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

As in the single‐continuum case, the referentialor Lagrangian velocityof is

To find the velocity of constituent at a fixed spatial point at time , we first find the particle that occupies at time , then compute the spatialor Eulerian velocity:

We associate with each constituent a material derivative, which gives the time rate of change following a fixed particle . For functions of , the material derivative is simply the partial derivative with respect to :

For functions of , an application of the chain rule similar to that employed in Section 2.1for simple continua yields

As with single continua, we assign to each constituent a mass density such that the mass of the constituent in any measurable region of three‐dimensional space at time is

Engineers call the bulk densityof constituent ; it gives the mass of the constituent per unit of total volume in the continuum.

In the context of porous media, this last observation prompts a discussion of two different categories of multiconstituent continua. The first, which we call multiphase continuaor immiscible continua, includes materials for which microscopic observation reveals continuum‐scale interfaces that affect the mechanics. Figure 2.12illustrates the idea schematically. An example of this type of continuum is water‐saturated sandstone. In this porous medium, there exists a continuum‐scale interface between the rock and the fluid, but in most sandstones, the geometry of the interface is observable only at scales smaller than about m.

One way to think of this type of continuum is to regard macroscopic observation as a spatial averaging process. In this view, at each point in space we replace detailed properties of the material with properties averaged over a representative elementary volume( REV) having a characteristic radius, as Figure 2.12illustrates [121]. In the case of water‐saturated sandstone, if the radius of the REV ranges over values comparable to typical rock‐grain diameters, then the fraction of the REV occupied by fluid oscillates as the radius increases. The oscillation arises because, over this range of radii, the inclusion or exclusion of individual grains results in significant changes in the value of the average.

Figure 2.12 Sketch of a fluid‐saturated porous medium showing three possible representative elementary volumes.

For the concept of a multiconstituent continuum to furnish a useful model of the porous medium, there must exist a range of REV radii—typically exceeding several rock‐grain diameters—in which the fraction of the REV occupied by fluid exhibits a stable value, as drawn in Figure 2.13. Henceforth, we assume that the porous medium possesses a range of REV radii satisfying this condition. We also assume that this range includes radii that are small compared with the macroscopic scale of observation, so that it is reasonable to model the porous medium as a set of overlapping continua.

Under this assumption, we assign to each constituent a volume fraction . This function gives the fraction of any region of three‐dimensional space occupied by material from the constituent as