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Mathematical Modeling of Pollutant Transport in Groundwater. Rajesh Srivastava Department of Civil Engineering IIT Kanpur. Outline of the Talk Sources Processes Modelling Applications. Sources of GW Pollution Irrigation Landfills Underground Storage tanks Industry. Advection - PowerPoint PPT Presentation
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Mathematical Modeling ofMathematical Modeling ofPollutant Transport in GroundwaterPollutant Transport in Groundwater
Rajesh SrivastavaDepartment of Civil Engineering
IIT Kanpur
Outline of the TalkOutline of the Talk
•SourcesSources•ProcessesProcesses•ModellingModelling•ApplicationsApplications
Sources of GW PollutionSources of GW Pollution
•Irrigation•Landfills•Underground Storage tanks•Industry
Advection• Mass transport due to the flow of the water• The direction and rate of transport coincide
with that of the groundwater flow.
Diffusion • Mixing due to concentration gradients
Dispersion • Mechanical mixing due to movement of
fluids through the pore space
Dispersion
• Spreading of mass due to– Velocity differences
within pores – Path differences due to the
tortuosity of the pore network.
Position in Pore
Ve
loci
ty
Pore SpacesPore Spaces
Gas Gas
Mobile/flowing liquid
Stagnant or Immobile liquid Intra-particle pores
Figure: Courtesy Sylvie Bouffard, Biohydrometallurgy group, Vancouver 12 18
Brief Chronology
Unsaturated flow equation by Richards (1931)Coats and Smith (1964) proposed dead-end pores in oil wellsEquilibrium reactive transport theories proposedBreakthrough curves with pronounced tailings observedNon-equilibrium models developedGoltz and Roberts (1986) physical non-equilibrium modelBrusseau et al. (1989) developed MPNESlow and Fast Transport model developed by Kartha (2008)
Experimental Setup
Time
C/C
o
0
1
Start
Time
C/C
o
0
1
Start
INFLOW A OUTFLOW B
A B
Conservation of Liquid Mass
.l ll l lu S
t
where Sl is source/sink term.
l l lu K h
ˆl
ll
Ph z g
ˆl l l
l
ku P g
l g cP P P
lr satk k k
Hydraulic conductivity
Darcy velocity in unsaturated porous medium
Hydraulic head based on elevation head z
Darcy velocity
Liquid pressure in unsaturated conditions
Intrinsic permeability in unsaturated conditions
ˆl l
ll
k gK
•Relation between suction pressure, liquid pressure, and liquid saturation•Relation between relative permeability and liquid saturation
Brooks-Corey and van Genuchten Relations
1le l r rEffective saturation is given as
Gas pressure Pg is considered zero, therefore l cP P
B.C. - Model V.G. Model
Suction pressure
Relative Permeability
11* 1l
c le
gP
21/1 (1 )lr le lek
1c b leP P
23
lr lek
• van Genuchten equations
11
21/
11
1
1 1
l rle
r
r le lek
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.001 0.01 0.1 1 10
Suction (cm)
Wat
er C
on
ten
t
0.0001
0.001
0.01
0.1
1
0.001 0.01 0.1 1 10
Suction
Rel
. Hyd
. Con
d.
Transport ModelReactive advective-dispersiveReactive advective-dispersive equation
Here we use multi-process non-equilibrium equations.
MPNE model
Liquid exists in mobile and immobile phase.Solid in contact with mobile and immobile liquid.Instantaneous sorption mechanism between liquids and solids.Rate-limited sorption mechanism between liquids and solids.
*. .l ll l l l
CR u C D C
t
MPNE Equations
1m ml m m
S CF K
t t
221m
m m l m m m
Sk F K C S
t
1im iml im im
S CF K
t t
2
21imim im l im im im
Sk F K C S
t
1 21 1im im iml im b b l m im
C S Sf f C C
t t t
*2 0
.
. 1
ml l im m l b m m l l m
l l im m l m im b m m l m m m l
CC f F K u C
t t
D C C C f k F K C S S C
Where, Si - concentration of metal in sorbed phase (i.e. solid), Ki - adsorption coefficient, ki - sorption rate, α - mass transfer rate between mobile and immobile liquid, Fi - fraction for instantaneous sorption, f - fraction of sorption site in contact with mobile liquid.
