Evaluation of the Reduction in Uncertainty Obtained by Conditioning a 3D Stochastic Channel to Multiwell Pressure Data

A stochastic channel embedded in a background facies is conditioned to data observed at wells. The background facies is a fixed rectangular box. The model parameters consist of geometric parameters that describe the shape, size, and location of the channel, and permeability and porosity in the channel and nonchannel facies. We extend methodology previously developed to condition a stochastic channel to well-test pressure data, and well observations of the channel thickness and the depth of the top of the channel. The main objective of this work is to characterize the reduction in uncertainty in channel model parameters and predicted reservoir performance that can be achieved by conditioning to well-test pressure data at one or more wells. Multiple conditional realizations of the geometric parameters and rock properties are generated to evaluate the uncertainty in model parameters. The ensemble of predictions of reservoir performance generated from the suite of realizations provides a Monte Carlo estimate of the uncertainty in future performance predictions. In addition, we provide some insight on how prior variances, data measurement errors, and sensitivity coefficients interact to determine the reduction in model parameters obtained by conditioning to pressure data and examine the value of active and observation well data in resolving model parameters.     

Well Conditioning in a Fluvial Reservoir Model

This paper describes a method for conditioning an object model of a fluvial reservoir on facies observations. The channels are assumed parametrized at sections normal to their main channel direction. Projections of the observations on these sections generates a map suitable for drawing conditioning values. This map contains the information from every facies observation between two adjacent sections, enabling handling of any well path. Coupling between well observations is also discussed. The methodology is implemented and demonstrated in examples with complex wells.     

Generalized Sequential Gaussian Simulation on Group Size ν and Screen-Effect Approximations for Large Field Simulations

The modelling of spatial uncertainty in attributes of geological phenomena is frequently based on the stochastic simulation of Gaussian random fields. This paper presents a generalization of the sequential Gaussian simulation method founded upon the group decomposition of the posterior probability density function of a stationary and ergodic Gaussian random field into posterior probability densities of a set of groups of nodes of size . The method, which is termed generalized sequential Gaussian simulation on group size , relies computationally on sharing the neighborhood of adjacent nodes and simulates groups of nodes at a time, instead of the traditional node-by-node simulation. The new method is computationally efficient and suitable for simulation on large grids of nodes. Results suggest that, in terms of computing cost (time), an optimal size of a group is about 80% of the optimal neighborhood used for sequential Gaussian simulation and that computation can be up to 50 times faster than the regular sequential Gaussian method, with little loss in accuracy. The effectiveness of the method is assessed by using a general measure of accuracy, screen-effect approximation loss (SEA loss), defined herein as the mean-square difference between the simulated value posterior to the information in the neighborhood and the simulated value posterior to all information, and shown to be determined by the corresponding posterior variances. The results presented show that both the exponential and the spherical models perform well and can meet the less-than 5% relative SEA loss requirement for any grid setup using a relatively small neighborhood. The Gaussian covariance model was found to have a relatively high relative SEA loss in most cases.     

Conditioning Fractal (fBm/fGn) Porosity and Permeability Fields to Multiwell Pressure Data

Stochastic fractal (fGn and fBm) porosity and permeability fields are conditioned to given variogram, static (or hard), and multiwell pressure data within a Bayesian estimation framework. Because fGn distributions are normal/second-order stationary, it is shown that the Bayesian estimation methods based on the assumption of normal/second-order stationary distributions can be directly used to generate fGn porosity/permeability fields conditional to pressure data. However, because fBm is not second-order stationary, it is shown that such Bayesian estimation methods can be used with implementation of a pseudocovariance approach to generate fBm porosity/permeability fields conditional to multiwell pressure data. In addition, we provide methods to generate unconditional realizations of fBm/fGn fields honoring all variogram parameters. These unconditional realizations can then be conditioned to hard and pressure data observed at wells by using the randomized maximum likelihood method. Synthetic examples generated from one-, two-, and three-dimensional single-phase flow simulators are used to show the applicability of our methodology for generating realizations of fBm/fGn porosity and permeability fields conditioned to well-test pressure data and evaluating the uncertainty in reservoir performance predictions appropriately using these history-matched realizations.     

