Working With Simple Models to
Predict Contaminant Migration
Matt Small
U.S. EPA, Region 9, Underground Storage
Tanks Program Office
What is a Model?
• A systematic method for analyzing realworld 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
Source
Dissolved
Ground Water Flow Direction
Sources
Primary
Tanks
Piping
Spills
Secondary
Residual NAPL
Pathways
Soil
Vapors
Ground Water
Surface Water
Receptors
People
Animals, Fish
Ecosystems
Resources
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.
Ct  C0 e
K dh
v
ne dx
(  kt )
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
Input Parameters:
Ground Water Flow
•Processes Simulated
–Ground Water Flow
Rate, Seepage Velocity,
or Advection
•Input Parameters
–Hydraulic conductivity
Source
Plume Migration
due to Advection
–Gradient
–Aquifer thickness
–Aquitards/aquicludes
Ground Water Flow Direction
v
K dh
ne dx
C C
v

x t
Ground Water Flow Rate
Example Calculation
Ground Water Seepage Velocity (vs ) =
hydraulic conductivity x gradient
effective porosity
vs  
Ki
ne
 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.
vs 
Travel Time 
t1 =
Ki

ne
104
cm
ft
0.011
sec
ft
 ??
0.3
Distance ft
Ground Water Flow Rate ft
1,000 ft
 ?? years
ft
X
year
t2 =
year
1,000 ft
 ?? years
ft
X
year
Input Parameters: Retardation
•Processes Simulated
–Retarded
contaminant transport
–Adsorption and
desorption processes
–Interactions between
contaminants, soil, and
water
Source
•Input Parameters
–Fraction of organic
carbon
–Organic carbon
partitioning coefficient
–Soil bulk density
–Porosity
R = 1.1 For MTBE
R = 1.8 For Benzene
R = 1 For Advective Front
Ground Water Flow Direction
K d  f oc K oc
R  1
K d b

Retarded Ground Water Flow
Rate Example Calculation
Ground Water Flow Rate ft
Travel Time =
t1 =
year
Distance ft
1,000 ft
 264 years
ft
3.45
year
t2 =
1,000 ft
 2.6 years
ft
345
year
 R = 1.8 for benzene
 R = 1.1 for MTBE
t1, MTBE =1.1
t1, benz =1.8
1,000 ft
 290 years
ft
3.79
year
1,000 ft
 475 years
ft
3.79
year
t 2, MTBE =1.1
t 2, benz =1.8
1,000 ft
 2.9 years
ft
379
year
1,000 ft
 4.7 years
ft
379
year
Input Parameters: Dispersion
•Processes Simulated
–Macroscopic spatial
variability of hydraulic
conductivity
–Microscopic velocity
variations
Dy
Source
Dispersed
Plume
Non-Dispersed
Plume
Dx
•Input Parameters
–Ground water
seepage velocity
–Dispersivity
–Molecular diffusion
coefficient
Dz
Ground Water Flow Direction
Fick's Law  F  Dmolecular
dC
dx
Dmechanical   v
Dtotal  Dmolecular  Dmechanical
Input Parameters:
Biodegradation and Decay
•Processes Simulated
–Chemical
transformation and
decay
Decaying Front
–Biodegradation
–Volatilization
Retarded Front
Source
•Input Parameters
–Initial concentrations
–First order decay rate
or half life
Dissolved
Ground Water Flow Direction
Advective/Dispersive Front
(no decay or retardation)
Ct  C0 e
t1/ 2 
(  t )
ln 2

3-D Contaminant Fate and
Transport in Ground Water
C
C
2C
2C
 2C
R
 x
 Dx 2  Dx 2  Dx 2    C
t
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