Indicator Simulation Accounting for Multiple-Point Statistics

Quantitative data on geobodies are crucial for reservoir modelling. Although abundant quantitative data are available in the literature for siliciclastic depositional systems, equivalent data for carbonate systems are scarce. In this paper we introduce a new approach to the management of quantitative data on carbonate geobodies which is based on a hierarchical classification scheme. The classes to which a carbonate geobody are assigned are: (1) depo‐time (i.e. geological age); (2) depo‐system (i.e. type of carbonate platform); (3) depo‐zone (i.e. facies belt or zone); (4) depo‐shape (i.e. geometry of the geological body); (5) depo‐element (i.e. architectural elements present); and (6) depo‐facies (litho‐ and biofacies). This hierarchical classification is complemented by a set of rules for modifying depo‐shapes which refer to their spatial distribution and patterns of interaction.Based on this classification, an extensive database has been developed which can be used for 3D reservoir modelling. The database holds more than 600 case studies from outcrop analogues and the subsurface and also from satellite images of modern carbonate settings. The database can be used as the basis for a new workflow for reservoir modelling which uses multiple‐point statistics (MPS). MPS makes use of training images to capture and reproduce facies patterns and geometries during stochastic simulations.The application of this new approach is demonstrated by modelling a Cretaceous outcrop reservoir analogue from southern France. The use of MPS allows the generation of geologically realistic and complex facies distributions in the model based on the simplified training images.

Uncertainty quantification of structural and geotechnical parameter by geostatistical simulations applied to a stability analysis case study with limited exploration data

The development of effective algorithms for the simulation of facies in petroleum systems continues to be limited to techniques relying on linear spatial dependence; Truncated Gaussian Simulation, Plurigaussian Simulation and Sequential Indicator Simulation, in addition to pattern-based techniques based on Multiple Point Statistics. Recent research in the development of new geostatistical techniques for continuous variables has found that the requirement of linear spatial dependence is very rarely satisfied. Datasets from micrometre to kilometre scale have been analysed spatially and, in all cases, significant non-linearity in the spatial dependence structure have been observed. This paper replaces the Gaussian ‘engine’ in Truncated Gaussian Simulation and Plurigaussian Simulation, which requires linear spatial dependence, with a copula-based algorithm that is able to model non-linear dependence structures. The advantage is a facies model that does not contain the symmetric spatial artefacts that characterise Gaussian-based models. In flow simulation terms, this means that the connectivity (and directional connectivity) of different facies within the reservoir architecture to be modelled with greater fidelity, honouring the expected continuities of each of the facies elements. Furthermore, extending recent research by Horning and Bardossy on directional and order asymmetry for continuous variables to consider their application to facies models allows the development of facies-models that are able to represent the directional asymmetry found in sedimentary depositional systems. This allows a better representation of the distal and proximal elements in depositional systems as well as the opportunity to model complex asymmetries associated with meandering channel and estuaries.

Multiple techniques are available to construct three-dimensional reservoir models. This study uses comparative analysis to test the impact of applying four commonly used stochastic modeling techniques to capture geologic heterogeneity and fluid-flow behavior in fluvial-dominated deltaic reservoirs of complex facies architecture: (1) sequential indicator simulation; (2) object-based modeling; (3) multiple-point statistics (MPS); and (4) spectral component geologic modeling. A reference for comparison is provided by a high-resolution model of an outcrop analog that captures facies architecture at the scale of parasequences, delta lobes, and facies-association belts. A sparse, pseudosubsurface data set extracted from the reference model is used to condition models constructed using each stochastic reservoir modeling technique. Models constructed using all four algorithms fail to match the facies-association proportions of the reference model because they are conditioned to well data that sample a small, unrepresentative volume of the reservoir. Simulated sweep efficiency is determined by the degree to which the modeling algorithms reproduce two aspects of facies architecture that control sand-body connectivity: (1) the abundance, continuity, and orientation of channelized fluvial sand bodies; and (2) the lateral continuity of barriers to vertical flow associated with flooding surfaces. The MPS algorithm performs best in this regard. However, the static and dynamic performance of the models (as measured against facies-association proportions, facies architecture, and recovery factor of the reference model) is more dependent on the quality and quantity of conditioning data and on the interpreted geologic scenario(s) implicit in the models than on the choice of modeling technique.

