Plurigaussian Model Research Articles

Stochastic 3D Modeling of Three-Phase Microstructures for Predicting Transport Properties: A Case Study

We compare two conceptually different stochastic microstructure models, i.e., a graph-based model and a pluri-Gaussian model, that have been introduced to model the transport properties of three-phase microstructures occurring, e.g., in solid oxide fuel cell electrodes. Besides comparing both models, we present new results regarding the relationship between model parameters and certain microstructure characteristics. In particular, an analytical expression is obtained for the expected length of triple phase boundary per unit volume in the pluri-Gaussian model. As a case study, we consider 3D image data which show a representative cutout of a solid oxide fuel cell anode obtained by FIB-SEM tomography. The two models are fitted to image data and compared in terms of morphological characteristics (like mean geodesic tortuosity and constrictivity) as well as in terms of effective transport properties. The Stokes flow in the pore phase and effective conductivities in the solid phases are computed numerically for realizations of the two models as well as for the 3D image data using Fourier methods. The local and effective physical responses of the model realizations are compared to those obtained from 3D image data. Finally, we assess the accuracy of the two methods to predict permeability as well as electronic and ionic conductivities of the anode.

Nonparametric Geostatistical Simulation of Subsurface Facies: Tools for Validating the Reproduction of, and Uncertainty in, Facies Geometry

Delineation of facies in the subsurface and quantification of uncertainty in their boundaries are significant steps in mineral resource evaluation and reservoir modeling, which impact downstream analyses of a mining or petroleum project. This paper investigates the ability of nonparametric geostatistical simulation algorithms (sequential indicator, single normal equation and filter-based simulation) to construct realizations that reproduce some expected statistical and spatial features, namely facies proportions, boundary regularity, contact relationships and spatial correlation structure, as well as the expected fluctuations of these features across the realizations. The investigation is held through a synthetic case study and a real case study, in which a pluri-Gaussian model is considered as the reference for comparing the simulation results. Sequential indicator simulation and single normal equation simulation based on over-restricted neighborhood implementations yield the poorest results, followed by filter-based simulation, whereas single normal equation simulation with a large neighborhood implementation provides results that are closest to the reference pluri-Gaussian model. However, some biases and inaccurate fluctuations in the realization statistics (facies proportions, indicator direct and cross-variograms) still arise, which can be explained by the use of a single finite-size training image to construct the realizations.

On a pluri-Gaussian model for three-phase microstructures, with applications to 3D image data of gas-diffusion electrodes

A pluri-Gaussian model for three-phase microstructures is presented and relationships between model parameters and microstructure characteristics are discussed. In particular, analytical formulas for two-point coverage probability functions in terms of covariance functions of the underlying Gaussian random fields are considered, which allow for an efficient estimation of model parameters. The model is fitted to tomographic image data obtained by FIB-tomography, which represent porous gas-diffusion electrodes consisting of silver and polytetrafluorethylene. The considered type of electrode is used as oxygen depolarized cathode for the production of chlorine. In order to fit the microstructure model, the covariance functions of the Gaussian random fields are parameterized, which leads to a stochastic microstructure model with five parameters. It is shown that most microstructure characteristics of tomographic image data are well reproduced by the model despite the low number of model parameters. Finally, limitations of the model with respect to the fit of continuous phase size distributions are discussed. Combining stochastic microstructure modeling with numerical simulation of effective macroscopic properties will allow in future work for a model-based investigation of microstructure-property relationships for the considered gas-diffusion electrodes.

Data Assimilation in Truncated Plurigaussian Models: Impact of the Truncation Map

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.

Plurigaussian modeling of geological domains based on the truncation of non-stationary Gaussian random fields

The plurigaussian model is used in mining engineering, oil reservoir characterization, hydrology and environmental sciences to simulate the layout of geological domains in the subsurface, while reproducing their spatial continuity and dependence relationships. However, this model is well-established only in the stationary case, when the spatial distribution of the domains is homogeneous in space, and suffers from theoretical and practical impediments in the non-stationary case. To overcome these limitations, this paper proposes extending the model to the truncation of intrinsic random fields of order k with Gaussian generalized increments, which allows reproducing spatial trends in the distribution of the geological domains. Methodological tools and algorithms are presented to infer the model parameters and to construct realizations of the geological domains conditioned to existing data. The proposal is illustrated with the simulation of rock type domains in an ore deposit in order to demonstrate its applicability. Despite the limited number of conditioning data, the results show a remarkable agreement between the simulated domains and the lithological model interpreted by geologists, while the conventional stationary plurigaussian model turns out to be unsuccessful.

