Plurigaussian Simulation Research Articles

Geochemical anomaly separation based on geology, geostatistics, compositional data and local singularity analyses: A case study from the kuh panj copper deposit, Iran

This study combines geochemical anomaly separation with geostatistical approaches and compositional data analysis. To have a reasonable model for abnormal areas, suggesting the optimal drilling coordinates, one needs to consider some effective geological controllers on the mineralization such as lithology, alteration, and tectonic processes. The lithogeochemical samples at 608 locations were processed from the copper porphyry mineralization area of Kuh Panj, Iran, to illustrate the proposed approach. The heterogeneity of rock type domains was detected using contact analysis. Based on this, plurigaussian simulation made it possible to model the uneconomic dykes of the area. Copper-molybdenum grades and filler as geochemical components were transformed into multivariate normal data using isometric logarithmic ratio-flow anamorphosis. Hierarchical rock type-grade simulation reproduced the grade behavior at hard rock type boundaries. Selection of a dense and regular simulation grid minimized the smoothing effect caused by interpolation of the singularity index values. Finally, unsupervised machine learning identified the anomaly zone by clustering the results of the singularity analysis. Validation using drilling data and the geological map shows that the anomaly separation, considering the geological conditions and the compositional nature of the geochemical data, can provide a reliable model of the geochemical anomalies in the Kuh Panj deposit.

Using multiple-point geostatistics for geomodeling of a vein-type gold deposit

Geostatistical cascade modeling of Mineral Resources is challenging in vein-type gold deposits. The narrow shape and long-range features of these auriferous veins, coupled with the paucity of drill-hole data, can complicate the modeling process and make the use of two-point geostatistical algorithms impractical. Instead, multiple-point geostatistics techniques can be a suitable alternative. However, the most challenging part in implementing the MPS is to use a suitable training data set or training image (TI). In this paper, we suggest using the radial basis function algorithm to build a training image and the DeeSse algorithm, one of the multiple-point statistics (MPS) methods, to model two long-range veins in a gold deposit. It is demonstrated that DeeSse can replicate long-range vein features better than plurigaussian simulation techniques when there is a lack of conditioning data. This is shown by several validation processes, such as comparing simulation results with an interpretive geological block model and replicating geological proportions.

Stochastic Periodic Microstructures for Multiscale Modelling of Heterogeneous Materials

Plurigaussian simulation is a method of discrete random field generation that can be used to generate many complex geometries depicting real world structures. Whilst it is commonly applied at larger scales to represent geological phenomena, the highly flexible approach is suitable for generating structures at all scales. Here, an extension of plurigaussian simulation to periodic plurigaussian simulation (P-PGS) is presented, such that the resulting fields are periodic in nature. By using periodic Gaussian random fields as components of the method, periodicity is enforced in the generated structures. To substantiate the use of P-PGS in capturing complex heterogeneities in a physically meaningful way, the pore-scale microstructure of cement paste was represented such that its effective properties can be calculated through a computational homogenisation approach. The finite element method is employed to model the diffusion of heat through the medium under dry and saturated pore conditions, where numerical homogenisation is conducted to calculate the effective thermal conductivity of the medium. Comparison of the calculated values with experimental observations indicated that the generated microstructures are suitable for pore-scale representation, given their close match. A maximal error of 1.38% was observed in relation to the numerically determined effective thermal conductivity of mortar paste with air filled pores, and 0.41% when considering water filled pores. As the assumption of a periodic domain is often an underlying feature of numerical homogenisation, this extension of plurigaussian simulation enables a path for its integration into such computational schemes.Article HighlightsIntegrating P-PGS into numerical homogenisation frameworks enhances complex heterogeneous material representationThe flexibility of P-PGS enables a wide range of material microstructures to be represented accuratelyUse of the generated structures allows material properties to be estimated accurately through numerical homogenisation

Bridging element-free Galerkin and pluri-Gaussian simulation for geological uncertainty estimation in an ensemble smoother data assimilation framework

The facies distribution of a reservoir is one of the biggest concerns for geologists, geophysicists, reservoir modelers, and reservoir engineers due to its high importance in the setting of any reliable decision-making/optimization of field development planning. The approach for parameterizing the facies distribution as a random variable comes naturally through using the probability fields. Since the prior probability fields of facies come either from a seismic inversion or from other sources of geologic information, they are not conditioned to the data observed from the cores extracted from the wells. This paper presents a regularized element-free Galerkin (R-EFG) method for conditioning facies probability fields to facies observation. The conditioned probability fields respect all the conditions of the probability theory (i.e. all the values are between 0 and 1, and the sum of all fields is a uniform field of 1). This property achieves by an optimization procedure under equality and inequality constraints with the gradient projection method. The conditioned probability fields are further used as the input in the adaptive pluri-Gaussian simulation (APS) methodology and coupled with the ensemble smoother with multiple data assimilation (ES-MDA) for estimation and uncertainty quantification of the facies distribution. The history-matching of the facies models shows a good estimation and uncertainty quantification of facies distribution, a good data match and prediction capabilities.