Conditioning Geostatistical Operations to Nonlinear Volume Averages

The all-important process of data integration calls for algorithms that can handle secondary data often defined as nonlinear averages of the primary (hard) data over specific areas or volumes. It is suggested to approximate these nonlinear averages by linear averages of a nonlinear transform of the primary variable. Kriging of such nonlinear transforms, followed by the inverse transform, allows exact reproduction of all original data, both of point support and nonlinear volume averages. In a simulation mode, the previous cokriging provides the mean and variance of a conditional distribution from which to draw a simulated value, which is then backtransformed into a simulated value of the primary variable. The nonlinear averaged data values are then reproduced exactly. The direct sequential simulation algorithm adopted does not call for using any Gaussian distribution.     

An Initial Guess for the Levenberg–Marquardt Algorithm for Conditioning a Stochastic Channel to Pressure Data

A standard procedure for conditioning a stochastic channel to well-test pressure data requires the minimization of an objective function. The Levenberg–Marquardt algorithm is a natural choice for minimization, but may suffer from slow convergence or converge to a local minimum which gives an unacceptable match of observed pressure data if a poor initial guess is used. In this work, we present a procedure to generate a good initial guess when the Levenberg–Marquardt algorithm is used to condition a stochastic channel to pressure data and well observations of channel facies, channel thickness, and channel top depth. This technique yields improved computational efficiency when the Levenberg–Marquardt method is used as the optimization procedure for generating realizations of the model by the randomized maximum likelihood method.     

Well Conditioning in Object Models

This paper presents two object models with corresponding simulation algorithms, which aim to condition well data correctly while still converging in reasonable time. The first model is devoted to fluvial channels and the second one is mainly intended for smaller objects. To verify the conditioning, a method for validating well conditioning algorithms for object models is given. The purpose is to determine the extent to which the well conditioning introduces a bias in the models. To do this, we check that the double expectation of a parameter conditioned to wells is equal to the unconditional expectation. This method is applied to two different object models. Both the conditioning algorithms presented here give good results using this test.     

Conditioning Simulations of Gaussian Random Fields by Ordinary Kriging

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.     

距离法定量评价储层地质模型的不确定性

针对目前常用的储层地质模型不确定性评价方法存在的主要问题,提出了先用距离函数计算模型之间的差异,再以差异的大小来判断不确定性大小的方法.以WZ油田西区为例,采用相控物性参数建模技术,利用顺序高斯模拟方法建立渗透率的三维模型.对各种度量差异的距离函数进行对比研究和分析,结果显示曼哈顿距离函数和欧氏距离函数能较好地刻画模型之间的差异.选用欧氏距离函数计算模型之间的差异,其原理是先计算每两个模型之间相对应的每一网格节点的渗透率值差的平方和,然后取平方根,得到一个表征各模型之间差异的矩阵.根据该矩阵可得到各个模型之间的差异程度,差异越大,不确定性就越大.最后通过对比模型过井剖面图分析结果与距离矩阵分析结果,说明了本方法的正确性,结果显示该方法能有效评价随机模拟生成的储层地质模型的不确定性.     

Approximate local confidence intervals under change of support

Approximate local confidence intervals are constructed from uncertainty models in the form of the conditional distribution of the random variable Z given values of variables [Z_i, i=1,...,n]. When the support of the variable Z is any support other than that of the data, the conditional distributions require a change of support correction. This paper investigates the effect of change of support on the approximate local confidence intervals constructed by cumulative indicator kriging, class indicator kriging, and probability kriging under a variety of conditions. The conditions are generated by three simulated deposits with grade distributions of successively higher degree of skewness; a point support and two different block supports are considered. The paper also compares the confidence intervals obtained from these methods using the most used measures of confidence interval effectiveness.     

水文随机模拟进展

综述了近20年来水文随机模拟的新进展,包括三方面:①随机水文模型改进和创新;②水文随机模拟应用研究新进展;③水文随机模拟认识新进展。并指出了今后的研究重点:①对水文过程的重要物理特性和统计特性作深入的分析;②加强非参数模型和非线性模型的研究;③加强流域系统随机模型的研究;④加强建立模型时如何综合利用多种信息的研究;⑤加强模型的各种检验和合理分析。     

A Spatial-Temporal/3-D Model for Volcanic Hazard Assessment: Application to the Yucca Mountain Region, Nevada

We present a 3-D Poisson model that permits identification and quantification of volcanic phenomena distributed through space and evolving in time (i.e., spatiotemporal data). Specifically, the model: (1) is volcanologically informative in solving problems of volcanic risk/hazard which depends on the location and time of future events; (2) contains model fitting computation algorithms that are efficient; and (3) is flexible enough to handle a large class of volcanic risk/hazard studies. Furthermore, we apply the model fitting techniques developed in this paper to the volcanic data from the Yucca Mountain project to demonstrate a unified volcanic hazard analysis. This study also evaluates the sensitivity of the statistical models developed by experts who have addressed the volcanic hazard/risk assessment problem near the Yucca Mountain region.     

