Composition and microstructure of magma bodies from effective medium theory

Compressional and shear-wave velocities from gas hydrate bearing sediments: Examples from the India and Cascadia margins as well as Arctic permafrost regions

SUMMARY Knowledge of the reflectivity of the sediment-covered seabed is of significant importance to marine seismic data acquisition and interpretation as it governs the generation of reverberations in the water layer. In this context pertinent, but largely unresolved, questions concern the importance of the typically very prominent vertical seismic velocity gradients as well as the potential presence and magnitude of anisotropy in soft surficial seabed sediments. To address these issues, we explore the seismic properties of granulometric end-member-type clastic sedimentary seabed models consisting of sand, silt, and clay as well as scale-invariant stochastic layer sequences of these components characterized by realistic vertical gradients of the P- and S-wave velocities. Using effective media theory, we then assess the nature and magnitude of seismic anisotropy associated with these models. Our results indicate that anisotropy is rather benign for P-waves, and that the S-wave velocities in the axial directions differ only slightly. Because of the very high P -t oS-wave velocity ratios in the vicinity of the seabed our models nevertheless suggest thatS-wave triplications may occur at very small incidence angles. To numerically evaluate the P-wave reflection coefficient of our seabed models, we apply a frequency-slowness technique to the corresponding synthetic seismic wavefields. Comparison with analytical plane-wave reflection coefficients calculated for corresponding isotropic elastic half-space models shows that the differences tend to be most pronounced in the vicinity of the elastic equivalent of the critical angle as well as in the post-critical range. We also find that the presence of intrinsic anisotropy in the clay component of our layered models tends to dramatically reduce the overall magnitude of the P-wave reflection coefficient as well as its variation with incidence angle.

<p>Observations of P-wave (Vp) and S-wave (Vs) velocities in Antarctic and Greenland ice sheets show a strong decrease of 25% of Vs in their deep parts, while Vp remains approximately constant. The drastic Vs decrease corresponds to the basal “echo free zone”, where large-scale disturbances and strong preferred ice crystal orientation are found. According to Wittlinger and Farra (2014), the low Vs may be due to the presence of unfrozen liquids resulting from pre-melting at grain joints and/or melting of chemical solutions buried in ice. In this contribution we investigate the evolution of seismic velocity anisotropy during deformation of temperate ice by means of microdynamic numerical simulations. Temperate ice is modelled as a two-phase non-linear viscous aggregate constituted by a solid phase (ice polycrystal) and a liquid phase (melt). The viscoplastic full-field numerical approach (VPFFT-ELLE) (Lebensohn and Rollet, 2020) is used to calculate the mechanical response of the two-phase aggregate, which deforms purely by dislocation glide. Viscoplastic deformation is coupled with dynamic recrystallisation processes, such as grain boundary migration, intracrystalline recovery and polygonisation (Llorens et al., 2017), all driven by the reduction of surface and strain energies. The changes in P- and S-wave velocities are calculated with the AEH-EBSD software (Vel et al., 2016) from single crystal stiffness and microstructural measurements of crystal preferred orientations (CPO) during deformation. Regardless the amount of melt and intensity of recrystallisation, all simulations evolve from a fabric defined by randomly oriented c-axes to a c-axis preferred orientation (CPO) distribution approximately perpendicular to the shear plane.  For purely solid aggregates, the results show that the highest Vp and lowest Vs velocities are rapidly aligned with the CPO (at a shear strain of 1), and then evolve to a strong single maximum with progressive deformation. This alignment has been previously predicted in models, experiments and measured in ice core samples. When melt is present, the maximum and minimum seismic velocities are not aligned with the CPO and both Vp and Vs are considerably lower than in cases without melt.  However, if the bulk modulus of ice is assumed for the melt phase, the presence of melt produces a remarkable decrease in S-wave velocity while Vp is maintained constant. These results suggest that the decrease in S-wave velocity observed at the base of ice sheets could be explained by the presence of overpressured melt, which would be unconnected at triple grain junctions in the ice polycrystal.</p><p> </p><p>

