Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called physics-informed attention-based neural networks (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside the convex hull of the training set.
Full history match models in subsurface systems are challenging due to the large number of reservoir simulations required, and the need to preserve geological realism in matched models. This drawback increases significantly in big real fields due to the high heterogeneity of the geological models, the reservoir simulation computational time (which increases superlinearly). In this work, we propose a novel framework based on artificial intelligence to address these shortcomings. Our workflow is based on two main components: The first is the new combination of model order reduction techniques (e.g., principle component analysis (PCA), kernel-PCA (k-PCA)) and artificial intelligence for parameterizing complex three-dimensional (3D) geomodels, called "Geo-Net". Our new approach is able to create complex high dimensional heterogeneous reservoirs in seconds, providing better correspondence with the underlying geomodels, hard-data constraints and geological plausibility. The second component is a derivative-free optimization framework to complete the automatic history matching (AHM). This new approach allows us to perform local changes in the reservoir at the same time as we conserve geological plausibility. We have examined our methodology in a real field in Colombia. The Rubiales Oil Field is located in the Llanos Basin with original oil in place of around 6 billion barrels. The key finding here is that the Geo-Net is able to recreate the full geological workflow obtaining the same high order of statistics as traditional geo-statistical techniques. Nonetheless, our Geo-Net allows us to control the full process with a low-dimensional vector and reproduces the full geological workflow 10,000 times faster than commercial geo-statistical packages. Finally, the full optimization workflow has been applied to AHM. Results show an improvement with respect to best practice of traditional history match workflows.
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