observation is the mechanism of wave-induced pore fluid flow. Various theoretical models quantify dispersion and attenuation of normal incident compressional waves in finely layered porous media. Similar models of shear wave attenuation are not known, nor do general theories exist to predict wave-induced fluid flow effects in media with a more complex distribution of medium heterogeneities. By using finite element simulations of poroelastic relaxation, the total frequency-dependent complex stiffness tensor can be computed for a porous medium with arbitrary internal heterogeneity. From the stiffness tensor, velocity dispersion and frequency-dependent attenuation are derived for compressional and shear waves as a function of the angle of incidence. We apply our approach to the case of layered media and to that of an ellipsoidal poroelastic inclusion. In the case of the ellipsoidal inclusion, compressional and shear wave modes show significant attenuation, and the characteristic frequency dependence of the effect is governed by the spatiotemporal scale of the pore fluid pressure relaxation. In our anisotropic examples, the angle dependence of the attenuation is stronger than that of the velocity dispersion. It becomes clear that the spatial attenuation patterns show specific characteristics of wave-induced fluid flow, implying that anisotropic attenuation measurements may contribute to the inversion of fluid transport properties in heterogeneous porous media.

Anisotropic P-SV-wave dispersion and attenuation due to inter-layer flow in thinly layered porous rocks

Elastic upscaling of thinly layered rocks typically is performed using the established Backus averaging technique. Its poroelastic extension applies to thinly layered fluid-saturated porous rocks and enables the use of anisotropic effective medium models that are valid in the low- and high-frequency limits for relaxed and unrelaxed pore-fluid pressures, respectively. At intermediate frequencies, wave-induced interlayer flow causes attenuation and dispersion beyond that described by Biot’s global flow and microscopic squirt flow. Several models quantify frequency-dependent, normal-incidence P-wave propagation in layered poroelastic media but yield no prediction for arbitrary angles of incidence, or for S-wave-induced interlayer flow. It is shown that generalized models for P-SV-wave attenuation and dispersion as a result of interlayer flow can be constructed by unifying the anisotropic Backus limits with existing P-wave frequency-dependent interlayer flow models. The construction principle is exact and is ...