Plurigaussian modeling of non-stationary geological domains to assess geological uncertainty in a porphyry copper deposit

In mineral resources extraction projects, the morphology of geological domains that make up the ore types in a mineral deposit is essential for mine and plant planning and design. Modeling the layout of these domains and quantifying the uncertainty associated with such a layout can be performed via geostatistical simulation in order to provide multiple geological outcomes that are equally likely to occur. This work focuses on showing how plurigaussian simulation can help to model ore type domains exhibiting spatially varying probabilities of occurrence, through an application to a porphyry copper deposit in northwestern Iran. The results demonstrate the advantage of incorporating spatial variations of the domain proportions, in accordance with the geological interpretation of the deposit, to obtain geologically consistent and realistic outcomes.

Co-simulation of hydrofacies and piezometric data in the West Thessaly basin, Greece: A geostatistical application using the GeoSim R package

In the present study, we co-simulate hydrofacies and piezometric data in order to construct geostatistical realizations of underground geology in an area of the West Thessaly basin. This basin is of great importance in terms of sustainable water management and environmental perspective in Greece. Through Plurigaussian modeling, the hydrofacies are first transformed into Gaussian Random Fields. Then, a Linear Coregionalization Model is established to account for the dependencies between hydrofacies and the Normal scores of piezometric data. The effect of co-simulation shows an improvement of the facies transition probabilities in comparison with those of Plurigaussian simulation. For the purpose of this study, we use the GeoSim package in R developed by our team for the implementation of Plurigaussian simulation and co-simulation.

GeoSim: An R-package for plurigaussian simulation and Co-simulation between categorical and continuous variables

Plurigaussian simulation is widely used to model geological facies in geosciences and is predominantly applied in mineral deposits and petroleum reservoirs exploration. GeoSim package builds geostatistical models of categorical regionalized variables via conditional or unconditional Plurigaussian simulation and co-simulation. Co-simulation between Gaussian Random Fields representing the geological facies and other numerical variables accounting for auxiliary hydrological or geophysical quantities, is also available in this package with the definition of a linear coregionalization model. The algorithm is not restricted by the number of simulated facies and the number of truncated Gaussians, while parts of the code requiring heavy computations are compiled in C++ taking benefits of the integration between R and C++. In this work, we introduce the GeoSim package and demonstrate its capabilities. We present a 3D application focused on a lignite mine in Greece, where we investigate the Plurigaussian simulation and co-simulation of five geological facies (categorical variables) and the lower calorific value (continuous variable). The findings of our study highlight the significant benefits of Plurigaussian and co-simulation to capture the geological spatial heterogeneity.

Seismic Data Integration Workflow in Pluri-Gaussian Simulation: Application to a Heterogeneous Carbonate Reservoir in Southwestern Iran

Seismic Data Integration Workflow in Pluri-Gaussian Simulation: Application to a Heterogeneous Carbonate Reservoir in Southwestern Iran

Geostatistical Modeling of Overburden Lithofacies to Optimize Continuous Mining in the Ptolemais Lignite Mines, Greece

Lignite production in Greece is implemented mainly by the Public Power Corporation (PPC), with the higher production being in the Lignite Center of Western Macedonia. A continuous surface mining method is used in order to satisfy the high production needs combined with the necessity for selective mining; however, the occasional appearance of hard rock formations in the South Field mine overburdens was critical for the adoption of a discontinuous auxiliary method of rock mass removal, at these places, by explosives and large shovels. Furthermore, to minimize the delay of changing the machinery arrangements when a hard rock formation is met, an a priori knowledge of the spatial distribution of these rock masses would be catalytic. In this work, a plurigaussian simulation model of the overburden geological formations is developed in the South Field mine. This model could be used as a guide to schedule and optimize the overburden removal process. Validation of the model was affected in two ways: by direct comparing estimated to real cross-sections as observed on mine slopes or by correlating PPC’s recorded volumetric results to the average simulated hard rock percentages.

