Multiple-point Statistical Simulation Research Articles

Reconstruction of Missing GPR Data Using Multiple-Point Statistical Simulation

Reconstruction of Missing GPR Data Using Multiple-Point Statistical Simulation

Multiple-point statistics and non-colocational soft data integration

In geostatistics, available conditional information is typically categorized as either hard (with no uncertainty) or soft (associated with an uncertainty) data. 2-point based Gaussian (kriging) simulation methods have since their inception been able to account for hard and soft data. In contrast, multiple-point statistical (MPS) simulation methods, that allow quantifying more complex phenomena than Gaussian methods, have focused mostly on conditioning to hard data. The most widely used method for accounting for soft data is to consider only co-located soft data. Here we define conditional information data through a probability distribution and show how a general probabilistic solution, in the form of a posterior probability distribution, can be obtained, combining conditional categorical information with information from a categorical training image. We demonstrate how to compute the conditional distribution, as needed by most MPS-based sequential simulation algorithms, that allows accounting for both co- and non-co-located hard and soft data. Examples are provided comparing the direct computation of conditional statistics to using more computational demanding rejection sampling. In order to reduce the significant computational demands, a distance-based measure is proposed to increase the entropy of conditional data, based on the distance to the point being estimated and simulated.

Multiple-Point Statistical Simulation of rock fracture network as a key control on the hydrogeology and salinity: a case study from the Qarabagh area, West Azarbayjan Province, Iran

Modeling and characterization of the geometry and distribution of Rock Facture Networks (RFNs) are essential in applications such as hydrogeological or environmental evaluations. It is widely accepted that RFNs are potentially associated with the hydrogeological (thus salinity) characteristics of the surrounding environments. Despite the complexity and inaccessibility of RFNs, stochastic methods provide a functional framework to predict their characteristics in the subsurface. An efficient tool for modeling RFNs is the Discrete Fracture Network (DFN) which also includes a number of geostatistical techniques that consider spatial variability structure. The advantages of these techniques are: realistic results, ease of application, and uncertainty assessments. Multiple-point geostatistics/statistics (MPS) is a modern and effective geostatistical tool for realistically simulating RFNs. In the present study, we modeled the RFNs in a location near the Qarabagh area, in the western Urmia Lake; in this regard, we used the Single Normal Equation Simulation (SNESIM) algorithm of the MPS geostatistical method using Training Images (TIs) instead of variograms. The required datasets and information for this modeling was provided using the field measurements of the fracture orientations and dips, as well as the outcrop photographs. The outcomes of these models can be used in predicting the salinity distribution in the surrounding area . Therefore, through the SNESIM algorithm, TIs obtained from the outcrop photographs, and direct measurements, 100 RFN realizations were generated at each station. These realizations were then averaged to predict the locations with higher and lower fracture probabilities and to assess the general trend of the fracture distributions.

Assessing Multiple-Point Statistical Facies Simulation Behavior for Effective Conditioning on Probabilistic Data

Conditioning multiple-point statistical (MPS) facies simulation on dynamic flow data is complicated by the complex relation between flow responses and facies distribution. One way to incorporate feedback from the flow data into MPS simulation is by constructing (and updating) facies probability maps (soft data) and using the results to constrain the MPS simulation outputs, for example, through Single Normal Equation SIMulation (SNESIM) algorithm and the $$ \tau $$-model. The pattern-imitating behavior of MPS simulation (specifically, the SNESIM) has been shown to result in a sizable fraction of facies models that fail to effectively capture the correct location and spatial connectivity represented by the facies probability map. This paper presents two key observations that explains this behavior/outcome: (1) the facies patterns resulting from the SNESIM are primarily controlled by the outcome of the first few grid cells along the corresponding random path (early stages of the simulation); and (2) the facies outcomes in the remaining grid cells are dominated by extremely confident conditional probabilities extracted from the TI to generate the encoded connectivity patterns in the TI. Hence, the contribution of facies probability map become increasingly inconsequential as the simulation proceeds. This intrinsic property must be accounted for in developing data conditioning methods, especially when training images with large-scale connectivity patterns are considered. Two approaches for addressing this issue include adapting the random path at early stages to the key information in the probability map to properly reflect its impact, or treating high-probability events from the probability map as hard data. Examples are presented to illustrate this behavior, followed by a simple method to improve the effectiveness of integrating facies probability maps (and flow response data) into SNESIM.

