Simulation of Pollution Transport In Coastal Aquifers under Tidal

Movement

Amro Elfeki1, Gerard Uffink 2 and Sophie Lebreton2

1-King Abdulaziz University,

2- Delft University of Technology, Netherlands

• confined aquifer• upstream water level constant• downstream water level variable

• constant thickness• constant hydraulic conductivity Kover the depth• constant specific storage SS over

the depth• aquifer modelled in a 2D

horizontal plane

Investigate the impact of (tidal) oscillatory flow conditions on pollution transport at coastal aquifer

Study Objective

• Injection of inert solutes, • 2D homogeneous aquifer,• periodical fluctuations at the downstream boundary with a specified, amplitude and period,• instantaneous injection.

Study Scope

Flow model :• Hydraulic head• Velocity field

Transport model :• Concentrations • Contaminant plume characteristics

2 numerical models

Outlines

1. Flow model

2. Transport model

3. Verification of the models

4. Sensitivity analysis - influence of the period P- influence of the storativity S- influence of the amplitude of oscillation

5. Conclusions

Governing equation of the flow:

, , , , , ,, ,xx yy

h x y t h x y t h x y tS x y x yT Tt x x y y

Principle of the finite difference method :• discretization in space• discretization in time

Flow model : Finite difference method

where h hydraulic conductivity S the storativity or storage coefficient T=Kb the transmissivity

0

( , , ) 0 , (no-flow condition)

(0, , )( , , ) ( )

h x y t for x ynh y t hh d y t h t

1 1 1 1 1, ,1, , 1 1, , 1ij ij ij ij ij ijk k k k k ki j i ji j i j i j i j

F h A h B h C h D h E h

Solution by an iterative scheme : the conjugate gradient method h(x,y,t) x y

h hq K q Kx y

• Groundwater head :

• Darcy’s velocities :

0 50 100 150 200 250 300

-40

-20

0

Velocity field at a time t

Flow model : outputs

2 transport mechanisms :

Advection : this is the solute flux due to the flow of groundwater

Dispersion : this is due to the velocity variations

Transport model

Gaussian distribution of the concentration

x y xx xy yx yyC C C C C C CV V D D D Dt x y x x y y x y

This equation is not solved directly the random walk method is used

Principle of the random walk method: pollutant transport is modeled by using particles that are moved one by one to simulate advection and dispersion mechanisms.

Transport model : random walk

Governing equation of solute transport :

where C is the concentration Vx and Vy are pore velocities Dxx , Dyy , Dxy , Dyx are dispersion coefficients

i j*mij L ij L T

VVD = α V +D δ + α -α

V

Particle tracking random walk method

1 1

1 1

cos sin sin cos

. / . / . / . /

n n n np p x p p yL T L T

n n n np p x x y p p y y xL T L T

X X V t Z Z Y Y V t Z Z

X X V t Z V V Z V V Y Y V t Z V V Z V V

dispersive termadvective term

1 22 2xy yxx xp p x L T

D VD VX t t X t V t Z V t Z V tx y V V

1 22 2yx yy y xp p y L T

D D V VY t t Y t V t Z V t Z V tx y V V

The displacement is a normally distributed random variable, whose mean is the advective movement and whose deviation from the mean is the dispersive movement.

instantaneous injection + uniform flow

Transport model : example

Transport model : example

2 2

/( )( , , )4 4

( - - ( -) )exp -4 4

o

l x t x

o ox

l x t x

HMC x y t t tV V

x t yVX Y t tV V

0

/ ( )( )

1 ( ( () )exp( ) ( )

o

x l t

t 2 2o ox

l x t x

HMC x, y,t = 4 V

x - - t y -VX Y - + d t 4 t 4 tV V

Main outputs : • concentration• displacement of the center of mass and

• longitudinal variance σxx2

• lateral variance σyy2

• longitudinal and lateral macrodispersion2 2

,1 12 2XX YY

XX YYt tD D

Transport model : outputs

X Y

Fluctuating water level at the downstream boundary :

time step 0.5 day

20

2,cosh / -cos /

cos sinh / cos / sinh / cos /

-sin cosh / sin / sinh / cos /

sin sinh / cos / cosh / sin /

cos cosh / sin / cosh / sin / ]

hh x td l d l

t x l x l d l d l

t x l x l d l d l

t x l x l d l d l

t x l x l d l d l

TPl = πSwith

l is the penetration length

• Upstream water level: 0 m Downstream level : 5 cos(2πt/10) • Aquifer characteristics: length d=200m Storativity S=0.01

