Multiple-point Statistics Research Articles

Application of multiple-point geostatistics on modelling groundwater flow and transport in a cross-bedded aquifer (Belgium)

Sedimentological processes often result in complex three-dimensional subsurface heterogeneity of hydrogeological parameter values. Variogram-based stochastic approaches are often not able to describe heterogeneity in such complex geological environments. This work shows how multiple-point geostatistics can be applied in a realistic hydrogeological application to determine the impact of complex geological heterogeneity on groundwater flow and transport. The approach is applied to a real aquifer in Belgium that exhibits a complex sedimentary heterogeneity and anisotropy. A training image is constructed based on geological and hydrogeological field data. Multiple-point statistics are borrowed from this training image to simulate hydrofacies occurrence, while intrafacies permeability variability is simulated using conventional variogram-based geostatistical methods. The simulated hydraulic conductivity realizations are used as input to a groundwater flow and transport model to investigate the effect of small-scale sedimentary heterogeneity on contaminant plume migration. Results show that small-scale sedimentary heterogeneity has a significant effect on contaminant transport in the studied aquifer. The uncertainty on the spatial facies distribution and intrafacies hydraulic conductivity distribution results in a significant uncertainty on the calculated concentration distribution. Comparison with standard variogram-based techniques shows that multiple-point geostatistics allow better reproduction of irregularly shaped low-permeability clay drapes that influence solute transport.

Multiple-point statistical prediction on fracture networks at Yucca Mountain

In many underground nuclear waste repository systems, such as Yucca Mountain project, water flow rate and amount of water seepage into the waste emplacement drifts are mainly determined by hydrological properties of fracture network in the surrounding rock mass. Natural fracture network system is not easy to describe, especially with respect to its connectivity which is critically important for simulating the water flow field. In this paper, we introduced a new method for fracture network description and prediction, termed multi-point-statistics (MPS). The process of Multi-point Statistical method is to record multiple-point statistics concerning the connectivity patterns of fracture network from a known fracture map, and to reproduce multiple-scale training fracture patterns in a stochastic manner, implicitly and directly. It is applied to fracture data to study flow field behavior at Yucca Mountain waste repository system. First, MPS method is used to create fracture network with original fracture training image from Yucca Mountain dataset. After we adopt a harmonic and arithmetic average method to upscale the permeability to a coarse grid, THM simulation is carried out to study near-field water flow in surrounding rock of waste emplacement drifts. Our study shows that connectivity or pattern of fracture network can be grasped and reconstructed by Multi-Point-Statistical method. In theory, it will lead to better prediction of fracture system characteristics and flow behavior. Meanwhile, we can obtain variance from flow field, which gives us a way to quantify uncertainty of models even in complicated coupled THM simulation. It indicates that Multi-Point Statistics is a potential method to characterize and reconstruct natural fracture network in a fractured rock mass with advantages of quantifying connectivity of fracture system and its simulation uncertainty simultaneously.

Solution of Multiple-Point Statistics to Extracting Information from Remotely Sensed Imagery

Two phenomena of similar objects with different spectra and different objects with similar spectrum often result in the difficulty of separation and identification of all types of geographical objects only using spectral information. Therefore, there is a need to incorporate spatial structural and spatial association properties of the surfaces of objects into image processing to improve the accuracy of classification of remotely sensed imagery. In the current article, a new method is proposed on the basis of the principle of multiple-point statistics for combining spectral information and spatial information for image classification. The method was validated by applying to a case study on road extraction based on Landsat TM taken over the Chinese Yellow River delta on August 8, 1999. The classification results have shown that this new method provides overall better results than the traditional methods such as maximum likelihood classifier (MLC).

A SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM

The new multiple-point geostatistical algorithm (FILTERSIM), which can handle both categorical and continuous variable training images, is implemented in the SGeMS software. The spatial patterns depicted by the training image are first summarized into a few filter scores; then classified into pattern groups in the filter score space. The sequential simulation approach proceeds by associating each conditioning data event to a closest pattern group using some distance function. A training pattern is then sampled from that group and pasted back onto the simulation grid. Local multiple-point statistics carried by patterns are captured from the training image, and reproduced in the simulation realizations. Hence complex multiple-scale geological structures can be re-constructed in the simulation grid, conditional to a variety of sub-surface data such as well data and seismic survey.

