tracers for flow and mass transport philip bedient rice university 2004

Tracers for Flow and Mass Transport

Philip Bedient

Rice University

2004

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

Transport of Contaminants

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

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

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

€

∂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

2-D Computed Plume Map

Advection and Dispersion

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

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

€

∞

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

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

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

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)

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.

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

Domenico and Schwartz (1990)

VerticalSource

Plume at time t

x

z

y

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

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

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

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

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

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

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

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

PLAN VIEW OF TANK

Flow

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

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

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

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

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)