Alessandro Comunian

3D multiple-point statistics simulation using 2D training images

One of the main issues in the application of multiple-point statistics (MPS) to the simulation of three-dimensional (3D) blocks is the lack of a suitable 3D training image. In this work, we compare three methods of overcoming this issue using information coming from bidimensional (2D) training images. One approach is based on the aggregation of probabilities. The other approaches are novel. One relies on merging the lists obtained using the impala algorithm from diverse 2D training images, creating a list of compatible data events that is then used for the MPS simulation. The other (s2Dcd) is based on sequential simulations of 2D slices constrained by the conditioning data computed at the previous simulation steps. These three methods are tested on the reproduction of two 3D images that are used as references, and on a real case study where two training images of sedimentary structures are considered. The tests show that it is possible to obtain 3D MPS simulations with at least two 2D training images. The simulations obtained, in particular those obtained with the s2Dcd method, are close to the references, according to a number of comparison criteria. The CPU time required to simulate with the method s2Dcd is from two to four orders of magnitude smaller than the one required by a MPS simulation performed using a 3D training image, while the results obtained are comparable. This computational efficiency and the possibility of using MPS for 3D simulation without the need for a 3D training image facilitates the inclusion of MPS in Monte Carlo, uncertainty evaluation, and stochastic inverse problems frameworks.

Probability Aggregation Methods in Geoscience

The need for combining different sources of information in a probabilistic framework is a frequent task in earth sciences. This is a need that can be seen when modeling a reservoir using direct geological observations, geophysics, remote sensing, training images, and more. The probability of occurrence of a certain lithofacies at a certain location for example can easily be computed conditionally on the values observed at each source of information. The problem of aggregating these different conditional probability distributions into a single conditional distribution arises as an approximation to the inaccessible genuine conditional probability given all information. This paper makes a formal review of most aggregation methods proposed so far in the literature with a particular focus on their mathematical properties. Exact relationships relating the different methods is emphasized. The case of events with more than two possible outcomes, never explicitly studied in the literature, is treated in detail. It is shown that in this case, equivalence between different aggregation formulas is lost. The concepts of calibration, sharpness, and reliability, well known in the weather forecasting community for assessing the goodness-of-fit of the aggregation formulas, and a maximum likelihood estimation of the aggregation parameters are introduced. We then prove that parameters of calibrated log-linear pooling formulas are a solution of the maximum likelihood estimation equations. These results are illustrated on simulations from two common stochastic models for earth science: the truncated Gaussian model and the Boolean. It is found that the log-linear pooling provides the best prediction while the linear pooling provides the worst.

Modeling fine-scale geological heterogeneity-examples of sand lenses in tills

Sand lenses at various spatial scales are recognized to add heterogeneity to glacial sediments. They have high hydraulic conductivities relative to the surrounding till matrix and may affect the advective transport of water and contaminants in clayey till settings. Sand lenses were investigated on till outcrops producing binary images of geological cross-sections capturing the size, shape and distribution of individual features. Sand lenses occur as elongated, anisotropic geobodies that vary in size and extent. Besides, sand lenses show strong non-stationary patterns on section images that hamper subsequent simulation. Transition probability (TP) and multiple-point statistics (MPS) were employed to simulate sand lens heterogeneity. We used one cross-section to parameterize the spatial correlation and a second, parallel section as a reference: it allowed testing the quality of the simulations as a function of the amount of conditioning data under realistic conditions. The performance of the simulations was evaluated on the faithful reproduction of the specific geological structure caused by sand lenses. Multiple-point statistics offer a better reproduction of sand lens geometry. However, two-dimensional training images acquired by outcrop mapping are of limited use to generate three-dimensional realizations with MPS. One can use a technique that consists in splitting the 3D domain into a set of slices in various directions that are sequentially simulated and reassembled into a 3D block. The identification of flow paths through a network of elongated sand lenses and the impact on the equivalent permeability in tills are essential to perform solute transport modeling in the low-permeability sediments.

