simulation of solute transport under oscillating groundwater flow in homogeneous aquifers
TRANSCRIPT
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Simulation of Solute Transport under Oscillating Groundwater Flow in
Homogeneous Aquifers
Amro Elfeki, Gerard Uffink
and Sophie Lebreton
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• 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
The goal of the study :investigate the impact of transient flow conditions on solute transport in porous media
Case study
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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
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Flow model :• Hydraulic head• Velocity field
Transport model :• Concentrations • Contaminant plume characteristics
2 numerical models
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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
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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
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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
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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 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
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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 :
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Transport model : example
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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
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Fluctuating water level at the downstream boundary :
time step 0.5 day
02
hh 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 ]
TPl = πS
ana lytica l solution 1 day ana lytica l solution 2.5 days ana lytica l solution 5 daysana lytica l solution 7.5 days ana lytica l solution 10 daysnum erical solution 11 days num erical solution 12.5 days num erical solution 15 daysnum erical solution 17.5 days num erical 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
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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
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Influence of the period P
Aquifer response to periodic forcing :
At the downstream boundary :h(t)=5 cos( 2πt/10)
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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
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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
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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-ΔVS=ΔA.Δh
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Influence of the storativity SS=0.1S=0.01S=0.001S=0.0001
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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
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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
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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
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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
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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
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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
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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