Numerical Solution for Unsaturated Flow
The mass conservation equation is solved for liquid pressureImplicit finite-difference method is used
, ,
, ,, ,
, ,
, ,
1, 1
, ,1, 1, , , ,
1, 1 1, 1
ˆ 1
i j k
q i j kq i j k
i j k
i j k
n s nl l l
i j kn sl i j k l i j k
n s n sl lp b
l l qq l q
R St
P Pkg A
x
, ,
, , , , , ,
, ,
1,
1, 1 1, 1, 1
1,
i j k
i j k i j k qi j k
qi j k
n sln s n s n s
l l ln sq l
dRR R P
dP
, , , , , ,
1, 1 1, 1, 1
i j k i j k i j k
n s n s n sl l lP P P
Residual form of conservation of mass equation for liquid
Taylor’s series expansion of residual equation will lead to the following form
Pressure values updated at each iteration step
Numerical Solution for MPNE Transport
Conservation of mass for metal is solved for concentration in liquidImplicit finite-difference in time step used for formulationsResidual formulation obtained for concentration in mobile liquid
, ,
, ,
, , 2
1 1 1
10 , ,
1 1 12 ,1
1
11
q i j k
m q i j k
q q i j k im im
im
n n n nm m m m bn
C l l l im l b m m i j k l q l qq q
bn n n n n n nl l m q l m l m C im S im i jn
q C
C C C CR S C f F K d u A
t x
u C A C C A C A SA
,
121
, ,
11
1
k
n nim im im l im imn
b m m l m m i j kim
S k t F K Cf k F K C
k t
1 1
2 2
1(1 )
1n n n
m m m m l m mm
S S k t F K Ck t
The finite-difference formulation for sorbed concentration is
The residual formulation for solute concentration in mobile liquid is:
Updated Concentration is 1 1n n nm m mC C C
Taylor’s series expansion of the above residual equation
1 1m
m m
nCn n n
C C mnq m
dRR R C
dC
Verification of the Numerical Model
FLOW
(Compared with VG’s Flow Model and Kuo et al. (1989) Infiltration Model)
Inflow qt = 3 cm/d
10 cm
Water Table
15
0
cm
ksat 5.905×10-9 cm2
ε 0.45
σr 0.22
α* 0.025 cm
λ 0.394
Δz 1 cm
Δt 100 s
ρb 1.360 g.cm-3 α 8.681×10-7 s-1
θ 0.473 km 7.673×10-4 s-1
q 5.914×10-4 cm.s-1 kim 7.673×10-4 s-1
dz 0.34 cm Km 0.429 cm3.g-1
L 30.0 cm Kim 0.416 cm3.g-1
T0 7.672 days (662861 s) f 0.929
Fm 0.5 Fim 0.5
MPNE Transport
30 c
m
Input Parameters
Concept of Slow and Fast TransportMovement of liquids is heterogeneousLiquid flow is conceptualized as slow and fast zonesMultiple sources of non-equilibrium solute interactions occurs between solids and different liquids 4
IImmobile
LiquidCim and σim
IISlow Liquid
Csl and σsl
IIIFast LiquidCfs and σfs
IV
Instant Sorption
Site,Sim1
V
Rate – limited
Sorption Site,Sim2
VI
Instant Sorption
Site,Ssl1
VII
Rate-limited
Sorption Site,Ssl2
Kim kimKsl ksl
αim αsf
Conservation of solute mass
*0. .fs
l fs l fs fs fs f l fs fs l sf fs sl
Cu C D C S C C C
t
• In slow liquid
*0
2
.
.
1
sll sl b l sl sl l sl sl
sl l sf sl fs l im sl im l sl sl sl
b sl sl l sl sl sl
Cf F K u C
t
S C C C C C D C
f k F K C S
• Solute mass conservation in fast liquid
Conservation of solute mass….
