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