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- Volume 68, Issue 2, 2020
Geophysical Prospecting - Volume 68, Issue 2, 2020
Volume 68, Issue 2, 2020
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Low‐frequency anisotropy in fractured and layered media
Authors Shuo Pang and Alexey StovasABSTRACTComputing effective medium properties is very important when upscaling data measured at small scale. In the presence of stratigraphic layering, seismic velocities and anisotropy parameters are scale and frequency dependent. For a porous layer permeated by aligned fractures, wave‐induced fluid flow between pores and fractures can also cause significant dispersion in velocities and anisotropy parameters. In this study, we compare the dispersion of anisotropy parameters due to fracturing and layering at low frequencies. We consider a two‐layer model consisting of an elastic shale layer and an anelastic sand layer. Using Chapman's theory, we introduce anisotropy parameters dispersion due to fractures (meso‐scale) in the sand layer. This intrinsic dispersion is added to anisotropy parameters dispersion induced by layering (macro‐scale) at low frequencies. We derive the series coefficients that control the behaviour of anisotropy parameters at low frequencies. We investigate the influences of fracture length and fracture density on fracturing effect, layering effect and combined effect versus frequency and volume fraction of sand layer. Numerical modelling results indicate that the frequency dependence due to layering is not always the dominant effect of the effective properties of the medium. The intrinsic dispersion is not negligible compared with the layering effect while evaluating the frequency‐dependent properties of the layered medium.
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The attenuated Ricker wavelet basis for seismic trace decomposition and attenuation analysis
More LessABSTRACTThe widely used wavelets in the context of the matching pursuit are mostly focused on the time–frequency attributes of seismic traces. We propose a new type of wavelet basis based on the classic Ricker wavelet, where the quality factor Q is introduced. We develop a new scheme for seismic trace decomposition by applying the multi‐channel orthogonal matching pursuit based on the proposed wavelet basis. Compared with the decomposition by the Ricker wavelets, the proposed method could use fewer wavelets to represent the seismic signal with fewer iterations. Besides, the quality factor of the subsurface media could be extracted from the decomposition results, and the seismic attenuation could be compensated expediently. We test the availability of the proposed methods on both synthetic seismic record and field post‐stack data.
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Research Note: Frequency domain orthogonal projection filtration of surface microseismic monitoring data
Authors Anton V. Azarov, Aleksander S. Serdyukov and Denis N. GapeevABSTRACTWe address the problem of increasing the signal‐to‐noise ratio during surface microseismic monitoring data processing. Interference from different seismic waves causes misleading results of microseismic event locations. Ground‐roll suppression is particularly necessary. The standard noise suppression techniques assume regular and dense acquisition geometries. Many pre‐processing noise suppression algorithms are designed for special types of noise or interference. To overcome these problems, we propose a novel general‐purpose filtration method. The goal of this method is to amplify only the seismic waves that are excited in the selected target area and suppress all other signals. We construct a linear projector onto a frequency domain data subspace, which corresponds to the seismic emission of the target area. The novel filtration method can be considered an extension of the standard frequency–wavenumber flat wave filtration method for non‐flat waves and arbitrary irregular receiver‐position geometries. To reduce the effect of the uncertainty of the velocity model, we suggest using additional active shot data (typically the perforation shots), which provide static travel time corrections for the target area. The promising prospects of the proposed method are confirmed by synthetic and semi‐synthetic data processing.
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Estimation of a non‐stationary prior covariance from seismic data
Authors Rasmus Bødker Madsen, Thomas Mejer Hansen and Henning OmreABSTRACTNon‐stationarity in statistical properties of the subsurface is often ignored. In a classical linear Bayesian inversion setting of seismic data, the prior distribution of physical parameters is often assumed to be stationary. Here we propose a new method of handling non‐stationarity in the variance of physical parameters in seismic data. We propose to infer the model variance prior to inversion using maximum likelihood estimators in a sliding window approach. A traditional, and a localized shrinkage estimator is defined for inferring the prior model variance. The estimators are assessed in a synthetic base case with heterogeneous variance of the acoustic impedance in a zero‐offset seismic cross section. Subsequently, this data is inverted for acoustic impedance using a non‐stationary model set up with the inferred variances. Results indicate that prediction as well as posterior resolution is greatly improved using the non‐stationary model compared with a common prior model with stationary variance. The localized shrinkage predictor is shown to be slightly more robust than the traditional estimator in terms of amplitude differences in the variance of acoustic impedance and size of local neighbourhood. Finally, we apply the methodology to a real data set from the North Sea basin. Inversion results show a more realistic posterior model than using a conventional approach with stationary variance.
