Multiple-point Statistics Methods Research Articles

Reconstructing Three-dimensional geological structures by the Multiple-point statistics method coupled with a deep neural network: A case study of a metro station in Guangzhou, China

A credible large-scale three-dimensional (3D) geological model constructed from a small amount of local high-density data provides a vital data infrastructure for the design and construction of metro lines. The multiple-point statistics (MPS) method has successfully generated 3D geological models in many fields. However, the global cognition of the spatial relations between geological objects is difficult to reconstruct using a local optimization strategy with moving templates in the MPS simulation. This study presented an MPS simulation approach for constructing 3D geological structures with two-dimensional (2D) cross-sections, where the MPS process was coupled with a fully connected deep neural network (FCDNN) simulation. The core idea of the proposed method was that the global features of each geological object were generated from the trained FCDNN, the input parameter of which was the subsurface depth. Then, the entire model was constructed on the initial model using the following iterative MPS process: the semantic relationship of geological objects was extracted from 2D cross-sections and used as a constraint in the modeling process, which ensured the order of the stratigraphic sequence and rationality of the geological structure in the final model. The joint objective function of the proposed method was composed of the loss function of the FCDNN and conventional MPS objective function constrained by the spatial semantics of geological objects. The concrete example in the Jiangtai Road Metro Station of Guangzhou Line 11, China illustrated that the details of the complex geological structures could be obtained accurately by the presented method. The global features from the trained FCDNN provided the basic geometry for the final model, which overcame the limitations of the conventional MPS method.

A nearest neighbor multiple-point statistics method for fast geological modeling

Multiple-point statistics (MPS) is a powerful method to generate realistic geological models. Given a training image as a prior model, the program iteratively reproduces spatial patterns in the simulation grid. However, running speed becomes a limitation to practical applications. MPS has to handle complicated and high-dimensional structures at the cost of simulation time. With the objective to accelerate geostatistical modeling with categorical variables, we propose a nearest neighbor simulation (NNSIM) method. Several k-nearest neighbor (kNN) classifiers are incorporated into MPS framework. First, we identify representative patterns with a prototype selection method. Different from existing MPS programs, our method selects training patterns according to their influences on simulation quality. A pattern subset of small size has a positive effect on searching time. Second, a teacher-student architecture is suggested to improve the pattern subset. In order to address missing data, our program augments the subset with key patterns during simulation. A cosine distance metric is applied to compare the original dataset and pattern subset. Third, our program organizes patterns with a ball tree. Pattern groups with low similarity are dynamically removed to fulfill fast search. We examine the proposed NNSIM by a benchmark channel simulation, a 2D flume model, and a 3D sandstone modeling. Many quantitative approaches are employed to evaluate geometrical and physical properties. The experimental results indicate that our NNSIM significantly improves the computational efficiency while exhibits comparable simulation quality to traditional MPS programs.

Stochastic simulation of deltas based on a concurrent multi-stage VAE-GAN model

The stochastic simulation of deltas has always been one of the significant problems in the numerical simulation of reservoirs. Three problems of deltaic simulations are non-stationary geological characteristics in training images (TIs), insufficient quantity of TIs in certain regions and unavailable reuse of training parameters and models even after the first training, which have caused big challenges for the stochastic simulation of deltas especially in the circumstances of only heterogeneous TIs are available and the simulation quantity is quite large. As one of the important numerical methods for the simulation of deltas, multiple-point statistics (MPS) obtains the statistical characteristics through the patterns in TIs to perform simulation. However, due to the non-stationarity of deltaic TIs, the traditional MPS cannot extract the non-stationary characteristics of TIs smoothly and is incapable of reusing the extracted probability information, making the simulation process quite time-consuming if multiple simulations are performed successively. Thanks to the powerful feature extraction capability brought by deep learning, the delta simulation possibly will be largely improved. The generative adversarial network (GAN) is an important deep learning method for image generation, but it needs large-quantity training samples and its training process is unstable, but another famous neural network variational auto-encoder (VAE) is more stable. On the other hand, VAE-generated images are often a little blurred but GAN-generated images are clearer. Therefore, based on VAE, GAN and a multi-stage idea, a concurrent multi-stage VAE-GAN model is proposed for the stochastic simulation of deltas to hopefully address the three problems in deltaic simulation by concurrently performing multi-stage simulation based on GAN and VAE. The comparison of our method with some typical MPS methods and deep learning methods has shown its good performance in deltaic simulation.