Integration of soft data into multiple-point statistical simulation: re-assessing the probability conditioning method for facies model calibration

Conditioning multiple-point statistical (MPS) facies simulation on dynamic flow data is complicated by the complex relationship between discrete facies distribution and the flow response data. The process can be facilitated by using the probability conditioned method (PCM), in which an ensemble-based data assimilation approach is used to infer, from observed flow data, a facies probability map that can guide conditional simulation from a training image (TI). We discuss important aspects of flow model calibration based on facies probability maps and propose several improvements to the original PCM, including: (1) using a statistically consistent approach for inferring facies probability maps from updated ensemble of continuous hydraulic properties; (2) assigning spatially variable weights to simulation grids based on relative confidence in the information content of observed data; and (3) considering spatial patterns instead of marginal facies probabilities in constructing probability maps. We illustrate that, in simulating categorical facies, the pattern-imitating behavior of MPS can have a significant impact on soft data conditioning. In particular, in the Single Normal Equation SIMulation (SNESIM) algorithm, the initial points that are visited on the random path tend to determine the main global facies connectivity. Once these initial points are populated with simulated values, the conditional probabilities (inferred from the TI) for the subsequent cells on the random path tend to be very close to 0 and 1. Hence, the facies types in the remaining cells are known almost deterministically, making the soft data inconsequential at the later stages of simulation. This behavior suggests that, without compensating for this effect, the facies probability map is not effectively integrated into the SNESIM simulation, an issue that is reflected in the generated realizations. Several numerical experiments are presented and discussed to illustrate this behavior and its impact on soft-data conditioning.

Multiple-point statistical simulation of the ore boundaries for a lateritic bauxite deposit

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.

Contributions to uncertainty related to hydrostratigraphic modeling using multiple-point statistics

Abstract. Forecasting the flow of groundwater requires a hydrostratigraphic model, which describes the architecture of the subsurface. State-of-the-art multiple-point statistical (MPS) tools are readily available for creating models depicting subsurface geology. We present a study of the impact of key parameters related to stochastic MPS simulation of a real-world hydrogeophysical dataset from Kasted, Denmark, using the snesim algorithm. The goal is to study how changes to the underlying datasets propagate into the hydrostratigraphic realizations when using MPS for stochastic modeling. This study focuses on the sensitivity of the MPS realizations to the geophysical soft data, borehole lithology logs, and the training image (TI). The modeling approach used in this paper utilizes a cognitive geological model as a TI to simulate ensemble hydrostratigraphic models. The target model contains three overall hydrostratigraphic categories, and the MPS realizations are compared visually as well as quantitatively using mathematical measures of similarity. The quantitative similarity analysis is carried out exhaustively, and realizations are compared with each other as well as with the cognitive geological model. The results underline the importance of geophysical data for constraining MPS simulations. Relying only on borehole data and the conceptual geology, or TI, results in a significant increase in realization uncertainty. The airborne transient electromagnetic SkyTEM data used in this study cover a large portion of the Kasted model area and are essential to the hydrostratigraphic architecture. On the other hand, the borehole lithology logs are sparser, and 410 boreholes were present in this study. The borehole lithology logs infer local changes in the immediate vicinity of the boreholes, thus, in areas with a high degree of geological heterogeneity, boreholes only provide limited large-scale structural information. Lithological information is, however, important for the interpretation of the geophysical responses. The importance of the TI was also studied. An example was presented where an alternative geological model from a neighboring area was used to simulate hydrostratigraphic models. It was shown that as long as the geological settings are similar in nature, the realizations, although different, still reflect the hydrostratigraphic architecture. If a TI containing a biased geological conceptualization is used, the resulting realizations will resemble the TI and contain less structure in particular areas, where the soft data show almost even probability to two or all three of the hydrostratigraphic units.

