Ackerer, P.

Bildstein, O.

Hassane Maina, F.

English

Bildstein, Olivier

HAL-CEA

Simultaneous estimation of groundwater recharge and hydrodynamic parameters for groundwater flow modeling

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Simultaneous estimation of groundwater recharge and
hydrodynamic parameters for groundwater flow
modeling
F. Hassane Maina, P. Ackerer, O. Bildstein
To cite this version:
F. Hassane Maina, P. Ackerer, O. Bildstein. Simultaneous estimation of groundwater recharge and
hydrodynamic parameters for groundwater flow modeling. SAMO 2016 - International Conference on
Sensitivity Analysis of Model Output, Nov 2016, Le Tampon, France. ￿cea-02435102￿
SIMULTANEOUS ESTIMATION OF 
GROUNDWATER RECHARGE AND 
HYDRODYNAMIC PARAMETERS 
FOR GROUNDWATER FLOW 
MODELING 
29/09/2016 1 
Fadji HASSANE MAINA (Université de Strasbourg) 
Philippe ACKERER (Université de Strasbourg) 
Olivier BILDSTEIN (CEA Cadarache) 
 
 
INTRODUCTION 
www.ec.gc.ca 1 
INTRODUCTION 
29/09/2016 
 Groundwater (aquifers): 3rd freshwater reservoir on the planet       
 
 Replenished essentially by precipitation through groundwater recharge 
 
 
 . s
H
S K H q
t

  

q K H  
. s
H
S q q
t

 

2 
INTRODUCTION 
29/09/2016 
 Groundwater (aquifers): 3rd freshwater reservoir on the planet       
 
 Replenished essentially by precipitation through groundwater recharge 
 
 The main available water resources for many countries 
 
 Threatened by pollution and over-exploited  
 
 Usually described by models           Simulations of piezometric levels (Hydraulic heads) 
 
 Physical Models are widely used 
 
1. Continuity equation (mass conservation): 
 
 
2. Darcy’s law (computation of fluxes): 
 
 
  Diffusivity equation (flow equation): 
 
Values of H knowing S, K and qs    S and K are constant, qs includes 
groundwater recharge 
 
 . s
H
S K H q
t

  

q K H  
. s
H
S q q
t

 

2 
29/09/2016 
 Groundwater flow models: Sophisticated and 
accurate 
 Recharge modeling: Challenge 
o Complex hydrological component 
 
Groundwater recharge depends on: 
 
 Climatic conditions 
 Vegetation 
 Soil and root zone 
 Unsaturated zone 
 
 
GROUNDWATER RECHARGE 
Recharge : water transfer from precipitation to groundwater 
BRGM 
3 
29/09/2016 
 Groundwater flow models: Sophisticated and 
accurate 
 Recharge modeling: Challenge 
o Complex hydrological component 
 
Groundwater recharge depends on: 
 
 Climatic conditions 
 Vegetation 
 Soil and root zone 
 Unsaturated zone 
 
Values can not be directly measured    
 
 
Estimation 
 Direct methods: lysimeters, TDR 
 Empirical methods 
 Tracers 
 Mathematical models 
 
GROUNDWATER RECHARGE 
Recharge : water transfer from precipitation to groundwater 
BRGM 
3 
RECHARGE ESTIMATION:  
COUPLED MODEL 
29/09/2016 
Simulation of flow in both unsaturated and saturated zones 
 
 
 
4 
 
Runoff 
Precipitations, Evapotranspiration 
Infiltration 
Flow in the Unsaturated Zone  
Interactions between 
Atmosphere and Soil 
and Hydrodynamic in 
the Unsaturated Zone : 
Nash 
Groundwater Flow: 
Diffusivity equation 
Recharge 
Flow in the Saturated Zone  
Coupling 
Hydraulic Heads 
NASH MODEL 
Hydrodynamic in the unsaturated zone 
 
 
   


t
Rn Tau
e
t
Tau
t
Tau
Rn
 







 

1
1
Root 
Zone  11 1 1
0
Recharge
k
k
n t
n k n
t
k
Pe t d 
  

   
Nash parameters: RUMAX, RN, TAU 
Water transfer in the 
unsaturated zone: 
delay and spreading 
Modeling : propagation of a unit pulse signal 
Threshold : RUMAX 
ET Rainfall 
if RUMAX is reached 
 
the excess water (Pe) goes into the unsaturated zone 
5 
Infiltration 
Recharge 
Time 
Water  
Level 
ET depends on soil 
saturation 
 Model uses constants called parameters  
 Aquifers are highly heterogeneous   Parameters are not known accurately or even 
unknown in natural environments  
 