Application of turning bands technique to simulate values of copper ore deposit parameters in Rudna mine (Lubin-Sieroszowice region in SW part of Poland)

ABSTRACT

The turning bands simulation is a valuable and highly useful tool in solving various geological-mining, environmental and geological-engineering problems when it is essential to determine the uncertainty of the estimates of simulated values Z_s (realizations) and assess the risk. This paper presents an investigative methodology and the results of calculations connected with the use of conditional turning bands simulation and bundled indicator kriging, making it possible to analyse the risk at different levels of uncertainty in the solution of optimization of the exploitation problems encountered in the mining of the polymetallic copper ore deposits in the Lubin-Sieroszowice region (Foresudetic monocline, the SW part of Poland). Examples of the evaluation of simulated values Z_s and probability P average values Z* of the deposit parameters within the block located in the Rudna Mine (the block R-3) area are provided.     

地质条件不确定性下TBM管片结构失事概率计算

不良地质条件是影响TBM施工隧洞管片结构安全的重要因素。综合考虑围岩地质条件和衬砌结构的不确定性,提出了一种定量分析TBM管片结构失事概率的新方法。在基于Markov过程估计隧洞沿程地质岩性变化概率的基础上,建立了隧洞任意位置处管片选型不匹配的概率模型;考虑围岩和管片参数的不确定性,采用随机有限元方法计算某一类型管片在不同围岩下的失事概率;由此,采用全概率公式,可计算隧洞沿程任意位置处管片结构的失事概率。结合实际工程,针对施工期工况,确定了该隧洞管片沿程的失事概率、最大失事概率及其所对应的位置等,为管片选型、优化设计及TBM施工期的风险防范提供了依据。     

Correcting the Smoothing Effect of Estimators: A Spectral Postprocessor

The postprocessing algorithm introduced by Yao for imposing the spectral amplitudes of a target covariance model is shown to be efficient in correcting the smoothing effect of estimation maps, whether obtained by kriging or any other interpolation technique. As opposed to stochastic simulation, Yao's algorithm yields a unique map starting from an original, typically smooth, estimation map. Most importantly it is shown that reproduction of a covariance/semivariogram model (global accuracy) is necessarily obtained at the cost of local accuracy reduction and increase in conditional bias. When working on one location at a time, kriging remains the most accurate (in the least squared error sense) estimator. However, kriging estimates should only be listed, not mapped, since they do not reflect the correct (target) spatial autocorrelation. This mismatch in spatial autocorrelation can be corrected via stochastic simulation, or can be imposed a posteriori via Yao's algorithm.     

Hydraulic Effects of Shales in Fluvial-Deltaic Deposits: Ground-Penetrating Radar, Outcrop Observations, Geostatistics, and Three-Dimensional Flow Modeling for the Ferron Sandstone, Utah

Ground-penetrating radar (GPR) surveys, outcrop measurements, and cores provide a high-resolution 3D geologic model to investigate the hydraulic effects of shales in marine-influenced lower delta-plain distributary channel deposits within the Cretaceous-age Ferron Sandstone at Corbula Gulch in central Utah, USA. Shale statistics are computed from outcrop observations. Although slight anisotropy was observed in mean length and variogram ranges parallel and perpendicular to pale of low , the anisotropy is not statistically significant and the estimated mean length is 5.4 m. Truncated Gaussian simulation was used to create maps of shales that are placed on variably dipping stratigraphic surfaces interpreted from high-resolution 3D GPR surveys, outcrop interpretations, and boreholes. Sandstone permeability is estimated from radar responses calibrated to permeability measurements from core samples. Experimentally designed flow simulations examine the effects of variogram range, shale coverage fraction, and trends in shale coverage on predicted upscaled permeability, breakthrough time, and sweep efficiency. Approximately 1500 flow simulations examine three different geologic models, flow in the 3 coordinate directions, 16 geostatistical parameter combinations, and 10 realizations for each model. ANOVA and response models computed from the flow simulations demonstrate that shales decrease sweep, recovery, and permeability, especially in the vertical direction. The effect on horizontal flow is smaller. Flow predictions for ideal tracer displacements at Corbula Gulch are sensitive to shale-coverage fraction, but are relatively insensitive to twofold variations in variogram range or to vertical trends in shale coverage. Although the hydraulic effects of shale are statistically significant, the changes in flow responses rarely exceed 20%. As a result, it may be reasonable to use simple models when incorporating analogous shales into models of reservoirs or aquifers.     