Hydrocarbon reservoirs generally are composed of rocks with pores and cracks that are filled with different fluids: gas, water, oil, etc. The fluids filling the pores have a significant impact on the effective characteristics of the reservoir. Methods of Effective Medium Theory (EMT) can simulate the effective elastic properties of porous-fractured media, such as rocks. The practical applications of EMT methods for fractured porous rocks are important for fracking, reservoir property prediction using seismic data and understanding the AVO properties of reservoirs. To date, a huge number of different EMT approaches exist and it is very difficult to choose the best one. Each of the most popular EMT methods for the modeling of effective elastic properties of reservoir rocks, of course, generate different results. However, different conclusions on method’s applicability are drawn from the differences in the experimental and theoretical results. The main goal of the paper is to find the most optimal method of effective medium theory. It can be shown that the T-matrix approach is the most general, modern and adequate for the rock physics applications. A practical application of the T-matrix approach was developed for the inverse problem with a special choice of comparison body. Also, other theories exist for simulation the effective proprieties of rocks: poroelasticity, contact models etc. The EMT has a close connection with these theories. A connection between the Effective Medium Theory and Poroelasticity are shown. Our conclusions are verified on experimental data. As an example of practical utilization of the T-matrix approach mathematical models for the effective elastic properties of carbonate reservoir rocks including oolitic limestone are constructed.

A deterministic technique is described that predicts the dynamic (i.e., frequency-dependent) effective properties of inhomogeneous media. Situations are considered in which a host medium sustaining a distribution of spherical inhomogeneities is either fluid or solid. In the Rayleigh region (viz., a/λ < 1) in which most effective medium theories (EMTs) hold, inhomogeneities of any shape can be assumed spherical without much error. However, assuming them to be penetrable implies that they realistically admit interior fields, which are coupled to the exterior ones by stress and displacement boundary conditions. The outcome of dynamic EMTs is a set of expressions to predict the frequency dependence of the effective (sound) wave speed c̄eff, and the effective (sound) wave attenuation ᾱeff in the mixture. These quantities emerge from the real and imaginary parts of the effective wavenumber k̄eff, which are connected by Kramers-Kronig type relations, as has been discussed by Dr. Beltzer. An important consideration for the practical usefulness of an EMT is its simplicity. Often, the resulting predictive expressions are so cumbersome that they can only be evaluated by means of quite formidable computer codes. The principal EMT to the discussed here, yields closed-form simple predictive expressions for c̄eff and ᾱeff, which, in many instances, are valid over significantly broad frequency bands. These expressions reduce to (and contain) almost all existing earlier results for static (f→0) cases. The (present form of this) EMT is constructed by exploiting the presence of the monopole resonance of the inhomogeneities, and thus it works best in the cases in which this resonance is dominant. This EMT has to be further refined to account for cases where the dipole resonance is dominant. In this regard, Dr. Martinez, in a later lecture, will discuss an approach to more accurately deal with heavy/rigid inclusions in a rubberlike matrix. [Work supported by NSWC and ONR.]

The low porosity and permeability shale are nowadays known as self-resourcing reservoirs. In the unique organic shales, TOC has a signifi cant contribution to the elastic properties of rocks. TOC behaves like porosity to a density log and effects in decreasing density. To reduce the uncertainty due to TOC and mineral variability effect, a quantitative interpretation of shale reservoirs should be done properly to obtain the best image of shale systems. In this study, we built rock-physics templates (RPT) to esti mate seismic response by defi ning the relationship between total organic carbon (TOC) and effective elastic properties of shale reservoirs of a data set from South Sumatera Basin, Indonesia. RPT is carried out by incorporating the amount of organic matter into shale pore space as a solid-fi lling inclusion. Moreover, shale porosity is assumed to be fully water-saturated determined by the in-situ conditions. We have estimated the general distribution of pore geometry by investigating aspect ratio from the dataset. A solid background of shale from several different minerals is estimated by using effective medium theory. Properties of porous rocks for solid pore infi ll are estimated from a generalization of Brown-Korringa Equation. Effective elastic properties of bulk rock frame fi lled with a fl uid are obtained from Gassmann equations. Results show that increasing the TOC volumes generally reduces both P-wave and S-wave velocities, acoustic impedance, and density. On the contrary, the vp/vs ratio increased as the impact of immature organic matter which will be more affecting shale rigidity than its compressibility.

