共查询到20条相似文献，搜索用时 31 毫秒
1.
Gilles Bourgault 《Mathematical Geosciences》2012,44(8):1005-1037
The likelihood of Gaussian realizations, as generated by the Cholesky simulation method, is analyzed in terms of Mahalanobis distances and fluctuations in the variogram reproduction. For random sampling, the probability to observe a Gaussian realization vector can be expressed as a function of its Mahalanobis distance, and the maximum likelihood depends only on the vector size. The Mahalanobis distances are themselves distributed as a Chi-square distribution and they can be used to describe the likelihood of Gaussian realizations. Their expected value and variance are only determined by the size of the vector of independent random normal scores used to generate the realizations. When the vector size is small, the distribution of Mahalanobis distances is highly skewed and most realizations are close to the vector mean in agreement with the multi-Gaussian density model. As the vector size increases, the realizations sample a region increasingly far out on the tail of the multi-Gaussian distribution, due to the large increase in the size of the uncertainty space largely compensating for the low probability density. For a large vector size, realizations close to the vector mean are not observed anymore. Instead, Gaussian vectors with Mahalanobis distance in the neighborhood of the expected Mahalanobis distance have the maximum probability to be observed. The distribution of Mahalanobis distances becomes Gaussian shaped and the bulk of realizations appear more equiprobable. However, the ratio of their probabilities indicates that they still remain far from being equiprobable. On the other hand, it is observed that equiprobable realizations still display important fluctuations in their variogram reproduction. The variance level that is expected in the variogram reproduction, as well as the variance of the variogram fluctuations, is dependent on the Mahalanobis distance. Realizations with smaller Mahalanobis distances are, on average, smoother than realizations with larger Mahalanobis distances. Poor ergodic conditions tend to generate higher proportions of flatter variograms relative to the variogram model. Only equiprobable realizations with a Mahalanobis distance equal to the expected Mahalanobis distance have an expected variogram matching the variogram model. For large vector sizes, Cholesky simulated Gaussian vectors cannot be used to explore uncertainty in the neighborhood of the vector mean. Instead uncertainty is explored around the n-dimensional elliptical envelop corresponding to the expected Mahalanobis distance. 相似文献
2.
The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology. 相似文献
3.
Tingting Yao Craig Calvert Tom Jones Glen Bishop Yuan Ma Lincoln Foreman 《Mathematical Geology》2006,38(1):51-62
Spectral simulation has gained application in building geologic models due to the advantage of better honoring the spatial continuity of petrophysical properties, such as reservoir porosity and shale volume. Distinct from sequential simulation methods, spectral simulation is a global algorithm in the sense that a global density spectrum is calculated once and the inverse Fourier transform is performed on the Fourier coefficient also only once to generate a simulation realization. The generated realizations honor the spatial continuity structure globally over the whole field instead of only within a search neighborhood, as with sequential simulation algorithms. However, the disadvantage of global spectral simulation is that it traditionally cannot account for the local information such as the local continuity trends, which are often observed in reservoirs and hence are important to be accounted for in geologic models. This disadvantage has limited wider application of spectral simulation in building geologic models. In this paper, we present ways of conditioning geologic models to the relevant local information. To account for the local continuity trends, we first scale different frequency components of the original model with local-amplitude spectrum ratios that are specific to the local trend. The sum of these scaled frequency components renders a new model that displays the desired local continuity trend. The implementation details of this new method are discussed and examples are provided to illustrate the algorithm. 相似文献
4.
Teacher''s Aide Variogram Interpretation and Modeling 总被引：13，自引：0，他引：13
The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology. 相似文献
5.
Johannes Bruining Diederik van Batenburg Larry W. Lake An Ping Yang 《Mathematical Geology》1997,29(6):823-848
Random field generators serve as a tool to model heterogeneous media for applications in hydrocarbon recovery and groundwater
flow. Random fields with a power-law variogram structure, also termed fractional Brownian motion (fBm) fields, are of interest
to study scale dependent heterogeneity effects on one-phase and two-phase flow. We show that such fields generated by the
spectral method and the Inverse Fast Fourier Transform (IFFT) have an incorrect variogram structure and variance. To illustrate
this we derive the prefactor of the fBm spectral density function, which is required to generate the fBm fields. We propose
a new method to generate fBm fields that introduces weighting functions into the spectral method. It leads to a flexible and
efficient algorithm. The flexibility permits an optimal choice of summation points (that is points in frequency space at which
the weighting function is calculated) specific for the autocovariance structure of the field. As an illustration of the method,
comparisons between estimated and expected statistics of fields with an exponential variogram and of fBm fields are presented.
For power-law semivariograms, the proposed spectral method with a cylindrical distribution of the summation points gives optimal
results. 相似文献
6.
