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.

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

The theory of the dynamic bulk modulus, K(ω), of a porous rock, whose saturation occurs in patches of 100% saturation each of two different fluids, is developed within the context of the quasi-static Biot theory. The theory describes the crossover from the Biot–Gassmann–Woods result at low frequencies to the Biot–Gassmann–Hill result at high. Exact results for the approach to the low and the high frequency limits are derived. A simple closed-form analytic model based on these exact results, as well as on the properties of K(ω) extended to the complex ω-plane, is presented. Comparison against the exact solution in simple geometries for the case of a gas and water saturated rock demonstrates that the analytic theory is extremely accurate over the entire frequency range. Aside from the usual parameters of the Biot theory, the model has two geometrical parameters, one of which is the specific surface area, S/V, of the patches. In the special case that one of the fluids is a gas, the second parameter is a different, but also simple, measure of the patch size of the stiff fluid. The theory, in conjunction with relevant experiments, would allow one to deduce information about the sizes and shapes of the patches or, conversely, to make an accurate sonic-to-seismic conversion if the size and saturation values are approximately known.

Theory of Frequency Dependent Acoustics in Patchy-Saturated Porous Media

Gas in an unconsolidated sand reservoir encased in shale often results in a dramatic increase in amplitude of the seismic reflection from the shale/gas-sand interface. Unfortunately, reflection amplitude appears not to vary linearly with water (brine) saturation, and thus cannot be used to estimate gas quantity. Previously presented theoretical velocity computations for a Tertiary sedimentary section, which demonstrate that compressional-wave velocity in an unconsolidated gas sand varies nonlinearly with brine saturation, qualitatively agree with laboratory velocity measurements on a sand specimen composed of pure quartz grains. However, significant departure of measured and theoretical velocities at high brine saturation indicates that the technique for partially saturating the sand specimen by flowing a gas-brine mixture through the specimen does not provide a sufficiently uniform distribution. The gas preferentially seeks larger pores. In a subsequent experiment on a specimen composed of spherical glass beads of nearly uniform size, the previous, as well as a modified, fluid injection technique was used. For the latter, brine only was injected into the pore space previously filled with a mixture of gas and brine in nearly equal proportions. This resulted in a more uniform distribution of the gas-brine mixture. For approximately equal brine saturations, this modified technique more » resulted in a measured compressional-wave velocity approximately one-half of the velocity measured for the previously used fluid injection technique. This result implies that if the gas-brine mixture is uniformly distributed in a reservoir, the fluid compressibility is the weighted-by-volume average of the constituent compressibilities. « less

Effect of brine-gas mixture on velocity in an unconsolidated sand reservoir. [Laboratory velocity measurements in unconsolidated sand at various brine saturations]

A major cause of seismic attenuation in fluid-saturated rocks is the flow of the pore fluid induced by the passing wave. At sonic and ultrasonic frequencies, attenuation appears to be dominated by the local (pore-scale) flow between pores of different shapes and orientations. A simple squirt flow model is developed in which all of the parameters can be independently measured or estimated from measurements. The pore space of the rock is assumed to consist of stiff porosity and compliant (or soft) pores present at grain contacts. The effect of isotropically distributed compliant pores is modeled by considering pressure relaxation in a disk-shaped gap between adjacent grains. This derivation gives the complex and frequency-dependent effective bulk and shear moduli of a rock, in which the compliant pores are liquid saturated and stiff pores are dry. The resulting squirt model is consistent with Gassmann’s and Mavko–Jizba equations at low and high frequencies, respectively. The magnitude of attenuation and dis...

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A simple model for squirt-flow dispersion and attenuation in fluid-saturated granular rocks.

