| Abstract: |
| We study the motion of miscible liquids in porous media. Injection of a less viscous fluid into a more viscous one produces an instability known as viscous fingering. This phenomenon is described by a multidimensional PDE system consisting of mass conservation, incompressibility, and Darcy law. In heterogeneous media, the displacement pattern forms a regular structure. In homogeneous media, it is more chaotic, and displays an intriguing regime of intermediate concentration.
To improve estimates for the mixing-zone size, we study a related model of gravity-driven fingering based on the incompressible porous media equation with diffusion. We represent the medium as a system of vertical tubes. For the simplest case of 2 tubes we were able to find cascades of travelling waves [1]. For the case of infinitely many tubes we demonstrate the existence of an infinite cascade of travelling waves. Speed of travelling waves could be interpreted as speed of viscous fingers and back front propagation. The result in the simplified model suggests that existing estimates for original multi-dimensional problems could be improved.
[1] Petrova Yu., Tikhomirov S., Efendiev Ya. Propagating terrace in a two-tubes model of gravitational fingering. SIAM Journal on Mathematical Analysis, 57 (2025), 30-64. |
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