Multiple-point Geostatistical Simulation Research Articles

Generative Adversarial Networks to incorporate the Training Image uncertainty in multiple-point statistics simulation

Multiple-point geostatistics (MPS) algorithms have been extensively used for geostatistical reservoir modeling. These techniques are known for building geostatistical simulations with curvilinear features, which are crucial for applications in flow simulation. One challenge in using MPS algorithms is obtaining a representative Training Image (TI). The result is that often one TI is used to condition the entire set of geostatistical simulations, and the TI uncertainty is disregarded. If the TI is uncertain, this uncertainty should be considered in the geostatistical simulations. Otherwise, the spatial uncertainty represented by the geostatistical simulations is underestimated. Incorporating the TI uncertainty requires building a data set or catalog of plausible TIs. In this context, we propose to use Generative Adversarial Networks (GANs) to build this catalog of TIs prior to the MPS simulation. Each TI generated by GAN is used to condition one realization built by the chosen MPS algorithm. The proposed workflow combines the strength of GAN to generate TIs with the strength of MPS algorithms to consider several sources of information to condition the simulations. A case study is shown, where the methodology is compared against the traditional MPS workflow, which uses a single TI. The results show that the proposed method successfully incorporated the TI uncertainty into the geostatistical workflow. The proposed method built numerical models with higher uncertainty and variability and, thus, provides a more realistic quantification of the space of uncertainty.

Multiple-Point Geostatistical Simulation of Nonstationary Sedimentary Facies Models Based on Fuzzy Rough Sets and Spatial-Feature Method

Summary 3D simulation of sedimentary facies using seismic data is vital for reservoir evaluation and estimation of oil and gas reserves. As a result of the high nonstationarity of sedimentary facies and the highly nonlinear characteristics of seismic attributes, the mapping relationship with sedimentary facies has certain ambiguities and uncertainties that affect the modeling results of sedimentary facies. Multipoint geostatistics (MPS) has proven to be an effective technique for modeling subsurface geological bodies. However, the conventional MPS only deals with stationary applications, and its capability of revealing the distribution of sedimentary facies is thereby limited. In addition, the task of seismic attributes selection, which significantly affects the performance of the modeling method in simulating the distribution of sedimentary facies, is difficult because the relationship between sedimentary facies and seismic attributes is complex. This article presents a nonstationary modeling method for simulating the distribution of sedimentary facies, which is featured by the multiscale spatial feature of patterns. In particular, the spatial location of the patterns is introduced as auxiliary information in the classification and simulation processes. The method incorporates multiscale results during the modeling procedure. Patterns from the multicategory training images (TIs) are classified by the optimized workflow. The seismic attribute selection is achieved by using fuzzy-rough sets. The proposed simulation method is verified by two typical TIs, followed by applications to predict the actual distribution of sedimentary facies. Compared with the filter-based pattern simulation (FILTERSIM) approach, the proposed simulation method is applicable for revealing detailed subsurface models, especially under complex geological conditions and limited information.

Multiple-Point Geostatistics Simulation of the Natural Gas Hydrate Distribution at the Pore Scale

Heat and mass transfer in a natural gas hydrate reservoir is strongly affected by the hydrate pore habit. Some present simplified models used in numerical simulation of a hydrate-bearing sediment are difficult to catch features of the hydrate distribution in sediments precisely. Using an improved multiple-point geostatistics simulation method, this study generates a series of images of porous media that reproduce the hydrate distribution information on a training image. The improved algorithm completely solves the problem that the hydrate blocks the boundary and makes the variogram and visual properties similar to the training image. Some typical phenomena of hydrate formation appear in the results, such as Ostwald ripening and the shift of hydrate pore habit from grain coating toward pore filling. The feasibility of predicting the pore habit of a hydrate with time using this novel method is proven by the verification case. This study offers a new approach to predict the distribution behavior of a hydrate, and this algorithm may be applied widely by scanning training images from different reservoirs.

