Modeling dispersion under unsteady groundwater flow conditions

Modeling dispersion under unsteady
groundwater flow conditions
Sophie Lebreton
supervised by Amro Elfeki
and Gerard Uffink

• 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

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

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 model
4. Sensitivity analysis
- influence of the period P
- influence of the storativity S
- influence of the amplitude of oscillation
5. Simulations in heterogeneous aquifers
6. Application to a field tracer test, the “MADE1” experiment

• Hydraulic head h: z
Ph= g
(in meters)
This is the water level measured in an observation well :
Flow model : Definitions
• Hydraulic conductivity K :
This is the ability of an aquifer to conduct water through it under hydraulic
gradients. K has the dimension of a velocity (m/day).

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



   




Finite difference formulation of the flow equation :
Implicit scheme :
1 1 1 1 1
, ,1, , 1 1, , 1ij ij ij ij ij ij
k k k k k k
i 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)

The head h is known at each node and at each time
step.
Darcy’s law :
Velocities qx(x,y,t) and qy (x,y,t) are known at
each node and at each time step.
x y
h hq K q K
x 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 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

Particles are moved following the particle motion equation :
{
{
dispersive stepadvectice step
dispersive stepadvectice step
n+1 n
x xp p
n+1 n
y yp p
+ +
+ +
X = X V Δt
Y = Y V Δt
Z
Z
14 2 43
14 2 43
Transport model : random walk
advective and dispersive steps Two individual random paths with 10 steps each

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 :

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

Analytical solution for a sudden drop :
  
                          
  

1
n+1
2
n=
-1Hx 2H nπx nπh x,t = - sin exp - t
π nd d βd
0.5 day, analytical solution
1 day, analytical solution
2 days, analytical solution
0.5 day, numerical solution
1 day, numerical solution
2 days, numerical solution
• Upstream water level: 0 m
Drop : - 10 m
• Aquifer characteristics:
length d=200m
Storativity S=0.01.
time step 0.05 daytime step 0.5 day
Comparison with analytical solutions

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 17.5 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

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

Aquifer 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.

penetration length l=100m
d/l=1 (aquifer length d=100m)
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

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

S=0.1
S=0.01
S=0.001
S=0.0001

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

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

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

Spatial distribution of K is modelled as a log-normal distribution.
3 characteristics :
• a mean hydraulic conductivity <K>
• a standard deviation σK
• a correlation length λ
Simulations in heterogeneous aquifers
0 50 100 150 200 250 300
-40
-20
0
1.2
1.6
2
2.4
2.8
3.2
3.6
0 50 100 150 200 250 300
-40
-20
0
0.2
0.8
1.4
2
2.6
3.2
3.8
Map of ln(K)
( arithmetic mean of K =10 m/day and standard deviation σK =5m/day )
λ=2m
λ=40m

0 100 200 300 400
Time (days)
0
100
200
300
400
CentroidDisplacement
inX-direction(m)
Correlation Length= 1 m(unsteady)
Correlation Length= 1 m (steady)
Correlation Length= 2 m(steady)
3 correlation lengths are tested : 1, 2 and 3 m
Ergodic condition : the plume must discover within one period most heterogenities
of the aquifer by traveling at least a distance equal to 20 λ. Thus if λ remains small,
this condition is fulfilled. Moreover, we inject a large pollutant source that cover the
total width of the aquifer.

0 100 200 300 400
Time (days)
0
100
200
300
400
VarianceinX-direction(m2)
0 100 200 300 400
Time (days)
-2
-1
0
1
2
3
Macro-dispersion(m2/day)
Correlation Length= 1 m (unsteady)
0
0.1
1
10
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
Correlation Length = 1 m
50 days 150 days 300 days
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
0 50 100 150 200 250 300 350 400 450 500
-40
-20
0
Steady ConditionsUnsteady Conditions
• Transient case oscillate
around steady state.
• Strong variation in the
plume variance when λ
increases: phenomenon of
channeling appears.

