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Compositional data, as they typically appear in geochemistry in terms of concentrations of chemical elements in soil samples, need to be expressed in log-ratio coordinates before applying the traditional statistical tools if the relative structure of the data is of primary interest. There are different possibilities for this purpose, like centered log-ratio coefficients, or isometric log-ratio coordinates. In both the approaches, geometric means of the compositional parts are involved, and it is unclear how measurement errors or detection limit problems affect their presentation in coordinates. This problem is investigated theoretically by making use of the theory of error propagation. Due to certain limitations of this approach, the effect of error propagation is also studied by means of simulations. This allows to provide recommendations for practitioners on the amount of error and on the expected distortion of the results, depending on the purpose of the analysis.

Andre Journel joined Stanford University in 1978, and his program grew quickly to include a dozen students from the USA, Canada, Europe, and South Africa. He was instrumental in organizing the Second International Geostatistical Conference (Tahoe ’83), during which 13 papers were presented that can be linked to his group. Out of these 13 papers, 9 were mining-related, with 7 on recoverable reserves, 2 on uncertainty, 2 on conditional simulation, and 3 on nonparametric geostatistics. A significant research effort at the time was therefore directed at change of support, global and local recoveries, and uncertainty, but future trends could also be identified, such as nonparametric geostatistics and conditional simulation. This paper is a practical review of conditional simulation as a tool to improve mineral resource estimation in the areas of uncertainty, classification, and mining selectivity or dilution, based on the authors’ experience. Some practical considerations for conditional simulation are briefly discussed. Four case studies from the early 1990s to the late 2010s are presented to illustrate some solutions and challenges encountered when dealing with real-world commercial projects.

This paper considers the problem of conditional stochastic simulation in geostatistics. A new type of conditioning method that generalizes the well-known two-step conditioning approach is introduced. The distinctive feature of the new method is that it solves the modeling problem for multiple stochastic fields in a general setup. A generalized kriging procedure is developed in the paper, using linear combinations of the simulated fields as input data instead of the commonly used separate fields’ values. Although the new conditioning method was developed initially in the framework of a particular approach to geostatistical simulation (the spectral method), it can be applied in many more general settings of conditional simulation of stationary stochastic fields of arbitrary dimension. The workings of the method and its applicability are illustrated with the results of several numerical experiments, including simulation on real oil field data.

In a mining routine, the uncertainty assessment of resources can be evaluated through stochastic simulation methods that allow the characterisation of local probability density functions on point or block support and, consequently, the spatial uncertainty of grades per ore type. After the characterisation of mean grades and uncertainty per block, the main role of mine planning consists of characterising the time scheduling of reserves to arrive at a mining sequence. This paper seeks to transform the estimated block grades and uncertainty into the temporal production flow of average grades and consequent temporal uncertainty. The most straightforward approach consists of calculating the mining sequence for each of the simulated realisations of blocks, followed by accessing the uncertainty of each period in each sequence, and finally opting for the optimal sequence, according to an objective function. However, this approach needs to retain the N simulated models and calculate the mining sequence for each one, which can be a cumbersome task, particularly if the dimension of the block model is high. Hence, this paper proposes two methods: one using a Gaussian mixture model, and the other using a quantile interpolation mixture model to aggregate each period’s block production forecast by converting each block’s static uncertainty into production dynamic uncertainty. This allows for the period production uncertainty to be used as an optimisation parameter in mine planning routines. A test on the Neves Corvo mine synthetic case study is presented, demonstrating the applicability of this method in the context of an internal blending strategy or a selective mining approach.

The development of deep geothermal energy may be at risk from the economic point of view if the estimated deep thermal field is far from its real state. The strong heterogeneity of geological units makes it challenging to perform a reliable estimation on the construction of the deep thermal field over a large region. Additionally, the thermal properties of rocks, such as thermal conductivity and the radiogenic element concentration, whether they are from laboratory measurements or inversed from well logs, may strongly control the deep thermal field at a local scale. In this paper, the thermal conductivities of rocks from the Trois-Rivieres region in the Saint Lawrence Lowlands sedimentary basin in eastern Canada are obtained from two methods: (i) direct experimental measurement and (ii) indirect inversion method using well logs, including gamma ray, neutron porosity, density, and photoelectric absorption factor. The spatial distribution of subsurface temperature in the study area in the Trois-Rivieres region is numerically investigated by considering four case studies that include different values (minimum, average, and maximum) of the thermal properties by applying the Underworld simulator. The results show that thermal properties play a large role in controlling the subsurface temperature distribution and heat flux. The temperature difference can reach 15 °C in the basement, caused by the difference in thermal properties in the Trois-Rivieres region. The highest heat flux is found in the Trenton–Black River–Chazy groups, and the lowest heat flux is in the Potsdam group, which also has the highest thermal conductivity. Vertical heat flux does not change linearly with depth but is highly related to the thermal properties of specific geological formations. Furthermore, it does not have a positive correlation with the vertical temperature changes. This demonstrates that the assessment of the potential of deep geothermal energy depending merely on the surface heat flux may greatly overestimate or underestimate the geothermal capacity. Construction of the thermal models based on the integrated thermal properties from both the experimental measurement and well logs in this paper is useful in reducing the exploration risk associated with the utilization of deep geothermal energy.

In order to speed up the development and utilization of hydrothermal energy, it is essential to assess the potential of geothermal resources in petroliferous basins. In this paper, the distribution of reservoirs (aquifers) and the characteristics of geothermal fields have been studied systematically based on geological, geophysical, well drilling, temperature, and sample test data obtained from the major petroliferous basins of China. It has been found that some of the porous sandstone formations in these petroliferous basins are major geothermal reservoirs and are extensively thick and widely distributed. In general, the geothermal gradient in China is higher in the eastern basins and lower in the western basins. On average, the geothermal gradient is above 30 °C/km in the Bohai Bay, Songliao, and Subei basins. The geothermal resource abundance is also higher in eastern China and in the Beibuwan basin in southern China, and the geothermal source forming condition is better, followed by the Ordos, Qaidam, and Sichuan basins in Central China. Other potential basins include the Tarim and Junggar basins in western China, where the geothermal gradient ranges between 21 and 22 °C/km, on average. In this paper, three methods, stochastic simulation, unit volumetric, and analogy, were used for the assessment of geothermal resources. Using the stochastic simulation and unit volumetric methods, the geothermal resources, annual recovered geothermal resources, geothermal water resources, and thermal energy of water in 11 basins or blocks of up to 4000 m deep were calculated. Grading evaluation criteria were established by considering the heterogeneity of geothermal reservoirs. The results showed that the petroliferous basins are very rich in geothermal resources. The annual recoverable resources reach 1626.8 × 106 tons of standard coal, in which grade I, grade II, and grade III resources are 641.9 × 106, 298.6 × 106, and 686.3 × 106 tons of standard coal, respectively. The results demonstrate that the development and utilization of geothermal energy in oilfields has a huge potential for industrial production and family use, and a great significance for the development of green oilfields. With the high demand of heat, the eastern oilfields with high geothermal resource abundance should be the first to be considered for the production and utilization of geothermal energy, followed by the central and western oilfields.