Comparison of indicator kriging, conditional indicator simulation and multiple-point statistics used to model slate deposits

The resources in an ornamental slate deposit can be estimated using geostatistical estimation techniques applied to information collected from drill cores. The result, however, is a smooth approximation that fails to take account of the natural variability in mineralization, which is fundamental to proper design and evaluation of the financial viability of a mining deposit. Geostatistical simulation techniques are more useful in this respect, as they reflect different realizations of the reality and better reflect natural mineral dispersion. In this work, we evaluate the resources of a slate quarry by comparing the results obtained using two geostatistical techniques—indicator kriging (ik) and sequential indicator simulation (sisim)—with the results obtained using the single normal equation simulation (snesim) technique based on multiple-point statistics (mps), analyzing their usefulness in evaluating the financial risk derive from uncertainty in regard to knowledge of the deposit. Our results indicate that although the multiple-point statistics approach produces models that are closer to reality than the models produced by the geostatistical techniques, the simulation relies in part on information obtained via indicator kriging.

Using multiple-point statistics to build geologically realistic reservoir models: the MPS/FDM workflow

Abstract Building geologically realistic reservoir models that honour well data and seismic-derived information remains a major challenge. Conventional variogram-based modelling techniques typically fail to capture complex geological structures while object-based techniques are limited by the amount of conditioning data. This paper presents new reservoir facies modelling tools that improve both model quality and efficiency relative to traditional geostatistical techniques. Geostatistical simulation using Multiple-Point Statistics (MPS) is an innovative depositional facies modelling technique that uses conceptual geological models as training images to integrate geological information into reservoir models. Replacing the two-point statistic variogram with multiple-point statistics extracted from a training image enables to model non-linear facies geobody shapes such as sinuous channels, and to capture complex spatial relationships between multiple facies. In addition, because the MPS algorithm is pixel-based, it can handle a large amount of conditioning data, including many wells, seismic data, facies proportion maps and curves, variable azimuth maps, and interpreted geobodies, thus reducing the uncertainty in facies spatial distribution. Facies Distribution Modelling (FDM) is a new technique to generate facies probability cubes from user-digitized depositional maps and cross-sections, well data, and vertical facies proportion curves. Facies probability cubes generated by FDM are used as soft constraints in MPS geostatistical modelling. They are critical, especially with sparse well data, to ensure that the spatial distribution of the simulated facies is consistent with the depositional facies interpretation of the field. A workflow combining MPS and FDM has been successfully used in Chevron to model important oilfield assets in both shallow- and deep-water depositional environments. Sedimentary environments can be characterized by a succession of deposition of elements, or rock bodies, through time. These elements are traditionally grouped into classes, commonly named ‘depositional facies’, based on their lithology, petrophysical properties, and biological structures. For example, the typical depositional facies encountered in fluvial environments are high permeability sand channels, with levées and crevasse splays, having a more variable range of permeability and net-to-gross ratio, within a background of low permeability shaley facies.

Reconstruction of Existing Reservoir Model for Its Calibration to Dynamic Data

The increase in computer power and the recent developments in history-matching can motivate the reexamination of previously built reservoir models. To save the time of engineers and the CPU time, four distinct algorithms, which allow for rebuilding an existing reservoir model without restarting the reservoir study from scratch, were formulated. The algorithms involve techniques such as optimization, relaxation, Wiener filtering, or sequential reconstruction. They are used to identify a stochastic function and a set of random numbers. Given the stochastic function, the random numbers yield a realization that is close to the existing reservoir model. Once the random numbers are known, the existing reservoir model can be submitted to a new history-matching process to improve the data fit or to account for newly collected data. A practical implementation is presented within the context of facies reservoirs. This article focuses on a previously built facies reservoir model. Although the simulation procedure is unknown to the authors, a set of random numbers are identified so that when provided to a multiple-point statistics simulator, a realization very close to the existing reservoir model is obtained. A new history-matching procedure is then run to update the existing reservoir model and to integrate the fractional flow rates measured in two producing wells drilled after the building of the existing reservoir model.

