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

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)

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

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

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

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)

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

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

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

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

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

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

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)

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)

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

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)

Backward Travel Time PDF and CDF 5 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)

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

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

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

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

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

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 =

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

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 = 203.1 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

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

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 = 203.1 mg/L C = 150 mg/L 2 C = 5110 mg/L

Backward Location PDF for 1962 (Neupauer and Wilson, 2005) 1 scale in km 3 4 C = 58 mg/L C = 203.1 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

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

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

Backward Travel Time PDFs 1 3 4 2 T

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)

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)

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

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

Prioritizing Remediation Activities Warden Gulch

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

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

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

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)

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

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

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

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

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

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

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

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

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

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)