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Imaging of elastic seismic data by least‐squares reverse time migration with weighted L2‐norm multiplicative and modified total‐variation regularizations
Authors Zhiming Ren and Zhenchun LiABSTRACTLeast‐squares reverse time migration has the potential to yield high‐quality images of the Earth. Compared with acoustic methods, elastic least‐squares reverse time migration can effectively address mode conversion and provide velocity/impendence and density perturbation models. However, elastic least‐squares reverse time migration is an ill‐posed problem and suffers from a lack of uniqueness; further, its solution is not stable. We develop two new elastic least‐squares reverse time migration methods based on weighted L2‐norm multiplicative and modified total‐variation regularizations. In the proposed methods, the original minimization problem is divided into two subproblems, and the images and auxiliary variables are updated alternatively. The method with modified total‐variation regularization solves the two subproblems, a Tikhonov regularization problem and an L2‐total‐variation regularization problem, via an efficient inversion workflow and the split‐Bregman iterative method, respectively. The method with multiplicative regularization updates the images and auxiliary variables by the efficient inversion workflow and nonlinear conjugate gradient methods in a nested fashion. We validate the proposed methods using synthetic and field seismic data. Numerical results demonstrate that the proposed methods with regularization improve the resolution and fidelity of the migration profiles and exhibit superior anti‐noise ability compared with the conventional method. Moreover, the modified‐total‐variation‐based method has marginally higher accuracy than the multiplicative‐regularization‐based method for noisy data. The computational cost of the proposed two methods is approximately the same as that of the conventional least‐squares reverse time migration method because no additional forward computation is required in the inversion of auxiliary variables.
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Widely linear denoising of multicomponent seismic data
Authors Breno Bahia and Mauricio D. SacchiABSTRACTSeismic data processing is a challenging task, especially when dealing with vector‐valued datasets. These data are characterized by correlated components, where different levels of uncorrelated random noise corrupt each one of the components. Mitigating such noise while preserving the signal of interest is a primary goal in the seismic‐processing workflow. The frequency‐space deconvolution is a well‐known linear prediction technique, which is commonly used for random noise suppression. This paper represents vector‐field seismic data through quaternion arrays and shows how to mitigate random noise by proposing the extension of the frequency‐space deconvolution to its hypercomplex version, the quaternion frequency‐space deconvolution. It also shows how a widely linear prediction model exploits the correlation between data components of improper signals. The widely linear scheme, named widely‐linear quaternion frequency‐space deconvolution, produces longer prediction filters, which have enhanced signal preservation capabilities shown through synthetic and field vector‐valued data examples.
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Reverse time migration using water‐bottom‐related multiples
Authors Yanbao Zhang, Yike Liu, Xuejian Liu and Xiaopeng ZhouABSTRACTReverse time migration of multiples can be used to construct subsurface structures where primaries cannot illuminate well. However, the images generated using multiples suffer from severe artefacts due to the cross‐talks created by interference among unrelated multiples. We developed a migration approach using water‐bottom‐related multiples to reduce these cross‐talk artefacts. This approach first isolates primaries from the original data and predicts water‐column primaries. The nth‐order water‐column multiples can be obtained by auto‐convolving the water‐column primaries n times, followed by convolving the nth‐order water‐column multiples with the primaries to extract the (n+1)th‐order water‐bottom‐related multiples. The approach takes the nth‐order water‐column multiples as the secondary source and regards the (n+1)th‐order water‐bottom‐related multiples as the receiver wavefield, followed by a cross‐correlation imaging condition. Numerical examples from synthetic and field data sets reveal that our approach can provide images with substantially fewer cross‐talk artefacts than conventional reverse time migration using multiples, as well as greatly improving shallow imaging compared with reverse time migration of primaries.
