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- Volume 67, Issue 5, 2019
Geophysical Prospecting - Volume 67, Issue 5, 2019
Volume 67, Issue 5, 2019
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Slowness‐domain kinematical characteristics for horizontally layered orthorhombic media. Part I: Critical slowness match
Authors Igor Ravve and Zvi KorenABSTRACTKinematical characteristics of reflected waves in anisotropic elastic media play an important role in the seismic imaging workflow. Considering compressional and converted waves, we derive new, azimuthally dependent, slowness‐domain approximations for the kinematical characteristics of reflected waves (radial and transverse offsets, intercept time and traveltime) for layered orthorhombic media with varying azimuth of the vertical symmetry planes. The proposed method can be considered an extension of the well‐known ‘generalized moveout approximation’ in the slowness domain, from azimuthally isotropic to azimuthally anisotropic models. For each slowness azimuth, the approximations hold for a wide angle range, combining power series coefficients in the vicinity of both the normal‐incidence ray and an additional wide‐angle ray. We consider two cases for the wide‐angle ray: a ‘critical slowness match’ and a ‘pre‐critical slowness match’ studied in Parts I and II of this work, respectively. For the critical slowness match, the approximations are valid within the entire slowness range, up to the critical slowness. For the ‘pre‐critical slowness match’, the approximations are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. The critical slowness match is particularly effective when the subsurface model includes a dominant high‐velocity layer where, for nearly critical slowness values, the propagation in this layer is almost horizontal. Comparing the approximated kinematical characteristics with those computed by numerical ray tracing, we demonstrate high accuracy.
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Slowness‐domain kinematical characteristics for horizontally layered orthorhombic media. Part II: Pre‐critical slowness match
Authors Zvi Koren and Igor RavveABSTRACTPart II of this paper is a direct continuation of Part I, where we consider the same types of orthorhombic layered media and the same types of pure‐mode and converted waves. Like in Part I, the approximations for the slowness‐domain kinematical characteristics are obtained by combining power series coefficients in the vicinity of both the normal‐incidence ray and an additional wide‐angle ray. In Part I, the wide‐angle ray was set to be the critical ray (‘critical slowness match’), whereas in Part II we consider a finite long offset associated with a given pre‐critical ray (‘pre‐critical slowness match’). Unlike the critical slowness match, the approximations in the pre‐critical slowness match are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. Moreover, for the pre‐critical slowness match, there is no need to distinguish between the high‐velocity layer and the other, low‐velocity layers. The form of the approximations in both critical and pre‐critical slowness matches is the same, where only the wide‐angle power series coefficients are different. Comparing the approximated kinematical characteristics with those obtained by exact numerical ray tracing, we demonstrate high accuracy. Furthermore, we show that for all wave types, the accuracy of the pre‐critical slowness match is essentially higher than that of the critical slowness match, even for matching slowness values close to the critical slowness. Both approaches can be valuable for implementation, depending on the target offset range and the nature of the subsurface model. The pre‐critical slowness match is more accurate for simulating reflection data with conventional offsets. The critical slowness match can be attractive for models with a dominant high‐velocity layer, for simulating, for example, refraction events with very long offsets.
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Fast estimation of prestack common reflection surface parameters
ABSTRACTWe present a method for fast estimation of finite offset common reflection surface parameters. Firstly, the derivatives with respect to offset are derived from the velocity guide. Secondly, we apply structure tensors to extract the derivatives with respect to midpoint from stacked common offset sections. Finally, the mixed derivative is estimated using a one‐parametric semblance search. The proposed method is compared to the global five‐parametric semblance search and the pragmatic sequential two‐parametric semblance search on one synthetic and one real data set. The experiments show that the proposed method is more robust against noise than the pragmatic search and have comparable robustness with the global search. The proposed method smoothes parameter estimates in a local window, and the window size is set to give the best trade‐off between detail and robustness. Since the proposed method is dependent on a velocity guide, the quality of the other parameter estimates may be influenced by any inaccuracies in the guide. The main advantage of the proposed method is the computational efficiency. When compared with a gridded implementation of the semblance search, the proposed method is 10 and 400 times faster than the pragmatic and global search. Alternative search strategies significantly reduce the computational cost of the global search. However, since more than 99% of the computational cost of the proposed method comes from the semblance search to estimate the mixed derivative, it is expected that such techniques also reduce the computational cost for the proposed method.
