Solute Movement and Solute Dispersion

When the soil solution has a specific solute of interest in it – say a salt, a fertilizer, a pollutant, or a tracer – it can be useful to know how much it disperses.  In a classical sense, this amounts to asking 1) how fast does it move relative to the mean soil water flow, and 2) how much does it spread out as it moves?  Whether we’re working with fertilizers or pollutants, these are important questions.

An observation: if we establish steady water flow through a soil column, then at a particular time (call it t=0) we introduce a tracer (say, a salt) into the water, here’s what we’ll see in the column’s outflow:

If we know the size and porosity of the sample, and the flow rate, we can “normalize” the flow to units of “pore volume” – that is, 1 PV is the volume of water needed to precisely replace the water currently in the saturated sample.  This makes it easier to compare across different sample sizes and flow rates.

Likewise, we can replace the actual salt concentration by a normalized concentration, usually written as C/C₀.  This lets us compare across experiments with different concentrations (or even different solutes).

So now we have this:

If the salt had flowed through the sample with no dispersion, then the concentration would have jumped from 0 to C₀ at precisely 1 PV.  This extreme – which exists only in our minds – is called “piston flow” (Why only in our minds?  Because we can’t “turn off” diffusion, and velocity is always faster at the center of the pore than at the edges).  But in the real world, the solute arrival is dispersed around the mean, hence the term “dispersion”.

What are the mechanisms of dispersion?  An obvious one is diffusion, which is simply the random movement of molecules, as Einstein showed.  If you put, say, a drop of dye into a beaker of water, the dye will slowly diffuse, eventually reaching a uniform concentration everywhere in the beaker.  In mathematical terms, this is known as Fick’s 2^nd law (after the German physical chemist Fick):

, that is, the change in concentration over time, at a given point, is proportion to the slope of the concentration gradient, with movement from high concentration to low.  Of course, molecules can’t sense the gradient to know where to go: this “law” is simply (as Einstein showed) a mathematical description of the macroscopic consequence of random motion.

If we ran the same experiment at several different flow rates, and if diffusion were the only mechanism of dispersion, we’d expect to see the following:

-- because slow flow has more time, so there would be more diffusion and hence more spreading.

But in fact, for flow through a soil we see the opposite: faster flow has GREATER dispersion.  This leads us to suspect that diffusion is not the main mechanism involved.  So what are other candidate mechanisms of dispersion, and what are useful ways to think about dispersion as a whole?

The classical analysis follows from the research of Sir Geoffrey Taylor, who studied dispersion of solute in a liquid flowing through a tube (G. I. Taylor was a famous physicist whose work touched on many of the big issues in physics during the 20^th century.  This particular paper, “Dispersion of soluble matter in solvent flowing slowly through a tube,” Proc. Roy. Soc. CCXIX: 186-203, 1953, is probably his best known in hydrology, but by no means his greatest contribution to science).

Taylor examined the case of injection of a small quantity of ink into a tube in which water was flowing at a constant rate.  He measured the dispersion by measuring the absorption, by the ink, of light shining through the tube.  Knowing that flow through a tube has a parabolic velocity profile (Poiseuille’s law; see Fig. 7.2 in Hillel), he might have expected to see an asymmetrical breakthrough curve like this:

which is what would result if the velocity profile were the ONLY mechanism.  But instead Taylor saw a symmetrical, diffusion-like BTC.  He concluded that the symmetry came about through an interaction between convection (the mass movement, with a parabolic velocity profile), and diffusion.  He also noted that the dispersion due to (convection + diffusion) was less than that due to convection alone (although way more than that due to diffusion alone): in other words, a random process (diffusion) was decreasing the total randomness (dispersion) of the system!

You can download here the demo program I showed in class.  Remember that it is for dispersion between two parallel plates, rather than dispersion in a tube.

The key to Taylor’s analysis is that diffusion has the net result that molecules do not stay in one streamline (the path that would be followed by an individual molecule in the absence of diffusion) forever.  Molecules near the wall (velocity = 0) will diffuse toward the center, and those at the center will diffuse toward the wall.  This reduces the spread of solute that would have resulted if there had been convection only.  In Taylor’s words, “The time necessary for appreciable effects to appear, owing to convective transport, is long compared with the ‘time of decay’ during which radial variations of concentration are reduced to a fraction of their initial value through the action of molecular diffusion.”

In the simple case of tube, the velocity profile is known.  In a soil, we suppose that 1) some streamlines are longer than others, due to a distribution of tortuosity, and 2) some streamlines flow faster than others, due to differences in pore size.  These two can be combined by the concept of travel time, which is what the X-axis of a BTC really means.  So I’ll restate Taylor’s comment as a condition rather than a conclusion: dispersion will appear diffusion-like if there is sufficient (diffusive) mixing between short- and long-travel time pathways.  Corollary: if there is insufficient mixing, dispersion will be primarily due to advection and probably won’t look diffusive.