Resource estimation of mineral deposits requires spatial modelling of orebody boundaries based on a set of exploration borehole data. Given lateritic bauxite deposits, the spacing between the boreholes is often determined based on the grade continuity. As a result, the selected drill spacing might not capture the underlying (true) lateral variability apparent in the orebody boundaries. The purpose of this study is to investigate and address the limitations imposed by such problems in lateritic metal deposits through multiple-point statistics (MPS) framework. Rather than relying on a semivariogram model, we obtain the required structural information from the footwall topographies exposed after previous mining operations. The investigation utilising the MPS was carried out using the Direct Sampling (DS) MPS algorithm. Two historical mine areas along with their mined-out surfaces and ground penetrating radar surveys were incorporated as a bivariate training image to perform the MPS simulations. In addition, geostatistical simulations using the Turning Bands method were also performed to make the comparison against the MPS results. The performances were assessed using several statistical indicators including higher-order spatial cumulants. The results have shown that the DS can satisfactorily simulate the orebody boundaries by using prior information from the previously mined-out areas.

A new uncertainty analysis method based on geostatistical simulation and a case study

Despite its flexibility when compared with parametric methods, there are significant problems with sequential indicator simulation (SIS) for continuous variables. These come from the successive construction of the local conditional distributions by indicator kriging and the use of Monte–Carlo simulation for sampling the simulated values from such distributions. Some problems of SIS have been addressed satisfactorily; order relation deviations and the extrapolation of the distribution tails are manageable. The most important outstanding challenges are: (1) there is no resolution within a grade range, and (2) there is no correlation between different grade ranges. These problems lead to an over-smoothing of grades and a reduced, but misleading, uncertainty when point scale simulations are average up at the selective mining unit size. Attempting to account for the interclass correlation by cokriging fails because there is no adequate model for fitting the complete matrix of indicator variograms. Also, attempting to improve the within class resolution by using the class edge distance function does not provide a complete solution. In regard of these unresolved challenges, the practitioner is cautioned about the use of SIS of continuous variables for Resource Evaluation. Instead, a combination of categorical variables simulation methods (including SIS) for modeling the mineralisation domains, and Gaussian simulation of grades within these domains is recommended.

Removing the assumption of conditional independence from Bayesian decision models by using artificial neural networks: Some practical techniques and a case study

O objetivo deste trabalho foi avaliar o desempenho da simulação sequencial gaussiana (SSG) e da simulação sequencial indicatriz (SSI) na modelagem da incerteza das predições do K disponível em área de cana-de-açúcar, e comparar as simulações com o método já consagrado de krigagem ordinária (KO). Uma malha amostral com 626 pontos foi instalada em área de 200 ha, no Município de Tabapuã, em São Paulo. As simulações reproduziram a variabilidade dos dados amostrais de K disponível, enquanto a KO superestimou os baixos teores de K e subestimou os altos. O mapa de desvio-padrão obtido a partir da KO mostrou menor variação ao longo da área de estudo, quando comparado aos mapas obtidos a partir das simulações. A SSI obteve acurácia 22% superior à obtida pela SSG, na modelagem da função de distribuição condicional do K. As simulações apresentam maior eficiência que a KO para modelar incerteza na distribuição espacial do K. A SSI apresenta melhor desempenho que a SSG na estimativa dos teores de K disponível, em área de cana-de-açúcar.