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Pre‐stack Kirchhoff depth migration local space‐shift imaging condition: synthetic and data examples
Authors G.S. O'Brien, S.J. Delaney and M. IgoeABSTRACTExtracting accurate common image angle gathers from pre‐stack depth migrations is important in the generation of any incremental uplift to the amplitude versus angle attributes and seismic inversions that can lead to significant impacts in exploration and development success. The commonly used Kirchhoff migration outputs surface common offset image gathers that require a transformation to angle gathers for amplitude versus angle analysis. The accuracy of this transformation is one of the factors that determine the robustness of the amplitude versus angle measurements. Here, we investigate the possibility of implementing an extended imaging condition, focusing on the space‐lag condition, for generating subsurface reflection angle gathers within a Kirchhoff migration. The objective is to determine if exploiting the spatial local shift imaging condition can provide any increase in angle gather fidelity relative to the common offset image gathers. The same restrictions with a ray‐based approach will apply using the extended imaging condition as both the offset and extended imaging condition method use travel times derived from solutions to an Eikonal equation. The aims are to offer an alternative ray‐based method to generate subsurface angle gathers and to understand the impact on the amplitude versus angle response. To this end, the implementation of the space‐shift imaging condition is discussed and results of three different data sets are presented. A layered three‐dimensional model and a complex two‐dimensional model are used to assess the space shift image gathers output from such a migration scheme and to evaluate the seismic attributes relative to the traditional surface offset common image gathers. The synthetic results show that the extended imaging condition clearly provides an uplift in the measured amplitude versus angle over the surface offset migration. The noise profile post‐migration is also improved for the space‐lag migration due to the double summation inside the migration. Finally, we show an example of a space‐lag gather from deep marine data and compare the resultant angle gathers with those generated from an offset migration and a time‐shift imaging condition Kirchhoff migration. The comparison of the real data with a well log shows that the space‐lag result is a better match to the well compared to the time‐lag extended imaging condition and the common offset Kirchhoff migration. Overall, the results from the synthetics and real data show that a Kirchhoff migration with an extended imaging condition is capable of generating subsurface angle gathers with an incremental improvement in amplitude versus angle fidelity and lower noise but comes at a higher computational cost.
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Curvelet reconstruction of non‐uniformly sampled seismic data using the linearized Bregman method
Authors Hua Zhang, Su Diao, Wei Chen, Guangnan Huang, Hongxing Li and Min BaiABSTRACTSeismic data reconstruction, as a preconditioning process, is critical to the performance of subsequent data and imaging processing tasks. Often, seismic data are sparsely and non‐uniformly sampled due to limitations of economic costs and field conditions. However, most reconstruction processing algorithms are designed for the ideal case of uniformly sampled data. In this paper, we propose the non‐equispaced fast discrete curvelet transform‐based three‐dimensional reconstruction method that can handle and interpolate non‐uniformly sampled data effectively along two spatial coordinates. In the procedure, the three‐dimensional seismic data sets are organized in a sequence of two‐dimensional time slices along the source–receiver domain. By introducing the two‐dimensional non‐equispaced fast Fourier transform in the conventional fast discrete curvelet transform, we formulate an L1 sparsity regularized problem to invert for the uniformly sampled curvelet coefficients from the non‐uniformly sampled data. In order to improve the inversion algorithm efficiency, we employ the linearized Bregman method to solve the L1‐norm minimization problem. Once the uniform curvelet coefficients are obtained, uniformly sampled three‐dimensional seismic data can be reconstructed via the conventional inverse curvelet transform. The reconstructed results using both synthetic and real data demonstrate that the proposed method can reconstruct not only non‐uniformly sampled and aliased data with missing traces, but also the subset of observed data on a non‐uniform grid to a specified uniform grid along two spatial coordinates. Also, the results show that the simple linearized Bregman method is superior to the complex spectral projected gradient for L1 norm method in terms of reconstruction accuracy.
