Teacher's Aide Variogram Interpretation and Modeling

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.     

Teacher''s Aide Variogram Interpretation and Modeling

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.     

井震联合建模技术在萨尔图油田B1DD区块的应用及其效果

为解决以往油藏描述工作单孔资料在深度和广度上存在明显局限性的问题,提出了一套井震联合地质建模的方法：首先以地震精细解释的层位数据和断层数据为约束,结合测井资料建立构造模型;然后采用地震反演体的概率面在平面上作为约束,纵向上在测井曲线上分类统计各微相的概率分布曲线,在变差函数分析的基础上,采用序贯指示模拟算法模拟出储层微相的空间展布,建立沉积相模型;最后采用相控技术和地质统计学理论,分析参数区域化变量的特征,建立储层属性模型。在大庆萨尔图油田的B1DD区块进行验证的结果表明,井震资料的联合应用揭示了大量的小断层,使断点组合率提高了15%,大于2 m的砂岩预测符合率达到90%以上,提高了井间断层和砂岩的预测精度,进而提高了地质模型、尤其是井间模型的精度。     

地质统计学反演及其在吉林扶余油田储层预测中的应用

地质统计学反演方法将随机建模技术与常规地震反演相结合,有效地综合地质、测井和三维地震数据,可以更加精确地描述储层的变化.在执行地质统计学反演前,首先应用稀疏脉冲约束反演,了解储层的大致分布,以求取子波和水平变差函数.地质统计学反演从井点出发,井间以原始地震数据作为硬数据,通过随机模拟的产生井间波阻抗,然后将波阻抗转换成反射系数,并用确定性反演方法求得的子波褶积产生地震道,通过反复迭代直至合成地震道与原始地震数据达到一定程度的匹配,反演结果是多个等概率的波阻抗数据体实现.反演结果符合输入数据的地质统计学特征并受地质模型的约束,它综合了测井的垂向分辨率高和地震的横向分辨率高的优势,结果的多个实现用于不确定性评价.     

缝洞型碳酸盐岩储层多类多尺度建模方法研究:以塔河油田四区奥陶系油藏为例

碳酸盐岩缝洞型储层历经多期构造运动以及强烈的风化、剥蚀和淋滤作用,储集空间类型多样,形态极不规则且随机分布,导致储层三维空间描述困难,现有的碎屑岩储层建模方法难以直接借鉴。文中提出碳酸盐岩缝洞型储层应该按照大型洞穴、溶蚀孔洞、大尺度裂缝、小尺度裂缝的"多类多尺度建模"的基本思路。以钻井和地震识别成果作为大型洞穴确定性数据,以地震波阻抗的大型洞穴发育概率体作为井间约束数据,在垂向岩溶分带和平面古地貌分区的岩溶相控下,采用具有趋势的序贯指示模拟方法,建立大型洞穴离散分布模型;以大型洞穴分布作为"相控"约束条件,以井孔解释的溶蚀孔洞作为硬数据,采用序贯指示模拟算法,建立溶蚀孔洞随机分布模型;根据蚂蚁体地震属性自动拾取的断裂信息,人机交互补充和修正地震解释断层数据,建立确定性的大尺度裂缝离散分布模型;基于大尺度裂缝离散分布模型建立井间裂缝发育概率体,根据井孔裂缝密度、裂缝产状,结合退火模拟和基于目标的示性点过程模拟方法,建立小尺度裂缝离散分布模型。以塔河油田四区奥陶系缝洞型油藏为例,建立研究区缝洞型储集体空间展布模型,再现缝洞型储层的结构形态。     

基于马蒙算法地质统计学反演的缝洞储集体预测

塔河油田缝洞储集体空间非均质性强,基于地震资料的确定性预测方法表征缝洞储集体时存在多解性. 采用一种基于马蒙算法的地质统计学反演方法实现了缝洞储集体的定量预测并作出不确定性评价. 首先从地质、地球物理角度获取目标区储集体发育的先验信息,再进行地质统计学参数的实验调试,最终利用地质统计学反演获得储集体岩相类型结果及波阻抗结果的多个实现,并对预测结果作出不确定性评价. 该方法提供了一种地震数据主导的缝洞储集体定量预测手段,能够较好地解决溶洞储集体纵向深度归位、地震预测成果不确定评价问题,对于该类油藏的地质建模表征具有指导意义.     

Conditioning Surface-Based Geological Models to Well and Thickness Data

Geostatistical simulation methods aim to represent spatial uncertainty through realizations that reflect a certain geological concept by means of a spatial continuity model. Most common spatial continuity models are either variogram, training image, or Boolean based. In this paper, a more recent spatial model of geological continuity is developed, termed the event, or surface-based model, which is specifically applicable to modeling cases with complex stratigraphy, such as in sedimentary systems. These methods rely on a rule-based stacking of events, which are mathematically represented by two-dimensional thickness variations over the domain, where positive thickness is associated with deposition and negative thickness with erosion. Although it has been demonstrated that the surface-based models accurately represent the geological variation present in complex layered systems, they are more difficult to constrain to hard and soft data as is typically required of practical geostatistical techniques. In this paper, we develop a practical methodology for constraining such models to hard data from wells and thickness data interpreted from geophysics, such as seismic data. Our iterative methodology relies on a decomposition of the parameter optimization problem into smaller, manageable problems that are solved sequentially. We demonstrate this method on a real case study of a turbidite sedimentary basin.     

