1. Tracers for Flow and Mass Transport
Philip Bedient
Rice University
2004

2. Transport of Contaminants
Transport theory tries to explain the rate and extent of migration of chemicals from known source areas
Source concentrations and histories must be estimated and are often not well known
Velocity fields are usually complex and can change in both space and time
Dispersion causes plumes to spread out in x and y
Some plumes have buoyancy effects as well

3. Transport of Contaminants

4. What Drives Mass Transport: Advection and Dispersion
Advection is movement of a mass of fluid at the average seepage velocity, called plug flow
Hydrodynamic dispersion is caused by velocity variations within each pore channel and from one channel to another
Dispersion is an irreversible phenomenon by which a miscible liquid (the tracer) that is introduced to a flow system spreads gradually to occupy an increasing portion of the flow region

5. Advection and Dispersion in a Soil Column
Source Spill t = 0
Conc = 100 mg/L
Longitudinal Dispersion t = t1
n = Vv/Vt
porosity
Advection t = t1 C t

6. Contaminant Transport in 1-DFx
Fx + (dFx/dx) dx y z
Fx = total mass per area transported in x direction
Fy = total mass per area transported in y direction
Fz = total mass per area transported in z direction

7. Substituting in Fx for the x direction only yields
Accumulation Dispersion Advection
C = Concentration of Solute [M/L3]
D = Dispersion Coefficient [L2/T]
V = Velocity in x Direction [L/T]

8. 2-D Computed Plume Map
Advection and Dispersion

9. Analytical 1-D, Soil Column
Developed by Ogata and Banks, 1961
Continuous Source
C = Co at x = 0 t > 0
C (x, ) = 0 for t > 0

10. Error Function - Tabulated Fcn
Erf (0) = 0
Erf (3) = 1
Erfc (x) = 1 - Erf (x)
Erf (–x) = – Erf (x) x
Erf(x)
Erfc(x) 1
.25
.276
.724
.50
.52
.48
1.0
.843
.157
2.0
.995
.005
Erf x

11. Contaminant Transport Equation
C = Concentration of Solute [M/L3]
DIJ = Dispersion Coefficient [L2/T]
B = Thickness of Aquifer [L]
C’ = Concentration in Sink Well [M/L3]
W = Flow in Source or Sink [L3/T]
n = Porosity of Aquifer [unitless]
VI = Velocity in ‘I’ Direction [L/T]
xI = x or y direction

12. Analytical Solutions of Equations
Closed form solution, C = C ( x, y, z, t)
Easy to calculate, can often be done on a spreadsheet
Limited to simple geometries in 1-D, 2-D, or 3-D
Limited to simple sources such as continuous or instantaneous or simple combinations
Requires aquifer to be homogeneous and isotropic
Error functions (Erf) or exponentials (Exp) are usually involved

13. Numerical Solution of Equations
Numerically -- C is approximated at each point of a computational domain (may be a regular grid or irregular)
Solution is very general
May require intensive computational effort to get the desired resolution
Subject to numerical difficulties such as convergence problems and numerical dispersion
Generally, flow and transport are solved in separate independent steps (except in density-dependent or multi-phase flow situations)

14. Domenico and Schwartz (1990)
Solutions for several geometries (listed in Bedient et al. 1999, Section 6.8).
Generally a vertical plane, constant concentration source. Source concentration can decay.
Uses 1-D velocity (x) and 3-D dispersion (x,y,z)
Spreadsheets exist for solutions.
Dispersion = axvx, where ax is the dispersivity (L)
BIOSCREEN (1996) is handy tool that can be downloaded.

15. BIOSCREEN Features Answers how far will a plume migrate?
Answers How long will the plume persist?
A decaying vertical planar source
Biological reactions occur until the electron acceptors in GW are consumed
First order decay, instantaneous reaction, or no decay
Output is a plume centerline or 3-D graphs
Mass balances are provided

16. Domenico and Schwartz (1990)y
Plume at time t
Vertical
Source x z

17. Domenico and Schwartz (1990)
For planar source from -Y/2 to Y/2 and 0 to Z Y
Flow x Z
Geometry

18. Instantaneous Spill in 2-D
Spill source C0 released at x = y = 0, v = vx
First order decay l and release area A
2-D Gaussian Plume moving at velocity V

19. Breakthrough Curves
2 dimensional Gaussian Plume

20. Tracer Tests
Aids in the estimation of average hydraulic conductivity between sampling locations
Involves the introduction of a non-reactive chemical species of known concentration
Average seepage velocities can be calculated from resulting curves of concentration vs. time using Darcy’s Law

21. What can be used as a tracer?
An ideal tracer should:
1. Be susceptible to quantitative determination
2. Be absent from the natural water
3. Not react chemically or be absorbed
4. Be safe in drinking water
5. Be inexpensive and available
Examples:
Bromide, Chloride, Sulfates
Radioisotopes
Water-soluble dyes

22. Hour 14 Hour 43 Hour 85 Hour 8 Hour 30 Hour 55 Hour 79
The first line was very close to the gravel and was influenced by that. Also, there may have been some areas of hard pack that spilt the flow in the that region.
30 hours after the 1st line peaked—the 2nd line peaked
40 hours after that—the 3rd line peaked
If you extend the third curve out till it hits the x-axis, the area under the 2nd and 3rd curves are equal, I.e. bromide mass is conserved
Hour 8
Hour 30
Hour 55
Hour 79

23. Bromide Tracer Front - ECRS
Inlet
Outlet 1 2 3 4 5 6 7 8 21 22 23 24 25 26 27 28 9 10 12 14 16 18 20 11 13 15 17 19
Black t= 40 hrs
Red t= 85 hrs

24. New Experimental Tank
5000 mg/L Bromide tracer in advance of ethanol test
Pumped into 6 wells for 7 hour injection period
Pumping rate of 360 mL/min was maintained
Background water flow rate was mL/min

25. PLAN VIEW OF TANK
Flow

26. Line A Shallow

27. Line B Intermediate

28. Line E Center

29. Line I Shallow

30. July 2004 New Tank prior to 95E test
(5.5 ft to 9.5 ft down tank)