Geothermal energy is a clean energy source that can potentially mitigate greenhouse gas emissions, as its use can lead to a lower mitigation cost. However, research on the economic impacts of the geothermal industry is scarce. This paper describes the effect of the geothermal industry, its economic input and output, using Beijing as a case study. This paper adopts the input–output model. The results show that the demand for and input use of the geothermal sector vary greatly across industrial sectors: electricity, heat production, the supply industry and general equipment manufacturing have the greatest direct consumption coefficient for the geothermal industry. When considering direct and indirect demand, it is clear that the geothermal industry has a great effect on different industrial sectors in diverse ways. Its influence coefficient and sensitivity coefficient are 1.2167 (ranked 11th) and 1.2293 (ranked 8th), respectively, revealing that it exerts obvious demand-pulling and supply-pushing effects on the regional economy.

Characterization of field-scale reservoir connectivity is critical for production optimization and field development planning. The information content of the data corresponding to different physical processes, coupled with the discrepancy in production data and geologic model resolutions, complicate the estimation of global field connectivity. Due to its diffusive nature, pressure varies smoothly in the reservoir, a characteristic that can be adequately captured using coarse-scale description of the connectivity in the reservoir model. On the other hand, flowrate and watercut information are relatively more localized and require higher-resolution models to adequately describe their variations. Hence, pressure and flowrate data carry information pertinent to their respective scales. In this paper, a fast workflow is presented for estimating large-scale reservoir connectivity from static reservoir pressure measurements. For this purpose, a coarse-scale grid system with Delaunay triangulation is developed to represent the global reservoir connectivity. Flow-based upscaling is applied to create the initial coarse-scale static simulation models from geological or simulation models. The ensemble Kalman filter is then applied to calibrate the initial models against static pressure data and to identify field-scale connectivity parameters that control the global pressure distribution, such as aquifer strength, continuity/discontinuity in reservoir properties, and fault transmissibilities. This approach enables fast characterization of field-scale reservoir connectivity from pressure data with a coarser description of the reservoir connectivity, which offers a parameterization level that is commensurate with the resolution and information content of the static pressure measurements. The proposed calibration approach can serve as preconditioning to generate initial reservoir models with consistent global connectivity patterns for full-scale history matching. To illustrate the feasibility and performance of the method, examples are drawn from real field cases in which static pressure data are integrated to infer global reservoir connectivity maps. In this paper, the resulting connectivity maps are used to obtain facies probability maps to generate initial facies models for full-scale history matching.

Bayesian uncertainty quantification of reservoir prediction is a significant area of ongoing research, with the major effort focussed on estimating the likelihood. However, the prior definition, which is equally as important in the Bayesian context and is related to the uncertainty in reservoir model description, has received less attention. This paper discusses methods for incorporating the prior definition into assisted history-matching workflows and demonstrates the impact of non-geologically plausible prior definitions on the posterior inference. This is the first of two papers to deal with the importance of an appropriate prior definition of the model parameter space, and it covers the key issue in updating the geological model—how to preserve geological realism in models that are produced by a geostatistical algorithm rather than manually by a geologist. To preserve realism, geologically consistent priors need to be included in the history-matching workflows, therefore the technical challenge lies in defining the space of all possibilities according to the current state of knowledge. This paper describes several workflows for Bayesian uncertainty quantification that build realistic prior descriptions of geological parameters for history matching using support vector regression and support vector classification. In the examples presented, it is used to build a prior description of channel dimensions, which is then used to history-match the parameters of both fluvial and deep-water reservoir geostatistical models. This paper also demonstrates how to handle modelling approaches where geological parameters and geostatistical reservoir model parameters are not the same, such as measured channel dimensions versus affinity parameter ranges of a multi-point statistics model. This can be solved using a multilayer perceptron technique to move from one parameter space to another and maintain realism. The overall workflow was implemented on three case studies, which refer to different depositional environments and geological modelling techniques, and demonstrated the ability to reduce the volume of parameter space, thereby increasing the history-matching efficiency and robustness of the quantified uncertainty.

This paper presents an overview of results for the geostatistical analysis of collocated multivariate data sets, whose variables form a composition, where the components represent the relative importance of the parts forming a whole. Such data sets occur most often in mining, hydrogeochemistry and soil science, but the results gathered here are relevant for any regionalised compositional data set. The paper covers the basic definitions, the analysis of the spatial codependence between components, mapping methods of cokriging and cosimulation honoring compositional constraints, the role of pre- and post-transformations such as log-ratios or multivariate normal score transforms, and block-support upscaling. The main result is that multivariate geostatistical techniques can and should be performed on log-ratio scores, in which case the system data-variograms-cokriging/cosimulation is intrinsically consistent, delivering the same results regardless of which log-ratio transformation was used to represent them. Proofs of all statements are included in an appendix.

During a conventional multiple-point statistics simulation, the algorithm may not find a matched neighborhood in the training image for some unsimulated pixels. These pixels are referred to as the dead-end pixels; the existence of the dead-end pixels means that multiple-point statistics simulation is not a simple sequential simulation. In this paper, the multiple-point statistics simulation is cast as a combinatorial optimization problem, and the efficient backtracking algorithm is developed to solve this optimization problem. The efficient backtracking consists of backtracking, forward checking, and conflict-directed backjumping algorithms that are introduced and discussed in this paper. This algorithm is applied to simulate multiple-point statistics properties of some synthetic training images; the results show that no anomalies occurred in any of the produced realizations as opposed to previously published methods for solving the dead-end pixels. In particular, in simulating a channel system, all the channels generated by this method are continuous, which is of paramount importance in fluid flow simulation applications. The results also show that the presence of hard data does not degrade the quality of the generated realizations. The presented method provides a robust algorithmic framework for performing MPS simulation.

This paper describes a proposed method for clustering attributes on the basis of their spatial variability and the uncertainty of cluster membership. The method is applied to geometallurgical domaining in mining applications. The main objective of geometallurgical clustering is to ensure consistent feed to a processing plant by minimising transitions between different types of feed coming from different domains (clusters). For this purpose, clusters should contain not only similar geometallurgical characteristics but also be located in as few contiguous and compact spatial locations as possible so as to maximise the homogeneity of ore delivered to the plant. Most existing clustering methods applied to geometallurgy have two problems. Firstly, they are unable to differentiate subsets of attributes at the cluster level and therefore cluster membership can only be assigned on the basis of exactly identical attributes, which may not be the case in practice. Secondly, as they do not take account of the spatial relationships they can produce clusters which may be spatially dispersed and/or overlapped. In the work described in this paper a new clustering method is introduced that integrates three distinct steps to ensure quality clustering. In the first step, fuzzy membership information is used to minimise compactness and maximise separation. In the second step, the best subsets of attributes are defined and applied for domaining purposes. These two steps are iterated to convergence. In the final step a graph-based labelling method, which takes spatial constraints into account, is used to produce the final clusters. Three examples are presented to illustrate the application of the proposed method. These examples demonstrate that the proposed method can reveal useful relationships among geometallurgical attributes within a clear and compact spatial structure. The resulting clusters can be used directly in mine planning to optimise the ore feed to be delivered to the processing plant.