 
PARAMETERS ESTIMATION 
6 
 Model uses constants called parameters  
 Aquifers are highly heterogeneous   Parameters are not known accurately or even 
unknown in natural environments  
 Available data: meteorological and piezometric levels 
 
 
PARAMETERS ESTIMATION 
6 
0.00
50.00
100.00
150.00
200.00
250.00
300.00
294
296
298
300
302
304
306
308
R
a
in
fa
ll (m
m
) 
H
y
d
ra
u
li
c
 H
e
a
d
s
 (
m
) 
 Model uses constants called parameters  
 Aquifers are highly heterogeneous   Parameters are not known accurately or even 
unknown in natural environments  
 Available data: meteorological and piezometric levels 
 
 
PARAMETERS ESTIMATION 
Simultaneous estimation of 
groundwater flow and recharge 
parameters from hydraulic heads 
variations 
Recharge is estimated by 
calibrating piezometric levels 
6 
0.00
50.00
100.00
150.00
200.00
250.00
300.00
294
296
298
300
302
304
306
308
R
a
in
fa
ll (m
m
) 
H
y
d
ra
u
li
c
 H
e
a
d
s
 (
m
) 
GLOBAL SENSITIVITY ANALYSIS  
 Correlation between recharge and storage capacity (S and qs) 
Recharge and storage capacity lead to the same effect 
29/09/2016 
q K H   Determine q et K knowing H ! 
 Calibration is performed over time and space 
 But… recharge is variable over time while the storage capacity is constant 
When                               Determination of K  
When                Determination of q knowing K 
 
 Global sensitivity analysis 
Assess the calibration approach and guide the parameter estimation 
Recharge 
Storage 
capacity 
0 K H  
0q 
0q 
High recharge and high storage  Low recharge and low storage  
7 
ΔH ΔH ΔH 
Uncertain parameters: variability ranges known       Quantify the effects of parameters uncertainty 
 
Global sensitivity analysis: focuses on the output uncertainty over the entire range of values of 
the input parameters both single and in combination with one another  
 
 
 
 
 
 
 
 
 
 
 
GLOBAL SENSITIVITY ANALYSIS 
29/09/2016 
Variance based approach:  
Sobol Indices 
Surrogate model: Chaos polynomial expansion 
8 
q K H  
q and K are parameters and H 
the state variable 
q and K are unknowns 
  0.0 0.5 q 5 110 10K     
n simulations with 
different values of 
q and K 
n values of 
H 
Study the effects 
of variations of q 
and K on H 
29/09/2016 
SOBOL INDICES :  
CHAOS POLYNOMIAL EXPANSION 
Let us consider a model with y as output and n input parameters, X the parameters vector 
 
 ANOVA decomposition : 
 
 
  
 
 
 
Orthogonality of f  
 
 
Sensitivity (Sobol) Indices :  
 
 
Computation of f                 Polynomial expansion 
 
 Chaos polynomial expansion 
 
     0 1,2,..., 1 2
1 1
, ...... , ,.....,
n n
i i ij i j n n
i j
y f f x f x x f x x x
 
     
         0 , ,, , ....s K RN RN K S KH H f S f K f RN f RN K f S K      
;
iji
i ij
VV
S S
V V
 
 1
0
,......,j j n
j
y a x x



1,....,
1 1
.........
n n
i ij n
i j
V V V V
 
    
fi : contribution of single parameter i to the output 
fij : contribution of the combined effect of parameters i and j to the output  
x : variable 
𝚿 :  orthogonal polynomial (Legendre, Hermite) 
a : coefficients determined by least squares regression 9 
GLOBAL SENSITIVITY ANALYSIS 
29/09/2016 
 A subdomain of the Upper Rhine alluvial aquifer  
widely studied at Université de Strasbourg 
 
 Average rainfall per year: 1100 mm 
 Average of evapotranspiration per year: 900 mm 
 
 Simulate variations of piezometric levels during 1500 days 
 Sensitivity analysis performed over the last 450 days 
 
