Computational Subsurface Hydrology

This book is concerned with the mathematical description and numerical modeling of subsurface media. It is about the subsurface media that control the movement of fluids (including water, nonaqueous liquids, and gas), the migration of chemicals, the transfer of heat, and the deformation of media. It is about the physical laws that describe the flux of fluid, heat, and chemicals, and the relationship between stress and strain. It is about the chemical reactions along with fluid flows. It is about the biological interaction within the flow and thermal domain and among chemical constituents. It is about numerical methods needed to conduct simulations of both fluid flows and advection-dominant transport. In short, the study of the subsurface system is the investigation of major processes occurring in the subsurface and the interplay of these processes with the media through which they occur. Understanding the mechanisms controlling the occurrence of these processes and their interplay is the ultimate goal of this book because it provides a method for the prediction of the occurrence of these processes in the media. To make this goal possible, accurate numerical methods to efficiently and accurately approximate mathematical descriptions are of ultimate importance. Extensive coverages of finite element methods used in fluid flows and hybrid Lagrangian-Eulerian approaches best suited to deal with advection-dominant transport are included.

Numerical methods, as shall be described in this book, are merely tools used to enable one to replace differential equations governing the subsurface processes with approximation sets of algebraic equations or matrix equations, which are subsequently solved using the methods of linear algebra and requiring the manipulation of computers (Fig. 2.1). If the differential equations were solved exactly by analytic procedures, the solution would appear as some combination of mathematical functions. Subsequent interest in the value of the solution at various locations within a domain of interest would require that the functions be evaluated. Often, when the functions are of a complex form, the computer must be used to determine the values of the function at the points of interest. In many cases the analytical solution will be in terms of an infinite series or some transcendental functions that can be evaluated only approximately. Nevertheless, it is often possible to control the accuracy of the evaluation by careful use of the computer. The steps outlined above do require some facility with number manipulation on the computer and do yield an approximate value of the solution at points of interest. However, the actual steps involve numerical evaluation of an analytical solution to a differential equation rather than numerical solution to the differential equation. The differences between these concepts is the presence of an exact analytical expression as an intermediate step in the former case and the use of an approximation to the differential equation in the latter case.

First, a formal mathematical derivation of transient multidimensional flow and solute transport in subsurface media will be presented in this chapter. We attempt to include all important steps in the development to account for the assumptions and their bases and to incorporate boundary conditions as much as possible. Based on (1) the continuity of fluid, (2) the continuity of solid, (4) the motion of fluid (Darcy’s law), (4) the equation of the state, (5) the law of consolidation of media, (6) conservation of energy, and (7) the principle of mass balance, the governing equations can be derived for the distribution of pressure, temperature, and solute concentration for single-phase flow in subsurface media. Then, finite-element modeling of single-fluid phase flows under saturated and saturated-unsaturated conditions will be presented.