This paper demonstrates the usefulness of approaching the dynamic study of the precise positioning of a network of permanent global positioning system (GPS) stations through functional data analysis. The displacement data for each GPS station, obtained from observations of the global navigation satellite system, are a discrete sample of the positioning curve. The aim of this paper is to reconstruct the original functions in order to use them as functional data. In the method presented in this paper, the geodetic series are obtained first by processing the GPS data with respect to a reference station. Second, for each station, a cleaning process is applied to eliminate the values considered as outliers, and the missing values are imputed by using a Kalman filter. Finally, the original functions are reconstructed by using smoothing techniques and by evaluating several bases of functions. Moreover, these functions are treated with statistical techniques for functional data. This procedure is applied to the permanent stations of the south of the Iberian peninsula and the north of Africa (SPINA) network. The topocentric series: east, north and up are analysed. In the analysis of the positioning curves, there is observed a synchronized behaviour of the functions in those periods of time with important seismic activity. This behaviour also appears in the analysis of the second principal component of the East and up dimensions. Furthermore, the first two principal components of the East coordinate enable us to make a classification of the stations in the SPINA network. The classification made is consistent with the previous knowledge of the tectonic plates in the studied area.

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.

The applied context of this paper is the exploration for petroleum resources, like petroleum accumulations in different deposits, where the most promising deposits are likely to be drilled first, based on some size indicators, so-called creaming. The paper explores creaming models in the context of sampling with probabilities in proportion to size, for which a lognormal size distribution has nice analytical features. It departs from the traditional paradigm in petroleum resource assessment. Instead of conceiving a finite population being depleted over time in a decaying fashion with respect to size, the situation is studied within the framework of independent observations (infinite population) and an exploration maturity-dependent creaming factor. The theoretical and practical consequences for inference on the parent population and the probabilities and expectations linked to future discoveries are studied. The theory applies to the issue of remaining sizes of petroleum resources to be found within different future discovery horizons on the mature part of the Norwegian Continental Shelf. The aim is to obtain reasonable and useful predictions, and not to provide the best possible explanation of the exploratory behavior itself.

Preserving plausible geological features when updating facies models is still one of the main challenges with ensemble-based history matching. This is particularly evident for fields with complex geological description (e.g., fluvial channels). There is an impressive amount of research published in the last few years about this subject. However, it appears that there is no definitive solution and both, academia and industry, are looking for practical and robust methods. Among the parameterizations traditionally investigated for history matching, the principal component analysis (PCA) of the prior covariance matrix is an efficient alternative to represent models described by two-point statistics. However, there are some recent developments extending PCA-based parameterizations for models described by multiple-point statistics. The first part of this paper presents an investigation on PCA-based schemes for parameterizing channelized facies models for history matching with ensemble-based methods. The following parameterizations are tested: standard PCA, two alternative implementations of kernel PCA and optimization-based PCA. In the second part of the paper, the optimization-based PCA is modified to allow the use of covariance localization and adapted for simultaneously adjusting the facies type and the permeability values within each facies when history matching production data with an ensemble-based method.

This paper presents a model that can simulate early rock-forming processes, including the influence of the initial packing of the grains on the subsequent rearrangement that occurs as a consequence of pressure-induced grain damage. The paper is concerned with the behaviour of assemblies of loose grains and the mechanics of early lithification. Consider the concept of shear-induced negative dilatancy, where any shear deformation has a tendency to produce densification even at very low pressures. As shear deformation progresses, positive dilatancy starts to contribute and at the critical state the two effects balance. This concept is encapsulated within the mathematics of the model. The model building scheme is first outlined and demonstrated using a hard particle model. Then, the concept of ‘self cancelling shear deformations’ that contribute to the shear–volume coupling but not to the macroscopic shear deformation is explained. The structure of the hard particle model is modified to include low levels of damage at the grain contacts. A parameter that describes bonding between the grains and possible damage to those bonds is incorporated into a term that, depending on its magnitude, also accounts for frictional resistance between unbonded grains. This parameter has the potential to develop with time, increasing compressive stress, or in response to evolving chemical concentrations. Together these modifications allow densification in the short term, and the formation of sedimentary rocks in the long term, by pressure alone, to be simulated. Finally, simulations using the model are compared with experimental results on soils.

International audience; Building a 3D geological model from field and subsurface data is a typical task in geological studies involving natural resource evaluation and hazard assessment. However, there is quite often a gap between research papers presenting case studies or specific innovations in 3D modeling and the objectives of a typical class in 3D structural modeling, as more and more is implemented at universities. In this paper, we present general procedures and guidelines to effectively build a structural model made of faults and horizons from typical sparse data. Then we describe a typical 3D structural modeling workflow based on triangulated surfaces. Our goal is not to replace software user guides, but to provide key concepts, principles, and procedures to be applied during geomodeling tasks, with a specific focus on quality control.

Modern approaches for the spatial simulation of categorical variables are largely based on multi-point statistical methods, where a training image is used to derive complex spatial relationships using relevant patterns. In these approaches, simulated realizations are driven by the training image utilized, while the spatial statistics of the actual sample data are ignored. This paper presents a data-driven, high-order simulation approach based on the approximation of high-order spatial indicator moments. The high-order spatial statistics are expressed as functions of spatial distances that are similar to variogram models for two-point methods, while higher-order statistics are connected with lower-orders via boundary conditions. Using an advanced recursive B-spline approximation algorithm, the high-order statistics are reconstructed from the available data and are subsequently used for the construction of conditional distributions using Bayes’ rule. Random values are subsequently simulated for all unsampled grid nodes. The main advantages of the proposed technique are its ability to (a) simulate without a training image to reproduce the high-order statistics of the data, and (b) adapt the model’s complexity to the information available in the data. The practical intricacies and effectiveness of the proposed approach are demonstrated through applications at two copper deposits.

The important goals of mineral exploration geochemistry are detection and identification of underlying mineralization. This paper deals with element concentration data analyzed of surface geochemical samples acquired from soil horizons or plants. A new unsupervised procedure is proposed for this purpose when the samples have been taken on a regular or irregular grid in the area under investigation. The methodology is based on Generalized Additive Model fits on element concentration data. Then new data points are taken of the surface of the smooth fits across the entire sampling area as a regular grid. Pairwise log-ratios of elements are then calculated of these points, and curvature of the log-ratio pairs is computed. High curvature indicates abrupt spatial changes, which could point at locations of mineralized zones. A measure called c-value evaluates the overall curvature and thus serves as an importance measure of a log-ratio pair. The methodology is tested on two real surface geochemical data sets collected in areas with known underlying mineralization, and the results confirm existing knowledge of the underlying mineralization.