Risk-based methodology to optimize geotechnical site investigations in tunnel projects

The scope of geotechnical investigations in tunnel projects is generally driven by the allocated resources rather than the expected variability in the ground conditions. Uncertainties in ground conditions may lead to poor decisions in project planning and active risk management. This paper presents a systematic and rational methodology to identify priority locations of additional geotechnical investigations for soft ground tunneling applications based on tunnel risks, site conditions, and project-related constraints. The methodology is applied to the cutter tool wear risk and quantifies the uncertainty in soil abrasivity index (SAI), the relevant geotechnical parameter. Preliminary geotechnical investigation data and site conditions from an actual tunnel project in an urban environment are used to illustrate the proposed methodology and demonstrate its effectiveness. Pluri-Gaussian simulation and sequential Gaussian simulation are used to characterize the 3D spatial variability in soil units and soil abrasivity, respectively, along the tunnel alignment. The study integrates geospatial assessments of SAI uncertainty and consequences of tool wear to develop an R index map that delineates the impact of uncertainty in tool wear rate. Project constraints of drilling accessibility and budget are incorporated in the R index map to find locations of additional geotechnical investigations. The study simulates a virtual sampling of additional boreholes and quantifies the reduction in uncertainty in tool wear rate and tunnel boring machine (TBM) intervention locations. Additional investigation at priority locations is found to reduce cutterhead intervention location uncertainty by approximately 90 rings (160 m). Further, integrating the uncertainty in tool travel distance is found to influence the cutterhead intervention location uncertainty by approximately 40 rings.

Hierarchical reservoir lithofacies and acoustic impedance simulation: Application to an oil field in SW of Iran

In this study, a hierarchical simulation algorithm was implemented to produce 3D models of lithofacies and acoustic impedance (AI) in an oil field in southwestern Iran. Hierarchical simulation is a powerful method that takes into account the spatial distribution and hard contact relationships of continuous variables within the categorical variables in reservoir characterization. To implement the hierarchical simulation algorithm in this study, pluri-Gaussian simulation was used to construct the layouts of one categorical variable (lithofacies domains) and turning bands simulation was used to produce one continuous variable (AI models) within each simulated domain. The dataset that is available to test the proposed method includes the lithofacies and AI logs of seven wells. For validation of the proposed model, one of wells was excluded from the initial dataset to check the accuracy of results. The algorithm was implemented on a gridded model to produce 50 realizations of 3D lithofacies and AI models. The statistical analyses of the produced models verified the capabilities of hierarchical simulation in reproducing the lithofacies domains and AI values related to each domain. The simulated AI models were compared with the seismic inversion model to corroborate the reliability of the results. In addition, to show the superiority of the hierarchical approach, the results were compared to the results of direct AI modeling in which the AI models were generated without considering any lithofacies data. The comparisons in the blind well indicated a high correlation between the modeled values and the true AI. The correlation coefficient between average AI models of the hierarchical approach and the true AI was 0.85. Whereas, this value was 0.75 for the average of directly simulated AI models and 0.74 for the seismic inversion, respectively. Histogram analysis also verified the capability of the hierarchical modeling in reproducing the statistical properties of the original data. The resulting AI model obtained using the method proposed in this study also showed a high compatibility with the deterministic seismic inversion cube.

A new methodology based on finite element method (FEM) for generation of the probability field of rock types from subsurface

In this paper, we present a novel methodology by employing the finite element method (FEM) for creating probability fields of rock types (facies) present in a reservoir. The method is developed for a hydrocarbon reservoir, which, situated thousands of meters underground, has a high uncertainty regarding its geology. However, from the exploration phase, are collected facies observations (hard data) about the rock types present at some sparse locations of the reservoir domain. These observations are projected into a continuous probabilistic space, which is the environment of the FEM. The novelty consists of the fact that the probability field of a facies type is viewed as a deformation field of a plate. For each facies type, the FEM constructs a deformation profile which will be a probability field. The fields generated with the FEM verify the conditions of the probability theory without post-regularization techniques. We test the methodology for a two-dimensional reservoir model with three facies types. We also show the applicability of the new method by introducing the probability fields in the truncated pluri-Gaussian simulation method for simulating facies fields which are the inputs in the assisted history matching of the reservoir.

