Periodic Reporting for period 1 - MiDiROM (Deep learning enhanced numerical simulations of mixed-dimensional models for subsurface flow)
Reporting period: 2022-01-16 to 2024-01-15
This project focused on using the subsurface as a geo-thermal energy storage site for the green energy transition. The design and operation of such engineering practices is particularly challenging because it is not possible to observe the underground flow systems directly, and we must instead rely on mathematical models. We therefore developed accurate and efficient computational methods to model subsurface flow systems, with a specific focus on the influence of cracks and faults in the rock.
Two main objectives were targeted by the project. First, we developed numerical methods known as reduced order models for flows in fractured rock. Such models can capture the changes in flow due to different material properties and can therefore be used to rapidly simulate multiple scenarios. The second objective was to construct and analyze models for flow systems by using deep learning algorithms. The main challenge in this context was to ensure that the trained neural networks provide flow fields that satisfy physical laws, such as the law of mass conservation. We reached this objective by proposing a solution technique that exploits the underlying mathematical structure of the problem.
Two main objectives were targeted by the project. First, we developed numerical methods known as reduced order models for flows in fractured rock. Such models can capture the changes in flow due to different material properties and can therefore be used to rapidly simulate multiple scenarios. The second objective was to construct and analyze models for flow systems by using deep learning algorithms. The main challenge in this context was to ensure that the trained neural networks provide flow fields that satisfy physical laws, such as the law of mass conservation. We reached this objective by proposing a solution technique that exploits the underlying mathematical structure of the problem.
To model subsurface flows, we investigated a combination of techniques from the fields of mixed-dimensional modeling, reduced order modeling, and trained neural networks. A specific focus was put on the preservation of physical conversation laws at each step of the solving process. As a secondary research theme, we investigated a formulation of Stokes flow which led to a new multi-point vorticity mixed finite element method. We successfully extended these ideas to the Biot equations for poroelasticity, which models the interaction between flow and elasticity in a porous medium. Third, we developed new domain decomposition methods for Stokes-Darcy problems based on flux-mortar methods. Finally, we discovered a connection between the Čech-de-Rham complex and coupled problems such as multi-porosity models.
In parallel with the mathematical aspects of the project, we contributed to an open-source software package where all numerical experiments from the project can be easily reproduced. This code base, called PyGeoN, is designed to construct discretization methods and solvers for mixed-dimensional problems by providing access to the block matrix structure of the problem. The code availability allows for its usage by the scientific community and further developments.
In total, the project has generated seven scientific articles that have been submitted to peer-reviewed journals. Our findings were presented at seven international conferences, including the SIAM Conference on Mathematical and Computational Issues in the Geosciences and Computational Methods in Water Resources. Moreover, we discussed the results with experts in the field at six academic workshops such as SuPreNum: Structure Preserving Numerical Methods and FRAME: Fractured media; numerical methods for fluid flow and mechanics.
In parallel with the mathematical aspects of the project, we contributed to an open-source software package where all numerical experiments from the project can be easily reproduced. This code base, called PyGeoN, is designed to construct discretization methods and solvers for mixed-dimensional problems by providing access to the block matrix structure of the problem. The code availability allows for its usage by the scientific community and further developments.
In total, the project has generated seven scientific articles that have been submitted to peer-reviewed journals. Our findings were presented at seven international conferences, including the SIAM Conference on Mathematical and Computational Issues in the Geosciences and Computational Methods in Water Resources. Moreover, we discussed the results with experts in the field at six academic workshops such as SuPreNum: Structure Preserving Numerical Methods and FRAME: Fractured media; numerical methods for fluid flow and mechanics.
We developed and implemented the first reduced basis method for parameterized mixed-dimensional problems, capable of rapidly generating simulations for multiple scenarios. We then used the underlying differential structure to construct a three-step solution procedure that ensures that the flow field is locally mass conservative, regardless of the accuracy of the reduced basis method. The numerical error from the solver is thus concentrated in the constitutive law, typically Darcy's law, instead. These procedures were then generalized for other constrained partial differential equations and were shown to be effective in combination with deep learning techniques as well. Our mass conservative deep learning methods were significantly more accurate than other techniques when tested on non-linear flow problems.
Secondly, we constructed and analyzed a new mixed finite element method for Stokes flow with only one degree of freedom per facet and cell of the mesh. The same approach was then used to model the displacement and pressure in Biot poroelasticity. Linear convergence was shown theoretically and numerically in both cases, and the methods produce satisfactory results for well-known benchmark problems. Third, we developed domain decomposition methods using MPFA finite volume schemes for Darcy flows and the MAC-scheme for Stokes flow. These schemes, based on flux-mortar methods, were shown to satisfy a priori error estimates and proved efficient in practice on realistic model problems. Finally, we made a new connection between a class of coupled problems and the Čech-de-Rham complex. This perspective provides a new insight into the structure of these problems which helps in their analysis.
Secondly, we constructed and analyzed a new mixed finite element method for Stokes flow with only one degree of freedom per facet and cell of the mesh. The same approach was then used to model the displacement and pressure in Biot poroelasticity. Linear convergence was shown theoretically and numerically in both cases, and the methods produce satisfactory results for well-known benchmark problems. Third, we developed domain decomposition methods using MPFA finite volume schemes for Darcy flows and the MAC-scheme for Stokes flow. These schemes, based on flux-mortar methods, were shown to satisfy a priori error estimates and proved efficient in practice on realistic model problems. Finally, we made a new connection between a class of coupled problems and the Čech-de-Rham complex. This perspective provides a new insight into the structure of these problems which helps in their analysis.