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?