共查询到20条相似文献，搜索用时 62 毫秒
1.
Simulation of subsurface heterogeneity is important for modeling subsurface flow and transport processes. Previous studies have indicated that subsurface property variations can often be characterized by fractional Brownian motion (fBm) or (truncated) fractional Levy motion (fLm). Because Levy-stable distributions have many novel and often unfamiliar properties, studies on generating fLm distributions are rare in the literature. In this study, we generalize a relatively simple and computationally efficient successive random additions (SRA) algorithm, originally developed for generating Gaussian fractals, to simulate fLm distributions. We also propose an additional important step in response to continued observations that the traditional SRA algorithm often generates fractal distributions having poor scaling and correlation properties. Finally, the generalized and modified SRA algorithm is validated through numerical tests. 相似文献
2.
3.
Sedimentary deposits are often characterized by various distinct facies, with facies structure relating to the depositional
and post-depositional environments. Permeability (k) varies within each facies, and mean values in one facies may be several
orders of magnitude larger or smaller than those in another facies. Empirical probability density functions (PDFs) of log(k)
increments from multi-facies structures often exhibit properties well modeled by the Levy PDF, which appears unrealistic physically.
It is probable that the statistical properties of log(k) variations within a facies are very different from those between
facies. Thus, it may not make sense to perform a single statistical analysis on permeability values taken from a mix of distinct
facies. As an alternative, we employed an indicator simulation approach to generate large-scale facies distributions, and
a mono-fractal model, fractional Brownian motion (fBm), to generate the log(k) increments within facies. Analyses show that
the simulated log(k) distributions for the entire multi-facies domain produce apparent non-Gaussian log(k) increment distributions
similar to those observed in field measurements. An important implication is that Levy-like behavior is not real in a statistical
sense and that rigorous statistical measures of the log(k) increments will have to be extracted from within each individual
facies.
Electronic Publication 相似文献
4.
Histograms of observations from spatial phenomena are often found to be more heavy-tailed than Gaussian distributions, which
makes the Gaussian random field model unsuited. A T-distributed random field model with heavy-tailed marginal probability density functions is defined. The model is a generalization
of the familiar Student-T distribution, and it may be given a Bayesian interpretation. The increased variability appears cross-realizations, contrary
to in-realizations, since all realizations are Gaussian-like with varying variance between realizations. The T-distributed random field model is analytically tractable and the conditional model is developed, which provides algorithms
for conditional simulation and prediction, so-called T-kriging. The model compares favourably with most previously defined random field models. The Gaussian random field model
appears as a special, limiting case of the T-distributed random field model. The model is particularly useful whenever multiple, sparsely sampled realizations of the
random field are available, and is clearly favourable to the Gaussian model in this case. The properties of the T-distributed random field model is demonstrated on well log observations from the Gullfaks field in the North Sea. The predictions
correspond to traditional kriging predictions, while the associated prediction variances are more representative, as they
are layer specific and include uncertainty caused by using variance estimates. 相似文献
5.
G. Bourgault 《Mathematical Geology》1997,29(3):315-334
Parametric geostatistical simulations such as LU decomposition and sequential algorithms do not need Gaussian distributions.
It is shown that variogram model reproduction is obtained when Uniform or Dipole distributions are used instead of Gaussian
distributions for drawing i. i.d. random values in LU simulation, or for modeling the local conditional probability distributions
in sequential simulation. Both algorithms yield simulated values with a marginal normal distribution no matter if Gaussian,
Uniform, or Dipole distributions are used. The range of simulated values decreases as the entropy of the probability distribution
decreases. Using Gaussian distributions provides a larger range of simulated normal score values than using Uniform or Dipole
distributions. This feature has a negligible effect for reproduction of the normal scores variogram model but have a larger
impact on the reproduction of the original values variogram. The Uniform or Dipole distributions also produce lesser fluctuations
among the variograms of the simulated realizations. 相似文献
6.
