Physical Sciences and Mathematics Commons^™

A Heterogeneous Multiscale Method For Poroelasticity, Paul M. Delgado

Open Access Theses & Dissertations

In this Thesis, we develop and analyze a heterogeneous multiscale model for coupled fluid flow and solid deformation in porous media based on operator splitting and finite volume method. The splitting method results in two elliptic multiscale PDE's in the form of a reaction diffusion equation and a linear elasticity equation. We extend our previous multiscale method from 1D to higher dimensions and develop new approaches for the inclusion of mixed boundary conditions and source terms. We derive an error estimate for our multiscale method and analyze the stability of our splitting method. We also test the effectiveness of our …

A Block Precondtioner For A Mixed Finite Element Method For Biot;S Equations, Maranda Lee Bean

Open Access Theses & Dissertations

In this Thesis, we explore the solution methods for the linear system resulting from a mixed finite element method applied to the Biot's consolidation model. This model describes the coupled interactions between a porous solid and the fluid contained within it. Specifically, we use a method developed by Yi [Numer. Methods for PDEs, 29(5), pp. 1749-1777] that expands Biot's system to include fluid pressure, solid displacement, fluid flux and total stress as primary unknowns.

As the resulting linear system is a large, sparse, saddle point system, we attempt to solve this system via a Schur complement preconditioned iterative method. Using …

A Block Operator Splitting Method For Heterogeneous Multiscale Poroelasticity, Paul M. Delgado

Open Access Theses & Dissertations

Traditional models of poroelastic deformation in porous media assume relatively homogeneous material properties such that macroscopic constitutive relations lead to accurate results. Many realistic applications involve heterogeneous material properties whose oscillatory nature require multiscale methods to balance accuracy and efficiency in computation.

The current study develops a multiscale method for poroelastic deformation based on a fixed point iteration based operator splitting method and a heterogeneous multiscale method using finite volume and direct stiffness methods. To characterize the convergence

of the operator splitting method, we use a numerical root finding algorithm to determine a threshold surface in a non-dimensional parameter space …