Model adaptation in a discrete fracture network: existence of solutions and numerical strategies

TL;DR

This paper introduces an adaptive model for fluid flow in fractures that dynamically selects the appropriate physical regime based on velocity thresholds, supported by theoretical existence proofs and a novel numerical algorithm.

Contribution

It presents a new adaptive strategy for modeling flow regimes in fractures, with proven existence of solutions and a one-dimensional algorithm for interface tracking.

Findings

Existence of weak solutions established for the model.

The proposed algorithm effectively tracks flow regime interfaces.

Numerical experiments demonstrate the model's behavior in fracture networks.

Abstract

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.