Adding Local Accuracy to Direct Sequential Simulation 总被引：3，自引：0，他引：3
Jef Caers 《Mathematical Geology》2000,32(7):815-850
Geostatistical simulations are globally accurate in the sense that they reproduce global statistics such as variograms and histograms. Kriging is locally accurate in the minimum local error variance sense. Building on the concept of direct sequential simulation, we propose a fast simulation method that can share these opposing objectives. It is shown that the multiple-point entropy of the resulting simulation is related to the univariate entropy of the local conditional distributions used to draw simulated values. Adding local accuracy to conditional simulations does not detract much from variogram reproduction and can be used to increase multiple-point entropy. The methods developed are illustrated using a case study. 相似文献
7.
Gilles Bourgault 《Mathematical Geosciences》2014,46(7):841-868
A multivariate probability transformation between random variables, known as the Nataf transformation, is shown to be the appropriate transformation for multi-Gaussian kriging. It assumes a diagonal Jacobian matrix for the transformation of the random variables between the original space and the Gaussian space. This allows writing the probability transformation between the local conditional probability density function in the original space and the local conditional Gaussian probability density function in the Gaussian space as a ratio equal to the ratio of their respective marginal distributions. Under stationarity, the marginal distribution in the original space is modeled from the data histogram. The stationary marginal standard Gaussian distribution is obtained from the normal scores of the data and the local conditional Gaussian distribution is modeled from the kriging mean and kriging variance of the normal scores of the data. The equality of ratios of distributions has the same form as the Bayes’ rule and the assumption of stationarity of the data histogram can be re-interpreted as the gathering of the prior distribution. Multi-Gaussian kriging can be re-interpreted as an updating of the data histogram by a Gaussian likelihood. The Bayes’ rule allows for an even more general interpretation of spatial estimation in terms of equality for the ratio of the conditional distribution over the marginal distribution in the original data uncertainty space with the same ratio for a model of uncertainty with a distribution that can be modeled using the mean and variance from direct kriging of the original data values. It is based on the principle of conservation of probability ratio and no transformation is required. The local conditional distribution has a variance that is data dependent. When used in sequential simulation mode, it reproduces histogram and variogram of the data, thus providing a new approach for direct simulation in the original value space. 相似文献
8.
Conditional Simulation of Random Fields with Bivariate Gamma Isofactorial Distributions 总被引：5，自引：0，他引：5
Xavier Emery 《Mathematical Geology》2005,37(4):419-445
This work focuses on a random function model with gamma marginal and bivariate isofactorial distributions, which has been applied in mining geostatistics for estimating recoverable reserves by disjunctive kriging. The objective is to widen its use to conditional simulation and further its application to the modeling of continuous attributes in geosciences. First, the main properties of the bivariate gamma isofactorial distributions are analyzed, with emphasis in the destructuring of the extreme values, the presence of a proportional effect (higher variability in high-valued areas), and the asymmetry in the spatial correlation of the indicator variables with respect to the median threshold. Then, we provide examples of stationary random functions with such bivariate distributions, for which the shape parameter of the marginal distribution is half an integer. These are defined as the sum of squared independent Gaussian random fields. An iterative algorithm based on the Gibbs sampler is proposed to perform the simulation conditional to a set of existing data. Such ‘multivariate chi-square’ model generalizes the well-known multigaussian model and is more flexible, since it allows defining a shape parameter which controls the asymmetry of the marginal and bivariate distributions. 相似文献
9.
We present a general, flexible, and fast neural network approach to the modeling of a conditional distribution of a discrete random variable, given a continuous or discrete random vector. Although many more applications of the neural net technique could be envisioned, the aim is to apply the developed methodology to the integration of seismic data into reservoir models. Many geostatistical methods for integrating seismic data rely on a screening assumption of further away seismic events by the colocated seismic datum. Such assumption makes the task of modeling cross-covariances and local conditional distributions much easier. In many cases, however, the seismic data exhibit distinct and locally varying spatial patterns of continuity related to geological events such as channels, shale bodies, or fractures. The previous screening assumption prevents recognizing and hence utilizing these patterns of seismic data. In this paper we propose to relate seismic data to facies or petrophysical properties through a colocated window of seismic information instead of the single colocated seismic datum. The variation of seismic data from one window to another is accounted for. Several examples demonstrate that using such a window improves the predictive power of seismic data. 相似文献
10.
