Multiple-point Statistics Research Articles

Smart determination of borehole number and locations for stability analysis of multi-layered slopes using multiple point statistics and information entropy

Subsurface stratigraphy of multi-layered slopes is essential and crucial for slope stability analysis. It is usual practice for engineers to interpret stratigraphic boundaries separating different soil layers using both site investigation data and prior knowledge of local geology, but such practice might encounter significant challenge when the site data are very limited. In addition, uncertainty in stratigraphic boundaries has not been explicitly or quantitatively considered in planning of site investigation (e.g., determination of borehole number and locations). There lacks a quantitative and objective tool to determine the optimal locations and number of boreholes for slope stability analysis while accounting for stratigraphic uncertainty. In this study, a smart sampling strategy based on multiple point statistics and information entropy is proposed for delineation of slope subsurface stratigraphy and planning of geotechnical boreholes. It is a data-driven approach that enables an ensemble of prior knowledge within a training image using multiple point statistics. The proposed method not only provides evolution of the most probable interpolation from sparse measurements and the associated interpolation uncertainties, but also adaptively determines the optimal locations of boreholes. Effectiveness of the proposed method is illustrated and validated through both a simulation example and a real case. It is found that the data-driven framework can automatically identify locations of largest interpolation uncertainty within a multi-layered slope conditional on its outcrops, and that the associated stratigraphic uncertainty gradually reduces as borehole number increases. More importantly, the optimal number and locations of boreholes required for slope stability analysis are adaptively determined by the proposed method.

Multiple-Point Statistics Simulation Models: Pretty Pictures or Decision-Making Tools?

Abundant literature has been produced for the last two decades about multiple-point statistics simulation, or MPS. The idea behind MPS is very simple: reproduce patterns from a 2D, or most often a 3D, training image that displays the type of geological heterogeneity deemed to be relevant to the reservoir or field under study, while honoring local data. Replicating an image is a traditional computer science problem. Thus, it should come as no surprise if a growing number of publications on MPS borrow ideas and techniques directly from computer vision and machine learning to improve the reproduction of training patterns. However, quoting Andre Journel, “Geostatistics is not about generating pretty pictures.” Models have a purpose. For example, in oil and gas applications, reservoir models are used to estimate hydrocarbon volumes and book reserves, run flow simulations to forecast hydrocarbon production and ultimate recovery, and make decisions about field development or optimal well drilling locations. Specific key features such as the extent and connectivity of shale barriers may have a major impact on the reservoir performance forecasts and the field development decisions to be made. Those key features that need to be captured in the model, along with the available subsurface data and constraints of the project, should be the primary drivers in selecting the most appropriate modeling techniques and options to obtain reliable results and make sound decisions. In this paper, the practitioners’ point of view is used to evaluate alternative MPS implementations and highlight remaining gaps.

3D Geological Image Synthesis From 2D Examples Using Generative Adversarial Networks

Generative Adversarial Networks (GAN) are becoming an alternative to Multiple-point Statistics (MPS) techniques to generate stochastic fields from training images. But a difficulty for all the training image based techniques (including GAN and MPS) is to generate 3D fields when only 2D training data sets are available. In this paper, we introduce a novel approach called Dimension Augmenter GAN (DiAGAN) enabling GANs to generate 3D fields from 2D examples. The method is simple to implement and is based on the introduction of a random cut sampling step between the generator and the discriminator of a standard GAN. Numerical experiments show that the proposed approach provides an efficient solution to this long lasting problem.