[1] The finite element method is used to solve Biot's equations of consolidation in the displacement-pressure (u − p) formulation. We compute one-dimensional (1-D) and two-dimensional (2-D) numerical quasi-static creep tests with poroelastic media exhibiting mesoscopic-scale heterogeneities to calculate the complex and frequency-dependent P wave moduli from the modeled stress-strain relations. The P wave modulus is used to calculate the frequency-dependent attenuation (i.e., inverse of quality factor) and phase velocity of the medium. Attenuation and velocity dispersion are due to fluid flow induced by pressure differences between regions of different compressibilities, e.g., regions (or patches) saturated with different fluids (i.e., so-called patchy saturation). Comparison of our numerical results with analytical solutions demonstrates the accuracy and stability of the algorithm for a wide range of frequencies (six orders of magnitude). The algorithm employs variable time stepping and an unstructured mesh which make it efficient and accurate for 2-D simulations in media with heterogeneities of arbitrary geometries (e.g., curved shapes). We further numerically calculate the quality factor and phase velocity for 1-D layered patchy saturated porous media exhibiting random distributions of patch sizes. We show that the numerical results for the random distributions can be approximated using a volume average of White's analytical solution and the proposed averaging method is, therefore, suitable for a fast and transparent prediction of both quality factor and phase velocity. Application of our results to frequency-dependent reflection coefficients of hydrocarbon reservoirs indicates that attenuation due to wave-induced flow can increase the reflection coefficient at low frequencies, as is observed at some reservoirs.

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Quasi-static finite element modeling of seismic attenuation and dispersion due to wave-induced fluid flow in poroelastic media

Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks

Saturation of porous rocks with a mixture of two fluids has a substantial effect on seismic-wave propagation. In particular, partial saturation causes significant attenuation and dispersion of the propagating waves due to the mechanism of wave-induced fluid-flow. Such flow arises when a passing wave induces different fluid pressures in regions of rock saturated by different fluids. Most models of attenuation and dispersion due to mesoscopic heterogeneities imply that fluid heterogeneities are distributed in a regular way. However, recent experimental studies show that mesoscopic heterogeneities have less idealized distributions and that the distribution itself affects attenuation and dispersion. Based on an approximation for the coherent wavefield in random porous media, we develop a model which assumes a continuous distribution of fluid heterogeneities. As this continuous random media approach assumes that there will be a distribution of different patch sizes, it is expected to be better suited to modelling experimental data. We also show how to relate the random functions to experimentally measurable parameters.

Seismic attenuation in porous rocks with random patchy saturation

Velocity-saturation Relation For Partially Saturated Rocks With Fractal Pore Fluid Distribution

Physical properties of many reservoir rocks can be modelled using the concept of poroelasticity. Many reservoir rocks, in addition to the network of pores, contain larger fractures or cracks. Galvin and Gurevich (2006) solved the single scattering problem for a crack in a poroelastic medium and then estimated the effective properties for a distribution of cracks. However this problem requires the solution of a Fredholm integral equation of the 2 kind which in general has no analytical solution for intermediate frequencies. We propose a simple analytical approximation of this solution using the branching function approach. Quantitative comparison shows good agreement between the two solutions. Our analytical solution exhibits a relaxation peak at a frequency where the fluid diffusion length is of the order of the crack

https://www.math.purdue.edu/~santos/research/fracture_references/1_fractures_seg08.pdf

A simple approximation for seismic attenuation and dispersion in a fluid-saturated porous rock with aligned fractures

Mesoscopic heterogeneities often occur on a mesoscopic scale, that is scale which is greater than pore-scale but less than wavelength scale. Presence of mesoscopic fluid patches in a porous rock may cause a substantial phase velocity dispersion and attenuation. This is a result of wave induced fluid flow, which arises when a passing wave induces spatial gradients in fluid pressure. Attenuation and dispersion arising from mesoscopic heterogeneities is affected by the spatial distribution of saturating fluids. Here we compare theoretical models for attenuation and dispersion which utilize a 3D random and periodic distribution of fluid heterogeneities. In particular, the periodic model proposed by Johnson (2001) is reinterpreted within the context of the random model. Good agreement between estimates of attenuation and phase velocity is obtained showing that with the right choice of parameters Johnson’s model can describe random as well as periodic distribution of fluid patches.

Julianna Toms

Papers

Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks

Seismic attenuation in porous rocks with random patchy saturation

Velocity-saturation Relation For Partially Saturated Rocks With Fractal Pore Fluid Distribution

A simple approximation for seismic attenuation and dispersion in a fluid-saturated porous rock with aligned fractures

Attenuation and Dispersion in Partially Saturated Porous Rock - Random vs Periodic Models