A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model

We study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is employed to impose weakly this condition. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. A series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the sensitivity of the model with respect to its parameters

Mathematical and Numerical Modeling of Fluid-Poroelastic Structure Interaction

The focus of this thesis is on finite element computational models for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. We assume that the free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. We further impose equilibrium and kinematic conditions along the interface between two regions. As we employ the mixed Darcy formulation, continuity of flux condition becomes of the essential type and we use a Lagrange multiplier method to impose weakly this condition. The thesis consists of three major parts. First, we investigate a Lagrange multiplier method for the linear Stokes--Biot model under the assumption of Newtonian fluid. We perform a stability and error analysis for the semi-discrete continuous-in-time and the fully discrete formulations, that indicate optimal order of convergence. We proceed with performing a series of numerical experiments, designed to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the sensitivity of the model with respect to its parameters. In the second part, we present a nonlinear extension of the model, applicable to modeling non-Newtonian fluids. More precisely, we focus on the quasi-Newtonian fluids that exhibit a shear-thinning property. We establish existence and uniqueness of the solution of two alternative formulations of the proposed method in both fully continuous and semi-discrete continuous-in-time settings, and derive the error bounds for the formulation that appears more appealing from the computational point of view. We conclude with numerical tests, verifying theoretical findings and illustrating behavior of the method. Lastly, we discuss coupling of the Stokes--Biot model with an advection--diffusion equation for modeling transport of chemical species within the fluid, which we discretize using the non-symmetric interior penalty Galerkin method. We discuss the stability and convergence properties of the scheme, and provide extensive numerical studies showing applicability of the method to modeling fluid flow in an irregularly shaped fractured reservoir with physical parameters

A nonlinear Stokes–Biot model for the interaction of a non-Newtonian fluid with poroelastic media

We develop and analyze a model for the interaction of a quasi-Newtonian free fluid with a poroelastic medium. The flow in the fluid region is described by the nonlinear Stokes equations and in the poroelastic medium by the nonlinear quasi-static Biot model. Equilibrium and kinematic conditions are imposed on the interface. We establish existence and uniqueness of a solution to the weak formulation and its semidiscrete continuous-in-time finite element approximation. We present error analysis, complemented by numerical experiments