# Mathematical Modeling of Pollutant Transport in Groundwater Rajesh Srivastava Department of Civil Engineering IIT Kanpur.

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Mathematical Modeling of Pollutant Transport in Groundwater Rajesh Srivastava Department of Civil Engineering IIT Kanpur

Outline of the Talk SourcesSources ProcessesProcesses ModellingModelling ApplicationsApplications

Sources 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 Velocity

Pore Spaces Gas Mobile/ flowing liquid Stagnant or Immobile liquid Intra-particle pores Figure: Courtesy Sylvie Bouffard, Biohydrometallurgy group, Vancouver 12 1812 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) developed MPNE  Slow 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 where S l is source/sink term. 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

Relation between suction pressure, liquid pressure, and liquid saturation Relation between relative permeability and liquid saturation Brooks-Corey and van Genuchten Relations Effective saturation is given as Gas pressure P g is considered zero, therefore B.C. - ModelV.G. Model Suction pressure Relative Permeability

van Genuchten equations

Transport Model Reactive advective-dispersive 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, S i - concentration of metal in sorbed phase (i.e. solid), K i - adsorption coefficient, k i - sorption rate, α - mass transfer rate between mobile and immobile liquid, F i - 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: Updated Concentration is Taylor’s series expansion of the above residual equation

Verification of the Numerical Model FLOW (Compared with VG’s Flow Model and Kuo et al. (1989) Infiltration Model) Inflow q t = 3 cm/d 10 cm Water Table 150 cm k sat 5.905×10 -9 cm 2 ε0.45 σrσr 0.22 α*α* 0.025 cm λ0.394 ΔzΔz1 cm ΔtΔt100 s

ρbρb 1.360 g.cm -3 α8.681×10 -7 s -1 θ0.473kmkm 7.673×10 -4 s -1 q5.914×10 -4 cm.s -1 k im 7.673×10 -4 s -1 dzdz 0.34 cmKmKm 0.429 cm 3.g -1 L30.0 cmK im 0.416 cm 3.g -1 T0T0 7.672 days (662861 s)f0.929 FmFm 0.5F im 0.5 MPNE Transport 30 cm Input Parameters

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 44 I Immobile Liquid C im and σ im II Slow Liquid C sl and σ sl III Fast Liquid C fs and σ fs IV Instant Sorption Site, S im1 V Rate – limited Sorption Site, S im2 VI Instant Sorption Site, S sl1 VII Rate- limited Sorption Site, S sl2 K im k im K sl k sl α im α sf

Conservation of solute mass In slow liquid Solute mass conservation in fast liquid

Conservation of solute mass…. 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

fast moving The implicit finite-difference form of metal mass conservation in fast moving liquid in a FD cell is: slow moving The implicit finite-difference form of metal mass conservation in slow moving liquid in a FD cell is: immobile The implicit finite-difference form of metal mass conservation in immobile liquid in a FD cell is: 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…. The linear system of equations is solved Update concentration terms:

Numerical Model Validation….. Verification and Evaluation (Brusseau et. al., 1989) Bulk density1.36 g.cm -3 Porosity0.473 Inflow rate5.11 cm.d -1 Dispersivity0.34 cm Column height30.0 cm Immobile saturation0.071 Sorption coefficient K sl 0.429 cm 3.g -1 Sorption coefficient K im 0.416 cm 3.g -1 Sorption rate0.663 d -1 Mass transfer rate α im 0.075 d -1 Instantaneous sorption fraction0.50 Pulse duration7.67 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

Location map of Nauriyakhera IDA, Kanpur, U.P. ~ 5 km 2

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.

Total Chromium (mg/l) in groundwater - Nauriyakhera IDA, Kanpur March 2005

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

oSensitivity Analysis conducted to assess influence of model input parameter on output. oParameters considered are – α, k m and k im Recovery curves Influence of α MPNE Model Sensitivity Analyses of MPNE parameters

Influence of k m & k im 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 method 2.5 m 1.5 m 0.5 m SFT Parameters k sl = 4.98×10 -6 s -1 (σ sl ) max = 0.065 α sf = 2.875×10 -7 s -1 Grid Spacing  Horizontal Direction = 1.72 cm  Vertical Direction = 1.69 cm Average concentration of metal in the outflow is computed as

Sensitivity Analyses of SFT Parameters SFT Model Influence of α sf α sf has considerable influence in breakthroughs and recovery of metal after the irrigation is stopped Breakthrough curves Recovery Curves

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