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Wavelet transform‐based seismic facies classification and modelling: application to a geothermal target horizon in the NE German Basin
Authors Klaus Bauer, Ben Norden, Alexandra Ivanova, Manfred Stiller and Charlotte M. KrawczykABSTRACTAt the geothermal test site near Groß Schönebeck (NE German Basin), a new 3D seismic reflection survey was conducted to study geothermal target layers at around 4 km depth and 150°C. We present a workflow for seismic facies classification and modelling which is applied to a prospective sandstone horizon within the Rotliegend formation. Signal attributes are calculated along the horizon using the continuous Morlet wavelet transform. We use a short mother wavelet to allow for the temporal resolution of the relatively short reflection signals to be analysed. Time‐frequency domain data patterns form the input of a neural network clustering using self‐organizing maps. Neural model patterns are adopted during iterative learning to simulate the information inherent in the input data. After training we determine a gradient function across the self‐organizing maps and apply an image processing technique called watershed segmentation. The result is a pattern clustering based on similarities in wavelet transform characteristics. Three different types of wavelet transform patterns were found for the sandstone horizon. We apply seismic waveform modelling to improve the understanding of the classification results. The modelling tests indicate that thickness variations have a much stronger influence on the wavelet transform response of the sandstone horizon compared with reasonable variations of seismic attenuation. In our interpretation, the assumed thickness variations could be a result of variable paleo‐topography during deposition of predominantly fluvial sediments. A distinct seismic facies distribution is interpreted as a system of thicker paleo‐channels deposited within a deepened landscape. The results provide constraints for the ongoing development of the geothermal test site.
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S‐wave in 2D acoustic transversely isotropic media with a tilted symmetry axis
More LessABSTRACTIn an acoustic transversely isotropic medium, there are two waves that propagate. One is the P‐wave and another one is the S‐wave (also known as S‐wave artefact). This paper is devoted to analyse the S‐wave in two‐dimensional acoustic transversely isotropic media with a tilted symmetry axis. We derive the S‐wave slowness surface and traveltime function in a homogeneous acoustic transversely isotropic medium with a tilted symmetry axis. The S‐wave traveltime approximations in acoustic transversely isotropic media with a tilted symmetry axis can be mapped from the counterparts for acoustic transversely isotropic media with a vertical symmetry axis. We consider a layered two‐dimensional acoustic transversely isotropic medium with a tilted symmetry axis to analyse the S‐wave moveout. We also illustrate the behaviour of the moveout for reflected S‐wave and converted waves.
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A field experiment of walkaway distributed acoustic sensing vertical seismic profile in a deep and deviated onshore well in Japan using a fibre optic cable deployed inside coiled tubing
Authors Yuki Kobayashi, Yuto Uematsu, Shimpei Mochiji and Ziqiu XueABSTRACTA two‐dimensional walkaway vertical seismic profiling survey using distributed acoustic sensing was conducted at an onshore site in Japan. The maximum depth and the deviation of the observation well were more than 4,000 m and 81 degrees, respectively. Among the several methods for installing fibre optic cables, we adopted the inside coiled tubing method, in which coiled tubing containing a fibre optic cable is deployed. The signal‐to‐noise ratio of the raw shot gather was low, possibly due to poor coupling between the fibre optic cable and the subsurface formation resulting from the fibre optic cable deployment method and the existence of considerable tubewave noise. Nevertheless, direct P‐wave arrivals, P–P reflections and P–S converted waves exhibited acceptable signal‐to‐noise ratios after careful optimization of gauge length for distributed acoustic sensing optical processing and the application of carefully parameterized tubewave noise suppression. One of the challenges in current distributed acoustic sensing vertical seismic profile data processing is the separation of P‐ and S‐waves using only one‐component measurements. Hence, we applied moveout correction using two‐dimensional ray tracing. This process effectively highlights only reflected P‐waves, which are used in subsequent subsurface imaging. Comparison with synthetic well seismograms and two‐dimensional surface seismic data confirms that the final imaging result has a sufficiently high quality for subsurface monitoring. We acquired distributed acoustic sensing vertical seismic profile data under both flowing conditions and closed conditions, in which the well was shut off and no fluid flow was allowed. The two imaging results are comparable and suggest the possibility of subsurface imaging and time‐lapse monitoring using data acquired under flowing conditions. The results of this study suggest that, by adopting the inside coiled tubing method without drilling a new observation well, more affordable distributed acoustic sensing vertical seismic profile monitoring can be achieved in fields such as CO2 capture and storage and unconventional shale projects, where monitoring costs have to be minimized.
