Myron B. Allen, III - The Mathematics of Fluid Flow Through Porous Media

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.

At the macroscopic scale of observation, greater than about m, a natural porous medium such as sandstone is a complex mixture of solids and fluids, separated by interfaces whose geometries are often too small for humans to discern without aid. This book focuses mainly on the macroscopic scale. However, viewed at the microscopic scale, say – m, the solids and fluids in a porous medium appear as distinct continua, separated by observable interfaces. We begin with the mechanics of these simple continua. Section 2.5extends the discussion to the mechanics of multiconstituent continua, applicable at the macroscopic scale of observation.

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].

In continuum mechanics, the term bodyrefers to a collection of particles, sometimes called material points. A subset of the body that is a body in its own right is a partof the body. We assign to each body a reference configuration, which associates with the body a region in three‐dimensional Euclidean space. In the reference configuration, each particle in the body has a position , unique to that particle, as shown in Figure 2.1. The vector serves as a label, called the referentialor Lagrangiancoordinates of the particle. As with a person's home address, from a strictly logical point of view the particle need not ever occupy the point . That said, in some applications it is useful to choose the reference configuration in a way that associates each particle with a position that it occupies at some prescribed time, for example .

Figure 2.1 A reference configuration of a body, showing the referential coordinates used to label a particle according to its position in the reference configuration.

The central aim of kinematics is to describe the trajectories of particles, that is, to determine the position in three‐dimensional Euclidean space that each particle occupies at every time . For this purpose we assume that there exists a one‐parameter family of vector‐valued functions, time being the parameter, that has the following properties.

1 The vector , having dimension L, gives the spatial position of the particle at time .

2 At each time , the function of the referential coordinates is one‐to‐one, onto, and continuously differentiable with respect to .

3 Also at each fixed time , has a continuously differentiable inverse such that . That is, tells us which particle occupies the spatial position at time .

4 For each value of the coordinate , the function is twice continuously differentiable with respect to .

The function is the deformationof the body, illustrated in Figure 2.2. We call the vector the spatialor Euleriancoordinates of the particle at time .

Figure 2.2 The deformation mapping the reference configuration onto the body's configuration at time .