  Investigate spatial and temporal variations of sensitivity 
indices 
 
 
 1500 simulations of Quasi Monte Carlo 
 
 
Field based sensitivity analysis 
10 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
 Nash parameters for recharge: RUMAX, RN, TAU 
 Aquifer parameters: K and S 
 Parameters defined by zones (Zonation)                20 parameters 
Mean and Variance of the hydraulic head 
RN and TAU Permeability Storage capacity 
11 
RUMAX 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
29/09/2016 
 Nash parameters for recharge: RUMAX, RN, TAU 
 Aquifer parameters: K and S 
 Parameters defined by zones (Zonation)                20 parameters 
170
165
160
155
150
145
140
10 km
6
8
9
2
1
4
10
5
3
7
11
12
13
14
15
Frame 001  31 Mar 2015  Zonation
100.1
90.1
80.1
70.1
60.1
50.1
40.1
30.1
20.1
10.1
0.1
10 km
6
8
9
2
1
4
10
5
3
7
11
12
13
14
15
Frame 001  08 Jun 2016  ZonationMean and Variance of the hydraulic head 
Mean Variance 
 Hydraulic heads range between 140 and 165 m 
 Variance ranges between 0 and 100: the imposed boundaries have the lowest variance 
11 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
s
Time (days)
 S1
 K3
 RN3
 RUMAX1
Point 1
29/09/2016 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
Influential parameters: S, K, RUMAX, RN 
12 
Temporal variation of sensitivity indices at point 1 
29/09/2016 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
Influential parameters: S, K, RUMAX, RN 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
R
e
c
h
a
rg
e
 (
m
m
/d
a
y
)
Time (days)
Recharge
12 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
s
Time (days)
 S1
 K3
 RN3
 RUMAX1
Point 1
0 100 200 300 400 500
140,6
140,8
141,0
H
y
d
ra
u
lic
 H
e
a
d
s
 (
m
)
Time (days)
H
Temporal variation of sensitivity indices at point 1 
29/09/2016 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
Influential parameters: S, K, RUMAX, RN 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
R
e
c
h
a
rg
e
 (
m
m
/d
a
y
)
Time (days)
Recharge
12 
0 00 200 300 400 500
0,1
0,2
0,3
0,4
0,5
0,6
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
 o
f 
K
3
Time (days)
S
K3
0 100 200 300 400 500
0,1
0,2
0,3
0,4
0,5
0,6
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
 o
f 
S
1
Time (days)
S
S1
Decrease of piezometric level 
 
 
Decrease of the influence of 
permeability 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
s
Time (days)
 S1
 K3
 RN3
 RUMAX1
Point 1
0 100 200 300 400 500
140,6
140,8
141,0
H
y
d
ra
u
lic
 H
e
a
d
s
 (
m
)
Time (days)
H
Temporal variation of sensitivity indices at point 1 
Increase of piezometric level 
 
 
Increase of the influence of 
storage capacity 
29/09/2016 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
R
e
c
h
a
rg
e
 (
m
m
/d
a
y
)
Time (days)
Recharge
0 100 200 300 400 500
0,0
0,1
0,2
0,3
0,4
0,5
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
 o
f 
R
N
Time (days)
S
RN
13 
Temporal variation of sensitivity indices at point 1 
RN: water transfer in the unsaturated zone 
(delay and spreading) 
RUMAX: threshold parameter 
29/09/2016 
 
 
If RU>RUMAX 
 
 
 
Decrease of the sensitivity 
of RUMAX 
 
 
 
Increase of the recharge 
 
 
 
Increase of the sensitivity of 
RN 
 
 
 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
R
e
c
h
a
rg
e
 (
m
m
/d
a
y
)
Time (days)
Recharge
0 100 200 300 400 500
0,0
0,1
0,2
0,3
0,4
0,5
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
 o
f 
R
N
Time (days)
S
RN
0 100 200 300 400 500
0,00
0,05
0,10
0,15
0,20
0,25
 Ru
 Rumax
W
a
te
r 
L
e
v
e
l 
(m
)
Time (days)
Temporal Variation of RU and RUMAX
13 
0 100 200 300 400 500
0,0
0,1
0,2
0,3
0,4
0,5
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
 o
f 
R
U
M
A
X
Time (days)
SRUMAX
Temporal variation of sensitivity indices at point 1 
RN: water transfer in the unsaturated zone 
(delay and spreading) 
RUMAX: threshold parameter 
29/09/2016 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
R
e
c
h
a
rg
e
 (
m
m
/d
a
y
)
Time (days)
Recharge
0 100 200 300 400 500
0,0
0,1
0,2
0,3
0,4
0,5
T
o
ta
l 
S
e
n
s
it
iv
it
y
 I
n
d
ic
e
 o
f 
R
N
Time (days)
S
RN
13 
Temporal variation of sensitivity indices at point 1 
RN: water transfer in the unsaturated zone 
(delay and spreading) 
RUMAX: threshold parameter 
29/09/2016 
 
 
If RU>RUMAX 
 
 
 
Decrease of the sensitivity 
of RUMAX 
 
 
 
Increase of the recharge 
 
 
 
Increase of the sensitivity of 
RN 
 
 
 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
R
e
c
h
a
rg
e
 (
m
m
/d
a
y
)
Time (days)
Recharge
13 
Temporal variation of sensitivity indices at point 1 
RN: water transfer in the unsaturated zone 
(delay and spreading) 
RUMAX: threshold parameter 
29/09/2016 
Sensitivity maps over time 
GLOBAL SENSITIVITY ANALYSIS:  
NASH MODEL 
S1 
RUMAX3 14 
RUMAX3: sensitive from the 1st to 185th day 
S1: influential between the 185th and the 346th day 
 