Updating cover type maps using sequential indicator simulation

Flow simulation studies require an accurate model of the reservoir in terms of its sedimentological architecture. Pixel-based reservoir modeling techniques are often used to model this architecture. There are, however, two problem areas with such techniques. First, several statistical parameters have to be provided whose influence on the resulting model is not readily inferable. Second, conditioning the models to relevant geological data that carry great uncertainty on their own adds to the difficulty of obtaining reliable models and assessing model reliability. The Sequential Indicator Simulation (SIS) method has been used to examine the impact of such uncertainties on the final reservoir model. The effects of varying variogram types, frequencies of lithology occurrence, and the gridblock model orientation with respect to the sedimentological trends are illustrated using different reservoir modeling studies. Results indicate, for example, that the choice of variogram type can have a significant impact on the facies model. Also, reproduction of sedimentological trends and large geometries requires careful parameter selection. By choosing the appropriate modeling strategy, sedimentological principles can be translated into the numerical model. Solutions for dealing with such issues and the geological uncertainties are presented. In conclusion, each reservoir modeling study should begin by developing a thorough quantitative sedimentological understanding of the reservoir under study, followed by detailed sensitivity analyses of relevant statistical and geological parameters.

ABSTRACTGeostatistical modelling of grades in mineral deposits often requires the simulation of multiple related variables that have sum and fraction constraints. Sum constraints occur when the sum of some variables may not exceed or must equal a given constant. Fraction constraints occur when a variable may not exceed another variable. In this case, the geostatistical simulations should reproduce the histograms, variograms, multivariate relationships and honour the constraints. We present a methodology for the geostatistical simulation of multiple related variables that considers sum and fraction constraints. The methodology is illustrated in a case study. The data were obtained from a bauxite deposit. First, the original variables were re-expressed as ratios. Second, the variables were transformed using the Projection Pursuit Multivariate Transform (PPMT). Then the PPMT transformed variables were simulated independently using sequential Gaussian simulation and back-transformed. The simulations reproduced the histograms, variograms and bivariate relationships and honoured the sum and fraction constraints.

Regional scale inventories of heavy metal concentrations in soil increasingly are being done to evaluate their global patterns of variation. Sometimes these global pattern evaluations reveal information that is not identified by more detailed studies. Geostatistical methods, such as stochastic simulation, have not yet been used routinely for this purpose in spite of their potential. To investigate such a use of geostatistical methods, we analyzed a data set of 14,674 copper and 12,441 cadmium observations in the topsoil of Flanders, Belgium, covering 13,522 km2. Outliers were identified and removed, and the distributions were spatially declustered. Copper was analyzed using sequential Gaussian simulation, whereas for cadmium we used sequential indicator simulation because of the large proportion (43%) of censored data. We complemented maps of the estimated values with maps of the probability of exceeding a critical sanitation threshold for agricultural land use. These sets of maps allowed the identification of regional patterns of increased metal concentrations and provided insight into their potential causes. Mostly areas with known industrial activities (such as lead and zinc smelters) could be delineated, but the effects of shells fired during the First World War were also identified.

A fundamental task for petroleum exploration decision-making is to evaluate the uncertainty of well outcomes. The recent development of geostatistical simulation techniques provides an effective means to the generation of a full uncertainty model for any random variable. Sequential indicator simulation has been used as a tool to generate alternate, equal-probable stochastic models, from which various representations of uncertainties can be created. These results can be used as input for the quantification of various risks associated with a wildcat drilling program or the estimation of petroleum resources. A simple case study is given to demonstrate the use of sequential indicator simulation. The data involves a set of wildcat wells in a gas play. The multiple simulated stochastic models are then post-processed to characterize various uncertainties associated with drilling outcomes.

Multiple-point geostatistical simulation has recently become popular in stochastic hydrogeology, primarily because of its capability to derive geologically reasonable patterns and multivariate distributions from a training image and conditioning training image data to multiple hard and soft data sources. This article resents, evaluates and contrasts the results of using multiplepoint geostatistical simulation for producing geologically realistic models of a Quaternary inner-alpine aquifer. Borehole data, expert-designed geological profiles and training image data were subject to conditional simulation with the Single Normal Equation Simulation (SNESIM) algorithm, one of the most widely used Matrix Product State (MPS) algorithms. The sensitivity of model predictions to the training image and hard as well as soft data input was evaluated. Modeling results indicate that soft conditioning in MPS is a convenient and efficient way for integrating secondary data such as geological expert drawings. Keywords: Multiple point statistics; Secondary data integration; Aquifer heterogeneity; Quaternary over-deepened alpine basin