Spatially distributed and varying natural phenomena encountered in geoscience and engineering problem solving are typically
incompatible with Gaussian models, exhibiting nonlinear spatial patterns and complex, multiple-point connectivity of extreme
values. Stochastic simulation of such phenomena is historically founded on second-order spatial statistical approaches, which
are limited in their capacity to model complex spatial uncertainty. The newer multiple-point (MP) simulation framework addresses
past limits by establishing the concept of a training image, and, arguably, has its own drawbacks. An alternative to current
MP approaches is founded upon new high-order measures of spatial complexity, termed “high-order spatial cumulants.” These
are combinations of moments of statistical parameters that characterize non-Gaussian random fields and can describe complex
spatial information. Stochastic simulation of complex spatial processes is developed based on high-order spatial cumulants
in the high-dimensional space of Legendre polynomials. Starting with discrete Legendre polynomials, a set of discrete orthogonal
cumulants is introduced as a tool to characterize spatial shapes. Weighted orthonormal Legendre polynomials define the so-called
Legendre cumulants that are high-order conditional spatial cumulants inferred from training images and are combined with available
sparse data sets. Advantages of the high-order sequential simulation approach developed herein include the absence of any
distribution-related assumptions and pre- or post-processing steps. The method is shown to generate realizations of complex
spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data. In
addition, it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated
realizations, whereas the training images are only used to fill in high-order relations that cannot be inferred from data.
Compared to the MP framework, the proposed approach is data-driven and consistently reconstructs the lower-order spatial complexity
in the data used, in addition to high order. 相似文献
11.
We have analyzed the spatial distribution of galaxies from the latest release of the Sloan Digital Sky Survey of galactic
redshifts (SDSS DR7), applying the complete correlation function (conditional density), two-point conditional density (cylinder),
and radial density methods. Our analysis demonstrates that the conditional density has a power-law form for scales lengths
0.5–30 Mpc/h, with the power-law corresponding to the fractal dimension D = 2.2 ± 0.2; for scale lengths in excess of 30 Mpc/h, it enters an essentially flat regime, as is expected for a uniform distribution of galaxies. However, in the analysis applying
the cylinder method, the power-law character with D = 2.0 ± 0.3 persists to scale lengths of 70 Mpc/h. The radial density method reveals inhomogeneities in the spatial distribution of galaxies on scales of 200 Mpc/h with a density contrast of two, confirming that translation invariance is violated in the distribution of galaxies to 300
Mpc/h, with the sampling depth of the SDSS galaxies being 600 Mpc/h. 相似文献
12.
P. Seferou P. Soupios N. N. Kourgialas Z. Dokou G. P. Karatzas E. Candasayar N. Papadopoulos V. Dimitriou A. Sarris M. Sauter 《Hydrogeology Journal》2013,21(6):1219-1234
An integrated approach for monitoring the vertical transport of a solute into the subsurface by using a geophysical method and a simulation model is proposed and evaluated. A medium-scale (1 m3) laboratory tank experiment was constructed to represent a real subsurface system, where an olive-oil mill wastewater (OOMW) spill might occur. High-resolution cross-hole electrical resistivity tomography (ERT) was performed to monitor the OOMW transport. Time-lapse ERT images defined the spatial geometry of the interface between the contaminated and uncontaminated soil into the unsaturated and saturated zones. Knowing the subsurface characteristics, the finite element flow and transport model FEFLOW was used for simulating the contaminant movement, utilizing the ERT results as a surrogate for concentration measurements for the calibration process. A statistical analysis of the ERT measurements and the corresponding transport model results for various time steps showed a good agreement between them. In addition, a sensitivity analysis of the most important parameters of the simulation model (unsaturated flow, saturated flow and transport) was performed. This laboratory-scale study emphasizes that the combined use of geophysical and transport-modeling approaches can be useful for small-scale field applications where contaminant concentration measurements are scarce, provided that its transferability from laboratory to field conditions is investigated thoroughly. 相似文献
13.
Direct Multiple-Point Geostatistical Simulation of Edge Properties for Modeling Thin Irregularly Shaped Surfaces 总被引：3，自引：1，他引：2
Thin, irregularly shaped surfaces such as clay drapes often have a major control on flow and transport in heterogeneous porous
media. Clay drapes are often complex, curvilinear three-dimensional surfaces and display a very complex spatial distribution.