Correction to: Enhanced Multiple-Point Statistical Simulation with Backtracking, Forward Checking and Conflict-Directed Backjumping

In the original version of this article, unfortunately Figure 9 was wrong due to a typesetting mistake. The original article has been corrected.

Enhanced Multiple-Point Statistical Simulation with Backtracking, Forward Checking and Conflict-Directed Backjumping

During a conventional multiple-point statistics simulation, the algorithm may not find a matched neighborhood in the training image for some unsimulated pixels. These pixels are referred to as the dead-end pixels; the existence of the dead-end pixels means that multiple-point statistics simulation is not a simple sequential simulation. In this paper, the multiple-point statistics simulation is cast as a combinatorial optimization problem, and the efficient backtracking algorithm is developed to solve this optimization problem. The efficient backtracking consists of backtracking, forward checking, and conflict-directed backjumping algorithms that are introduced and discussed in this paper. This algorithm is applied to simulate multiple-point statistics properties of some synthetic training images; the results show that no anomalies occurred in any of the produced realizations as opposed to previously published methods for solving the dead-end pixels. In particular, in simulating a channel system, all the channels generated by this method are continuous, which is of paramount importance in fluid flow simulation applications. The results also show that the presence of hard data does not degrade the quality of the generated realizations. The presented method provides a robust algorithmic framework for performing MPS simulation.

Pilot points method for conditioning multiple-point statistical facies simulation on flow data

Abstract We propose a new pilot points method for conditioning discrete multiple-point statistical (MPS) facies simulation on dynamic flow data. While conditioning MPS simulation on static hard data is straightforward, their calibration against nonlinear flow data is nontrivial. The proposed method generates conditional models from a conceptual model of geologic connectivity, known as a training image (TI), by strategically placing and estimating pilot points. To place pilot points, a score map is generated based on three sources of information: (i) the uncertainty in facies distribution, (ii) the model response sensitivity information, and (iii) the observed flow data. Once the pilot points are placed, the facies values at these points are inferred from production data and then are used, along with available hard data at well locations, to simulate a new set of conditional facies realizations. While facies estimation at the pilot points can be performed using different inversion algorithms, in this study the ensemble smoother (ES) is adopted to update permeability maps from production data, which are then used to statistically infer facies types at the pilot point locations. The developed method combines the information in the flow data and the TI by using the former to infer facies values at selected locations away from the wells and the latter to ensure consistent facies structure and connectivity where away from measurement locations. Several numerical experiments are used to evaluate the performance of the developed method and to discuss its important properties.

Multiple-point statistical simulation for hydrogeological models: 3-D training image development and conditioning strategies

Abstract. Most studies on the application of geostatistical simulations based on multiple-point statistics (MPS) to hydrogeological modelling focus on relatively fine-scale models and concentrate on the estimation of facies-level structural uncertainty. Much less attention is paid to the use of input data and optimal construction of training images. For instance, even though the training image should capture a set of spatial geological characteristics to guide the simulations, the majority of the research still relies on 2-D or quasi-3-D training images. In the present study, we demonstrate a novel strategy for 3-D MPS modelling characterized by (i) realistic 3-D training images and (ii) an effective workflow for incorporating a diverse group of geological and geophysical data sets. The study covers an area of 2810 km2 in the southern part of Denmark. MPS simulations are performed on a subset of the geological succession (the lower to middle Miocene sediments) which is characterized by relatively uniform structures and dominated by sand and clay. The simulated domain is large and each of the geostatistical realizations contains approximately 45 million voxels with size 100 m × 100 m × 5 m. Data used for the modelling include water well logs, high-resolution seismic data, and a previously published 3-D geological model. We apply a series of different strategies for the simulations based on data quality, and develop a novel method to effectively create observed spatial trends. The training image is constructed as a relatively small 3-D voxel model covering an area of 90 km2. We use an iterative training image development strategy and find that even slight modifications in the training image create significant changes in simulations. Thus, this study shows how to include both the geological environment and the type and quality of input information in order to achieve optimal results from MPS modelling. We present a practical workflow to build the training image and effectively handle different types of input information to perform large-scale geostatistical modelling.