0 50 100 150 200 250 300
-40
-20
0
0 50 100 150 200 250 300
-40
-20
0
0 50 100 150 200 250 300
-40
-20
0
0
5
10
15
20
25
30
35
40
0 10 20 30 40
K
0
0.2
0.4
pdf
0 10 20 30 40
K
0
0.04
0.08
0.12
pdf
0 10 20 30 40
K
0
0.04
0.08
0.12
pdf
3 standard deviations are tested : 1, 5 and 10 m/day
when σK increases :
values of K lower than the mean <K> are more probable to
be present in the medium
contrast in K increases
σK=1 m/day
σK=5 m/day
σK=10m/day

heterogeneous aquifer
standard deviation = 1m
0 40 80 120 160
time (days)
0
100
200
300
meandisplacementinthex-direction(m)
0 40 80 120 160
time (days)
0
100
200
300
400
longitudinalvariance(m2)
0 40 80 120 160
time (days)
170
180
190
200
210
220
230
lateralvariance(m2
)
0 40 80 120 160
time (days)
-2
0
2
4
6
DXX(m2
/day)
when σK increases :
• Slowing down of the plume
• Enhancement of the
longitudinal spreading
• Enhancement of the lateral
spreading

0 20 40 60 80
time (days)
-8
-4
0
4
8
xsteady
-xunsteady
0 20 40 60 80
time (days)
-8
-4
0
4
8
(xx
2steady
-(xx
2unsteady
0 20 40 60 80
time (days)
-0.8
-0.4
0
0.4
0.8
lateralvariance(m2)
(yy
2steady
-(yy
2unsteady
0 20 40 60 80
time (days)
-8
-4
0
4
8
xsteady
-xunsteady
0 20 40 60 80
time (days)
-8
-4
0
4
8
12
(xx
2steady
-(xx
2unsteady
0 20 40 60 80
time (days)
-2
-1
0
1
2
lateralvariance(m2)
(yy
2steady
-(yy
2unsteady
0 20 40 60 80
time (days)
0
40
80
120
160
steady state
unsteady state
Unsteady – steady = temporal variation ?
In steady state heterogeneity
In unsteady state heterogeneity + temporal variation

Description of the experiment :
Injection of a non-reactive tracer (bromide) in an real heterogeneous aquifer
Depth-averaged distribution of K
Application to a field tracer test : “ Made1 experiment ’’
Observed concentration and cross-section of K

Description of the field data :
Depth-averaged bromide concentration
Application to a field tracer test : “ Made1 experiment ’’
A lot of uncertainty on:
• the spatial distribution of K
• the values of αL and αT
• the storage coefficient S
Location of the measurements of K

0 50 100 150 200 250 300
-150
-100
-50
0
0
2.5
10
40
70
100
140
180
220
0 20 40 60 80 100 120 140 160
-40
-20
0
0.78
1.3
2
3.5
6
10
16
26
43
71
116
0 50 100 150 200 250
-100
-80
-60
-40
-20
0
0
4.3
43
430
injection point
Spatial distribution of K : grid map
where nodes are assigned a value of
K
Case 1. depth averaged map
Case 2. depth averaged map
Case 3. distribution of K at elevation
59m
Data for the numerical simulation

0
50
100
150
200
250
-100-80-60-40-200
62
62.1
62.2
62.3
62.4
62.5
62.6
62.7
62.8
62.9
63
Case 1
Simulations under steady
state
Near the injection point :
closely spaced contours
Far from the injection point :
widely spaced contours
Case 2 Case 3
Comparison between
observed and simulated
head field

Simulations under steady state
Observed concentrations :
Depth-averaged bromide concentration

0 20 40 60 80 100 120 140 160 180 200 220 240 260
-100
-80
-60
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200 220 240 260
-100
-80
-60
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200 220 240 260
-100
-80
-60
-40
-20
0
49 days
279 days
503 days
0 20 40 60 80 100 120 140 160 180 200 220 240 260
-100
-80
-60
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200 220 240 260
-100
-80
-60
-40
-20
0
0 20 40 60 80 100 120 140 160 180 200 220 240 260
-100
-80
-60
-40
-20
0
0.1
1
10
100
49 days
279 days
503 days
Case 1
small dispersivities αL and αT large dispersivities αL and αT

0 20 40 60 80 100 120 140 160
-50
-40
-30
-20
-10
0
0 20 40 60 80 100 120 140 160
-50
-40
-30
-20
-10
0
0 20 40 60 80 100 120 140 160
-50
-40
-30
-20
-10
0
49 days
279 days
503 days
0 20 40 60 80 100 120 140 160
-50
-40
-30
-20
-10
0
0 20 40 60 80 100 120 140 160
-50
-40
-30
-20
-10
0
0 20 40 60 80 100 120 140 160
-50
-40
-30
-20
-10
0
0.1
1
10
100
49 days
279 days
503 days
Case 2

49 days
279 days
503 days
0 50 100 150 200 250 300
-150
-100
-50
0
0 50 100 150 200 250 300
-150
-100
-50
0
0 50 100 150 200 250 300
-150
-100
-50
0
49 days
0 50 100 150 200 250 300
-150
-100
-50
0
0 50 100 150 200 250 300
-150
-100
-50
0
0 50 100 150 200 250 300
-150
-100
-50
0
279 days
503 days
0.1
1
10
100
Case 3

