Pattern-Based History Matching Maintains Consistency for Complex-Facies Reservoirs
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A challenging problem of automated history-matching work flows is ensuring that, after applying updates to previous models, the resulting history-matched models remain consistent geologically. This is particularly challenging in formations with complex connectivity patterns. In this work, the authors introduce a novel machine-learning approach with the aim of preserving the main connectivity patterns of previous reservoir models during history matching of complex geologic formations.
The authors introduce a machine-learning algorithm to incorporate discrete connectivity patterns in history matching of complex-geologic-facies models. This is achieved by splitting the introduced history-matching optimization problem into two iterative subproblems: a continuous approximation of the solution that is obtained by solving a regularized least-squares inversion (while maintaining the expected connectivity of the patterns) followed by a machine-learning-based mapping of the continuous solution to the discrete feasible set defined by previous models. The second step involves a machine-learning approach that uses offline training to implement the mapping. The offline learning process uses the k-nearest neighbor (k‑NN) algorithm to construct local pattern (feature) vectors and compare them with the feature vectors in the training data set. For each spatial template, the feature vectors with the smallest distance in the learning data set are selected and their corresponding label vectors (i.e., multivariate discrete patterns) are identified and stored. Once all local patterns are scanned and processed using a defined template size, an aggregation step is applied on the overlapping templates to incorporate the multipoint statistics patterns collectively in assigning discrete labels to each gridblock.
History Matching With Facies-Feasibility Constraint. An indirect method is developed for solving the constrained minimization problem by defining a two-step alternating-directions method. In this approach, the first step uses a standard gradient-based method to find an approximate continuous solution to the problem and the second step maps the resulting continuous solution onto the feasible set while ensuring that the updated solution remains close to the continuous solution from the first step. These two steps are repeated until no further improvement in the data match is obtained. A machine-learning algorithm is used to implement the mapping in the second step.
Mapping Onto the Feasible Set. The authors use a supervised-learning approach to implement the mapping in Step 2 of the solution approach. The k-NN method is implemented in this paper as a simple classification technique in which the decision about the label of a feature vector is based on a distance measure defined between the feature vector and the learning data set. Using a predefined distance measure, the algorithm selects feature vectors with minimum distances from the training data set and assigns the most-frequent label corresponding to these feature vectors as the output label.
The authors incorporate the statistical information in the previously mentioned data on the basis of local pattern-matching, which is implemented in two stages. In the first stage, using a specified template size, the given continuous image is scanned and the corresponding feature vector (local patterns inside the defined template) is computed.
In the second step, an aggregation approach is adopted to combine all the stored facies instances in each model cell to represent the conditional distribution for the cells. The facies type in each cell is then taken to be the discrete facies with the highest frequency. The resulting map is passed to Step 1 to continue the iterations. The feature vectors do not need to be the spatial descriptions of the patterns and could be defined on the basis of various factors, including computation. For instance, one class of feature vectors can be the projected coefficients of the patterns onto a user-defined subspace. The feature vector could also be defined by considering a subset of grid cells within the template (either randomly or deterministically) to improve the computational complexity.
In the learning stage, a circular template is used to scan previous image realizations to generate corresponding segments of the continuous and discrete input images. In this work, the feature vectors are created by extracting the exact gridblock values inside the neighborhood defined by the templates.
Fig. 1 presents a schematic of the first stage of the segmentation procedure. The classification approach, which is based on the k-NN classifier, explores the feature space in the training data set to find the feature vectors that have the smallest distance to each scanned portion and stores the corresponding discrete label vectors for the scanned region.
After the initial classification and storing of the label vectors, an aggregation (or voting) step is used to combine all the facies assignments to each cell and decide about the discrete facies type for each gridblock. In the aggregation approach for a given cell, the information from scanning different regions is combined to generate the conditional facies probability. While alternative methods can be applied to assign facies types on the basis of the stored labels, in this paper, the facies with the highest frequency is assigned to each gridblock. The aggregation step ensures that facies assignment to each gridblock accounts for the spatial statistics (and connectivity patterns) from an extended neighborhood (approximately twice the size of the template in each direction). Moreover, aggregation leads to far more samples for each grid cell, which increases the accuracy and spatial consistency of the method substantially. Aggregation improves the connectivity by exploiting additional spatial information about the extended local neighborhood. Furthermore, when aggregation is used, less sensitivity is observed because many overlapping templates cover the same cell in the model.
The authors present two numerical experiments to examine the performance of the developed approach for inference of rock-facies distribution. The first example is a straight-ray travel-time tomographic inversion, which leads to a linear inverse problem and is used to demonstrate the performance of the proposed method. In the second experiment, a two-phase-flow history-matching problem is considered wherein a 3D facies map is estimated from pressure and flow-rate data. In these examples, a random subsampling approach by scanning only 20% of the gridblocks is used for implementing the k-NN classification step.
In history-matching problems, ensuring that the solutions are plausible geologically can be a difficult constraint to enforce when the expected connectivity patterns are complex and hard to preserve during model updating. For instance, updating meandering fluvial channels to match the observed data, while preserving their shape and connectivity pattern, is difficult to achieve using classical history-matching techniques.
In this paper, machine-learning techniques were used to develop a framework for automatic history matching of complex facies models while maintaining their complex connectivity patterns. Specifically, a feasible set for model parameters was defined that must be observed during model updating. By use of examples from fluvial systems, the feasible set was described with a large number of previous model realizations that summarize the expected spatial statistics of the solution. To implement the feasibility constraint, a regularized least-square formulation was presented in which the regularization imposes a complex feasibility constraint. Using an alternative-directions method of optimization, the authors split the objective function into two sequential optimization subproblems. In the first problem, a continuous model calibration was solved to use dynamic flow data to infer the approximate connectivity patterns without honoring the feasibility constraint. In the second step, this solution was mapped onto the feasible set by using the k-NN algorithm as a supervised machine-learning technique. By use of a series of increasingly complex numerical experiments, the performance of the proposed approach in honoring complex previous models was established.
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Pattern-Based History Matching Maintains Consistency for Complex-Facies Reservoirs
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10 April 2019
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