3D multiple-point statistics simulations of the Roussillon Continental Pliocene aquifer using DeeSse

Abstract. This study introduces a novel workflow to model the heterogeneity of complex aquifers using the multiple-point statistics algorithm DeeSse. We illustrate the approach by modeling the Continental Pliocene layer of the Roussillon aquifer in the region of Perpignan (southern France). When few direct observations are available, statistical inference from field data is difficult if not impossible and traditional geostatistical approaches cannot be applied directly. By contrast, multiple-point statistics simulations can rely on one or several alternative conceptual geological models provided using training images (TIs). But since the spatial arrangement of geological structures is often non-stationary and complex, there is a need for methods that allow to describe and account for the non-stationarity in a simple but efficient manner. The main aim of this paper is therefore to propose a workflow, based on the direct sampling algorithm DeeSse, for these situations. The conceptual model is provided by the geologist as a 2D non-stationary training image in map view displaying the possible organization of the geological structures and their spatial evolution. To control the non-stationarity, a 3D trend map is obtained by solving numerically the diffusivity equation as a proxy to describe the spatial evolution of the sedimentary patterns, from the sources of the sediments to the outlet of the system. A 3D continuous rotation map is estimated from inferred paleo-orientations of the fluvial system. Both trend and orientation maps are derived from geological insights gathered from outcrops and general knowledge of processes occurring in these types of sedimentary environments. Finally, the 3D model is obtained by stacking 2D simulations following the paleo-topography of the aquifer. The vertical facies transition between successive 2D simulations is controlled partly by the borehole data used for conditioning and by a sampling strategy. This strategy accounts for vertical probability of transitions, which are derived from the borehole observations, and works by simulating a set of conditional data points from one layer to the next. This process allows us to bypass the creation of a 3D training image, which may be cumbersome, while honoring the observed vertical continuity.

Quantitative Analysis for the Reconstruction of Porous Media Using Multiple-Point Statistics

The pore structure reconstruction of the porous media is of great importance to the research of mechanisms of fluid flow in porous media. To capture the large-scale patterns in the pore space, the multiple-point statistical technique is generally adopted for porous media reconstruction. Commonly, two different schemes, i.e., the single-grid scheme and the multiple-grid scheme, can be applied for simulation realization. The selection between this two schemes and a proper data template size have thus become a new research issue, and the performance of the characteristic reproduction of the training image using this two schemes must be quantified. In this paper, a series of multiple-point statistics simulation basing on a 2D micro-CT sandstone image are proceeded using both single- and multiple-grid schemes, and different data templates are adapted for porous media reconstruction. Further, to quantify the impact of the computational schemes and setting of the data template to the simulation realizations, a number of measurements considering the pore diameter, porosity, connectivity, and permeability are implemented to fully analyze the results obtained. Results show that by using the single-point statistical method, a large template is necessary to reproduce large-scale structures. The multiple-grid template method may bring great benefits to simulation efficiency over the simple data template method, as well as the recovery of the pore long-range geometric features and seepage characteristics. With the extension of the template for the multiple-grid scheme, the simulation results show lack of variations to some extent.

Missing Data Imputation for Multisite Rainfall Networks: A Comparison between Geostatistical Interpolation and Pattern-Based Estimation on Different Terrain Types

AbstractMissing rainfall data are a major limitation for distributed hydrological modeling and climate studies. Practitioners need reliable approaches that can be employed on a daily basis, often with too limited data in space to feed complex predictive models. In this study we compare different automatic approaches for missing data imputation, including geostatistical interpolation and pattern-based estimation algorithms. We introduce two pattern-based approaches based on the analysis of historical data patterns: (i) an iterative version of K-nearest neighbor (IKNN) and (ii) a new algorithm called vector sampling (VS) that combines concepts of multiple-point statistics and resampling. Both algorithms can draw estimations from variably incomplete data patterns, allowing the target dataset to be at the same time the training dataset. Tested on five case studies from Denmark, Australia, and Switzerland, the algorithms show a different performance that seems to be related to the terrain type: on flat terrains with spatially homogeneous rain events, geostatistical interpolation tends to minimize the average error, while in mountainous regions with nonstationary rainfall statistics, data mining can recover better the rainfall patterns. The VS algorithm, requiring minimal parameterization, turns out to be a convenient option for routine application on complex and poorly gauged terrains.

