The Boundary Element Method for Groundwater Flow

The study of groundwater and porous media flow has become important during the last years in fields as different as civil engineering, aquifer managing and petroleum engineering. The fundamental equations describing these physical phenomena are well known [5], [6] and before the computers allowed them to be solved numerically, many authors developed analytical solutions for special cases [57], [66]. Nowadays, analytical solutions are very important because they allow to check the numerical methods. Different numerical methods have been developed to solve numerically porous media flow problems. The finite difference method (FDM), the finite element method (FEM) and the boundary element method (BEM) have successfully been applied to solve the equations. The application of these three methods to the computation of groundwater flow has shown that the BEM is the most suitable, since it allows the most efficient handling of free surfaces, infinite flow domains and singularities. Besides, the BEM gives more accurate results than the two other methods because the weighting functions used by this method are analytical solutions of the governing equations. In this book, the application of the BEM to the different types of groundwater flows, especially to those described by the Laplace equation, will be discussed.

The subject of this book is the study of steady and unsteady porous media flow. In order to study a physical problem, one can describe it by a mathematical model. In the present case, this process leads to the Laplace equation (see previous chapter) which is one of the fundamental equations of engineering analysis.

The boundary element method was developed at the University of Southampton by combining the methodology of the finite element method with the boundary integral method. The first international conference devoted to the boundary element method took place in 1978 at Southampton [7]. Since that time, many books have been published ([8], [9], [10],) and the numerous contributions to the annual conferences like BEM and BETECH show the rapid development of the new method for all the engineering fields. In this chapter, the weighed residual method will be used to develop the boundary element method for the case of anisotropic Laplace problems. The weighed residual method is the most general technique, because it can also be applied to develop the finite difference method and the finite element method for instance.

After the introduction to the boundary element method in the previous chapter, the numerical integration of the integrals (3.32) and (3.33) will be considered here, ie.

In order to check the accuracy of the boundary element method, the method will be applied to several examples for which analytical solutions are available. In spite of the importance of these comparisons when developing new software, only few examples comparing numerical results with analytical solutions have been presented in the literature [7], [41], [64], [12], [75], [55], [15]. The aim of the present chapter is the study of some numerical aspects of the boundary element method, like the comparison between the singular and the Gauss integration method, the choice of the number of integration points and the discretization of the boundaries.

The division of the domain into sub-regions is a very useful technique particularly in two cases. First, if the domain is divided in several parts having different properties, and second, if the domain has a complicated geometry. In this second case, it will give more accurate results if one subdivides the domain into several sub-regions rather than considers the whole domain without divisions. The sub-regions are separated by interfaces along which both the potential (u) and the normal velocity (q) are unknown.

In this chapter, the boundary element method is applied to solve steady porous media flows described by the Laplace equation. In the present case, the potential u = Y + p/p has to satisfy the Laplace equation. The potential contains two terms of which the first represents the influence of the gravity and the second the effect of the water pressure. In some special cases, the influence of the pressure is negligible. This case will be analysed in a later chapter.

In this chapter, the boundary element method will be applied to solve unsteady porous media flows described by the Laplace equation. In the present case, as in the steady state case, the potential u = Y + p/p has to satisfy the Laplace equation. The potential contains two terms of which the first represents the influence of the gravity and the second the influence of the water pressure. Thus, the potential is independent of time and consequently, only the boundary conditions take the transient character of the flow into account.

In this chapter, infiltration problems of water from canals or rivers through porous media soils to underlying groundwater will be studied. The study of these infiltrations is very important in areas where the agriculture is dependent on irrigation. In fact, infiltration and evaporation produce water losses that have to be taken into account to develop a water management project.

The subject of the studies presented in this book is the application of the boundary element method to two dimensional potential problems described by the Laplace equation, with special reference to the study of free surface groundwater problems.