Assisted History Matching of Channelized Models by Use of Pluri-Principal-Component Analysis

SummaryAssisted history matching (AHM) of a channelized reservoir is still a very-challenging task because it is very difficult to gradually deform the discrete facies in an automated fashion, while preserving geological realism. In this paper, a pluri-principal-component-analysis (PCA) method, which supports PCA with a pluri-Gaussian model, is proposed to reconstruct geological and reservoir models with multiple facies. PCA extracts the major geological features from a large collection of training channelized models and generates gridblock-based properties and real-valued (i.e., noninteger-valued) facies. The real-valued facies are mapped to discrete facies indicators according to rock-type rules (RTRs) that determine the fraction of each facies and neighboring connections between different facies. Pluri-PCA preserves the main (or principal) features of both geological and geostatistical characteristics of the prior models. A new method is also proposed to automatically build the RTRs with an ensemble of training realizations. An AHM work flow is developed by integrating pluri-PCA with a derivative-free optimization algorithm. This work flow is validated on a synthetic model with four facies types and a real-field channelized model with three facies types, and it is applied to update both the facies model and the reservoir model by conditioning to production data and/or hard data. The models generated by pluri-PCA preserve the major geological/geostatistical descriptions of the original training models. This has great potential for practical applications in large-scale history matching and uncertainty quantification.

Morphological modelling of three-phase microstructures of anode layers using SEM images.

A general method is proposed to model 3D microstructures representative of three-phases anode layers used in fuel cells. The models are based on SEM images of cells with varying morphologies. The materials are first characterized using three morphological measurements: (cross-)covariances, granulometry and linear erosion. They are measured on segmented SEM images, for each of the three phases. Second, a generic model for three-phases materials is proposed. The model is based on two independent underlying random sets which are otherwise arbitrary. The validity of this model is verified using the cross-covariance functions of the various phases. In a third step, several types of Boolean random sets and plurigaussian models are considered for the unknown underlying random sets. Overall, good agreement is found between the SEM images and three-phases models based on plurigaussian random sets, for all morphological measurements considered in the present work: covariances, granulometry and linear erosion. The spatial distribution and shapes of the phases produced by the plurigaussian model are visually very close to the real material. Furthermore, the proposed models require no numerical optimization and are straightforward to generate using the covariance functions measured on the SEM images.

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

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

Morphological Modelling of a Metal Foam Supported SOFC Configuration

The EVOLVE cell, named after a running FP7 project, is an innovative SOFC concept that integrates advanced materials, providing multiple functionalities (electrochemical activity, ion or/and electron conduction, gas diffusion…), and an inventive anode current collector made of a NiCrAl preoxidized foam impregnated with a conductive perovskite. Therefore, this anode combines the beneficial characteristics of MSC and ASC technologies, and strong improvements in terms of reliability and durability (inter-diffusion, Ni coarsening and agglomeration, sulphur tolerance, mechanical robustness and chemical stability under redox cycling in temperature) are expected. In order to enhance the electrode efficiency, the 3D microstructure needs to be precisely described at the microscopic scale by a morphology model. An optimized microstructure (controlled pores size, morphology and connectivity, grains size distribution, 3D tortuosity and 3D percolation of phases, volume fraction of phases, final thickness of components …) is closely related to the starting parameters (nature and quantity of starting powder, binder, shaping aids…) and to the control of the shaping process parameters (sintering treatment and atmosphere). It should meet the best compromise between a good electrocatalytic activity and a low ohmic resistivity while ensuring a stisfactory long term thermomechanical stability on the life time of the device. Based on a mathematical morphology approach applied on symmetrical LST/CGO anode layers, the present work aims at showing how a morphology model can be establised and 3D microstructural data relevant for shaping and performances can be derived from back-scattered electrons (BSE) SEM observations. For each sample, a series of images were used as input information for the microstructural modelling. All the original images were first filtered to remove the noise, thresholded and finally a series of morphological operators (openings, surface openings, closings and reconstruction with markers) were applied to remove artefacts. A 3D pluri-Gaussian model has been generated from 2D images, and then computed to predict volume fraction of the 3 phases, 3D tortuosity, gas permeation and ionic conductivities. These results were used for the optimization of the shaping process that includes the integration of the metal foam.

Morphological Modelling of a Metal Foam Supported SOFC Configuration

The present paper refers to an innovative SOFC concept named EVOLVE after a running FP7 project which integrates advanced materials and an inventive anode current collector made from a NiCrAl preoxidized foam impregnated with a conductive perovskite. Even if the concept is the beneficial combination of Metal supported cell (MSC) and Anode supported cell (ASC) technologies, strong improvements in terms of reliability and durability are expected. In order to enhance anode efficiency, the 3D microstructure needs to be precisely described at the microscopic scale. ISTEC has produced LST-GDC active anode layers using screen printing process. Back-scattered electrons (BSE) SEM images that contain relevant microstructure information have been realized by Centre des Matériaux. All the original images have been enhanced by Centre de Morphologie Mathématique using a series of morphological operators to remove artefacts. Finally, a 3D pluri-Gaussian model has been generated from 2D images, and then computed to predict volume fraction of the 3 phases, 3D tortuosity, gas permeation and ionic conductivities.