Comparison with analytical solutions

TPl =πSPenetration length :

l is the factor that controls the propagation of oscillations withinthe aquifer.

When the period P increases, the penetration length increases

Influence of the period P

Influence of the period PAquifer response to periodic forcing : At the downstream boundary :

h(t)=5 cos( 2πt/10)

Head profiles along the aquifer length. The downstream water level is a cosine function with an amplitude of 5m and with different periods: 1, 5, 10 days. The length of the aquifer is 300m, the

storativity S=0.01.

pe n e tra tion le n g th l=10 0 m

d/l=1 (aqu ifer length d=100m )d/l=3 (aqu ifer length d=300m )d/l=6 (aqu ifer length d=600m ) Conclusion

When the period P increases :• propagation of oscillations increases• amplitude increases

• d aquifer length• l penetration length

d/l determine the head profile within the aquifer

Influence of the period P

Influence of the storativity SS=0.1S=0.01S=0.001S=0.0001

Simulation Example

For high storativity : - small amplitude - delay of the response

- high variations of the velocity near the downstream boundary

steady sta te unsteady sta te S =0.1unsteady sta te S =0.01unsteady sta te S =0.001unsteady sta te S =0.0001

Influence of the storativity S

3 amplitudes of oscillations are tested : 1, 3 and 20 m

Average head gradient variation 0.003

Average head gradient variation 0.01

Average head gradient variation 0.07

Influence of the amplitude

Small amplitude no significant difference with steady state

Large amplitude oscillations around steady state

Influence of the amplitudesteady sta te head d iffe rence 20msteady sta te head d iffe rence 3msteady sta te head d iffe rence 1munsteady sta te am plitude 20m unsteady sta te am plitude 3m unsteady sta te am plitude 1m

conclusions

Sensitivity analysis enables to conclude that :1. The model provides a good representation of the hydraulic head

variations.

2. The response of the aquifer to periodic fluctuations is controlled by the ratio,

When the penetration length l is large with respect to the length of the aquifer d, the propagation of oscillations within the aquifer is significant.

3. Oscillatory flow conditions have an impact, only if the amplitude of oscillations is large. Otherwise, results are very close to steady state

2d/l = πSd /TP

conclusions

Sensitivity analysis enables to conclude that :1. The model provides a good representation of the hydraulic head

variations.

2. The response of the aquifer to periodic fluctuations is controlled by the ratio,

When the penetration length l is large with respect to the length of the aquifer d, the propagation of oscillations within the aquifer is significant.

3. Oscillatory flow conditions have an impact, only if the amplitude of oscillations is large. Otherwise, results are very close to steady state.

2d/l = πSd /TP

conclusions

Sensitivity analysis enables to conclude that :1. The model provides a good representation of the hydraulic head

variations.

2. The response of the aquifer to periodic fluctuations is controlled by the ratio,

When the penetration length l is large with respect to the length of the aquifer d, the propagation of oscillations within the aquifer is significant.

3. Oscillatory flow conditions have an impact, only if the amplitude of oscillations is large. Otherwise, results are very close to steady state.

2d/l = πSd /TP

conclusions

Sensitivity analysis enables to conclude that :1. The model provides a good representation of the hydraulic head

variations.

2. The response of the aquifer to periodic fluctuations is controlled by the ratio,

When the penetration length l is large with respect to the length of the aquifer d, the propagation of oscillations within the aquifer is significant.

3. Oscillatory flow conditions have an impact, only if the amplitude of oscillations is large. Otherwise, results are very close to steady state

2d/l = πSd /TP