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Study on the full‐waveform inversion strategy for 3D elastic orthorhombic anisotropic media: application to ocean bottom cable data
Authors Ju‐Won Oh and Tariq AlkhalifahABSTRACTThe multi‐parameter full waveform inversion is an essential tool to estimate subsurface anisotropic properties in a reservoir that may have complex geological behaviour requiring an elastic orthorhombic medium description. For such elastic orthorhombic media, finding a proper inversion strategy to mitigate parameter trade‐off and reduce the Null space is crucial considering the large number of parameters describing the medium. We apply our recently developed strategy for orthorhombic medium inversion on synthetic and real ocean bottom cable data, and find the most efficient inversion strategy. At first, we analyse the trade‐off patterns in three elastic orthorhombic parameterizations for a hockey‐puck‐shaped model. We, then, compare the performance of these orthorhombic parameterizations on a 3D synthetic ocean bottom cable data, which are obtained from a channel‐shaped model. By interpreting the inverted models based on analytic radiation patterns of the nine elastic orthorhombic parameters in each parameterization, we observe that parameterizations, which have one P‐wave velocity, one S‐wave velocity and seven dimensionless parameters, can be optimal to recover subsurface isotropic properties in the early stages of inversion. Then, we show that the choice of three anisotropic parameters and their deviations along the horizontal plane helps us mitigate the complexity of the multi‐parameter inversion by decoupling anisotropic features, which have vertical and horizontal symmetric axes, respectively. This decoupling of isotropic and anisotropic properties enables us to perform the multi‐parameter anisotropic inversion in a multi‐stage manner. In addition, for the marine acquisition, we show that the number of parameters can be reduced from 9 to 4, which makes the multi‐parameter inversion more practical. Finally, we apply the elastic orthorhombic inversion to real ocean bottom cable data.
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A new parameterization for generalized moveout approximation, based on three rays
Authors Mohammad Mahdi Abedi and Alexey StovasABSTRACTA simple and accurate traveltime approximation is important in many applications in seismic data processing, inversion and modelling stages. Generalized moveout approximation is an explicit equation that approximates reflection traveltimes in general two‐dimensional models. Definition of its five parameters can be done from properties of finite offset rays, for general models, or by explicit calculation from model properties, for specific models. Two versions of classical finite‐offset parameterization for this approximation use traveltime and traveltime derivatives of two rays to define five parameters, which makes them asymmetrical. Using a third ray, we propose a balance between the number of rays and the order of traveltime derivatives. Our tests using different models also show the higher accuracy of the proposed method. For acoustic transversely isotropic media with a vertical symmetry axis, we calculate a new moveout approximation in the generalized moveout approximation functional form, which is explicitly defined by three independent parameters of zero‐offset two‐way time, normal moveout velocity and anellipticity parameter. Our test shows that the maximum error of the proposed transversely isotropic moveout approximation is about 1/6 to 1/8 of that of the moveout approximation that had been reported as the most accurate approximation in these media. The higher accuracy is the result of a novel parameterization that do not add any computational complexity. We show a simple example of its application on synthetic seismic data.
