In this study, we utilize deep neural networks to approximate operators of a nonlinear partial differential equation (PDE), within the Operator-Based Linearization (OBL) simulation framework, and discover the physical space for a physics-based proxy model with reduced degrees of freedom. In our methodology, observations from a high-fidelity model are utilized within a supervised learning scheme to directly train the PDE operators and improve the predictive accuracy of a proxy model. The governing operators of a pseudo-binary gas vaporization problem are trained with a transfer learning scheme. In this two-stage methodology, labeled data from an analytical physics-based approximation of the operator space are used to train the network at the first stage. In the second stage, a Lebesgue integration of the shocks in space and time is used in the loss function by the inclusion of a fully implicit PDE solver directly in the neural network's loss function. The Lebesgue integral is used as a regularization function and allows the neural network to discover the operator space for which the difference in shock estimation is minimal. Our Physics-Informed Machine Learning (PIML) methodology is demonstrated for an isothermal, compressible, two-phase multicomponent gas-injection problem. Traditionally, neural networks are used to discover hidden parameters within the nonlinear operator of a PDE. In our approach, the neural network is trained to match the shocks of the full-compositional model in a 1D homogeneous model. This training allows us to significantly improve the prediction of the reduced-order proxy model for multi-dimensional highly heterogeneous reservoirs. With a relatively small amount of training, the neural network can learn the operator space and decrease the error of the phase-state classification of the compositional transport problem. Furthermore, the accuracy of the breakthrough time prediction is increased therefore improving the usability of the proxy model for more complex cases with more nonlinear physics.

The energy transition is inevitable since approximately two-thirds of the current global GHG emissions are related to energy production. Subsurface can provide a great opportunity for innovative low-carbon energy solutions such as geothermal energy production, hydrogen storage, carbon capture, and sequestration, etc. Well and borehole operations play an important role in all these applications. In order to operate wells intelligently, there must be a robust simulation technology that captures physics and the expected production scenario. In this study, we design a numerical framework for predictive simulation and monitoring of injection and production wells based on the general multi-segment well model. In our simulation model, wells are segmented into connected control volumes similar to the finite-volume discretization of the reservoir. Total velocity serves as an additional nonlinear unknown and it is constrained by the momentum equation. Moreover, transforming nonlinear governing equations for both reservoir and well into linearized equations benefits from operator-based linearization (OBL) techniques and reduce further the computational cost of simulation. This framework was tested for several complex physical kernels including thermal compositional multiphase reactive flow and transport. The proposed model was validated using a comparison with analytic and numerical results.

Simulation of CO₂ utilization and storage (CCUS) in subsurface reservoirs with complex heterogeneous structures requires a model that captures multiphase compositional flow and transport. Accurate simulation of these processes necessitates the use of stable numerical methods that are based on an implicit treatment of the flux term in the conservation equation. Due to the complicated thermodynamic phase behavior, including the appearance and disappearance of multiple phases, the discrete approximation of the governing equations is highly nonlinear. Consequently, robust and efficient techniques are needed to solve the resulting nonlinear system of algebraic equations. In this study, we present a powerful nonlinear solver based on a generalization of the trust-region technique for compositional multiphase flows. The approach is designed to embed a newly introduced Operator-Based Linearization technique and is grounded on the analysis of multi-dimensional tables related to parameterized convection operators. We split the parameter space of the nonlinear problem into a set of trust regions where the convection operators preserve the second-order behavior (i.e., they remain positive or negative definite). We approximate these trust regions in the solution process by detecting the boundary of convex regions via analysis of the directional derivative. This analysis is performed adaptively while tracking the nonlinear update trajectory in the parameter space. The proposed nonlinear solver locally constrains the update of the overall compositions across the boundaries of convex regions. We tested the performance of the proposed nonlinear solver for various scenarios. In many cases, our approach yields an improved behavior of the nonlinear solution in comparison to state-of-the-art solvers.

The energy transition is inevitable since approximately two-third of the current global emission is due to energy production. Subsurface can provide a great opportunity for innovative lowcarbon energy solutions such as geothermal energy production, hydrogen storage, carbon capture, and sequestration, etc. Well and borehole operations play an important role in all these applications. In order to operate wells intelligently, there must be a robust simulation technology that captures physics and the expected production scenario. We design a numerical framework for predictive simulation and monitoring of injection and production wells based on the general multi-segment well model. Our simulation model is based on the general unstructured grid framework in which the wells are segmented similar to finite-volume discretization of reservoir. Total velocity serves as an additional nonlinear unknown and it is constrained with the momentum equation. Moreover, transforming nonlinear governing equations for both reservoir and well into operator form benefits from operator-based linearization (OBL) techniques and reduce further the computational cost related to linearization. This framework was tested for several complex physical kernels including thermal compositional multiphase reactive flow and transport. The proposed model was validated using a comparison with analytic and numerical results.

Simulation of compositional problems in hydrocarbon reservoirs with complex heterogeneous structure requires adopting stable numerical methods that rely on an implicit treatment of the flux term in the conservation equation. The discrete approximation of convection term in governing equations is highly nonlinear due to the complex properties complemented with a multiphase flash solution. Consequently, robust and efficient techniques are needed to solve the resulting nonlinear system of algebraic equations. The solution of the compositional problem often requires the propagation of the displacement front to multiple control volumes at simulation timestep. Coping with this issue is particularly challenging in complex subsurface formations such as fractured reservoirs. In this study, we present a robust nonlinear solver based on a generalization of the trust-region technique to compositional multiphase flows. The approach is designed to embed a newly introduced Operator-Based Linearization technique and is grounded on the analysis of multi-dimensional tables related to parameterized convection operators. We segment the parameter-space of the nonlinear problem into a set of trust regions where the convection operators maintain the second-order behaviour (i.e., they remain positive or negative definite). We approximate these trust regions in the solution process by detecting the boundary of convex regions via analysis of the directional derivative. This analysis is performed adaptively while tracking the nonlinear update trajectory in the parameter-space. The proposed nonlinear solver locally constraints the updating of the overall compositions across the boundaries of convex regions. Besides, we enhance the performance of the nonlinear solver by exploring diverse preconditioning strategies for compositional problems. The proposed nonlinear solution strategies have been validated for both miscible and immiscible gas injection problems of practical interest.