Co-simulating Total and Soluble Copper Grades in an Oxide Ore Deposit

In oxide copper deposits, the acid soluble copper represents the fraction of total copper recoverable by heap leaching. Two difficulties often complicate the joint modeling and simulation of total and soluble copper grades: the inequality constraint linking both grade variables and the sampling design for soluble copper grade, which may be preferential and cause biases in sample statistics. A methodology is presented in order to accurately estimate the total and soluble copper grade bivariate distribution, based on an explicit modeling of the conditional distributions of soluble copper grade. Co-simulation is then realized by converting the copper grades into Gaussian random fields, through stepwise conditional transformation, and by fitting a coregionalization model while accounting for the preferential sampling design. The proposed approach is illustrated through an application to an ore deposit located in northern Chile.     

Risk quantification with combined use of lithological and grade simulations: Application to a porphyry copper deposit

The uncertainty in the recoverable tonnages and grades in a mineral deposit is a key factor in the decision-making process of a mining project. Currently, the most prevalent approach to model the uncertainty in the spatial distribution of mineral grades is to divide the deposit into domains based on geological interpretation and to predict the grades within each domain separately. This approach defines just one interpretation of the geological domain layout and does not offer any measure of the uncertainty in the position of the domain boundaries and in the mineral grades. This uncertainty can be evaluated by use of geostatistical simulation methods. The aim of this study is to evaluate how the simulation of rock type domains and grades affects the resources model of Sungun porphyry copper deposit, northwestern Iran. Specifically, three main rock type domains (porphyry, skarn and late-injected dykes) that control the copper grade distribution are simulated over the region of interest using the plurigaussian model. The copper grades are then simulated in cascade, generating one grade realization for each rock type realization. The simulated grades are finally compared to those obtained using traditional approaches against production data.     

Simulation of mineral grades with hard and soft conditioning data: application to a porphyry copper deposit

This work deals with the geostatistical simulation of mineral grades whose distribution exhibits spatial trends within the ore deposit. It is suggested that these trends can be reproduced by using a stationary random field model and by conditioning the realizations to data that incorporate the available information on the local grade distribution. These can be hard data (e.g., assays on samples) or soft data (e.g., rock-type information) that account for expert geological knowledge and supply the lack of hard data in scarcely sampled areas. Two algorithms are proposed, depending on the kind of soft data under consideration: interval constraints or local moment constraints. An application to a porphyry copper deposit is presented, in which it is shown that the incorporation of soft conditioning data associated with the prevailing rock type improves the modeling of the uncertainty in the actual copper grades.     

A probabilistic parametrization for geological uncertainty estimation using the ensemble Kalman filter (EnKF)

In the past years, many applications of history-matching methods in general and ensemble Kalman filter in particular have been proposed, especially in order to estimate fields that provide uncertainty in the stochastic process defined by the dynamical system of hydrocarbon recovery. Such fields can be permeability fields or porosity fields, but can also fields defined by the rock type (facies fields). The estimation of the boundaries of the geologic facies with ensemble Kalman filter (EnKF) was made, in different papers, with the aid of Gaussian random fields, which were truncated using various schemes and introduced in a history-matching process. In this paper, we estimate, in the frame of the EnKF process, the locations of three facies types that occur into a reservoir domain, with the property that each two could have a contact. The geological simulation model is a form of the general truncated plurigaussian method. The difference with other approaches consists in how the truncation scheme is introduced and in the observation operator of the facies types at the well locations. The projection from the continuous space of the Gaussian fields into the discrete space of the facies fields is realized through in an intermediary space (space with probabilities). This space connects the observation operator of the facies types at the well locations with the geological simulation model. We will test the model using a 2D reservoir which is connected with the EnKF method as a data assimilation technique. We will use different geostatistical properties for the Gaussian fields and different levels of the uncertainty introduced in the model parameters and also in the construction of the Gaussian fields.     

Comparison of machine learning methods for copper ore grade estimation

In this study, machine learning methods such as neural networks, random forests, and Gaussian processes are applied to the estimation of copper grade in a mineral deposit. The performance of these methods is compared to geostatistical techniques, such as ordinary kriging and indicator kriging. To ensure that these comparisons are realistic and relevant, the predictive accuracy is estimated on test instances located in drill holes that are different from the training data. The results of an extensive empirical study in the Sarcheshmeh porphyry copper deposit in Southeastern Iran illustrate that specially designed Gaussian processes with a symmetric standardization of the spatial location inputs and an anisotropic kernel yield the most accurate predictions. Furthermore, significant improvements are obtained when, besides location, information on the rock type is included in the set of predictor variables. This observation highlights the importance of carrying out detailed studies of the geological composition of the deposit to obtain more accurate ore grade predictions.     