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3D diffraction imaging of fault and fracture zones via image spectral decomposition of partial images
Authors M.I. Protasov, K.G. Gadylshin, V.A. Tcheverda and A.P. PravduhinABSTRACTThe paper presents 3D diffraction imaging based on the spectral decomposition of a different combination of selective or partial images. These images are obtained by the pre‐stack asymmetric migration procedure, which is weighted data summation. Spectral decomposition is done in the Fourier domain with respect to spatial dip and azimuth angles. Numerical examples with the application of different workflows for the synthetic and real data examples demonstrate detailed reliable reconstruction of the fractured zones and reliable reconstruction of fracture orientation on synthetic and real 3D data examples.
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Topographic elastic least‐squares reverse time migration based on vector P‐ and S‐wave equations in the curvilinear coordinates
Authors Yingming Qu, Zhe Guan and Zhenchun LiABSTRACTElastic least‐squares reverse time migration has been applied to multi‐component seismic data to obtain high‐quality images. However, the final images may suffer from artefacts caused by P‐ and S‐wave crosstalk and severe spurious diffractions caused by complex topographic surface conditions. To suppress these crosstalk artefacts and spurious diffractions, we have developed a topographic separated‐wavefield elastic least‐squares reverse time migration algorithm. In this method, we apply P‐ and S‐wave separated elastic velocity–stress wave equations in the curvilinear coordinates to derive demigration equations and gradient formulas with respect to P‐ and S‐velocity. For the implementation of topographic separated‐wavefield elastic least‐squares reverse time migration, the wavefields, gradient directions and step lengths are all calculated in the curvilinear coordinates. Numerical experiments conducted with the two‐component data synthetized by a three‐topographic‐layer with anomalies model and the Canadian Foothills model are considered to verify our method. The results reveal that compared with the conventional method, our method promises imaging results with higher resolution and has a faster residual convergence speed. Finally, we carry out numerical examples on noisy data, imperfect migration velocity and inaccurate surface elevation to analyse its sensitivity to noise, migration velocity and surface elevation error. The results prove that our method is less sensitive to noise compared with the conventional elastic least‐squares reverse time migration and needs good migration velocities as other least‐squares reverse time migration methods. In addition, when implementing the proposed method, an accurate surface elevation should be obtained by global positioning system to yield high‐quality images.
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Poynting and polarization vectors based wavefield decomposition and their application on elastic reverse time migration in 2D transversely isotropic media
Authors Yongming Lu, Qiancheng Liu, Jianfeng Zhang, Kai Yang and Hui SunABSTRACTWith the progress in computational power and seismic acquisition, elastic reverse time migration is becoming increasingly feasible and helpful in characterizing the physical properties of subsurface structures. To achieve high‐resolution seismic imaging using elastic reverse time migration, it is necessary to separate the compressional (P‐wave) and shear (S‐wave) waves for both isotropic and anisotropic media. In elastic isotropic media, the conventional method for wave‐mode separation is to use the divergence and curl operators. However, in anisotropic media, the polarization direction of P waves is not exactly parallel to the direction of wave propagation. Also, the polarization direction of S‐waves is not totally perpendicular to the direction of wave propagation. For this reason, the conventional divergence and curl operators show poor performance in anisotropic media. Moreover, conventional methods only perform well in the space domain of regular grids, and they are not suitable for elastic numerical simulation algorithms based on non‐regular grids. Besides, these methods distort the original wavefield by taking spatial derivatives. In this case, a new anisotropic wave‐mode separation scheme is developed using Poynting vectors. This scheme can be performed in the angle domain by constructing the relationship between group and polarization angles of different wave modes. Also, it is performed pointwise, independent of adjacent space points, suitable for parallel computing. Moreover, there is no need to correct the changes in phase and amplitude caused by the derivative operators. By using this scheme, the anisotropic elastic reverse time migration is more efficiently performed on the unstructured mesh. The effectiveness of our scheme is verified by several numerical examples.
