Geological Controls of Variograms in a Complex Carbonate Reservoir, Eastern Province, Saudi Arabia

To derive meaningful results from a geostatistical study, it is extremely important to establish the relationship between geology and variograms. However, it is not always easy to establish such a relationship, mainly because of the inadequacy of quantitative or qualitative information. Large amounts of reliable well-log porosity data and the detailed geological information from a complex carbonate reservoir located in the Eastern Province of Saudi Arabia provided an ideal case study of the relationships between geology and variograms. The reservoir under consideration is in the form of a gently dipping, elongated anticline and consists of three productive zones which have been subdivided into several lithologically distinct layers. Results, which indicate an excellent match between geology and porosity variograms, are summarized as follows: (1) use of stratigraphic distances resulted in considerable improvement in the behavior of variograms, (2) structurally controlled geometrical anisotropy is the most obvious feature of variograms, and (3) layers consisting of more complex lithologic framework provided variograms with relatively greater nugget variances and shorter ranges. These results provide further insight into geology and form a meaningful basis for estimation and simulation of the reservoir. They also could be used as a general guide in the geostatistical study of similar carbonate reservoirs.     

Teacher''s Aide: Geologic Characteristics of Hole-Effect Variograms Calculated from Lithology-Indicator Variables

Variograms calculated from binary variables, such as from two lithologies, tend to show sinusoidal forms with decreasing amplitudes for increasing lag distances. This cyclicity is observed often when analyzing drill-hole data for rock sequences with alternating lithologies, and the variograms are thus labeled hole-effect variograms. Such variograms show a variety of forms: (1) Low to moderate variation in lithologic-body dimensions causes variograms to have strong cyclicity with decaying amplitude. (2) Variograms with one or more peaks and troughs usually result from a binary variable for which lithologies are about equally abundant but possibly large variations exist in the size of lithologic bodies. (3) Variograms show poor cyclicity if one lithology has highly variable body sizes and the other has moderately variable body dimensions. (4) Variograms that attain a plateau at short lag distances represent extremely high or low sandstone fraction, high variability in size of the most abundant lithology, and low variability in the other. Information about the dimensions of lithologic bodies makes it possible to approximate characteristics of the variogram of the lithology variable without numerous wells. Conversely, a hole-effect variogram of lithology may be used to estimate lithologic dimensions.     

Teacher's Aide: Geologic Characteristics of Hole-Effect Variograms Calculated from Lithology-Indicator Variables

Variograms calculated from binary variables, such as from two lithologies, tend to show sinusoidal forms with decreasing amplitudes for increasing lag distances. This cyclicity is observed often when analyzing drill-hole data for rock sequences with alternating lithologies, and the variograms are thus labeled “hole-effect variograms.” Such variograms show a variety of forms: (1) Low to moderate variation in lithologic-body dimensions causes variograms to have strong cyclicity with decaying amplitude. (2) Variograms with one or more peaks and troughs usually result from a binary variable for which lithologies are about equally abundant but possibly large variations exist in the size of lithologic bodies. (3) Variograms show poor cyclicity if one lithology has highly variable body sizes and the other has moderately variable body dimensions. (4) Variograms that attain a plateau at short lag distances represent extremely high or low sandstone fraction, high variability in size of the most abundant lithology, and low variability in the other. Information about the dimensions of lithologic bodies makes it possible to approximate characteristics of the variogram of the lithology variable without numerous wells. Conversely, a hole-effect variogram of lithology may be used to estimate lithologic dimensions.     

Variograms and their role in interpretation of soft iron ore types

Difficulties are encountered in the interpretation of variograms from mixed populations. A case history study of a soft iron ore body capped by laterite demonstrates that the variations observed in variograms for different elements are caused mainly by intermixing of different lithologic units which are interstratified. In the case of soft iron ore, occasional bands of clay and blue dust affect significantly the variations observed for iron, silicon, and aluminum. In the future, it is recommended that the variograms produced reflect a single population.     

Scale Matching with Factorial Kriging for Improved Porosity Estimation from Seismic Data

When seismic data and porosity well logs contain information at different spatial scales, it is important to do a scale-matching of the datasets. Combining different data types with scale mismatch can lead to suboptimal results. A good correlation between seismic velocity and rock properties provides a basis for integrating seismic data in the estimation of petrophysical properties. Three-dimensional seismic data provides an unique exhaustive coverage of the interwell reservoir region not available from well data. However, because of the limitations of measurement frequency bandwidth and view angles, the seismic image can not capture the true seismic velocity at all spatial scales present in the earth. The small-scale spatial structure of heterogeneities may be absent in the measured seismic data. In order to take best advantage of the seismic data, factorial kriging is applied to separate the small and large-scale structures of both porosity and seismic data. Then the spatial structures in seismic data which are poorly correlated with porosity are filtered out prior to integrating seismic data into porosity estimation.     

