Ebook A Least-Squares Mixed Finite Element Method For Biot’s Consolidation Problem In Porous Media
In recent years, a lot of effort has been dedicated to the theoretical and numerical treatment of models for fluid flow and deformation in porous media. Our model will be based on the classical phenomenological approach by Biot [2] using the concept of effective stresses. Modern formulations based on multiphase mixture theories were developed in the last 30 years, see the monograph by de Boer [12] for an overview. A lot of current research activity concerned with the numerical treatment of such coupled problems is devoted to the extension to two-phase flow and elasto-plastic deformation models.
The classical Biot consolidation problem assumes a fully saturated porous medium and a linearly elastic material law leading to a linear parabolic system. We restrict our attention to this linear problem since our aim is to analyze a least-squares mixed finite element method in the simplest possible situation. The novelty of our least-squares approach is the introduction of approximation spaces for the fluid flux and the stress tensor in addition to the primary variables fluid pressure and displacement field.
The numerical treatment of Biot’s consolidation problem by the Taylor-Hood finite element spaces was studied by Murad and Loula in [16] and [17]. This work was continued with a detailed analytical investigation in the contribution by Murad, Thomée and Loula [18]. A general reference for the use of the finite element method for the numerical simulation of fluid flow and deformation processes in porous media is the monograph by Lewis and Schrefler [15]. A nonlinearly elastic material law for fluid-saturated porous solids is considered in Ehlers and Eipper [13]. Wang and Kolditz [21] investigate the numerical treatment of elasto-plastic material behavior using a Drucker-Prager model for some two-dimensional plane strain problems under fully saturated conditions. Wieners, Ammann, Diebels and Ehlers in [22] consider saturated porous medium flow in three spatial dimensions where the material behavior of the porous skeleton is assumed to be elasto-viscoplastic. Taylor-Hood elements are used in [22] for the approximation of the displacement field and fluid pressure.
Our purpose in this paper is the derivation and study of a least-squares finite element approach to the coupled mechanical and flow problem. The least squares approach introduces finite element spaces for the approximation of all the process variables involved in our model. In the case of fluid flow in deformable porous media these consist of fluid pressure and flux as well as displacement field and stress tensor. At first sight this appears to increase the computational work by introducing more variables compared to standard approaches involving only pressure and displacements. However, the introduction of these variables does, in general, lead to significant simplifications at various stages of the solution algorithm. Firstly, the variables coupling the flow and deformation processes are used directly to describe the problem which makes it straightforward to derive the underlying variational formulation.
In this way, all the variables of interest can be approximated directly and, for fluid flux and for the stress tensor, more accurately than by postprocessing from the results of the standard formulation. Moreover, the finite element spaces used for approximating the different process variables can be chosen independently since no inf-sup compatibility conditions are required in the least-squares finite element approach. Finally, the local evaluation of the least-squares functional provides an a posteriori error estimator at no additional cost.
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