Simulation of cement raw material deposits using plurigaussian technique

Abstract Plurigaussian simulation is a powerful and very effective technique for modelling subsurface rock type domain distribution and in-situ mining reserve analysis. Modelling of subsurface to reveal the rock type distribution plays a key role for raw material extraction planning and plant operations such as extraction, transportation and comminution strategies. Because, the raw material distribution defines the plant operations and final product quality (cement modulus). This study addresses the application of plurigaussian simulation technique to reveal the subsurface rock type distribution of a cement raw material deposit in Turkey. The rock type domains include the limestone, clayey limestone, marl and sandstone which are the basic four rock type classes effecting the cement modulus in the field. The simulation process is carried out using these four rock type data on a determined grid system. A series of tests are made for the validation of the plurigaussian simulation. As a result, the rock type distributions are presented as both 2D-3D graphics and tabulated. The limestone is found as a dominant rock type in the deposit. The marl – a natural clinker - is another widespread raw material in the field and is found interbedded with limestone across the study field. The unwanted sandstone existence exhibited a sparse distribution in reserve body. The results indicated that, the deposit can provide the required raw material for the plant, showing the localised rock type distribution. A detailed raw material extraction planning and scheduling may be made using the results of this study.

Lithofacies uncertainty modeling in a siliciclastic reservoir setting by incorporating geological contacts and seismic information

Deterministic modeling lonely provides a unique boundary layout, depending on the geological interpretation or interpolation from the hard available data. Changing the interpreter’s attitude or interpolation parameters leads to displacing the location of these borders. In contrary, probabilistic modeling of geological domains such as lithofacies is a critical aspect to providing information to take proper decision in the case of evaluation of oil reservoirs parameters, that is, applicable for quantification of uncertainty along the boundaries. These stochastic modeling manifests itself dramatically beyond this occasion. Conventional approaches of probabilistic modeling (object and pixel-based) mostly suffers from consideration of contact knowledge on the simulated domains. Plurigaussian simulation algorithm, in contrast, allows reproducing the complex transitions among the lithofacies domains and has found wide acceptance for modeling petroleum reservoirs. Stationary assumption for this framework has implications on the homogeneous characterization of the lithofacies. In this case, the proportion is assumed constant and the covariance function as a typical feature of spatial continuity depends only on the Euclidean distances between two points. But, whenever there exists a heterogeneity phenomenon in the region, this assumption does not urge model to generate the desired variability of the underlying proportion of facies over the domain. Geophysical attributes as a secondary variable in this place, plays an important role for generation of the realistic contact relationship between the simulated categories. In this paper, a hierarchical plurigaussian simulation approach is used to construct multiple realizations of lithofacies by incorporating the acoustic impedance as soft data through an oil reservoir in Iran.

Calibration of categorical simulations by evolutionary gradual deformation method

Methods to simulate facies (or categorical) fields are numerous. However, calibration of simulated facies fields to large-scale or dynamic data still remains an important challenge due to the discrete nature of the fields, the non-linearity of the response with respect to the facies fields, and the non-derivability of the objective function used in calibration. A new gradual deformation method (GDM) is presented and tested for the calibration of facies realizations obtained by patch-multipoint simulation (MPS). The proposed method borrows ideas from pluriGaussian simulation, evolutionary algorithms, and GDM. Various test cases are considered: proportion maps, section of seismic amplitudes, inlet to outlet travel time along the shortest path, and water-cut curves obtained with a flow simulator. Both conditional/unconditional MPS simulations and 2D/3D problems are considered. In all studied test cases, the new GDM approach has provided excellent calibration to the target variables. The method is general as it can be used in conjunction with any facies simulator.

Geochemical modeling of the Cu-Ag deposits from the Lubin-Głogów Copper District (Poland) supported by lithological modeling

Abstract The most important reasons for the limited credibility of 3D geochemical models of the Cu-Ag deposits (The Lubin-Głogów Copper District) are unfavorable statistical characteristics of the main metals (Cu, Ag), and the unfavorable structure of their variability with a relatively poorly marked non-random component of variability. It is hoped that the accuracy of geochemical models can be increased by previous lithological modeling of a given deposit. This is due to significant differences in mean Cu content in both the main lithological series (carbonates, shales and sandstones) and individual lithological units within them, which justifies separate geochemical modeling of each. The paper presents the results of the comparison of the reliability of 3D modeling of Cu content carried out in both individual and main lithological units of the Cu-Ag Rudna deposit (The Lubin- Głogów Copper District). The 3D lithological models were made using ordinary kriging (main lithological units) and Plurigaussian simulation (individual lithological units). The 3D estimation of Cu content within lithological models was carried out using the ordinary kriging (OK ) and squared inverse distance (ID2) methods. The verification of the accuracy of Cu estimates in the 3D model using spot samples of the test data set has shown only a slight increase in the accuracy of the estimates of Cu content within the individual lithological units of the shale series compared to the estimates of Cu content in the whole main lithology. In most cases, 3D models of Cu content carried out using the ordinary kriging method are slightly more accurate than the analogous squared inverse distance weighting method.