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Target‐oriented imaging using extended image volumes: a low‐rank factorization approach
Authors Rajiv Kumar, Marie Graff, Ivan Vasconcelos and Felix J. HerrmannABSTRACTImaging in geological challenging environments has led to new developments, including the idea of generating reflection responses by means of interferometric redatuming at a given target datum in the subsurface, when the target datum lies beneath a complex overburden. One way to perform this redatuming is via conventional model‐based wave‐equation techniques. But those techniques can be computationally expensive for large‐scale seismic problems since the number of wave‐equation solves is equal to two times the number of sources involved during seismic data acquisition. Also conventional shot‐profile techniques require lots of memory to save full subsurface extended image volumes. Therefore, we can only form subsurface image volumes in either horizontal or vertical directions. To exploit the information hidden in full subsurface extended image volumes, we now present a randomized singular value decomposition‐based approach built upon the matrix probing scheme, which takes advantage of the algebraic structure of the extended imaging system. This low‐rank representation enables us to overcome both the computational cost associated with the number of wave‐equation solutions and memory usage due to explicit storage of full subsurface extended image volumes employed by conventional migration methods. Experimental results on complex geological models demonstrate the efficacy of the proposed methodology and allow practical reflection‐based extended imaging for large‐scale five‐dimensional seismic data.
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Complex frequency‐shifted multi‐axial perfectly matched layer for frequency‐domain seismic wavefield simulation in anisotropic media
Authors Zhencong Zhao and Jingyi ChenABSTRACTSeismic anisotropy has an important influence on seismic data processing and interpretation. Although the frequency‐domain seismic wavefield simulation has a problem of solving the large scale linear sparse matrix due to the computational limitations, it has some advantages over the time‐domain seismic wavefield simulation including efficient inversion using only a limited number of frequency components and easy implementation of multiple sources. To accurately simulate seismic wave propagation in the frequency domain, we also need to choose the absorbing boundary conditions to absorb artificial reflections from edges of the model as we do in the time domain. Compared with the classical boundary conditions including the perfectly matched layer and complex frequency‐shifted perfectly matched layer, the complex frequency‐shifted multi‐axial perfectly matched layer has been proven to effectively suppress the unwanted reflections at grazing incidence and solve the instability problem in the time‐domain seismic numerical modelling in anisotropic elastic media. In this paper, we propose to extend the complex frequency‐shifted multi‐axial perfectly matched layer absorbing boundary condition to the frequency‐domain seismic wavefield simulation in anisotropic elastic media. To test the validity of our proposed algorithm, we compare the results (snapshots and seismograms) of the frequency‐domain seismic wavefield simulation with those of the time‐domain modelling. The model studies indicate that the complex frequency‐shifted multi‐axial perfectly matched layer absorbing boundary condition is stable in the frequency‐domain seismic wavefield simulation in anisotropic media, and provides better absorbing performance than the complex frequency‐shifted perfectly matched layer boundary condition.
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Wave refocusing for linear diffractor using the time‐reversal principle
Authors Shemer Keydar and Evgeny LandaABSTRACTOne of the problems encountered in a variety of near‐surface investigations is detecting and mapping localized heterogeneities. The heterogeneities may be classified under two kinds of objects: (1) a point diffractor that can be considered as an approximation of a small quasi‐isometric, such as small karstic cavities and caves; (2) a linear diffractor roughly approximating an elongated object, such as a tube or fault plane. The point and linear diffractors generate two types of seismic diffraction: tip and edge waves, respectively. During the last few decades, different methods were proposed by many researchers for detecting these heterogeneities utilizing seismic waves diffracted by them. An alternative method for detecting point diffractors using a time‐reversal principle combined with focusing analysis is proposed in this study: we present an extension of the time‐reversal method for linear diffractors. It consists of a coherent summation of seismic energy along edge‐diffraction traveltimes. Real data examples show the feasibility and efficiency of the proposed method.
