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Ground Water journal
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Technical Commentary

ON THE MISUSE OF THE SIMPLEST TRANSPORT MODEL


by Ernesto Baca

Consider the following scenario. You are a groundwater professional. A client comes in, gives you a site investigation report, and asks for your professional opinion regarding a contaminant plume's travel time from the source location to the receptor, assuming a continuous release of contaminants.

Being a groundwater professional, you get the plume map and you measure the distance from the source to the receptor, for example, from a pit to the tip of the plume as measured at a monitoring well located at the prop erty boundary. Let's say this distance (x) is 300 m. In reading the site investigation report, you learn the following values for site parameters: a hydraulic conductivity (K) of 3000 m/yr (i.e., 10-2 cm/sec), a hydr aulic gradient (i) of 0.002 m/m, an effective porosity (n) of 0.25, a retardation factor (R) of 2, and a contaminant decay of about zero. Furthermore, the vertical travel time from the pit to the groundwater is on the order of 1 week.

Great, now you get your calculator and a piece of paper and using the average linear groundwater velocity (v) equation you calculate the travel time (t). The average linear groundwater velocity is:
v = K*i/n = (3000 m/yr)*(0.002 m/m)/(0.25) = 24 m/yr.

Since the retardation factor is 2, you figure the contaminant travels at half of the calculated velocity or 12 m/yr (i.e., vc = (24 m/yr)/2 = 12 m/yr).

You can now estimate the travel time (t) by dividing the travel distance (x) by the contaminant velocity (vc) or:
t = x/vc = (300 m)/(12 m/yr) = 25 yrs.

You tell your client that the contaminant took about 25 years to reach the property boundary, and you would be WRONG.

Let's try an advective-dispersive one-dimensional analytical model to demonstrate the problem with this analysis. The model used (Ogata, 1970) is presented as equation 9.5 in Freeze and Cherry's book (1979). At this hypothetical site, we assume a dispersivity (a) of 30 m, (taking 10% of the plume length as the estimate for dispersivity). The dispersion coefficient (D) is approximately equa l to the dispersivity times the average linear velocity, in this case the dispersion coefficient is 720 m2/yr (= 30 m * 24 m/yr). Since this model does not account for retardation, the contaminant velocity will be used, i.e., vc = 12 m/yr, to calculate the travel time. Using the Ogata model to solve for the travel time, you will find that it takes about 9.2 years for the tip of the plume (here defined as 1% of the source concentration, C0) to reach the property bound ary, 300 m downgradient from the source. This travel time estimate is about 16 years shorter than the original estimate! In 25 years the travel distance of the tip of the plume would be more than 610 m downgradient of the source, or more than twice the di stance observed.

Why is this common scenario wrong? Lets go back to basics. The scenario outlined above describes a plume that has only two major transport components, advection and dispersion. The travel time calculated from the equat ion above (v = K*i/n) only accounts for the advection component of this transport. By measuring the plume from the source location to the tip of the plume, one is implicitly considering both advective and dispersive transport processes. Therefore, the lar ger the dispersive effects the more your estimate will be off.

How can we avoid making such a mistake in the future? The mistake here was in assuming that the travel distance of the tip of the plume divided by time is equal to the average linear velocity. The relationship of trave l distance divided by time being equal to the average linear velocity is only true at the estimated location of the mid-level concentration. In this case, the travel distance should be measured from the location of the initial release, the downgradient ed ge of the pit, to the estimated location of the mid-level concentration point of the plume (i.e., C=0.5*C0 or C/C0=0.5), not the tip of the plume. In this example, one should be able to estimate that the mid-level concentration point is located at approximately 110 m (i.e., x = t * vc = 9.2 yrs * 12 m/yr) from the pit and still have the tip of the plume at 300 m from the pit. It is important to realize that, from the mid-level concentration point, or inflection point, to the tip of the plume contaminants move faster than the calculated average linear contaminant velocity.

There is nothing new about these concepts. For example, Freeze and Cherry (1979) explain these basic concepts in Chapter 9 of their book. What is surprising is how many professionals overlook the details of the circumstances under which these calculations can be used and, therefore, misuse these simple calculations. These remarks are just a reminder to be careful on how these "back-of -the-envelope" calculations are used and interpreted.


References

Freeze, R. A. and J. A. Cherry. 1979. Groundwater. Prentice-Hall, Englewood Cliffs, NJ. 604 pp.

Ogata, Akio. 1970. Theory of dispersion in a granular medium. U.S. Geological Survey Professional Paper 411-I, p. 134.


CORRECTIONó
Technical Commentary

"On The Misuse Of The Simplest Transport Model," by Ernesto Baca, Ground Water, July-August 1999, v. 37, n. 4, p. 483.