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A comparison of seismic saltbody interpretation via neural networks at sample and pattern levels
Authors Haibin Di and Ghassan AlRegibABSTRACTSaltbodies are important subsurface structures that have significant implications for hydrocarbon accumulation and sealing in petroleum reservoirs, and accurate saltbody imaging and delineation is now greatly facilitated with the availability of three‐dimensional seismic surveying. However, with the growing demand for larger survey coverage and higher imaging resolution, the size of seismic data is increasing dramatically. Correspondingly, manual saltbody interpretation fails to offer an efficient solution, particularly in exploration areas of complicated salt intrusion history. Recently, artificial intelligence is attracting great attention from geoscientists who desire to utilize the popular machine learning technologies for evolving the interpretational tools capable of mimicking an experienced interpreter's intelligence. This study first implements two popular machine learning tools, the multi‐layer perceptron and the convolutional neural network, for delineating seismic saltbodies at sample and pattern levels, respectively, then compares their performance through applications to the synthetic SEAM seismic volume, and moreover tentatively investigates what contributes to the better convolutional neural network delineation. Specifically, the multi‐layer perceptron scheme is capable of efficiently utilizing an interpreter's knowledge by selecting, pre‐conditioning and integrating a set of seismic attributes that best highlight the target saltbodies, whereas the convolutional neural network scheme makes it possible for saltbody delineation directly from seismic amplitude and thus significantly reduces the dependency on attribute selection from interpreters. It is concluded that the better performance from the convolutional neural network scheme results from two factors. First, the convolutional neural network builds the mapping relationship between the seismic signals and the saltbodies using the original seismic amplitude instead of manually selected seismic attributes, so that the negative impact of using less representative attributes is virtually eliminated. Second and more importantly, the convolutional neural network defines, learns and identifies the saltbodies by utilizing local seismic reflection patterns, so that the seismic noises and processing artefacts of distinct patterns are effectively identified and excluded.
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Towards large‐scale stochastic refraction tomography: a comparison of three evolutionary algorithms
Authors Keurfon Luu, Mark Noble, Alexandrine Gesret and Philippe ThierryABSTRACTThe main goal of this study is to assess the potential of evolutionary algorithms to solve highly non‐linear and multi‐modal tomography problems (such as first arrival traveltime tomography) and their abilities to estimate reliable uncertainties. Classical tomography methods apply derivative‐based optimization algorithms that require the user to determine the value of several parameters (such as regularization level and initial model) prior to the inversion as they strongly affect the final inverted model. In addition, derivative‐based methods only perform a local search dependent on the chosen starting model. Global optimization methods based on Markov chain Monte Carlo that thoroughly sample the model parameter space are theoretically insensitive to the initial model but turn out to be computationally expensive. Evolutionary algorithms are population‐based global optimization methods and are thus intrinsically parallel, allowing these algorithms to fully handle available computer resources. We apply three evolutionary algorithms to solve a refraction traveltime tomography problem, namely the differential evolution, the competitive particle swarm optimization and the covariance matrix adaptation–evolution strategy. We apply these methodologies on a smoothed version of the Marmousi velocity model and compare their performances in terms of optimization and estimates of uncertainty. By performing scalability and statistical analysis over the results obtained with several runs, we assess the benefits and shortcomings of each algorithm.
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De‐aliased and de‐noise Cadzow filtering for seismic data reconstruction
Authors Weilin Huang, Deshan Feng and Yangkang ChenABSTRACTCadzow filtering is currently considered as one of the most effective approaches for seismic data reconstruction. The basic version of Cadzow filtering first reorders each frequency slice of the seismic data (to be reconstructed) to a block Hankel/Toeplitz matrix, and then implements a rank‐reduction operator, that is truncated singular value decomposition, to the Hankel/Toeplitz matrix. However, basic Cadzow filtering cannot deal with the problem of recovering regularly missing data (up‐sampling) in the case of strongly aliased energy, because the regularly missing data will mix with signals in the singular spectrum. To solve this problem, it has been proposed to precondition the reconstruction of high‐frequency components using information from the low‐frequency components which are less aliased. In this paper, we further extend the de‐aliased Cadzow filtering approach to reconstruct regularly sampled seismic traces from the noisy observed data by modifying the reinserting operation, in which the high‐frequency components are projected onto the sub‐space spanned by several singular vectors of the low‐frequency components. At each iteration, the filtered data are weighted to the original data as a feedback. The weighting factor is related to the background noise level and changes with iteration. The feasibility of the proposed technique is validated via two‐dimensional, three‐dimensional and five‐dimensional synthetic data examples, as well as two‐dimensional post‐stack and three‐dimensional pre‐stack field data examples. The results demonstrate that the proposed technique can effectively interpolate regularly sampled data and is robust in noisy environments.
