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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

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Mathematical Modeling of Pollutant Transport in Groundwater

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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


  • Mass transport due to the flow of the water

  • The direction and rate of transport coincide with that of the groundwater flow.


  • Mixing due to concentration gradients


  • Mechanical mixing due to movement of fluids through the pore space


Position in Pore


  • Spreading of mass due to

    • Velocity differences within pores

    • Path differences due to the tortuosity of the pore network.

Stagnant or Immobile liquid

Pore Spaces

Mobile/flowing 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 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)developedMPNE

  • Slow and Fast Transport model developed by Kartha (2008)













Experimental Setup



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

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

  • van Genuchten equations

  • 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.

MPNE Equations

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 pressure

  • Implicit finite-difference method is used

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 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:

Taylor’s series expansion of the above residual equation

Updated Concentration is

Inflow qt = 3 cm/d

10 cm

Water Table

Verification of the Numerical Model


(Compared with VG’s Flow Model and Kuo et al. (1989) Infiltration Model)

150 cm

MPNE Transport

30 cm

Input Parameters


Immobile Liquid

Cim and σim


Slow Liquid

Csland σsl


Fast Liquid

Cfsand σfs








Rate – limited Sorption Site,



Rate-limited Sorption Site,



Instant Sorption Site,



Instant Sorption Site,


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

Conservation of solute mass

  • Solute mass conservation in fast liquid

  • In slow liquid

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


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:

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 Taylor’s series approximation.

The linear system of equations is solved

Update concentration terms:

Verification and Evaluation(Brusseau et. al., 1989)

Numerical Model Validation…..

Brusseau, M.L., Jessup, R.E., Rao, P.S.C.: Modeling the transport of solutes….. Water Resources Research 25 (9), 1971 – 1988 (1989)


Central Pollution Control Board Lucknow

National Geophysical Research Institute Hyderabad

Industrial Toxicology Research CentreLucknow

Indian Institute of Technology Kanpur

~ 5 km2

Location map of Nauriyakhera IDA, Kanpur, U.P.

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/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

Observation Wells in Nauriyakhera IDA, Kanpur, U.P.

March 2005

Total Chromium (mg/l) in groundwater - Nauriyakhera IDA, Kanpur

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.

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

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.

Mine Pit


Sprinklers or wobblers

Leach pad

Recovery Plant

Pregnant solution pond

Barren Solution Pond

Types of leaching

  • Underground in-situ leaching

  • Tank leaching

  • Heap leaching

  • Pressure leaching


  • Impermeable leach pad

  • Liners

  • Crushed metal ore

  • Irrigation system

  • Pregnant solution pond

  • Barren solution pond

Components of

a heap

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

MPNE Model

Sensitivity Analyses of MPNE parameters

  • Sensitivity Analysis conducted to assess influence of model input parameter on output.

  • Parameters considered are – α, km and kim

Influence of α

Recovery curves

MPNE Model - Sensitivity Analyses..

Higher recovery and higher peaks for cases having

higher sorption rates

Influence of km & kim

Breakthrough Curves Recovery Curves

MPNE Model

Effect of variation in irrigation

Recovery Curves

Outflow Curves

Higher recovery of metal at

slower irrigation rate

Breakthrough Curves

Two Dimensional Heap Leaching by SFT method

1.5 m

SFT Parameters

ksl = 4.98×10-6 s-1

(σsl)max = 0.065

αsf = 2.875×10-7 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

SFT Model

Influence of αsf

Sensitivity Analyses of SFT Parameters

Breakthrough curves

αsf has considerable influence in breakthroughs and recovery of metal after the irrigation is stopped

Recovery Curves

Thank You !


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