应用随机反演改善储层表征的过程

为了适当地完成储层表征的过程,一个有效的方法就是把现场所有可以利用的信息融合成一个一致性的模型。在实际生产中实现这种融合并非简单的任务,所以有必要运用如地震反演等特殊方法。应用地震反演可以使测井数据和地震数据的有效结合成为可能,并且可以得到一个模型,该模型在预测过程中可通过流体数字模拟来验证。地震反演可以通过多种方法进行,主要分为两大类:一类是确定性方法(其代表是回归反演和约束稀疏脉冲反演),另一类是随机方法(其代表是地质统计学反演)。在本次研究中,通过随机反演结果和确定性反演结果的对比展示了随机反演是如何改进储层表征过程的。事实上,随机反演,可以运用较高的采样率(和储层模型的网格大小相接近),来产生一个更可靠的模型。随机反演的另一个好处就是随机方法可产生一些基本的统计测量值来改进解释精度,并且在储层表征过程中能生成大量的实现,从而使储层模型的不确定性研究成为可能。     

Using non-Gaussian distributions in geostatistical simulations

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.     

Sedimentary Microfacies and Porosity Modeling of Deep-Water Sandy Debris Flows by Combining Sedimentary Patterns with Seismic Data: An Example from Unit I of Gas Field A, South China Sea

Sandy debris flow deposits are present in Unit I during Miocene of Gas Field A in the Baiyun Depression of the South China Sea. The paucity of well data and the great variability of the sedimentary microfacies make it difficult to identify and predict the distribution patterns of the main gas reservoir, and have seriously hindered further exploration and development of the gas field. Therefore, making full use of the available seismic data is extremely important for predicting the spatial distribution of sedimentary microfacies when constructing three-dimensional reservoir models. A suitable reservoir modeling strategy or workflow controlled by sedimentary microfacies and seismic data has been developed. Five types of seismic attributes were selected to correlate with the sand percentage, and the root mean square (RMS) amplitude performed the best. The relation between the RMS amplitude and the sand percentage was used to construct a reservoir sand distribution map. Three types of main sedimentary microfacies were identified: debris channels, fan lobes, and natural levees. Using constraints from the sedimentary microfacies boundaries, a sedimentary microfacies model was constructed using the sequential indicator and assigned value simulation methods. Finally, reservoir models of physical properties for sandy debris flow deposits controlled by sedimentary microfacies and seismic inversion data were established. Property cutoff values were adopted because the sedimentary microfacies and the reservoir properties from well-logging interpretation are intrinsically different. Selection of appropriate reservoir property cutoffs is a key step in reservoir modeling when using simulation methods based on sedimentary microfacies control. When the abnormal data are truncated and the reservoir properties probability distribution fits a normal distribution, microfacies-controlled reservoir property models are more reliable than those obtained from the sequence Gauss simulation method. The cutoffs for effective porosity of the debris channel, fan lobe, and natural levee facies were 0.2, 0.09, and 0.12, respectively; the corresponding average effective porosities were 0.24, 0.13, and 0.15. The proposed modeling method makes full use of seismic attributes and seismic inversion data, and also makes the property data of single-well depositional microfacies more conformable to a normal distribution with geological significance. Thus, the method allows use of more reliable input data when we construct a model of a sandy debris flow.     

A Fast Approximation for Seismic Inverse Modeling: Adaptive Spatial Resampling

Seismic inverse modeling, which transforms appropriately processed geophysical data into the physical properties of the Earth, is an essential process for reservoir characterization. This paper proposes a work flow based on a Markov chain Monte Carlo method consistent with geology, well-logs, seismic data, and rock-physics information. It uses direct sampling as a multiple-point geostatistical method for generating realizations from the prior distribution, and Metropolis sampling with adaptive spatial resampling to perform an approximate sampling from the posterior distribution, conditioned to the geophysical data. Because it can assess important uncertainties, sampling is a more general approach than just finding the most likely model. However, since rejection sampling requires a large number of evaluations for generating the posterior distribution, it is inefficient and not suitable for reservoir modeling. Metropolis sampling is able to perform an equivalent sampling by forming a Markov chain. The iterative spatial resampling algorithm perturbs realizations of a spatially dependent variable, while preserving its spatial structure by conditioning to subset points. However, in most practical applications, when the subset conditioning points are selected at random, it can get stuck for a very long time in a non-optimal local minimum. In this paper it is demonstrated that adaptive subset sampling improves the efficiency of iterative spatial resampling. Depending on the acceptance/rejection criteria, it is possible to obtain a chain of geostatistical realizations aimed at characterizing the posterior distribution with Metropolis sampling. The validity and applicability of the proposed method are illustrated by results for seismic lithofacies inversion on the Stanford VI synthetic test sets.     

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.     

Is the ocean floor a fractal?