Ice volume and basal topography estimation using geostatistical methods and ground-penetrating radar measurements: application to the Tsanfleuron and Scex Rouge glaciers, Swiss Alps

Abstract. Ground-penetrating radar (GPR) is widely used for determining mountain glacier thickness. However, this method provides thickness data only along the acquisition lines, and therefore interpolation has to be made between them. Depending on the interpolation strategy, calculated ice volumes can differ and can lack an accurate error estimation. Furthermore, glacial basal topography is often characterized by complex geomorphological features, which can be hard to reproduce using classical interpolation methods, especially when the field data are sparse or when the morphological features are too complex. This study investigates the applicability of multiple-point statistics (MPS) simulations to interpolate glacier bedrock topography using GPR measurements. In 2018, a dense GPR data set was acquired on the Tsanfleuron Glacier (Switzerland). These data were used as the source for a bedrock interpolation. The results obtained with the direct-sampling MPS method are compared against those obtained with kriging and sequential Gaussian simulations (SGSs) on both a synthetic data set – with known reference volume and bedrock topography – and the real data underlying the Tsanfleuron Glacier. Using the MPS modeled bedrock, the ice volume for the Scex Rouge and Tsanfleuron glaciers is estimated to be 113.9 ± 1.6 million cubic meters. The direct-sampling approach, unlike the SGS and kriging, allowed not only an accurate volume estimation but also the generation of a set of realistic bedrock simulations. The complex karstic geomorphological features are reproduced and can be used to significantly improve for example the precision of subglacial flow estimation.

Conditioning Multiple‐Point Statistics Simulation to Inequality Data

AbstractStochastic modeling is often employed in environmental sciences for the analysis and understanding of complex systems. For example, random fields are key components in uncertainty analysis or Bayesian inverse modeling. Multiple‐point statistics (MPS) provides efficient simulation tools for simulating fields reproducing the spatial statistics depicted in a training image (TI), while accounting for local or block conditioning data. Among MPS methods, the direct sampling algorithm is a flexible pixel‐based technique that consists in first assigning the conditioning data values (so‐called hard data) in the simulation grid, and then in populating the rest of the simulation domain in a random order by successively pasting a value from a TI cell sharing a similar pattern. In this study, an extension of the direct sampling method is proposed to account for inequality data, that is, constraints in given cells consisting of lower and/or upper bounds for the simulated values. Indeed, inequality data are often available in practice. The new approach involves the adaptation of the distance used to compare and evaluate the match between two patterns to account for such constraints. The proposed method, implemented in the DeeSse code, allows generating random fields both reflecting the spatial statistics of the TI and honoring the inequality constraints. Finally examples of topography simulations illustrate and show the capabilities of the proposed method.

A method for generating geomodels conditioned to well data with high net:gross ratios but low connectivity

The sand connectivity in object- and pixel-based models is inevitably controlled by the proportion of sand present, and these methods seem unable to generate models that reproduce systems with low connectivity at high net:gross ratios. A new workflow is described which addresses this limitation and permits poorly connected facies models conditioned to well data to be built. The approach combines the compression algorithm with multiple-point statistics (MPS) modelling. Geometrically transformed wells and appropriately scaled training images provide the inputs to the MPS modelling. The inverse transformation is applied to the resultant MPS model, leading to the creation of reservoir geomodels with realistic, user-defined connectivity while also honouring well data. The approach is described and validated using a range of models. Considerations of other potential workflows using different types of training image suggest that application of the compression algorithm may be necessary in general to achieve models with realistic connectivity using the simplest and most widely-available pixel-based MPS method (the SNESIM algorithm).

