1. Principles and Applications of Backward-in-Time Modeling of Contaminants in the Environment
Roseanna M. Neupauer
Department of Civil, Environmental, and Architectural Engineering
University of Colorado
March 26, 2008

2. Motivation Source Characterization
TCE is observed
in wells
Where did it
come from?
When was it
released?
Flow
direction
How much TCE
was released?
(Alaska Department of Environmental Conservation)

3. Motivation Capture Zone Delineation
Water supply wells
Where does their
water come from?

4. Motivation Aquifer Vulnerability
How sensitive is
the pesticide
concentration here …
… to a pesticide
application here?
… or here?
… or here?
… or here?
© Minnesota Pollution Control Agency

5. Motivation Groundwater Age Simulations
Water sampled in well
When did the water
recharge the aquifer?
(Stoner et al., 1997)

6. Motivation Remediation Prioritization
Abandoned Mines
Which mine contributes the most
contamination to the water bodies?
How long does it take for the mine
contamination to reach the water bodies?
(Ryan and Reynolds, 2003)

7. Forward Modeling vs. Backward Modeling
Where is the contamination going?
Where did the contamination come from?
Flow
direction

8. Forward Modeling vs. Backward Modeling
Information is known (or assumed to be known) about the source
Information is known about the present state of contamination
Flow
direction

9. Forward Modeling vs. Backward Modeling
One or a few sources
Many possible receptors
Many possible sources
One or a few receptors
Flow
direction

10. Forward Modeling vs. Backward Modeling
Models transport from the source to the receptors
Models transport from the receptor to the possible sources
Flow
direction

11. Forward Modeling vs. Backward Modeling
Concentrations
Probability density function
Flow
direction

12. Forward Contaminant Transport= -
(vqC) (q D ÑC ) Ñ
- lqC + qI CI - qoC Rq
Co(x) =(M/q)d(x-xo)
Contour interval: C = 0.1 g/m3
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume

13. Forward Location PDF, fX(x;t)= -
(vqC) (q D ÑC ) Ñ
- lqC + qI CI - qoC Rq
Co(x) =(M/q)d(x-xo)
Contour interval: C = 0.1 g/m3; fX = 1 x 10-5 m-2
C(x,t)
dC(x)
fX(x;t)= q =
q(x) M
dM(xo)

14. Forward Location PDF, fX(x;t)
Co(x) =(M/q)d(x-xo) xo xw
Contour interval: C = 0.1 g/m3; fX = 1 x 10-5 m-2
dC(x)
Sensitivity of concentration, C,
at any X to source mass, M, at xo
fX(x;t) =
q(x)
dM(xo)

15. Backward Location PDF, fX(x;t)
fo(x) =d(x-xo) xw
dC(xw)
fX(x;t) =
q(x)
dM(x)

16. Backward Location PDF, fX(x;t)
fo(x) =d(x-xo) xw
dC(xw)
Sensitivity of concentration, C,
at xw to source mass, M, at any X
fX(x;t) =
q(x)
dM(x)

17. Backward Travel Time PDF and CDF5
x 10-5
8.64
0.36
5.78
0.24
PDF fT (d-1)
CDF FT (-)
2.88
0.12
dCf(xw)
Sensitivity of flux concentration, Cf,
to source mass, M
fT(t;x) = Q(x)
dM(x)

18. Forward and Backward Models
dM(xo)
fX(x;t)=
dC(x)
q(x)
Forward Equation C t Rq = Ñ (q D ÑC ) - Ñ
(vqC)
- lqC + qI CI - qoC
dM(x)
fX(x;t)=
dC(xw)
q(x)
Backward (Adjoint) Equation
 y t ¶h ¶C Rq + = Ñ ( qD Ñy ) Ñ
(v q y)
- lqy - qI y +
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume y = adjoint state (sensitivity) t = backward time

19. Forward and Backward Models
Forward Equation C t Rq Ñ (q D ÑC ) - = Ñ
(vqC)
- lqC + qI CI - qoC
Backward (Adjoint) Equation
 y t ¶h ¶C Rq = Ñ ( qD Ñy ) + Ñ
(v q y)
- lqy - qI y +
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume y = adjoint state (sensitivity) t = backward time

20. Forward and Backward Models
Forward Equation C t Rq Ñ (q D ÑC ) - = Ñ
(vqC)
- lqC + qI CI - qoC
Backward (Adjoint) Equation
 y t ¶h ¶C Rq = Ñ ( qD Ñy ) + Ñ
(v q y)
- lqy - qI y +
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume y = adjoint state (sensitivity) t = backward time