Geostatistical modeling of the geological uncertainty in an iron ore deposit

This paper addresses the problem of quantifying the joint uncertainty in the grades of elements of interest (iron, silica, manganese, phosphorus and alumina), loss on ignition, granulometry and rock types in an iron ore deposit. Sampling information is available from a set of exploration drill holes. The methodology considers the construction of multiple rock type outcomes by plurigaussian simulation, then outcomes of the quantitative variables (grades, loss on ignition and granulometry) are constructed by multigaussian joint simulation, accounting for geological domains specific to each quantitative variable as well as for a stoichiometric closure formula linking these variables. The outcomes are validated by checking the reproduction of the data distributions and of the data values at the drill hole locations, and their ability to measure the uncertainty at unsampled locations is assessed by leave-one-out cross validation.Both the plurigaussian and multigaussian models offer much flexibility to the practitioner to face up to the complexity of the variables being modeled, in particular: (1) the contact relationships between rock types, (2) the geological controls exerted by the rock types over the quantitative variables, and (3) the cross-correlations and stoichiometric closure linking the quantitative variables. In addition to this flexibility, the use of efficient simulation algorithms turns out to be essential for a successful application, due to the high number of variables, data and locations targeted for simulation.

Multiple imputation framework for data assignment in truncated pluri-Gaussian simulation

Truncated pluri-Gaussian simulation (TPGS) is suitable for the simulation of categorical variables that show natural ordering as the TPGS technique can consider transition probabilities. The TPGS assumes that categorical variables are the result of the truncation of underlying latent variables. In practice, only the categorical variables are observed. This translates the practical application of TPGS into a missing data problem in which all latent variables are missing. Latent variables are required at data locations in order to condition categorical realizations to observed categorical data. The imputation of missing latent variables at data locations is often achieved by either assigning constant values or spatially simulating latent variables subject to categorical observations. Realizations of latent variables can be used to condition all model realizations. Using a single realization or a constant value to condition all realizations is the same as assuming that latent variables are known at the data locations and this assumption affects uncertainty near data locations. The techniques for imputation of latent variables in TPGS framework are investigated in this article and their impact on uncertainty of simulated categorical models and possible effects on factors affecting decision making are explored. It is shown that the use of single realization of latent variables leads to underestimation of uncertainty and overestimation of measured resources while the use constant values for latent variables may lead to considerable over or underestimation of measured resources. The results highlight the importance of multiple data imputation in the context of TPGS.

Grid-free petroleum reservoir characterization with truncated pluri-Gaussian simulation: Hekla case study

A new geostatistical grid-free simulation (GFS) method has been developed recently that represents simulated continuous attributes of the natural phenomena, such as stratigraphic surface boundaries or petrophysical properties, as an analytical function of the coordinates of the simulation locations. Thus, GFS resolves challenges related to model regridding, increasing resolution around already simulated locations and integration of newly available data in a consistent manner. The present paper contains further developments in simulation of categorical variables, such as facies, in a grid-free fashion based on the truncated pluri-Gaussian simulation (TPG) paradigm. The resultant simulation engine allows the entire reservoir system to be represented as an analytical stochastic function: that is, values of any reservoir properties are simulated on demand at requested locations in space. The selection of proper variograms of Gaussian continuous variables for simulation of categorical variables in a TPG framework is proposed through a methodology based on Monte Carlo simulation. The variogram models of the underlying Gaussian continuous variables are obtained by minimizing the difference between numerically computed and target indicator variograms. A local optimization approach is suggested for a fast precise derivation of the variograms. The stable variogram model leads to the closest fit to the experimental variograms of the continuous variables. The automatic establishment of a truncation mask based on multidimensional scaling to convert Gaussian continuous variables to categories is also explained. Finally, the proposed GFS algorithm for petroleum reservoir characterization is demonstrated in its full-scale applicability to the Hekla offshore petroleum reservoir located in the North Sea. The results look promising, and should be beneficial to petroleum reservoir modelling in practice.

Truncated Gaussian and derived methods

The interest of a digital model to represent the geological characteristics of the field is well established. However, the way to obtain it is not straightforward because this translation is necessarily a simplification of the actual field. This paper describes a stochastic model called truncated Gaussian simulations (TGS), which distributes a collection of facies or lithotypes over an area of interest. This method is based on facies proportions, spatial distribution and relationships, which can be easily tuned to produce numerous different textures. Initially developed for ordered facies, this model has been extended to complex organizations, where facies are not sequentially ordered. This method called pluri-Gaussian simulation (PGS) considers several Gaussian random functions, which can be correlated. PGS can produce a large variety of lithotype setups, as illustrated by several examples such as oriented deposits or high frequency layering.