1sl sll sl sl
S CF K
t t
221sl
sl sl l sl sl sl
Sk F K C S
t
• Rate of change of instantaneously sorbed solute mass
• Rate of change of rate-limited sorbed mass
• Solute mass conservation in immobile liquid
Similar instantaneous and rate-limited sorption exist for immobile liquid
2(1 ) 1 1im
l im b l im im b im im l im im im
l im sl im
Cf F K f k F K C S
t
C C
, ,
, ,
, ,
1
1, ,
1 1
1 10 , ,
1
1
q q i j k
q q i j k
q i j k
q
n nfs fs bn
l fs i j k l fs fs qq
n nfs fs fs b n n
l fs q q fs l sff s sl i j kq fs q
C Cu C A
t
u C Cd A S C C C
x
, ,
, ,
, ,
1
1 1 1 1 1, , 0 , ,
1 1
1 12 , ,1
q q i j k
q i j k
q i j k
n nsl sl n n n n n
l sl l b sl sl i j k l sl sl q sl l sf sl fs l im sl im i j kq
n nsl sln n
b sl sl l sl sl sl i j k l q sl qq
C Cf F K u C A S C C C C C
t
C Cf k F K C S d u A
x
q
The implicit finite-difference form of metal mass conservation in fast movingfast moving liquid in a FD cell is:
The implicit finite-difference form of metal mass conservation in slow movingslow moving liquid in a FD cell is:
The implicit finite-difference form of metal mass conservation in immobileimmobile liquid in a FD cell is:
1
1 1 1 121 1 1
n nim im n n n n
l im b im im b im im l im im im l im sl im
C Cf F K f k F K C S C C
t
FINITE-DIFFERENCE FORMULATION OF SFT MODEL
Residual equations are formed for the finite-difference equations for conservation of metal mass in fast and slow moving liquids.
Residual equations expanded using Taylor’s series approximation.
Formulations continued….
1,
1, 1 1, 11,
fs
fs fs
n sCn s n n s
C C fsn sfs
dRR R C
dC
1,
1, 1 1, 11,
sl
sl sl
n sCn s n n s
C C sln ssl
dRR R C
dC
The linear system of equations is solved
Update concentration terms:
1, 1 1, 1, 1n s n s n sfs fs fsC C C
1, 1 1, 1, 1n s n s n ssl sl slC C C
Numerical Model Validation…..
Verification and Evaluation (Brusseau et. al., 1989)
Bulk density 1.36 g.cm-3
Porosity 0.473
Inflow rate 5.11 cm.d-1
Dispersivity 0.34 cm
Column height 30.0 cm
Immobile saturation 0.071
Sorption coefficient Ksl 0.429 cm3.g-1
Sorption coefficient Kim 0.416 cm3.g-1
Sorption rate 0.663 d-1
Mass transfer rate αim 0.075 d-1
Instantaneous sorption fraction 0.50
Pulse duration 7.67 d
Brusseau, M.L., Jessup, R.E., Rao, P.S.C.: Modeling the transport of solutes….. Water Resources Research 25 (9), 1971 – 1988 (1989)
REMEDIATION OF GROUNDWATER POLLUTION DUE TO CHROMIUM IN NAURIA KHERA AREA OF
KANPUR
Central Pollution Control Board Lucknow
National Geophysical Research Institute Hyderabad
Industrial Toxicology Research Centre Lucknow
Indian Institute of Technology Kanpur
8 0 .2 5 5 8 0 .2 6 8 0 .2 6 5 8 0 .2 7
2 6 .4 3 5
2 6 .4 4
2 6 .4 4 5
2 6 .4 5
2 6 .4 5 5
Can
al
P an d u R iver
N au riy ak h era
N R
R o a d
R a il
N G R I / C P C B / IT R C / IIT -K
0
0
Location map of Nauriyakhera IDA, Kanpur, U.P.
~ 5 km2
CGWB Observations in Kanpur 1994-2000
• Cr 6+ found in groundwater generally exceed > 0.11 mg/l (Permissible Limit is 0.05 mg/l)
• Cr 6+ observed in Industrial areas in depth range of 15 – 40 m >10 mg/l
• Nauriakhera (Panki Thermal Power Plant Area) Cr 6+ 14 m - 8.0 mg/l15 m – 0.31 mg/l35 m – 7.0 mg/l40 m – 0.68 mg/l
• Used Chromite ore (Sodium Bichromate) dumped in pits and low lying areas cause of Cr pollution
• Persistence in the phreatic zone up to 40 m depth despite presence of thick clay zones
8 0 .2 5 5 8 0 .2 6 8 0 .2 6 5 8 0 .2 7
2 6 .4 3 5
2 6 .4 4
2 6 .4 4 5
2 6 .4 5
2 6 .4 5 5
1
23
4
5 678
9
1 011
1 2
1 3
1 41 5
1 6
1 7
1 8
1 92 02 12 22 32 4
2 52 6
2 7
2 8
3 0
3 13 2
Can
al
P an d u R iver
N au riy ak h era
N R
R o a d
R a il
O b serv a tio n W ell
N G R I / C P C B / IT R C / IIT -K
0
0
Observation Wells in Nauriyakhera IDA, Kanpur, U.P.