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.     

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.     

三维Kriging方法中的变异函数套合

三维属性建模是利用有限的采样数据, 通过插值或模拟的方法来重构地学属性在三维空间中的分布.将Kriging方法推广到三维空间, 从而演化为三维Kriging方法, 可以为三维属性建模提供可靠的手段.而三维Kriging方法面临的一大难题就是各向异性变异函数的套合.提出了一种简单通用的三维空间变异函数的套合方法.该方法以空间坐标基的变换为基础, 在套合时充分考虑轴向上变异差异的影响, 并由此提出各向异性变化率的概念; 论证了套合方法的可行性, 并通过地下水水质三维属性建模的实例对该方法进行了有效的验证.      

储层物性的空间结构特征分析与预测—以青海尕斯库勒油田E3^1油藏为例

储层物性受诸多地质因素的影响，经常表现出强烈的非均质性。这种非均质性在一定的空间尺度上往往具有明显的结构特点。本文以青海尕斯库勒油田E3^1油藏为例，讨论了地质统计学方法在油田储层特性空间结构特征分析与预测中的应用。采用变异函数定量描述了孔隙度和渗透率的空间分布结构特征，并在此基础上利用克立格法进行了最成插值预测。结果表明，研究区储层物性（孔隙度和渗透率）具有显著的空间结构性特点，变程一般在800－2000m之间；不同小层的储层物性具有不同的空间结构方向性。这种特征主要沉积相带空间展布的影响，各小层孔隙度和渗透率的实验半变异函数均可用具有块金效应的球状模型来拟合并进行预测。作为验证，本文还采用“多重趋势面”预测模型对储层的孔隙度和渗透率进行了预测分析。     

储层物性的空间结构特征分析与预测——以青海尕斯库勒油田E_3~1油藏为例

储层物性受诸多地质因素的影响,经常表现出强烈的非均质性。这种非均质性在一定的空间尺度上往往具有明显的结构特点。本文以青海尕斯库勒油田E_3~1油藏为例,讨论了地质统计学方法在油田储层物性空间结构特征分析与预测中的应用。采用变异函数定量描述了孔隙度和渗透率的空间分布结构特征,并在此基础上利用克立格法进行了最优插值预测。结果表明,研究区储层物性(孔隙度和渗透率)具有显著的空间结构性特点,变程一般在800～2000m之间;不同小层的储层物性具有不同的空间结构方向性。这种特征主要受沉积相带空间展布的影响,各小层孔隙度和渗透率的实验半变异函数均可用具有块金效应的球状模型来拟合并进行预测。作为验证,本文还采用“多重趋势面”预测模型对储层的孔隙度和渗透率进行了预测分析。     

Analysis of Time of Occurrence of Earthquakes: A Functional Data Approach

There is no single method available for estimating the seismic risk in a given area, and as a result most studies are based on some statistical model. If we denote by Z the random variable that measures the maximum magnitude of earthquakes per unit time, the seismic risk of a value m is the probability that this value will be exceeded in the next time units, that is, R(m)=P(Z>m). Several approximations can be made by adjusting different theoretical distributions to the function R, assuming different distributions for the magnitude of earthquakes. A related method used to treat this problem is to consider the difference between the times of occurrence of consecutive earthquakes, or inter-event times. The hazard function, or failure rate function, of this variable measures the instantaneous risk of occurrence of a new earthquake, supposing that the last earthquake happened at time 0. In this paper, we will consider the estimation of the variable that measures the inter-event time and apply nonparametric techniques; that is, we do not consider any theoretical distribution. Moreover, because the stochastic process associated with this variable can sometimes be non-stationary, we condition each time by the previous ones. We then work with a multidimensional estimation, and consider each multidimensional variable as a functional datum. Functional data analysis deals with data consisting of curves or multidimensional variables. Nonparametric estimation can be applied to functional data, to describe the behavior of seismic zones and their associated instantaneous risk. The applications of estimation techniques are shown by applying them to two different regions and data catalogues: California and southern Spain.     

