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Working With Simple Models to Predict Contaminant Migration
Matt SmallU.S. EPA, Region 9, Underground Storage
Tanks Program Office
What is a Model?• A systematic method for analyzing real-
world data and translating it into a meaningful simulation that can be used for system analysis and future prediction.
• A model should not be a “black box.”
Modeling Process
• Determine modeling objectives
• Review site conceptual model
• Compare mathematical model capabilities with conceptual model
• Model calibration
• Model application
Site Conceptual Model
Ground Water Flow DirectionGround Water Flow Direction
Dissolved
Source
Sources ReceptorsPathwaysPrimaryTanksPipingSpills
SecondaryResidual NAPL
SoilVaporsGround WaterSurface Water
PeopleAnimals, FishEcosystemsResources
Mathematical Model
• A mathematical Model is a highly idealized approximation of the real-world system involving many simplifying assumptions based on knowledge of the system, experience and professional judgment.
2+2=4 ( )0
kttC C e
e
K dhv
n dx
Model Assumptions
• Common simplifying assumptions– 2-Dimensional flow field (no flux in z direction)– Uniform flow field (1-D flow)– Uniform properties (homogenous conductivity)– Steady state flow (no change in storage)
Model Selection
• Select the simplest model that will fit the available data
Input Parameters
• Model input parameter values can be either variable, uncertain, or both.– Variable parameters are those for which a value can
be determined, but the value varies spatially or temporally over the model domain.
– Uncertain parameters are those for which a value cannot be accurately determined with available data.
• To evaluate variability and uncertainty we can use several possible values to describe a given input parameter and bound the model result.
Lumped Input parameters
• To simplify the mathematics, and quantify poorly understood (complex) natural phenomena, subsurface processes are typically described by five parameters:– source– velocity– retardation– dispersion– decay
Plume Migration due to Advection
Source
Input Parameters: Ground Water Flow
C Cvx t
Ground Water Flow DirectionGround Water Flow Direction
e
K dhv
n dx
•Processes Simulated–Ground Water Flow Rate, Seepage Velocity, or Advection
•Input Parameters–Hydraulic conductivity
–Gradient
–Aquifer thickness
–Aquitards/aquicludes
Ground Water Flow Rate Example Calculation
s
hydraulic conductivity x gradientGround Water Seepage Velocity (v ) =
effective porosity se
Kiv
n
Hydraulic conductivity (K) estimated to be between 10-2 and 10-4 cm/sec. Ground water gradient measured from ground water contour map 0.011 ft/ft. Effective Porosity estimated to be 30% or 0.3.
410 0.011sec
??0.3s
e
cm ft
Ki ftv
n
Distance ftTravel Time
ftGround Water Flow Rate year
1 2
1,000 ft 1,000 ftt = ?? t = ??
ft ftX X year year
years years
Input Parameters: Retardation
Ground Water Flow DirectionGround Water Flow Direction
R = 1.8 For BenzeneR = 1.8 For Benzene R = 1.1 For MTBER = 1.1 For MTBE
R = 1 For Advective FrontR = 1 For Advective Front
•Processes Simulated–Retarded contaminant transport
–Adsorption and desorption processes
–Interactions between contaminants, soil, and water
•Input Parameters–Fraction of organic carbon
–Organic carbon partitioning coefficient
–Soil bulk density
–Porosity
1 d bd oc oc
KK f K R
Source
Retarded Ground Water Flow Rate Example Calculation
ftGround Water Flow Rate yearTravel Time =
Distance ft
1 2
1,000 ft 1,000 ftt = 264 t = 2.6
ft ft3.45 345 year year
years years
R = 1.8 for benzene R = 1.1 for MTBE
1, benz 2, benz
1,000 ft 1,000 ftt =1.8 475 t =1.8 4.7
ft ft3.79 379 year year
years years
1, MTBE 2, MTBE
1,000 ft 1,000 ftt =1.1 290 t =1.1 2.9
ft ft3.79 379 year year
years years
Input Parameters: Dispersion
mechanicalD v
total molecular mechanicalD D D
Ground Water Flow DirectionGround Water Flow Direction
Dispersed Plume
Dx
Dz
Dy
Non-Dispersed Plume
Source
Fick's Law molecular
dCF D
dx
•Processes Simulated–Macroscopic spatial variability of hydraulic conductivity
–Microscopic velocity variations
•Input Parameters–Ground water seepage velocity
–Dispersivity
–Molecular diffusion coefficient
Input Parameters:Biodegradation and Decay
Ground Water Flow DirectionGround Water Flow DirectionAdvective/Dispersive Front Advective/Dispersive Front (no decay or retardation)(no decay or retardation)
Retarded FrontRetarded Front
Dissolved
Decaying FrontDecaying Front
Source
( )0
1/ 2
ln 2
ttC C e
t
•Processes Simulated–Chemical transformation and decay
–Biodegradation
–Volatilization
•Input Parameters–Initial concentrations
–First order decay rate or half life
3-D Contaminant Fate and Transport in Ground Water
2 2 2
2 2 2x x x x
C C C C CR D D D Ct x x y z
Numerical Model Example
Model Output
Making Regulatory Decisions
• What models can do:– Predict trends and directions of changes– Improve understanding of the system and
phenomena of interest– Improve design of monitoring networks– Estimate a range of possible outcomes or
system behavior in the future.
Making Regulatory Decisions
• What models CANNOT do:– Replace site data– Substitute for site-specific understanding of
ground water flow– Simulate phenomena the model wasn’t
designed for.– Represent natural phenomena exactly
– Predict unpredictable future events
– Eliminate uncertainty