MPS realization selection with an innovative LSTM tool

Multiple point statistics (MPS) methods are good tools to recreate the spatial heterogeneity of permeability in porous media. On the other hand, a reliable forecast of reservoir performance necessitates a large number of realistic permeability realizations which could be achieved by MPS techniques. However, the cost of numerical simulation on a large number of realizations, especially for screening, optimization or history matching purposes, is completely prohibitive. Emerging techniques in deep learning such as Long Short-Term Memory (LSTM) networks can provide robust proxy models replacing the numerical simulation with reasonable run-time and acceptable predictability. We train LSTM networks using some of realizations as input with a novel implementation. The LSTM network accepts the permeability values in a state of art manner, i.e. accepting the permeability values row by row. The network is trained with reservoir simulation outputs of corresponding MPS-generated realizations. In fact, the output values for training (the labels) are the reservoir simulation outputs for each realization which are obtained by an MPS method. In this way, MPS can be coupled with deep learning to find the realizations having the best match with a reference case.The architecture of the trained LSTM network is illustrated with details. The results show that the trained network is able to find realizations resulting in an adequate match with the reference case. These realizations not only exhibit fairly good prediction in future times of simulation due to MPS-derived permeability heterogeneity within but have an acceptable match with the historical reservoir performance. Our hybrid MPS-LSTM approach allows reducing the uncertainty in reservoir performance prediction. The reason is both meticulous heterogeneity modeling and deep-learning-assisted approach of realization screening. The proposed workflow could be a promising basis for history matching and uncertainty quantification calculation of reservoir.

Nonstationary stochastic rain type generation: accounting for climate drivers

Abstract. At subdaily resolution, rain intensity exhibits a strong variability in space and time, which is favorably modeled using stochastic approaches. This strong variability is further enhanced because of the diversity of processes that produce rain (e.g., frontal storms, mesoscale convective systems and local convection), which results in a multiplicity of space–time patterns embedded into rain fields and in turn leads to the nonstationarity of rain statistics. To account for this nonstationarity in the context of stochastic weather generators and therefore preserve the relationships between rainfall properties and climatic drivers, we propose to resort to rain type simulation. In this paper, we develop a new approach based on multiple-point statistics to simulate rain type time series conditional to meteorological covariates. The rain type simulation method is tested by a cross-validation procedure using a 17-year-long rain type time series defined over central Germany. Evaluation results indicate that the proposed approach successfully captures the relationships between rain types and meteorological covariates. This leads to a proper simulation of rain type occurrence, persistence and transitions. After validation, the proposed approach is applied to generate rain type time series conditional to meteorological covariates simulated by a regional climate model under an RCP8.5 (Representative Concentration Pathway) emission scenario. Results indicate that, by the end of the century, the distribution of rain types could be modified over the area of interest, with an increased frequency of convective- and frontal-like rains at the expense of more stratiform events.

Recursive convolutional neural networks in a multiple-point statistics framework

This work proposes a new technique for multiple-point statistics simulation based on a recursive convolutional neural network approach coined RCNN. The work focuses on methodology and implementation rather than performance to demonstrate the potential of deep learning techniques in geosciences. Two and three dimensional case studies are carried out. A sensitivity analysis is presented over the main RCNN structural parameters using a well-known training image of channel structures in two dimensions. The optimum parameters found are applied into image reconstruction problems using two other training images. A three dimensional case is shown using a synthetic lithological surface-based model. The quality of realizations is measured by statistical, spatial and accuracy metrics. The RCNN method is compared to standard MPS techniques and an improving framework is proposed by using the RCNNE-type as secondary information. Strengths and weaknesses of the methodology are discussed by reviewing the theoretical and practical aspects.