Multiple-Point Statistics for Training Image Selection

Selecting a training image (TI) that is representative of the target spatial phenomenon (reservoir, mineral deposit, soil type, etc.) is essential for an effective application of multiple-point statistics (MPS) simulation. It is often possible to narrow potential TIs to a general subset based on the available geological knowledge; however, this is largely subjective. A method is presented that compares the distribution of runs and the multiple-point density function from available exploration data and TIs. The difference in the MPS can be used to select the TI that is most representative of the data set. This tool may be applied to further narrow a suite of TIs for a more realistic model of spatial uncertainty. In addition, significant differences between the spatial statistics of local conditioning data and a TI may lead to artifacts in MPS. The utilization of this tool will identify contradictions between conditioning data and TIs. TI selection is demonstrated for a deepwater reservoir with 32 wells.

A library of training images for fluvial and deepwater reservoirs and associated code

Geostatistical algorithms that consider multiple-point statistics are becoming increasingly popular. These methods allow for the reproduction of complicated features beyond the commonly implemented variogram. In practice, it is not possible to infer many multiple-point statistics directly from the available data; therefore, it is common to borrow statistics from training images. A library of training images is developed for fluvial and deepwater depositional settings. These training images are based on object-based models, surface-based models and pseudo-genetic process mimicking (event-based) models. The training images represent a range of net-to-gross fractions and depositional styles. Associated code provides the ability to modify, format and tailor the training images and to extract multiple-point statistics. The training image library provides a source for multiple-point statistics, can be used in comparative flow studies and as an aid in scenario-based uncertainty studies.

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.

Conditional Simulation with Patterns

An entirely new approach to stochastic simulation is proposed through the direct simulation of patterns. Unlike pixel-based (single grid cells) or object-based stochastic simulation, pattern-based simulation simulates by pasting patterns directly onto the simulation grid. A pattern is a multi-pixel configuration identifying a meaningful entity (a puzzle piece) of the underlying spatial continuity. The methodology relies on the use of a training image from which the pattern set (database) is extracted. The use of training images is not new. The concept of a training image is extensively used in simulating Markov random fields or for sequentially simulating structures using multiple-point statistics. Both these approaches rely on extracting statistics from the training image, then reproducing these statistics in multiple stochastic realizations, at the same time conditioning to any available data. The proposed approach does not rely, explicitly, on either a statistical or probabilistic methodology. Instead, a sequential simulation method is proposed that borrows heavily from the pattern recognition literature and simulates by pasting at each visited location along a random path a pattern that is compatible with the available local data and any previously simulated patterns. This paper discusses the various implementation details to accomplish this idea. Several 2D illustrative as well as realistic and complex 3D examples are presented to showcase the versatility of the proposed algorithm.

3D Porosity Modeling of a Carbonate Reservoir Using Continuous Multiple-Point Statistics Simulation

Summary Training images are numerical representations of geological conceptual models that provide prior information on reservoir architecture. A new emerging geostatistical approach named multiple-point statistics (MPS) simulation allows extracting multiple-point structures from such training images and anchoring these structures to the data actually observed in the reservoir. By reproducing multiple-point statistics inferred from training images, MPS enables the modeling of complex curvilinear structures (e.g., sinuous channels) in a much more geologically realistic way than traditional two-point statistics (variogram-based) techniques. However, in the original MPS implementation, all MPS moments computed from the training image are exported to the reservoir model without processing, which allows simulating only categorical or discretized variables. This implementation is appropriate with clastic reservoirs for which, typically, depositional facies are simulated first using MPS, then porosity and permeability are distributed within each simulated facies using traditional variogram-based techniques. But for reservoirs, in particular in carbonate environments, where porosity and permeability trends/cycles are not closely tied to any facies distribution, simulating porosity/permeability directly using corresponding continuous training images appears to be a more suitable approach. In this paper, a new filter-based implementation of MPS, named filtersim, is proposed to reproduce features from continuous variable training images. First, a set of general filters is applied to the training image to transform (summarize) each training pattern into a set of scores accounting for different aspects of the pattern, such as north-south and east-west gradients and curvatures. The training patterns are classified in the score space and grouped into a small number of similarity classes. The simulation consists then of visiting each grid node along a random path, identifying the similarity class that best fits to the local conditioning data, and patching a pattern drawn from that selected similarity class. In our study, this new approach was applied to model the 3D porosity distribution of a carbonate reservoir in Kazakhstan. First, the original "categorical" MPS program snesim was used to model the two main reservoir regions, platform and slope, where the spatial porosity distribution is characterized by significantly different features. Interpreted well markers and seismic data were used to condition the modeling of these two regions. Then porosity was distributed in the platform region using the "continuous" filter-based MPS program filtersim, as described previously. The 3D training images used in that second step displayed porosity trends/cycles controlled by the type of geological sedimentation process believed to have occurred in the reservoir.

