Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.     

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。     

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.     

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。     

Substitution Random Fields with Gaussian and Gamma Distributions: Theory and Application to a Pollution Data Set

This paper presents random field models with Gaussian or gamma univariate distributions and isofactorial bivariate distributions, constructed by composing two independent random fields: a directing function with stationary Gaussian increments and a stationary coding process with bivariate Gaussian or gamma distributions. Two variations are proposed, by considering a multivariate directing function and a coding process with a separable covariance, or by including drift components in the directing function. Iterative algorithms based on the Gibbs sampler allow one to condition the realizations of the substitution random fields to a set of data, while the inference of the model parameters relies on simple tools such as indicator variograms and variograms of different orders. A case study in polluted soil management is presented, for which a gamma model is used to quantify the risk that pollutant concentrations over remediation units exceed a given toxicity level. Unlike the multivariate Gaussian model, the proposed gamma model accounts for an asymmetry in the spatial correlation of the indicator functions around the median and for a spatial clustering of high pollutant concentrations.     

Lithofacies modeling by multipoint statistics and economic evaluation by NPV volume for the early Cretaceous Wabiskaw Member in Athabasca oilsands area,Canada

The static modeling and dynamic simulation are essential and critical processes in petroleum exploration and development. In this study, lithofacies models for Wabiskaw Member in Athabasca, Canada are generated by multipoint statistics(MPS) and then compared with the models built by sequential indicator simulation(SIS). Three training images(Tls) are selected from modern depositional environments;the Orinoco River Delta estuary, Cobequid bay-Salmon River estuary, and Danube River delta environment. In order to validate lithofacies models, average and variance of similarity in lithofacies are calculated through random and zonal blind-well tests.In random six-blind-well test, similarity average of MPS models is higher than that of SIS model. The Salmon MPS model closely resembles facies pattern of Wabiskaw Member in subsurface. Zonal blind-well tests show that successful lithofacies modeling for transitional depositional setting requires additional or proper zonation information on horizontal variation, vertical proportion, and secondary data.As Wabiskaw Member is frontier oilsands lease, it is impossible to evaluate the economics from production data or dynamic simulation. In this study, a dynamic steam assisted gravity drainage(SAGD)performance indicator(SPIDER) on the basis of reservoir characteristics is calculated to build 3 D reservoir model for the evaluation of the SAGD feasibility in Wabiskaw Member. SPIDER depends on reservoir properties, economic limit of steam-oil ratio, and bitumen price. Reservoir properties like porosity,permeability, and water saturation are measured from 13 cores and calculated from 201 well-logs. Three dimensional volumes of reservoir properties are constructed mostly based on relationships among properties. Finally, net present value(NPV) volume can be built by equation relating NPV and SPIDER. The economic area exceeding criterion of US$ 10,000 is identified, and the ranges of reservoir properties are estimated. NPV-volume-generation workflow from reservoir parameter to static model provides costand time-effective method to evaluate the oilsands SAGD project.     

Conditional simulation of intrinsic random functions of order k

Conditional simulation of intrinsic random functions of orderk is a stochastic method that generates realizations which mimic the spatial fluctuation of nonstationary phenomena, reproduce their generalized covariance and honor the available data at sampled locations. The technique proposed here requires the following steps: (i) on-line simulation of Wiener-Levy processes and of their integrations; (ii) use of the turning-bands method to generate realizations in Rⁿ; (iii) conditioning to available data; and (iv) verification of the reproduced generalized covariance using generalized variograms. The applicational aspects of the technique are demonstrated in two and three dimensions. Examples include the conditional simulation of geological variates of the Crystal Viking petroleum reservoir, Alberta, Canada.     

