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
• Region 1:
Lag phase

Microbial Growth
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
log [X]
1
2
3
• Region 4
Decay phase,

substrate supply has been
exhausted
Time
4
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’
dC max CM t


dt
KC  C
•
•
•
•
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]
Monod Kinetics
•
First-order region,
C << KC the equation can
be approximated by
exponential decay
–dC
(C = C0e–kt)
dt
•
Center region, Monod
kinetics must be used
•
Zero-order region,
C >> KC, the equation
can be approximated by
linear decay
(C = C0 – kt)
dC

 max Mt
dt
Zero-order
region
Firstorder
region
C

dC kCMt

dt
KC
Modeling Monod Kinetics
• Reduction of concentration expressed as:
 C 
C  Mt max 
t
 Kc  C
•
•
•
•
•
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
Bioplume II Equation - Monod
• Including the previous equation for reaction
results in this advection-dispersion-reaction
equation:
 C 
C
 C
C
 Dx 2  v
 Mt  max 

t
x
x
Kc  C 
2
Multi-Species Monod Kinetics
• For multiple species, one must track the species
together, and the rate is dependent on the
concentrations of both species
 C  O 
C  Mt max 

t
Kc  C  Ko  O 
 C  O 
O  Mt max F 

t
 Kc  CKo  O 
Multi-Species
• Adding these equations to the advectiondispersion equation results in one equation for
each component (including microbes)
 C  O 
C
1

=
  (DC  vC)  Mt max 


t
Rc
Rc Kc  C Ko  O 
 C  O 
O
=   (DO  vO)  Mt  max F 
K  O 
t
K

C
 c
 o

 C  O  k cY (OC)
=
  (DMs - vMs )  M smax Y 
 bM s

 
t
Rm
Rm
Kc  C  Ko  O 
M s
1
• BIOPLUME III doesn’t model microbes
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)
Modeling
Instantaneous Biodegradation
• Excess Hydrocarbon: Hn > On/F
•
On+1 = 0
• Excess Oxygen:
•
On+1 = On - HnF
Hn+1 = Hn - On/F
Hn < On/F
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+ ––> SO42– ––> 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
Initial Hydrocarbon
Concentration
+
Reduced Hydrocarbon
Concentration
=
Background D.O.
Oxygen
Depletion
Reduced Oxygen
Concentration
Oxygen Utilization of Substrates
• Benzene:
C6H6 + 7.5O2 ––> 6CO2 + 3H2O
• Stoichiometric ratio (F) of oxygen to benzene
7.5 molO 2 32 mgO 2
1 molC 6 H6
F
1 molC 6 H 6 1 molO 2 (12  6 1 6) mgC 6 H 6
F  3.07 mgO 2 mgC 6 H 6
• Each mg/L of benzene consumes 3.07 mg/L of O2
Biodegradation in BIOPLUME II
A
A'
H
Without Oxygen
With
Oxygen
Zone of Treatment
Zone of Reduced
Hydrocarbon Concentrations
B
Background D.O.
B'
A
A'
D.O.
Background D.O.
Depleted
Oxygen
Zone of Oxygen
Depletion
Zone of Reduced
Oxygen Concentration
B
B'
Initial Contaminant Plume
Concentration
xx x
oo o
1.00e + 3
8.89e + 2
7.78e + 2
6.67e + 2
2.22e + 2
1.11e + 2
0.00e + 0
x Injection Well
o Production Well
Values represent upper limits
for corresponding color.
Model Parameters
Grid 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
Longitud inal Dispersivity
10 ft
Transverse Dispersivity
3 ft
Effective Porosity
0.3
Biodegrading Plume
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
4
7
9
9
5
2
0
0
0
0
0
0
0
0
0
0
0
0
1
6
38
71
97
104
90
54
19
4
1
0
0
0
0
0
0
0
0
0
1
11
123
1000
831
710
600
449
285
109
24
4
1
0
0
0
0
0
0
0
0
0
1
6
38
71
97
104
90
54
19
4
1
0
0
0
0
0
0
0
0
0
0
0
0
1
4
7
9
9
5
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Original Plume Concentration
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
3
4
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
3
7
11
8
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
4
12
20
11
4
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
3
8
13
8
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
3
5
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Plume after two years
Extraction Only - No Added O2
Plume Concentrations
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
7
6
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
15
10
3
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
8
7
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
5
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
9
8
3
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
5
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
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
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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
Biodegradation Models
Name
Description
Author
1
aerobic, microcolony, Monod
Molz, et al. (1986)
1
aerobic, Monod
Borden, et al. (1986)
X
1
analytical f ir st-order
Domenico (1987)
BIOID
1
aerobic and anaerobic, Monod
Srinivasan and Mercer (1988)
X
1
cometabolic, Monod
Semprini and McCarty (1991)
X
1
aerobic, anaerobic, nutrient
limitations, microcolony, Monod
Widdowson, et al. (1988)
X
1
aerobic, cometabolic, multiple
substrates, f ermentative, Monod
Celia, et al. (1989)
BIOSCREEN
1
analytical f ir st-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)
X
BIOPLUME
Dimension
Dehalogenation of PCE
• PCE (perchloroethylene
or tetrachloroethylene)
Cl
Cl
Cl
• TCE (trichloroethylene)
Cl
H
• DCE (cis-, trans-,
H
C
and
Cl
1,1-dichloroethylene
• VC (vinyl chloride)
Cl
C C
PCE
TCE
C C
H
Cl
Cl
Cl
H
Cl
C C
C
Cl
DCE's
C C
H
Cl
H
H
C C
Cl
H
Cl
VC
H
H
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 firstorder 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
TCE
DCE
VC
–k1–>
–k2–>
–k3–>
–k4–>
TCE
DCE
VC
Ethene