Training Images from Process-Imitating Methods

The lack of a suitable training image is one of the main limitations of the application of multiple-point statistics (MPS) for the characterization of heterogeneity in real case studies. Process-imitating facies modeling techniques can potentially provide training images. However, the parameterization of these process-imitating techniques is not straightforward. Moreover, reproducing the resulting heterogeneous patterns with standard MPS can be challenging. Here the statistical properties of the paleoclimatic data set are used to select the best parameter sets for the process-imitating methods. The data set is composed of 278 lithological logs drilled in the lower Namoi catchment, New South Wales, Australia. A good understanding of the hydrogeological connectivity of this aquifer is needed to tackle groundwater management issues. The spatial variability of the facies within the lithological logs and calculated models is measured using fractal dimension, transition probability, and vertical facies proportion. To accommodate the vertical proportions trend of the data set, four different training images are simulated. The grain size is simulated alongside the lithological codes and used as an auxiliary variable in the direct sampling implementation of MPS. In this way, one can obtain conditional MPS simulations that preserve the quality and the realism of the training images simulated with the process-imitating method. The main outcome of this study is the possibility of obtaining MPS simulations that respect the statistical properties observed in the real data set and honor the observed conditioning data, while preserving the complex heterogeneity generated by the process-imitating method. In addition, it is demonstrated that an equilibrium of good fit among all the statistical properties of the data set should be considered when selecting a suitable set of parameters for the process-imitating simulations.

High resolution multi-facies realizations of sedimentary reservoir and aquifer analogs.

Geological structures are by nature inaccessible to direct observation. This can cause difficulties in applications where a spatially explicit representation of such structures is required, in particular when modelling fluid migration in geological formations. An increasing trend in recent years has been to use analogs to palliate this lack of knowledge, i.e., exploiting the spatial information from sites where the geology is accessible (outcrops, quarry sites) and transferring the observed properties to a study site deemed geologically similar. While this approach is appealing, it is difficult to put in place because of the lack of access to well-documented analog data. In this paper we present comprehensive analog data sets which characterize sedimentary structures from important groundwater hosting formations in Germany and Brazil. Multiple 2-D outcrop faces are described in terms of hydraulic, thermal and chemical properties and interpolated in 3-D using stochastic techniques. These unique data sets can be used by the wider community to implement analog approaches for characterizing reservoir and aquifer formations.

A reassessment of the Lower Namoi Catchment aquifer architecture and hydraulic connectivity with reference to climate drivers

We demonstrate the need for better representations of aquifer architecture to understand hydraulic connectivity and manage groundwater allocations for the ∼140 m-thick alluvial sequences in the Lower Namoi Catchment, Australia. In the 1980s, an analysis of palynological and groundwater hydrograph data resulted in a simple three-layer stratigraphic/hydrostratigraphic representation for the aquifer system, consisting of an unconfined aquifer overlying two semi-confined aquifers. We present an analysis of 278 borehole lithological logs within the catchment and show that the stratigraphy is far more complex. The architectural features and the net-to-gross line-plot of the valley-filling sequence are best represented by a distributive fluvial system, where the avulsion frequency increases at a slower rate than the aggradation rate. We also show that an improved understanding of past climates contextualises the architectural features observable in the valley-filling sequence, and that the lithofacies distribution captures information about the impact of climate change during the Neogene and Quaternary. We demonstrate the correlation between climate and the vertical lithological succession by correlating the sediment net-to-gross ratio line-plot with the marine benthic oxygen isotope line-plot – a climate change proxy. Pollens indicate that there was a transition from a relatively wet climate in the mid–late Miocene to a drier climate in the Pleistocene, with a continuing drying trend until present. Groundwater is currently extracted from the sand and gravel belts associated with the high-energy wetter climate. However, some of these channel belts are disconnected from the modern river and flood zone. We show that the cutoff between the hydraulically well- and poorly connected portions of the valley-filling sequence matches the connectivity threshold expected from a fluvial system.