Quantitative Analysis of Porosity Heterogeneity: Application of Geostatistics to Borehole Images

High levels of heterogeneity in many carbonate reservoirs have raised concerns about the validity and relevance of small-scale measurements from core plugs and high-resolution logs. While the measurements themselves may be accurate, they may not be representative of the average formation properties. A related question is one of reconciling the measurements made in small volume of investigation data (e.g., core plugs), with the measurements from relatively large volume of investigation data (e.g., wireline logs). This paper presents a technique to quantitatively describe the porosity heterogeneity in a borehole at the scale of several tenths of an inch. The method involves treating high-resolution borehole imagery as a 2D sample from a 3D data volume, and applying geostatistical analysis to these data. We compute the experimental semi-variogram and upscale its range and sill to larger (several inches) scales of measurement to predict the impact of heterogeneity on conventional core plug and logging tool porosity measurements. The resulting dispersion variance between the different measurement scales support the interpretation, application and comparison of these porosity measurements. This technique was applied to an Early Cretaceous carbonate reservoir in Abu Dhabi. We found that the scale of the heterogeneity is typically less than 1– 2in., so that while significant heterogeneity is observed at the core plug and smaller scales of measurement, the larger-volume logging tool measurements smooth out the heterogeneity and show considerably less variability. The differences between porosity measured in core plugs can be completely accounted for by this upscaling effect.     

Scales of Reservoir Heterogeneities and Impact of Seismic Resolution on Geostatistical Integration

Seismic measurements may be used in geostatistical techniques for estimation and simulation of petrophysical properties such as porosity. The good correlation between seismic and rock properties provides a basis for these techniques. Seismic data have a wide spatial coverage not available in log or core data. However, each seismic measurement has a characteristic response function determined by the source-receiver geometry and signal bandwidth. The image response of the seismic measurement gives a filtered version of the true velocity image. Therefore the seismic image cannot reflect exactly the true seismic velocity at all scales of spatial heterogeneities present in the Earth. The seismic response function can be approximated conveniently in the spatial spectral domain using the Born approximation. How the seismic image response affects the estimation of variogram. and spatial scales and its impact on geostatistical results is the focus of this paper. Limitations of view angles and signal bandwidth not only smooth the seismic image, increasing the variogram range, but also can introduce anisotropic spatial structures into the image. The seismic data are enhanced by better characterizing and quantifying these attributes. As an exercise, examples of seismically assisted cokriging and cosimulation of porosity between wells are presented.     

Variograms of Order ω: A Tool to Validate a Bivariate Distribution Model

The multigaussian model is used in mining geostatistics to simulate the spatial distribution of grades or to estimate the recoverable reserves of an ore deposit. Checking the suitability of such model to the available data often constitutes a critical step of the geostatistical study. In general, the marginal distribution is not a problem because the data can be transformed to normal scores, so the check is usually restricted to the bivariate distributions. In this work, several tests for diagnosing the two-point normality of a set of Gaussian data are reviewed and commented. An additional criterion is proposed, based on the comparison between the usual variogram and the variograms of lower order: the latter are defined as half the mean absolute increments of the attribute raised to a power between 0 and 2. This criterion is then extended to other bivariate models, namely the bigamma, Hermitian and Laguerrian models. The concepts are illustrated on two real data-sets. Finally, some conditions to ensure the internal consistency of the variogram under a given model are given.     

Seismic reflectivity of detachment faults of the Iberian and Tethyan distal continental margins based on geological and petrophysical data

Low-angle detachment faults are key to our understanding of the tectonic evolution of magma-poor rifted continental margins. In seismic images of present-day rifted margins the identification and interpretation of such features is, however, notoriously difficult and ambiguous. We address this problem by studying the structure and seismic response of such faults through a synoptic interpretation of petrophysical data and geological evidence from the distal segments of the present-day West Iberian and the ancient Tethyan margins. On the basis of the geologically well-constrained remnants of the Tethyan margins, which are spectacularly preserved and exposed in the Alps of Eastern Switzerland, vertical profiles at four key geological settings of a typical magma-poor rifted margin are constructed and their synthetic seismic responses are compared to the observed seismic data from corresponding locations in the present-day Iberian margin. The seismic structure of these profiles is considered as the sum of deterministic large-scale and the stochastic small-scale components. Both components are analyzed for all pertinent lithologies. The large-scale structures are derived from laboratory measurements on samples from both, the West Iberian and Tethyan margins, whereas the small-scale fluctuations are constrained predominantly on the basis of well-log data from the Iberian margin. Different realizations of the simulated stochastic small-scale velocity fluctuations illustrate the potential variability of impedance contrasts and its impact on the seismic response from lithological interfaces and fault structures. Our results indicate that the nature of the seismic response from low-angle detachment faults is largely determined through the fracture-healing behavior of the surrounding rocks. Geological evidence from the exposed fragments of the Tethyan margins indicate that fracture-healing is generally well developed in crustal lithologies, but largely absent in mantle lithologies. It is for this reason that low-angle, intra-crustal detachment faults tend to be seismically undetectable. Conversely, crust–mantle detachments have a complex and variable seismic response, depending on the nature of the damaged zone and on the frequency content of the seismic data. These model-based inferences are consistent with the available evidence from the present-day Iberian passive margin and thus open new perspectives for the interpretation of the corresponding seismic images.     