Hybrid geological modeling: Combining machine learning and multiple-point statistics

Accurately modeling and constructing a geologically realistic subsurface model remains an outstanding problem as the morphology controls the flow behaviors. Particularly, one of the pattern-based methods, namely cross-correlation based simulation, has been proved to be an effective way to reconstruct a realistic model, at both small and large scales. However, conditioning to point data in the large-scale problems is still a crucial issue in these algorithms, since there is always a trade-off between the quality of the realizations and the degree of point data reproduction. Specifically, it is not practical to build a training image (TI) which includes all the possibilities and variabilities. Therefore, finding a pattern that can represent the point data and, at the same time, preserving the connectivities is difficult. This leads to producing highly-connected realizations with a significant mismatch or poor models with a reasonable degree of point data reproduction. To accurately reproduce the densely distributed hard data, pixel-based methods can also produce some unrealistic artifacts around the hard data. In this paper, to overcome this challenge, however, we use pattern-based methods as they often produce more disconnected geobodies when dealing with dense hard data, and proposed a hybrid algorithm using the pattern-based methods and convolutional neural network (CNN). The trained CNN model is utilized to improve the quality of conditioning to point data for the original realizations generated by the pattern-based algorithm. As such, the mismatch locations are identified, and the same regions are used in the training of CNN to mimic the procedure through which a missing region can be filled. To evaluate the performance of the proposed hybrid algorithm, it is tested on cases with different dimensions and different numbers of facies. Then, the newly improved realizations are compared with the initial realizations generated by the pattern-based algorithm. The comparison is also conducted by the flow simulation test. And it indicates that the proposed hybrid algorithm can better reproduce the point data, while the connectivities are better preserved.

Comparison of various 3D pore space reconstruction methods and implications on transport properties of nanoporous rocks

Understanding fluid flow and transport within clay rock is essential for predicting caprock integrity in underground gas storage or as host rock in deep radioactive waste storage. The connectivity and topology of the nanopore space, which drive the transfer mechanisms of these materials, are still poorly known and direct 3D imaging is particularly challenging. In this work, we investigate and compare different stochastic reconstruction approaches based on two-point and multiple-point statistics (MPS) methods and using information from 2D training images for 3D volume rendering at submicron scale. A particular emphasis is given to the maximal critical distance of sampling between two consecutive 2D images which is necessary to obtain a coherent 3D reconstruction of the nanopore structure. We assess how these realizations honour various crucial transport properties of material, namely permeability, effective diffusion and longitudinal dispersion. Morphological features such as pore volume, specific surface, Euler characteristic and tortuosity are used to analyze the results. The methods are employed on a synthetic clay of nanometric porosity for which FIB-SEM images are available. Results indicate that the 3DA-MPS and weighted-3DA-MPS approaches are the most suited for preserving pore space features and transport properties, the choice depending on the level of conditioning data available.

Indicator-based data assimilation with multiple-point statistics for updating an ensemble of models with non-Gaussian parameter distributions

Ground-water flow and the transport of contaminants in groundwater systems are strongly controlled by geologic heterogeneities. Because, the data available to model these subsurface heterogeneities are generally sparse, the observed groundwater flow, piezometric head at well locations, or the concentration profile of solutes provide valuable additional information about these complex subsurface systems. Regardless, there is likely to be considerable uncertainty associated with the predictions of the subsurface heterogeneities. This has motivated the development of several ensemble-based schemes for assimilating flow response data into predictions of subsurface heterogeneities. Most ensemble-based data assimilation methods including ensemble Kalman filter (EnKF) or indicator-based data assimilation (InDA), used for assimilation of dynamic data into geologic models, utilize statistics in the form of covariances calculated using the ensemble of models to perform model updates. These covariance based updates do not preserve the complex spatial characteristics of geologic structures that are better represented using multiple-point statistics. Additionally, in EnKF, the mismatch between the observed and the simulated values are assumed to be linearly related to the parameter updates, and the distribution of the state variable is assumed to be multi-Gaussian. The spatial distribution of the primary variables like facies can be non-Gaussian because of the spatial continuity exhibited by channel facies (sands) and non-channel facies (clay), that have distinctly different properties. Also, the relationship between the mismatch and the parameter updates can be strongly non-linear. In indicator-based data assimilation (InDA) method the indicator transforms of parameter and mismatch variables are used, which affords the treatment of these variables as bi-variate non-Gaussian. Furthermore, in the indicator transformed space, restrictive linear assumption can also be lifted as the indicator transform is invariant under non-linear transformations. However, the updates are still governed by the bivariate interactions between the state variables. A multiple-point extension of InDA is proposed in this paper which utilizes existing multiple-point simulation algorithms like single normal equation simulation (SNESIM) in combination with InDA, to preserve the spatial characteristics of the geologic model while at the same time honor the observed flow or transport response. The proposed method is applied to a synthetic groundwater system with complex channel distributions. The channel features of the final updated ensemble of models is shown to converge towards the reference model and the ensemble flow responses are also shown to match the reference flow response.