Fracture set properties such as orientation, spacing, trace length, and waviness tend to be spatially correlated. These properties can be efficiently simulated by spectral analysis procedures that take advantage of the computational speed of the fast Fourier transform. The covariance function of each property to be simulated is obtained from the variogram function estimated from mapped fracture set data and is typically referenced to the mean vector of the set. Simulation procedures for normally and exponentially distributed data involve generating uncorrelated Fourier coefficients that are assigned proper variance according to the spectral density, which is the Fourier transform of the covariance function. These coefficients are then reverse Fourier transformed to produce simulated set properties that have the desired variance and variogram function. 相似文献
7.
Statistical characterization and stochastic modeling of pore networks in relation to fluid flow 总被引：18，自引：0，他引：18
R. D. Hazlett 《Mathematical Geology》1997,29(6):801-822
8.
Djuro Novakovic Christopher D. White Rucsandra M. Corbeanu William S. Hammon III Janok P. Bhattacharya George A. McMechan 《Mathematical Geology》2002,34(7):857-893
Ground-penetrating radar (GPR) surveys, outcrop measurements, and cores provide a high-resolution 3D geologic model to investigate the hydraulic effects of shales in marine-influenced lower delta-plain distributary channel deposits within the Cretaceous-age Ferron Sandstone at Corbula Gulch in central Utah, USA. Shale statistics are computed from outcrop observations. Although slight anisotropy was observed in mean length and variogram ranges parallel and perpendicular to pale of low
, the anisotropy is not statistically significant and the estimated mean length is 5.4 m. Truncated Gaussian simulation was used to create maps of shales that are placed on variably dipping stratigraphic surfaces interpreted from high-resolution 3D GPR surveys, outcrop interpretations, and boreholes. Sandstone permeability is estimated from radar responses calibrated to permeability measurements from core samples. Experimentally designed flow simulations examine the effects of variogram range, shale coverage fraction, and trends in shale coverage on predicted upscaled permeability, breakthrough time, and sweep efficiency. Approximately 1500 flow simulations examine three different geologic models, flow in the 3 coordinate directions, 16 geostatistical parameter combinations, and 10 realizations for each model. ANOVA and response models computed from the flow simulations demonstrate that shales decrease sweep, recovery, and permeability, especially in the vertical direction. The effect on horizontal flow is smaller. Flow predictions for ideal tracer displacements at Corbula Gulch are sensitive to shale-coverage fraction, but are relatively insensitive to twofold variations in variogram range or to vertical trends in shale coverage. Although the hydraulic effects of shale are statistically significant, the changes in flow responses rarely exceed 20%. As a result, it may be reasonable to use simple models when incorporating analogous shales into models of reservoirs or aquifers. 相似文献
9.
Geostatistical Seismic Inversion with Direct Sequential Simulation and Co-simulation with Multi-local Distribution Functions 总被引：1，自引：0，他引：1
Rúben Nunes Amílcar Soares Leonardo Azevedo Pedro Pereira 《Mathematical Geosciences》2017,49(5):583-601
Stochastic sequential simulation is a common modelling technique used in Earth sciences and an integral part of iterative geostatistical seismic inversion methodologies. Traditional stochastic sequential simulation techniques based on bi-point statistics assume, for the entire study area, stationarity of the spatial continuity pattern and a single probability distribution function, as revealed by a single variogram model and inferred from the available experimental data, respectively. In this paper, the traditional direct sequential simulation algorithm is extended to handle non-stationary natural phenomena. The proposed stochastic sequential simulation algorithm can take into consideration multiple regionalized spatial continuity patterns and probability distribution functions, depending on the spatial location of the grid node to be simulated. This work shows the application and discusses the benefits of the proposed stochastic sequential simulation as part of an iterative geostatistical seismic inversion methodology in two distinct geological environments in which non-stationarity behaviour can be assessed by the simultaneous interpretation of the available well-log and seismic reflection data. The results show that the elastic models generated by the proposed stochastic sequential simulation are able to reproduce simultaneously the regional and global variogram models and target distribution functions relative to the average volume of each sub-region. When used as part of a geostatistical seismic inversion procedure, the retrieved inverse models are more geologically realistic, since they incorporate the knowledge of the subsurface geology as provided, for example, by seismic and well-log data interpretation. 相似文献
10.
G. Bourgault 《Mathematical Geology》1996,28(6):723-734
A combination of factorial kriging and probability field simulation is proposed to correct realizations resulting from any simulation algorithm for either too high nugget effect (noise) or poor histogram reproduction. First, a factorial kriging is done to filter out the noise from the noisy realization. Second, the uniform scores of the filtered realization are used as probability field to sample the local probability distributions conditional to the same dataset used to generate the original realization. This second step allows to restore the data variance. The result is a corrected realization which reproduces better target variogram and histogram models, yet honoring the conditioning data. 相似文献
11.
