Post on 08-Jan-2017
transcript
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