Quasi-static poroelastic equations as a symmetric positive system and its numerical approximation

Abstract

This dissertation is concerned with the equations of linear poroelasticity and numerical simulation in the framework of symmetric positive systems. Physical systems arising in geomechanics, hydrology, soil mechanics, reservoir engineering, biomedical engineering etc. are modeled with linear poroelasticity equations. The purpose of this dissertation is to present well-posedness results and numerical analysis techniques using the framework of symmetric positive systems for the variants of poroelasticity equations.

Symmetric positive system, commonly known as Friedrich's system is a system of first order partial differential equations (PDEs) with symmetry and positivity properties. A PDE, that can be written in this framework is well-posed and such PDE can be numerically solved easily. We will exploit these properties of Friedrich's systems in our model problem of poroelasticity.

We consider a quasi-static poroelasticity model with two sets of different base variables. First we consider fluid content ($\eta$), rotation variables ($w_{ij}$) and pressure gradients ($p_{x_i}$). With those variables, the original PDE (or its arbitrary purturbation) is written in a symmetric positive form and subsequently a least square finite element analysis, followed by numerical simulation results is presented. In the second case, we choose stress components ($\sigma_{ij}$), displacement variables ($u_i$), pressure ($p$), pressure gradients ($p_{x_i}$). A scaling technique is used after semi-descretization in order to ensure the sufficient condition for positivity. We have successfully applied this technique for a wide varieties of rocks. Finally, a least square finite element method has been employed to find its numerical solutions.