基于精细岩相约束的致密低渗储层参数建模研究——以红河油田92井区长812致密砂岩储层为例

鄂尔多斯盆地红河油田属于低孔、超低渗岩性油气藏.红河油田长8储层主要发育辫状河三角洲前缘水下分流河道和分流间湾两种沉积微相,并且在水下分流河道沉积微相中储层参数变化范围大,非均质性强,用常规的单一沉积相控建模方法无法对储层参数空间分布的非均质性进行充分表征.为了精确描述其储层参数及展布特征,以红河油田92井区长8₁²致密砂岩储层为例,在地质、岩心、测井等资料的基础上,应用基于精细岩相约束的储层参数建模方法建立了研究区三维地质模型,即先用沉积相发育模式约束建立沉积微相模型,再对水下分流河道沉积微相进行岩相细分,在沉积微相模型和不同岩相概率体的双重控制下建立岩相模型,然后以岩相模型为约束建立储层参数模型.结果表明,该方法建立的储层参数模型具有较高的可靠性,与地质认识符合较好,为该区油藏数值模拟提供了精确的地质模型,并为研究区下一步开发方案调整提供了地质依据.     

基于精细岩相约束的致密低渗储层参数建模研究——以红河油田92井区长812致密砂岩储层为例

鄂尔多斯盆地红河油田属于低孔、超低渗岩性油气藏.红河油田长8储层主要发育辫状河三角洲前缘水下分流河道和分流间湾两种沉积微相，并且在水下分流河道沉积微相中储层参数变化范围大，非均质性强，用常规的单一沉积相控建模方法无法对储层参数空间分布的非均质性进行充分表征.为了精确描述其储层参数及展布特征，以红河油田92井区长8₁²致密砂岩储层为例，在地质、岩心、测井等资料的基础上，应用基于精细岩相约束的储层参数建模方法建立了研究区三维地质模型，即先用沉积相发育模式约束建立沉积微相模型，再对水下分流河道沉积微相进行岩相细分，在沉积微相模型和不同岩相概率体的双重控制下建立岩相模型，然后以岩相模型为约束建立储层参数模型.结果表明，该方法建立的储层参数模型具有较高的可靠性，与地质认识符合较好，为该区油藏数值模拟提供了精确的地质模型，并为研究区下一步开发方案调整提供了地质依据.     

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.     

Positive definiteness is not enough

Geostatisticians know that the mathematical functions chosen to represent spatial covariances and variograms must have the appropriate type of positive definiteness, but they may not realize that there are restrictions on the types of covariances and variograms that are compatible with particular distributions. This paper gives some examples showing that (1) the spherical model is not compatible with the multivariate lognormal distribution if the coefficient of variation is 2.0 or more (even in 1-D), and (2) the Gaussian covariance and several other models are not compatible with indicator random functions. As these examples concern quite different types of random functions, it is clear that there is a general problem of compatibility between spatial covariance models (or variograms) and a specified multivariate distribution. The problem arises with all distributions except the multivariate normal, and not just the two cited here. The need for a general theorem giving the necessary and sufficient conditions for a covariance or a variogram to be compatible with a particular distribution is stressed.     

Simulation of anisotropic heterogeneous near-well flow using MPFA methods on flexible grids

Control-volume multipoint flux approximations (MPFA) are discussed for the simulation of complex near-well flow using geometrically flexible grids. Due to the strong non-linearity of the near-well flow, a linear model will, in general, be inefficient. Instead, a model accounting for the logarithmic pressure behavior in the well vicinity is advocated. This involves a non-uniform refinement of the grid in the radial direction. The model accounts for both near-well anisotropies and heterogeneities. For a full simulation involving multiple wells, this single-well approach can easily be coupled with the reservoir model. Numerical simulations demonstrate the convergence behavior of this model using various MPFA schemes under different near-well conditions for single-phase flow regimes. Two-phase simulations support the results of the single-phase simulations.     

Comparing sequential Gaussian and turning bands algorithms for cosimulating grades in multi-element deposits

Stochastic simulation is increasingly used to map the spatial variability in the grades of elements of interest and to assess the uncertainty in the mineral resources and ore reserves. The practical implementation requires specifying a stochastic model, which describes the spatial distribution of the grades, and an algorithm to construct realizations of these grades, viewed as different possible outcomes or scenarios. In the case of the Gaussian random field model, a variety of algorithms have been proposed in the past decades, but their ability to reproduce the model statistics is often unequal. In this paper, we compare two such algorithms, namely the turning bands and the sequential algorithms. The comparison is hold through a synthetic case study and a real case study in a porphyry copper deposit located in southeastern Iran, in which it is of interest to jointly simulate the copper, molybdenum, silver, lead and zinc grades. Statistical testing and graphical validations are realized to check whether or not the realizations reproduce the features of the true grades, in particular their direct and cross variograms. Sequential simulation based on collocated cokriging turns out to poorly reproduce the cross variograms, while turning bands proves to be accurate in all the analyzed cases.     