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Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method
Authors Edvaldo S. Araujo and Reynam C. PestanaABSTRACTWe derive a governing second‐order acoustic wave equation in the time domain with a perfectly matched layer absorbing boundary condition for general inhomogeneous media. Besides, a new scheme to solve the perfectly matched layer equation for absorbing reflections from the model boundaries based on the rapid expansion method is proposed. The suggested scheme can be easily applied to a wide class of wave equations and numerical methods for seismic modelling. The absorbing boundary condition method is formulated based on the split perfectly matched layer method and we employ the rapid expansion method to solve the derived new perfectly matched layer equation. The use of the rapid expansion method allows us to extrapolate wavefields with a time step larger than the ones commonly used by traditional finite‐difference schemes in a stable way and free of dispersion noise. Furthermore, in order to demonstrate the efficiency and applicability of the proposed perfectly matched layer scheme, numerical modelling examples are also presented. The numerical results obtained with the put forward perfectly matched layer scheme are compared with results from traditional attenuation absorbing boundary conditions and enlarged models as well. The analysis of the numerical results indicates that the proposed perfectly matched layer scheme is significantly effective and more efficient in absorbing spurious reflections from the model boundaries.
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Seismic wavefield modelling in two‐phase media including undulating topography with the modified Biot/squirt model by a curvilinear‐grid finite difference method
Authors Shang‐bei Yang, Chao‐ying Bai and Stewart GreenhalghABSTRACTIn this paper, we deduced the corresponding first‐order velocity–stress equation for curvilinear coordinates from the first‐order velocity–stress equation based on the modified Biot/squirt model for a two‐dimensional two‐phase medium. The equations are then numerically solved by an optimized high‐order non‐staggered finite difference scheme, that is, the dispersion relation preserving/optimization MacCormack scheme. To implement undulating free‐surface topography, we derive an analytical relationship between the derivatives of the particle velocity components and use the compact finite‐difference scheme plus a traction‐image method. In the undulating free surface and the undulating subsurface interface of two‐phase medium, the complex reflected wave and transmitted wave can be clearly recognized in the numerical simulation results. The simulation results show that the curvilinear‐grid finite‐difference method, which uses a body‐conforming grid to describe the undulating surface, can accurately reduce the numerical scattering effect of seismic wave propagation caused by the use of ladder‐shaped grid to fit the surfaces when undulating topography is present in a two‐phase isotropic medium.
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A rock physics model to map gross lithology using compaction
Authors Jeremy Gallop, Olena Babak and Kun LiuABSTRACTDifferential compaction has long been used by seismic interpreters to infer subsurface geology using knowledge of the relative compaction of different types of sediments. We outline a method to infer the gross fraction of shale in an interval between two seismic horizons using sandstone and shale compaction laws. A key component of the method involves reconstruction of a smooth depositional horizon by interpolating decompacted thicknesses from well control. We derive analytic formulae for decompaction calculations using known porosity–stress relations and do not employ discrete layer iterative methods; these formulae were found to depend not only upon the gross fraction of shale but also on the clay content of the shales and the thickness of the interval. The relative merits of several interpolation options were explored, and found to depend upon the structural setting. The method was successfully applied to an oil sands project in Alberta, Canada.
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Effects of pore fluids on quasi‐static shear modulus caused by pore‐scale interfacial phenomena
More LessABSTRACTIt is evident from the laboratory experiments that shear moduli of different porous isotropic rocks may show softening behaviour upon saturation. The shear softening means that the shear modulus of dry samples is higher than of saturated samples. Shear softening was observed both at low (seismic) frequencies and high (ultrasonic) frequencies. Shear softening is stronger at seismic frequencies than at ultrasonic frequencies, where the softening is compensated by hardening due to unrelaxed squirt flow. It contradicts to Gassmann's theory suggesting that the relaxed shear modulus of isotropic rock should not depend upon fluid saturation, provided that no chemical reaction between the solid frame and the pore fluid. Several researchers demonstrated that the shear softening effect is reversible during re‐saturation of rock samples, suggesting no permanent chemical reaction between the solid frame and the pore fluid. Therefore, it is extremely difficult to explain this fluid–rock interaction mechanism theoretically, because it does not contradict to the assumptions of Gassmann's theory, but contradicts to its conclusions. We argue that the observed shear softening of partially saturated rocks by different pore fluids is related to pore‐scale interfacial phenomena effects, typically neglected by the rock physics models. These interface phenomena effects are dependent on surface tension between immiscible fluids, rock wettability, aperture distribution of microcracks, compressibility of microcracks, porosity of microcracks, elastic properties of rock mineral, fluid saturation, effective stress and wave amplitude. Derived equations allow to estimate effects of pore fluids and saturation on the shear modulus and mechanical strength of rocks.