Variogram-based stochastic approaches are also often not able to describe the spatial distribution of clay drapes since complex,
curvilinear, continuous, and interconnected structures cannot be characterized using only two-point statistics. Multiple-point
geostatistics aims to overcome the limitations of the variogram. The premise of multiple-point geostatistics is to move beyond
two-point correlations between variables and to obtain (cross) correlation moments at three or more locations at a time using
training images to characterize the patterns of geological heterogeneity. Multiple-point geostatistics can reproduce thin
irregularly shaped surfaces such as clay drapes, but this is often computationally very intensive. This paper describes and
applies a methodology to simulate thin, irregularly shaped surfaces with a smaller CPU and RAM demand than the conventional
multiple-point statistical methods. The proposed method uses edge properties for indicating the presence of thin irregularly
shaped surfaces. Instead of pixel values, edge properties indicating the presence of irregularly shaped surfaces are simulated
using snesim. This method allows direct simulation of edge properties instead of pixel properties to make it possible to perform multiple-point
geostatistical simulations with a larger cell size and thus a smaller computation time and memory demand. This method is particularly
valuable for three-dimensional applications of multiple-point geostatistics. 相似文献
14.
We investigate the effects of using different types of statistical distributions (lognormal, gamma, and beta) to characterize the variability of Young’s modulus of soils in random finite element analyses of shallow foundation settlement. We use a two-dimensional linear elastic, plane-strain, finite element model with a rigid footing founded on elastic soil. Poisson’s ratio of the soil is considered constant, and Young’s modulus is characterized using random fields with extreme values of the scale of fluctuation. We perform an extensive sensitivity analysis to compare the distributions of computed settlements when different types of statistical distributions of Young’s modulus, different coefficients of variation of Young’s modulus, and different scales of fluctuation of the random field of Young’s modulus are considered. A large number of realizations are employed in the Monte Carlo simulations to investigate the influence of the tails of the statistical distributions under study. Results indicate the type of distribution considered for characterization of the random field of Young’s modulus can have a significant impact on computed settlement results. In particular, considering different types of distributions of Young’s modulus can lead to more than 600% differences on computed mean settlements for cases with high coefficient of variation and large scale of fluctuation of Young’s modulus. The effect of considering different types of distributions is reduced, but not completely eliminated, for smaller coefficients of variation of Young’s modulus (because the differences between distributions decrease) and for small values of the scale of fluctuation of Young’s modulus (because of an identified “averaging effect”). 相似文献
15.
To simulate geological models comprising several litho-types—or facies—we need first to estimate their proportions, which are often poorly known. The corresponding uncertainties can be modelled using a Bayesian approach for inverting the multinomial distribution. The result obtained is known as the Dirichlet distribution. It can be simulated by decomposition into independent conditional distributions. Application of the model is extended to the case of nonstationary proportions and, with some approximation, to the case of correlated spatial data. The mathematical developments presented in the appendices provide a more precise and general definition of the distribution, several decomposition formulae into independent variables, the determination of remarkable stability properties, and the resulting consequences for the conditional and marginal distributions. 相似文献
16.
Most approaches in statistical spatial prediction assume that the spatial data are realizations of a Gaussian random field.
However, this assumption is hard to justify for most applications. When the distribution of data is skewed but otherwise has
similar properties to the normal distribution, a closed skew normal distribution can be used for modeling their skewness.
Closed skew normal distribution is an extension of the multivariate skew normal distribution and has the advantage of being
closed under marginalization and conditioning. In this paper, we generalize Bayesian prediction methods using closed skew
normal distributions. A simulation study is performed to check the validity of the model and performance of the Bayesian spatial
predictor. Finally, our prediction method is applied to Bayesian spatial prediction on the strain data near Semnan, Iran.
The mean-square error of cross-validation is improved by the closed skew Gaussian model on the strain data. 相似文献
17.