There are ambiguities and uncertainties in the recognition of gas hydrate seismic reflections and in quantitative predictions of physical information of natural gas hydrate reservoirs from seismic data. Rock physical modelling is a bridge that transforms the seismic information of geophysical observations into physical information, but traditional rock physics models lack descriptions of reservoir micro-structures and pore-filling materials. Considering the mineral compositions and pore micro-structures of gas hydrates, we built rock physical models for load-bearing and pore-filling gas hydrate-bearing sediments, describe the mineral compositions, pore connectivity and pore shape using effective media theory, calculated the shear properties of pore-filling gas hydrates using Patchy saturation theory and Generalized Gassmann theory, and then revealed the quantitative relation between the elastic parameters and physical parameters for gas hydrate-bearing sediments. The numerical modelling results have shown that the ratios of P-wave and S-wave velocities decrease with hydrate saturation, the P-wave and S-wave velocities of load-bearing gas hydrate-bearing sediments are more sensitive to hydrate saturation, sensitivity is higher with narrower pores, and the ratios of the P-wave and S-wave velocities of pore-filling gas hydrate-bearing sediments are more sensitive to shear properties of hydrates at higher hydrate saturations. Theoretical analysis and practical application results showed that the rock physical models in this paper can be used to calculate the quantitative relation between macro elastic properties and micro physical properties of gas hydrate-bearing sediments, offer shear velocity information lacking in well logging, determine elastic parameters that have more effective indicating abilities, obtain physical parameters such as hydrate saturation and pore aspect ratios, and provide a theoretical basis and practical guidance for gas hydrate quantitative predictions.

By integrating Gassmann equations with the dry rock frame model, which is called the DEM analytical model, derived from the differential effective medium theory, this paper presents a method for pore structure evaluation and shear wave velocity prediction from the effective pore aspect ratios inverted from P‐wave (and S‐wave) velocity. First, we build a link among P‐wave and S‐wave velocity, density, porosity, saturation and mineral compositions through Gassmann equations and the DEM analytical model. Then, by assuming that the porous rock contains spheroidal pores with a single aspect ratio, we apply nonlinear global optimization algorithm to find the best estimate for the effective pore aspect ratio in terms of minimizing the error between theoretical predictions and experimental measurements. Finally, we estimate the S‐wave velocity from the inverted effective pore aspect ratios. The experimental and well logging examples show that the inverted aspect ratio can be used to accurately indicate the pore structure of the reservoir. For fracture reservoirs, such as granite, its pore aspect ratio is typically less than 0.025, and for pore reservoirs such as sandstone, the pore aspect ratio is usually greater than 0.08. Comparison of the inversion of pore aspect ratio between only from P‐wave and from both P‐wave and S‐wave indicates that the pore aspect ratios from these two methods are almost the same. And, the predicted shear‐velocity only from P‐wave is in good agreement with the measured one. These results show that our method of the shear‐velocity prediction by using the pore aspect ratio inverted from P‐wave is effective.

ABSTRACTThe effective medium theory based on the Hertz–Mindlin contact law is the most popular theory to relate dynamic elastic moduli (or elastic velocities) and confining pressure in dry granular media. However, many experimental results proved that the effective medium theory predicts pressure trends lower than experimental ones and over‐predicts the shear modulus. To mitigate these mispredictions, several evolutions of the effective medium theory have been presented in the literature. Among these, the model named modified grain contact theory is an empirical approach in which three parametric curves are included in the effective medium theory model. Fitting the parameters of these curves permits to adjust the pressure trends of the Poisson ratio and the bulk modulus. In this paper, we present two variations of the modified grain contact theory model. First, we propose a minor modification in the fitting function for the porosity dependence of the calibration parameters that accounts for non‐linearity in the vicinity of the critical porosity. Second, we propose a major modification that reduces the three‐step modified grain contact theory model to a two‐step model, by skipping the calibration parameter–porosity fit in the model and directly modelling the calibration parameter–pressure relation. In addition to an increased simplicity (the fitting parameters are reduced from 10 to 6), avoiding the porosity fit permits us to apply the model to laboratory data that are not provided with accurate porosity measurements. For this second model, we also estimate the uncertainty of the fitting parameters and the elastic velocities. We tested this model on dry core measurements from literature and we verified that it returns elastic velocity trends as good as the original modified grain contact theory model with a reduced number of fitting parameters. Possible developments of the new model to add predictive power are also discussed.

