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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.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractOur interest lies in extending the streamline method compositional simulation. In this paper we develop improved mappings to and from streamlines that are necessary to obtain reliable predictions of gas injection processes. Our improved mapping to streamlines utilizes a piecewise linear representation of saturations on the background grid in order to minimize numerical smearing. Our strategy for mapping saturations from streamlines to the background grid is based on Kriging. We test our improvements to the streamline method using a simple model for miscible flooding based on incompressible, Darcy flow. Results indicate that our mappings offer improved resolution and reduce mass balance errors relative to the standard mappings. Our mappings also require fewer streamlines to achieve a desired level of accuracy. In compositional cases where the computational cost of a streamline solve is high, we anticipate that this will lead to significant improvement in the efficiency of streamlinebased simulation.

Capacitance-Resistance (CR) models have received renewed interest in the past few years as a fast alternative to reservoir simulation to model and predict complex water or gas floods in mature reservoirs. Using an analogy between reservoirs and electrical systems, CR models represent the interactions between wells through analytical solutions to an equivalent capacitor-resistor circuit. CR models do not require a geologic model and can be built with only production and injection data. When modeling fields with numerous wells and a long history, traditional reservoir simulation workflows are extremely time-consuming. The simplicity of CR models make them extremely attractive to quickly model and predict the behavior of these complex reservoirs. Current CR models are able to represent accurately the behavior of reservoirs under strong water or gas floods, where the injection is the main driving mechanism for production. In such cases, the production rates are strongly correlated to the injection rates and CR model are ideally suited to decipher these interactions. However, most reservoirs start with a period of primary depletion or many are exploited under a weak injection strategy, for which CR models are not ideally suited. Here, we propose to combine decline-curve (DC) analysis with a CR model in order to solve this shortcoming. Using the superposition principle, the contribution of primary depletion to production is represented by DC and the contribution of injection is represented by the CR model. After presenting the formulation and implementation of our DC-CR model, we demonstrate its performance on a deep naturally fractured carbonate reservoir under hydrocarbon gas and nitrogen injection. The reservoir has over 30 years of production history: 23 years of primary depletion and 8 years of gas and nitrogen injection. Using a one-year blind test, we demonstrate that the model is able to accurately predict the reservoir behavior.