Stochastic inversion combining seismic data, facies properties, and advanced multiple-point geostatistics

We proposed a novel seismic inversion approach that integrates the physical properties of litho-facies, and geophysical data, within a multiple-point geostatistical framework to reduce the uncertainty in predictions of litho-facies spatial arrangement away from wells or control points. The litho-facies groups (rock-type) in the well locations are defined and conditioned to the distribution of elastic properties, including the values of P-wave velocity (Vp) and facies density (ρ) within the borehole. A conceptual geological model (training image) is utilized within a wavelet-based multiple-point geostatistical simulation (WAVESIM) algorithm to generate litho-facies realizations. In our inversion algorithm, the forward model is created by implementing the bivariate Kernel density estimation (KDE) technique of the litho-facies properties (Vp and ρ) that are distributed within the well locations. We utilize an iterative inversion framework, where a particular number of elastic properties (Vp and ρ) for each WAVESIM realization are drawn. For each generated set of the WAVESIM realizations, we simulate reflectivity series by KDE that are convolved with the seismic wavelet to create synthetic seismograms. Then, using a normalized, cross-correlation function we select the realization that provides the best-match between synthetic seismogram and the input seismic data. Our inversion technique is successfully applied to synthetic and field datasets. Our results demonstrate the efficiency of our inversion to characterize highly heterogeneous reservoirs in a reasonable computational time and control the connectivity between the channels.

Stochastic simulation of facies using deep convolutional generative adversarial network and image quilting

Sedimentary facies simulation is one of the essential works in sedimentary environment analysis and reservoir characterization. The traditional facies simulation method is based on geostatistics. However, the traditional two-point geostatistics-based facies simulation method cannot characterize complex facies structures. Most multiple-point geostatistical simulation methods are unable to flexibly generate abundant geologic patterns. To address these shortcomings, we develop an intelligent method to automatically simulate sedimentary facies according to the training image provided by geologists. The method can learn efficient representations for complex facies architectures and obtain the simulation results in a larger area, rather than being limited to an area with the same size as the training image. First, we construct a deep convolutional generative adversarial network to extract the high-dimensional features of facies. Then, a large number of specific patterns are randomly generated based on these features. Thus, the diversity of geologic patterns is improved. Finally, the patterns are spliced together to obtain possible facies maps by using an improved image quilting algorithm. A model test is described and analyzed to demonstrate the effectiveness and reliability of the new method. The results are consistent with the actual situation in the aspect of variability and continuity. The method is also applied to non-stationary geological facies unconditional simulation. The successful application indicates that the method is able to learn the features of non-stationary geological phenomena, showing the practicability of the proposed methods.

Quantitative 3-D Model of Carbonate Reef and Shoal Facies Based on UAV Oblique Photogrammetry Data: A Case Study of the Jiantanba Outcrop in West Hubei, China

Aiming at the problem of insufficient data acquisition for steep carbonate outcrops, we used unmanned aerial vehicle oblique photogrammetry to quantitatively evaluate Jiantianba outcrops from a global perspective. This method can quickly and flexibly acquire, process and interpret outcrop data, establish the three-dimensional digital outcrop model and quantitative reservoir geological knowledge database. Through the fine analysis and comparative study of multiple outcrop sections, we use lithofacies combination, hierarchical interface, sequential indicator stochastic simulation and multiple-point geostatistical simulation methods to establish a three-dimensional multi-point statistical model of the outcrop area. The results show that the model of the Jiantianba carbonate outcrop has three-dimensional coordinates that correspond to oblique photograph image data, allowing for both the accurate location of any point and the measurement of the rock body, and thus providing a base for studying the stratigraphic architecture of the outcrop. Through fine-scale geological and statistical analyses of the geological parameters of three geologic sections, we established a database and model of a typical outcrop, including the types, continuity and scale of lithology, as well as different lithological combinations and the geological evolution of the region. And we established a geological model and quantitative geological database via sequential indicator simulation, using virtual well location, lithological combination, stratigraphic correlation, three-dimensional stratigraphic and lithological models of a typical outcrop. Subsequently, we analyzed the training images of different microfacies based on these models combined with the outcrop geometry. Finally, we established a three-dimensional geological model of reef–shoal facies that is more in line with our current understanding of the geology of the outcrop area by multi-point geostatistical simulations.