Hierarchical Geostatistical Analysis of an Experimental Stratigraphy

A hierarchical geostatistical analysis is conducted on a high-resolution, multiscale hydraulic conductivity (ln K) map, created by scaling up an experimental stratigraphy. Unlike a previous study which evaluates ln K variograms within individual depositional environments, this study analyzes deposits (or samples) that incorporate multiple depositional environments. Based on conductivity cutoffs selected from a global ln K histogram, an indicator map is created to divide the deposits into 4 categories: sand, silty sand, clayey silt, and clay (Hierarchy-I). Based on facies and facies assemblage types selected using geological criteria, two more indicator maps are created at a higher hierarchy (Hierarchy-II) to divide the deposits into 14 units and 2 units, respectively. For each sample, its experimental ln K variogram is decomposed into 4 auto- and cross-transition component variograms. The decomposition characteristics are then evaluated against the underlying heterogeneity and specific division rule. The analysis reveals that: (1) ln K cutoffs (sand contents of the physical stratigraphy) can be used to distinguish the shifts in dominant deposition mode; (2) sample univariate modes depend on the choice of hierarchical division; (3) sample variograms exhibit smooth-varying correlation structures (exponential-like variograms are observed in samples with a large variance in mean facies length); (4) the decomposition characteristics are sensitive to the division based on conductivity cutoffs, but not sensitive to the division based on depositional environment (For all samples, with appropriate division, the sample variogram is closely approximated by the sum of the cross-transition component variograms.); and (5) at the Hierarchy-II level, the 2-unit division gives similar decomposition characteristics as the 14-unit division. For the select samples, parsimony in hierarchical division is achieved at the facies assemblage scale.     

Multivariable variogram and its application to the linear model of coregionalization

In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1.     

二态序列变差函数的定理及初步地质应用

提出并证明了一个关于二态序列变差函数的定理,即二元平稳标准化正态序列的变差函数与其以零为水平的0—1二元序列变差函数的关系式。该定理为通过对定性地质变量的空间序列变异性研究以达到对原定量地质变量空间变异性的研究提供了可能与方便。文中进行了数字仿真分析,并提供了一个地质方面的初步应用。     

天然气水合物地震空白带现象正演模型研究

空白带现象是人工地震方法识别含天然气水合物沉积地层的重要标志之一。针对水合物沉积物的 3种微观模式 ,首先采用度量振幅空白程度的振幅比方法 ,计算并分析了水合物饱和度不随孔隙度变化 ,以及水合物的饱和度随孔隙度变化 (即CGHC和VGHC)两种情况 ,然后对上述两种情况设计了地震模型 ,合成了水合物沉积物地震纪录 ,获得了地震空白带现象。研究结果表明 :水合物沉积物饱和度的变化是振幅空白的主要原因。对孔隙度和饱和度都不固定的复杂情况 ,沉积物的岩性存在多解性 ,需用各种方法的多元信息进行对比研究 ,才能更好地解释空白带的产生及其特征。     

Integrating Large-Scale Soft Data by Simulated Annealing and Probability Constraints

Interpretation of geophysical data or other indirect measurements provides large-scale soft secondary data for modeling hard primary data variables. Calibration allows such soft data to be expressed as prior probability distributions of nonlinear block averages of the primary variable; poorer quality soft data leads to prior distributions with large variance, better quality soft data leads to prior distributions with low variance. Another important feature of most soft data is that the quality is spatially variable; soft data may be very good in some areas while poorer in other areas. The main aim of this paper is to propose a new method of integrating such soft data, which is large-scale and has locally variable precision. The technique of simulated annealing is used to construct stochastic realizations that reflect the uncertainty in the soft data. This is done by constraining the cumulative probability values of the block average values to follow a specified distribution. These probability values are determined by the local soft prior distribution and a nonlinear average of the small-scale simulated values within the block, which are all known. For each realization to accurately capture the information contained in the soft data distributions, we show that the probability values should be uniformly distributed between 0 and 1. An objective function is then proposed for a simulated annealing based approach to enforce this uniform probability constraint. The theoretical justification of this approach is discussed, implementation details are considered, and an example is presented.