Analog‐based meandering channel simulation

Characterizing the complex geometries and the heterogeneity of the deposits in meandering river systems is a long-standing issue for the 3-D modeling of alluvial formations. Such deposits are important sources of accessible groundwater in alluvial aquifers throughout the world and also play a major role as hydrocarbons reservoirs. In this paper, we present a method to generate meandering river centerlines that are stochastic, geologically realistic, connected, and conditioned to local observations or global geomorphological characteristics. The method is based on fast 1-D multiple-point statistics in a transformed curvilinear domain: the succession in directions observed in a real-world meandering river (the analog) is considered as statistical model for multiple-point statistics simulation. The integration of local data is accomplished by an inverse procedure ensuring that the channels pass through a given set of locations while conserving the high-order spatial characteristics of an analog. The methodology is applied on seven real-world case studies. This work demonstrates the flexibility and the applicability of multiple-point statistics outside the standard paradigm that considers the simulation of a 2-D or 3-D variable with spatial coordinates.

Parameterization of training images for aquifer 3‐D facies modeling integrating geological interpretations and statistical inference

We develop a stochastic approach to construct channelized 3-D geological models constrained to borehole measurements as well as geological interpretation. The methodology is based on simple 2-D geologist-provided sketches of fluvial depositional elements, which are extruded in the 3rd dimension. Multiple-point geostatistics (MPS) is used to impair horizontal variability to the structures by introducing geometrical transformation parameters. The sketches provided by the geologist are used as elementary training images, whose statistical information is expanded through randomized transformations. We demonstrate the applicability of the approach by applying it to modeling a fluvial valley filling sequence in the Maules Creek catchment, Australia. The facies models are constrained to borehole logs, spatial information borrowed from an analogue and local orientations derived from the present-day stream networks. The connectivity in the 3-D facies models is evaluated using statistical measures and transport simulations. Comparison with a statistically equivalent variogram-based model shows that our approach is more suited for building 3-D facies models that contain structures specific to the channelized environment and which have a significant influence on the transport processes.

Methods for Exploring Uncertainty in Groundwater Management Predictions

Models of groundwater systems help to integrate knowledge about the natural and human system covering different spatial and temporal scales, often from multiple disciplines, in order to address a range of issues of concern to various stakeholders. A model is simply a tool to express what we think we know. Uncertainty, due to lack of knowledge or natural variability, means that there are always alternative models that may need to be considered. This chapter provides an overview of uncertainty in models and in the definition of a problem to model, highlights approaches to communicating and using predictions of uncertain outcomes and summarises commonly used methods to explore uncertainty in groundwater management predictions. It is intended to raise awareness of how alternative models and hence uncertainty can be explored in order to facilitate the integration of these techniques with groundwater management.

Hierarchical simulation of aquifer heterogeneity: implications of different simulation settings on solute-transport modeling

