Free Boundary Problems

Theory and Applications

Ioannis Athanasopoulos, Jose Francisco Rodrigues, George Makrakis

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9781584880189
• eBook ISBN: 9781351447133
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Free boundary problems arise in an enormous number of situations in nature and technology. They hold a strategic position in pure and applied sciences and thus have been the focus of considerable research over the last three decades. Free Boundary Problems: Theory and Applications presents the work and results of experts at the forefront of current research in mathematics, material sciences, chemical engineering, biology, and physics. It contains the plenary lectures and contributed papers of the 1997 International Interdisciplinary Congress proceedings held in Crete. The main topics addressed include free boundary problems in fluid and solid mechanics, combustion, the theory of filtration, and glaciology. Contributors also discuss material science modeling, recent mathematical developments, and numerical analysis advances within their presentations of more specific topics, such as singularities of interfaces, cusp cavitation and fracture, capillary fluid dynamics of film coating, dynamics of surface growth, phase transition kinetics, and phase field models. With the implications of free boundary problems so far reaching, it becomes important for researchers from all of these fields to stay abreast of new developments. Free Boundary Problems: Theory and Applications provides the opportunity to do just that, presenting recent advances from more than 50 researchers at the frontiers of science, mathematics, and technology.

Part 1.

Plenary Lectures

L. BADEA, R. E. EWING, AND J. WANG*

A Study of Free Boundary Problems of Fluid Flow in Porous Media by Mixed Methods

Abstract

In this article the flow of fluids in porous media is studied as a free or moving boundary problem by the mixed method. In particular, a new weak formulation for the problem of seepage of fluids through a porous media is discussed and analyzed mathematically and numerically. The new formulation is in a mixed form and is suitable for the use of mixed finite element methods in the numerical approximation. It is proved that the weak formulation and its finite element discretization have a solution which can be approximated by a sequence of regularized problems.

1 Introduction

In this paper, we are concerned with the free or moving boundary value problem in the study of the flow of fluids through a heterogeneous porous media. Such problems are important in many branches of science and engineering. For example, in the areas of soil science, agricultural engineering, and groundwater hydrology, the movement of fluids and their dissolved components in both saturated and unsaturated soils is an important environmental consideration. In petroleum engineering, improved recovery of oil and gas is based on simulation of multiphase and multicomponent fluid transport in deep rocks. In both application areas, mass transfer across phase boundaries is an important consideration which can be discussed in the context of free or moving boundary problems.

Another important application of the free or moving boundary problem is water seepage through a dam, or rain water creeping through an unsaturated zone. The underlying physics of the petroleum and seepage problems are very similar. For comparison, assume that there are two fluids flowing simultaneously in the porous medium. In unsaturated flow, these fluids are water and air; while in the petroleum problem, the fluids are assumed to be water and oil. Relevant material properties, including the capillary pressure and relative permeability are assumed to be known.

Free boundary problems are also seen in other areas of the petroleum industry such as basin simulation. The research of basin simulation is important because it attempts to determine when and where organic matter was put down and how hydrocarbons may have been produced. This research requires treating the top layer of rock as a boundary which varies in time. Due to the deposition and erosion of materials, the shifting of land masses, and natural processes which change solid organic matter into hydrocarbons, the precise definition of the rock strata and the surface of the basin, at any instant in time is not known a priori. In the description and modelling of the basin evolution, free boundaries are necessary.

For simplicity of presentation, we restrict ourselves to the 2-dimensional flow of fluids through a dam in which the free or moving boundary divides the saturated soil from the unsaturated part. This phenomena can be characterized by using the Richards equation [22]; similar phenomena in the petrol...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Part 1. Plenary Lectures
• Part 2. Mathematical Developments of Free Boundary Problems
• Part 3. Free Boundary Problems in Fluid Mechanics
• Part 4. Phase Change in Material Science
• Part 5. Computational Methods and Numerical Analysis