The topographic structure of the ocean bottom is investigated at different scales of resolution to answer the question: Can the seafloor be described as a fractal process? Methods from geostatistics, the theory of regionalized variables, are used to analyze the spatial structure of the ocean floor at different scales of resolution. The key to the analysis is the variogram criterion: Self-similarity of a stochastic process implies self-similarity of its variogram. The criterion is derived and proved here: it also is valid for special cases of self-affinity (in a sense adequate for topography). It has been proposed that seafloor topography can be simulated as a fractal (an object of Hausdorff dimension strictly larger than its topological dimension), having scaling properties (self-similarity or self-affinity). The objective of this study is to compare the implications of these concepts with observations of the seafloor. The analyses are based on SEABEAM bathymetric data from the East Pacific Rise at 13°N/104°W and at 9°N/104°W and use tracks that run both across the ridge crest and along the ridge flank. In the geostatistical evaluation, the data are considered as a stochastic process. The spatial continuity of this process is described by variograms that are calculated for different scales and directions. Applications of the variogram criterion to scale-dependent variogram models yields the following results: Although the seafloor may be a fractal in the sense of the definition involving the Hausdorff dimension, it is not self-similar, nor self-affine (in the given sense). Mathematical models of scale-dependent spatial structures are presented, and their relationship to geologic processes such as ridge evolution, crust formation, and sedimentation is discussed.     

Conditional curvilinear stochastic simulation using pixel-based algorithms

In geology, structures displaying differing local directions of continuity are widespread, a typical example being a flusial depositional system. Conventional pixel-based geostatistical algorithms, may fail to reproduce such curvilinear structures. Conversely, object-based algorithms can reproduce curvilinear shapes but are difficult to condition to dense local data. Local depositional directions as obtained from dipmeter data. 3D seismic data, and geological interpretation represent critical information. An improved pixel-based geostatistical algorithm is proposed to account for such directional information. Case studies demonstrate the potential and limitations of the algorithm.     

Is the ocean floor a fractal?

The topographic structure of the ocean bottom is investigated at different scales of resolution to answer the question: Can the seafloor be described as a fractal process? Methods from geostatistics, the theory of regionalized variables, are used to analyze the spatial structure of the ocean floor at different scales of resolution. The key to the analysis is the variogram criterion: Self-similarity of a stochastic process implies self-similarity of its variogram. The criterion is derived and proved here: it also is valid for special cases of self-affinity (in a sense adequate for topography). It has been proposed that seafloor topography can be simulated as a fractal (an object of Hausdorff dimension strictly larger than its topological dimension), having scaling properties (self-similarity or self-affinity). The objective of this study is to compare the implications of these concepts with observations of the seafloor. The analyses are based on SEABEAM bathymetric data from the East Pacific Rise at 13°N/104°W and at 9°N/104°W and use tracks that run both across the ridge crest and along the ridge flank. In the geostatistical evaluation, the data are considered as a stochastic process. The spatial continuity of this process is described by variograms that are calculated for different scales and directions. Applications of the variogram criterion to scale-dependent variogram models yields the following results: Although the seafloor may be a fractal in the sense of the definition involving the Hausdorff dimension, it is not self-similar, nor self-affine (in the given sense). Mathematical models of scale-dependent spatial structures are presented, and their relationship to geologic processes such as ridge evolution, crust formation, and sedimentation is discussed.     

Modelling Spatial Variability Along Drainage Networks with Geostatistics

Local characteristics of drainage networks such as cross-section geometry and hydraulic roughness coefficient, influence surface water transfers and must be taken into account when assessing the impact of human activities on hydrological risks. However, as these characteristics have not been available till now through remote sensing or hydrological modelling, the only available methods are interpolation or simulation based on scarce data. In this paper we propose a statistical model based on geostatistics that allows us to take account of both the spatial distribution and spatial uncertainties. To do this, we modify the geostatistical framework to suit directed tree supports corresponding to drainage network structures. The stationarity concept is specified assuming conditional independence between parts of the network; variogram fitting and modelling are then modified accordingly. A sequential multi Gaussian simulation procedure going upstream along the network is proposed. We illustrate this approach by studying the width of an 11-km long artificial drainage network in the south of France.     

不同变差函数变程下随机反演与随机模拟对比分析

随机模拟和随机反演是两种比较常用的较精细的储层预测方法,而变差函数是随机模拟、随机反演的核心技术。以喇嘛甸油田北北二区块SⅢ层段为研究对象,选用300 m×300 m、600 m×600 m、800 m×800 m、1 200 m×1 200 m不同变程变差函数,对随机模拟和随机反演结果对比分析,探讨不同变程对预测结果的影响。结果显示：随着变程的增大,随机模拟砂体结果由离散发展成连续片状分布,而随机反演砂体分布整体趋势一致,只有局部薄层砂体受到较小程度的影响;通过盲井验证,随机反演较随机模拟,其预测的砂体与井点的符合率更高。分析认为,随机反演较随机模拟多了井间地震的约束,减少了砂体分布的不确定性和多解性,提高了井间砂体的预测精度,便于在油田开发阶段推广应用。     

Gradual Deformation and Iterative Calibration of Sequential Stochastic Simulations

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.     

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.