Adaptive Conditioning of Multiple-Point Statistical Facies Simulation to Flow Data with Probability Maps

Multiple-point statistics (MPS) provides a flexible grid-based approach for simulating complex geologic patterns that contain high-order statistical information represented by a conceptual prior geologic model known as a training image (TI). While MPS is quite powerful for describing complex geologic facies connectivity, conditioning the simulation results on flow measurements that have a nonlinear and complex relation with the facies distribution is quite challenging. Here, an adaptive flow-conditioning method is proposed that uses a flow-data feedback mechanism to simulate facies models from a prior TI. The adaptive conditioning is implemented as a stochastic optimization algorithm that involves an initial exploration stage to find the promising regions of the search space, followed by a more focused search of the identified regions in the second stage. To guide the search strategy, a facies probability map that summarizes the common features of the accepted models in previous iterations is constructed to provide conditioning information about facies occurrence in each grid block. The constructed facies probability map is then incorporated as soft data into the single normal equation simulation (snesim) algorithm to generate a new candidate solution for the next iteration. As the optimization iterations progress, the initial facies probability map is gradually updated using the most recently accepted iterate. This conditioning process can be interpreted as a stochastic optimization algorithm with memory where the new models are proposed based on the history of the successful past iterations. The application of this adaptive conditioning approach is extended to the case where multiple training images are proposed as alternative geologic scenarios. The advantages and limitations of the proposed adaptive conditioning scheme are discussed and numerical experiments from fluvial channel formations are used to compare its performance with non-adaptive conditioning techniques.

GPU-accelerated Direct Sampling method for multiple-point statistical simulation

Geostatistical simulation techniques have become a widely used tool for the modeling of oil and gas reservoirs and the assessment of uncertainty. The Direct Sampling (DS) algorithm is a recent multiple-point statistical simulation technique. It directly samples the training image (TI) during the simulation process by calculating distances between the TI patterns and the given data events found in the simulation grid (SG). Omitting the prior storage of all the TI patterns in a database, the DS algorithm can be used to simulate categorical, continuous and multivariate variables. Three fundamental input parameters are required for the definition of DS applications: the number of neighbors n, the acceptance threshold t and the fraction of the TI to scan f. For very large grids and complex spatial models with more severe parameter restrictions, the computational costs in terms of simulation time often become the bottleneck of practical applications.This paper focuses on an innovative implementation of the Direct Sampling method which exploits the benefits of graphics processing units (GPUs) to improve computational performance. Parallel schemes are applied to deal with two of the DS input parameters, n and f. Performance tests are carried out with large 3D grid size and the results are compared with those obtained based on the simulations with central processing units (CPU). The comparison indicates that the use of GPUs reduces the computation time by a factor of 10X–100X depending on the input parameters. Moreover, the concept of the search ellipsoid can be conveniently combined with the flexible data template of the DS method, and our experimental results of sand channels reconstruction show that it can improve the reproduction of the long-range connectivity patterns.

GPU-based SNESIM implementation for multiple-point statistical simulation

Among techniques applied to categorical variables simulation, multiple-point statistical simulation is widely used because of its non-iterative characteristic and powerful capability of curvilinear features reproduction. In current implementations, the multiple-point statistics (MPS) are inferred from the training image by storing all the observed patterns scanned by data templates of a certain size within a data structure, either a tree used by Single Normal Equation Simulation Algorithm (SNESIM) or a list used by IMPALA. This type of algorithms has the advantage of being fast to be applied, but it presents some critical limitations. In particular, the data structure is extremely memory demanding. For large-scale problems with numerous patterns, large data templates cannot be used. Therefore, complex structures are then difficult to be simulated.A GPU computing scheme for SNESIM is proposed for multiple point statistical simulation in this paper. Taking advantage of powerful computing capability of GPU's many-core architecture, parallel operations are applied to each simulation grid node, which is the most time-consuming portion among entire simulation process. This scheme requires fixed size memory, so it is independent of inference of data template size, which is especially important for large-scale problems. The simulation results based on a NVIDIAGTX680 device can obtain about 15× speedup than on an Intel Core i3 540 CPU, which demonstrates the efficiency of the scheme against search-tree based implementation of SNESIM contained in SGeMS software.

