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Tracers for Flow and Mass Transport
Philip Bedient
Rice University
2004
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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
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Transport of Contaminants
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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
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Advection and Dispersionin a Soil Column
Source Spill t = 0Conc = 100 mg/L
Longitudinal Dispersion t = t1
Advection t = t1
C
t
n = Vv/Vt
porosity
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Contaminant Transport in 1-D
Fx = total mass per area transported in x direction
FxFx + (dFx/dx) dx
Fy = total mass per area transported in y direction
Fz = total mass per area transported in z direction
z
y
€
Fx = vxnC − nDx
∂C
∂x
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€
∂C
∂t= D
∂2C
∂x 2
⎛
⎝ ⎜
⎞
⎠ ⎟−V
∂C
∂x
C = Concentration of Solute [M/L3]D = Dispersion Coefficient [L2/T]V = Velocity in x Direction [L/T]
Accumulation Dispersion Advection
€
Inflow − Outflow = n ∂C
∂t
⎛
⎝ ⎜
⎞
⎠ ⎟dxdydz
Substituting in Fx for the x direction only yields
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2-D Computed Plume Map
Advection and Dispersion
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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
CC0
=0.5
erfcx−vt2 Dt
⎛ ⎝ ⎜
⎞ ⎠ ⎟
+expvxD
⎛ ⎝ ⎜ ⎞
⎠ ⎟ erfc
x+vt2 Dt
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
⎫
⎬ ⎪ ⎪
⎭ ⎪ ⎪
€
∞
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Error Function - Tabulated Fcn
€
Erf (x) =2
πe−u2
du0
x
∫
Erf (0) = 0Erf (3) = 1Erfc (x) = 1 - Erf (x)Erf (–x) = – Erf (x)
x Erf(x) Erfc(x)0 0 1
.25 .276 .724
.50 .52 .48
1.0 .843 .157
2.0 .995 .005Erf
x
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Contaminant Transport Equation
€
∂CB
∂t=
∂
∂xI
BDIJ
∂C
∂xJ
⎛
⎝ ⎜
⎞
⎠ ⎟−
∂
∂xI
BCVI( ) −′ C W
n
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
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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
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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)
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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 = xvx, where x is the dispersivity (L)• BIOSCREEN (1996) is handy tool that can be
downloaded.
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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
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Domenico and Schwartz (1990)
VerticalSource
Plume at time t
x
z
y
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Domenico and Schwartz (1990)
€
C x,y,z, t( )C0
=1
8
⎛
⎝ ⎜
⎞
⎠ ⎟erfc
x − vt
2 α xvt
⎡
⎣ ⎢
⎤
⎦ ⎥
erfy + Y 2
2 α y x
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥- erf
y −Y 2
2 α y x
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
erfz + Z
2 α z x
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥- erf
z − Z
2 α z x
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
For planar source from -Y/2 to Y/2 and 0 to Z
Flow x
Y
Z
Geometry
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Instantaneous Spill in 2-D
€
C x, y,z, t( ) =C0A
4(πt)(DxDy )1/ 2•
exp[(x − vt)2
4Dxt−
y 2
4Dyt− λ t]
Spill source C0 released at x = y = 0, v = vx
First order decay and release area A
2-D Gaussian Plume moving at velocity V
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Breakthrough Curves
Predicted Rice ECRS Tracer Test w/ ????? cm
0
50
100
150
200
250
300
350
400
0 10 20 30 40 50 60 70 80 90
Time (hours)
Concentration (mg/L)
C-1
C-2
C-3
C-4
C-5
C-6
C-7
C-8
2 dimensional Gaussian Plume
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Tracer Tests• Aids in the estimation of average hydraulic
conductivity between sampling locations• Involves the introduction of a non-reactive
chemical species of knownconcentration
• Average seepage velocities can be calculated from resulting curves of concentration vs. time using Darcy’s Law
Predicted Rice ECRS Tracer Test w/ ????? cm
0
50
100
150
200
250
300
350
400
0 10 20 30 40 50 60 70 80 90
Time (hours)
Concentration (mg/L)
C-1
C-2
C-3
C-4
C-5
C-6
C-7
C-8
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What can be used as a tracer?• An ideal tracer should:
1. Be susceptible to quantitative determination2. Be absent from the natural water3. Not react chemically or be absorbed4. Be safe in drinking water5. Be inexpensive and available
• Examples:– Bromide, Chloride, Sulfates– Radioisotopes– Water-soluble dyes
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Tracer Test Results from Locations Down the Centerline in Rice ECRS
0
100
200
300
400
500
600
700
800
900
0 20 40 60 80 100 120
Time (hours)
Concentration (mg/L)
Line 21
Line 22
Line 23
Line 24
Line 25
Line 26
Line 27
Line 28
Hour 14 Hour 43
Hour 85
Hour 8 Hour 30 Hour 55 Hour 79
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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 Arrows @ t= 40 hrs
Red Arrows @ t= 85 hrs
Bromide Tracer Front - ECRSBromide Tracer Front - ECRS
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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 900-1000 mL/min
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PLAN VIEW OF TANK
Flow
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New Tank Bromide Tracer Test July 2004
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 5 10 15 20 25 30 35
Time (Hours)
0.5A
1A
2A
4A
Line A Shallow
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New Tank Bromide Tracer Test July 2004
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 5 10 15 20 25 30 35
Time (Hours)
Bromide Concentration (mg/L)
1B
2B
4B
Line B Intermediate
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New Tank Bromide Tracer Test July 2004
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 5 10 15 20 25 30 35
Time (Hours)
Bromide Concentration (mg/L)
0.5E
1E
2E
4E
Line E Center
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New Tank Bromide Tracer Test July 2004
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 5 10 15 20 25 30 35
Time (Hours)
Bromide Concentration (mg/L)
0.5I
1I
2I
4I
Line I Shallow
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Lines Time 2 Time 1 Distance (ft) Gradient Seepage Velocity (ft/hr) Vs (ft/day) Vs (m/day) K (ft/hr) K (ft/day) K (cm/sec)
0.5A to 1A 4 3 0.5 0.027778 0.500 12.000 3.658 5.580 133.92 4.724E-021A to 2A 5 4 1 0.027778 1.000 24.000 7.315 11.160 267.84 9.449E-02
2A to 4A 13 5 2 0.027778 0.250 6.000 1.829 2.790 66.96 2.362E-02
0.5E to 1E 6 4 0.5 0.027778 0.250 6.000 1.829 2.790 66.96 2.362E-021E to 2E 8 6 1 0.027778 0.500 12.000 3.658 5.580 133.92 4.724E-02
2E to 4E 12 8 2 0.027778 0.500 12.000 3.658 5.580 133.92 4.724E-02
0.5I to 1I 8 6 0.5 0.027778 0.250 6.000 1.829 2.790 66.96 2.362E-021I to 2I 10 8 1 0.027778 0.500 12.000 3.658 5.580 133.92 4.724E-02
2I to 4I 14 10 2 0.027778 0.500 12.000 3.658 5.580 133.92 4.724E-02
1B to 2B 10 5 1 0.027778 0.200 4.800 1.463 2.232 53.57 1.890E-022B to 2B 17 10 2 0.027778 0.286 6.857 2.090 3.189 76.53 2.700E-02
July 2004 New Tank prior to 95E test July 2004 New Tank prior to 95E test (5.5 ft to 9.5 ft down tank)(5.5 ft to 9.5 ft down tank)