Numerical Modeling of Biodegradation

Analytical and Numerical Methods

By

Philip B. Bedient

Modeling Biodegradation

• Three main methods for modeling biodegradation

Monod kinetics

First-order decay

Instantaneous reaction

• Methods can be used where appropriate for aerobic, anaerobic, hydrocarbon, or chlorinated

Microbial Growth• Region 1: Lag phase

microbes are adjusting to the new substrate (food source)

• Region 2 Exponential growth phase,

microbes have acclimated to the conditions

• Region 3 Stationary phase,

limiting substrate or electron acceptor limits the growth rate

• Region 4 Decay phase,

substrate supply has been exhausted

Time

log [X]32 41

Monod Kinetics

• The rate of biodegradation or biotransformation is generally the focus of environmental studies

• Microbial growth and substrate consumption rates have often been described using ‘Monod kinetics’

• C is the substrate concentration [mg/L]

• Mt is the biomass concentration [mg/ L]

• µmax is the maximum substrate utilization rate [sec -1]

• KC is the half-saturation coefficient [mg/L]

−dCdt

=μmaxCMt

KC +C

Monod Kinetics• First-order region,

C << KC the equation can be approximated by exponential decay (C = C0e–kt)

• Center region, Monod kinetics must be used

• Zero-order region, C >> KC, the equation can be approximated by linear decay (C = C0 – kt)

–dCdt

C

First-orderregion

Zero-orderregion

−dCdt

=kCMt

KC

−dCdt

=μmaxMt

Modeling Monod Kinetics

• Reduction of concentration expressed as:

• Mt = total microbial concentration• µmax = maximum contaminant utilization rate per

mass of microorganisms

• KC = contaminant half-saturation constant• ∆t = model time step size• C = concentration of contaminant

ΔC=MtμmaxC

Kc +C

⎛

⎝ ⎜

⎞

⎠ ⎟ Δt

Bioplume II Equation - Monod

• Including the previous equation for reaction results in this advection-dispersion-reaction equation:

∂C∂t

=Dx∂2C

∂x2 −v∂C∂x

−MtμmaxC

Kc +C

⎛

⎝ ⎜

⎞

⎠ ⎟

Multi-Species Monod Kinetics

• For multiple species, one must track the species together, and the rate is dependent on the concentrations of both species

ΔC=MtμmaxC

Kc +C

⎛

⎝ ⎜

⎞

⎠ ⎟

OKo +O

⎛

⎝ ⎜

⎞

⎠ ⎟ Δt

ΔO=MtμmaxF C

Kc +C

⎛

⎝ ⎜

⎞

⎠ ⎟

O

Ko +O

⎛

⎝ ⎜

⎞

⎠ ⎟ Δt

Multi-Species

• Adding these equations to the advection-dispersion equation results in one equation for each component (including microbes)

• BIOPLUME III doesn’t model microbes

∂C∂t

=1Rc

∇ ⋅ (D∇C −vC) −Mtμmax

Rc

CKc +C

⎛

⎝ ⎜

⎞

⎠ ⎟

OKo +O

⎛

⎝ ⎜

⎞

⎠ ⎟

∂O∂t

=∇ ⋅(D∇O−vO) −MtμmaxF C

Kc +C

⎛

⎝ ⎜

⎞

⎠ ⎟

OKo +O

⎛

⎝ ⎜

⎞

⎠ ⎟

∂Ms

∂t=

1

Rm∇ ⋅(D∇Ms - vMs)+MsμmaxY

C

Kc +C

⎛

⎝ ⎜

⎞

⎠ ⎟

O

Ko +O

⎛

⎝ ⎜

⎞

⎠ ⎟ +

kcY(OC)

Rm−bMs

Modeling First-Order Decay

• Cn+1 = Cn e–k∆t

• Generally assumes nothing about limiting substrates or electron acceptors

• Degradation rate is proportional to the concentration

• Generally used as a fitting parameter, encompassing a number of uncertain parameters

• BIOPLUME III can limit first-order decay to the available electron acceptors (this option has bugs)

ModelingInstantaneous Biodegradation

• Excess Hydrocarbon: Hn > On/F

• On+1 = 0 Hn+1 = Hn - On/F

• Excess Oxygen: Hn < On/F

• On+1 = On - HnF Hn+1 = 0

• All available substrate is biodegraded, limited only by the availability of terminal electron acceptors

• First used in BIOPLUME II - 1987

Sequential Electron Acceptor Models

• Newer models, such as BIOPLUME III, RT3D, and SEAM3D allow a sequential process - 1998

• After O2 is depleted, begin using NO3–

• Continue down the list in this order

O2 ––> NO3– ––> Fe3+ ––> SO4

2– ––> CO2

Superposition of Components

• Electron donor and acceptor are each modeled separately (advection/dispersion/sorption)

• The reaction step is performed on the resulting plumes

• Each cell is treated independently

• Technique is called Operator Splitting

Principle of Superposition

Background D.O.

Initial HydrocarbonConcentration

Reduced OxygenConcentration

OxygenDepletion

Reduced HydrocarbonConcentration

+ =

Oxygen Utilization of Substrates

• Benzene: C6H6 + 7.5O2 ––> 6CO2 + 3H2O

• Stoichiometric ratio (F) of oxygen to benzene

• Each mg/L of benzene consumes 3.07 mg/L of O2

F=7.5 molO2

1 molC6H6

32 mgO2

1 molO2

1 molC6H6

(12•6+1•6) mgC6H6

F=3.07 mgO2 mgC6H6

Biodegradation in BIOPLUME II

A A'

B B'

Zone of TreatmentZone of ReducedHydrocarbon Concentrations

Background D.O.

