2021/03/02 by Alessio Fumagalli, Fumagalli, Alessio, Francesco S. Patacchini +2
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #cs.NA #math.NA
paper · pdf · doi:10.48550/arxiv.2103.01668
arxiv created 2021/03/02 · openalex publication_date 2021/03/02 · arxiv updated 2021/03/03 · openalex created_date 2021/03/15 · openalex updated_date 2026/07/28
Fractures are normally present in the underground and are, for some physical processes, of paramount importance. Their accurate description is fundamental to obtain reliable numerical outcomes useful, e.g., for energy management. Depending on the physical and geometrical properties of the fractures, fluid flow can behave differently, going from a slow Darcian regime to more complicated Brinkman or even Forchheimer regimes for high velocity. The main problem is to determine where in the fractures one regime is more adequate than others. In order to determine these low-speed and high-speed regions, this work proposes an adaptive strategy which is based on selecting the appropriate constitutive law linking velocity and pressure according to a threshold criterion on the magnitude of the fluid velocity itself. Both theoretical and numerical aspects are considered and investigated, showing the potentiality of the proposed approach. From the analytical viewpoint, we show existence of weak solutions to such model under reasonable hypotheses on the constitutive laws. To this end, we use a variational approach identifying solutions with minimizers of an underlying energy functional. From the numerical viewpoint, we propose a one-dimensional algorithm which tracks the interface between the low- and high-speed regions. By running numerical experiments using this algorithm, we illustrate some interesting behaviors of our adaptive model on a single fracture and small networks of intersecting fractures.