Periodic Reporting for period 1 - MulPHys (A multiscale approach to unsaturated flow in porous media with Preisach hysteresis)
Période du rapport: 2024-02-01 au 2026-01-31
Experimental evidence indicates that fluid filtration through unsaturated porous media exhibits a hysteretic behavior, originating at a microscopic level from surface tension at the point of contact between water and air in the pores. As a result, the pressure-saturation constitutive relation turns out to be of hysteresis type, accurately described with the Preisach operator by a thorough fitting procedure.
The main objective of the MulPHys project is to expand the knowledge about Preisach hysteresis for fluid filtration and build new mathematical models for unsaturated porous media, employing a multiscale approach.
The suitability of the Preisach operator in describing the hysteretic behavior of unsaturated porous solids sparked an intense research effort in the community of experts in PDEs with hysteresis, with the goal of including the Preisach operator in mathematical models. In these endeavors, the presence of a microstructure was neglected, with the approach being directly macroscopic. Another branch of research has considered porous media as objects with a microstructure, and has derived the macroscopic description of fluid flow from local behavior. In this body of research, however, porous media are often assumed to be completely saturated so that no hysteresis can occur.
With MulPHys, we will fill the gap between these two research areas, employing multiscale techniques to provide a justification of the Preisach operator as the correct tool for describing filtration. Particular attention will be paid to including gravity effects and understanding solid-liquid interactions at the microscopic level. Numerical simulations and experimental data will be instrumental in achieving these objectives.
The main objective of the MulPHys project is to expand the knowledge about Preisach hysteresis for fluid filtration and build new mathematical models for unsaturated porous media, employing a multiscale approach.
The suitability of the Preisach operator in describing the hysteretic behavior of unsaturated porous solids sparked an intense research effort in the community of experts in PDEs with hysteresis, with the goal of including the Preisach operator in mathematical models. In these endeavors, the presence of a microstructure was neglected, with the approach being directly macroscopic. Another branch of research has considered porous media as objects with a microstructure, and has derived the macroscopic description of fluid flow from local behavior. In this body of research, however, porous media are often assumed to be completely saturated so that no hysteresis can occur.
With MulPHys, we will fill the gap between these two research areas, employing multiscale techniques to provide a justification of the Preisach operator as the correct tool for describing filtration. Particular attention will be paid to including gravity effects and understanding solid-liquid interactions at the microscopic level. Numerical simulations and experimental data will be instrumental in achieving these objectives.
A1. Building upon the "convexification method" for solving degenerate diffusion equations with a Preisach hysteresis operator, previously established by the fellow and the supervisor, the core activity during the project focused on extending this framework to capture more complex physical realities of porous media flow. The objective was the generalization of the existence theory to incorporate additional physical effects into the governing system:
- Inclusion of gravity effects (convective term).
- Incorporation of deformations (coupled mechanical systems).
- Extension to nonlinear diffusion (e.g. p-Laplacian type flux) to model non-Darcy flow.
The existence of weak solutions was established by combining the convexification technique with novel anisotropic embedding theorems in Orlicz spaces. The primary achievement was the successful extension of the convexification method to solve complex, coupled, and possibly degenerate systems.
A2. Another activity involved the Richards equation, focusing on the competitive interaction between two driving forces: capillarity and gravity. This involved applying advanced techniques from the theory of degenerate parabolic partial differential equations (PDEs) to establish an explicit functional relationship that governs the competition between the two driving forces. Using the method of traveling wave supersolutions, we computed an explicit upper bound for the solution’s support and provided a purely analytical characterization of the wetting front's evolution. Work included a rigorous mathematical derivation of analytical upper and lower asymptotic bounds for the speed of the moisture front propagation. The main achievement was the derivation of an explicit, rigorous criterion that describes the competition between gravity and capillarity, determining the asymptotic behavior of the moisture profile.
A critical phase of this activity involved transitioning from theoretical analysis to practical simulation. The primary task was the development and implementation of robust numerical algorithms combining time discretization and finite element method. The numerical simulations successfully reproduced and confirmed the behavior predicted by the model, particularly in describing the competition between transport and diffusion.
A3. A parallel activity involved addressing fundamental challenges in the mathematical analysis of functions defined on domains with complex topologies, which are often encountered when studying porous media at multiple scales. The specific activity was the construction of suitable extensions for manifold-valued Sobolev functions defined on perforated domains, with the aim of preserving certain constraints (the "manifold”) while maintaining boundedness properties independently of the scale of the microperforations.
- Inclusion of gravity effects (convective term).