Particle-tracking simulation offers a fast and robust alternative to conventional numerical discretization techniques for modeling solute transport in subsurface formations. A common challenge is that the modeling scale is typically much larger than the volume scale over which measurements of rock properties are made, and the scale-up of measurements have to be made accounting for the pattern of spatial heterogeneity exhibited at different scales. In this paper, a statistical scale-up procedure developed in our previous work is adopted to estimate coarse-scale (effective) transition time functions for transport modeling, while two significant improvements are proposed: considering the effects of non-stationarity (trend), as well as unresolved (residual) heterogeneity below the fine-scale model. Rock property is modeled as a multivariate random function, which is decomposed into the sum of a trend (which is defined at the same resolution of the transport modeling scale) and a residual (representing all heterogeneities below the transport modeling scale). To construct realizations of a given rock property at the transport modeling scale, multiple realizations of the residual components are sampled. Next, a flow-based technique is adopted to compute the effective transport parameters: firstly, it is assumed that additional unresolved heterogeneities occurring below the fine scale can be described by a probabilistic transit time distribution; secondly, multiple realizations of the rock property, with the same physical size as the transport modeling scale, are generated; thirdly, each realization is subjected to particle-tracking simulation; finally, probability distributions of effective transition time function are estimated by matching the corresponding effluent history for each realization with an equivalent medium consisting of averaged homogeneous rock properties and aggregating results from all realizations. The proposed method is flexible that it does not invoke any explicit assumption regarding the multivariate distribution of the heterogeneity. 相似文献
18.
Modern geostatistical techniques allow the generation of high-resolution heterogeneous models of hydraulic conductivity containing millions to billions of cells. Selective upscaling is a numerical approach for the change of scale of fine-scale hydraulic conductivity models into coarser scale models that are suitable for numerical simulations of groundwater flow and mass transport. Selective upscaling uses an elastic gridding technique to selectively determine the geometry of the coarse grid by an iterative procedure. The geometry of the coarse grid is built so that the variances of flow velocities within the coarse blocks are minimum. Selective upscaling is able to handle complex geological formations and flow patterns, and provides full hydraulic conductivity tensor for each block. Selective upscaling is applied to a cross-bedded formation in which the fine-scale hydraulic conductivities are full tensors with principal directions not parallel to the statistical anisotropy of their spatial distribution. Mass transport results from three coarse-scale models constructed by different upscaling techniques are compared to the fine-scale results for different flow conditions. Selective upscaling provides coarse grids in which mass transport simulation is in good agreement with the fine-scale simulations, and consistently superior to simulations on traditional regular (equal-sized) grids or elastic grids built without accounting for flow velocities. 相似文献
19.
Aquifer properties, for example permeability and porosity, vary in space and may be characterized by their distributions. The property distribution is not totally random but shows some correlation structure. Because most of the values are not known, some rational method is required to generate credible aquifer distribution properties for inclusion in fluid transport models. This paper presents a numerically efficient method of generating geostatistical random fields, by the source Point Method (SPM). The SPM is a very efficient method and requires little computer time and relatively small data storage, as compared to other methods of generating random fields. In addition, the SPM is modified to include any desired amount of anisotropy in the property distribution of a system. By using conditional covariances, a formula for a two-dimensional anisotropic field is derived to prespecify the desired correlation length in any direction. Results show that for an anisotropic medium the correlation length can be pre-specified in any specific direction. 相似文献
20.
Masoud Monjezi Majid Rajabalizadeh Kashani Mohammad Ataei 《Arabian Journal of Geosciences》2013,6(1):123-128
Grade estimation is very important in designing open pits. In the process of grade estimation, underestimation can result in loss of economic ore, whereas overestimation would unnecessarily increase stripping ratio. Normally, kriging method, which suffers from underestimation and/or overestimation due to smoothing effect, is used for grade estimation. To overcome drawbacks of the kriging method, more efficient techniques such as conditional simulation can be applied. In this paper, utilizing sequential Gaussian conditional simulation, grade models were constructed for Sungun copper deposit situated in the North West of Iran. According to the obtained results, it was observed that conditional simulation can effectively cope with the weakness of kriging method. Also, it was observed that as compared to the kriging method, grade distribution, resulted from the conditional simulation, is almost identical to that of the real exploration data. Accordingly, using conditional simulation, the amount of mineable ore was significantly increased, and also, average net present value as the mines’ most important economic indicator was improved by 40%. 相似文献