21. Forward and Backward Models
Forward Equation C t Rq Ñ (q D ÑC ) - = Ñ
(vqC)
- lqC + qI CI - qoC
Backward (Adjoint) Equation
 y t ¶h ¶C Rq = Ñ ( qD Ñy ) + Ñ
(v q y)
- lqy - qI y +
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume y = adjoint state (sensitivity) t = backward time

22. Forward and Backward Models
Forward Equation C t Rq Ñ (q D ÑC ) - = Ñ
(vqC)
- lqC + qI CI - qoC
Backward (Adjoint) Equation
 y t ¶h ¶C Rq = Ñ ( qD Ñy ) + Ñ
(v q y)
- lqy - qI y +
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume y = adjoint state (sensitivity) t = backward time

23. Forward and Backward Models
Forward Equation C t Rq Ñ (q D ÑC ) - = Ñ
(vqC)
- lqC + qI CI - qoC
Backward (Adjoint) Equation
 y t ¶h ¶C Rq Ñ ( qD Ñy ) + = Ñ
(v q y)
- lqy - qI y +
Location PDF
dM(x)
dC(xw,t)
dCf(xw,t)
y =
h =
C(x,t) d(x-xw) d(t)
fX(x;t) =q(x)y(x,t)
Travel Time PDF
fT(t;x) =|Q(x)|y(x,t)
Cf(x,t) d(x-xw) d(t)
h =

24. Applications of Backward-in-Time Modeling
Source characterization (location, release time):
Unconditioned location and travel time PDFs
Conditioned location and travel time PDFs
Capture zone delineation: travel time CDF
Groundwater age dating: travel time CDF
Remediation Prioritization:
Marginal sensitivity of concentration to source mass
Travel time CDF

25. Source Location Identification (MMR TCE Plume; 1997 5 mg/L contour)1
scale in km 3 4
Scale in miles
C = 58 mg/L
C = mg/L
C = 150 mg/L 2
C = 5110 mg/L
Model provided by Chunmiao Zheng; data provided by Jacobs Engineering Group
Model provided by Chunmiao Zheng; data provided by Jacobs Engineering Group

26. Backward Governing Equation
 y t = +
(v q y) ( qD Ñy ) Ñ
- lqy - qI y + Rq ¶h ¶C
Location PDF
h =
C(x,t) d(x-xw) d(t)
fX(x;t) =q(x)y(x,t)
One simulation for each observation

27. Backward Location PDF for 1962 (Neupauer and Wilson, 2005)
Sample Location
Contour Interval: 2 x 10-7 m-2 1
scale in km
Contour: 2 x 10-7 m-2 1
scale in km 3 4
C = 58 mg/L
C = mg/L
C = 150 mg/L 2
C = 5110 mg/L

28. Backward Location PDF for 1962 (Neupauer and Wilson, 2005)1
scale in km 3 4
C = 58 mg/L
C = mg/L
C = 150 mg/L 2
C = 5110 mg/L
Contour Interval: 2 x 10-7 m-2
Sample Location 1
scale in km
Contour: 2 x 10-7 m-2
Multiple-detection Location PDF
Contour Interval: 2 x 10-6 m-2

29. Conditioned Location PDF (Neupauer and Lin, 2006)4 3
Contour Interval: 2 x 10-7 m-2
Sample Location 1
scale in km
Contour: 2 x 10-7 m-2
Conditioned Location PDF
Contour Interval: 2 x 10-6 m-2

30. Backward Governing Equation
 y t = +
(v q y) ( qD Ñy ) Ñ
- lqy - qI y + Rq ¶h ¶C
Travel Time PDF
h =
Cf(x,t) d(x-xw) d(t)
fT(t;x) =Q(x)y(x,t)
One simulation for each observation

31. Backward Travel Time PDFs1 3 4 2 T

32. Capture Zone Delineation
Reinjection Well
Extraction Well
Infiltration
Trench
TCE Plume
Plume boundary: 5 mg/L
Scale in km
Groundwater Remediation System
Massachusetts Military Reservation
(model provided by Chunmiao Zheng)

33. Backward Governing Equation
 y t = +
(v q y) ( qD Ñy ) Ñ
- lqy - qI y + Rq ¶h ¶C
Travel Time PDF
h =
Cf(x,t) d(x-xw) d(t)
Travel Time CDF
h =
Cf(x,t) d(x-xw)
FT(t;x) =|Q(x)|y(x,t)

