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

(2.24)

Exercise 2.13 Row‐reduce Eq . (2.24) to deduce that is a free variable, for which one may choose any value, and .

Arbitrarily picking yields the single dimensionless variable ; all other dimensionless variables for this problem must be multiples of this product.

The calculation in Exercise 2.13 shows that any relationship equivalent to Eq. (2.22)but involving only dimensionless variables has the form . Solutions to such an equation are constant values of . Setting for a generic constant , we conclude that Stokes drag has the form

(2.25)

This result is consistent with that of Stokes's original calculation, except that we have an undetermined constant instead of .

To anticipate the constitutive theory of flows in porous media, discussed in Chapter 3, observe that the drag on the solid particle in Eq. (2.25)is proportional to the fluid velocity and the fluid viscosity, and it involves a geometric factor . This result suffices for the derivation pursued in Section 3.1.

The mechanics discussed so far cannot distinguish among the various solid and fluid bodies that make up a porous medium. To accommodate mixtures of different types of materials, such as the solid and fluid in a porous medium, we must adopt additional physics.

The first step in extending the mechanics of single continua is to consider a set of bodies , , called constituents. For example, in a porous medium, rock and water can be constituents. We postulate that each spatial point can be occupied by particles from every constituent. In this sense, the bodies constitute overlapping continua. This postulate clearly fails at scales of observation at which the constituents appear to occupy distinct regions of space. But for many natural porous media found in Earth's subsurface, the postulate yields reasonable models at scales of observation greater than about m.

Paralleling the development for single continua, for each constituent , we fix a reference configuration that assigns, to each particle in , a point in three‐dimensional space. The vector serves as a label for the particle. We denote by the region in three‐dimensional Euclidean space occupied by all of these vectors for the constituent .

We also associate with each constituent a one‐parameter family of mappings from to three‐dimensional Euclidean space such that:

1 The vector , having dimension L, gives the spatial position of the particle at time , as illustrated in Figure 2.11.

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

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

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

We call the deformationof constituent .

Figure 2.11 A reference configuration and the deformation at times and for constituent in a multiconstituent continuum.