Geostatistically consistent history matching of 3D oil-and-gas reservoir models

An approach for geostatistically consistent matching of 3D flow simulation models and 3D geological models is proposed. This approach uses an optimization algorithm based on identification of the parameters of the geostatistical model (for example, the variogram parameters, such as range, sill, and nugget effect). Here, the inverse problem is considered in the greatest generality taking into account facies heterogeneity and the variogram anisotropy. The correlation dependence parameters (porosity-to-log permeability) are clarified for each single facies.     

Directional Wavelets and a Wavelet Variogram for Two-Dimensional Data

This paper describes two new approaches that can be used to compute the two-dimensional experimental wavelet variogram. They are based on an extension from earlier work in one dimension. The methods are powerful 2D generalizations of the 1D variogram that use one- and two-dimensional filters to remove different types of trend present in the data and to provide information on the underlying variation simultaneously. In particular, the two-dimensional filtering method is effective in removing polynomial trend with filters having a simple structure. These methods are tested with simulated fields and microrelief data, and generate results similar to those of the ordinary method of moments variogram. Furthermore, from a filtering point of view, the variogram can be viewed in terms of a convolution of the data with a filter, which is computed fast in O(NLogN) number of operations in the frequency domain. We can also generate images of the filtered data corresponding to the nugget effect, sill and range of the variogram. This in turn provides additional tools to analyze the data further.     

变差函数球状模型的自动拟合与实现

根据变差函数的随机性和空间结构性,综合利用变差函数计算方法和加权线性规划拟合方法,分别拟合出各主要方向上的模型参数,再根据各向异性情况进行结构套合,实现了变差函数的计算及球状模型的自动拟合.针对样本中存在特异值的情况,算法中还提供了相对变差函数方法,有效地抑制了特异值对变差函数的影响,保证了球状模型拟合的精度.本算法在VC++6.0中实现,利用拟合出的模型,对样本区域进行插值得到网格文件,调用Surfer8.0绘制了等值线图.通过交叉验证和综合法验证,表明拟合度较高.     

Estimating Variogram Uncertainty

The variogram is central to any geostatistical survey, but the precision of a variogram estimated from sample data by the method of moments is unknown. It is important to be able to quantify variogram uncertainty to ensure that the variogram estimate is sufficiently accurate for kriging. In previous studies theoretical expressions have been derived to approximate uncertainty in both estimates of the experimental variogram and fitted variogram models. These expressions rely upon various statistical assumptions about the data and are largely untested. They express variogram uncertainty as functions of the sampling positions and the underlying variogram. Thus the expressions can be used to design efficient sampling schemes for estimating a particular variogram. Extensive simulation tests show that for a Gaussian variable with a known variogram, the expression for the uncertainty of the experimental variogram estimate is accurate. In practice however, the variogram of the variable is unknown and the fitted variogram model must be used instead. For sampling schemes of 100 points or more this has only a small effect on the accuracy of the uncertainty estimate. The theoretical expressions for the uncertainty of fitted variogram models generally overestimate the precision of fitted parameters. The uncertainty of the fitted parameters can be determined more accurately by simulating multiple experimental variograms and fitting variogram models to these. The tests emphasize the importance of distinguishing between the variogram of the field being surveyed and the variogram of the random process which generated the field. These variograms are not necessarily identical. Most studies of variogram uncertainty describe the uncertainty associated with the variogram of the random process. Generally however, it is the variogram of the field being surveyed which is of interest. For intensive sampling schemes, estimates of the field variogram are significantly more precise than estimates of the random process variogram. It is important, when designing efficient sampling schemes or fitting variogram models, that the appropriate expression for variogram uncertainty is applied.     

Kriging with imprecise (fuzzy) variograms. II: Application

The geostatistical analysis of soil liner permeability is based on 20 measurements and imprecise prior information on nugget effect, sill, and range of the unknown variogram. Using this information, membership functions for variogram parameters are assessed and the fuzzy variogram is constructed. Both kriging estimates and estimation variances are calculated as fuzzy numbers from the fuzzy variogram and data points. Contour maps are presented, indicating values of the kriged permeability and the estimation variance corresponding to selected membership values called levels.     

