Geostatistics has gained popularity as a quantitative tool to generate multiple geological models, or realizations, that honor a given statistical structure and various types of measured and interpreted data. Two key approaches have traditionally been followed: a pixel based approach such as sequential indicator simulation based on correlation variograms, and an object based (Boolean) approach in which large objects representing geobodies are inserted into the reservoir grid. Both of these approaches, although widely used in the practice have limitations for reproducing geologically realistic volumes, especially when trying to incorporate a user-defined, conceptual geological model. Initial algorithms generate geologically realistic realizations by using these training images to obtain conditional probabilities needed in a stochastic simulation framework. More recent pattern-based geostatistical algorithms attempt to improve the accuracy of the training image pattern reproduction. In these approaches, the training image is used to construct a pattern database. Consequently, sequential simulation will be carried out by selecting a pattern from the database and pasting it onto the simulation grid. One of the shortcomings of the present algorithms is the lack of a unifying framework for classifying and modeling the patterns from the training image. In this research an entirely different approach will be taken towards geostatistical modeling. A novel, principled and unified technique for pattern analysis and generation that ensures computational efficiency and enables a straightforward incorporation of domain knowledge is obtained. The outlook of the workflow is shown below: In the developed methodology, patterns scanned from the training image are represented as points in a Cartesian space using multi-dimensional scaling. The idea behind this mapping is to use distance functions as a tool for analyzing variability between all the patterns in a training image. These distance functions can be tailored to the application at hand. Next, by significantly reducing the dimensionality of the problem and using kernel space mapping, an improved pattern classification algorithm is obtained. An improved pattern continuity and data-conditioning capability is observed in the generated realizations for both continuous and categorical variables. I show how the proposed methodology is much less sensitive to the user-provided parameters, and at the same time has the potential to reduce computational time significantly.
The details of the algorithm and many specific algorithmic issues will not be discussed here. However, two different results are shown to illustrate the effectiveness of the proposed methodology in multiple-point statistics. They all show better pattern reproductivity within the results. The patterns and the features that was inherently provided within a training image are almost perfectly reproduced. The connectivity of the channels or sinusoidal features within the generated realization are well above expectation. This rise to the new era of possibilities in geomodeling and reservoir characterization within petroleum industry. MPS algorithms needs to be programmed so that companies and interested engineers could be able to easily use them. Here, a C++ software implementation of the MPS algorithm, innovated during the PhD, is done. The name of the algorithm is called DisPAT. This algorithm is a pattern-based multiple-point geostatistical routine, where not only is superior to its predecessors in terms of pattern reproductivity and more geologically realistic results, but also, is orders of magnitude faster than them. The goals of DisPAT algorithm is summarized below:
The Stanford Geostatistical Modeling Software (SGeMS) is an open-source computer package for solving problems involving spatially related variables. It provides geostatistics practitioners with a user-friendly interface, an interactive 3-D visualization, and a wide selection of algorithms. Users can perform complex tasks using the embedded Python scripting language, and new algorithms can be developed using the SGeMS plug-in mechanism. SGeMS is the first software to provide algorithms for multiple-point statistics. The SGeMS package provides a versatile toolkit for Earth Sciences graduates and researchers, as well as practitioners of environmental, mining and petroleum engineering. Better and more realistic models should not require an increase in user-set parameters (a disadvantage of filtersim) and a much simpler method should work just as well. I believe the same to be true for pattern analysis. Therefore, in this DisPAT algorithm, a new mathematical framework for modeling patterns of natural images, representing subsurface heterogeneities, was obtained. Distance-based methods for mapping the space of patterns, through the mathematical theory of multi-dimensional scaling, was introduced. In distance-based modeling, many of the tasks usually performed in multiple-point geostatistical algorithms can be carried out in a surprisingly simple yet powerful way. A speed comparison with two different training images is provided below. The first one is simple 2D training image and the second one is a more complex 3D training image. All different MPS algorithms have been tested on the same machine with 2.66 GHz CPU and 2 GB of RAM. These clearly show the superiority of DisPAT over other MPS routines. DisPAT: Distance-based Pattern Simulator Feature Identification and Template Size Selection in Training Images
In order to generate realizations in any geostatistical simulation, some form of statistical prior model is required. For example, in sequential Gaussian simulation, a mean and a variogram alone are sufficient to generate realizations. Similarly, in multiple-point geostatistics, one acquires the statistics from a conceptual training image. However in two-point statistics, the variogram range is implicitly taken into account by the formulation. But in multiple-point modeling, the relevant statistics are implicitly defined through the choice of the size of the template. In probabilistic approaches, such as snesim, this corresponds to the search neighborhood for the calculation of probabilities, and in pattern-based approaches, such as simpat and filtersim, it corresponds to the size of the patterns that are used to generate realizations. So far, selection of an optimal template size, associated with the training image, has been made with cumbersome trial-and-errors by analyzing the reproduction of patterns and large-scale structures in the final simulated realization. In the next section, an algorithm that automatically selects the optimal template size is provided.
