Fundamentals of Fluid Flow in Porous Media
The Equation of Continuity in Porous Media: Solutions to the One-Dimensional Convection-Dispersion Model
For a single solute and no source/sink, Eq. (5‑56) becomes:
The following assumptions are implicit in eq. (5‑68):
- homogeneous porous medium of constant cross-section;
- bulk flow in the axial direction at constant interstitial velocity;
- constant fluid density;
- constant dispersion coefficient;
- incompressible porous medium;
- uniform concentration distribution in the direction perpendicular to flow (i.e., time is “long” enough for the convection-dispersion model to hold);
- no solute sources or sinks.
Eq. (5‑68) is a second order partial differential equation, requiring two boundary and an initial condition for its solution.
When the space variable x is translated such that the transformed distance x’ becomes the distance from the flood front rather than the distance from the inlet of the porous medium, eq. (5‑68) reduces to the one-dimensional unsteady diffusion or heat conduction equation, also known as Fick’s second law (Taylor, 1953; Nunge and Gill, 1970):
x’ = x – vt
vt = distance of the flood front from the porous medium entrance.
This equation has been solved analytically for a variety of initial and boundary conditions (see, for example, Carslaw and Jaeger, 1959; Özişik, 1980).
Consider an infinite porous medium at zero initial concentration, with a step change in inlet concentration (c0):
|t = 0||x’ ≥ 0||c = 0||(5‑70)|
|x’ < 0||c = c0|
|t > 0||x’ → -∞||c → c0|
|x’ → +∞||c → 0|
The solution to eq. (5‑69) with initial and boundary conditions (5‑70) is (Danckwerts, 1953; Brigham et al., 1961):
c / co = normalized concentration,
er ƒ(r) = error function of a variable r:
Some properties of the error function are