Reproduction of the Mean, Variance, and Variogram Model in Spectral Simulation

The application of spectral simulation is gaining acceptance because it honors the spatial distribution of petrophysical properties, such as reservoir porosity and shale volume. While it has been widely assumed that spectral simulation will reproduce the mean and variance of the important properties such as the observed net/gross ratio or global average of porosity, this paper shows the traditional way of implementing spectral simulation yields a mean and variance that deviates from the observed mean and variance. Some corrections (shift and rescale) could be applied to generate geologic models yielding the observed mean and variance; however, this correction implicitly rescales the input variogram model, so the variogram resulting from the generated cases has a higher sill than the input variogram model. Therefore, the spectral simulation algorithm cannot build geologic models honoring the desired mean, variance, and variogram model simultaneously, which is contrary to the widely accepted assumption that spectral simulation can reproduce all the target statistics. However, by using Fourier transform just once to generate values at all the cells instead of visiting each cell sequentially, spectral simulation does reproduce the observed variogram better than sequential Gaussian simulation. That is, the variograms calculated from the generated geologic models show smaller fluctuations around the target variogram. The larger the generated model size relative to the variogram range, the smaller the observed fluctuations.     

重磁场的变差函数特征及应用

用变差函数研究重磁场的区域变化特征.变差函数的变程反映重磁场的相干范围, 块金效应反映随机干扰, 基台值反映变异程度.重磁场的理论模拟说明: 重力场的相干范围大于磁场, 重磁场变程主要取决于场源深度, 浅源重磁场变差函数近似为球状模型或指数模型, 深源重磁场近似为连续性更好的高斯模型.磁场场源深度近似等于变程的一半, 重力场场源深度近似等于变程的四分之一.湖北大冶铁矿垂直分量磁异常具有几何各向异性, 北西-南东走向, 变差函数推测磁铁矿平均深度为250m.磁异常小波多尺度分解细节和逼近部分磁场具有协调几何各向异性, 变差函数的各阶场源深度估计结果与功率谱估计结果吻合.      

基于Ruger近似式预测裂缝型储层的可行性研究

叠前纵波方位各向异性检测裂缝是目前应用最为广泛的方法,该方法主要是利用AVAZ地震资料椭圆拟合的长轴与短轴信息来评价裂缝,该方法在具体应用时,认为裂缝型储层AVAZ响应特征的变化仅由各向异性参数决定,而没有考虑非各向异性参数的影响,从而引起裂缝评价的多解性。为提高该技术的裂缝评价精度,从模型分析入手,将模型中各向异性参数及岩性参数设置为概率密度分布函数,然后采用Monte Carlo随机方法进行叠前AVAZ正演模拟。首先,对各向异性参数对椭圆拟合的影响进行简单分析,得出各向异性参数γ对椭圆扁率B/A与各向异性因子B影响最大、δ次之、ε最小。然后,进一步重点分析不同标准差的速度与密度模型的B/A与B响应特征,得出地层的速度与密度的变化在较小范围内时,该技术评价裂缝才具有一定的可行性,其中,地层纵波和横波速度的变化（尤其是纵波速度）对叠前AVAZ响应影响较大,地层密度几乎没有影响,故当横向岩性变化较大时,该方法在评价裂缝型储层时的结果具有一定的不确定性。     

The variogram sill and the sample variance

The relationship between the sill of the variogram and the sample variance is explored. The common practice of using the sample variance as an estimate of the variogram sill is questioned, and a conceptual framework for determining the appropriateness of this heuristic is constructed.     

A nonparametric view of generalized covariances for kriging

Fitting trend and error covariance structure iteratively leads to bias in the estimated error variogram. Use of generalized increments overcomes this bias. Certain generalized increments yield difference equations in the variogram which permit graphical checking of the model. These equations extend to the case where errors are intrinsic random functions of order k, k=1, 2, ..., and an unbiased nonparametric graphical approach for investigating the generalized covariance function is developed. Hence, parametric models for the generalized covariance produced by BLUEPACK-3D or other methods may be assessed. Methods are illustrated on a set of coal ash data and a set of soil pH data.     