The fine-scale heterogeneity of porous media affects the large-scale transport of solutes and contaminants in groundwater and it can be reproduced by means of several geostatistical simulation tools. However, including the available geological information in these tools is often cumbersome. A hierarchical simulation procedure based on a binary tree is proposed and tested on two real-world blocks of alluvial sediments, of a few cubic meters volume, that represent small-scale aquifer analogs. The procedure is implemented using the sequential indicator simulation, but it is so general that it can be adapted to various geostatistical simulation tools, improving their capability to incorporate geological information, i.e., the sedimentological and architectural characterization of heterogeneity. When compared with a standard sequential indicator approach on bi-dimensional simulations, in terms of proportions and connectivity indicators, the proposed procedure yields reliable results, closer to the reference observations. Different ensembles of three-dimensional simulations based on different hierarchical sequences are used to perform numerical experiments of conservative solute transport and to obtain ensembles of equivalent pore velocity and dispersion coefficient at the scale length of the blocks (meter). Their statistics are used to estimate the impact of the variability of the transport properties of the simulated blocks on contaminant transport modeled on bigger domains (hectometer). This is investigated with a one-dimensional transport modeling based on the Kolmogorov-Dmitriev theory of branching stochastic processes. Applying the proposed approach with diverse binary trees and different simulation settings provides a great flexibility, which is revealed by the differences in the breakthrough curves.RésuméL’hétérogénéité à échelle fine des milieux poreux affecte le transport de solutés et de contaminants à grande échelle dans les eaux souterraines et elle peut être reproduite au moyen de nombreux outils de simulation géostatistique. Cependant, il est souvent difficile d’intégrer l’information géologique disponible avec ces outils. Une procédure de simulation hiérarchique basée sur un arbre binaire est proposée et testée sur deux blocs réels de sédiments alluviaux, d’un volume de quelques mètres cubes, qui représentent des analogues d’aquifère à petite échelle. La procédure est implémentée en utilisant la simulation d’un indicateur séquentiel. Néanmoins, elle est suffisamment générale pour pouvoir être adaptée à de nombreux outils de simulation géostatistique, améliorant ainsi leur facilité à incorporer de l’information géologique, i.e., la caractérisation de l’hétérogénéité en termes sédimentologique et d’architecture. Lorsqu’elle est comparée à une approche d’indicateur séquentiel standard pour des simulations en 2D, la procédure proposée fournit des résultats fiables, plus près des observations de référence, en termes d’indicateurs de proportions et de connectivité. Différents ensembles de simulations 3D, basées sur différentes séquences hiérarchiques, sont utilisées pour réaliser des expériences numériques de transport conservatif de solutés et pour obtenir des ensembles de vitesse de pore équivalente et de coefficient de dispersion à l’échelle de longueur des blocs (mètre). Leurs statistiques sont utilisées pour estimer l’impact de la variabilité des propriétés de transport des blocs simulés sur le transport de contaminants modélisé sur de plus grands domaines (hectomètre). Ceci est investigué avec un modèle de transport 1D basé sur la théorie de Kolmogorov-Dmitriev de processus stochastiques en branches. L’application de l’approche proposée avec différents arbres binaires et différentes configurations de simulation fournit une grande flexibilité, qui est mise en lumière par les différences des courbes de restitution.ResumenLa heterogeneidad a escala fina de loa medios porosos afecta el transporte a gran escala de los solutos y contaminantes en el agua subterránea y puede ser reproducida por medio de varias herramientas de simulación geoestadística. Sin embargo, la inclusión de la información geológica disponible en estas herramientas es a menudo engorrosa. Se propone y se prueba un procedimiento de simulación jerárquica basada en un árbol binario y en dos bloques de sedimentos aluviales de existencia real de un volumen de unos pocos metros cúbicos, que representan una analogía del acuífero en pequeña escala. El procedimiento se implementa utilizando un indicador de la simulación secuencial, pero es tan general que puede ser adaptado a diversas herramientas de simulación geoestadística, mejorando su capacidad para incorporar la información geológica, es decir, la caracterización sedimentológica y arquitectónica de la heterogeneidad. Cuando se los compara con el enfoque de un indicador secuencial estándar en simulaciones bidimensionales, en términos de proporciones e indicadores de conectividad, el