Groundwater flow varies spatiotemporally under many real-world situations, different from the natural gradient experiments in Chapter 10. This chapter presents field experiments that explore the role of velocity variation at the local- and large-scale solute migration in the aquifer and reveal difficulties in characterizing the aquifer, monitoring, and predicting solute transport even in a small-scale aquifer.

Widely used, parsimonious, practical well-mixed models are presented in this chapter for lakes water quality analyses. In addition, the volume-average, ensemble average, and stochastic concepts implicitly rooted in these models are explained and emphasized.

With the rapid development of unmanned aerial vehicle (UAV), extensive attentions have been paid to UAV-aided data collection in wireless sensor networks. However, it is very challenging to maintain the information freshness of the sensor nodes (SNs) subject to the UAV’s limited energy capacity and/or the large network scale. This chapter introduces two modes of data collection: single and continuous data collection with the aid of UAV, respectively. In the former case, the UAVs are dispatched to gather sensing data from each SN just once according to a preplanned data collection strategy. To keep information fresh, a multistage approach is proposed to find a set of data collection points at which the UAVs hover to collect and the age-optimal flight trajectory of each UAV. In the later case, the UAVs perform data collection continuously and make real-time decisions on the uploading SN and flight direction at each step. A deep reinforcement learning (DRL) framework incorporating the deep Q-network (DQN) algorithm is proposed to find the age-optimal data collection solution subject to the maximum flight velocity and energy capacity of each UAV. Numerical results are presented to show the effectiveness of the proposed methods in different scenarios.

The Age of Information metric, which is a measure of the freshness of a continually updating piece of information as observed at a remote monitor, has been studied for a variety of different update monitoring systems. In this chapter, we introduce three network control mechanisms for controlling the age, namely, buffer size, packet deadlines, and packet management. In the case of packet deadlines, we analyze update monitoring system for the cases of a fixed deadline and a random exponential deadline and derive closed-form expressions for the average age. We also derive a closed-form expression for the optimal average deadline for the random exponential case.

Chapter 4 introduces the molecular diffusion concept and Fick’s Law to explain the mixing phenomena at a small-scale CV in the distributed models rather than the large CV of the well-mixed model. For this purpose, it begins with describing diffusion phenomena, then formulating Fick’s law and developing the diffusion equation. Subsequently, examining the random velocity of Brownian particles and their pure random walk, we articulate the probabilistic nature of the molecular diffusion process and the reason why Fick’s Law is an ensemble mean law. Next, analytical solutions to the diffusion equation for various types of inputs are introduced. The advection-dispersion equation (ADE) formulation then follows, which couples the effect of fluid motion at fluid continuum scale and random motion of fluid molecules at the molecular scale to quantify solute migration. Likewise, we present analytical solutions to the ADE for several input forms and discuss snapshots and breakthroughs for different input forms.

This chapter discusses two different approaches to describing fluid flow: a Lagrangian approach (following a fluid element as it moves) and a Eulerian approach (watching fluid pass through a fixed volume in space). Understanding each of these descriptions of flow is needed to fully understand the dynamics of fluids. This chapter is devoted to diving into the differences between the two descriptions of fluid motion. Understanding this chapter will help tremendously in the understanding of the upcoming chapters when the Navier–Stokes equations and energy equation are discussed. This chapter will introduce the material derivative. It is extremely important to understand this derivative before the Navier–Stokes equations themselves are tackled.

Field tracer experiments in Borden aquifer in Canada and the Macrodispersion Experiment (MADE) Site, Mississippi, are reviewed. Both experiments injected tracers in aquifers and monitored their movements over fields at ten to hundred meters under natural gradient conditions. The behaviors of the tracer plumes at these two sites are distinctly different because the Borden aquifer is statistically homogeneous, and the aquifer of MADE is statistically heterogeneous. As a result, the validity of the classical ADE and non-Fickian dual-domain models becomes a contentious debate and deserves articulation of the differences between the ensemble mean nature of the models and the observations in one realization. The two experiments provided opportunities for understanding the limitations of applying solute transport theories and mathematical models based on soil-column experiments to real-world scenarios where heterogeneity is multi-scales, and groundwater flow varies spatiotemporally. Ignorance of the differences in scale of dominant heterogeneity and the observation, model, and interest scales is to blame. We explore and discuss the strengths and weaknesses of the theories and models.

In the previous chapter, the forces acting on a moving fluid element were exhaustively studied. Using Newtons second law of motion, the Navier–Stokes equations for both compressible and incompressible flows were obtained. This chapter uses an alternative approach to developing the Navier–Stokes equations. Namely, by starting from a Eulerian description (as opposed to a Lagrangian description), the integral form and conservation form of the Navier–Stokes equations are developed. The continuity and Navier–Stokes equations in its various forms are tabulated and reviewed in this chapter. This chapter ends by solving some very simple, yet common, problems involving the incompressible Navier–Stokes equations.

This chapter introduces numerical methods, including 1) Finite Difference Approach, 2) Methods of characteristics (Eulerian-Lagrangian), and 3) Finite Element Approach for solving the ADE applicable to multidimensional, variable velocity, irregular boundary, and initial conditions. However, only one- and two-dimension examples are illustrated for convenience. Once the algorithms are understood, they can be expanded to other situations with ease.