Regional Geochemical Anomaly Identification Based on Multiple-Point Geostatistical Simulation and Local Singularity Analysis—A Case Study in Mila Mountain Region, Southern Tibet

The smoothing effect of data interpolation could cause useful information loss in geochemical mapping, and the uncertainty assessment of geochemical anomaly could help to extract reasonable anomalies. In this paper, multiple-point geostatistical simulation and local singularity analysis (LSA) are proposed to identify regional geochemical anomalies and potential mineral resources areas. Taking Cu geochemical data in the Mila Mountain Region, southern Tibet, as an example, several conclusions were obtained: (1) geochemical mapping based on the direct sampling (DS) algorithm of multiple-point geostatistics can avoid the smoothing effect through geochemical pattern simulation; (2) 200 realizations generated by the direct sampling simulation reflect the uncertainty of an unsampled value, and the geochemical anomaly of each realization can be extracted by local singularity analysis, which shows geochemical anomaly uncertainty; (3) the singularity-quantile (S-Q) analysis method was used to determine the separation thresholds of E-type α, and uncertainty analysis was carried out on the copper anomaly to obtain the anomaly probability map, which should be more reasonable than the interpolation-based geochemical map for geochemical anomaly identification. According to the anomaly probability and favorable geological conditions in the study area, several potential mineral resource targets were preliminarily delineated to provide direction for subsequent mineral exploration.

Hybrid parallel framework for multiple-point geostatistics on Tianhe-2: A robust solution for large-scale simulation

Multiple-point geostatistical (MPS) simulation methods have attracted an enormous amount of attention in earth and environmental sciences due to their ability to enhance extraction and synthesis of heterogeneous patterns. To characterize the subtle features of heterogeneous structures and phenomena, large-scale and high-resolution simulations are often required. Accordingly, the size of simulation grids has increased dramatically. Since MPS is a sequential process for each grid unit along a simulation path, it results in severe computational consumption. In this work, a new hybrid parallel framework is proposed for the case of MPS simulation on large areas with enormous amount of grid cells. Both inter-node-level and intra-node-level parallel strategies are combined in this framework. To maintain the quality of the realizations, we implement a conflict control method adapting to the Monte-Carlo process. Also, an optimization method for the simulation information is embedded to reduce the inter-node communication overhead. A series of synthetic tests were used to verify the availability and performance of the proposed hybrid parallel framework. The results corroborate that the proposed framework can efficiently achieve the high-resolution reproduction and characterization of complex structures and phenomena in earth sciences.

Seismic Lithofacies Distribution Modeling Using the Single Normal Equation Simulation (SNESIM) Algorithm of Multiple-Point Geostatistics in Upper Assam Basin, India

Recently, multiple-point geostatistical simulation gained much attention for its role in spatial reservoir characterization/modeling in geosciences. Accurate lithofacies modeling is a critical step in the characterization of complex geological reservoirs. In this study, multiple-point facies geostatistics based on the SNESIM algorithm integrated with the seismic modeling technique is used as an efficient reservoir modeling approach for lithofacies modeling of the fluvial Tipam formation in the Upper Assam Basin, India. The Tipam formation acts as a potential reservoir rock in the Upper Assam Basin, India. Due to the basin geological complexity and limitation in seismic resolution, many discontinuities in depositional channels in this fluvial depositional environment have been identified using conventional lithofacies mapping. This study combines three techniques to reproduce continuity of the lithofacies for better reservoir modeling. The first is simultaneous prestack inversion for inverting prestack gathers with angle-dependent wavelets into seismic attributes. A cross-plot of P-impedance and VP/VS ratio from well-log data was used to classify the different reservoir lithofacies such as hydrocarbon sand, brine sand, and shale. The second is the Bayesian approach that incorporates probability density functions (PDFs) of non -parametric statistical classification with seismic attributes for converting the seismic attributes into lithofacies volume and the probability volumes of each type lithofacies. The third technique is multiple-point geostatistical simulation (MPS) using the Single Normal equation Simulation (SNESIM) algorithm applied to training images and probability volumes as constraints for a better lithofacies model. These integrated study results proved that MPS could improve reservoir lithofacies characterization.