Geologic uncertainty in a regulatory environment: An example from the potential Yucca Mountain nuclear waste repository site 总被引：1，自引：0，他引：1
Regulatory geologists are concerned with predicting the performance of sites proposed for waste disposal or for remediation of existing pollution problems. Geologic modeling of these sites requires large-scale expansion of knowledge obtained from very limited sampling. This expansion induces considerable uncertainty into the geologic models of rock properties that are required for modeling the predicted performance of the site.One method for assessing this uncertainty is through nonparametric geostatistical simulation. Simulation can produce a series of equiprobable models of a rock property of interest. Each model honors measured values at sampled locations, and each can be constructed to emulate both the univariate histogram and the spatial covariance structure of the measured data. Computing a performance model for a number of geologic simulations allows evaluation of the effects of geologic uncertainty. A site may be judged acceptable if the number of failures to meet a particular performance criterion produced by these computations is sufficiently low. A site that produces too many failures may be either unacceptable or simply inadequately described.The simulation approach to addressing geologic uncertainty is being applied to the potential high-level nuclear waste repository site at Yucca Mountain, Nevada, U.S.A. Preliminary geologic models of unsaturated permeability have been created that reproduce observed statistical properties reasonably well. A spread of unsaturated groundwater travel times has been computed that reflects the variability of those geologic models. Regions within the simulated models exhibiting the greatest variability among multiple runs are candidates for obtaining the greatest reduction in uncertainty through additional site characterization. 相似文献
12.
Based on the algorithm for gradual deformation of Gaussian stochastic models, we propose, in this paper, an extension of this method to gradually deforming realizations generated by sequential, not necessarily Gaussian, simulation. As in the Gaussian case, gradual deformation of a sequential simulation preserves spatial variability of the stochastic model and yields in general a regular objective function that can be minimized by an efficient optimization algorithm (e.g., a gradient-based algorithm). Furthermore, we discuss the local gradual deformation and the gradual deformation with respect to the structural parameters (mean, variance, and variogram range, etc.) of realizations generated by sequential simulation. Local gradual deformation may significantly improve calibration speed in the case where observations are scattered in different zones of a field. Gradual deformation with respect to structural parameters is necessary when these parameters cannot be inferred a priori and need to be determined using an inverse procedure. A synthetic example inspired from a real oil field is presented to illustrate different aspects of this approach. Results from this case study demonstrate the efficiency of the gradual deformation approach for constraining facies models generated by sequential indicator simulation. They also show the potential applicability of the proposed approach to complex real cases. 相似文献
13.
Stochastic fractal (fGn and fBm) porosity and permeability fields are conditioned to given variogram, static (or hard), and multiwell pressure data within a Bayesian estimation framework. Because fGn distributions are normal/second-order stationary, it is shown that the Bayesian estimation methods based on the assumption of normal/second-order stationary distributions can be directly used to generate fGn porosity/permeability fields conditional to pressure data. However, because fBm is not second-order stationary, it is shown that such Bayesian estimation methods can be used with implementation of a pseudocovariance approach to generate fBm porosity/permeability fields conditional to multiwell pressure data. In addition, we provide methods to generate unconditional realizations of fBm/fGn fields honoring all variogram parameters. These unconditional realizations can then be conditioned to hard and pressure data observed at wells by using the randomized maximum likelihood method. Synthetic examples generated from one-, two-, and three-dimensional single-phase flow simulators are used to show the applicability of our methodology for generating realizations of fBm/fGn porosity and permeability fields conditioned to well-test pressure data and evaluating the uncertainty in reservoir performance predictions appropriately using these history-matched realizations. 相似文献
14.
在地下水污染模拟预报中,弥散参数是很难确定的一个模型参数。因实验室小尺度弥散规律一般不能用于大尺度弥散过程,而野外示踪试验却耗资大、周期长,限制了其实用性。文中利用随机数值模拟手段、基于随机理论的蒙特卡罗方法及序贯高斯模拟技术来生成渗透系数随机场,并研究渗透系数对数场的方差、相关长度以及变异函数类型在不同尺度上对纵向弥散度的影响,进而建立纵向弥散度与随机分布渗透系数场的方差和相关长度的统计定量关系,并与Gelhar理论计算结果进行比较。数值模拟结果表明,经过一定迁移距离后纵向弥散度与随机分布渗透系数对数场的方差和相关长度具有良好的线性统计关系,与Gelhar理论公式表达的关系类型类似。但对于较大的方差,纵向弥散度模拟结果明显大于Gelhar理论计算值,而对于较大相关长度在迁移距离不很大时,纵向弥散度模拟结果明显小于Gelhar理论计算值。本研究可为野外大尺度地下水污染预报模型中水动力弥散参数的确定提供方法借鉴。 相似文献
15.
