Convergence and numerical simulations of the continuous data assimilation for Biot's poroelasticity system

Real-world applications often suffer from incomplete or uncertain data, which can undermine the accuracy of physical models. In our work, we tackle this challenge by applying continuous data assimilation (CDA) to Biot's poroelasticity system, a model that captures how fluid-saturated porous materials, such as soils or biological tissues, deform under stress. We utilize a nudging term in the conservation of mass equation to integrate sparse observational data directly into the pressure field. We then use an iterative fixed-stress method to decouple the pressure and displacement problem, making the process both stable and efficient. Our approach employs finite element methods along with an interpolation strategy that adapts to different data resolutions. Several numerical tests illustrate that the proposed method reliably drives the model's predictions towards the true state, even in complex scenarios involving heterogeneous permeability or nonlinear permeability, and including a three dimensional setup. This study is the first to rigorously apply CDA to Biot's poroelasticity system, offering a promising avenue for more accurate predictive modeling in fields ranging from geomechanics to biomechanics.