Fundamentals of Fluid Flow in Porous Media

 

Chapter 5

Miscible Displacement

The Equation of Continuity in Porous Media

Eqs (5‑54) to (5‑56) are usually derived by applying the fluid continuum approach to an element of bulk fluid, for example to the fluid continuum filling a pipe or the pore space in a porous medium. An elementary volume inside the bulk fluid is denoted “microscopic control volume.” In the case of porous media, a mathematical description of the pore and flow geometry is too complex to be modeled, and it is difficult to define boundary conditions for the complex solid/fluid boundaries. A continuum approach on a coarser level is therefore applied to describe fluid flow in porous media (Bear, 1972; Rumer, 1972).

The multiphase porous medium is replaced by a fictitious continuum, in which the values of the properties of the continuum can be assigned to any point in the medium (solid or fluid) and are continuous functions of space and time. This leads to the definition of the “macroscopic control volume”.

The macroscopic control volume in a continuum representing a porous medium must be much larger than an individual pore or grain, and much smaller than the entire flow domain, and the porosity of the macroscopic control volume must be representative of the porous medium as a whole. The properties of the macroscopic control volume are neither solid (grain) nor pore space properties, but are averages.

The equation of continuity for porous media can be derived by applying a mass balance over a macroscopic control volume and replacing some of the variables with quantities more applicable to porous media. The molecular diffusion coefficient (Do), is replaced by the dispersion coefficient K_L , a tensor, and the velocity u  by the apparent linear velocity v , defined as:

Where,

Q = volumetric flow rate (volume per unit time),

A = porous medium cross-sectional area (length squared),

φ = porosity.

The resulting equation is of the same form as that for bulk fluids:

A more rigorous derivation of the equation of continuity for porous media uses the procedure of spacial averaging (Bear, 1972; Rumer, 1972). In such an approach, every velocity and concentration is expressed as the sum of the average value taken over the macroscopic control volume and the spacial variation at any point inside the macroscopic control volume. The expressions for the velocities and concentrations and their variations are substituted into the equation of continuity for a fluid continuum (Equation 3), and the resulting equation is averaged over the macroscopic control volume to obtain Equation 6.

Carrying out the averaging procedure, as described by Bear (1972), rather than using the simpler approach of writing a mass balance over the macroscopic control volume, reveals information on the structure of the dispersion coefficient: the dispersion of a substance during flow in porous media results from molecular diffusion in the direction of flow, coupled with transverse molecular diffusion due to the velocity profiles (as in capillary tubes, see below), and mechanical mixing arising from velocity variations due to the complex nature of the pore structure. The velocity variations are not easily measured, and the dispersion coefficient has therefore frequently been correlated with more easily measurable quantities such as the apparent linear velocity and some characteristic length of the porous medium, for example particle diameter or porous medium length. The complex pore structure causes flow in the porous network to deviate from the mean direction of flow at the pore level. This effect can be described by the flow tortuosity, a tensor quantity (Bear, 1972). Mechanical mixing due to flow is different in different directions. The dispersion coefficient, a measure of mixing during flow, is therefore anisotropic, hence is a tensor quantity.

In one-dimensional flow, with constant ρ and K_L , Eq. (5‑58)  becomes:

This equation is commonly referred to as the one-dimensional convection-dispersion equation.