3D multiple-point geostatistical simulation of joint subsurface redox and geological architectures

Abstract. Nitrate contamination of subsurface aquifers is an ongoing environmental challenge due to nitrogen (N) leaching from intensive N fertilization and management on agricultural fields. The distribution and fate of nitrate in aquifers are primarily governed by geological, hydrological and geochemical conditions of the subsurface. Therefore, we propose a novel approach to modeling both geology and redox architectures simultaneously in high-resolution 3D (25m×25m×2m) using multiple-point geostatistical (MPS) simulation. Data consist of (1) mainly resistivities of the subsurface mapped with towed transient electromagnetic measurements (tTEM), (2) lithologies from borehole observations, (3) redox conditions from colors reported in borehole observations, and (4) chemistry analyses from water samples. Based on the collected data and supplementary surface geology maps and digital elevation models, the simulation domain was subdivided into geological elements with similar geological traits and depositional histories. The conceptual understandings of the geological and redox architectures of the study system were introduced to the simulation as training images for each geological element. On the basis of these training images and conditioning data, independent realizations were jointly simulated of geology and redox inside each geological element and stitched together into a larger model. The joint simulation of geological and redox architectures, which is one of the strengths of MPS compared to other geostatistical methods, ensures that the two architectures in general show coherent patterns. Despite the inherent subjectivity of interpretations of the training images and geological element boundaries, they enable an easy and intuitive incorporation of qualitative knowledge of geology and geochemistry in quantitative simulations of the subsurface architectures. Altogether, we conclude that our approach effectively simulates the consistent geological and redox architectures of the subsurface that can be used for hydrological modeling with nitrogen (N) transport, which may lead to a better understanding of N fate in the subsurface and to future more targeted regulation of agriculture.

Choosing between Gaussian and MPS simulation: the role of data information content\u2014a case study using uncertain interpretation data points

Integration of geophysical data with information such as from boreholes and expert domain knowledge is often performed as cognitive or explicit geological modeling when generating deterministic geological models of the subsurface. However, such cognitive geological models lack the ability to express the uncertainty of layer boundaries. To remedy the shortcomings of this strategy we propose a novel stochastic methodology combining the efforts of probabilistic data integration and cognitive modeling. We treat geological interpretation points from the cognitive model as uncertain “soft” data. These data are then combined with analogous geology in a probabilistic model. We test two ways of combining and sampling from such a probabilistic model. Firstly, a high-entropy setup based on Gaussian distributions simulation. Secondly, lower entropy (and conceivable more realistic) geological solutions are obtained from multiple-point geostatistics (MPS). We apply both ways of solving the problem at a study site near Horsens, Denmark, where airborne transient electromagnetic measurements, seismic data, and borehole information are available and interpreted in cognitive modeling. We explain the complete framework for integrating the uncertain interpretations in geostatistical simulation. Results show that both the Gaussian simulation and multiple-point geostatistical approach allows satisfactory simulations of uncertain geological interpretations and are consistent with prior geological knowledge. Our results suggest that the number of uncertain data points and their information content play a pivotal role in selecting the most appropriate simulation method for the given framework. MPS simulations allow connectivity in scenarios with few data points due to the low entropy of the model. When the number of soft data increases, Gaussian simulation is less prone to produce simulation artifacts, faster to apply, and could be considered preferential to using MPS simulation.