8 0 .2 5 5 8 0 .2 6 8 0 .2 6 5 8 0 .2 7
2 6 .4 3 5
2 6 .4 4
2 6 .4 4 5
2 6 .4 5
2 6 .4 5 5
0 .5 3 5
3 .2 7 51 .3 0 4
0 .7
4 .5 7 45 .1 8 74 .7 6 50 .5 7 4
0 .0 4
0 .70
1 .4 4
0 .8 3 5
3 .7 4 30
0
0 .4 7 8
11 .6 5
0 .0 0 40 .5 8 300 .40 .4 0 40
0 .4 8 70 .0 9 6
0 .2 5 5
0 .0 9
0
00 .0 5 7
Can
al
P an d u R iver
N au riy ak h era
N R
N G R I / C P C B / IT R C / IIT -K
0
0
Total Chromium (mg/l) in groundwater - Nauriyakhera IDA, Kanpur
March 2005
Can
a l
P an d u R iver
N au riyak h era
N R
N G R I / C P C B / IT R C / IIT -K
m g/l
T - C r (m g /l)P o st M o n soo n
20 0 4
0
2
4
6
8
1 0
Total Chromium (mg/l) in groundwater -Nauriyakhera IDA, Kanpur
Fence Diagram – Nauriyakhera IDA, Kanpur
Total Chromium Plume from Source after 10 years
Total Chromium Plume from Source after 40 years
Application to Heap Leaching
• Heap leaching is a simple, low-cost method of recovering precious metals from low-grade ores.
• Ore is stacked in heaps over an impermeable leaching-pad.
• Leach liquid is irrigated at the top
• Liquid reacts with metal and dissolves it.
• Dissolved metal collected at the bottom in the leaching pad.
• Traditional methods of gold extraction viz - ore sieving, washing, etc. are obsolete and uneconomical.
• Pyro-metallurgy is highly costly and non-viable for low-grade ores.
• Leaching is the only process to extract metallic content from the low-grade ores.
• Among leaching methods – Heap leaching is most economical
Why Heap Leaching ?
Why we are interested in Heap Leaching?
• Heaps are generally stacked in unsaturated conditions.
• The dissolution reaction occurs in the presence of oxygen.
• The flow of liquid and metals inside the heaps are governed by principles of flow and solute transport through porous medium
• Solving unsaturated flow equations and reactive transport equations enables us to model heap leaching process.
Types of leaching
Underground in-situ leaching
Tank leaching Heap leaching Pressure leaching
Components of a heap
Impermeable leach pad Liners Crushed metal ore Irrigation system Pregnant solution pond Barren solution pond
ORE PREPARATION
Recovery Plant
Mine Pit
Sprinklers or wobblers
Pregnant solution pond Barren Solution Pond
Leach pad
Heap
Effluent outflow into the leaching pad
Average outflow Cumulative outflow
The average outflow gradually attains steady state Sudden decrease in outflow on stoppage of irrigation Rate of recovery reduced after stoppage
MPNE Model
o Sensitivity Analysis conducted to assess influence of model input parameter on output.
o Parameters considered are – α, km and kim
Recovery curves
Influence of α
MPNE Model
Sensitivity Analyses of MPNE parametersSensitivity Analyses of MPNE parameters
Influence of km & kim
Higher recovery and higher peaks for cases having higher sorption rates
MPNE Model - Sensitivity Analyses..
Breakthrough Curves Recovery Curves
Effect of variation in irrigation
Outflow Curves
Recovery Curves
Breakthrough Curves
Higher recovery of metal at slower irrigation rate
MPNE Model
Two Dimensional Heap Leaching by SFT methodTwo Dimensional Heap Leaching by SFT method
2.5 m
1.5 m
0.5 m
SFT Parameters
ksl = 4.98×10-6 s-1
(σsl)max = 0.065αsf = 2.875×10-7 s-1
Grid Spacing Horizontal Direction = 1.72 cm Vertical Direction = 1.69 cm
1
1 N fs f sl sli
avg li i
C u C u AC
N uA
Average concentration of metal in the outflow is computed as
Sensitivity Analyses of SFT ParametersSensitivity Analyses of SFT ParametersSFT Model
Influence of Influence of ααsfsf
αsf has considerable influence in breakthroughs and recovery of metal after the irrigation is stopped
Breakthrough curves
Recovery Curves
Thank You !Thank You !
Questions?Questions?