The Theoretical Links Between Sequential Gaussian Simulation, Gaussian Truncated Simulation, and Probability Field Simulation

Sequential Gaussian simulation (sgsim), Gaussian truncated simulation (gtsim), and probability field simulation (pfsim) are three algorithms frequently used for conditional stochastic simulation. They were developed independently and are seen as different algorithms in applications. This paper establishes that gtsim and pfsim can be bridged by a simple quantile transform between Gaussian and uniform distributions. As for the sgsim algorithm, the normal score back-transform can be seen as a series of truncations of the simulated Gaussian field. All three algorithms are shown to be applicable to both continuous and categorical variables. In practice, gtsim can be most often replaced by the more CPU-efficient pfsim algorithm.     

A Multistep Methodology for Building a Stochastic Model of Gold Grades in the Disseminated and Complex Deposit of Casas Novas in Alentejo,Southern Portugal

This paper proposes a multistep approach for creating a 3D stochastic model of gold grades in a complex disseminated auriferous deposit located in the Alentejo region, southern Portugal. The approach involves the following steps: (i) the creation of a 3D low‐resolution vector object model of two geological domains that better discriminate gold grades; (ii) the definition of three intervals for gold grades (low, intermediate, and high values); (iii) the estimation of the probability of each grid block belonging to each interval constrained by the geological domains; (iv) the calculation of local conditional cumulative distribution functions (CCDFs) of gold grades, for each block, taking into account the geological domains and the borehole data; and finally (v) the simulation of images of gold grades. The results demonstrate the usefulness and accuracy of the proposed procedure, as they are in compliance with the conceptual model, the distribution laws are conditioned by the geological domains, and the transition of grades between domains is continuous.     

从端点走向连续:河流沉积模式研究进展述评

从河道类型的划分、河床演变与河型转换、河道沉积与河流砂体的建筑结构要素、河漫滩沉积、季节性河流与分支河流体系、河流沉积相模式、河流沉积学研究技术与方法等方面对国内外河流沉积模式的研究进展进行了综述,认为近十年来河流沉积学的理论和方法都发生了重要的变化。地貌学家、沉积学家和工程师认识到河道形态是连续可变的,而不是只有4~40多个端点类型。河床的演变受河床比降、流量变幅、河岸沉积物粒度构成、气候、植被以及构造沉降速率等多方面的影响。垂向剖面分析法难以对古河流类型做出正确的判断,运用建筑结构要素分析法重建河道内大型底形的地貌形态是河型判别和河流相模式重建的正确方法。河漫滩是河流沉积事件记录最为齐全的部位,对河漫滩、天然堤和泛滥平原沉积层序的研究能够揭示更多古河流沉积过程以及古环境、古气候和古生物方面的信息。对季节性河流、受季风强烈影响地区的河流、以及不同气候带河流所发育的独特沉积构造和建筑结构要素的研究不断增加。分支河流体系的概念得到越来越多的应用,但也得到不少质疑。我国学者应当注重对现代河流地貌形态和沉积过程的观察,把河床演变学的定量方法与沉积学的观点、理论和资料相结合,利用露头、三维地震资料和探地雷达技术建立河流砂体内部建筑结构信息数据库,加强对古河流河漫滩和泛滥平原的沉积过程、特征及其控制因素的研究,加强对不同构造和气候条件下河流沉积的差异性研究,不断发展河流沉积学研究技术,加强河流沉积学实验室建设和研究队伍建设,加强国际交流与合作,使我国河流沉积学为国家经济社会发展提供更加有力和有效的支撑,为推动国际河流沉积学发展做出中国人自己的贡献。