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Analysis of attenuation and dispersion of propagating wave due to the coexistence of three fluid phases in the pore volume
Authors Qazi Adnan Ahmad, Guochen Wu, Zong Zhaoyun, Wu Jianlu, Li Kun, Du Tianwei and Nasir KhanABSTRACTIn exploration geophysics, the efforts to extract subsurface information from wave characteristics exceedingly depend on the construction of suitable rock physics model. Analysis of different rock physics models reveals that the strength and magnitude of attenuation and dispersion of propagating wave exceedingly depend on wave‐induced fluid flow at multiple scales. In current work, a comprehensive analysis of wave attenuation and velocity dispersion is carried out at broad frequency range. Our methodology is based on Biot's poroelastic relations, by which variations in wave characteristics associated with wave‐induced fluid flow due to the coexistence of three fluid phases in the pore volume is estimated. In contrast to the results of previous research, our results indicate the occurrence of two‐time pore pressure relaxation phenomenon at the interface between fluids of disparate nature, that is, different bulk modulus, viscosity and density. Also, the obtained results are compatible with numerical results for the same 1D model which are accounted using Biot's poroelastic and quasi‐static equation in frequency domain. Moreover, the effects of change in saturation of three‐phase fluids were also computed which is the key task for geophysicist. The outcomes of our research reveal that pore pressure relaxation phenomenon significantly depends on the saturation of distinct fluids and the order of saturating fluids. It is also concluded that the change in the saturation of three‐phase fluid significantly influences the characteristics of the seismic wave. The analysis of obtained results indicates that our proposed approach is a useful tool for quantification, identification and discrimination of different fluid phases. Moreover, our proposed approach improves the accuracy to predict dispersive behaviour of propagating wave at sub‐seismic and seismic frequencies.
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Estimation of the 3D correlation structure of an alluvial aquifer from surface‐based multi‐frequency ground‐penetrating radar reflection data
Authors Zhiwei Xu, James Irving, Kyle Lindsay, John Bradford, Peimin Zhu and Klaus HolligerABSTRACTKnowledge about the stochastic nature of heterogeneity in subsurface hydraulic properties is critical for aquifer characterization and the corresponding prediction of groundwater flow and contaminant transport. Whereas the vertical correlation structure of the heterogeneity is often well constrained by borehole information, the lateral correlation structure is generally unknown because the spacing between boreholes is too large to allow for its meaningful inference. There is, however, evidence to suggest that information on the lateral correlation structure may be extracted from the correlation statistics of the subsurface reflectivity structure imaged by surface‐based ground‐penetrating radar measurements. To date, case studies involving this approach have been limited to 2D profiles acquired at a single antenna centre frequency in areas with limited complementary information. As a result, the practical reliability of this methodology has been difficult to assess. Here, we extend previous work to 3D and consider reflection ground‐penetrating radar data acquired using two antenna centre frequencies at the extensively explored and well‐constrained Boise Hydrogeophysical Research Site. We find that the results obtained using the two ground‐penetrating radar frequencies are consistent with each other, as well as with information from a number of other studies at the Boise Hydrogeophysical Research Site. In addition, contrary to previous 2D work, our results indicate that the surface‐based reflection ground‐penetrating radar data are not only sensitive to the aspect ratio of the underlying heterogeneity, but also, albeit to a lesser extent, to the so‐called Hurst number, which is a key parameter characterizing the local variability of the fine‐scale structure.
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Volumes & issues
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Volume 72 (2023 - 2024)
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Volume 71 (2022 - 2023)
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Volume 70 (2021 - 2022)
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Volume 69 (2021)
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Volume 68 (2020)
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