This paper presents the application of nonempirical effective medium theories to describe composite mixtures of spherical inclusions within a host continuum. It is shown that the most common effective medium theories collapse into Bruggeman's (1935) asymmetric formula when they are implemented in an iterative scheme to extend their validity to higher volume fractions. Comparisons of DC and 4-GHz data show that of all the formulas Bruggeman's asymmetric formula corresponds best with experiment for large differences between the complex permittivities of the host and inclusion materials. Permeability values are also formulated and compared with experiment and a simple scheme is considered to extend the effective medium theories herein to a description of the diamagnetic effect of induced current in metal spherical inclusions.

The present paper consists in two parts, determined by the historical emerging production of Dipole Sonic Imager (DSI) measurements and results in the early 1990’s. The DSI data were processed following two methods simultaneously developed in France and in USA by Schlumberger. In the first part the early dipole sonic S-wave velocity results obtained in late 1993 are confronted with the other borehole data obtained in the scientific borehole MM-1, entirely cored and extensively logged, as part of the comprehensive scientific project named Géologie Profonde de la France (GPF), conducted by the Bureau de Recherches Géologiques et Minières (BRGM, i.e. the French Geological Survey), in Ardèche, southern France. In 1994, José Perrin summarized and integrated all the borehole information including the preliminary results from an azimuthal “rotation scan” of S-wave sonic slowness determination method quickly developed in Schlumberger-France and aiming at detecting only the presence of S-wave velocity anisotropy in a first step. The initial results were presented to the French industrial logging community in April 1994, prior to the commercialization of any S-wave splitting computer detection routine applied to dipole sonic data. The second part focuses on the comparison of the dipole sonic S-wave anisotropy detection results from two methods produced at a later time by Schlumberger, namely: a) results from the commercial S-wave anisotropy detection routine based on cross energy minimization, obtained in October 1994, and b) principal S-wave azimuth results sorted from the “rotation scan” azimuthal method, produced in 1995 and further improved in July 1997. After discussing the discrepancies of the principal fast S-wave azimuth derived from the two methods with diverse specialists in Schlumberger, over several years, and on a spare time basis, the authors expose constructive explanations in the present paper. A limited overview of the latest dipole sonic data processing developments has also been attempted to better understand the differing S-wave birefringence results obtained in MM-1, suggesting that the rock formation in the immediate borehole vicinity, up to three times the borehole radius, may not be homogeneous along the borehole depth depending on the local geological context. Besides, the Fast Azimuth split S-wave (FAZ) fits with the strike of major regional faults and parallel to the maximal horizontal palaeo-stress, which happens to be nearly orthogonal to the local present stress direction accepted by the geologists! The present case study suggests that the S-wave anisotropy results ought to become more reliable, mainly on the accuracy and precision of the FAZ. Additionally, the efficiency of the semblance parameter for S-wave attenuation anisotropy detection is pondered, where no S-wave velocity anisotropy is detected over the dipole sonic receiver array.

Accurate and detailed models of the seismic velocity structure of gas hydrate‐bearing sediments may be determined by careful analysis of controlled source seismic data. However, interpretation of these velocities in terms of hydrate saturation of the pore space has hitherto relied on semiempirical formulae and/or simple effective medium theory. We develop a rigorous theoretical scheme to relate the seismic properties of a clay‐rich hydrate‐bearing sediment to its porosity, mineralogy, microstructural features and hydrate saturation. We consider separately the two possible end‐members for the distribution of hydrate in the pore space: (1) hydrates are unconnected and located in the pore voids without appreciable grain contact and (2) connected hydrates are forming cement binding around the grains. The scheme is transversely isotropic, to allow for anisotropy due to alignment of clay platelets, and is based on a combination of a self‐consistent approximation, a differential effective medium theory, and a method of smoothing for crystalline aggregates. We have applied the scheme to lithological and seismic velocity data from Ocean Drilling Program Site 995 on the Blake Ridge (southeastern U.S. continental margin) to make estimates of the hydrate saturation. It was found that the hydrates are probably unconnected, and their volume concentration varies between ∼0% at 100 m below the seabed and ∼9% at 400 m depth, just above the “bottom simulating reflector”, if the clay platelet orientation distribution resembles the function we have used.