A general concept of entropy is used to find an optimal template size. Entropy is a statistical measure of randomness that can be used to characterize the texture of the training image. Modeling Upscaling Errors of Ensembles Using Distance-Based Methods
In order to now learn the model, we also incorporate the permeability fields of both fine-scale and coarse-scale models. However, we will only look at the principal components. For all the models, we will apply PCA on the permeability fields. We assume that this capture significant variability between them. By differencing the important PCAs of fine and coatse models, one is able to define the error in upscaling with respect to the feature seen within the differences, i.e. the high-permeability streaks existing within one part of the model could be causing the errors.
Upscaling techniques are widely used to render flow simulation possible at an affordable cost. There exist a variety of upscaling techniques. Upscaling techniques in general can be classified according to the size of the region on which the flow simulation is performed in order to determine the coarse-scale properties. Purely local methods solve local fine-scale flow problems only on the target coarse-grid block, and local boundary conditions need to be assumed. Extended local methods include some neighboring fine-scale cells to reduce the impact of local boundary conditions. Global upscaling considers global fine-scale flow simulation on the entire global domain and computes the upscaled quantities from the global fine-scale solutions. It eliminates the need of local boundary conditions, but requires flow simulation on the highresolution grid. In between, local-global upscaling procedures incorporates global coarse-scale flow into local upscaling calculations, providing more accurate coarsescale models than local upscaling methods while reducing the computational cost in global upscaling. Figure above shows the MDS space of all fine-scale and coarse-scale models. The red ones show the fine-scale and the yellow coarse-scale. As can be seen there is a systematic error in upscaling. The purpose here is to leanr this error model. However, unlike other algorithms, we also learn exactly where upscaling error is emenating from. In other words, depending on the permeability of the field, or different features within the reservoir model, upscaling technique causes systematic errors within. In order to model this, we use distance-based methods over coarse-scale runs, and then select the most representative set of models that can capture the differences within the responses as widely as possible. This will ensure that the models selected are as diverse as possible. Next, we run fine-scale flow simulation on those mosels and obtain their results. Here is a representation of the selected models responses with respect to the coarse-scale results.
Permeability Anisotropy from Flow Simulation (Well & Interpretation tests)
This project was to obtain the permeability field of the Cerereide field in Montpollier, France. It was an aquifer and the whole process was two-fold: to obtain the permeability field for furthur studies, and to find the systematic methodology on how to accomplish this.
Computer Programming of a Water-Flooding Simulator with Visual Basic In this project a water-flooding simulator was coded using visual basic. This Bachelor thesis was oriented towards calculating the differential equations governing thw fuild flow and coding them in visual basic. the results were compared with commercial simulators and an excellent match were obtained, showing accuracy and speed of the coded algorithm. Journal Paper:Honarkhah, M and Caers, J, 2010, Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling, Mathematical Geosciences, 42: 487 - 517 (best paper award IAMG 09) doi:10.1007/s11004-010-9276-7 MPS Algorithm:DispAT algorithm will be available online shortly. The algorithm comes as a plugin for SGeMS software. Improved functionality, parameter-free learning, extremely fast multiple-point geostatistics simulations are some of the advantages of the DisPAT plugin. |