Multiple-point geostatistical simulation based on conditional conduction probability

Multiple-point geostatistical (MPS) simulation can enhance extraction and synthesis of various information in earth and environmental sciences. In particular, it is able to characterize the complex spatial structures of heterogeneous phenomena more accurately. In this paper, we propose a new MPS simulation method based on conditional conduction probability, namely the CCPSIM algorithm, to mitigate the uncertainty of MPS realizations. In CCPSIM, the simulated nodes will be treated differently from the original samples. The probability distributions of the simulated nodes will be used as prior conditions to calculate the probability distributions of the following nodes, and the prior conditions will be conducted during the whole simulation process. 2D and 3D synthetic tests are used to verify the applicability and advantages of CCPSIM. The results confirm that CCPSIM is able to reproduce spatial patterns of heterogeneous structures presented in categorical training images, and it reduces the uncertainty of the MPS realizations caused by the undistinguished using of the original known samples and the simulated uncertain values.

CHDS: conflict handling in direct sampling for stochastic simulation of spatial variables

In recent years, multiple-point geostatistical (MPS) approaches have gained significant popularity for modeling subsurface heterogeneity in hydrogeological systems by employing a training image for describing the features of the target field. The most important challenges of MPS simulation methods include appropriate pattern reproduction and connectivity preservation, handling conditional data, and appropriately modeling the variability of real fields. Preserving connectivity of the patterns is of paramount importance, particularly in fluid flow modeling problems. During sequential simulation, if the algorithm produces a value (or patch) inconsistently with previously synthesized data, such conflicts will propagate in the realization and lead to poor pattern reproduction. Here, we have introduced a two-step simulation algorithm, where in the first phase, the coarse structure of the realization is synthesized with minimum conflicts by rejecting inconsistent patterns and allowing removing previously synthesized data, and in the second phase, the fine grid is simulated by ignoring the conflicts. Ignoring short-range inconsistencies in the fine simulation phase not only improves the algorithm’s convergence but also leads to higher variabilities without sacrificing the quality of the realizations. Convergence problems of traditional conflict-handling methods are further alleviated by a new distance reweighting strategy, which prevents cyclic deletions and resimulations. We have employed different statistical descriptors to evaluate our method in comparison with existing pixel and patch-based methods in conditional and unconditional modes. The proposed method shows outstanding results in terms of connectivity preservation, conditional data handling, and pattern innovation. Compared to traditional conflict-handling methods, the proposed method shows good convergence and histogram preservation.

Multiple point geostatistical simulation with adaptive filter derived from neural network for sedimentary facies classification

Abstract Sedimentary facies distribution is a vital reference for oil and gas exploration in an offshore area. One of remaining questions in a marine exploration area is to acquire an accurate sedimentary facies map with limited loggings and a set of indirect information. Previous studies are largely based on geological parameters and structures for estimating the sedimentary distributions, while some other researches employ machine learning algorithm to participate the classification tasks based on accessible datasets. However, the application of either method alone still cannot overcome the information limitation or the low accuracy of final results. This study is a kind of attempt to introduce machine learning algorithm into traditional multiple point geostatistical simulation (MPS) with seismic and logging information for increasing the applicability of multiple point geostatistical simulation and solving the previous mentioned problems of sedimentary classification. The filter-generation auto-encoder (FAE) using in this work is constructed by an encoder, a decoder, and a classifier, which are designed for generating adaptive filters of MPS, maintaining the stability of data extraction, and introducing the constraints of sedimentary facies, separately. For applying these adaptive filters into MPS, the distance function need to be modified to meet the requirement of indirect information without prototypes. Then the multi-distances function is able to measure the relationship among a known pattern, a nearby layer, related simulated points, and seismic attributes. After that, the sedimentary maps can be obtained by this method with great continuity in horizontal and vertical direction at the mean time, and its effectivity has been verified by the loggings in this area. Furthermore, the 3-D sedimentary facies map have valuable meanings for both petroleum exploration and geological engineering in an area with limited information.

