Mathematical Modeling of

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Mathematical Modeling of Pollutant Transport in
Groundwater
Rajesh Srivastava Department of Civil
Engineering IIT Kanpur
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• Outline of the Talk
• Sources
• Processes
• Modelling
• Applications

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• Sources of GW Pollution
• Irrigation
• Landfills
• Underground Storage tanks
• Industry

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• 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

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Dispersion

• Spreading of mass due to
• Velocity differences within pores
• Path differences due to the tortuosity of the
  pore network.

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Pore Spaces
Stagnant or Immobile liquid
Mobile/flowing liquid
Intra-particle pores
Gas
Figure Courtesy Sylvie Bouffard,
Biohydrometallurgy group, Vancouver 12 18
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Brief Chronology

• Unsaturated flow equation by Richards (1931)
• Coats and Smith (1964) proposed dead-end pores in
  oil wells
• Equilibrium reactive transport theories proposed
• Breakthrough curves with pronounced tailings
  observed
• Non-equilibrium models developed
• Goltz and Roberts (1986) physical non-equilibrium
  model
• Brusseau et al. (1989) developed MPNE
• Slow and Fast Transport model developed by Kartha
  (2008)

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Experimental Setup
INFLOW A
OUTFLOW B
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Conservation of Liquid Mass
where Sl is source/sink term.
Darcy velocity in unsaturated porous medium
Hydraulic head based on elevation head z
Hydraulic conductivity
Darcy velocity
Liquid pressure in unsaturated conditions
Intrinsic permeability in unsaturated conditions
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Brooks-Corey and van Genuchten Relations

• Relation between suction pressure, liquid
  pressure, and liquid saturation
• Relation between relative permeability and liquid
  saturation

Effective saturation is given as
Gas pressure Pg is considered zero, therefore
B.C. - Model V.G. Model
Suction pressure
Relative Permeability
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• van Genuchten equations

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• Transport Model
• Reactive 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.

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MPNE Equations
Where, Si - concentration of metal in sorbed
phase (i.e. solid), Ki - adsorption
coefficient, ki - sorption rate, a - mass
transfer rate between mobile and immobile liquid,
Fi - fraction for instantaneous sorption, f
- fraction of sorption site in contact with
mobile liquid.
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• Numerical Solution for Unsaturated Flow
• The mass conservation equation is solved for
  liquid pressure
• Implicit finite-difference method is used

Residual form of conservation of mass equation
for liquid
Taylors series expansion of residual equation
will lead to the following form
Pressure values updated at each iteration step
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• Numerical Solution for MPNE Transport
• Conservation of mass for metal is solved for
  concentration in liquid
• Implicit finite-difference in time step used for
  formulations
• Residual formulation obtained for concentration
  in mobile liquid

The finite-difference formulation for sorbed
concentration is
The residual formulation for solute concentration
in mobile liquid is
Taylors series expansion of the above residual
equation
Updated Concentration is
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Verification of the Numerical Model FLOW
(Compared with VGs Flow Model and Kuo et al.
(1989) Infiltration Model)
150 cm
ksat 5.90510-9 cm2
e 0.45
sr 0.22
a 0.025 cm
? 0.394
?z 1 cm
?t 100 s
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MPNE Transport
30 cm
Input Parameters
?b 1.360 g.cm-3 a 8.68110-7 s-1
? 0.473 km 7.67310-4 s-1
q 5.91410-4 cm.s-1 kim 7.67310-4 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
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Concept of Slow and Fast Transport

• Movement of liquids is heterogeneous
• Liquid flow is conceptualized as slow and fast
  zones
• Multiple sources of non-equilibrium solute
  interactions occurs between solids and different
  liquids 4

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Conservation of solute mass

• Solute mass conservation in fast liquid

• In slow liquid

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Conservation of solute mass.

• Rate of change of instantaneously sorbed
  solute mass

• Rate of change of rate-limited sorbed mass

Similar instantaneous and rate-limited sorption
exist for immobile liquid

• Solute mass conservation in immobile liquid

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FINITE-DIFFERENCE FORMULATION OF SFT MODEL
The implicit finite-difference form of metal mass
conservation in fast moving liquid in a FD cell
is
The implicit finite-difference form of metal mass
conservation in slow moving liquid in a FD cell
is
The implicit finite-difference form of metal mass
conservation in immobile liquid in a FD cell is
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Formulations continued.
Residual equations are formed for the
finite-difference equations for conservation of
metal mass in fast and slow moving
liquids. Residual equations expanded using
Taylors series approximation.
The linear system of equations is solved Update
concentration terms
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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 aim 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)
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REMEDIATION OF GROUNDWATER POLLUTION DUE TO
CHROMIUM IN NAURIA KHERA AREA OF KANPUR
Central Pollution Control Board Lucknow National
Geophysical Research Institute Hyderabad Industria
l Toxicology Research Centre Lucknow Indian
Institute of Technology Kanpur
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5 km2
Location map of Nauriyakhera IDA, Kanpur, U.P.
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• CGWB Observations in Kanpur 1994-2000
• Cr 6 found in groundwater generally exceed gt
  0.11 mg/l
• (Permissible Limit is 0.05 mg/l)
• Cr 6 observed in Industrial areas in depth range
  of 15 40 m gt10 mg/l
• Nauriakhera (Panki Thermal Power Plant Area) Cr
  6
• 14 m - 8.0 mg/l
• 15 m 0.31 mg/l
• 35 m 7.0 mg/l
• 40 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

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Observation Wells in Nauriyakhera IDA, Kanpur,
U.P.
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March 2005
Total Chromium (mg/l) in groundwater -
Nauriyakhera IDA, Kanpur
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Total Chromium (mg/l) in groundwater
-Nauriyakhera IDA, Kanpur
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Fence Diagram Nauriyakhera IDA, Kanpur

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Total Chromium Plume from Source after 10 years
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Total Chromium Plume from Source after 40 years
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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.

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Why Heap Leaching ?

• 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

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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.

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Types of leaching

• Underground in-situ leaching
• Tank leaching
• Heap leaching
• Pressure leaching

Heap

• Impermeable leach pad
• Liners
• Crushed metal ore
• Irrigation system
• Pregnant solution pond
• Barren solution pond

Components of a heap
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MPNE Model

• 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

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MPNE Model
Sensitivity Analyses of MPNE parameters

• Sensitivity Analysis conducted to assess
  influence of model input parameter on output.
• Parameters considered are a, km and kim

Influence of a
Recovery curves
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• Influence of km kim

MPNE Model - Sensitivity Analyses..
Higher recovery and higher peaks for cases having
higher sorption rates
Breakthrough Curves
Recovery Curves
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MPNE Model

• Effect of variation in irrigation

Recovery Curves
Outflow Curves
Higher recovery of metal at slower irrigation
rate
Breakthrough Curves
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Two Dimensional Heap Leaching by SFT method
1.5 m
SFT Parameters ksl 4.9810-6 s-1 (ssl)max
0.065 asf 2.87510-7 s-1
0.5 m

• Grid Spacing
• Horizontal Direction 1.72 cm
• Vertical Direction 1.69 cm

2.5 m
Average concentration of metal in the outflow is
computed as
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SFT Model

• Sensitivity Analyses of SFT Parameters

Influence of asf

Breakthrough curves

asf has considerable influence in breakthroughs
and recovery of metal after the irrigation is
stopped
Recovery Curves
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• Thank You !
• Questions?