Zone of ReducedOxygen Concentration

Zone of OxygenDepletion

A A'

H

Without Oxygen

B B'

D.O.

Background D.O.

DepletedOxygen

WithOxygen

Initial Contaminant Plume

x x

o o

Concentrationx

8.89e + 2 o Production Well7.78e + 26.67e + 22.22e + 21.11e + 2

1.00e + 3

0.00e + 0o

x

Values represent upper limitsfor corresponding color.

Injection Well

Model ParametersGrid Size 20 x 20 cells

Cell Size 50 ft x 50 ft

Transmissivity 0.002 ft2/sec

Thickness 10 ft

Hydraulic Gradient .001 ft/ft

Longitudinal Dispersivity 10 ft

Transverse Dispersivity 3 ft

Effective Porosity 0.3

Biodegrading Plume

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 1 11 1 0 0 00 0 0 6 123 6 0 0 00 0 1 38 1000 38 1 0 00 0 4 71 831 71 4 0 00 0 7 97 710 97 7 0 00 1 9 104 600 104 9 1 00 0 9 90 449 90 9 0 00 0 5 54 285 54 5 0 00 0 2 19 109 19 2 0 00 0 0 4 24 4 0 0 00 0 0 1 4 1 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 1 1 1 1 1 0 00 0 2 3 4 3 2 0 00 0 3 7 12 8 3 1 00 0 4 11 20 13 5 0 00 0 2 8 11 8 2 0 00 0 0 2 4 2 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

Original Plume Concentration Plume after two years

Extraction Only - No Added O2

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 2 6 2 0 0 00 0 3 7 15 8 3 0 00 0 2 6 10 7 1 0 00 0 0 1 3 1 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

Plume Concentrations

Plume after two years Plume after two years

O2 Injected at 20 mg/L O2 Injected at 40 mg/L

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 1 2 9 3 1 0 00 0 1 5 8 5 1 0 00 0 0 1 3 1 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

Biodegradation Models

• Bioscreen -GSI

• Biochlor - GSI

• BIOPLUME II and III - Bedient & Rifai

• RT3D - Clement

• MT3D MS• SEAM 3D

Name Dimension Description Author

X 1 aerobic, microcolony, Monod Molz, et al. (1986)

BIOPLUME 1 aerobic, Monod Borden, et al. (1986)

X 1 analytical first-order Domenico (1987)

BIOID 1 aerobic and anaerobic, Monod Srinivasan and Mercer (1988)

X 1 cometabolic, Monod Semprini and McCarty (1991)

X 1aerobic, anaerobic, nutrientlimitations, microcolony, Monod

Widdowson, et al. (1988)

X 1aerobic, cometabolic, multiplesubstrates, fermentative, Monod

Celia, et al. (1989)

BIOSCREEN 1 analytical first-order, instantaneous Newell, et al. (1996)

BIOCHLOR 1 analytical Aziz, et al. (1999)

BIOPLUME II 2 aerobic, instantaneous Rifai, et al. (1988)

X 2 Monod MacQuarrie, et al. (1990)

X 2 denitrification Kinzelbach, et al. (1991)

X 2 Monod, biofilm Odencrantz, et al. (1990)

BIOPLUME III 2 aerobic and anaerobic Rifai, et al. (1997)

RT3D 3 aerobic and anaerobic Clement (1998)

Biodegradation Models

Dehalogenation of PCE

• PCE (perchloroethylene or tetrachloroethylene)

• TCE (trichloroethylene)

• DCE (cis-, trans-, and 1,1-dichloroethylene

• VC (vinyl chloride)

C C

Cl Cl

Cl Cl

C C

Cl H

H Cl

C C

Cl H

Cl Cl

C C

H H

Cl H

C C

H H

Cl Cl

C C

Cl H

Cl H

PCE

TCE

DCE's

VC

Dehalogenation

• Dehalogenation refers to the process of stripping halogens (generally Chlorine) from an organic molecule

• Dehalogenation is generally an anaerobic process, and is often referred to as reductive dechlorination

R–Cl + 2e– + H+ ––> R–H + Cl–

• Can occur via dehalorespiration or cometabolism• Some rare cases show cometabolic dechlorination

in an aerobic environment

Chlorinated Hydrocarbons

• Multiple pathways• Electron donor – similar to hydrocarbons• Electron acceptor – depends on human-added electron

donor• Cometabolic

• Mechanisms hard to define• First-order decay often used due to uncertainties in

mechanism

Modeling Dechlorination

• Few models specifically designed to simulate dechlorination

• Some general models can accommodate dechlorination

• Dechlorination is generally modeled as a first-order biodegradation process

• Often, the first dechlorination step results in a second compound that must also be dechlorinated

Sequential Dechlorination

• Models the series of dechlorination steps between a parent compound and a non-hazardous product

• Each compound will have a unique decay constant• For example, the reductive dechlorination of PCE

requires at least four constants• PCE –k1–> TCE

• TCE –k2–> DCE

• DCE –k3–> VC

• VC –k4–> Ethene