The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

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

The state of the art in monitoring and verification—Ten years on

Computational physics has become an essential research and interpretation tool in many fields. Particularly in reservoir geophysics, ultrasonic and seismic modeling in porous media is used to study the properties of rocks and to characterize the seismic response of geologic formations. We provide a review of the most common numerical methods used to solve the partial differential equations describing wave propagation in fluid-saturated rocks, i.e., finite-difference, pseudospectral, and finite-element methods, including the spectral-element technique. The modeling is based on Biot-type theories of dynamic poroelasticity, which constitute a general framework to describe the physics of wave propagation. We explain the various techniques and discuss numerical implementation aspects for application to seismic modeling and rock physics, as, for instance, the role of the Biot diffusion wave as a loss mechanism and interface waves in porous media.

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Computational poroelasticity — A review

Differenttheoreticalandlaboratorystudiesonthepropagation ofelasticwavesinrealrockshaveshownthatthepresenceofheterogeneities larger than the pore size but smaller than the predominant wavelengths mesoscopic-scale heterogeneities may producesignificantattenuationandvelocitydispersioneffectson seismic waves. Such phenomena are known as “mesoscopic effects” and are caused by equilibration of wave-induced fluid pressure gradients.We propose a numerical upscaling procedure to obtain equivalent viscoelastic solids for heterogeneous fluidsaturated rocks. It consists in simulating oscillatory compressibility and shear tests in the space-frequency domain, which enable us to obtain the equivalent complex undrained plane wave and shear moduli of the rock sample. We assume that the behavior of the porous media obeys Biot’s equations and use a finiteelementproceduretoapproximatethesolutionsoftheassociated boundary value problems. Also, because at mesoscopic scales rock parameter distributions are generally uncertain and of stochastic nature, we propose applying the compressibility and sheartestsinaMonteCarlofashion.Thisfacilitatesthedefinition of average equivalent viscoelastic media by computing the moments of the equivalent phase velocities and inverse quality factors over a set of realizations of stochastic rock parameters described by a given spectral density distribution.We analyzed the sensitivity of the mesoscopic effects to different kinds of heterogeneities in the rock and fluid properties using numerical examples.Also,theapplicationoftheMonteCarloprocedureallowed us to determine the statistical properties of phase velocities and inverse quality factors for the particular case of quasi-fractal heterogeneities.

Equivalent viscoelastic solids for heterogeneous fluid-saturated porous rocks

Fil: Rubino, Jorge German. Universite de Lausanne; Suiza. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

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Do seismic waves sense 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.

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Seismoacoustic signatures of fracture connectivity

SUMMARY

Using a numerical approach, we explore wave-induced fluid flow effects in partially saturated porous rocks in which the gas–water saturation patterns are governed by mesoscopic heterogeneities associated with the dry frame properties. The link between the dry frame properties and the gas saturation is defined by the assumption of capillary pressure equilibrium, which in the presence of heterogeneity implies that neighbouring regions can exhibit different levels of saturation. To determine the equivalent attenuation and phase velocity of the synthetic rock samples considered in this study, we apply a numerical upscaling procedure, which permits to take into account mesoscopic heterogeneities associated with the dry frame properties as well as spatially continuous variations of the pore fluid properties. The multiscale nature of the fluid saturation is taken into account by locally computing the physical properties of an effective fluid, which are then used for the larger-scale simulations. We consider two sets of numerical experiments to analyse such effects in heterogeneous partially saturated porous media, where the saturation field is determined by variations in porosity and clay content, respectively. In both cases we also evaluate the seismic responses of corresponding binary, patchy-type saturation patterns. Our results indicate that significant attenuation and modest velocity dispersion effects take place in this kind of media for both binary patchy-type and spatially continuous gas saturation patterns and in particular in the presence of relatively small amounts of gas. The numerical experiments also show that the nature of the gas distribution patterns is a critical parameter controlling the seismic responses of these environments, since attenuation and velocity dispersion effects are much more significant and occur over a broader saturation range for binary patchy-type gas–water distributions. This analysis therefore suggests that the physical mechanisms governing partial saturation should be accounted for when analysing seismic data in a poroelastic framework. In this context, heterogeneities associated with the dry frame properties, which do not play important roles in wave-induced fluid flow processes per se, should be taken into account since they may determine the kind of gas distribution pattern taking place in the porous rock.

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Seismic attenuation and velocity dispersion in heterogeneous partially saturated porous rocks

When a seismic wave travels through a fluid-saturated porous reservoir containing aligned fractures, it induces oscillatory fluid flow between the fractures and the embedding background medium. Although there are numerous theoretical models for quantifying the associated seismic attenuation and velocity dispersion, they rely on certain assumptions, such as infinitesimal fracture thickness and dilute fracture concentration, which usually do not hold in real reservoirs. The objective of this work is to overcome some of these limitations and, therefore, improve the applicability of the available theoretical models. To do so, we extend existing models to the finite fracture thickness case for P-waves propagating perpendicular to the fracture plane using the so-called branching function approach. We consider three types of fractures, namely, periodically- and randomly-spaced planar fractures, as well as penny-shaped cracks. The extended unified model is then tested by comparing with corresponding numerical sim...

Seismic dispersion and attenuation in saturated porous rocks with aligned fractures of finite thickness: Theory and numerical simulations - part 1: P-wave perpendicular to the fracture plane

J. Germán Rubino

Papers

Equivalent viscoelastic solids for heterogeneous fluid-saturated porous rocks

Do seismic waves sense fracture connectivity

Seismoacoustic signatures of fracture connectivity

Seismic attenuation and velocity dispersion in heterogeneous partially saturated porous rocks

Seismic dispersion and attenuation in saturated porous rocks with aligned fractures of finite thickness: Theory and numerical simulations - part 1: P-wave perpendicular to the fracture plane