Simulation of geo-domains accounting for chronology and contact relationships: application to the Río Blanco copper deposit

The plurigaussian model is increasingly used for simulating geo-domains and quantifying geological uncertainty in the subsurface. However, because they rely on the truncation of only two Gaussian random fields, the current implementations of this model are often restricted in the number of geo-domains that can be simulated and in their contact relationships. A solution to overcome these restrictions is to increase the number of underlying Gaussian random fields. Such an approach yields a very flexible model, able to reproduce the contact relationships between geo-domains in agreement with their chronology (i.e., such that younger geo-domains cross-cut the older ones), as well as the geo-domain proportions and spatial correlation structure. The proposed approach is applied to a dataset from the Rio Blanco copper deposit in the Chilean Central Andes, in which it is of interest to simulate the layout of seven rock units (andesite, granitoid, tourmaline breccia, monolithic breccia, magmatic breccia, porphyry and pipe). The results are used to map the probabilities of occurrence of the rock units and to identify the sectors where the interpreted rock model is uncertain.

Joint Simulation of Grade and Rock Type in a Stratabound Copper Deposit

This work deals with the joint simulation of copper grade (as a continuous regionalized variable) and rock type (as a categorical variable) in Lince–Estefania deposit, located in northern Chile. The region under study is heterogeneous, containing three main rock types (intrusive, andesite and breccia bodies) with different copper grade distributions. To perform joint simulation, the multi-Gaussian and pluriGaussian models are used in a combined form. To this end, three auxiliary Gaussian random fields are considered, one for simulating copper grade, up to a monotonic transformation, and two for simulating rock types according to a given truncation rule. Furthermore, the dependence between copper grade and rock types is reproduced by considering cross correlations between these Gaussian random fields. To investigate the benefits of the joint simulation algorithm, copper grade and rock types are also simulated by the traditional cascade approach and the results are compared. It is shown that the cascade approach produces hard boundaries, that is, abrupt transitions of copper grades when crossing rock-type boundaries, a condition that does not exist in the study area according to the contact analysis held on the available data. In contrast, the joint simulation approach produces gradual transitions of the copper grade near the rock-type boundaries and is more suited to the actual data.

Conditioning Truncated Pluri-Gaussian Models to Facies Observations in Ensemble-Kalman-Based Data Assimilation

The truncated pluri-Gaussian model is a powerful tool for representing realistic spatial distributions of facies in reservoir characterization. It is suitable for generating stochastic three-dimensional facies realizations with complex vertical and lateral relationships such as are observed in algal mound behavior. Truncated pluri-Gaussian realizations account for anisotropies and relative proportions of the facies. Despite their advantages, truncated pluri-Gaussian models have not been extensively used in data assimilation techniques such as ensemble-Kalman-based algorithms. One of the major limitations encountered in the existing implementations is the difficulty of preserving facies observations at well locations through the data assimilation procedure arising even for weakly correlated data. In this work, the problem of maintaining consistency of realizations with facies is solved by merging the data assimilation algorithm (Levenberg–Marquardt ensemble randomized maximum likelihood) with an interior-point method suitable of inequality constraints. The iterative ensemble smoother is effective at assimilating highly nonlinear production data, while the interior-point method takes into account the inequality constraints on the Gaussian model variables during the data assimilation. The formulation uses an objective function that includes a model-mismatch term, a data-mismatch term and a boundary penalization function. The method allows for its approximate version neglecting model-mismatch. The algorithm was tested on a three-dimensional synthetic reservoir mimicking the algal mounds shapes of an outcrop in the Paradox Basin of Utah and includes a large number of strongly correlated facies data and significant change of petrophysical properties for different facies. It resulted in good decrease in the data-mismatch while preserving the mound structure realism and variability in the ensemble.

Simulation of the lately injected dykes in an Iranian porphyry copper deposit using the plurigaussian model

Ore deposits are usually composed of rock units or facies with different grade distributions and complex spatial structures. Being able to simulate the spatial layout of these facies are essential to have a comprehensive mining plan and an accurate resources and reserves evaluation. Modelers are faced with a set of challenges when creating the facies model such as: reproducing the facies proportions and spatial continuity as well as the topological contacts between facies, capturing post depositional overprinting, and honoring the data obtained from drill holes. Plurigaussian simulation (PGS) is a geostatistical approach that allows covering these challenges. This study addresses the application of PGS to Sungun porphyry copper deposit (Iran), in order to simulate the layout of three facies: mineralized porphyry and skarn and non-mineralized dykes. The aim of this study is to construct numerical models in which the dyke structures reflect the evolution observed in the geology.