Reconstruction of Three-Dimensional Porous Media Using Deep Transfer Learning

The reconstruction of porous media is widely used in the study of fluid flows and engineering sciences. Some traditional reconstruction methods for porous media use the features extracted from real natural porous media and copy them to realize reconstructions. Currently, as one of the important branches of machine learning methods, the deep transfer learning (DTL) method has shown good performance in extracting features and transferring them to the predicted objects, which can be used for the reconstruction of porous media. Hence, a method for reconstructing porous media is presented by applying DTL to extract features from a training image (TI) of porous media to replace the process of scanning a TI for different patterns as in multiple-point statistical methods. The deep neural network is practically used to extract the complex features from the TI of porous media, and then, a reconstructed result can be obtained by transfer learning through copying these features. The proposed method was evaluated on shale and sandstone samples by comparing multiple-point connectivity functions, variogram curves, permeability, porosity, etc. The experimental results show that the proposed method is of high efficiency while preserving similar features with the target image, shortening reconstruction time, and reducing the burdens on CPU.

Vector Field‐Based Simulation of Tree‐Like Nonstationary Geostatistical Models

AbstractIn this work, a new nonstationary multiple‐point geostatistical methodology called vector field‐based simulation is presented. This method generates multiple realizations of the phenomena being modeled, whose spatial continuity is described by a training image. It can be applied in reservoir characterization and can be used to study the behavior of hydrogeological systems. The main novelty of this work is that nonstationarity is addressed through a vector field inferred from the training image. This field represents the directions in which the image develops and controls the multiple‐point statistics methods (MPS) simulation. Additionally, our simulation is adapted to support hard data conditioning. The method is tested, and the realizations exhibit a similar morphology as the training image but with substantial variability. Furthermore, in the results it can be seen that the connectivity presented in the training image is preserved.

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.

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.

Nested multiresolution hierarchical simulated annealing algorithm for porous media reconstruction.

Microstructure strongly influences flow and transport properties of porous media. Flow and transport simulations within porous media, therefore, requires accurate three-dimensional (3D) models of the pore and solid phase structure. To date, no imaging method can resolve all relevant heterogeneities from the nano- to the centimeter scale within complex heterogeneous materials such as clay, reservoir rocks (e.g., travertine, chalk, ...), hardened cement paste, and concrete. To reconstruct these porous materials it is thus necessary to merge information from different 2D and potentially 3D imaging methods. One porous media reconstruction methodology that has been around for at least two decades is simulated annealing (SA). However, realizations with SA typically suffer an artificially reduced long-range connectivity, while multiphase reconstructions are not feasible in most cases because of a prohibitive computational burden. To solve these problems we propose a hierarchical multiresolution and multiphase simulated annealing algorithm. To decrease the computational cost of multiphase simulation, our algorithm sequentially simulates one phase after another, in a hierarchical way, which enables handling multimodal distributions and topological relations. Building upon recent work, our algorithm improves long-range connectivity and CPU efficiency by simulating larger particles using a coarser resolution that is subsequently refined compared to standard SA; our proposed extension not only offers the possibility to perform multiphase reconstruction but also allows us (i) to improve binary reconstruction quality, as quantified, e.g., by multiple-point histograms by up to one order of magnitude and (ii) to achieve an overall speed-up. The proposed algorithm is also shown to outperform the direct sampling multiple-point statistics method for the generation of cement paste microstructure with respect to both generation time and quality.

MPS-APO: a rapid and automatic parameter optimizer for multiple-point geostatistics

Multiple-point statistics (MPS) have been widely used in Earth and environmental sciences because of their ability to generate realistic stochastic realizations of complex natural processes. The spatial patterns and statistical information required for MPS modeling are represented by a training image. However, each MPS algorithm has a specific set of parameters that have a direct impact on the quality of pattern reproduction and should be chosen prior to the modeling. While there are some general guidelines for some MPS algorithms, a general parameter interference methodology is currently lacking. To date, the common practice for finding optimal parameters is to carry out a sensitivity analysis, which can be cumbersome especially in complex applications. In this study, we propose the MPS Automatic Parameter Optimizer (MPS-APO), a generic method based on stochastic optimization to rapidly approximate optimal parameters for any MPS method and different types of settings. The MPS-APO formulates an objective function that quantifies spatial pattern reproduction for each set of parameters. The Simultaneous Perturbation Stochastic Approximation (SPSA) optimization method is used because of its computational efficiency, and also its ability to cope with the stochastic nature of the objective function. The optimization proceeds in two steps. The first step aims to optimize the parameters for the best quality regardless of computational cost. When no more improvement can be achieved, the second step minimizes the CPU cost without degrading the spatial structures reproduction attained in the first step. In this study, MPS-APO is performed on different pixel-based and patch-based MPS methods: SNESIM, FILTERSIM, Direct Sampling and Image Quilting. Test cases show that MPS-APO is a useful heuristic to automatically approximate optimal parameters for good patterns reproduction with minimal computational cost. Therefore, it can help non-expert users and increase the usability of MPS methods for practical applications.