2D investigation of water permeability in hydrate-bearing sediment with coexisting hydrate growth habits

AbstractPermeability variation in the presence of hydrates plays an important role in hydrate dissociation and may lead to extraction issues. Therefore, a wide range of pore-scale studies considering the hydrate growth habit have been conducted by various researchers. Here, we use microscopic training images of hydrate-bearing sediment to replicate the micromorphology of hydrates and coexisting hydrate growth habits (pore filling and coating) into 2D reconstruction models by multiple point statistics (MPS). Accounting for the geometric differences between each realisation and the different ratios between coating hydrate and pore-filling hydrate, permeability simulation is performed by the lattice Boltzmann Method (LBM). Results show that subtle geometric differences of hydrates may result in great changes in permeability for a given hydrate saturation for the same hydrate growth habit. The Kozeny–Carman model for single hydrate growth habit can also be used for the hydrate-bearing sediment with coexisting hydrate growth habits if the ratio of these two hydrate saturations is about 4, where the effect of coating or pore-filling hydrate can be neglected. This reconstruction model agrees well with experimental results and can be easily implemented in reservoir simulators for complex hydrate distributions.

Nonparametric and data-driven interpolation of subsurface soil stratigraphy from limited data using multiple point statistics

An essential task in many geotechnical projects is delineation of subsurface soil stratigraphy from scatter measurements. Geotechnical engineers often use their knowledge on local geology and interpret soil strata boundaries by linear interpolation of measured data. This usual practice may encounter difficulties when interpreting complex deposits, particularly when measurements are limited. In this study, a novel nonparametric, data-driven method based on multiple point statistics (MPS) is proposed to interpolate subsurface soil stratigraphy from sparse measurements. MPS may be formulated as Bayesian supervised machine learning, which adaptively learns high-order spatial information (e.g., curvilinear features of soil layers) using sparse measurements obtained in a specific site and training image that reflects pre-existing engineering knowledge on similar geological settings. The proposed method is the first ever purely data-driven method (i.e., without using any pre-specified parametric functions) for geotechnical site characterization. The proposed method is illustrated by a simulated example and real data from a reclamation site in Hong Kong. The proposed method not only accurately interpolates the subsurface soil stratigraphy from sparse measurements, but also quantifies uncertainty associated with the interpolation. Effects of governing parameters in the proposed method are explicitly investigated, and parameters appropriate for subsurface soil stratigraphy are identified.

Reconstruction of shale image based on Wasserstein Generative Adversarial Networks with gradient penalty

