Chapter 1
Modeling

1.1 Introduction

This book uses analytical modeling to provide the reader with insights into the fate and transport of solutes in groundwater. In this chapter, we begin by introducing what we mean by analytical modeling, solute transport, and groundwater. We describe models and modeling, define some common modeling terms, and present the fundamental mathematical model that is used to simulate the flow of water in a porous medium. We also include some example model applications, showing how modeling may be used to help us understand the behavior of systems.

To begin, what is a model? Simply put, a model is any approximation of reality, based on simplifications and assumptions. Thus, a model may be a small-scale depiction of reality (a physical model), a mental model or set of ideas/theories as to how reality works (a conceptual model), a network of resistors and capacitors that use electricity to simulate a real system (an analog model), or a set of mathematical equations that are used to describe reality (a mathematical model). In this book, we focus on mathematical models. Specifically, we use partial differential equations (PDEs) to model reality, for as Seife (2000, p. 119) noted in his book Zero: The Biography of a Dangerous Idea, ".nature. speaks in differential equations." Thus, when we subsequently talk about models, we will be talking about one or more PDEs, along with their associated initial and boundary conditions, which, based on various assumptions, are being used to approximate reality.

Having defined what a model is, we need to think about the purpose of modeling; how are models used? Essentially, models have two basic purposes: (1) making predictions and (2) facilitating understanding. Newton's model of gravitational attraction, which allows us to forecast the motion of the planets, and Einstein's model of the mass/energy relationship, which allows us to estimate the energy released in a nuclear explosion, are examples of model applications for predictive purposes. Modeling for understanding, though, is at least as important as using models to make predictions. Especially when modeling a real system that has many unknowns and much uncertainty associated with it, such as the subsurface, it may be extremely difficult to make good predictions. In a classic study, Konikow (1986) conducted a postaudit to see how satisfactorily a well-calibrated model, based on 40 years of data, predicted the response of water levels in an aquifer to pumping over a subsequent 10-year period. The correlation between observed and model-predicted water levels was "poor." The study concluded that ".the predictive accuracy of . models does not necessarily represent their primary value. Rather, they provide a means to assess and assure consistency within and between (1) concepts of the governing processes, and (2) data describing the relevant coefficients. In this manner, a model helps . improve understanding.." In this book, we focus on the use of modeling to improve understanding. The models presented in later chapters are gross simplifications of reality and have little predictive value, except in a general qualitative sense. However, the model applications that are presented hopefully provide the reader with important insights into how governing processes and parameter values, which quantify the magnitude of those processes, affect the response of chemical contaminants that are being transported in the complex subsurface environment.

We use the classic definition of groundwater: "the subsurface water that occurs beneath the water table in soils and geologic formations that are fully saturated" (Freeze and Cherry, 1979). These geologic strata that contain water are sometimes referred to as aquifers, but the word aquifer is generally reserved for geologic strata that not only contain water but can also transmit or yield appreciable quantities of water. Since the modeling that we are discussing is not predicated on having a minimum transmissivity, we most often refer to the material through which water is flowing simply as a hydrogeologic unit or as a porous medium where we understand that the medium is of geologic origin. The water table is the division between the unsaturated zone and the saturated zone. As we are looking at groundwater, we are concerned with flow below the water table.

Groundwater is not pure water; it contains a variety of solutes, which may occur naturally or be of anthropogenic origin (i.e., human-made). Generally, the majority of the naturally occurring solutes are ionic in form, with natural organic matter concentrations low in comparison. However, much of the current focus on groundwater contamination is due to the presence of hazardous solutes, both organic and inorganic, of anthropogenic origin. The solutes found in groundwater cover a wide range of chemical species. In this text, we take a generic approach, using simple models that simulate the physical, chemical, and biological processes that affect the fate and transport of no particular solute. In this way, we hope the text achieves its goal of providing the reader with insights into how governing processes and parameter values, which quantify the magnitude of those processes, affect the response of chemical contaminants in the subsurface.

1.2 Definitions

To make sure we are all speaking the same language, let's present some terminology that is used throughout this text. We have already defined a model as a set of differential equations with boundary and/or initial conditions. We therefore refer to "the model" or "the model equations", synonymously. Often, the solution to the model equations is also referred to, imprecisely, as the model. We try to be precise and explicitly refer to the model solution (or system response or value of the dependent variable as a function of space and time), rather than using the term model. Similarly, the computer code that is used to obtain the model solution is sometimes referred to as the model. Here, we use the term model code to refer to the set of computer instructions or program that is used to solve the model equations. The model code may utilize analytical, semianalytical, or numerical methods to solve the model equations (Javandel et al., 1984). As the title of this book suggests, we present analytical solutions to the model equations. Analytical solutions have a couple of important advantages: (1) they are mathematically exact and do not involve approximating the model equations as numerical methods do and (2) computer codes can evaluate the solutions quickly. The main disadvantage of these analytical solution methods, which was alluded to earlier, is that the model equations and initial/boundary conditions that are being solved must be "simple." Typically, this means that the PDEs must be linear and that the parameters in the PDEs used to quantify the processes being modeled, as well as the PDE initial and boundary conditions, are either constant or described by simple relationships (e.g., an initial or boundary condition described by a trigonometric function). Such limitations mean that the system that is being modeled is either homogenous in space and constant in time or else changes in space and time are easy to express mathematically. Obviously, conditions in the subsurface are far from homogeneous or easily expressed, and many processes are most appropriately described by nonlinear equations. Nevertheless, for our purposes (remember, we are focused on using models to gain understanding, not to make predictions), these simplifications are acceptable, and, in fact, helpful.

1.3 A Simple Model - Darcy's Law and Flow Modeling

1.3.1 Darcy's Law

In 1856, Henri Darcy, a French engineer, published his findings that the rate of flow of water through a porous material was proportional to the hydraulic gradient. Hydraulic gradient is defined as the change in hydraulic head (h) with distance, where, if one assumes the density of water is constant in space, head is a measure (in units of length) of the potential energy of water at a point in space. In three dimensions, Darcy's law is

where [L-T-1] is the specific discharge or Darcy velocity, a vector that describes the magnitude and direction of flow per unit area perpendicular to the flow direction, is the hydraulic gradient, a vector describing the magnitude and direction of the steepest change in head with distance ( where are unit vectors in the x-, y-, and z-directions, respectively), and K [L-T-1] is a constant of proportionality, known as the hydraulic conductivity. In the version of Darcy's law shown in Equation (1.1), we are assuming that hydraulic conductivity is a single constant, independent of location (i.e., the porous medium is assumed to be homogeneous) and flow direction (so, the medium is said to be isotropic). More complex versions of Darcy's law allow the value of hydraulic conductivity to vary with both location (a heterogeneous medium) and flow direction (anisotropic medium). Note that for an isotropic medium, Equation (1.1) indicates that the direction of flow is in the direction of the largest decrease in hydraulic gradient (hence, the minus sign on the right-hand side of the equation).

We see that Darcy's law is a very simple model, yet it provides us with some important insights. This simple PDE tells us that if we want to predict the...