A practical guide to performing multiple-point statistical simulations with the Direct Sampling algorithm

The Direct Sampling (DS) algorithm is a recently developed multiple-point statistical simulation technique. It directly scans the training image (TI) for a given data event instead of storing the training probability values in a catalogue prior to simulation. By using distances between the given data events and the TI patterns, DS allows to simulate categorical, continuous and multivariate problems. Benefiting from the wide spectrum of potential applications of DS, requires understanding of the user-defined input parameters. Therefore, we list the most important parameters and assess their impact on the generated simulations. Real case TIs are used, including an image of ice-wedge polygons, a marble slice and snow crystals, all three as continuous and categorical images. We also use a 3D categorical TI representing a block of concrete to demonstrate the capacity of DS to generate 3D simulations. First, a quantitative sensitivity analysis is conducted on the three parameters balancing simulation quality and CPU time: the acceptance threshold t, the fraction of TI to scan f and the number of neighbors n. Next to a visual inspection of the generated simulations, the performance is analyzed in terms of speed of calculation and quality of pattern reproduction. Whereas decreasing the CPU time by influencing t and n is at the expense of simulation quality, reducing the scanned fraction of the TI allows substantial computational gains without degrading the quality as long as the TI contains enough reproducible patterns. We also illustrate the quality improvement resulting from post-processing and the potential of DS to simulate bivariate problems and to honor conditioning data. We report a comprehensive guide to performing multiple-point statistical simulations with the DS algorithm and provide recommendations on how to set the input parameters appropriately.

A Comparison of Methods for the Stochastic Simulation of Rock Fractures

Methods reported in the literature for rock fracture simulations include approaches based on stochastic geometry, multiple-point statistics and a combination of geostatistics for fracture density and object-based modelling for fracture geometries. The advantages and disadvantages of each of these approaches are discussed with examples. By way of review, the authors begin with the geostatistical indicator simulation method, based on the truncated–Gaussian algorithm; this is followed by multiple-point statistical simulation and then the stochastic geometry approach, which is based on marked point process simulation. A new approach, based on pluriGaussian structural simulation, is then introduced. The new approach incorporates in the simulation the spatial correlation between different sets of fractures, which in general, is very difficult, if not impossible, to accomplish in the three methods reviewed. Each simulation method is summarised together with detailed simulation procedures for each. A published two-dimensional fracture dataset is used as a means of assessing the performance of each simulation method and of demonstrating the concepts discussed in the text.

Using the Snesim program for multiple-point statistical simulation

Traditionally, there are two mainstream avenues for geostatistical modeling: pixel-based two-point simulation and object-based simulation. Each is good at either data conditioning or reproducing geological shapes, but none is good at both. Multiple-point simulation combines the strengths of these two avenues. As an advanced pixel-based technique, it inherits the flexibility of pixel-based techniques by building the model one pixel at a time, hence data conditioning is easily achieved; it is also capable of reproducing curvilinear geological shapes through borrowing multiple-point statistics from a training image. The snesim code provides such a multiple-point simulation program. A training image is used to represent the prior geological knowledge, which is scanned to obtain the conditioning probability values for the central node belonging to a facies category given any multiple-point conditioning data event. These training probability values are stored in a search tree a prior to simulation. Then in a sequential simulation mode, at each uninformed node, according to its specific conditioning data event a probability value is retrieved from the search tree, and a value is simulated from it. There are many input parameters to the snesim program, the impact of which might not be immediately clear to people who are not familiar with the code. In this paper, we aim at bringing important aspects of this program and providing practical guidelines to using the program. Sensivity analyses are performed on the important input parameters. The results are analyzed and recommendations are provided on how to set these parameters appropriately.