Enduring effect of permeability texture for enhancing accuracy and reducing uncertainty of reservoir fluid flow through porous media

Modeling reservoir permeability is one of the crucial tasks in reservoir simulation studies. Traditionally, it is done by kriging-based methods. More rigorous modeling of the permeability results in more reliable outputs of the reservoir models. Recently, a new category of geostatistical methods has been used for this purpose, namely multiple point statistics (MPS). By this new category of permeability modeling methods, one is able to predict the heterogeneity of the reservoir permeability as a continuous variable. These methods consider the direction of property variation in addition to the distances of known locations of the property. In this study, the reservoir performance of a modified version of the SPE 10 solution project as a pioneer case is used for investigating the efficiency of these methods and paralleling them with the kriging-based one. In this way, the permeability texture concept is introduced by applying some MPS methods. This study is accomplished in the conditions of real reservoir dimensions and velocities for the whole reservoir life. A continuous training image is used as the input of calculation for the permeability modeling. The results show that the detailed permeability of the reservoir as a continuous variable makes the reservoir simulation show the same fluid front movement and flooding behavior of the reservoir similar to the reference case with the same permeability heterogeneity. Some MPS methods enable the reservoir simulation to reproduce the fluid flow complexities such as bypassing and oil trapping during water flooding similar to the reference case. Accordingly, total oil production is predicted with higher accuracy and lower uncertainty. All studied cases are identical except for the permeability texture. Even histograms and variograms of permeabilities for the studied reservoir are quite similar, but the performance of the reservoir shows that kriging-based method results have slightly less accuracy than some MPS methods. Meanwhile, it results in lower uncertainty in outputs for this water flooding case performance.

Combining Clustering Methods With MPS to Estimate Structural Uncertainty for Hydrological Models

This study presents a novel expansion of the clay-fraction (CF)/resistivity clustering method aiming at developing realizations of subsurface structures based on multiple point statistics (MPS). The CF-resistivity clustering method is used to define a data driven training image (TI) for MPS simulations. By combining this TI with uncertainty estimates obtained from correlation between the resistivity models and the unique categories in the TI, subsurface realizations are generated honoring all geophysical and lithological data including any equivalent resistivity layer sequences. The generated subsurface realizations were calibrated in a steady state groundwater model. Forecasts of well catchment zones were derived based on two wells located in areas with different levels of structural uncertainty. The catchment probability maps derived from the structural realizations were compared with the well catchment forecasted by a deterministic subsurface structure, and we are able to capture this catchment within the estimated uncertainties. We believe that this study is the first to combine MPS methods with a complete data driven workflow going directly from lithological and geophysical data to realizations of the subsurface structures. The main benefits of this is that it is data driven, fast, reproducible, and transparent.

A pattern-matching method for flow model calibration under training image constraint

Modern geostatistical modeling techniques are developed to simulate complex geologic connectivity patterns (e.g., curvilinear fluvial systems) at the grid-level using a training image (TI) that encodes multiple-point statistical (MPS) information. A challenging aspect of using MPS methods is conditioning the resulting models on nonlinear flow data. We develop a pattern-matching method for calibration of MPS-based facies models subject to the TI constraint. Since the exact statistical information in the TI can only be expressed empirically, flow data conditioning and pattern matching are carried out in two iterative steps, using an alternating-direction algorithm. Flow data integration is formulated through a regularized least-squares by taking advantage of learned k-SVD sparse parametrization and l1-norm sparsity-promoting regularization methods. The TI constraint is enforced through a MPS-based pattern-matching algorithm that uses the identified model calibration solution to generate a corresponding facies model that is consistent with the TI. The pattern-matching algorithm uses a local search template to scan the TI to find facies patterns with smallest distances from the corresponding local patterns in the parameterized approximate solution. The identified patterns for each location in the model are stored and used to estimate local conditional probabilities for assigning the facies types to each grid cell. The resulting solution is passed to the flow data conditioning step as a regularization term to perform the next iteration. The process is repeated until the MPS facies model provides an acceptable match to the data. Numerical experiments are presented to evaluate the performance of the pattern-matching method for calibration of complex facies models.

