Quantifying the multiscale hydraulic heterogeneity in aquifers and their effects on solute transport is the task of this chapter. Using spatial statistics, we explain how to quantify spatial variability of hydraulic properties or parameters in the aquifer using the stochastic or random field concept. In particular, we discuss spatial covariance, variogram, statistical homogeneity, heterogeneity, isotropy, and anisotropy concepts. Field examples complement the discussion. We then present a highly parameterized heterogeneous media (HPHM) approach for simulating flow and solute transport in aquifers with spatially varying hydraulic properties to meet our interest and observation scale. However, our limited ability to collect the needed information for this approach promotes alternatives such as Monte Carlo simulation, zonation, and equivalent homogeneous media (EHM) approaches with macrodispersion approaches. This chapter details the EHM with the macordispersion concept.

This chapter develops the Navier–Stokes equations using a Lagrangian description. In doing so, the concept of a stress tensor and its role in the overall force balance on a fluid element is discussed. In addition, the various terms in the stress tensor as well as the individual force terms in the Navier–Stokes equations are investigated. The chapter ends with a discussion on the incompressible Navier–Stokes equations.

This chapter serves as an introduction to the concept of conservation and how conservation principles are used in fluid mechanics. The conservation principle is then applied to mass and an equation known as the continuity equation is developed. Various mathematical operations such as the dot product, the divergence, and the divergence theorem are introduced along the way. The continuity equation is discussed and the idea of an incompressible flow is introduced. Some examples using mass conservation are also given.

In this chapter, a concept known as scaling is introduced. Scaling (also known as nondimensionalization) is essentially a form of dimensional analysis. Dimensional analysis is a general term used to describe a means of analyzing a system based off the units of the problem (e.g. kilogram for mass, kelvin for temperature, meter for length, coulomb for electric change, etc.). The concepts of this chapter, while not entirely about the fluid equations per se, is arguably the most useful in understanding the various concepts of fluid mechanics. In addition, the concepts discussed within this chapter can be extended to other areas of physics, particularly areas that are heavily reliant on differential equations (which is most of physics and engineering).

This chapter introduces simple graphical methods to estimate advection velocity and dispersivity of solute migration through soil columns, using one-dimensional ADE presented in previous chapters. Methods of spatial and temporal moments are also introduced for solute concentration breakthroughs in one-dimensional transport and snapshots of the multi-dimensional solute migrations, respectively. Unlike automatic nonlinear regression analysis, these methods use physical insights and analytical solutions to illustrate logical approaches to estimate these parameters. The automatic regression analysis (such as Microsoft Excel introduced in Chapter 1) may find the parameters that fit the solution to the data well. However, the parameter values may not be physically possible if the estimation problem is poorly constrained (see examples in Chapter 11).

In addition to the continuity equation, there is another very important equation that is often employed alongside the Navier–Stokes equations: the energy equation. The energy equation is required to fully describe compressible flows. This chapter guides the student through the development of the energy equation, which can be an intimidating equation. A discussion on diffusion and its interplay with advection is also included, leading to the idea of a boundary layer. The chapter ends with the addition of the energy equation in shear-driven and pressure-driven flows.

High-resolution imaging of solute movement at pore scale in core samples and numerical simulations are presented to demonstrate the effects of pore-scale velocity variations, neglected in Darcy’s velocity, on the spread of solutes. Advection and dispersion equations (ADE) for solute transport in variably saturated media are thus formulated. Then, Peclet number analysis relates the dispersion coefficient to dispersivity --- the solute transport property of a porous medium -- and Darcy’s velocity. Well-controlled laboratory soil column experiments and related numerical experiments are examined to illustrate the validity and weakness of ADE. They also show that dispersion in porous media is an ensemble mean description of the effects of pore-scale velocity variations neglected by Darcy law. This chapter presents the dead-end pores, mobile-immobile zones, or dual-domain models, explains their ensemble mean nature, and discusses their pros and cons. Lastly, it formulates the ADE for the reactive solute in porous media.

This chapter develops the water balance equation and its solutions for various inputs to study temporal water fluctuations in the groundwater system. This equation is applied to a field aquifer for estimating the parameters and recharge. Subsequently, the well-mixed model for the solute and its analytical solutions for various input forms are developed. Further, the chapter discusses the hydraulic response, water retention, and chemical response time, pertinent to understanding energy propagation, solute advection, and mixing concepts. The application of the groundwater model to highway deicing salt application follows. The model for reactive chemical solutions comes next, considering chemical decay, first-order equilibrium, and nonequilibrium reactions. Their effects on solution output are discussed. We then introduce the Monte Carlo simulation, sensitivity analysis, and first-order analysis based on the well-mixed model to address uncertainty in the model outputs due to unknown parameters and inputs. Finally, its application to groundwater reservoirs’ effects on buffering acid rains in a lake is presented.

This chapter is the most crucial part of the book, the fundamental building block of the concept of dispersion in porous media and macrodispersion in the field-scale aquifers. It unravels the myth of macrodispersion, anomalous dispersion, scale-dependent dispersion, dual-domain, and other recently developed dispersion models for solute transport in aquifers (Chapters 9 and 10). This chapter first explains how the concept of dispersion evolves from molecular diffusion to account for the effects of fluid-dynamics-scale velocity variation in solute migration in a pipe. The relationship between dispersion and the concentration averaged over the cross-section of a pipe is visited. Further, this chapter illustrates the molecular and fluid-dynamics-scale velocity variations, the interaction between diffusion and dispersion, and scale issues associated with the shear flow dispersion. Finally, it discusses the limitations of extending Fick’s law for molecular-scale velocity variations to describe the effects of fluid-dynamics-scale velocity variations.

ADE with macrodispersion (Fickian) and non-Fickian models rely on the aquifer-scale ensemble mean concept. They satisfy our needs only in the ensemble mean sense, rather than at one field site unless the tracer cloud reaches ergodicity or the tracer plume has experienced enough heterogeneity. The applicability of current theories to an aquifer (one realization of the ensemble) is inadequate at our interest and observation scales. Such scale issues demand a high-resolution delineation of the multi-scale heterogeneity in the aquifer. This chapter introduces new technologies (hydraulic and geophysical tomographic surveys) that minimize reliance on the large-scale ensemble mean models. Most importantly, it demonstrates the effects of interaction between different regional-scale velocities on mixing, dilution, and dispersion---a testimony of the importance of dominant large-scale flow on solute transport. This chapter, in essence, promotes a better understanding of solute migration in field-scale geologic media to minimize our prediction uncertainty.

This chapter first reviews the linear first-order non-homogeneous ordinary differential equation. Introduction to statistics and stochastic processes follows. Afterward, it explains the stochastic fluid continuum concept, associated control volume, and spatial- and ensemble-representative control volume concepts. It then uses the well-known solute concentration definition as an example to elucidate the volume- and spatial-, ensemble-average, and ergodicity concepts. This chapter is to provide basic mathematics and statistics knowledge necessary to comprehend the themes of this book. Besides, this chapter’s home works demonstrate the power of the widely available Microsoft Excel for scientific investigations.