Applied Mathematics in Hydrogeology

Tien-Chang Lee

• Publisher: CRC Press
• Year: 2019
• ISBN:
  9781351465977
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

As introduced in Dr. Lee's 10-week class, Applied Mathematics in Hydrogeology is written for professionals and graduate students who have a keen interest in the application of mathematics in hydrogeology.
Its first seven chapters cover analytical solutions for problems commonly encountered in the study of quantitative hydrogeology, while the final three chapters focus on solving linear simultaneous equations, finite element analysis, and inversion for parameter determination.
Dr. Lee provides various equation-solving methods that are of interest to hydrogeologists, geophysicists, soil scientists, and civil engineers, as well as applied physicists and mathematicians. In the classroom, this same information will help students realize how familiar equations in hydrogeology are derived-an important step toward development of a student's own mathematical models.
Unlike other applied mathematics books that are structured according to systematic methodology, Applied Mathematics in Hydrogeology emphasizes equation-solving methods according to topics. Hydrogeological problems and governing differential equations are introduced, including hydraulic responses to pumping in confined and unconfined aquifers, as well as transport of heat and solute in flowing groundwater.

Chapter 1

CONSERVATION EQUATIONS

Heat transfer, fluid flow, and solute transport are governed by the principles of conservation of energy, momentum, and mass. Mathematically, these conservation principles, together with empirical laws, can be expressed as a set of partial differential equations. Subject to initial and boundary conditions as well as appropriate source functions, the equations can be solved analytically or numerically to interpret observations or predict certain phenomena.

In this chapter we will show how seemingly different physical processes can be modeled by differential equations that have the same functional forms. We will attempt to derive each equation with a different method.

1.1 Heat Conduction

1.1.1 Heat Energy

Heat flux is the rate of heat energy transferred through a unit area across the transport path. It is a vector and has a dimension of energy per unit area per unit time, [W m⁻²]. The heat flux can be conductive (diffusive), convective, or radiative. In this section, we consider heat conduction problems only.

Let the positive x-coordinate be in the direction of heat flux. Given heat flux q at location x in a one-dimensional continuum, the heat flux at neighboring location x + Δx is,

Figure 1.1: Heat flux q_x+Δx at x + Δx as extrapolated by the Taylor series from q_x at x, i.e., q_x+Δx ≈ q_x + (∂q/∂x)Δx.

$q_x+\Delta x=q_x+{\left(\frac{\partial q}{\partial x}\right)}_x\Delta x,$

(1.1)

which follows the first-order Taylor theorem (Figure 1.1). This relation means that if a functional value q_x and its slope ∂q_x/∂x at x are given, the functional value q_x+Δx at x + Δx can be extrapolated from the given value by following the slope to the desired location.

If q_x+Δx > q_x, the material in the space between x + Δx and x loses energy by the amount of ΔE per unit volume over time interval Δt. This energy change is accompanied by temperature change ΔT. The two changes are empirically related by

$\Delta E=\rho c_{\text{v}}\Delta T$

(1.2)

where the proportional constant ρc_v is ...