Toward more realistic models of reservoir by cutting-edge characterization of permeability with MPS methods and deep-learning-based selection

Different sources of data are used to construct a reliable model of reservoir for oil/gas production. This model ought to be matched with the production history of reservoir and also show reliable predictions for future performance. To this end, permeability modeling (characterization of heterogeneity) is crucially important which is proved to be done by Multiple Point Statistics (MPS) recently. Furthermore, deep learning methods are massively used as a promising tool for regression applications. In this study, one MPS method is employed for generating the reservoir realizations. Realizations, alongside their simulation outputs, are utilized for training a convolutional deep network. In this manner, MPS is joined with deep learning to find the most appropriate realization(s) of the reservoir based on the fluid flow simulation. Moreover, unseen MPS realizations as well as another MPS realizations are used to verify the selection ability of trained network. The detailed architecture of convolutional network is illustrated in this study.The purpose of training this network and combination with MPS is to generate the matched realization(s) in history period that also show acceptable reservoir behavior in the future times of reservoir simulation. After training, the actual production data of selected realizations are obtained by simulation the reservoir for history and also future times. The results show that selected realizations efficiently capture the trend of reference behavior. Although these realizations lack identical permeability values, they have same texture of permeability (permeability heterogeneity). Meanwhile, they show acceptable match in reservoir simulation outputs. By proposed workflow, the uncertainty of permeability modeling is considered more exhaustively. It is done by selecting the realizations from enormous possible realizations dataset and providing a deep learning tool which is capable for screening quite large number of realizations. Interesting finding is satisfactory behavior of realization(s) in both history and future periods of reservoir performance.

A new methodology to train fracture network simulation using multiple-point statistics

Abstract. Natural fracture network characteristics can be establishes from high-resolution outcrop images acquired from drone and photogrammetry. Such images might also be good analogues of subsurface naturally fractured reservoirs and can be used to make predictions of the fracture geometry and efficiency at depth. However, even when supplementing fractured reservoir models with outcrop data, gaps will remain in the model and fracture network extrapolation methods are required. In this paper we used fracture networks interpreted from two outcrops from the Apodi area, Brazil, to present a revised and innovative method of fracture network geometry prediction using the multiple-point statistics (MPS) method. The MPS method presented in this article uses a series of small synthetic training images (TIs) representing the geological variability of fracture parameters observed locally in the field. The TIs contain the statistical characteristics of the network (i.e. orientation, spacing, length/height and topology) and allow for the representation of a complex arrangement of fracture networks. These images are flexible, as they can be simply sketched by the user. We proposed to simultaneously use a set of training images in specific elementary zones of the Apodi outcrops in order to best replicate the non-stationarity of the reference network. A sensitivity analysis was conducted to emphasise the influence of the conditioning data, the simulation parameters and the training images used. Fracture density computations were performed on selected realisations and compared to the reference outcrop fracture interpretation to qualitatively evaluate the accuracy of our simulations. The method proposed here is adaptable in terms of training images and probability maps to ensure that the geological complexity in the simulation process is accounted for. It can be used on any type of rock containing natural fractures in any kind of tectonic context. This workflow can also be applied to the subsurface to predict the fracture arrangement and fluid flow efficiency in water, geothermal or hydrocarbon fractured reservoirs.

Improved multipoint statistics method for reconstructing three-dimensional porous media from a two-dimensional image via porosity matching.