- Incorporation of deformations (coupled mechanical systems).
- Extension to nonlinear diffusion (e.g. p-Laplacian type flux) to model non-Darcy flow.
The existence of weak solutions was established by combining the convexification technique with novel anisotropic embedding theorems in Orlicz spaces. The primary achievement was the successful extension of the convexification method to solve complex, coupled, and possibly degenerate systems.
A2. Another activity involved the Richards equation, focusing on the competitive interaction between two driving forces: capillarity and gravity. This involved applying advanced techniques from the theory of degenerate parabolic partial differential equations (PDEs) to establish an explicit functional relationship that governs the competition between the two driving forces. Using the method of traveling wave supersolutions, we computed an explicit upper bound for the solution’s support and provided a purely analytical characterization of the wetting front's evolution. Work included a rigorous mathematical derivation of analytical upper and lower asymptotic bounds for the speed of the moisture front propagation. The main achievement was the derivation of an explicit, rigorous criterion that describes the competition between gravity and capillarity, determining the asymptotic behavior of the moisture profile.
A critical phase of this activity involved transitioning from theoretical analysis to practical simulation. The primary task was the development and implementation of robust numerical algorithms combining time discretization and finite element method. The numerical simulations successfully reproduced and confirmed the behavior predicted by the model, particularly in describing the competition between transport and diffusion.
A3. A parallel activity involved addressing fundamental challenges in the mathematical analysis of functions defined on domains with complex topologies, which are often encountered when studying porous media at multiple scales. The specific activity was the construction of suitable extensions for manifold-valued Sobolev functions defined on perforated domains, with the aim of preserving certain constraints (the "manifold”) while maintaining boundedness properties independently of the scale of the microperforations.
The main results for activities A1, A2, and A3 from the previous section are:
R1. This body of work successfully moves beyond the limitations of play-type models by developing an existence theory for systems using the Preisach operator, which provides a rigorous mathematical guarantee for the boundedness of the saturation variable. This required overcoming standard existence theory to prove the well-posedness of strongly degenerate systems with hysteresis that are possibly coupled with gravity effects and mechanical deformations. This generalization addresses a long-standing gap in the state of the art for modeling unsaturated porous media and provides the necessary mathematical foundation for realistic, fully coupled modeling of moisture transport and soil mechanics.
R2. In contrast to purely numerical treatments, our work offers a rigorous analytical perspective rooted in the equation’s structure to determine the effects of the competition between gravity and capillarity. A precise understanding of the propagation speed of the moisture front — especially under the influence of gravity — is critical for predicting drainage and infiltration patterns.
The developed code and simulation results provide a verified computational tool for studying moisture front dynamics under realistic conditions, directly linking the analytical findings to quantitative simulation data.
R3. This work provides essential mathematical tools for the challenging field of homogenization, that is, the process of deriving macroscale (effective) equations from microscale physics in a porous structure. Specifically, starting from the microscopic description, our approach provides a straightforward way to preserve certain constraints (the "manifold”) in the limiting (macroscopic) model. This ensures the foundational mathematical rigor behind the modeling effort, addressing analytical challenges that underpin the well-posedness of equations used in porous media modeling.
R1. This body of work successfully moves beyond the limitations of play-type models by developing an existence theory for systems using the Preisach operator, which provides a rigorous mathematical guarantee for the boundedness of the saturation variable. This required overcoming standard existence theory to prove the well-posedness of strongly degenerate systems with hysteresis that are possibly coupled with gravity effects and mechanical deformations. This generalization addresses a long-standing gap in the state of the art for modeling unsaturated porous media and provides the necessary mathematical foundation for realistic, fully coupled modeling of moisture transport and soil mechanics.
R2. In contrast to purely numerical treatments, our work offers a rigorous analytical perspective rooted in the equation’s structure to determine the effects of the competition between gravity and capillarity. A precise understanding of the propagation speed of the moisture front — especially under the influence of gravity — is critical for predicting drainage and infiltration patterns.
The developed code and simulation results provide a verified computational tool for studying moisture front dynamics under realistic conditions, directly linking the analytical findings to quantitative simulation data.
R3. This work provides essential mathematical tools for the challenging field of homogenization, that is, the process of deriving macroscale (effective) equations from microscale physics in a porous structure. Specifically, starting from the microscopic description, our approach provides a straightforward way to preserve certain constraints (the "manifold”) in the limiting (macroscopic) model. This ensures the foundational mathematical rigor behind the modeling effort, addressing analytical challenges that underpin the well-posedness of equations used in porous media modeling.