New variogram modeling method using MGGP and SVR

A critical step for kriging in geostatistics is estimation of the variogram. Traditional variogram modeling comprise of the experimental variogram calculation, appropriate variogram model selection and model parameter determination. Selecting of the variogram model and fitting of model parameters is the most controversial aspect of geostatistics. Shapes of valid variogram models are finite, and sometimes, the optimal shape of the model can not be fitted, leading to reduced estimation accuracy. In this paper, a new method is presented to automatically construct a model shape and fit model parameters to experimental variograms using Support Vector Regression (SVR) and Multi-Gene Genetic Programming (MGGP). The proposed method does not require the selection of a variogram model and can directly provide the model shape and parameters of the optimal variogram. The validity of the proposed method is demonstrated in a number of cases.     

Modelling of anisotropic coal swelling and its impact on permeability behaviour for primary and enhanced coalbed methane recovery

Coal swelling/shrinkage during gas adsorption/desorption is a well-known phenomenon. For some coals the swelling/shrinkage shows strong anisotropy, with more swelling in the direction perpendicular to the bedding than that parallel to the bedding. Experimental measurements performed in this work on an Australian coal found strong anisotropic swelling behaviour in gases including nitrogen, methane and carbon dioxide, with swelling in the direction perpendicular to the bedding almost double that parallel to the bedding. It is proposed here that this anisotropy is caused by anisotropy in the coal's mechanical properties and matrix structure. The Pan and Connell coal swelling model, which applies an energy balance approach where the surface energy change caused by adsorption is equal to the elastic energy change of the coal solid, is further developed to describe the anisotropic swelling behaviour incorporating coal property and structure anisotropy. The developed anisotropic swelling model is able to accurately describe the experimental data mentioned above, with one set of parameters to describe the coal's properties and matrix structure and three gas adsorption isotherms. This developed model is also applied to describe anisotropic swelling measurements from the literature where the model was found to provide excellent agreement with the measurement. The anisotropic coal swelling model is also applied to an anisotropic permeability model to describe permeability behaviour for primary and enhanced coalbed methane recovery. It was found that the permeability calculation applying anisotropic coal swelling differs significantly to the permeability calculated using isotropic volumetric coal swelling strain. This demonstrates that for coals with strong anisotropic swelling, anisotropic swelling and permeability models should be applied to more accurately describe coal permeability behaviour for both primary and enhanced coalbed methane recovery processes.     

Teacher''s Aide: Modeling Hole-Effect Variograms of Lithology-Indicator Variables

Common variogram models, such as spherical or exponential functions, increase monotonically with increasing lag distance. On the other hand, a hole-effect variogram typically exhibits sinusoidal waves that form peaks and troughs, thereby conveying the cyclicity of the underlying phenomenon. In order to incorporate this cyclicity into a stochastic simulation, hole effects in the experimental variogram must be fitted appropriately. In this paper, we recommend use of several multiplicative-composite variogram models to fit hole-effect experimental variograms. These consist of a cosine function to provide wavelength and phase of cyclicity, multiplied by a monotonic model (e.g., spherical) to attenuate amplitudes of the cyclical peaks and troughs. These composite models can successfully fit experimental lithology-indicator variograms that contain a range of cyclicities, although experimental variograms with poor cyclicity require special considerations.     

Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.     

Teacher's Aide: Modeling Hole-Effect Variograms of Lithology-Indicator Variables

Common variogram models, such as spherical or exponential functions, increase monotonically with increasing lag distance. On the other hand, a hole-effect variogram typically exhibits sinusoidal waves that form peaks and troughs, thereby conveying the cyclicity of the underlying phenomenon. In order to incorporate this cyclicity into a stochastic simulation, hole effects in the experimental variogram must be fitted appropriately. In this paper, we recommend use of several multiplicative-composite variogram models to fit hole-effect experimental variograms. These consist of a cosine function to provide wavelength and phase of cyclicity, multiplied by a monotonic model (e.g., spherical) to attenuate amplitudes of the cyclical peaks and troughs. These composite models can successfully fit experimental lithology-indicator variograms that contain a range of cyclicities, although experimental variograms with poor cyclicity require special considerations.     

Variogram models for regional groundwater geochemical data

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.     