procedimiento propuesto produce resultados confiables, más cercanos de las observaciones de referencia. Se usan diferentes conjuntos de simulaciones tridimensionales basadas en distintas secuencias jerárquicas para realizar experimentos numéricos de transporte conservativos de solutos y para obtener conjuntos de velocidad de equivalente poral y coeficientes de dispersión a escala de los bloques (metros). Sus estadísticas se utilizaron para estimar el impacto de la variabilidad de las propiedades de transporte de los bloques simulados sobre el transporte de contaminantes modelados en dominios más grandes (hectómetro). Esto se investigó con una modelización del transporte unidimensional basada en la teoría de Kolmogorov-Dmitriev de los procesos estocásticos ramificados. La aplicación del enfoque propuesto con distintos árboles binarios y diferentes configuraciones de simulación proporciona una gran flexibilidad, que está ilustrada por las diferencias en las curvas de rotura.摘要多孔介质小尺度异质性影响着地下水中溶质和污染物大规模迁移,可通过几个地质统计模拟工具再现此情景。然而,包含现有地质信息的这些工具常常是笨重的。本文提出了基于二元树的分等级模拟程序,并在两个现实生活中几立方的冲积沉积地块中进行了测试,这几立方的冲积沉积地块代表着小尺度的含水层类似物。采用连续指示物模拟完成了模拟过程,但过程非常普通,以至于可以用到各种地质统计模拟工具中,提高综合地质信息的能力,即沉积学和建筑学上的异质性描述。在与二维模拟标准的连续指示物方法相比,在比例和连通性指示物方面,提出的程序可得到可靠的结果,更接近参考观测值。基于不同分等级序列的三维模拟不同的总结果用于进行保守的溶质迁移数值实验,并获取地块标尺长度(米)相等孔隙速度和弥散系数的总结果。其统计结果用于估算模拟地块迁移特性的变化性对较大域(百米)污染物迁移的影响。利用基于分支随机程序Kolmogorov-Dmitriev理论的一维迁移模型对此进行了研究。采用各种各样的二元树和不同的模拟背景应用所提出的方法提供了很大的灵活性,而此灵活性通过突破曲线的差别得到显现。RiassuntoL’eterogeneità dei mezzi porosi a scala fine influenza il trasporto a larga scala di soluti e contaminanti negli acquiferi, e può essere riprodotta tramite diversi metodi di simulazione geostatistica. Tuttavia, non è sempre semplice includere informazioni geologiche in questi metodi di simulazione. Si propone quindi una procedura di simulazione gerarchica basata sul concetto di albero binario, e la si verifica su due blocchi di sedimenti alluvionali aventi un volume di pochi metri cubi e che rappresentano analoghi di acquiferi a piccola scala. La procedura è implementata tramite la simulazione sequenziale con indicatori, ma il principio è generale e può essere adattato ad altri metodi di simulazione geostatistica, migliorandone la capacità di includere informazioni geologiche legate per esempio alla caratterizzazione sedimentologica e architetturale dell’eterogeneità. Confrontata con una procedura di simulazione sequenziale con indicatori standard in termini di proporzioni e di indicatori di connettività, la procedura proposta restituisce risultati affidabili e più vicini alle osservazioni di riferimento. Diversi ensemble di simulazioni tridimensionali, basate su diverse sequenze gerarchiche, sono utilizzati in simulazioni numeriche di trasporto conservativo e per ottenere ensemble di velocità di poro equivalente e di coefficiente di dispersione alla scala dei blocchi (metri). Le loro statistiche sono usate per stimare l’impatto della variabilità delle proprietà di trasporto dei blocchi simulati sul trasporto di inquinanti modellato su domini più grandi (ettometri). Queste stime sono effettuate tramite simulazioni monodimensionali di trasporto basate sulla teoria dei processi stocastici ramificati di Kolmogorov-Dmitriev. Applicare l’approccio proposto utilizzando alberi binari differenti e diverse configurazioni di simulazione risulta in una notevole flessibilità, messa in evidenza dalle differenze osservate nelle curve di restituzione.ResumoA heterogeneidade em pequena escala do meio poroso afeta o transporte de solutos e contaminantes nas águas subterrâneas em grande escala e estes podem ser reproduzidos por diversas ferramentas de simulação geoestatísticas. Entretanto, incluir as informações geológicas disponíveis nestas ferramentas é frequentemente trabalhoso. Um procedimento de simulação hierárquica baseado em uma árvore binaria é proposta e testada em dois blocos de sedimentos aluviais reais com volume de poucos metros cúbicos, que representam aquíferos em pequena escala análogos. O procedimento é implementado usando simulação sequencial indicadora, porém isto é tão genérico que pode ser adaptado para diversas ferramentas geoestatísticas, melhorando a capacidade de incorporar as informações geológicas, por exemplo, a caracterização sedimentológica e arquitetural da heterogeneidade. Quando comparada com uma abordagem tradicional de simulação sequencial indicadora em bidimensionais, em termos de indicadores de proporções e conectividade, o procedimento proposto apresenta resultados confiáveis, próximos as observações de referência. Diferentes conjuntos de simulações tridimensionais baseadas em diferentes sequências hierárquicas são usados para gerar experimentos numéricos de transporte de solutos conservativos e para obter conjuntos de velocidade equivalente em poros e coeficientes de dispersão na escala do comprimento dos blocos (metros). Suas estatísticas são usadas para estimar o impacto da variabilidade das propriedades de transporte dos blocos simulados no trans