Distributed parameter processes are challenging when it comes to modeling and control. Steam assisted gravity drainage (SAGD), used for in-situ extraction and recovery of oil sands bitumen, is a large scale distributed parameter process. The analysis of control relevant properties like controllability and observability enables to address the problem of control of steam chamber growth and sensor placement. We present a data driven and computationally affordable technique to assess the controllability and observability of the SAGD steam chamber dynamics in a structural perspective by exploiting the underlying interaction amongst different regions of the reservoir. A reservoir simulator is used to gather the data, and density-based clustering combined with Granger causality is used to develop a directed graph through which the structural controllability and observability of the SAGD process is characterized. This paper presents a detailed procedure and results for the sensor and actuator locations for partial and full controllability and observability to validate the discussed approach using the data acquired from the CMG-STARS simulator.
Algebraic connectivity is one way to quantify graph connectivity, which in turn gauges robustness as a network. In this paper, we consider the problem of maximizing algebraic connectivity both locally and globally overall simple, undirected, unweighted graphs with a given number of vertices and edges. We pursue this optimization by equivalently minimizing the largest eigenvalue of the Laplacian of the 'complement graph'. We establish that the union of complete subgraphs are largest eigenvalue local minimizer graphs. Further, under sufficient conditions satisfied by the edge/vertex counts, we prove that this union of complete components graphs are, in fact, Laplacian largest eigenvalue global maximizers; these results generalize the ones in the literature that are for just two components. These sufficient conditions can be viewed as quantifying situations where the component sizes are either 'quite homogeneous' or some of them are relatively 'negligibly small,' and thus generalize known results of homogeneity of components. While a conjecture about global optimality of complete bipartite graphs' from the literature continues to remain open, assuming appropriate constraints we prove the conjecture and also formulate/prove a variant of this claim. We finally relate this central optimization problem in this paper with the Discrete Fourier Transform (DFT) and circulant graphs/matrices.
Algebraic connectivity is one way to quantify graph connectivity, which in turn gauges robustness as a network. In this paper, we consider the problem of maximising algebraic connectivity both local and globally over all simple, undirected, unweighted graphs with a given number of vertices and edges. We pursue this optimization by equivalently minimizing the largest eigenvalue of the Laplacian of the 'complement graph'. We establish that the union of complete subgraphs are largest eigenvalue "local" minimizer graphs. Further, under sufficient conditions satisfied by the edge/vertex counts we prove that this union of complete components graphs are, in fact, Laplacian largest eigenvalue "global" maximizers; these results generalize the ones in the literature that are for just two components. These sufficient conditions can be viewed as quantifying situations where the component sizes are either 'quite homogeneous' or some of them are relatively 'negligibly small', and thus generalize known results of homogeneity of components. We finally relate this optimization with the Discrete Fourier Transform (DFT) and circulant graphs/matrices.
A closed-loop optimal control strategy is extensively used in all engineering disciplines to achieve desired control while maximizing performance. Balancing the economics and safety in petroleum reservoirs calls for a closed-loop control scheme in its operation. With the objective of maximizing the oil production rate (OP) and tracking the factor of safety (FoS) within the safety limit, a stochastic optimization-based model predictive controller (MPC) formulation is proposed in this paper. In this work, we build a deterministic proxy model for the OP and polynomial chaos expansion (PCE)-based model for the FoS with the well bottom hole pressure (well BHP) as the inputs using the CMG-STARS and FLAC3D simulator data. These models are used to formulate the MPC to determine the optimal dynamic well bottom hole pressure (MOP). Results are compared with a fixed MOP and to a static measure directly based on the PCE model of the FoS.
Abstract Reservoir modelling tools have played a significant role in designing the subsurface fluid injection, such as CO2 enhanced oil recovery (EOR). However, these models are computationally expensive; they require extensive geological and engineering data that often are not available in the early phase of carbon utilization and storage projects. This work presents the development of fast predictive models and optimization methodologies to quickly evaluate the CO2 EOR and storage operations in mature oil fields. Considerable experience with CO2 EOR and storage has been gained by the petroleum industry. In particular, the Weyburn-Midale project (Canada) is a comprehensive case to show how an oil reservoir could securely store CO2. Employing the Weyburn-Midale project, we developed, trained and tested several types of proxy models in multiple scenarios to assess the performance of the miscible CO2 flood in recovering residual oil, increasing the ultimate oil recovery factor while maximizing the permanent CO2 storage. The history matching of the Weyburn-Midale CO2 EOR model involved 216 well histories (producers and injectors) from 1964 to 2006 using a compositional reservoir simulator. The predominant exploitation scheme was based on an inverted nine-spot pattern waterflooding, water alternating CO2, and consequently CO2 injection. Two simulation data sets were employed at different periods of 1956 through 2006, and 2007 through 2025. Among several proxy models, an artificial neural network (ANN) model proved to accurately estimate features of interest, namely fluid production (oil, water, gas), fluid injection (water, CO2) and the amount of CO2 stored in the reservoir. Additionally, an autoregressive exogenous input (ARX) model was implemented to predict the future outputs in response to a future input. Inspection of the relative estimation error and the model fitness score showed that the proxy model was capable of rapidly reproducing the trend in the validation set satisfactorily. Lastly, we evaluated the transfer of learning from a proxy model, trained to the Weyburn-Midale field (Canada), to assess the performance of CO2 EOR in another mature oil reservoir in Europe (Romania). The application of proxy models under geological and operation uncertainties offers huge reduction in computational time and engineering data requirements. The results from the Weyburn-Midale case study deliver critical insights into the analysis of many process factors and modeling techniques intended to assess the economic limits and long-term performance of CO2 EOR and storage in mature oil fields.
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