Lesson 4 - Physical Processes Controlling Mass Transport: Diffusion and Dispersion

In the previous Lesson, we noted that when a chemical is transported through a tube as in our simple example, we would not be able to explain its breakthrough curve using only advection.  We noted that the breakthrough curve would be dispersed.

In order to understand how we describe and quantify this, let’s simplify our tube example even further. In particular, let’s assume we inject a slug of mass (uniformly over the entire cross-sectional area) at one end of the tube, but do not allow any water to flow through the tube at all. That is, both ends of the tube are closed and there is no advection. What would happen to the concentration of the chemical in the tube?

What would happen, of course, is that the molecules would diffuse across the tube (due to the random thermal motion of the molecules) until eventually the entire tube was at a uniform concentration.  These random movements can be described statistically, and the evolution of the concentration in the tube can be described using Fick’s Second Law:

where D_m is the molecular diffusion coefficient (dimensions of L2/T). Molecular diffusion through water is a very slow process (diffusion of the length of a meter would take on the order of 10 years). Nevertheless, as we will discuss in later Units, this process can be critical for some applications, particularly those where engineered diffusive barriers are part of the design and the models have time scales of interest that are very large (e.g., radioactive waste management applications).

So let’s return to our tube example again, and now consider both advection and diffusion. If we combine the diffusive and advective terms, our equation for concentration in the tube looks like this:

Even though we have now included an additional term in our equation, because diffusion is such a slow process, we would find that our simulated dispersion would still be quite low (the breakthrough curve if we solved this equation would be, for all practical purposes, still a vertical line).  The dimensionless Peclet number can be used to compare the importance of advection to diffusion:

where v is the advective velocity, L is the distance of interest and D_m is the diffusion coefficient. Hence, over a distance of a meter, and assuming a diffusion coefficient of 1E-9 m2/s (a typical value for most chemicals), the advective velocity would need to be 1E-9 m/s in order for advection and diffusion to be comparable!  This indicates that diffusion is generally only of interest in the absence of advection.

So if diffusion does not cause the dispersion we said we would see in an actual tube, what does cause it? There are actually two other processes that can cause dispersion (and these, when present, are generally many orders of magnitude greater than molecular diffusion).  Mathematically these are treated in the same manner as molecular diffusion. These processes are turbulent (or eddy) diffusion and mechanical dispersion.

Turbulent water (or air) motions consist of constantly changing swirls of fluid of different sizes, referred to as eddies. These are small-scale fluctuations superimposed on the large-scale movement of the fluid. Although we cannot describe these movements exactly, they can be described statistically in the same way that molecular diffusion can be described statistically. That is, turbulent diffusion has the effect of carrying mass in the decreasing direction of concentration (in exactly the same manner as molecular diffusion does).

For flow through porous media, the fluid is generally moving much more slowly and is not turbulent. But the groundwater must take detours around the various solid particles, and these detours result in mixing, referred to as mechanical dispersion (or hydrodynamic dispersion). This kind of dispersion can also take place at a larger scale than that of the individual particles, as water may flow around large-scale regions of less permeable material (referred to as macrodisperson).  Like molecular diffusion and turbulent diffusion, mechanical dispersion has the effect of carrying mass in the decreasing direction of concentration.

The total effective Fickian transport coefficient is the sum of these three processes:

where D_m is the molecular diffusion coefficient, E is the eddy diffusion coefficient and D is the mechanical dispersion coefficient (and all three have dimensions of L2/T). 

Note: This equation is presented generically here, and typically only one process would be applicable and/or significant for a particular system.  Generally, both E and D would not be present in the same system.  In surface waters (e.g., a river or lake), D would not apply, E would be dominant and D_m would be insignificant. In a porous medium, only D and D_m would apply (and D_m may or may not be insignificant). Moreover, as we will see in later Units, in a porous medium, this equation is not exactly correct for diffusion (D_m would be multiplied by some additional parameters to adjust for the presence of the porous medium).

The molecular diffusion coefficient is a function of the fluid, the size of the chemical diffusing and the temperature.  As we shall see in a subsequent Unit, when diffusing through a porous medium, the properties of the porous medium (e.g., porosity, tortuosity, saturation) also affect the value. Eddy diffusion coefficients and mechanical dispersion coefficients depend not on the properties of the chemical, but on site-specific flow conditions (and hence can vary with time and location and be anisotropic). They also tend to be scale-dependent (i.e., the values grow as the size of the system increases due to processes such as macrodispersion mentioned above).

Focusing on transport through porous media, D is often approximated as:

where α is the (longitudinal) dispersivity (dimensions of L) and v is the seepage velocity (dimension of L/T), defined, assuming the system is saturated, as Q/(nA), where Q is the volumetric flow rate, A is the cross-sectional area perpendicular to flow and n is the porosity of the saturated porous medium. As noted above, the dispersivity is often treated as scale-dependent.

As will be discussed in great detail in subsequent Units, GoldSim can easily represent all of these physical transport processes (advection, dispersion and diffusion).

Now that we have discussed the major physical processes we will simulate in GoldSim, in the next Lesson we will consider some chemical processes.