The effective medium theory (EMT) proposed by Kirkpatrick (1973) is known to be a useful tool to describe the conducting properties of heterogeneous media. We check the validity of the theory on several probability distribution functions. The accuracy of the theory is inferred by comparing the results given by the EMT to those obtained when using an exact method based on the resolution of two‐dimensional conducting networks. It turns out that the EMT approximation is quite good for quasi‐uniform distributions of conducting elements, whereas large discrepancies appear for highly contrasted distributions such as decreasing exponential functions. The reason is that ensemble averaging and spatial averaging are not equivalent for broad, nonuniform distributions. Therefore the effective medium theory may not be appropriate for the calculation of the transport properties of rocks which exhibit this type of distribution. The nature of the distribution varies, however, depending on the rock type and the length scale of interest. Critical path analysis can be a complementary tool to deal with highly heterogeneous conducting media. The accuracy of such a technique appears to be better than that of the EMT approximation. But it depends also on the representativity of the input data. Finally, an example of data relative to Fontainebleau Sandstone is developed in this paper. It is concluded that for this rock, network theory is the most accurate tool for calculating transport properties from statistically averaged observations on two‐dimensional sections.

This paper presents a thorough examination of rock physics, discipline central to integrating geophysical observation and the underlying physical properties of geological materials. The research addresses systematically how seismic waves interact with rocks, providing valuable information about subsurface lithology, fluid saturation, pore pressure, and stress regime. The approach brings together theoretical models, empirical correlations, and sophisticated computational methods. We use known rock physics models, such as effective medium theories and Gassmann's fluid substitution, together with a detailed consideration of Petrophysical data from well logs and core measurements to comprehend the main controls on seismic properties like porosity, mineralogy, fluid type, and effective stress. Seismic response predictions and estimation of the effect of different parameters are made using forward modeling and sensitivity analyses, with special focus on the diagnostic capability of Poisson's ratio and VP/VS ratio for\ fracture and fluid detection. The paper also discusses micro-scale observations using Digital Rock Physics (DRP), explaining the significant role of pore structure on macroscopic elastic properties. The revolutionary contribution of machine learning and artificial intelligence is also emphasized, showing their effectiveness in predicting petrophysical properties and lithofacies classifying from seismic attributes. While recognizing the progress, the topic covers the existing challenges, such as uncertainty quantification, complex anisotropy, and the requirement for strong time-lapse and geomechanical integration. Finally, this book emphasizes the critical importance of rock physics in minimizing subsurface uncertainty and maximizing decision-making for a wide range of applications in energy, environmental science, and engineering.

Recent progress in micro-scale three-dimensional (3D) characterization techniques such as focused ion beam – scanning electron microscopy (FIB-SEM) and X-ray nanotomography has brought unprecedented opportunities in linking material microstructure to performance and properties. The link between solid oxide fuel cell (SOFC) electrode microstructure and performance is sometimes established by effective medium theory, where effective performance parameters are calculated from a representative volume element (RVE), an adequate volume that statistically represents global microstructure. However, different from performance, degradation and failure have not been addressed by mean parameters from effective medium theory. With the advent of modern 3D characterization techniques, there are new approaches in linking microstructure and degradation. This work aims to emphasize distributions (and outliers), rather than the mean, in better establishing the link between microstructure and degradation, failure, and performance. Using an open source finite element framework (MOOSE, Idaho National Laboratory) on a high performance computing platform (Joule, National Energy Technology Laboratory), a numerical transport and reaction model is applied to microstructural meshes to compute local distributions of electrochemical parameters. The microstructural meshes are obtained from the plasma sourced FIB-SEM characterization of a commercial SOFC electrode. The RVE size with respect to the computed distributions is determined based on examining the distributions from a series of microstructural meshes with various sizes. To validate the distributions, the averaged numbers from the distributions are compared with the effective medium theories in the literature. Finally, features such as the tails of the distributions and the local gradients of the electrochemical parameters are highlighted and discussed.