Reconstructing a three-dimensional (3D) structure from a single two-dimensional training image (TI) is a challenging issue. Multiple-point statistics (MPS) is an effective method to solve this problem. However, in the traditional MPS method, errors occur while statistical features of reconstruction, such as porosity, connectivity, and structural properties, deviate from those of TI. Due to the MPS reconstruction mechanism that the voxel being reconstructed is dependent on the reconstructed voxel, it may cause error accumulation during simulations, which can easily lead to a significant difference between the real 3D structure and the reconstructed result. To reduce error accumulation and improve morphological similarity, an improved MPS method based on porosity matching is proposed. In the reconstruction, we search the matching pattern in the TI directly. Meanwhile, a multigrid approach is also applied to capture the large-scale structures of the TI. To demonstrate its superiority over the traditional MPS method, our method is tested on different sandstone samples from many aspects, including accuracy, stability, generalization, and flow characteristics. Experimental results show that the reconstruction results by the improved MPS method effectively match the CT sandstone samples in correlation functions, local porosity distribution, morphological parameters, and permeability.

Hydrostratigraphic modeling using multiple-point statistics and airborne transient electromagnetic methods

Abstract. Creating increasingly realistic groundwater models involves the inclusion of additional geological and geophysical data in the hydrostratigraphic modeling procedure. Using multiple-point statistics (MPS) for stochastic hydrostratigraphic modeling provides a degree of flexibility that allows the incorporation of elaborate datasets and provides a framework for stochastic hydrostratigraphic modeling. This paper focuses on comparing three MPS methods: snesim, DS and iqsim. The MPS methods are tested and compared on a real-world hydrogeophysical survey from Kasted in Denmark, which covers an area of 45 km2. A controlled test environment, similar to a synthetic test case, is constructed from the Kasted survey and is used to compare the modeling results of the three aforementioned MPS methods. The comparison of the stochastic hydrostratigraphic MPS models is carried out in an elaborate scheme of visual inspection, mathematical similarity and consistency with boreholes. Using the Kasted survey data, an example for modeling new survey areas is presented. A cognitive hydrostratigraphic model of one area is used as a training image (TI) to create a suite of stochastic hydrostratigraphic models in a new survey area. The advantage of stochastic modeling is that detailed multiple point information from one area can be easily transferred to another area considering uncertainty. The presented MPS methods each have their own set of advantages and disadvantages. The DS method had average computation times of 6–7 h, which is large, compared to iqsim with average computation times of 10–12 min. However, iqsim generally did not properly constrain the near-surface part of the spatially dense soft data variable. The computation time of 2–3 h for snesim was in between DS and iqsim. The snesim implementation used here is part of the Stanford Geostatistical Modeling Software, or SGeMS. The snesim setup was not trivial, with numerous parameter settings, usage of multiple grids and a search-tree database. However, once the parameters had been set it yielded comparable results to the other methods. Both iqsim and DS are easy to script and run in parallel on a server, which is not the case for the snesim implementation in SGeMS.

Accelerating simulation for the multiple-point statistics algorithm using vector quantization.

Multiple-point statistics (MPS) is a prominent algorithm to simulate categorical variables based on a sequential simulation procedure. Assuming training images (TIs) as prior conceptual models, MPS extracts patterns from TIs using a template and records their occurrences in a database. However, complex patterns increase the size of the database and require considerable time to retrieve the desired elements. In order to speed up simulation and improve simulation quality over state-of-the-art MPS methods, we propose an accelerating simulation for MPS using vector quantization (VQ), called VQ-MPS. First, a variable representation is presented to make categorical variables applicable for vector quantization. Second, we adopt a tree-structured VQ to compress the database so that stationary simulations are realized. Finally, a transformed template and classified VQ are used to address nonstationarity. A two-dimensional (2D) stationary channelized reservoir image is used to validate the proposed VQ-MPS. In comparison with several existing MPS programs, our method exhibits significantly better performance in terms of computational time, pattern reproductions, and spatial uncertainty. Further demonstrations consist of a 2D four facies simulation, two 2D nonstationary channel simulations, and a three-dimensional (3D) rock simulation. The results reveal that our proposed method is also capable of solving multifacies, nonstationarity, and 3D simulations based on 2D TIs.