As an essential step in reservoir characterization, seismic stratigraphic interpretation is often dependent on the development of powerful computer-based interpretation tools that can simulate the intelligence of experienced interpreters. With the success of machine/deep learning applications in many aspects of geoscience in recent decades, geophysicists have become more dedicated to exploring seismic big data in a smarter and more sophisticated way to better image subsurface reservoirs/structures. In this paper, a specific U-shaped fully convolutional network (U-Net) is established for automatic seismic stratigraphic interpretation. Specifically, this task is formulated as a semantic segmentation problem by identifying strata at the pixel level and classifying each pixel in the image into a specific stratum category. An experiment using the Netherlands F3 seismic dataset suggests that, compared with previously established deep learning models requiring a large number of training sets, the proposed U-Net method can achieve high evaluation indicators and better stratum segmentations in the case of a limited training set. During the test, the proposed U-Net model outperforms the Bayesian neural network (BNN) model for seismic stratum segmentation with regard to the training time, prediction speed, and segmentation accuracy. These results indicate the great potential of using U-Net-based deep learning for intelligent seismic stratigraphic interpretation.

Dissolution is a common diagenetic effect in carbonate formations. Vugs caused by dissolution significantly impact carbonate reservoir quality by affecting the porosity and permeability of the reservoir. However, without core and image logs, the identification and classification of vugs using wireline logs only is challenging, because logging tool responses reflect a mixed effect of changing mineral/fluid composition and diagenetic features. This paper presents a data-driven approach using neural networks to identify vugs and classify vug facies based on vug size. The purpose is to predict wells/intervals with limited measurements by machine learning models trained with core data from key wells. The input features for vug identification are conventional well logs (i.e., gamma ray, resistivity, neutron/density porosity, photoelectric factor, and acoustic slowness) from the Cambrian-Ordovician Arbuckle formation, Kansas. Two classification labels are used as the prediction target for the neural networks: (1) a binary vuggy index derived from nuclear magnetic resonance (NMR) measurements using a cutoff on T2 distribution, which presents the proportion of large pores over the total porosity, and (2) vug size labels from depth-by-depth core visual descriptions. A one-hidden-layer shallow neural network is compared against deep neural networks, including structures such as one-dimensional convolutional layers (1-D CNN) and long short-term memory (LSTM) layers. Results suggest that using a combination of multi-mineral analysis results and original well logs will increase the prediction accuracy of vug facies. Shallow and deep neural networks show a similar ability to identify vugs, with average accuracy of around 80%. However, to predict vug-size-based facies labels, deep neural networks outperform shallow neural networks, with overall accuracy improved by as much as 10%. The proposed method shows that deep neural networks (1-D CNN and LSTM) are reliable tools for vug facies prediction.

The level to which data are aggregated or smoothed can impact analytical and predictive modeling results. This paper discusses findings regarding such impacts on estimating change points in production flow regimes of horizontal hydraulically fractured shale oil wells producing from the middle member of the Bakken Formation. Change points that signal transitions in flow regimes are important because they subsequently affect estimates of ultimate recovery from wells producing from shale plays. Extending our earlier work, we employ two different statistical approaches, Bacon–Watts Bayesian regression and nonlinear constrained least squares regression, and a designed computational experiment to estimate the time of transition from the transient to the boundary-dominated flow regime for 14 different wells using daily production data rather than aggregated monthly data, as previously considered. The daily data were also smoothed to reduce noise. Computational experiments suggest that both statistical approaches can lead to plausible estimates of the transition point under different data aggregation or smoothing regimes, but that daily data are likely too granular to produce credible estimates. Although the expected value of transition points using smoothed daily data and monthly disaggregated data are generally comparable, the confidence intervals bounding the estimates based on smoothed daily data are generally wider. Our results not only inform the operational practices of oil producers engaged in economic evaluation of their shale resources and additional play development activities, but also the activities of petroleum research groups, government agencies, and financial organizations seeking to improve the trustworthiness of resource projections.

Real geo-data are three-dimensional (3D) and spatially varied, but measurements are often sparse due to time, resource, and/or technical constraints. In these cases, the quantities of interest at locations where measurements are missing must be interpolated from the available data. Several powerful methods have been developed to address this problem in real-world applications over the past several decades, such as two-point geo-statistical methods (e.g., kriging or Gaussian process regression, GPR) and multiple-point statistics (MPS). However, spatial interpolation remains challenging when the number of measurements is small because a suitable covariance function is difficult to select and the parameters are challenging to estimate from a small number of measurements. Note that a covariance function form and its parameters are key inputs for some methods (e.g., kriging or GPR). MPS is a non-parametric simulation method that combines training images as prior knowledge for sparse measurements. However, the selection of a suitable training image for continuous geo-quantities (e.g., soil or rock properties) faces certain difficulties and may become increasingly complicated when the geo-data to be interpolated are high-dimensional (e.g., 3D) and exhibit non-stationary (e.g., with unknown trends or non-stationary covariance structure) and/or anisotropic characteristics. This paper proposes a non-parametric approach that systematically combines compressive sensing and variational Bayesian inference for statistical interpolation of 3D geo-data. The method uses sparse measurements and their locations as the input and provides interpolated values at unsampled locations with quantified interpolation uncertainty as the output. The proposed method is illustrated using a series of numerical 3D examples, and the results indicate a reasonably good performance.

Abundant literature has been produced for the last two decades about multiple-point statistics simulation, or MPS. The idea behind MPS is very simple: reproduce patterns from a 2D, or most often a 3D, training image that displays the type of geological heterogeneity deemed to be relevant to the reservoir or field under study, while honoring local data. Replicating an image is a traditional computer science problem. Thus, it should come as no surprise if a growing number of publications on MPS borrow ideas and techniques directly from computer vision and machine learning to improve the reproduction of training patterns. However, quoting Andre Journel, “Geostatistics is not about generating pretty pictures.” Models have a purpose. For example, in oil and gas applications, reservoir models are used to estimate hydrocarbon volumes and book reserves, run flow simulations to forecast hydrocarbon production and ultimate recovery, and make decisions about field development or optimal well drilling locations. Specific key features such as the extent and connectivity of shale barriers may have a major impact on the reservoir performance forecasts and the field development decisions to be made. Those key features that need to be captured in the model, along with the available subsurface data and constraints of the project, should be the primary drivers in selecting the most appropriate modeling techniques and options to obtain reliable results and make sound decisions. In this paper, the practitioners’ point of view is used to evaluate alternative MPS implementations and highlight remaining gaps.

