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a mathematical perspective, the study of fluid flows in porous media offers fertile ground for inquiry into PDEs more generally. In particular, this book employs many broadly applicable concepts in the theory of PDEs, including:

      1 Mass and momentum balance laws

      2 Variational principles

      3 Fundamental solutions

      4 The principle of superposition

      5 Similarity methods

      6 Stability analysis

      7 The method of characteristics and jump conditions.

      Where possible, the narrative introduces these topics in the simplest possible settings before applying them to more complicated problems.

      Topic 1, covered in Chapter 2, deserves comment. Few PDE texts at this level discuss balance laws in the detail pursued here. However, it is hard to build intuition about porous‐medium flows without knowing the principles from which they arise. The balance laws furnish those principles. On the other hand, a completely rigorous study of balance laws for fluids flowing in porous media would require a monograph‐length treatment in its own right. Chapter 2 reflects an attempt to weigh the importance of fundamental principles against the need for a concise explanation of how the governing PDEs emerge from basic laws of physics. The references offer suggestions for deeper inquiry.

      We frequently refer to PDEs according to a classification system inherited from the algebra of quadratic equations. The utility of this system becomes more apparent as one becomes more familiar with examples. For now, it suffices to review the system for second‐order PDEs in two independent variables having the form

(1.1)

      Here,

, , and are functions of the independent variables and , which we can replace with and in time‐dependent problems; is the unknown solution; and denotes a function of five variables that describes the lower‐order terms in the PDE.

The highest-order terms determine the classification. Thediscriminant of Eq. (1.1) is

, which is a function of . Equation (1.1) is

       hyperbolic at any point of the ‐plane where ;

       parabolic at any point of the ‐plane where ;

       elliptic at any point of the ‐plane where .

      Extending this terminology, we say that a first‐order PDE of the form

      is hyperbolic at any point

where

.

Exercise 1.1 Verify the following classifications, where

and

are real‐valued with

:

Mathematicians associate the wave equation with time‐dependent processes that exhibit wave‐like behavior, the heat equation with time‐dependent processes that exhibit diffusive behavior, and the Laplace equation with steady‐state processes. These associations arise from applications, some of which this book explores, reinforced by theoretical analyses of the three exemplars in Exercise 1.1. For more information about the classification of PDEs, see [65, Section 2‐6].

1.3 Dimensions and Units

      In contrast to most texts on pure mathematics, in this book physical dimensions play an important role. We adopt the basic physical quantities length, mass, and time, having physical dimensions

,

, and

, respectively. All other physical quantities encountered in this book—except for one case involving temperature in Chapter 7 —are derived quantities, having physical dimensions that are products of powers of Скачать книгу