甘肃省合作市德乌鲁岩体及外围金多金属成矿区成矿地质特征

德乌鲁岩体及外围金多金属成矿区位于中秦岭海西褶皱带西段。区内矿产主要有金、锑、铜、铅、锌;矿化类型主要有蚀变岩型金(锑)矿化类型和矽卡岩型的铜砷、铜铅锌多金属矿化类型。各矿点在空间分布上,显示等间距分布的基本特征,受近南北向分布的断层蚀变破碎带控制。矿化蚀变主要有黄铁矿化、电气石化、硅化、绢云母化、毒砂化、辉锑矿化、次生蚀变以褐铁矿化、高岭土化。本文结合德乌鲁岩体及外围区域成矿特点、矿床分布规律、成矿地质特征、控矿因素,从本区的大地构造背景及区域构造活动特点等两个方面对区内成矿构造破碎蚀变带的形成进行了力学成因研究,并提出了本区今后找矿的主要方向。     

A generalized probabilistic approach for slope analysis

Summary A new probabilistic approach is introduced for slope stability analysis, which is general in types of variable distributions and correlations or dependency between variables, and flexible enough to include any adverse impact analysis for blasting vibrations and groundwater conditions.The material strength within a slope area, given in terms of the internal friction angle (ø) and cohesion (c), is randomized in the bivariate joint probability analysis. To be a completely general engineering method, the new probabilistic approach employs the random variable transformation technique: the Hermite model of the Gaussian transformation function, which transforms the experimental histogram of shear strength parameters to the standard Gaussian distribution (=0, ²=1.0).Because a binormal joint probability is analysed on the true probability region projected on the plane of the Gaussian transformed variables, it is an exact solution of slope stability based on the available sample data. No assumption on the shape of the experimental histogram or independency between two random variables is made as in the current probability methods of slope analysis.     

Joint High-Order Simulation of Spatially Correlated Variables Using High-Order Spatial Statistics

Joint geostatistical simulation techniques are used to quantify uncertainty for spatially correlated attributes, including mineral deposits, petroleum reservoirs, hydrogeological horizons, environmental contaminants. Existing joint simulation methods consider only second-order spatial statistics and Gaussian processes. Motivated by the presence of relatively large datasets for multiple correlated variables that typically are available from mineral deposits and the effects of complex spatial connectivity between grades on the subsequent use of simulated realizations, this paper presents a new approach for the joint high-order simulation of spatially correlated random fields. First, a vector random function is orthogonalized with a new decorrelation algorithm into independent factors using the so-termed diagonal domination condition of high-order cumulants. Each of the factors is then simulated independently using a high-order univariate simulation method on the basis of high-order spatial cumulants and Legendre polynomials. Finally, attributes of interest are reconstructed through the back-transformation of the simulated factors. In contrast to state-of-the-art methods, the decorrelation step of the proposed approach not only considers the covariance matrix, but also high-order statistics to obtain independent non-Gaussian factors. The intricacies of the application of the proposed method are shown with a dataset from a multi-element iron ore deposit. The application shows the reproduction of high-order spatial statistics of available data by the jointly simulated attributes.     

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.     

Geological modeling of rock type domains in the Balya (Turkey) lead-zinc deposit using plurigaussian simulation

Mineral resource evaluation requires defining geological rock-type domains. The traditional simulation methods have serious limitations for applications to large numbers of domains, which have complex contact relations. Plurigaussian simulation is an effective method which can be applied, in a simple way, to any number of domains, using both local and global geological information to infer the distributions of rock types. This work not only presents the application of the plurigaussian simulation method to the Balya lead-zinc deposit, but also assesses the spatially varying rock type proportions, and accounts for uncertainties between them. These parameters are extremely important for mining deposits, since the mineralizations of interest generally occur only in certain rock types. Furthermore, being able to model the different geological rock types is vital to good mine operations, production planning, and management. The results indicate that the plurigaussian method correctly reproduces the different orientations of the individual rock types, as seen in drill holes, and the proportion of each rock type, even if this varies in space.     

Geostatistics for Power Models of Gaussian Fields

This paper introduces geostatistical approaches (i.e., kriging estimation and simulation) for a group of non-Gaussian random fields that are power algebraic transformations of Gaussian and lognormal random fields. These are power random fields (PRFs) that allow the construction of stochastic polynomial series. They were derived from the exponential random field, which is expressed as Taylor series expansion with PRF terms. The equations developed from computation of moments for conditional random variables allow the correction of Gaussian kriging estimates for the non-Gaussian space. The introduced PRF geostatistics shall provide tools for integration of data that requires simple algebraic transformations, such as regression polynomials that are commonly encountered in the practical applications of estimation. The approach also allows for simulations drawn from skewed distributions.     