Modeling a mineral microstructure accurately in three dimensions can render realistic mineralogical patterns which can be used for three-dimensional processing simulations and calculation of three-dimensional mineral quantities. The present study introduces a flexible approach to model the microstructure of mineral material composed of a large number of facies. The common plurigaussian method, a valuable approach in geostatistics, can account for correlations within each facies and in principle be extended to correlations between the facies. Assuming stationarity and isotropy, founded on a new description of this model, formulas for first- and second-order characteristics, such as volume fraction, correlation function and cross-correlation function can be given by a multivariate normal distribution. In this particular situation, based on first- and second-order statistics, a fitting procedure can be developed which requires only numerical inversion of several one-dimensional monotone functions. The paper describes the whole workflow. The covariance structure is quickly obtained from two-dimensional particle pixel images using Fourier transform. Followed by model fitting and sampling, where the resulting three-dimensional microstructure is then efficiently represented by tessellations. The applicability is demonstrated for the three-dimensional case by modeling the microstructure from a Mineral Liberation Analyzer image data set of an andesitic basalt breccia.

Obtaining representative samples is of the uttermost importance for modelling mineral deposits. On one hand, Gy’s formula makes it possible to compute relative error variance for a specified target grade on the stringent assumption of perfect homogenization. On the other hand, duplicates (field, rejects and pulp) are now routinely used in QA/QC. No direct links have been established between Gy’s relative variance and the main duplicates statistics used in QA/QC. This paper presents a simple multiplicative error model that makes it possible to predict from Gy’s formula the standard deviation of duplicates differences and duplicates correlation for any distribution. Assuming lognormality, a test is derived to predict whether the sample preparation procedure is likely to meet the target proportion of duplicates with half absolute relative difference (HARD) of less than 10%. The test and the predicted statistics can be used to predict duplicates’ QA/QC results and make necessary adjustments early, before any QA/QC data is acquired. The relationships are tested and validated by simulations and a series of deposit data extracted from NI43-101 published reports.

Closed terrain depressions are geomorphic features that show scale invariance over the wide range of scales on which they occur, from metres to thousands of kilometres. They are also global features that are present on all continents. Morphometric analyses of these depressions are now possible because of the availability of global digital elevation models and advances in methods for mapping closed depressions. Here, a simple and automatic but efficient method is proposed for mapping closed terrain depressions. The GTOPO30 global digital elevation model, with a spatial resolution of around 1 km, was used to map the closed depressions of eight large test sites sampled from all inhabited continents. The identified and delineated terrain depressions include lake watersheds together with many other terrain depressions including deserts, karst terrains and rift valleys. A fractal analysis shows that South America and Central Africa have similar, and also the largest, fractal dimensions; Central Europe and North Africa have intermediate fractal dimensions; and North America, Central Asia, the Near East and Australia have similar, low fractal dimensions. The methodology proposed in this paper can be used to identify areas of interest for which a more exhaustive analysis, using digital elevation models with better spatial resolution, can be conducted.

When originally developed, the Koval theory was meant to interpret the core-scale production of oil by a miscible displacement by solvent injection. An essential parameter in the theory is the Koval factor, KV, which combines both the viscosity contrast effect (E) and the heterogeneity effect (HK) as KV = E HK. The Koval method has done an excellent job of interpreting oil recovery; however, the physical origin of this simple model is not clear. To address the physical significance of these factors, this paper explains a novel approach to derive the Koval factor using petrophysical data and vertical equilibrium theory. The study examines core data from 112 vertical production wells from fields located in the United States and Algeria. The heterogeneity factor is estimated from the flow- and storage-capacity relations, according to the Koval theory. Then, using fractional flow theory and assuming vertical equilibrium, we derive a mathematical expression that conveys the physical interpretation of the Koval factor. The results indicate that the physical explanation of the Koval factor is that of a product of the mobility ratio and the heterogeneity of the reservoir. The Koval factor simplifies computations of oil recovery by transforming the original reservoir to one with uniform layers, in vertical equilibrium, and piston-like displacement within each using only the flow and storage capacities. The physical interpretation of the Koval factor could offer new insights to examine previous results obtained from the theory itself.

Surface ocean currents are often of interest in environmental monitoring. These vectorial data can be reasonably treated as a finite realization of a complex- valued random field, where the decomposition in modulus (current speed) and direction (current direction) of the current field is natural. Moreover, when observations are also available for different time points (other than at several locations), it is useful to evaluate the evolution of their complex correlation over time (rather than in space) and the corresponding modeling which is required for estimation purposes. This paper illustrates a first approach where the temporal profile of surface ocean currents is considered. After introducing the fundamental aspects of the complex formalism of a random field indexed in time, a new class of models suitable for including the temporal component is proposed and applied to describe the time-varying complex covariance function of current data. The analysis concerns ocean current observations, taken hourly on 30 April 2016 through high frequency radar systems at some stations located in the Northeastern Caribbean Sea. The selected complex covariance model indexed in time is used for estimation purposes and its reliability is confirmed by a numerical analysis.

Time-shift, one of the most popular time-lapse seismic attributes, has been widely used in dynamic reservoir characterization by linking it with pressure and geomechanical changes. Therefore, it is important to select appropriate calculation methods according to different time-lapse seismic data quality and time-shift magnitude. To date, there have been various published works comparing different time-shift calculation methods and discussing their advantages and disadvantages. However, most of these comparisons are based only on synthetic tests or single field applications. As the quality of time-lapse seismic data and time-shift magnitude can vary in different fields, one method may not work consistently well for each case. In this paper, a critical comparison of three different time-shift calculation techniques (Hale’s fast cross-correlation, Rickett’s non-linear inversion, and Whitcombe’s correlated leakage method) is provided. The three methods are applied to a set of synthetic data sets that are designed to account for various seismic noise and time-shift magnitudes. They are also applied to four real time-lapse seismic data sets from three North Sea fields. The calculated time-shift results are compared with the input (in synthetic tests) or the real observations from information such as seabed subsidence and compaction (in field applications). Both qualitative and quantitative comparisons are performed. At the end, each of the time-shift methods is evaluated based on different aspects, and the most appropriate method is suggested for each data scenario. All three time-shift methods are found to successfully measure time-shifts. However, Rickett’s non-linear inversion is the most outstanding method, as it gives smooth time-shifts with relatively good accuracy, and the derived time strains are more stable and interpretable.