Three-dimensional reconstruction of liquid phases in disordered mesopores usingin situsmall-angle scattering

Small-angle scattering of X-rays (SAXS) or neutrons is one of the few experimental methods currently available for thein situanalysis of phenomena in mesoporous materials at the mesoscopic scale. In the case of disordered mesoporous materials, however, the main difficulty of the method lies in the data analysis. A stochastic model is presented, which enables one to reconstruct the three-dimensional nanostructure of liquids confined in disordered mesopores starting from small-angle scattering data. This so-called plurigaussian model is a multi-phase generalization of clipped Gaussian random field models. Its potential is illustrated through the synchrotron SAXS analysis of a gel permeated with a critical nitrobenzene/hexane solution that is progressively cooled below its consolute temperature. The reconstruction brings to light a wetting transition whereby the nanostructure of the pore-filling liquids passes from wetting layers that uniformly cover the solid phase of the gel to plugs that locally occlude the pores. Using the plurigaussian model, the dewetting phenomenon is analyzed quantitatively at the nanometre scale in terms of changing specific interface areas, contact angle and specific length of the triple line.

Iterative algorithms for fitting a linear model of coregionalization

In geostatistical applications, automated or semi-automated procedures are often used for modeling the spatial correlation structure (simple and cross variograms) of multivariate data. This paper deals with the well-known linear model of coregionalization and presents three iterative algorithms to find out coregionalization matrices that minimize a weighted sum of the squared deviations between sample and modeled variograms. The first one is a variation of Goulard and Voltz’s proposal for variogram fitting with no constraint other than mathematical consistency. The second one uses simulated annealing for fitting subject to constraints on the simple variogram sills. The third one is a nonlinear least squares algorithm for fitting a plurigaussian model. In all three algorithms, the sample variogram matrices need not be entirely known for every lag vector, a situation of interest with heterotopic samplings. To demonstrate the capabilities of the proposed algorithms, a set of computer programs is provided and applied to case studies in mineral resources evaluation.

Simulation of geological domains using the plurigaussian model: New developments and computer programs

The plurigaussian model is currently used for simulating geological domains (facies) in petroleum reservoirs and mineral deposits, with the aim of assessing the uncertainty in the domain boundaries and of improving the geological controls in the characterization of quantitative attributes. This paper discusses the main aspects of the model and provides a set of computer programs to perform its inference and conditional simulation. Two types of conditioning information are allowed: hard data for which one has an exact knowledge of the actual domain at sample locations, and soft data consisting of inequality constraints on the local domain proportions (probabilities of occurrence) at control points chosen by the mining or reservoir geologist. An application to a Chilean porphyry copper deposit is finally presented, in which three Gaussian fields are used to simulate the spatial distribution of five mineralogical domains: gravels, leached capping, oxides, primary and secondary sulfides. The model is constructed so as to honor the topological contacts between mineralogical domains, their spatial continuity, the information logged at exploration drill holes, as well as the vertical proportion curves that indicate the mineralization profile with depth.

Ensemble Kalman filter for automatic history matching of geologic facies

In this paper we address two fairly difficult problems. The first is the problem of matching production data (in this case, production and injection rates) by adjustment of the locations of the geologic facies boundaries. The second is the use of a Kalman filter for updating the facies locations in the reservoir model. Traditional automatic history matching tools are not widely available for reservoirs with unknown facies boundaries, largely because of the complexity of developing software for computing the sensitivity of the data to model parameters, the lack of differentiability of facies type, and the high computational cost in generating multiple reservoir models that are conditional to given data. With careful definition of variables, the use of the ensemble Kalman filter (EnKF) minimizes those difficulties. First, the gradient does not need to be computed explicitly, the coding for the EnKF algorithm is easy and adaptable to any reservoir simulator on a plug-in basis. Second, an approximation to differentiability results from the correlation of variables. Third, the ensemble Kalman filter (EnKF) method takes one simulation run per reservoir model realization, and the simulations of the reservoir models in the ensemble are ideal for multiple-processor parallel computation. We use the truncated pluri-Gaussian model to generate random facies realizations. The geostatistical model is fully specified by the threshold truncation map and the covariance models for the two Gaussian random fields. The pluri-Gaussian model is well known but not widely used, partly because of the difficulty of generating conditional realizations. In the first example, we demonstrate the application of the EnKF to the problem of generating facies realizations conditional to observations at 18 wells on a 128 × 128 grid. In the second example, realizations of facies on a 50 × 50 grid, conditional to facies observations at the wells and to production and injection rates, are generated using the EnKF. In general, we found that application of the EnKF to the problem of adjusting facies boundaries to match production data was relatively straightforward and efficient.