Application of the EnKF and Localization to Automatic History Matching of Facies Distribution and Production Data

The performance of the ensemble Kalman filter (EnKF) for continuous updating of facies location and boundaries in a reservoir model based on production and facies data for a 3D synthetic problem is presented. The occurrence of the different facies types is treated as a random process and the initial distribution was obtained by truncating a bi-Gaussian random field. Because facies data are highly non-Gaussian, re-parameterization was necessary in order to use the EnKF algorithm for data assimilation; two Gaussian random fields are updated in lieu of the static facies parameters. The problem of history matching applied to facies is difficult due to (1) constraints to facies observations at wells are occasionally violated when productions data are assimilated; (2) excessive reduction of variance seems to be a bigger problem with facies than with Gaussian random permeability and porosity fields; and (3) the relationship between facies variables and data is so highly non-linear that the final facies field does not always honor early production data well. Consequently three issues are investigated in this work. Is it possible to iteratively enforce facies constraints when updates due to production data have caused them to be violated? Can localization of adjustments be used for facies to prevent collapse of the variance during the data-assimilation period? Is a forecast from the final state better than a forecast from time zero using the final parameter fields?To investigate these issues, a 3D reservoir simulation model is coupled with the EnKF technique for data assimilation. One approach to enforcing the facies constraint is continuous iteration on all available data, which may lead to inconsistent model states, incorrect weighting of the production data and incorrect adjustment of the state vector. A sequential EnKF where the dynamic and static data are assimilated sequentially is presented and this approach seems to have solved the highlighted problems above. When the ensemble size is small compared to the number of independent data, the localized adjustment of the state vector is a very important technique that may be used to mitigate loss of rank in the ensemble. Implementing a distance-based localization of the facies adjustment appears to mitigate the problem of variance deficiency in the ensembles by ensuring that sufficient variability in the ensemble is maintained throughout the data assimilation period. Finally, when data are assimilated without localization, the prediction results appear to be independent of the starting point. When localization is applied, it is better to predict from the start using the final parameter field rather than continue from the final state.     

A two‐dimensional coupled hydromechanical discontinuum model for simulating rock hydraulic fracturing

Within the framework of our discontinuous deformation analysis for rock failure algorithm, this paper presents a two‐dimensional coupled hydromechanical discontinuum model for simulating the rock hydraulic fracturing process. In the proposed approach, based on the generated joint network, the calculation of fluid mechanics is performed first to obtain the seepage pressure near the tips of existing cracks, and then the fluid pressure is treated as linearly distributed loads on corresponding block boundaries. The contribution of the hydraulic pressure to the initiation/propagation of the cracks is considered by adding the components of these blocks into the force matrix of the global equilibrium equation. Finally, failure criteria are applied at the crack tips to determine the occurrence of cracking events. Several verification examples are simulated, and the results show that this newly proposed numerical model can simulate the hydraulic fracturing process correctly and effectively. Although the numerical and experimental verifications focus on one unique preexisting crack, because of the capability of discontinuous deformation analysis in simulating block‐like structures, the proposed approach is capable of modeling rock hydraulic fracturing processes. Copyright © 2014 John Wiley & Sons, Ltd.     

基于饱和渗透系数空间变异结构的斜坡渗流及失稳特征

以往研究一般采用单随机变量方法（SRV）或基于水平或垂直方向波动范围生成的空间变异随机场来模拟岩土参数的空间变异性,对具有倾斜定向特征的空间变异随机场未有涉及.基于条件模拟相关理论和非侵入式随机有限元的理论框架,提出了利用序贯高斯模拟方法进行斜坡参数条件随机场模拟并运用有限元方法进行斜坡渗流和稳定性分析的方法.针对理想边坡,对各向同性和几何各向异性的共7种空间变异结构的饱和渗透系数（K_s）各进行了200次条件随机场模拟,基于条件随机场模拟结果进行了有限元渗流和稳定性计算,对每种空间变异结构多次计算结果进行了统计分析.结果表明:本文所提出的方法不仅再现了研究区域参数的空间二阶统计特性,通过设定变异函数参数进行不同空间变异类型、变异程度、变异定向性的随机场模拟,同时利用现场观测数据对随机场模拟结果进行条件限制,从而提高了随机场的赋值精度;K_s的空间变异结构对孔隙水压力的分布规律、地下水位线变化范围、稳定性系数和最危险滑动面分布特征均有一定程度的影响.本研究为库岸斜坡稳定性评价提供方法支撑.      

