How many transmissivity data are required to safely delineate a protection zone of a
braided alluvial aquifer when using kriging and upscaling?
J. Kerrou 1, P. Renard 1, I. Lunati 2, S. Madier 3, and H.J. Hendricks-Franssen 4
1
Center of Hydrogeology, University of Neuchâtel, Neuchâtel, Switzerland
2 Institute of Fluiddynamics, ETH Zurich, Zurich, Switzerland
Strasbourg National School for Water and Environment Engineering, Strasbourg, France
4 Institute of Hydromechanics and Water Resources Management, ETH Zurich, Zurich, Switzerland
CENTRE D’HYDROGÉOLOGIE
UNIVERSITÉ DE NEUCHÂTEL
3
1. Introduction
2. Methodology
Protection zones play a key role in the protection of water-supply wells against contamination by
pollutants from industry and agriculture. In Switzerland, the protection zones (SII) correspond to the 10days isochrone area around the well. To be accurate, the forecast of these zones requires a good
knowledge of the geometry, the hydrodynamic parameters, and the boundary conditions. However,
usually in practice, these parameters are poorly known and the model results suffer from uncertainty.
Therefore, the knowledge of subsurface heterogeneity is of prime importance to evaluate transit times of
pollutants from infiltrating surfaces or riverbanks to the water-supply wells. Many studies have shown the
critical impact of spatial heterogeneity on mass transport predictions (Dagan 1986). In this study, we test
the efficiency of kriging estimation by taking into account upscaling on a synthetic alluvial aquifer. Then
we investigate the effect of adding transmissivity data points on the accuracy of protection zone forecasts.
To estimate the worth of adding transmissivity data and upscaling effect on improving the accuracy of
the calculated protection zone, it is necessary to have a detailed solution (reference). This study
includes the following steps:
¾ A realistic bimodal transmissivity field is generated by combining deterministic and stochastic
elements. It represents a braided river alluvial aquifer environment;
¾ Steady-state reverse flow and advective-dispersive transport are solved in the reference T field
under a uniform gradient and a single pumping well to calculate the reference protection zone;
¾ Successive data sets of punctual transmissivities are randomly sampled from the T reference;
¾ Experimental variograms are calculated, modeled and used to krige a number of T fields;
¾ 9 successive stages of upscaling are applied on every estimated T field;
¾ Protection zones are calculated by solving again the steady-state reverse flow and advectivedispersive transport in each estimated T field (including the upscaled T fields);
¾ Calculated protection zones are compared with the reference and error indicators are calculated.
3. The reference
The transmissivity reference field is a 1000 m x 400 m 2D domain with a resolution of one meter. The
field has been constructed from an aerial photograph of a braided river system containing lenses and
channels. These two structures are then populated by MultiGaussian simulations. The results is a
bimodal distribution for ln(T) that displays an anisotropic experimental variogram. The integral scale i
(Gelhar 1993) is calculated in each direction: iX = 25.77 m and iY = 7.36 m.
-8
-6
-4
-2
0
0
ln(T)
6. T field estimation
100 200 300 400 500
Distance
(c)
(a) ln(T) distribution for the reference field; (b) ln(T) histogram; and (c) experimental reference
field variogram
5. Upscaling
According to Renard (1997), the tensorial renormalisation (Gautier and Noetinger 1997) is one of the most
accurate fast methods to calculate the equivalent permeability tensor. We therefore selected this technique
and applied it on a series of successive upscaled grids. A dimensionless factor Li represents the size of the
homogenized blocks normalized by the integral scale. As a point of comparison, we also calculate the
geometric mean of the sampled transmissivities and use it in a homogeneous field.
(a)
(b)
(c)
An experimental variogram
for each ln(T) set of samples
has been calculated.
Directional variograms (30°
and 45°…) were tested but
rejected because they didn’t
show anisotropy. A spherical
model with nugget effect has
been kept for all sets of data.
Specific model parameters
were inferred for every case
and cross-validated. The
interpolation was then
conducted by lognormal
kriging. The figure shows the
data histograms, variograms
models and the estimated T
field. Note that the channels of
the reference are reproduced
only when a high number of
samples is used.
ln(T)
400
0.15
-2.5
-3.5
200
-4.5
-5.0
0
0
400
200
600
800
ln(T)
400
0.08
-1.6
-2.4
-3.2
200
-4.1
-4.9
-5.7
0
0
200
400
600
800
-6
- Nb. S
- Std. Dev.
- Mean
- Min
- Max
Distance
0.00
-2.4
-3.4
-4.5
-5.6
-7
-5
-3
0
(c)
2
1
-1
0.08
- Nb. S
- Std. Dev.