Generative Adversarial Networks (GANs), as most popular artificial intelligence models in the current image generation field, have excellent image generation capabilities. Based on Wasserstein GANs with gradient penalty, this paper proposes a novel digital core reconstruction method. First, a convolutional neural network is used as a generative network to learn the distribution of real shale samples, and then a convolutional neural network is constructed as a discriminative network to distinguish reconstructed shale samples from real ones. Through this confrontation training method, realistic digital core samples of shale can be reconstructed. The paper uses two-point covariance function, Frechet Inception Distance and Kernel Inception Distance, to evaluate the quality of digital core samples of shale reconstructed by GANs. The results show that the covariance function can test the similarity between generated and real shale samples, and that GANs can efficiently reconstruct digital core samples of shale with high-quality. Compared with multiple point statistics, the new method does not require prior inference of the probability distribution of the training data, and directly uses noise vector to generate digital core samples of shale without using constraints of hard data in advance. It is easy to produce an unlimited number of new samples. Furthermore, the training time is also shorter, only 4 hours in this paper. Therefore, the new method has some good points compared with current methods. Cited as : Zha, W., Li, X., Xing, Y., He, L., Li, D. Reconstruction of shale image based on Wasserstein Generative Adversarial Networks with gradient penalty. Advances in Geo-Energy Research, 2020, 4(1): 107-114, doi: 10.26804/ager.2020.01.10

Non-parametric machine learning methods for interpolation of spatially varying non-stationary and non-Gaussian geotechnical properties

Spatial interpolation has been frequently encountered in earth sciences and engineering. A reasonable appraisal of subsurface heterogeneity plays a significant role in planning, risk assessment and decision making for geotechnical practice. Geostatistics is commonly used to interpolate spatially varying properties at un-sampled locations from scatter measurements. However, successful application of classic geostatistical models requires prior characterization of spatial auto-correlation structures, which poses a great challenge for unexperienced engineers, particularly when only limited measurements are available. Data-driven machine learning methods, such as radial basis function network (RBFN), require minimal human intervention and provide effective alternatives for spatial interpolation of non-stationary and non-Gaussian data, particularly when measurements are sparse. Conventional RBFN, however, is direction independent (i.e. isotropic) and cannot quantify prediction uncertainty in spatial interpolation. In this study, an ensemble RBFN method is proposed that not only allows geotechnical anisotropy to be properly incorporated, but also quantifies uncertainty in spatial interpolation. The proposed method is illustrated using numerical examples of cone penetration test (CPT) data, which involve interpolation of a 2D CPT cross-section from limited continuous 1D CPT soundings in the vertical direction. In addition, a comparative study is performed to benchmark the proposed ensemble RBFN with two other non-parametric data-driven approaches, namely, Multiple Point Statistics (MPS) and Bayesian Compressive Sensing (BCS). The results reveal that the proposed ensemble RBFN provides a better estimation of spatial patterns and associated prediction uncertainty at un-sampled locations when a reasonable amount of data is available as input. Moreover, the prediction accuracy of all the three methods improves as the number of measurements increases, and vice versa. It is also found that BCS prediction is less sensitive to the number of measurement data and outperforms RBFN and MPS when only limited point observations are available.

3D modeling of deepwater turbidite lobes: a review of the research status and progress

Deepwater turbidite lobe reservoirs have massive hydrocarbon potential and represent one of the most promising exploration targets for hydrocarbon industry. Key elements of turbidite lobes internal heterogeneity include the architectural hierarchy and complex amalgamations at each hierarchical level leading to the complex distribution of shale drapes. Due to limitation of data, to build models realistically honoring the reservoir architecture provides an effective way to reduce risk and improve hydrocarbon recovery. A variety of modeling techniques on turbidite lobes exist and can be broadly grouped into pixel-based, process-based, process-oriented, surface-based, object-based and a hybrid approach of two or more of these methods. The rationale and working process of methods is reviewed, along with their pros and cons. In terms of geological realism, object-based models can capture the most realistic architectures, including the multiple hierarchy and the amalgamations at different hierarchical levels. In terms of data conditioning, pixel-based and multiple-point statistics methods could honor the input data to the best degree. In practical, different methods should be adopted depending on the goal of the project. Such a review could improve the understanding of existing modeling methods on turbidite lobes and could benefit the hydrocarbon exploration activities of such reservoirs in offshore China.