Simulations under unsteady state
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
time (months)
0.001
0.002
0.003
0.004
0.005
gradientmagnitude
measured gradient
fitted seasonal component
simulation period
Transient flow conditions :
Fluctuations are imposed at the downstream boundary to recreate
the variations of the head gradient
Variation of the hydraulic head gradient magnitude

0 100 200 300 400 500 600
time (days)
-35
-30
-25
-20
-15
-10
meandisplacementinthey-direction(m)
0 100 200 300 400 500 600
time (days)
0.1
1
10
100
lateralvariance(m2)
0 100 200 300 400 500 600
time (days)
0
20
40
60
80
0 100 200 300 400 500 600
time (days)
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
Simulations under unsteady state small dispersivities
steady state
seasonal trend for S=0.04 (cosine)
measured gradient for S=0.04 (dots)
observed data
• transient simulations
and simulations under
steady state are close
• the spreading is
underestimated
Case 2

0 100 200 300 400 500 600
time (days)
-35
-30
-25
-20
-15
-10
meandisplacementinthey-direction(m)0 100 200 300 400 500 600
time (days)
10
20
30
40
50
60
Simulations under unsteady state large dispersivities
steady state
observed data
0 100 200 300 400 500 600
time (days)
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
• transient simulations
and simulations under
steady state are close
• better simulation of
the spreading
0 100 200 300 400 500 600
time (days)
1
10
100
lateralvariance(m2)
Case 2

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

conclusions
Application to the MADE1 experiment enables to
conclude that :
1. In this example, transient flow conditions don’t show much
difference with steady state conditions
2. The poor agreement between simulated and observed results can
be primarily attributed to uncertainty in the spatial distribution of K:
sparse data and depth-averaged values coarse map
A good knowledge of the geology and thus of the fine-scale
heterogeneity in the aquifer is necessary for simulations

conclusions
Application to the MADE1 experiment enables to conclude
that :
1. In this example, transient flow conditions don’t show much difference
with steady state conditions
2. The poor agreement between simulated and observed results can be
primarily attributed to uncertainty in the spatial distribution of K:
sparse data and depth-averaged values coarse map
A good knowledge of the geology and thus of the fine-scale
heterogeneity in the aquifer is necessary for simulations

Annexes
1 1
1 1
cos sin sin cos
. / . / . / . /
n n n n
p p x p p yL T L T
n n n n
p 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
    
 
         
         
6 4 4 4 44 7 4 4 4 4 486 7 8
dispersive termadvective term
    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
 
 
 
 
 
 
          
 
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

Annexes
“Courant condition” :
The distance traveled by a particle in one step must not exceed the size of the
cells: thus particles don’t jump over cells, and move continuously from one cell to
another.
trans
max
ΔxΔt
V
Time discretization of flow and transport
         
         
          
          
x x xt t t1 2 2 2 1
V t =V t 1-A +V t A with A = t-t / t -t

Annexes
dtflow=10 days dttrans=10 days (no interpolation)
dtflow=10 days dttrans= 5 days ( interpolation)
dtflow=10 days dttrans= 1 days ( interpolation)
dtflow= 5 days dttrans= 5 days (no interpolation)
dtflow= 1 days dttrans= 1 days (no interpolation)

Annexes
0 100 200 300 400 500 600
time (days)
-114
-112
-110
-108
-106
-104
meandisplacementiny-direction(m)
0 100 200 300 400 500 600
time (days)
1
10
100
lateralvariance(m2)
Case 3
steady state
observed data
0 100 200 300 400 500 600
time (days)
60
64
68
72
76
80
meandisplacementinx-direction(m)
0 100 200 300 400 500 600
time (days)
0.001
0.01
0.1
1
10
100
1000
10000
small dispersivities

Annexes
Case 3
large dispersivities
0 100 200 300 400 500 600
time (days)
-116
-114
-112
-110
-108
-106
-104
meandisplacementiny-direction(m)
steady state
observed data
0 100 200 300 400 500 600
time (days)
1
10
100
lateralvariance(m2)
0 100 200 300 400 500 600
time (days)
60
64
68
72
76
80
meandisplacementinx-direction(m)
0 100 200 300 400 500 600
time (days)
0.001
0.01
0.1
1
10
100
1000
10000

head at x=50m
head at x=100m
head at x=150m
head at x=200m
velocity at x=50m
velocity at x=100m
velocity at x=150m
Annexes

unsteady state: fluctuating boundary
steady state (upstream level=20m, downstream level=0m)
Annexes

Modeling dispersion under unsteady groundwater flow conditions

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