Although the concepts presented in this Technical Commentary are correct, the sample calculations related to the Ogata model are in numerical error. There are three points to be made at this time. The first point is the correction of the error, and the following two points are further explanations and clarifications of the calculations.

(1) In performing the calculations presented in the Technical Commentary, it was erroneously assumed that equation 9.5 in Freeze and Cherry (1979, p. 391) was correct. There is a misprint of that equation in some early printings of the book. The term that is incorrectly printed is in the argument of the exp(.) function. The erroneously printed equation shows this term as ‾v l / Dl. The correct argument should be ‾v l / Dl.

(2) Ogata's equation as shown in Freeze and Cherry's equation 9.5 has to be corrected for the fact that retardation is not equal to 1. To make this adjustment ‾v in eq uation 9.5 (or v in my Technical Commentary) should be substituted by ‾v divided by the retardation factor (or vc in my Technical Commentary). If the dispersion coefficient is made to be a function of the average linear velocity (v) and dispersivit y (α), as was done in this example, this term must also be adjusted the same way. This adjustment is consistent with, for example, van Genuchten and Alves' (1982) solution to their Case A1, which is the same as Ogata's equation but accounts for reta rdation. The corrected equation, using the notation used in the Technical Comment, is:

C/C0=0.5*[erfc((x - vc*t)/(2*sqrt(α*vc*t))) + exp(x/α)*erfc((x + vc*t)/(2*sqrt(α*vc*t)))]

where, sqrt(.) is the square root function, exp(.) the exponential function, and erfc(.) is the complimentary error function (see Freeze and Cherry, Appendix V, 1979).

Using this corrected equation and the original parameters, the tip of the plume will reach the boundary after 8.7 years and not 9.2 years as mentioned in the original comments. In 25 years the travel distance of the tip of the plume would be almost 630 m downgradient of the source. In this example, the travel distance from the pit (x), due to advective transport only is about 104 m (i.e., x = t * vc = 8.7 yrs * 12 m/yr). Dispersive effects place the tip of the plume at 300 m from the pit. The c oncept to remember is that the tip of the plume is located at a distance significantly different than 104 m. The tip of the plume is at 300 m or almost three times as far away from the source as would have been predicted by using the v=k*i/n relationship alone.

(3) The contaminant travel velocity (vc) generally describes the rate of movement of the mid-level concentration (C/C0=0.5) point in a one-dimensional plume such as Ogata's model. In cases where the plume is relatively short and/or a large dispersion coe fficient is involved, as in my example, a source boundary effect is noted. When there is a source boundary effect, the travel distance computed using the contaminant travel velocity (i.e., x = t * vc) corresponds to a level higher than the mid-lev el concentration. The deviation from the mid-level concentration is described by Ogata (1970), and is shown graphically in his Figure 3 as a function of the magnitude of the source boundary effect. Negligible boundary effects are noted in long plumes, p lumes with small dispersion, or a combination of these two conditions.


References

Freeze, R. A. and J. A. Cherry. 1979. Groundwater. Prentice-Hall, Englewood Cliffs, NJ. 604 pp.

Ogata, Akio. 1970. Theory of dispersion in a granular medium. U.S. Geological Survey Professional Paper 411-I, p. 134.

van Genuchten, M. Th., and W. J. Alves. 1982. Analytical Solutions of the One-Dimensional Convective-Dispersive Solute Transport Equation. U.S. Department of Agriculture, Technical Bulletin No. 1661.


Discussion of Papers

"On The Misuse Of The Simplest Transport Model," by Ernesto Baca, Ground Water, July-August 1999, v. 37, n. 4, p. 483.

DISCUSSION by P. Binning, Department of Civil, Surveying and Environmental Engineering, The University of Newcastle, Callaghan, N.S.W. 2308, Australia.

see Ground Water, January-February 2000, v. 38, n. 1, pp. 4-5.

AUTHOR'S REPLY by Ernesto Baca, Environmental Consultant, 3216 Georgetown, Houston, TX 77005.

The comments by Binning relate to the phrase "mid-level concentration" or, algebraically, the C/C0=0.5 point of the solution (using the notation presented in my Technical Commentary). Notice that in my Technical Commentary I refer to the "estimated" location of this point. The reason for this wording is because, although well aware that the location of the mid-level concentration is not exactly at vc * t (i.e., contaminant travel velocity times travel time), the focus of the paper was on the difference of the location of a point at or near the middle of the plume versus a point at the tip of the plume. Binning is absolutely correct in stating that there are differences between the location of the mid-level concentration point and that predicted by using vc * t. Binning's detailed explanation of these differences around the mid-level concentration point are welcomed. In fact, my original article contained a computational error and a Correction was submitted to Ground Water. As part of that Correction I took the opportunity to highlight the same facts that Binning reports (see item (3) of Correction in Ground Water, Nov.-Dec. 1999).


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