Multiple-point statistics using multi-resolution images

Multiple-point statistics (MPS) is a simulation technique allowing to generate images that reproduce the spatial features present in a training image (TI). MPS algorithms consist in sequentially filling a simulation grid such that patterns around the simulated values come from the TI. Following this principle, joint simulations of multiple variables can be handled and complex heterogeneous fields can be generated. However, inconsistent patterns are often found in the results and some spatial features can be difficult to reproduce. In this paper, a new MPS algorithm based on a multi-resolution representation of the TI is proposed to enhance the quality of the realizations. The method consists in first building a pyramid of images from the TI by successive convolution using Gaussian-like kernels. Secondly, a MPS simulation is done at the lowest resolution level. Then, the result is expanded to the next level of resolution (one rank higher) and used as a conditioning variable for a joint MPS simulation at that level. This last step is repeated up to the initial resolution, where the final simulation is retrieved. The method is implemented in the DeeSse code based on the direct sampling algorithm. Most of the features provided by the direct sampling (conditioning to hard data, uni- or multi-variate simulation of categorical and continuous variables, scaling and rotation of the training structures) are compatible with the proposed method and the usability is maintained. Finally, various examples show that in most of the situations, combining Gaussian pyramids with MPS allows to get results of better quality and in less time compared to direct MPS simulations.

Multiresolution Approach to Condition Categorical Multiple‐Point Realizations to Dynamic Data With Iterative Ensemble Smoothing

AbstractA new methodology is presented for the conditioning of categorical multiple‐point statistics (MPS) simulations to dynamic data with an iterative ensemble smoother (ES‐MDA). The methodology relies on a novel multiresolution parameterization of the categorical MPS simulation. The ensemble of latent parameters is initially defined on the basis of the coarsest‐resolution simulations of an ensemble of multiresolution MPS simulations. Because this ensemble is non‐multi‐Gaussian, additional steps prior to the computation of the first update are proposed. In particular, the parameters are updated at predefined locations at the coarsest scale and integrated as hard data to generate a new multiresolution MPS simulation. The performance of the methodology was assessed on a synthetic groundwater flow problem inspired from a real situation. The results illustrate that the method converges towards a set of final categorical realizations that are consistent with the initial categorical ensemble. The convergence is reliable in the sense that it is fully controlled by the integration of the ES‐MDA update into the new conditional multiresolution MPS simulations. Thanks to a massively reduced number of parameters compared to the size of the categorical simulation, the identification of the geological structures during the data assimilation is particularly efficient for this example. The comparison between the estimated uncertainty and a reference estimate obtained with a Monte Carlo method shows that the uncertainty is not severely reduced during the assimilation as is often the case. The connectivity is successfully reproduced during the iterative procedure despite the rather large distance between the observation points.

3D stochastic modeling framework for Quaternary sediments using multiple-point statistics: A case study in Minjiang Estuary area, southeast China

Multiple-point statistics (MPS) has shown promise in representing complicated subsurface structures, such as the sedimentary system. The Quaternary sedimentary system possesses prominent anisotropy characteristics. For a practical three-dimensional (3D) application, therefore, the critical challenges come not only from the difficulty of obtaining credible 3D training images, but also from the serious non-stationarity. Moreover, MPS-based simulations are usually performed on a regular Cartesian grid so that they cannot realistically reflect the actual shape and distribution of geological structures. In our work, an integrated MPS-based 3D modeling framework is presented by incorporating the characteristics of Quaternary sediments and the datasets obtained from geological exploration. The framework is embedded into a 3D modeling grid to achieve more precise visualization for the subsurface structures. Following the proposed workflow, we perform the 3D modeling practices for Quaternary sedimentary facies and Quaternary stratigraphic structures by using the datasets from a Quaternary geological survey project at a southeast coastal city in China. The multiple-scale models are constructed by performing the MPS simulations which take into account the constraints of structural conditions. The application verifies the rationality and applicability of the presented workflow where the various structural features are integrated into a unified 3D model to achieve a more realistic characterization.