Simulation of Solute Transport under Oscillating Groundwater Flow in

Homogeneous Aquifers

Amro Elfeki, Gerard Uffink

and Sophie Lebreton

• 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

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

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

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

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

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

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 :

Transport model : example

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

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

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 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.

Influence of the period P

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

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

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

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

head gradient variation 0.007

head gradient variation 0.002

head gradient variation 0.13

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. 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

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

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

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