- Mean
- Min
- Max
0
600
X(m)
800
1000
4
0.06
0.04
0.02
0.00
100 200 300 400 500
Distance
: 9954
: 1.895
: -5.178
: -9.249
: -0.854
-6.6
400
3
0
-9
-1.3
200
100 200 300 400 500
4
ln(T)
400
200
0
-2
-4
0.02
ln(T)
0
4
2
ln(T)
: 1000
: 1.903
: -5.141
: -8.813
: -0.854
0.04
X(m)
(b)
6
0
0.06
1000
Di1=Dx/ιX
[]
3.467
2.310
1.566
1.023
0.743
0.486
0.246
0.156
0.039
8
0.05
0.00
-8
1000
: 21
: -7.617
: -2.380
: -4.754
: 1.785
0.10
X(m)
(a)
- Nb. S
- Min
- Max
- Mean
- Std. Dev.
Variogram ln(T)
-10
(b)
Variogram ln(T)
1000
800
X(m)
1
0
Variogram ln(T)
600
400
200
0.00
Frequencies
0
(a)
0.02
3
2
Frequencies
-9.1
0.04
Average
distance Dx
[m]
89.33
59.54
40.36
26.37
19.16
12.51
6.33
4.02
1.00
Number of
samples
21
44
96
250
416
1000
4000
9954
400000
Frequencies
-7.4
4
Y(m)
-5.8
0
: 400000
: -9.837
: -0.798
: -5.171
: 1.891
Y(m)
-4.1
200
0.06
Successive data sets are taken randomly from the transmissivity
reference field. Average distances between samples are
calculated (table besides). We define Di the dimensionless
average distance between the samples normalized by the
integral scale. If it is greater than 1, sample points cannot
resolve the heterogeneity that has a length scale on the order of
the integral scale.
Y(m)
Frequencies
-2.4
Y(m)
- Nb. S
- Min
- Max
- Mean
- Std. Dev.
0.08
-0.8
Variogram ln(T)
ln(T)
400
4. Sampling the reference
3
2
1
0
-10
-8
-6
-4
ln(T)
-2
0
100 200 300 400 500
Distance
Histogram, variogram model and estimated (ln)T field for (a) 21, (b)
1000, and (c) 9954 samples
7. Protection zone calculation
The protection zone is calculated by solving the 2D steady-state reverse groundwater flow and advectivedispersive transport with the finite element code Feflow. The result is the distribution of average life
expectancies. We then select all the grid nodes having average life expectancies lower than 10 days. The
same method is applied for the reference and for all the estimated and upscaled T fields.
(a) Upscaled maximum principal value of the transmissivity; (b) upscaled anisotropy factor;
and (c) upscaled anisotropy angle. Calculated in this example for the reference field.
8. Error and accuracy norms
To quantify and compare the error we decided to use two error norms:
α = the percentage of the reference protection zone that has been correctly forecasted. (red area
normalized by red and green)
β = the percentage of the calculated zone that is protected for nothing normalized by the calculated zone
(blue area normalized by blue plus red)
9. Results and Discussion
The graphs of the accuracy and error norms α and β (below) and the maps (besides) have
shown that:
¾ When the average distance between samples is clearly greater than the integral scale
(Di>1) the forecasts are improving approximately linearly with the logarithm of Di
(graph-a-). When Di is less than 1, adding data points does not improve the forecasts.
¾ When Di>1, upscaling with the tensorial renormalization before calculating the
protection zones usually improves the forecasts if the size of the homogenized blocks is
greater than the integral scale (Li>1). The best accuracy is obtained with a completely
homogenized domain (graph-b,c&d-).
¾ A sensitivity study analyzing the effect of sample location showed that the previous
effect is confirmed on average (graph-d-) even if there are some particular cases for which
the error increases with upscaling (graph-c-).
¾ In all cases, the forecasts made with the geometric mean of the sample data were bad
because the geometric mean is ignoring anisotropy.
(a)
1.00
(c)
1.00
0.90
0.90
0.80
0.80
0.80
0.80
0.70
0.70
0.70
0.70
0.60
0.60
0.60
0.60
0.30
Li=0.076
Li=0.15
Li=0.3
Li=0.6
Li=1.21
0.30
0.20
0.10
0.20
0.10
0.00
10
1
0.1
Di
Di=3.46
Di=2.31
Di=1.56
Di=1.02
0.01
0.00
0.01
%
0.50
0.40
0.40
<α>
<α>+2σα
<α>−2σα
α
0.50
(d)
1.00
0.90
α
α
(b)
1.00
0.90
0.50
0.40
0.30
0.30
0.20
0.20
0.10
0.10
0.00
0.00
0.1
1
Li
10
100
0.50
0.40
0.01
0.1
Accuracy norm α
1
Li
10
100
0.01
0.1
1
Li
10
100
References
Dagan G. (1986) Statistical theory of groundwater flow and transport; pore to laboratory, laboratory to formation and formation to regional scale. Water
Resources Research. 22, 1205–1355
Gautier Y., Noetinger B. (1997) Preferential flow-paths detection for heterogeneous reservoirs using a new renormalisation technique. Porous Media, 26, 1-23
Gelhar L.W. (1993) Stochastic Subsurface Hydrology. Prentice Hall New Jersey
Renard P., Le Loc'h G., Ledoux E., De Marsily G. (2000) A fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous
media. Water Resources Research 36(12), 3567-3580