Dimensional Reduction of Pattern-Based Simulation Using Wavelet Analysis

A pattern-based simulation technique using wavelet analysis is proposed for the simulation (wavesim) of categorical and continuous variables. Patterns are extracted by scanning a training image with a template and then storing them in a pattern database. The dimension reduction of patterns in the pattern database is performed by wavelet decomposition at certain scale and the approximate sub-band is used for pattern database classification. The pattern database classification is performed by the k-means clustering algorithm and classes are represented by a class prototype. For the simulation of categorical variables, the conditional cumulative density function (ccdf) for each class is generated based on the frequency of the individual categories at the central node of the template. During the simulation process, the similarity of the conditioning data event with the class prototypes is measured using the L ₂-norm. When simulating categorical variables, the ccdf of the best matched class is used to draw a pattern from a class. When continuous variables are simulated, a random pattern is drawn from the best matched class. Several examples of conditional and unconditional simulation with two- and three- dimensional data sets show that the spatial continuity of geometric features and shapes is well reproduced. A comparative study with the filtersim algorithm shows that the wavesim performs better than filtersim in all examples. A full-field case study at the Olympic Dam base metals deposit, South Australia, simulates the lithological rock-type units as categorical variables. Results show that the proportions of various rock-type units in the hard data are well reproduced when similar to those in the training image; when rock-type proportions between the training image and hard data differ, the results show a compromise between the two.     

Direct Sequential Simulation and Cosimulation

Sequential simulation of a continuous variable usually requires its transformation into a binary or a Gaussian variable, giving rise to the classical algorithms of sequential indicator simulation or sequential Gaussian simulation. Journel (1994) showed that the sequential simulation of a continuous variable, without any prior transformation, succeeded in reproducing the covariance model, provided that the simulated values are drawn from local distributions centered at the simple kriging estimates with a variance corresponding to the simple kriging estimation variance. Unfortunately, it does not reproduce the histogram of the original variable, which is one of the basic requirements of any simulation method. This has been the most serious limitation to the practical application of the direct simulation approach. In this paper, a new approach for the direct sequential simulation is proposed. The idea is to use the local sk estimates of the mean and variance, not to define the local cdf but to sample from the global cdf. Simulated values of original variable are drawn from intervals of the global cdf, which are calculated with the local estimates of the mean and variance. One of the main advantages of the direct sequential simulation method is that it allows joint simulation of N _v variables without any transformation. A set of examples of direct simulation and cosimulation are presented.     

Continuous Facies Updating Using the Ensemble Kalman Filter and the Level Set Method

We present a methodology based on the ensemble Kalman filter (EnKF) and the level set method for the continuous model updating of geological facies with respect to production data. Geological facies are modeled using an implicit surface representation and conditioned to production data using the ensemble Kalman filter. The methodology is based on Gaussian random fields used to deform the facies boundaries. The Gaussian random fields are used as the model parameter vector to be updated sequentially within the EnKF when new measurements are available. We show the successful application of the methodology to two synthetic reservoir models.     

Simulation of mineral grades and classification of mineral resources by using hard and soft conditioning data: application to Sungun porphyry copper deposit

The presence of geological units with different grade characteristics mostly leads to problems during the grade modeling process. In special cases, if the area under study has units with small thickness and low grade with respect to the dominant unit of the area, it is difficult to reproduce different grade contents in these units in the simulated grade models because of the low thickness and lack of data in these units. In this study, the local moment constraints method, based on the definition of soft conditioning data reflecting geological knowledge, is investigated for improving simulated grade models under the mentioned conditions. This method is applied for grade simulation at the 1,750 m level of Sungun porphyry copper mine. The studied area is divided into two rock type domains: Sungun porphyry and Dyke. The Sungun porphyry unit is the dominant rock type in the considered area and has, on average, a higher copper grade, while dykes discontinue Sungun porphyry rock units sporadically and most of them are barren of mineralization. It is demonstrated that the use of soft conditioning data makes the simulated grade model closer to reality and improves the reproduction of grade contents considering the rock type units in the area. In the next step, the results obtained from conditional simulation are used for mineral resources classification. To this end, the conditional coefficient of variation is chosen as a criterion for measuring uncertainty and for defining the resources classes. Then, it is shown that uncertainty can be considerably reduced in the prepared models if soft data are considered; as a result, an increase in measured resource classification is observed.     

秦山三期核电工程反应堆地段节理裂隙模拟

以秦山三期核电工程反应堆地段作为研究对象 ,针对节理裂隙在岩体中的分布具有随机性的特点 ,运用随机方法对其分布规律进行了研究。根据实测的节理裂隙产状、间距及迹线长度 ,推断节理裂隙概率分布特征。同时充分考虑到节理裂隙分布的不均匀性和方向性等特点 ,采用非平稳态随机过程模拟节理裂隙间距 ,通过 Monte- Carlo方法得到等效的节理裂隙网络。根据统计和拟合的结果 ,对反应堆地段岩体的节理裂隙进行了评价。