Investigating extreme scenarios with multiple-point geostatistics and variance maximization

In many geoscience applications, the data extracted from environmental variables are very limited. Multiple-point geostatistical (MPS) approaches simulate these variables and associated uncertainties at unknown locations by using an exemplar model for the field, called the training image (TI). Existing MPS approaches aim at simulating the field in a way consistent with both available conditional data and TI properties. The inevitably limited size of the training database usually leads to an underestimated variability between different realizations as compared to the variability of the real phenomenon. Furthermore, in over-conditioned regions, patch-based methods often tend to paste the same patch in all realizations. In this paper, we suggest an optimization-based approach for MPS simulation that simulates a bunch of realizations simultaneously. In addition to maintaining consistency with both conditional data and TI properties, the proposed method aims at maximizing the variability between different realizations. Our experiments show that the proposed strategy enhances the variability of the realizations to better conform with real variabilities. The idea of targeting variance maximization can potentially be applied to other MPS simulation methods by simulating a bunch of realizations simultaneously with a constraint to avoid similar patterns at the same location in different realizations.

Conditional multiple-point geostatistical simulation for unevenly distributed sample data

To expand the applicability/versatility of multiple-point geostatistical (MPS) methods to the unevenly distributed sample data acquired from geological and environmental surveys, this paper presents a conditional MPS-based simulation method which considers the distribution characteristics of sample data adequately. In this work, we mainly focus on the improvement of two key steps in MPS methods, i.e. the selection of simulation paths and the construction of data events, aiming at mitigating the adverse effects of unevenly distributed conditioning data. First, a simulation path sensitive to the distribution density of informed samples is adopted to ensure that each simulation of the unknown nodes in a simulation grid is done from the location with the highest density of informed nodes around. Second, a stable data event is obtained by evenly extracting several informed nodes closest to the current node from each subarea. This improvement avoids the excessive concentration of the nodes in a data event, so that the nodes in an obtained data event are more evenly distributed around the current node. The two improvements are embedded into a widely used MPS method, the direct sampling. Several 2D and 3D synthetic experiments with categorical or continuous variables are used to test the proposed method. The results demonstrate their applicability in characterizing heterogeneous phenomena when faced with unevenly distributed conditioning data.

Analogue-based colorization of remote sensing images using textural information

Satellite images are richer than ever before. For example, new Landsat-8 images with their 11 bands carry much more information than older generations of satellites. These differences in spectral representation imply a major difficulty for assessing long-term land surface changes. The easiest solution is to reduce the information of the most recent product, for example by only keeping a subset of the Landsat-8 bands that matches old imagery. To avoid such loss of information, we propose a new method based on multiband spatial pattern matching. We are focusing on increasing the spectral resolution of archive satellite images to the same level of spectral resolution and coverage as modern imagery. Our method uses analogous scenes taken from modern satellites, which have conceptually the same role as the training images used in multiple-point geostatistics simulation. The spectral characteristics of the training image are then transferred to a target archive image, where new synthetic spectral bands are generated. A spatial pattern matching procedure is used to control this transfer, resulting in preservation of spatial and spectral coherence in the results. We illustrate the methodology on Landsat 8 and Corona imagery. The proposed method was benchmarked against other state-of-the-art colorization techniques, and it shows globally better results.

Multiple-point geostatistical simulation for mine evaluation with aeromagnetic data

Multiple-point geostatistical simulation (MPS) was applied to develop 3D ore models matched to surrounding geological information accompanying aeromagnetic data using a training image (TI). Conventional 3D geological models generated from a limited number of boreholes and other geological information may be useful for evaluating the mineral resources around the boreholes, while also bearing uncertainty regarding the evaluation of the ore body over the entire area. Geostatistical analysis accompanying the geophysical interpretation is adopted to reduce the uncertainty of the 3D ore model. Among the geostatistical methods, MPS based on a TI made from available geological information is chosen to simulate the configuration and distribution of the ore body according to the geological structure. The present study proposes a method for reducing the uncertainty of the 3D ore model, applying MPS for mine evaluation to create probabilistic ore models and analysing the correlation between the models and geophysical data. This method was applied to a metal mine located in Korea. Single normal equation simulation (SNESIM) was chosen as the simulation algorithm, and aeromagnetic data were used to support the analysis of simulated models. With comparison/analysis of the probabilistic ore model and geophysical data, the 3D geological model utilising MPS represented the configuration and distribution of the ore body well according to the geological structure. The SNESIM cluster results indicated high reliability for the final interpretation of the 3D models.

