Estimating runoff by analytical
modeling of groundwater levels
data by courtesy of NVE and met.no
Application of the Dupuit-Forchheimer
assumption for calculating
groundwater heads
http://folk.uio.no/nilsotto/
Nils-Otto Kitterød
The Norwegian Centre for Soil and Environmental Research
and University of Oslo, Dept. of Geoscience
nilsotto@geo.uio.no
Why runoff ??
in groundwater and climatic change context ??
Water balance: R = P – E + DS
DS if time 
P ~ f(atmc,T)
E ~ f(T,q)
R
point
values
average value
depending on
catchment scale
Changes in atmosphere circulation pattern  Runoff
P, E, T, q, grw.levels
Changes in atmosphere circulation pattern

Runoff
Challenge: Coupling Atmosphere Circulation Modeling to Runoff
- weakly to get ‘correct runoff’ or
- strongly which include the land – atmosphere coupling
Idea:
1) Given present observations of runoff (R) and groundwater levels (G)
2) Calibrate hydraulic conductivity (k) or transmissivity (kH) in a
groundwater model (f) that reproduce present observations of G
3) Validate f on independent R or G
4) Use paleo-ecological data to reconstruct groundwater levels (Gp)
5) Estimate paleo-runoff (Rp) by applying f on Gp
Problem: significant uncertainties of Gp
 develop simple but robust groundwater model (f)
Assumption: f is ~ constant ??
errosion rate will affect boundary conditions
include boundary location as a stochastic function
Paleo-data
The inland glacier
~10500 B.P.
(Andersen, B.G.,2000)
Sea level about 200 m
higher than present
sea level.
Average paleodischarge: ~3000m2/s
(Tuttle, K., 1998)
15 km
15 km
The Gardermoen
delta
Kettle lakes may be used as ’paleo-sensor’s for
changing groundwater levels (Stabell, B.)
Risa runoff
station
Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal
~30 kettle lakes
?
Bonntjern &
Svenskestutjern
Kettle
lakes
The Bonntjern
kettle lake
Groundwater levels,
Hauerseter
1974
low
groundwater
levels
1991
high
-17
Grw.level, m
-18
-19
-20
-21
-22
2002
1998
1994
1990
1986
1982
1978
1974
1970
1966
year
2000
medium
high
Groundwater levels
high
low
high
high
low
Idea:
estimate runoff (or net
precipitation) from ~ 6000 B.P.
– present given a record of
groundwater levels based on
paleo-studies of kettle lake
sediments
low
high
low
high
low
“The paleo-sensor hypothesis”,
Stabell, B. (2001)
Runoff
precipitation
1967
1968
1969
1970
1971
temp.
runoff
Note that variation of runoff is more related to
temperature than precipitation due to the importance of snow at Gardermoen
temp.
precipitation
runoff
Observations from
the Gardermoen
aquifer, Oslo.
Tuttle, K. (1997)
Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal
11000
paleo-distribution
channels shows
radial flow
10000
9000
8000
Trandum Delta
Paleo-portal
UTM-North
7000
Helgebostad Delta
Paleo-portal
6000
5000
4000
3000
2000
1000
4000
5000
6000
7000
8000
UTM-East
9000
10000
11000
12000
Gardermoen, grw. obs, eu89
cross-section
Tuttle, K. (1997)
m a.s.l.
Kettle- Ice-contact
hole slope
Delta Topsets
200
Delta Foresets
180
160
Delta Bottomsets
?
140
?
?
Bedrock
120
3000
West
Profile 1
2000
meters
1000
0
East
m a.s.l.
200
3000
2000
West
meters
1000
500
0
East
Tuttle, K. (1997)
100
Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal
”The Gardermoen
doughnut”
Yallahs River, Jamaica
N
Reading, H.G. (1998)
Darcy’s law + conservation of mass
 Poisson’s equation:
2
 F=-N(t)
R
Precipitation N, over an island/delta
with radius R:
r
N
The Dupuit-Forchheimer assumption: dF/dz = 0
Because R >> z
F = 1
k h2
2
phreatic aquifer:
F=
confined aquifer:
F= k Hf
N r2 + R2
4
+ F0
Risa runoff
station
Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal
River runoff
groundwater levels (1992-94) at the Trandum delta
some grw.levels from 1967-today Helgebostad delta
and meteorological data
Forward problem: find h given k
Inverse problem: determine k given h
h
inverse
Confidence
intervall
Notice
sensitivity
problems
k
forward
For the Gardermoen case, we suggest
Poisson’s equation as flow model:
2
 F=-N(t)
geology: 3d 2d
because of symmetry
around paleo-portals
and