Jinchi Chu 《Mathematical Geology》1996,28(7):923-936
Sequential Indicator Simulation (SIS), although widely used, is relatively slow, and requires tedious inference of a large number of indicator variogram models. SIS is designed only to estimate class proportions and to reproduce indicator variogram models; the statistics of the continuous attribute being simulated,z-histogram and variogram, may be poorly reproduced. Several implementations of the SIS algorithm are proposed resulting in better reproduction of statistics yet with better CPU performance. 相似文献
16.
A stationary specification of anisotropy does not always capture the complexities of a geologic site. In this situation, the
anisotropy can be varied locally. Directions of continuity and the range of the variogram can change depending on location
within the domain being modeled. Kriging equations have been developed to use a local anisotropy specification within kriging
neighborhoods; however, this approach does not account for variation in anisotropy within the kriging neighborhood. This paper
presents an algorithm to determine the optimum path between points that results in the highest covariance in the presence
of locally varying anisotropy. Using optimum paths increases covariance, results in lower estimation variance and leads to
results that reflect important curvilinear structures. Although CPU intensive, the complex curvilinear structures of the kriged
maps are important for process evaluation. Examples highlight the ability of this methodology to reproduce complex features
that could not be generated with traditional kriging. 相似文献
17.
The increasing use of unstructured grids for reservoir modeling motivates the development of geostatistical techniques to
populate them with properties such as facies proportions, porosity and permeability. Unstructured grids are often populated
by upscaling high-resolution regular grid models, but the size of the regular grid becomes unreasonably large to ensure that
there is sufficient resolution for small unstructured grid elements. The properties could be modeled directly on the unstructured
grid, which leads to an irregular configuration of points in the three-dimensional reservoir volume. Current implementations
of Gaussian simulation for geostatistics are for regular grids. This paper addresses important implementation details involved
in adapting sequential Gaussian simulation to populate irregular point configurations including general storage and computation
issues, generating random paths for improved long range variogram reproduction, and search strategies including the superblock
search and the k-dimensional tree. An efficient algorithm for computing the variogram of very large irregular point sets is developed for
model checking. 相似文献
18.
In previous studies, the groundwater flow models formulated for the Hat Yai Basin were conventional and deterministic because the geologic heterogeneity of the alluvial aquifer system in the basin had not yet been assessed. This paper describes an effort to develop hydrofacies models, such that the spatial variability of the aquifer system can be represented in a systematic way. Variogram parameters that characterize the alluvial aquifer heterogeneity were determined. Based on these variogram parameters, an indicator-based geostatistical approach was used to develop hydrofacies models using sequential indicator simulation. The hydrofacies models indicate three distinct aquifer units, namely Hat Yai, Khu Tao, and Kho Hong aquifers, which is in good agreement with a conceptual model, and incorporates spatial variability as observed in field data from borehole logs. The hydrofacies models can be used in groundwater modeling and simulations. 相似文献
19.
Four variogram models for regional groundwater geochemical data are presented. These models were developed from an empirical study of the sample variograms for more than 10 elements in groundwaters from two geologic regions in the Plainview quandrangle, Texas. A procedure is given for the estimation of the variogram in the isotropic and anisotropic case. The variograms were found useful for quantifying the differences in spatial variability for elements within a geologic unit and for elements in different geologic units. Additionally, the variogram analysis enables assessment of the assumption of statistical independence of regional samples which is commonly used in many statistical procedures. The estimated variograms are used in computation of kriged estimates for the Plainview quadrangle data. The results indicate that an inverse distance weighting model was superior for prediction than simple kriging with the particular variograms used. 相似文献
20.
Before optimal linear prediction can be performed on spatial data sets, the variogram is usually estimated at various lags and a parametric model is fitted to those estimates. Apart from possible a priori knowledge about the process and the user's subjectivity, there is no standard methodology for choosing among valid variogram models like the spherical or the exponential ones. This paper discusses the nonparametric estimation of the variogram and its derivative, based on the spectral representation of positive definite functions. The use of the estimated derivative to help choose among valid parametric variogram models is presented. Once a model is selected, its parameters can be estimated—for example, by generalized least squares. A small simulation study is performed that demonstrates the usefulness of estimating the derivative to help model selection and illustrates the issue of aliasing. MATLAB software for nonparametric variogram derivative estimation is available at http://www-math.mit.edu/~gorsich/derivative.html. An application to the Walker Lake data set is also presented. 相似文献