Improving Filtersim Simulation Algorithm of Multiple-Point Geostatistics Using New Training Image Based Adaptive Filters and Supervised Hard Data Conditioning

Summary Simulations based on Multiple-point statistics (MPS) form a class of newly developed techniques that have received great attention in recent years and are mainly used to map spatial complexity and heterogeneity of geological phenomenon. One of the most commonly used pattern based multiple-point geostatistical simulation algorithm is called Filtersim. In the conventional Filtersim algorithm, the detected patterns in training images are transformed into filter score space using a fixed number of filters that are neither dependent on the training images nor on the characteristics of patterns extracted from them. Through using principal component analysis in current study, a set of new filters are designed in such a way to include most structural information specific to the governing training image resulting in the selection of closer patterns in the filter score space. Comparing the results of applying our proposed filters to that of conventional Filtersim algorithm shows a significant improvement in recovering expected shapes and structural continuity in the final simulated realizations. Introduction MPS simulation is capable of reproducing complex geological patterns that cannot be modeled by two-point statistics moments such as variograms. One of the commonly used pattern-based approach, FILTERSIM, introduces filters to summarize high dimensional patterns into a filter score space. Then, it classifies the patterns into a limited number of classes. FILTERSIM uses six predefined filters, which are not specific to the Training Image. The aim of current research is to introduce new improvements to the initial FILTERSIM technique through defining new adaptive filters. Methodology and Approaches In contrast to original FILTERSIM method, our approach uses some filters which are designed specifically for any given Training Image using PCA analysis. In addition, our approach uses a combination of raster path and random partitioning methods in the course of simulation. The raster path approach results in simulations showing a good continuity of the patterns. The random partitioning is a new feature that allows better simulated realizations as compared to the straightforward use of the raster path. Results and Conclusions The newly proposed approach have been tested on some training images and the results have been compared with previous pattern-based algorithms. Through visual inspection and some discrepancy measures it is shown that the simulated realizations obtained by our approach are much closer to the training image than those obtained with the other methods. In particular, the continuity of the patterns is better preserved with the proposed method.

Grade and Tonnage Uncertainty Analysis of an African Copper Deposit Using Multiple-Point Geostatistics and Sequential Gaussian Simulation

Spatial uncertainty analysis is a complex and difficult task for orebody estimation in the mining industry. Conventional models (kriging and its variants) with variogram-based statistics fail to capture the spatial complexity of an orebody. Due to this, the grade and tonnage are incorrectly estimated resulting in inaccurate mine plans, which lead to costly financial decision. Multiple-point geostatistical simulation model can overcome the limitations of the conventional two-point spatial models. In this study, a multiple-point geostatistical method, namely SNESIM, was applied to generate multiple equiprobable orebody models for a copper deposit in Africa, and it helped to analyze the uncertainty of ore tonnage of the deposit. The grade uncertainty was evaluated by sequential Gaussian simulation within each equiprobable orebody models. The results were validated by reproducing the marginal distribution and two- and three-point statistics. The results show that deviations of volume of the simulated orebody models vary from − 3 to 5% compared to the training image. The grade simulation results demonstrated that the average grades from the different simulation are varied from 3.77 to 4.92% and average grade 4.33%. The results also show that the volume and grade uncertainty model overestimates the orebody volume as compared to the conventional orebody. This study demonstrates that incorporating grade and volume uncertainty leads to significant changes in resource estimates.