Scaling multiple-point statistics to different univariate proportions

Multiple-point statistics are used in geostatistical simulation to improve forecasting of responses that are highly dependent on the reproduction of complex features of the phenomenon. Often, complex features cannot be captured by conventional two-point simulation methods, based on the variogram. Inference of multiple-point statistics requires a training image that depicts the geological features of the geological setting being modelled. The proportions of facies in the training image may not match the target statistics of the final model. This is a problem because taking multiple point statistics from a training image also takes the univariate proportions, that is, the multiple point statistics contain all lower order statistics. There is a need to scale multiple-point statistics to different target univariate proportions. In other cases, locally varying facies proportions must be honored, but a single training image is available. The multiple-point statistics from the training image are scaled to the appropriate target univariate proportions of facies. An iterative scaling approach based on the expression for scaling multiple-point statistics in a purely random case is proposed. The implementation is illustrated through an example where it is shown that the proposed method lies between two extreme cases for a Boolean simulation, namely, the change in size of the objects and the change in their number of occurrences. A second example is presented to illustrate the potential use of this scaling procedure for nonstationary multiple-point geostatistical simulation.

The Necessity of a Multiple-Point Prior Model

Any interpolation, any hand contouring or digital drawing of a map or a numerical model necessarily calls for a prior model of the multiple-point statistics that link together the data to the unsampled nodes, then these unsampled nodes together. That prior model can be implicit, poorly defined as in hand contouring; it can be explicit through an algorithm as in digital mapping. The multiple-point statistics involved go well beyond single-point histogram and two-point covariance models; the challenge is to define algorithms that can control more of such statistics, particularly those that impact most the utilization of the resulting maps beyond their visual appearance. The newly introduced multiple-point simulation (mps) algorithms borrow the high order statistics from a visually and statistically explicit model, a training image. It is shown that mps can simulate realizations with high entropy character as well as traditional Gaussian-based algorithms, while offering the flexibility of considering alternative training images with various levels of low entropy (organized) structures. The impact on flow performance (spatial connectivity) of choosing a wrong training image among many sharing the same histogram and variogram is demonstrated.

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.

Quantifying geological uncertainty for flow and transport modeling in multi-modal heterogeneous formations

In this work, we address the problem of characterizing the heterogeneity and uncertainty of hydraulic properties for complex geological settings. Hereby, we distinguish between two scales of heterogeneity, namely the hydrofacies structure and the intrafacies variability of the hydraulic properties. We employ multiple-point geostatistics to characterize the hydrofacies architecture. The multiple-point statistics are borrowed from a training image that is designed to reflect the prior geological conceptualization. The intrafacies variability of the hydraulic properties is represented using conventional two-point correlation methods, more precisely, spatial covariance models under a multi-Gaussian spatial law. We address the different levels and sources of uncertainty in characterizing the subsurface heterogeneity, and explore their effect on groundwater flow and transport predictions. Typically, uncertainty is assessed by way of many images, termed realizations, of a fixed statistical model. However, in many cases, sampling from a fixed stochastic model does not adequately represent the space of uncertainty. It neglects the uncertainty related to the selection of the stochastic model and the estimation of its input parameters. We acknowledge the uncertainty inherent in the definition of the prior conceptual model of aquifer architecture and in the estimation of global statistics, anisotropy, and correlation scales. Spatial bootstrap is used to assess the uncertainty of the unknown statistical parameters. As an illustrative example, we employ a synthetic field that represents a fluvial setting consisting of an interconnected network of channel sands embedded within finer-grained floodplain material. For this highly non-stationary setting we quantify the groundwater flow and transport model prediction uncertainty for various levels of hydrogeological uncertainty. Results indicate the importance of accurately describing the facies geometry, especially for transport predictions.