34. Capture Zone Delineation (Neupauer and Wilson, 2004)
Groundwater Remediation System
Massachusetts Military Reservation
(model provided by Chunmiao Zheng)
10-yr probabilistic capture zone
Groundwater Remediation System
Massachusetts Military Reservation
TCE Plume
Infiltration
Trench
Extraction Well
Reinjection Well
Scale in km
Plume boundary: 5 mg/L

35. Groundwater Age Dating (Weissmann et al. 2002)10 5 km

36. Prioritizing Remediation Activities
Warden Gulch

37. Which mine is the major source?
 y t = +
(v q y) ( qD Ñy ) Ñ
- lqy - qI y + Rq ¶h ¶C
Marginal Sensitivity
h =
C (x,t) d(x-xw) d(t)
dC(t)|stream
y(x,t) = dM

38. Model Domain and Flow Field
Warden Gulch
Head contours: 2-m interval m
Hypothetical
mine sites
Sampling sites

39. Marginal sensitivity dC(t)|stream y(x,t) = dM Conservative solute
t=50 yr
t=125 yr
dC(t)|stream
y(x,t) = dM
Conservative solute
Contour intervals: 10-11,10-10,…10-7 m-3
t=50 yr
t=125 yr
Strontium, R=151

40. When will remediation have an impact?
 y t = +
(v q y) ( qD Ñy ) Ñ
- lqy - qI y + Rq ¶h ¶C
Travel Time CDF, Solute Age Distribution
h =
Cf(x,t) |at stream
FT(x,t) =|Q(x)|y(x,t)

41. Solute Age Distribution
t=50 yr
t=125 yr
Contour intervals: 10-4,10-3,10-2,10-1,0.95
t=50 yr
t=125 yr
t=250 yr
t=500 yr
t=250 yr
t=500 yr
Conservative solute
Strontium, R=151

42. Summary Backward-in-time modeling:
is useful if information is known about where contamination is now, and information is desired about where contamination was in the past
produces probability density functions that can be used to
Determine groundwater age or solute age
Delineate probabilistic capture zones
Identify characteristics of contaminant sources
uses standard numerical codes with some minor post-processing

43. Dispersion - C t (vqC) (q D ÑC ) Ñ - lqC + qI CI - qoC Rq =
- lqC + qI CI - qoC Rq
Mean groundwater flow direction
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume

44. Advection - C t (vqC) (q D ÑC ) Ñ - lqC + qI CI - qoC Rq = t=t1
- lqC + qI CI - qoC Rq
Dissolved contaminant
t=t1
t2>t1
groundwater flow path
t3>t2
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume

45. Sorption - C t (vqC) (q D ÑC ) Ñ - lqC + qI CI - qoC Rq = FLOW t=t1
- lqC + qI CI - qoC Rq
FLOW
t=t1
t2>t1
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume

46. Transformations - C t (vqC) (q D ÑC ) Ñ - lqC + qI CI - qoC Rq =
- lqC + qI CI - qoC Rq
MICROBE
t=t1
CO2
t2>t1
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume

47. Forward Governing EquationC t = -
(vqC) (q D ÑC ) Ñ
- lqC + qI CI - qoC Rq
Sources of contamination:
- natural recharge
- infiltration from rivers
- spills
- accidental releases
Sinks of contamination:
- discharge to rivers
- withdrawals at wells
C = concentration R = retardation coefficient t = time
q = porosity D = dispersion coefficient l = decay rate
v = groundwater velocity qI = inflow rate per unit volume CI = inflow concentration qO = outflow rate per unit volume

48. Multiple Observations
P fx(x;t,xwi)
i=1 N
fx(x;t,xw1,xw2, …,xwN) = N õ ó
P fx(x;t,xwi) dx
i=1
fx(x;t,xwi) = single observation location probability
fx(x;t ,xw1,xw2,…,xwN) = multiple observation
location probability

49. Why Use Concentrations?
xo = source location Mo = source mass

50. Conditioning on Measurements
fx (x;t|Ĉ (xw1,t1)…Ĉ (xwn,tn)) =
backward location PDF n a
P fx(x;t,xwi,ti) P(Ĉ(xwi,ti)| M,x) dM
i=1
source mass
Conditioned location PDF
measured concentration
PDF of measured concentration
P(Ĉ(xwi,ti)| M,x) ~ N(C(xwi,ti), s2)