青海木里三露天天然气水合物分布与储层特征

根据在青海木里三露天施工完成的14 口天然气水合物钻孔揭露的地质信息,从天然气水合物产出、赋存、空间展布3个方面介绍工作区天然气水合物基本特征。此外,文章还以岩性特征、空间展布、裂隙、岩石物性多元控制因素 间相互制约和促进的内在联系为桥梁建立了天然气水合物储层这一研究对象。从储层方面分析,受岩相的控制作用三露天天然气水合物在空间上的分布有一定的选择性、延展性和可追索性, 即其储层具有相对稳定性;三露天天然气水合物在储层内部的分布除了受有利储层产状变化限制外,还受到裂隙产状的控制,致使其在空间上分布既不规则、又往往突然消失或突然出现。     

Adding Local Accuracy to Direct Sequential Simulation

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.     

Quality mapping of the Ryytimaa dolomite in western Finland

The Ryytimaa dolomite formation in western Finland is mined for fertilizing purposes in agriculture. Ordinary kriging, indicator kriging, and indicator principal components simulation were used to map the dolomite quality. The fertilizing properties depend on the relative amounts of MgO and neutralizing CaO in dolomite. Based on the chemical composition the dolomite is divided into six quality classes according to the Finnish legislation. This quality classification is only partially in accordance with the three populations inferred from the distributions of the MgO, CaO, and neutralizing CaO contents. These populations probably represent compositionally different dolomite layers, now forming complicated fold interference structures that are visible on the kriged maps. The mapping of the quality, and thus, selective mining of the dolomite deposit is possible using the quality classification based on the three populations. In contrast, selective mining of the dolomite based on the legislation is difficult—at least with techniques used in this study.     

Use of indicator variograms for an enhanced spatial analysis

Flat variograms often are interpreted as representing a lack of spatial autocorrelation. Recent research in earthquake engineering shows that nearby field noise can substantially mask a prominent spatial autocorrelation and result in what appears to be a purely random spatial process. A careful selection of threshold in assigning an indicator function can yield an indicator variogram which reveals underlying spatial autocorrelation. Although this application involves use of seismic data, the results are relevant to geostatistical applications in general.     

Use of Variography in Permeability Characterization of Visually Homogeneous Sandstone Reservoirs with Examples from Outcrop Studies

Sandstones of different ages provide economically significant oil, gas, and water reservoirs. In sandstones where heterogeneities are not visually obvious, it is particularly difficult to predict the location of permeability barriers and the scale at which high and low permeability zones occur, yet this is critical in providing information on hydrocarbon reservoir performance. This study uses variogram analysis to investigate spatial variation in permeability in visually homogeneous reservoir sandstone successions. Air permeability measurements were taken using unsteady state probe permeametry following regular grid schemes with centimeter spacing. Spatial variation in permeability was characterized using omnidirectional and directional variograms. This study combines variography with geological interpretation to assess the degree of heterogeneity of permeability in visually homogeneous sandstone successions. Variography indicates spatial dependence and short-range variation at 1 cm grid spacings that is not apparent at a larger 5 cm grid spacing in the visually homogeneous sandstones studied. The range of the models fitted to the variograms provide a potentially important index of spatial variability in permeability for different depositional settings including aeolian, fluvial, shallow marine, and marine/mass- flow turbidite.     

油气地质模型中流体流动机制的模拟研究

通过数值模拟与物理模拟分别研究了油气在聚集后散失过程的流体流动机制。流体场的力学机制决定了油气在散失过程中的差异性以及油气层自身流动的独立性。研究表明:一个局部高点的石油泄漏对另一个高点石油保存影响不大或基本没有影响,虽一个高点泄漏而两个高点的天然气均能被产出,遭受破坏的聚集带油藏一般比较孤立而气藏却成组分布。这对同一聚集带上的油、气勘探有重要的指导意义。