 
Weak or even negligible correlations between the 
185th and the 346th day 
 
 
 
Identification of S 
 
 
 
 
During the first days 
 
 
 
Identification of RUMAX and RN knowing S 
 Determination of influential parameters 
 
        Nash: RUMAX and RN       
        Groundwater flow: K and S 
 
 Temporal variation of parameter sensitivity 
 
        Nash: saturation of the 1st reservoir (RUMAX) and recharge (RN) 
        Permeability: decrease of the hydraulic heads 
        Storage capacity: increase of the hydraulic heads 
  
 Weak interactions between aquifer and recharge parameters 
 
 
 
 
 
Confirmation of the feasibility of simultaneous estimation of recharge and 
groundwater flow parameters 
 29/09/2016 
q K H   K constant, q variable      q=0, détermination of K   
GLOBAL SENSITIVITY ANALYSIS: 
CONCLUSION 
15 
  
Thank you for 
your attention 
GLOBAL SENSITIVITY ANALYSIS 
29/09/2016 
Parameters Interactions 
Nash model       Richards model 
14 
0 100 200 300 400 500
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
V
a
ri
a
n
c
e
Temps (jour)
 RUMAX3
 RN1
 RN2
 RN3
 RN4
 K3
 S1
 S2
 Variance
Point 4
0 100 200 300 400 500
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
In
d
ic
e
s
 d
e
 S
e
n
s
ib
ili
té
 d
'O
rd
re
 2
Temps (jour)
 K3&S1
 RN3&S1
 RN2&K3
 RU3&S1
 RUMAX3&K2
 RUMAX3&RN2
Point 4
0 100 200 300 400 500
0,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,40
V
a
ri
a
n
c
e
Temps (jour)
 KS1
 
 K3
 S1
 Variance
Point 1
0 100 200 300 400 500
0,00
0,05
0,10
0,15
0,20
In
d
ic
e
s
 d
e
 S
e
n
s
ib
ili
té
 d
'O
rd
re
 2
Temps (jour)
 K3&S1
 HE1&S1
 HE1&K3
 S1
 KS1&S1
 KS1&K3
Point 1
0 100 200 300 400 500
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
V
a
ri
a
n
c
e
Temps (jour)
 RUMAX3
 RN1
 TAU1
 K2
 S2
 S3
 Variance
Point 2
0 100 200 300 400 500
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
In
di
ce
s 
de
 S
en
si
bi
lit
é 
d'
O
rd
re
 2
Temps (jour)
 K2&S3
 RN1&S3
 RN1&K2
 RN1 & TAU1
 RUMAX3&RN1
Point 2
PARAMETER ESTIMATION 
29/09/2016 
Diffusivity equation (Transient state with recharge ) 
 
 
Steady state without recharge 
 
 
Steady state with recharge 
 
 
Transient state without recharge  
 
 
14 
 . s
H
S K H q
t

  

 . 0  K H
 .   sK H q
 . 0

  

H
S K H
t
OVERESTIMATION OF RECHARGE 
29/09/2016 
14 
If the hydraulic heads decreases, the imposed 
pressure decreases, and the flux calculated is 
positive and strong   
 
If the hydraulic heads increases, the imposed 
pressure increases, and the flux calculated is 
negative   
 
Assumption of constant hydraulic head over decade 
1
1
2
( )  1

  
      
   
n n
n i i
i
i
i
h h
q K h z K h
z z
RESOLUTION OF RICHARDS 
EQUATION 
29/09/2016 
14  
Pression à l’état initial 
M
ét
h
o
d
es
 i
té
ra
ti
v
es
 :
 P
ic
a
rd
 o
u
 N
ew
to
n
 
Ajustement des 
paramètres 
Ajustement du pas 
de temps 
Calcul du pas de 
temps 
Résolution du 
système : Calcul de 
la pression 
Calcul du pas de 
temps suivant 
Calcul des 
paramètres 
Calcul des 
paramètres 
Calcul du pas de 
temps 
Résolution du 
système : Calcul de 
la pression 
Vérification des 
critères de 
convergence 
Vérification des 
critères de 
convergence 
M
éth
o
d
es n
o
n
 itéra
tiv
es 
Pression au pas de temps suivant 
h=f(K,θ), K= f(h,θ) et θ = f(h) 


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Simultaneous estimation of groundwater recharge and hydrodynamic parameters for groundwater flow modeling

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Archive Ouverte en Sciences de l'Information et de la Communication