Integration of renewable energy into mining and processing operations is becoming necessary as part of a strategy towards sustainability in the minerals industry. A solar photovoltaic plant along with a battery energy storage system (PV-BESS) can provide a long-term solution to cope with increasing energy costs, thus reducing the tension with other societal competing needs for a key resource such as clean energy. However, sizing these systems is challenging, in face of uncertain ore grindability and solar power availability. In this paper, we present an application of an integrated model to size the PV-BESS, where the variability of ore grindability is modeled using geostatistical tools, solar irradiance variability is captured using a Markov chain simulation model, and the entire system is optimized through linear stochastic optimization, considering a fixed mine schedule for the feed of a semiautogenous grinding (SAG) mill. The main goals are to minimize the costs associated with operating the SAG mill in the presence of a PV-BESS system, understand how the sizing and costs change under the influence of stochastic drivers, and reveal the potential for demand-side management in mines. The size and costs of the necessary infrastructure are determined for a model of estimated grindability and an ensemble of 50 simulated models of grindability, and compared against the sizing and cost of the ground truth. The effect of climate change on solar energy availability is accounted for by forecasting the ratio of excellent, good, and moderate days over bad days in terms of irradiance out to the year 2030. Finally, the effects of stockpiles and feed control according to the processing plant needs, namely a demand-side management approach, are evaluated to reveal the impact on the energy requirements and the sizing of the photovoltaic and storage system. The model is optimized considering a yearly cost function, with hourly resolution for the solar irradiance and hardness models. The results show that integrating solar power into the operation of a SAG mill has potential to reduce the total energy cost by 27%. Robustness against climate change can be achievedwith an increase in total cost of 1%. Finally, use of stockpiles to manage the ore supply to the mill and minimize the energy cost to process it results in a cost reduction of around 2%, which should offset the rehandling cost of managing the stockpiles.

Study of the characteristics and classification of geothermal gradients can effectively guide the exploration and development of geothermal resources. In this paper, we systematically compare the relationship between the distribution of the geothermal gradient and its basin structure, and numerically model the heat conduction course of typical geothermal–geological profiles. The results show that the distribution of the geothermal gradient in the Beijing–Tianjin–Hebei Plain exhibits the characteristics of east–west zonation, north–south segmentation, and vertical stratification. From west to east, the basin structure can be divided into five belts, viz. the western sag, the central bulge, the eastern sag, the Cangxian uplift, and the Cangdong sag belts. The heat flow and the geothermal gradient of the cap rocks in the basal bulge/uplift belts are higher than in the sag belts. From north to south, the basin structure of the Jizhong depression is subdivided into the northern, middle, and southern segments by the Xushui-Anxin and Hengshui Faults. It also shows an obvious distribution of low and high geothermal gradient interphases, which are similar to their upper and lower basement. Vertically, the geothermal gradient in the bulge/uplift belts exhibits an asymmetrical “mirror reflection” structure. Abnormal geothermal gradient distributions are observed, being confined to the basement uplift belts caused by the tectonic movement which led to the heterogeneity of the horizontal and vertical heat conduction capacity. Therefore, according to the corresponding relationship between the amplitude of the basement uplifting and the size of the geothermal gradient of the cap rock, the geothermal gradient of the Beijing–Tianjin–Hebei Plain can be divided into five categories, of high convex, convex, low convex, deep buried, and exposed types. Among these, the high convex and convex types are the primary targets for geothermal exploration and development.

The paper discusses the influence of various methods for splitting raw data into test and training subsets on the accuracy of the prediction of the spatial distribution of the variable for the model based on a multilayer perceptron. A comparison of models was performed taking into account both spatial heterogeneity and the spread of values of the modeled variable with a completely random splitting option. The study was based on the results obtained during soil screening of urbanized territories in the Russian subarctic zone, Novy Urengoy city and Noyabrsk city. The spatial distribution of the chemical element chromium (Cr) in the surface layer of the soil was modeled. For both surveyed areas, the models in which a controlled splitting method was used for training demonstrated more accurate results than models trained using completely random splitting.

Conditioning multiple-point statistical (MPS) facies simulation on dynamic flow data is complicated by the complex relation between flow responses and facies distribution. One way to incorporate feedback from the flow data into MPS simulation is by constructing (and updating) facies probability maps (soft data) and using the results to constrain the MPS simulation outputs, for example, through Single Normal Equation SIMulation (SNESIM) algorithm and the $$ \tau $$-model. The pattern-imitating behavior of MPS simulation (specifically, the SNESIM) has been shown to result in a sizable fraction of facies models that fail to effectively capture the correct location and spatial connectivity represented by the facies probability map. This paper presents two key observations that explains this behavior/outcome: (1) the facies patterns resulting from the SNESIM are primarily controlled by the outcome of the first few grid cells along the corresponding random path (early stages of the simulation); and (2) the facies outcomes in the remaining grid cells are dominated by extremely confident conditional probabilities extracted from the TI to generate the encoded connectivity patterns in the TI. Hence, the contribution of facies probability map become increasingly inconsequential as the simulation proceeds. This intrinsic property must be accounted for in developing data conditioning methods, especially when training images with large-scale connectivity patterns are considered. Two approaches for addressing this issue include adapting the random path at early stages to the key information in the probability map to properly reflect its impact, or treating high-probability events from the probability map as hard data. Examples are presented to illustrate this behavior, followed by a simple method to improve the effectiveness of integrating facies probability maps (and flow response data) into SNESIM.

Two common tasks when processing point cloud data sets are surface estimation and point cloud registration. In this paper, a statistical approach is developed to solve both of these problems simultaneously. In particular, a surface is estimated from a pair of unregistered three-dimensional scans of the same spatial region. In this method, one point cloud defines the fixed coordinate system, and a rigid transformation is applied to the second cloud. Observations from both scans are considered a single realization of a Gaussian process. The registration problem is solved by jointly optimizing the likelihood over the parameters specifying the domain transformation and the mean and covariance functions. Given parameter estimates, surface estimation follows using the spatial stochastic model. While other existent approaches do not account for registration uncertainty, the likelihood-based approach to solving the registration and surface estimation problems jointly allows uncertainty in registration to be propagated to the surface prediction variance. The new method is motivated and illustrated using a digital elevation model estimation problem near the Chalk Cliffs in Colorado. The method developed is compared against the popular iterative closest point method. The results of a simulation study show significant improvement in transformation parameter estimates using the statistical approach. In a cross-validation experiment with the Chalk Cliffs data, there is an $$18\%$$ reduction in predictive mean squared error using the likelihood method over iterative closest point.

High-order sequential simulation methods have been developed as an alternative to existing frameworks to facilitate the modeling of the spatial complexity of non-Gaussian spatially distributed variables of interest. These high-order simulation approaches address the modeling of the curvilinear features and spatial connectivity of extreme values that are common in mineral deposits, petroleum reservoirs, water aquifers, and other geological phenomena. This paper presents a new high-order simulation method that generates realizations directly at the block support scale conditioned to the available data at point support scale. In the context of sequential high-order simulation, the method estimates, at each block location, the cross-support joint probability density function using Legendre-like splines as the set of basis functions needed. The proposed method adds previously simulated blocks to the set of conditioning data, which initially contains the available data at point support scale. A spatial template is defined by the configuration of the block to be simulated and related conditioning values at both support scales, and is used to infer additional high-order statistics from a training image. Testing of the proposed method with an exhaustive dataset shows that simulated realizations reproduce major structures and high-order relations of data. The practical intricacies of the proposed method are demonstrated in an application at a gold deposit.