复杂多金属矿床空间变异性套合技术研究与应用

半变异函数是分析区域化变量空间结构的核心内容和有效工具。文章以云南大红山铜铁多金属矿床作为研究对象,基于DIMINE软件平台,通过Cu、TFe、SFe主要成矿元素的统计特征的分析,并构建了沿走向、倾向及厚度方向上的变异函数球状模型,从而确立矿体主要变化方向上的变异函数参数。分析显示各元素具有明显的空间结构性,Cu元素品位呈几何异向性,而TFe、SFe元素品位呈带状异向性,运用模型叠加的方法建立了各向异性的套合结构模型,结合矿体的实际赋存条件及产出特征,总结出各成矿元素总体沿水平方向具有较强的连续性,而在厚度上表现出逐层均匀变化的特征。根据成矿元素品位空间变异性分析结果,为矿床矿化规律的研究提供重要依据,为后续生产工作提供理论指导。     

Estimating transmissivity using empirical and geostatistical methods in the volcanic aquifers of Upper Awash Basin, central Ethiopia

Transmissivity (T) is a basic hydraulic parameter of an aquifer that is utilized in most groundwater flow equations to understand the flow dynamics and is generally estimated from pumping tests. However, the cost of performing a large number of aquifer tests is expensive and time consuming. The fact that specific capacity (S _c) is correlated with hydraulic flow properties of aquifers simplifies parameter estimation mainly because specific capacity values are more abundant in groundwater databases than values of transmissivity and they offer another approach to estimate hydraulic parameters of aquifers. In this study, an empirical relation is derived using 214 pairs of transmissivity and specific capacity values that are obtained from pumping tests conducted on water wells penetrating the complex volcanic aquifers of Upper Awash Basin, central Ethiopia. Linear and logarithmic regression functions have been performed and it is found that the logarithmic relationship predicting transmissivity from specific capacity data has a better correlation (R = 0.97) than the linear relationship (R = 0.79). The two parameters are log-normally distributed, in which the logarithmic relation is also better statistically justified than the linear relation. Geostatistical estimations of the transmissivity were made using different inputs and methods. Measured and supplemented transmissivity data obtained from estimates using the derived empirical relation were krigged and cokrigged, spherical and exponential models were fitted to the experimental variograms. The cross-validation results showed that the best estimation is provided using the kriging procedure, the transmissivity field represented by the measured transmissivity data and the experimental variogram fitted with the exponential model. Based on the geostatistical approach, the transmissivity map of the aquifer is produced, which will be used for groundwater flow modeling of the study area that will follow this analysis.     

The anisotropy of magnetic susceptibility in biotite, muscovite and chlorite single crystals

The anisotropy of magnetic susceptibility (AMS) of single crystals of biotite, muscovite and chlorite has been measured in order to provide accurate values of the magnetic anisotropy properties for these common rock-forming minerals. The low-field AMS and the high-field paramagnetic susceptibility are defined. For the high-field values, it is necessary to combine the paramagnetic deviatoric tensor obtained from the high-field torque magnetometer with the paramagnetic bulk susceptibility measured from magnetization curves of the crystals. This leads to the full paramagnetic susceptibility ellipsoid due to the anisotropic distribution of iron cations in the silicate lattice. The ellipsoid of paramagnetic susceptibility, which was obtained for the three phyllosilicates, is highly oblate in shape and the minimum susceptibility direction is subparallel to the crystallographic c-axes. The anisotropy of the susceptibility within the basal plane of the biotite has been evaluated and found to be isotropic within the accuracy of the instrumental measurements. The degree of anisotropy of biotite and chlorite is compatible with previously reported values while for muscovite the smaller than previously published values. The shape of the chlorite AMS ellipsoid for all the samples is near-perfect oblate in contrast with a wide distribution of oblate and prolate values reported in earlier studies. Reliable values are important for deriving models of the magnetic anisotropy where it reflects mineral fabrics and deformation of rocks.     

含水层渗透性空间分布的指示克立格估值

详细介绍了指示克立格估值计算的理论和方法。以指示变异函数为基本工具分析了华北某地区第四系含水层渗透性空间分布的结构特征,结果表明该地区含水层渗透性存在明显的各向异性特征。水平方向上,X轴方向的相关性较Y轴方向的好,Z轴的相关性最差。用指示克立格法对未采样点处进行估值,估值结果显示含水层渗透性由山前向滨海逐渐变低,在垂直方向上,渗透性变化不明显,浅部比深部略好;同时给出了估计精度,并认为对估计精度不高的区域可通过增加适当的工程加以控制。最后用交叉验证法对估值结果进行了检验,证明建立的指示变异函数模型合理且估值效果较好。这一实际应用表明指示克立格法可以很好地描述第四系含水层渗透性的空间分布规律。