0
z
(but without assuming qz = 0)
because L >> H,
 Dupuit-Forcheimer assumption
2d 1d
i.e.  or h is a function of r
hydrology: transient  steady state
Model:
purpose: E{k|obs}
from geology: 3d is overkill
from hydrology: transient flow is overkill
R2
h2
l
R1
h1
A) Steady state model for l  r < R1:
N
h2
Balance of mass:
i) QA = Npl2 – Npr2
ii) Qr = – QA
2pr
h1
r
R2 l
R1
Darcy’s law:
iii)
Qr = q = – k dh
dr
H
d FA
Qr = –
dr
Trick:
F = ½ kh2 , open
F = kHf , confined
2
N
l
i, ii and iii) dFA = 2 r dr – r dr
solution: FA = – N
4
r2
2
1
–R
+
N l2
2
ln
r
R1
+
F1
(iv)
N l2
2
ln
r
R2
+
F2
(v)
B) Same procedure for R2  r < l :
solution: FB = – N
4
r2
substitute l in iv and v, then FA
F=
N
4
r2
– N
4
r2
2
– R 1 – F1
2
2
– R2
– R 2 – F1
+
= FB = F
ln r – ln R2
ln R2 – ln R1
ln r – ln R1
ln R2 – ln R1
(vi)
Problem: does not fit observed l very well
N
h2
h1
r
l
R2 l
R1
blue is ’observations’
purple is the simple
doughnut equation
Solution: let H (or k) = f(r)
2
dA = N l dr – r dr
2k rH
H
N
H=H1 – a(r-R1)
2

1
r
R2
R1
H2
H1
Two (simple) integrals
to solve:

1
r(H0-ar) dr
(1)

r
(H0-ar) dr
(2)
Limitations of Dupuit-Forcheimer assumptions
For isotropic aquifer and constant H, the 3d effect is neglect able at
distance 1-2H from the boundaries, 2H(kh/kv)1/2 if anisotropic
What if H  constant?
H1
H2
The vertical gradient cannot
be ignored if H1>>H2, but for
H1/H2 < 10, max. relative
error is < 1%.
k = 1.73e-5 m/s
N = 1.267e-8 m/s  1mm/d
H1 = 1000 m, H2 = 100m, H’=0.5(H1+H2)
H=H1=H2=100m
k L
N H

L = 5100 –1000 m
100
3
k L
N H’

550
3
H2 is constant
analytical
numerical
H1 gradually opens up
H1 is constant
analytical
numerical
H2 gradually opens up
11000
observation
points
10000
9000
8000
Trandum Delta
Paleo-portal
UTM-North
7000
Helgebostad Delta
Paleo-portal
6000
5000
4000
3000
2000
1000
4000
5000
6000
7000
8000
UTM-East
9000
10000
11000
12000
Gardermoen, grw. obs, eu89
Results:
h=f(r,R1,R2,h1,h2,N,k1,k2)
k1,k2 unknown
=f(r,R1,R2, 1,2,N,k’,H1,H2)
H1,H2 unknown
grw. obs from 1992 -1994
“Validation” exercise:
R1= 336 m, h1=171.5 m
R2= 5100 m, h2=185 m
1) calculate optimal k1,k2
(or k’,H1,H2)
given R (or Neff) and
grw. obs (1992/94)
k1=2.89e-5 m/s
k2=5.69e-6 m/s
k’=1.73e-5 m/s
H1= 302 m
H2= 66 m
2) calculate R (or Neff)
given optimal k1, k2 and
a timeseries of groundwater levels
3) compare calculated R (or Neff) with independent observations of R (or Neff)
from the Risa runoff station
3) compare observed
and calculated R
overall
pattern OK
 steady state
is sufficient
under estimate
max values
over estimate
min values
grw. obs from L.Olimb
1967
In this case
all data is
available
except
evapotranspitation,
which can be calculated:
1968
1969
1970
temperature
E=P-R
E is expected to
correspond with
temprature
E = moving average B=70 days
1971
Conclusion:
why analytical modeling?
analytical solutions are at hand
focus the essence
easy sensitivity analysis
continuous in time and space
numerical modeling require details
discretization in time and space
time consuming
Climatic change:
The present is the key to the past
The past is the key to the future
Next step I:
Trandum Delta
Paleo-portal
Helgebostad Delta
Paleo-portal
Stochastich R1 and R2
sensitivity analysis
Next step II:
z
zn
high
low
(m a.m.s.l.)
high
high
low
low
high
low
z1
high
low
x,y
Levelling of kettle lakes
 paleo runoff (proxy data to paleo-climatic modeling)
Thank you!