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Zeta potential of porous rocks in contact with mixtures of monovalent and divalent electrolytes
Authors Luong Duy Thanh, Rudolf Sprik and Xuan Ca NguyenABSTRACTThe zeta potential is one of the most important parameters influencing the electrokinetic coupling. Most reservoir rocks are saturated or partially saturated by natural water containing various types of ions (mostly monovalent and divalent ions). Therefore, understanding how the zeta potential behaves for mixtures of electrolytes is very important. In this work, measurements of the zeta potential for four different silica‐based samples saturated by seven different mixtures of monovalent and divalent electrolytes are then carried out at a fixed ionic strength. It is seen that the magnitude of the measured zeta potential decreases with increasing divalent cation fraction. The experimental results are then explained by a model developed for mixtures of monovalent and divalent electrolytes. The result shows that the theoretical model is able to reproduce the main trend of the variation of the zeta potential with divalent cation fractions. Additionally, the model can fit the experimental data reported in literature well for reasonable values of the input parameters.
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Connectivity constrained segmentation of geologic features
Authors Zhining Liu, Chengyun Song, Bin She, Xingmiao Yao and Guangmin HuABSTRACTSegmentation of geologic features plays a significant role in seismic interpretation. Based on the segmentation results, interpreters can readily recognize the shape and distribution of geologic features in three‐dimensional space and conduct further quantitative analysis. Usually, there are mainly two steps for the segmentation of geologic features: the first step is to extract seismic attributes that can highlight the occurrence of geologic features, and the second step is to apply the segmentation algorithm on the seismic attribute volumes. However, the occurrence of geologic features is not always corresponding to the anomaly value on the seismic attribute volumes and vice versa because of several factors, such as noise in the seismic data, the limited resolution of seismic images and the limited effectiveness of the seismic attribute. Therefore, the segmentation results, which are generated solely based on seismic attributes, are not sufficient to give an accurate depiction of geologic features. Aiming at this problem, we introduce the connectivity constraint into the process of segmentation based the assumption that for one single geologic feature all of its components should be connected to each other. Benefiting from this global constraint, the segmentation results can precisely exclude the interference by false negatives on seismic attribute volumes. However, directly introducing the connectivity constraint into segmentation would face the risk that the segmentation results would deteriorate significantly because of false positives with relatively large area when the connectivity constraints are enforced. Therefore, based on the seismic attribute that highlights the boundary of geologic feature, we further propose a post‐processing technique, called pruning, to refine the segmentation results. By taking the segmentation of the channel as an example, we demonstrate that the proposed method is able to preserve the connectivity in the process of segmentation and generate better segmentation results on the field data.
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Semi‐automatic fault/fracture interpretation based on seismic geometry analysis
Authors Haibin Di and Ghassan AlRegibABSTRACTFault and fracture interpretation is a fundamental but essential tool for subsurface structure mapping and modelling from 3D seismic data. The existing methods for semi‐automatic/automatic fault picking are primarily based on seismic discontinuity analysis that evaluates the lateral changes in seismic waveform and/or amplitude, which is limited by its low resolution on subtle faults and fractures without apparent vertical displacements in seismic images. This study presents an innovative workflow for computer‐aided fault/fracture interpretation based on seismic geometry analysis. First, the seismic curvature and flexure attributes are estimated for highlighting both the major faults and the subtle fractures in a seismic volume. Then, fault probability is estimated from the curvature and flexure volumes for differentiation between the potential faults and non‐faulting features in the geometric attributes. Finally, the seeded fault picking is implemented for interpreting the target faults and fractures guided by the knowledge of interpreters to avoid misinterpretation and artefacts in the presence of faulting complexities as well as coherent seismic noises. Applications to two 3D seismic volumes from the Netherlands North Sea and the offshore New Zealand demonstrate the added values of the proposed method in imaging and picking the subtle faults and fractures that are often overlooked in the conventional seismic discontinuity analysis and the following fault‐interpretation procedures.
