Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model

TL;DR

This paper develops and analyzes a continuous data assimilation algorithm for the 3D Brinkman-Forchheimer-extended Darcy model, improving state estimates by integrating measurements through a feedback control mechanism.

Contribution

It introduces a novel data assimilation algorithm tailored for the 3D BFeD model and provides theoretical convergence analysis for the method.

Findings

Existence and uniqueness of solutions for the 3D BFeD system.

Convergence of the data assimilation algorithm.

Enhanced state estimation accuracy.

Abstract

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtaining improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the setting and convergence analysis of the data assimilation algorithm.