For example, the physical dimension of force
Analyzing the physical dimensions of quantities that arise in physical laws can yield surprisingly powerful mathematical results. Subsequent chapters exploit this concept many times.
Physical laws such as
Where practical, this book uses the Système Internationale (SI) as the preferred system of units. The current standards for time, length, and mass in the SI are as follows:
Time: One second (s) is the duration of 9 192 631 770 periods of the radiation emitted by the transition between the two hyperfine levels of the ground state of cesium‐133. This period of time is approximately 1/86 400 of one Earth day.
Length: One meter (m) is the distance traveled in a vacuum by light in 1/299 792 458 s. This distance is approximately times the distance from the Earth's geographic north pole to the equator along a great circle.
Mass: One kilogram (kg) is the mass required to fix the value of the Planck constant as kg , given the definition of one second and 1 m. This mass is approximately that of (1 liter) of water at room temperature and pressure.
In some cases, non‐SI units are more convenient for measuring physical quantities that arise in the bench‐ or field‐scale study of fluid flows in porous media. When these cases arise, we give the factor that enables conversion to SI units. The fact that scientists and engineers prefer non‐SI units in some instances highlights the inherently subjective nature of units: Humans tend to prefer standards that yield numerical values not far from 1 in our everyday experience. One advantage of using dimensionless quantities—a technique employed frequently in this book—is that we avoid this subjectivity.
1.4 Limitations in Scope
Three limitations in scope are worth noting. First, we treat only isothermal flows in porous media, that is, flows at constant temperature. This restriction conveniently allows us to ignore the energy balance equation in deriving governing PDEs. On the other hand, it also eliminates several types of flows that have important applications, including flows in geothermal reservoirs and thermal methods of enhanced oil recovery, such as steam flooding.
Also glaringly absent from the table of contents is the topic of flows in fractured porous media. Geoscientists correctly point out that most geologic porous media possess fractures, which exert significant influences on fluid flows. Yet the mathematics of flow in fractured porous media remains poorly delineated, owing not so much to the absence of mathematical models (see [21] for a recent overview and [8, 15, 86, 153] for prominent examples) but, more importantly, to the observation that fractures exist at many scales of observation. In some underground formations, one must know something about the geometry of individual fractures to model fluid flows accurately. In these settings, the modeler's challenge is to represent the discrete fracture system (or statistical realizations) on tractably coarse computational grids. In other geologic settings, it suffices to treat the pore network and the fracture network as overlapping porosity systems, and the challenge is to model how fluids move within and between them. This spectrum of modeling approaches deserves a monograph of its own.
Also missing from the topics covered here is a discussion of fluid flows in extremely flow‐resistant media, often but debatably referred to as nanodarcy flows but more properly characterized as non‐Darcy flows. Flows of this type have increased in practical importance during the past two decades, owing especially to vastly improved technologies for producing natural gas from shale formations when hydrocarbon commodity prices justify the costs. The physics here are complex, involving gas–rock interactions in interstices whose typical diameters approach the mean free path of the gas molecules. None of the classical macroscopic transport models—such as Darcy's law or Fick's law of diffusion—suffices by itself to capture these phenomena [37, 81]. One can hope that further advances in our understanding of these flows, analogous to the advances described above for classical Darcy flows, will yield more settled mathematical models in years to come.
2 Mechanics
2.1 Kinematics of Simple Continua
At the macroscopic scale of observation, greater than about
The first step is to establish the kinematics. This branch of mechanics provides a framework for describing the motions of continua geometrically, without reference to the forces that cause motion. The treatment here is an abbreviated version of material that appears in standard courses on continuum mechanics; for more details consult [4].
2.1.1 Referential and Spatial Coordinates
In continuum mechanics, the term body refers to a collection