A primary concern of the mining industry is to meet production targets, which are required and defined by customers. Deviations from these targets, in terms of quality and quantity, highly affect the economical aspect. Recently, an efficient resource model updating framework concept has been proposed aiming for the improvement of raw material quality control and process efficiency in any type of mining operation. The concept integrates online sensor measurements, obtained during production, into the resource model. In this way, due to the spatial variability, quality attributes of the blocks that will be produced in the next days or weeks are being updated based on real-time measurements. The concept has been applied in a lignite field with the aim of identifying local impurities in a lignite seam and to improve the prediction of coal quality attributes in neighbouring blocks. This paper investigates the added value of using the resource model updating framework by using the value of information analysis. The expected benefit of additional information (integration of the online sensor measurements into the resource model) is compared to a case where there is no additional information integrated into the process. These benefits are evaluated based on the economic impact determined by applying the resource model updating framework in mine planning.

Multiple point statistics (MPS) algorithms allow generation of random fields reproducing the spatial features of a training image (TI). Although many MPS techniques offer options to prescribe characteristics deviating from those of the TI (e.g., facies proportions), providing a TI representing the target features as well as possible is important. In this paper, methods for editing stationary images by applying a transformation—painting or warping—to the regions, similar to a representative pattern selected by the user in the image itself, are proposed. Painting simply consists in replacing image values, whereas warping consists in deforming the image grid (compression or expansion of similar regions). These tools require few parameters and are interactive: the user defines locally how the image should be modified, then the changes are propagated automatically to the entire image. Examples show the ability of the proposed methods to keep spatial features consistent within the entire edited image.

In mining operations, the time delay between grade estimations and decision-making based on those estimations can be substantial. This may lead to the scheduling of stopes mining that is based on information which is seriously out of date and, consequently, results in a substantial mined resources and reserves bias. To mitigate this gap between the grade estimation of an orebody and its exploitation, this paper proposes a method to quickly update resources and reserves that are integrated into the concept of real-time mining. The current standard for grade data collection in underground mines relies on a conventional chemical lab analysis of sparse drill hole or chip/face samples. The proposed methodology for the continuous and swift updating of mine resources and reserves requires a constant and rapid stream of measurements at the stopes. Consequently, this work proposes using portable X-ray fluorescence (XRF) devices to carry out the fast and abundant monitoring of ore grades. However, these fast data are highly uncertain; hence, the objective of this proposed method is to use the total data measurements and integrate their uncertainty into the resources modeling. The first step in the proposed methodology consists of creating a joint distribution function between “uncertain” XRF and the corresponding “hard” measurements based on empirical historical data. Then, the uncertainty of the XRF measurements is derived from these joint distributions through the conditional distribution of the real values applied to the known XRF measurement. The second step involves updating the reserves by integrating these uncertain XRF data, which are quantified by conditional distributions, into the grade characterization models. To achieve this, a stochastic simulation with point distributions is applied. An actual case study of a copper sulfide deposit illustrates the proposed methodology.

Models used for reservoir prediction are subject to various types of uncertainty, and interpretational uncertainty is one of the most difficult to quantify due to the subjective nature of creating different scenarios of the geology and due to the difficultly of propagating these scenarios into uncertainty quantification workflows. Non-uniqueness in geological interpretation often leads to different ways to define the model. Uncertainty in the model definition is related to the equations that are used to describe the modelled reality. Therefore, it is quite challenging to quantify uncertainty between different model definitions, because they may include completely different model parameters. This paper is a continuation of work to capture geological uncertainties in history matching and presents a workflow to handle uncertainty in the geological scenario (i.e. the conceptual geological model) to quantify its impact on the reservoir forecasting and uncertainty quantification. The workflow is based on inferring uncertainty from multiple calibrated models, which are solutions of an inverse problem, using adaptive stochastic sampling and Bayesian inference. The inverse problem is solved by sampling a combined space of geological model parameters and a space of reservoir model descriptions, which represents uncertainty across different modelling concepts based on multiple geological interpretations. The workflow includes building a metric space for reservoir model descriptions using multi-dimensional scaling and classifying the metric space with support vector machines. The proposed workflow is applied to a synthetic reservoir model example to history match it to the known truth case reservoir response. The reservoir model was designed using a multi-point statistics algorithm with multiple training images as alternative geological interpretations. A comparison was made between predictions based on multiple reservoir descriptions and those of a single one, revealing improved performance in uncertainty quantification when using multiple training images.

This paper presents an extended least squares framework for geospatial frame transformation problems with nonlinear models in the presence of noisy data. The goal is to set up an analytic scheme for obtaining transformed coordinates in the target frame with improved accuracy over the classic stepwise approach that is commonly used in practice. The proposed methodology is based on the joint spatial adjustment of all observed coordinates from the respective frames (initial and target) in conjunction with their stochastic model over all points of interest. This allows us to properly handle the cross-correlated noise in the initial coordinates between control and new points, and thus to overcome a crucial limitation of other operational approaches. In algebraic terms, the proposed approach relies on the weighted least squares adjustment of a stacked set of Gauss–Helmert models that can unequivocally describe any geospatial frame transformation over different groups of points. The analytic solution to this stacking problem is derived for the case of nonlinear parametric models, regardless of the structure of the error covariance matrices of the input data. The study focuses on computational aspects and the proper implementation of the solution algorithm via the Newton–Gauss iteration method. A numerical example is given to demonstrate the expected improvement in the statistical accuracy of the transformed coordinates under the proposed stacking approach.

Assimilation of production data into reservoir models for which the distribution of porosity and permeability is largely controlled by facies has become increasingly common. When the locations of the facies bodies must be conditioned to observations, the truncated plurigaussian model has been often shown to be a useful method for modeling as it allows gaussian variables to be updated instead of facies types. Previous experience has also shown that ensemble Kalman filter-like methods are particularly effective for assimilation of data into truncated plurigaussian models. In this paper, some limitations are shown of the ensemble-based or gradient-based methods when applied to truncated plurigaussian models of a certain type that is likely to occur for modeling channel facies. It is also shown that it is possible to improve the data match and increase the ensemble spread by modifying the updating step using an approximate derivative of the truncation map.