Now the observation that dispersion appears to increase at higher velocities makes sense: when velocity is increased, exchange between short-travel-time and long-travel-time streamlines decreases, and so the net spreading is greater.  Based upon these observations, it is an easy step to describe two common mathematical descriptions of dispersion in soils:

the ADE (Advection-Dispersion Equation), also called the CDE (Convection-Dispersion Equation):  This model is simply a mathematical statement of Taylor’s observations.  If the center of mass of the solute travels at the same speed as the mean water flux, and the spreading is normal (Gaussian, Fickian, gives a normal curve), then the concentration of solute at a point (i.e., what you’d see coming out the end of a soil column in a lab experiment) changes over time as

.  Notice that when spreading is diffusion-like, the width of the solute “plume” increases proportionately to the square root of time: this is a signature of a diffusive process.  The “D” in this equation is mathematically like diffusion, but it is higher than what would be observed by diffusion alone.  It is called the dispersion coefficient, but to emphasize that it includes both convective and diffusive mechanisms, some people will call it the diffusive-dispersive coefficient.  If the dispersion is seen to increase with velocity, it is often given as a composite term: D = a v.  This factor a, called the dispersivity, now has units of length, with typical values in the range of 1 meter.  Conceptually the ADE is similar enough to Taylor’s tube that we can think of it as just flow in a single tube.

the stream-tube model, popularized by Bill Jury as the “transfer function model,” and especially as his “Convective Lognormal Transfer” model, or CLT.  This is like the case of very fast flow in Taylor’s tube: there is not enough time for diffusive exchange between streamlines, so we can think of each streamline as acting independently.  If we know the velocity distribution, we can predict the dispersion (Jury’s CLT is a stream-tube model that posits a log-normal distribution of velocities).  Notice that in this case the center of mass of the solute still moves at the same velocity as the mean water flux, but the dispersion increases linearly with time, rather than with the square root of time.  This indicates that the process is NOT diffusive.  Conceptually we might think of this as flow in several parallel tubes of different diameters.

But (you might object) soils aren’t so uniform: often there are aggregates, macropores, and the like.  In such a case, we could think of the flow as taking place through only part of the soil (say the macropores), while the solute diffuses into and out of the aggregates.  Such a model is also commonly used to describe solute movement through soils.  Called the Mobile-Immobile Model, or MIM, this third model supposes that water flows through some fraction of the total porosity, say the mobile region or theta_m, and not through the rest, the immobile region (where theta = theta_m + theta_im).  Exchange between the mobile and immobile regions is through diffusion only, so we can think of this as being flow through a single tube with dead-ends, or rest stops, off it.  This diffusive exchange has the net effect of slowing down the solute movement – solute spends lots of time in immobile regions.  So solute doesn’t move at the same velocity as the water, but it scales the same: velocity is linear with distance.  This brings us to the idea of the retardation factor: if the solute moves more slowly than the water, we say it is retarded (slowed down), and stick a retardation factor into the ADE: .  Retardation can be caused by chemical reactions, such as sorption, as well as be diffusion into immobile regions.  In fact, the MIM can also describe so-called two-site sorption: if the mobile region is thought of as instantaneous sorption sites, and the immobile region as sorption sites with a time delay, the same math and BTCs result.

Recall that the ADE turns into a stream tube model when there is insufficient time for diffusive exchange.  This means that, as velocity increases, the spreading goes from being proportional to the square root of time, to being proportional to time.  Presumably there is a transition zone, within which the exponent moves from 0.5 to 1.  How do we describe this region?  A new dispersion model, developed in the 1970s for electron transport in amorphous silica and only recently applied to dispersion in hydrology, appears to work well in this region as well as in all the others (in other words, all 3 other models are effectively special cases of this more general model).  This model, called the Continuous Time Random Walk (CTRW), concentrates on the fact that solute hops between short-transit-time and long-transit-time states or streamlines: it you know the transition probability, you know something about the dispersion.  Another interesting thing about the CTRW is that it expresses dispersion in terms of a single factor, beta, that is like a dispersion coefficient in that it describes the extent of spreading, but also like a scaling exponent in that it tells you how the solute moves and spreads over time.  In the CTRW, the solute center of mass does not necessarily scale linearly with time, and the solute spreading scales with anything between time⁰ and time¹.  Check out the CTRW home page for more information.

To summarize the scaling:

Model	Solute center of mass	Solute Std. Dev.
ADE	t	Sqrt(t)
MIM	t	Sqrt(t)
CLT	t	t
CTRW	t^0 to t^1	t^0 to t^1

Why am I putting so much stress on this scaling?  Because:

· It’s currently an area of great debate and confusion, but if you learn it as a normal part of dispersion rather than as something strange or unusual, you’ll be better off.

· It’s how a physicist thinks about these things, and this is after all a physics class.

· If you’re off by a scalar – say, your dispersion coefficient is 20% low – that’s not too bad, and you still understand what is going on.  But if dispersion is increasing with t, while you predict that it increases with sqrt(t), you clearly don’t know what is happening.

· I’m writing a paper about it right now (Uh-oh, the secret’s out!).