Well-posedness and Fingering Patterns in $A + B \rightarrow C$ Reactive Porous Media Flow
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arXiv
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| Format: | Preprint |
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2025
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| author | Kundu, Sahil Maharana, Surya Narayan Mishra, Manoranjan |
| author_facet | Kundu, Sahil Maharana, Surya Narayan Mishra, Manoranjan |
| contents | The convection-diffusion-reaction system governing incompressible reactive fluids in porous media is studied, focusing on the \( A + B \to C \) reaction coupled with density-driven flow. The time-dependent Brinkman equation describes the velocity field, incorporating permeability variations modeled as an exponential function of the product concentration. Density variations are accounted for using the Oberbeck-Boussinesq approximation, with density as a function of reactants and product concentration. The existence and uniqueness of weak solutions are established via the Galerkin approach, proving the system's well-posedness. A maximum principle ensures reactant nonnegativity with nonnegative initial conditions, while the product concentration is shown to be bounded, with an explicit upper bound derived in a simplified setting. Numerical simulations are performed using the finite element method to explore reactive fingering instabilities and illustrate the effects of density stratification, differential product mobility, and two- or three-dimensionality. Two cases with initial flat and elliptic interfaces further demonstrate the theoretical result that solutions continuously depend on initial and boundary conditions. These theoretical and numerical findings provide a foundation for understanding chemically induced fingering patterns and their implications in applications such as carbon dioxide sequestration, petroleum migration, and rock dissolution in karst reservoirs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17943 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness and Fingering Patterns in $A + B \rightarrow C$ Reactive Porous Media Flow Kundu, Sahil Maharana, Surya Narayan Mishra, Manoranjan Analysis of PDEs The convection-diffusion-reaction system governing incompressible reactive fluids in porous media is studied, focusing on the \( A + B \to C \) reaction coupled with density-driven flow. The time-dependent Brinkman equation describes the velocity field, incorporating permeability variations modeled as an exponential function of the product concentration. Density variations are accounted for using the Oberbeck-Boussinesq approximation, with density as a function of reactants and product concentration. The existence and uniqueness of weak solutions are established via the Galerkin approach, proving the system's well-posedness. A maximum principle ensures reactant nonnegativity with nonnegative initial conditions, while the product concentration is shown to be bounded, with an explicit upper bound derived in a simplified setting. Numerical simulations are performed using the finite element method to explore reactive fingering instabilities and illustrate the effects of density stratification, differential product mobility, and two- or three-dimensionality. Two cases with initial flat and elliptic interfaces further demonstrate the theoretical result that solutions continuously depend on initial and boundary conditions. These theoretical and numerical findings provide a foundation for understanding chemically induced fingering patterns and their implications in applications such as carbon dioxide sequestration, petroleum migration, and rock dissolution in karst reservoirs. |
| title | Well-posedness and Fingering Patterns in $A + B \rightarrow C$ Reactive Porous Media Flow |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.17943 |