The three-dimensional high-resolution imaging of rock samples is the basis for pore-scale characterization of reservoirs. Micro X-ray computed tomography (µ-CT) is considered the most direct means of obtaining the three-dimensional inner structure of porous media without deconstruction. The micrometer resolution of µ-CT, however, limits its application in the detection of small structures such as nanochannels, which are critical for fluid transportation. An effective strategy for solving this problem is applying numerical reconstruction methods to improve the resolution of the µ-CT images. In this paper, a convolutional neural network reconstruction method is introduced to reconstruct high-resolution porous structures based on low-resolution µ-CT images and high-resolution scanning electron microscope (SEM) images. The proposed method involves four steps. First, a three-dimensional low-resolution tomographic image of a rock sample is obtained by µ-CT scanning. Next, one or more sections in the rock sample are selected for scanning by SEM to obtain high-resolution two-dimensional images. The high-resolution segmented SEM images and their corresponding low-resolution µ-CT slices are then applied to train a convolutional neural network (CNN) model. Finally, the trained CNN model is used to reconstruct the entire low-resolution three-dimensional µ-CT image. Because the SEM images are segmented and have a higher resolution than the µ-CT image, this algorithm integrates the super-resolution and segmentation processes. The input data are low-resolution µ-CT images, and the output data are high-resolution segmented porous structures. The experimental results show that the proposed method can achieve state-of-the-art performance.

High-order sequential simulation techniques for complex non-Gaussian spatially distributed variables have been developed over the last few years. The high-order simulation approach does not require any transformation of initial data and makes no assumptions about any probability distribution function, while it introduces complex spatial relations to the simulated realizations via high-order spatial statistics. This paper presents a new extension where a conditional probability density function (cpdf) is approximated using Legendre-like orthogonal splines. The coefficients of spline approximation are estimated using high-order spatial statistics inferred from the available sample data, additionally complemented by a training image. The advantages of using orthogonal splines with respect to the previously used Legendre polynomials include their ability to better approximate a multidimensional probability density function, reproduce the high-order spatial statistics, and provide a generalization of high-order simulations using Legendre polynomials. The performance of the new method is first tested with a completely known image and compared to both the high-order simulation approach using Legendre polynomials and the conventional sequential Gaussian simulation method. Then, an application in a gold deposit demonstrates the advantages of the proposed method in terms of the reproduction of histograms, variograms, and high-order spatial statistics, including connectivity measures. The C++ course code of the high-order simulation implementation presented herein, along with an example demonstrating its utilization, are provided online as supplementary material. Electronic supplementary material The online version of this article (10.1007/s11004-018-9741-2) contains supplementary material, which is available to authorized users.

In the paper, a novel inversion approach is used for the solution of the problem of factor analysis. The float-encoded genetic algorithm as a global optimization method is implemented to extract factor variables using open-hole logging data. The suggested statistical workflow is used to give a reliable estimate for not only the factors but also the related petrophysical properties in hydrocarbon formations. In the first step, the factor loadings and scores are estimated by Joreskog’s fast approximate method, which are gradually improved by the genetic algorithm. The forward problem is solved to calculate wireline logs directly from the factor scores. In each generation, the observed and calculated well logs are compared to update the factor population. During the genetic algorithm run, the average fitness of factor populations is maximized to give the best fit between the observed and theoretical data. By using the empirical relation between the first factor and formation shaliness, the shale volume is estimated along the borehole. Permeability as a derived quantity also correlates with the first factor, which allows its determination from an independent source. The estimation results agree well with those of independent deterministic modeling and core measurements. Case studies from Hungary and the USA demonstrate the feasibility of the global optimization based factor analysis, which provides a useful tool for improved reservoir characterization.

The widely used wavelet filtering technique holds potential to approach anomaly–background separation in geophysical and geochemical data processing. Wavelet statistics provide crucial information on such filtering methods. In general, conventional (Gaussian-type) statistical modeling is insufficient to adequately describe the heavily tailed and sharply peaked (at zero) distribution of the wavelet coefficients of irregular geo-anomaly patterns. This paper demonstrates that the cumulative (frequency) number of the wavelet coefficient yields a power-law scaling relationship with the coefficient based on wavelet transform of a fractal/singular measure. This wavelet coefficient–cumulative number power-law model is proven to be more flexible and appropriate than the Gaussian model for characterizing the scaling nature of the coefficient distribution. Accordingly, a fractal-based filtering technique is developed based on the wavelet statistical model to decompose mixed patterns into components based on the distinct self-similarities identified in the wavelet domain. The decomposition scheme of the fractal-based wavelet filtering method considers not only the coefficient frequency distribution but also the fractal spectrum of singularities and the self-similarity of real-world features. Finally, a synthetic data test and real applications from two metallogenic provinces of China are used to validate the proposed fractal filtering method for anomaly–background separation and identification of geophysical or geochemical anomalies related to mineralization and other geological features.

Realistic and accurate static geologic models are an essential element needed to predict the behavior of subsurface reservoirs and play an important role in petroleum engineering. Data used in the development of a static geologic model are gathered from various sources, such as seismic, log, and core data, each of them providing information on different physical properties of interest and with varying degrees of resolution. Compiling all data from various sources into a single representation of the subsurface formation of interest is a daily challenge for many petroleum geologists and engineers. This paper describes a framework to develop and select process-mimicking models that are consistent with available seismic attributes, namely impedance. Using a process-mimicking modeling package, 75 models of a fluvial meandering system are generated, one of which is chosen as the “true” model and masked thereafter. The implemented selection method relies on the degree of similarity in the histogram of representations of clusters of all possible patterns in the seismic impedance domain based on each process-mimicking model and that of the “true” model at several resolutions. The results demonstrate the effectiveness of the use of a weighted average divergence distance across multiple levels to select process-mimicking models that honor seismic data the best.

Mining operations face a decision regarding additional drilling several times during their lifetime. The two questions that always arise upon making this decision are whether more drilling is required and, if so, where the additional drill holes should be located. The method presented in this paper addresses both of these questions through an optimization in a multi-armed bandit (MAB) framework. The MAB optimizes the best infill drilling pattern while taking geological uncertainty into account by using multiple conditional simulations for the deposit under consideration. The proposed method is applied to a long-term, multi-element stockpile, which is a part of a gold mining complex. The stockpiles in this mining complex are of particular interest due to difficult-to-meet blending requirements. In several mining periods grade targets of deleterious elements at the processing plant can only be met by using high amounts of stockpiled material. The best pattern is defined in terms of causing the most material type changes for the blocks in the stockpile. Material type changes are the driver for changes in the extraction sequence, which ultimately defines the value of a mining operation. The results of the proposed method demonstrate its practical aspects and its effectiveness towards the optimization of infill drilling schemes.

The log ratio methodology converts compositional data, such as concentrations of chemical elements in a rock, from their original Aitchison geometry to interpretable real orthonormal coordinates, thereby allowing meaningful statistical processing and visualization. However, it must be taken into account that the original concentrations can be flawed by detection limit or imprecision problems that can severely affect the resulting coordinates. This paper aims to construct such orthonormal log ratio coordinates, called weighted pivot coordinates, that capture the relevant relative information about an original component and treat the redundant information in a controlled manner. Theoretical developments are supported by a thorough simulation study. Weighted pivot coordinates are then applied to the geochemical mapping of catchment outlet sediments from the National Geochemical Survey of Australia illustrating their advantage over possible alternatives.