Simulation of Solute Transport under Oscillating Groundwater Flow in Homogeneous Aquifers, Journal of Hydraulic Research, Vol. 45 (2), pp. 254-260.

1. Simulation of Solute Transport under
Oscillating Groundwater Flow in
Homogeneous Aquifers
Amro Elfeki, Gerard Uffink
and Sophie Lebreton

2. • confined aquifer
• upstream water level constant
• downstream water level variable
• constant thickness
• constant hydraulic conductivity K
over the depth
• constant specific storage SS over
the depth
• aquifer modelled in a 2D horizontal
plane
The goal of the study :
investigate the impact of transient flow conditions on solute
transport in porous media
Case study

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

4. Flow model :
• Hydraulic head
• Velocity field
Transport model :
• Concentrations
• Contaminant plume characteristics
2 numerical models

5. Outlines
1. Flow model
2. Transport model
3. Verification of the model
4. Sensitivity analysis
- influence of the period P
- influence of the storativity S
- influence of the amplitude of oscillation

6. Governing equation of the flow:
   
   
 , , , , , ,
, ,xx yy
h x y t h x y t h x y t
S 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 y
n
h y t h
h d y t h t



   




7. x y xx xy yx yy
C C C C C C CV V D D D D
t 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
VV
D = α V +D δ + α -α
V

8. Particles are moved following the particle motion equation :
Transport model : random walk
advective and dispersive steps Two individual random paths with 10 steps each
    1 22 2xy yxx x
p p x L T
D VD V
X t t X t V t Z V t Z V t
x y V V
 
 
 
 
 

          
 
    1 22 2yx yy y x
p p y L T
D D V V
Y t t Y t V t Z V t Z V t
x y V V
 
 
 
 
 
 
          
 

9. Transport model : algorithm
Algorithm :
• A mass of pollutant is injected at a given location in the aquifer
• The velocity field that prevails at time k (computed by the flow
model) is read
• All particles are moved one by one with an advective and a
dispersive step using the given velocity
• Particles are counted within each cell to compute the
concentration distribution
• The velocity field that prevails at time k+1 is read…
etc…
time k :
time k+1 :

10. Transport model : example

11. Main outputs :
• concentration
• displacement of the center of mass and
• longitudinal variance σxx
2
• lateral variance σyy
2
• longitudinal and lateral macrodispersion
2 2
,
1 1
2 2XX YY
XX YY
t t
D D
  
 
 
Transport model : outputs
X Y

12. Fluctuating water level at the downstream boundary :
time step 0.5 day
 
   
         
         
         
         


0
2
h
h x,t = ×
cosh d/l - cos d/l
cos ωt sinh x/l cos x/l sinh d/l cos d/l
-sin ωt cosh x/l sin x/l sinh d/l cos d/l
+sin ωt sinh x/l cos x/l cosh d/l sin d/l
+cos ωt cosh x/l sin x/l cosh d/l sin d/l ]
TP
l =
πS
analytical solution 1 day
analytical solution 2.5 days
analytical solution 5 days
analytical solution 7.5 days
analytical solution 10 days
numerical solution 11 days
numerical solution 12.5 days
numerical solution 15 days
numerical solution 17.5 days
numerical solution 20 days
with
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

13. TP
l =
πS
Penetration length :
l is the factor that controls the propagation of oscillations within
the aquifer.
When the period P increases, the penetration length increases
Influence of the period P

14. Influence of the period P
Aquifer response to periodic forcing :
At the downstream boundary :
h(t)=5 cos( 2πt/10)

15. 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.
Influence of the period P

16. penetration length l=100m
d/l=1 (aquifer length d=100m)
d/l=3 (aquifer length d=300m)
d/l=6 (aquifer 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

17. Storativity is the ability of the aquifer to store or release water:
For high storativity, the aquifer stores and releases a large
amount of water : fluctuations of the water level will be absorbed
by the porous media.
Influence of the storativity S
water
-ΔV
S=
ΔA.Δh

18. Influence of the storativity S
S=0.1
S=0.01
S=0.001
S=0.0001

19. For high storativity : - small amplitude
- delay of the response
- high variations of the velocity near the downstream
boundary
steady state
unsteady state S=0.1
unsteady state S=0.01
unsteady state S=0.001
unsteady state S=0.0001
Influence of the storativity S

20. 3 amplitudes of oscillations are tested : 1, 3 and 20 m
head gradient variation 0.007
head gradient variation 0.002
head gradient variation 0.13
Influence of the amplitude

21. Small amplitude no significant difference with steady state
Large amplitude oscillations around steady state
Influence of the amplitude
steady state head difference 20m
steady state head difference 3m
steady state head difference 1m
unsteady state amplitude 20m
unsteady state amplitude 3m
unsteady state amplitude 1m

22. 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. Transient flow conditions have an impact only if the amplitude of
oscillations is large. Otherwise, results are very close to steady
state.
4. Heterogeneity and temporal variations interact together in a
complex manner.
2d/l= πSd /TP

23. 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. Transient 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

24. 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. Transient 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

25. 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. Transient 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