On the relationship between indicators of geostatistical, flow and transport connectivity

Connectivity of high conductivity ( K) paths is important because it can lead to channeling, i.e. flow along preferential paths, which can reduce travel times very significantly. Nevertheless, limited effort has been devoted to defining the concept quantitatively. We propose and evaluate nine indicators of connectivity. Three account for the presence of flow connectivity, that is, the flow rate increase caused by preferential flow paths. Two account for the presence of transport connectivity, that is, the existence of fast paths allowing early solute arrival. The remaining four are statistical indicators based on two- and multiple-point statistics. We test these indicators on heterogeneous conductivity fields with different visual connectivity. The indicators of flow connectivity and one of the transport connectivity indicators succeed in identifying the increased presence of connected high- K features. The two-point statistical indicators fail to do so. The directional multi-point statistical indicator performs better. None of the statistical indicators correlate with the flow and transport indicators. We find only weak dependence between the flow and transport indicators. Our results suggest that transport connectivity is much less sensitive to barriers, which control flow connectivity. Instead, transport connectivity appears to be controlled by the existence of narrow, possibly discontinuous high- K paths. Therefore, we conclude that connectivity is a process-dependent concept.

Prediction of permeability for porous media reconstructed using multiple-point statistics

To predict multiphase flow through geologically realistic porous media, it is necessary to have a three-dimensional (3D) representation of the pore space. We use multiple-point statistics based on two-dimensional (2D) thin sections as training images to generate geologically realistic 3D pore-space representations. Thin-section images can provide multiple-point statistics, which describe the statistical relation between multiple spatial locations and use the probability of occurrence of particular patterns. Assuming that the medium is isotropic, a 3D image can be generated that preserves typical patterns of the void space seen in the thin sections. The method is tested on Berea sandstone for which a 3D image from micro-CT (Computerized Tomography) scanning is available and shows that the use of multiple-point statistics allows the long-range connectivity of the structure to be preserved, in contrast to two-point statistics methods that tend to underestimate the connectivity. Furthermore, a high-resolution 2D thin-section image of a carbonate reservoir rock is used to reconstruct 3D structures by the proposed method. The permeabilities of the statistical images are computed using the lattice-Boltzmann method (LBM). The results are similar to the measured values, to the permeability directly computed on the micro-CT image for Berea and to predictions using analysis of the 2D images and the effective medium approximation.

Properties and limitations of sequential indicator simulation

The sequential indicator algorithm is a widespread geostatistical simulation technique that relies on indicator (co)kriging and is applicable to a wide range of datasets. However, such algorithm comes up against several limitations that are often misunderstood. This work aims at highlighting these limitations, by examining what are the conditions for the realizations to reproduce the input parameters (indicator means and correlograms) and what happens with the other parameters (other two-point or multiple-point statistics). Several types of random functions are contemplated, namely: the mosaic model, random sets, models defined by multiple indicators and isofactorial models. In each case, the conditions for the sequential algorithm to honor the model parameters are sought after. Concurrently, the properties of the multivariate distributions are identified and some conceptual impediments are emphasized. In particular, the prior multiple-point statistics are shown to depend on external factors such as the total number of simulated nodes and the number and locations of the samples. As a consequence, common applications such as a flow simulation or a change of support on the realizations may lead to hazardous interpretations.

Pore space reconstruction using multiple-point statistics

The reconstruction of porous media is of great interest in a wide variety of fields, including earth science and engineering, biology, and medicine. To predict multiphase flow through geologically realistic porous media it is necessary to have a three-dimensional (3D) representation of the pore space. Multiple-point statistics were used, based on two-dimensional (2D) thin-sections as training images, to generate 3D pore space representations. The method was borrowed from geostatistical techniques that use pixel-based representations to reproduce large-scale patterns. Thin-section images can provide multiple-point statistics, which describe the statistical relation between multiple spatial locations. Assuming that the medium is isotropic, a 3D image can be generated that preserves typical patterns of the void space seen in the thin sections. The method is tested on Fontainebleau and Berea sandstones for which 3D images from micro-CT scanning are available. The use of multiple-point statistics predicts long-range connectivity of the structures (measured by local percolation probability) better than standard two-point statistics methods. The selection of multiple-point statistics is a key issue and is discussed in detail.