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A discontinuous Galerkin method for seismic wave propagation in coupled elastic and poroelastic media
Authors Yijie Zhang, Jinghuai Gao, Weimin Han and Yanbin HeABSTRACTNumerical simulation in coupled elastic and poroelastic media is important in oil and gas exploration. However, the interface between elastic and poroelastic media is a challenge to handle. In order to deal with the coupled model, the first‐order velocity–stress wave equations are used to unify the elastic and poroelastic wave equations. In addition, an arbitrary high‐order discontinuous Galerkin method is used to simulate the wave propagation in coupled elastic–poroelastic media, which achieves same order accuracy in time and space domain simultaneously. The interfaces between the two media are explicitly tackled by the Godunov numerical flux. The proposed forms of numerical flux can be used efficiently and conveniently to simulate the wave propagation at the interfaces of the coupled model and handle the absorbing boundary conditions properly. Numerical results on coupled elastic–poroelastic media with straight and curved interfaces are compared with those from a software that is based on finite element method and the interfaces are handled by boundary conditions, demonstrating the feasibility of the proposed scheme in dealing with coupled elastic–poroelastic media. In addition, the proposed method is used to simulate a more complex coupled model. The numerical results show that the proposed method is feasible to simulate the wave propagation in such a media and is easy to implement.
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Bulk moduli and seismic attenuation in partially saturated rocks: hysteresis of liquid bridges effect
More LessABSTRACTA key task of exploration geophysics is to find relationships between seismic attributes (velocities and attenuation) and fluid properties (saturation and pore pressure). Experimental data suggest that at least three different factors affect these relationships, which are not well explained by classical Gassmann, Biot, squirt‐flow, mesoscopic‐flow and gas dissolution/exsolution models. Some of these additional factors include (i) effect of wettability and surface tension between immiscible fluids, (ii) saturation history effects (drainage versus imbibition) and (iii) effects of wave amplitude and effective stress. We apply a new rock physics model to explain the role of all these additional factors on seismic properties of a partially saturated rock. The model is based on a well‐known effect in surface chemistry: hysteresis of liquid bridges. This effect is taking place in cracks, which are partially saturated with two immiscible fluids. Using our model, we investigated (i) physical factors affecting empirical Brie correlation for effective bulk modulus of fluid, (ii) the role of liquids on seismic attenuation in the low frequency (static) limit, (iii) water‐weakening effects and (iv) saturation history effects. Our model is applicable in the low frequency limit (seismic frequencies) when capillary forces dominate over viscous forces during wave‐induced two‐phase fluid flow. The model is relevant for the seismic characterization of immiscible fluids with high contrast in compressibilities, that is, for shallow gas exploration and CO2 monitoring.
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On the determinability of all of the self‐ and mutual resistances in a grounded electrode array
By S.L. ButlerABSTRACTIn the geophysical electrical resistivity method, arrays of electrodes are commonly deployed on the Earth's surface. In electrical power engineering applications, arrays of electrodes are often connected in parallel in order to reduce the total resistance to ground. The electrical characteristics of an array of N electrodes can be fully described by N self‐resistances and mutual resistances for a total of independent parameters which represent the coefficients of an N by N symmetric matrix. Typically, certain linear combinations of the mutual resistances are measured during a geophysical electrical resistivity survey while protocols of specific measurements are used to determine self‐resistances. In this contribution, I investigate whether it is possible to determine all of the self‐ and mutual resistances for an array and hence capture all possible information. I assume that measurements of potential and current can be made at each electrode and that any combination of series and parallel current injections between the electrodes can be made. I show that all resistances can only be determined uniquely if current and voltage reference electrodes that are external to the array are used. If only an external current electrode or only an external voltage reference is used then only independent measurements can be made and the system is underdetermined. If only electrodes within the array itself are available only, independent measurements can be made and the system is strongly underdetermined. The procedure outlined here also gives a general prescription for calculating the completeness and redundancy of any given array.
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