Realistic Forecasting
of Groundwater Level,
Based on the Eigenstructure
of Aquifer Dynamics
Vince Bidwell
Lincoln Environmental
Lincoln Ventures Ltd
New Zealand
What is eigenstructure?
- an analogy
Music:
• Sound of a drum
is a mixture (eigenvector)
of the modal frequencies (eigenvalues)
Aquifer:
• Groundwater level response to recharge
is a mixture (eigenvector)
of water storage modes (eigenvalues)
Water storage modes
ki - discharge coefficient (eigenvalue)
discharge = ki x storage
Recharge
k2
k1
g1
g2
k3
g3
gi - weighting coefficient (eigenvector)
Aquifer
storage
Piezometric
effect
Aquifer eigenstructure properties
• Water storages are conceptual, not physical
• Discharge coefficients (eigenvalues) are the same
everywhere in the aquifer, and are related to
aquifer properties
• Weighting coefficients (eigenvectors) depend on
observation location and spatial pattern of
recharge
• Piezometric response to land surface recharge is
the most time-variable, and is usually dominated
by the smallest eigenvalues
Conceptual model of piezometric
response to land surface recharge
Land surface recharge (LSR)
kv(x,y)
Vadose zone
storage
k2
k1
g1(x,y)
g2(x,y)
k3
Aquifer
storage
g3(x,y)
Piezometric effect of LSR
relative to river recharge effect d(x,y)
Model assumptions
• Piezometric effects of river recharge are steady,
but spatially variable
• Spatial pattern of land surface recharge is fixed,
but the magnitude is unsteady
• Groundwater abstractions are unknown, and
considered to be part of the time-varying model
error
From conceptual model to
forecast equation
• Conceptual model of piezometric response to
land surface recharge is a linear dynamic system
• System structure is described by z-transforms
• Z-transform model, with a noise term, is
expressed as an ARMAX (Box-Jenkins)
stochastic difference equation for forecasting
groundwater level
• Forecast equations have previous forecast errors
as an additional input
Model calibration
• Parameters of the conceptual model are
calibrated, because of structural independence
and physical realism
• Noise term has one autocorrelation parameter
• Objective function is minimisation of meansquare forecasting error
• Calibration with the “solver” tool in excel
• Avoids some identification and calibration issues
in conventional use of ARMAX equations
Application example
• Observation well in a 2000 km2 aquifer, Central
Canterbury Plains, New Zealand
• Land surface recharge (one, monthly series)
calculated from daily water balance model
• Forecasts of groundwater level convert to
forecasts of low flow in river supplied by aquifer,
under drought conditions
• Effect of unknown groundwater abstraction is
incorporated into the forecast equation as inputs
of previous forecast errors
3000
44
2500
42
2000
40
River flow
38
1500
36
1000
34
500
32
0
May-95
30
Oct-95
Apr-96
Oct-96
Apr-97
Oct-97
Apr-98
Oct-98
Groundwater level
40 km
Apr-99
Oct-99
Apr-00
Observed
Predicted
1-mth forecast
Land surface recharge
-6
Groundwater depth (m)
-7
200
-8
150
-9
-10
-11
100
-12
-13
50
-14
-15
1995
0
1996
1997
1998
1999
2000
Land surface recharge (mm/mth)
250
-5
Summary
• Eigenstructure approach expresses the dynamic
behaviour of an aquifer as a simple linear system
• System parameters are physically realistic,
related to aquifer properties, and structurally
independent
• System can be expressed as a stochastic
difference equation for real-time forecasting
• Calibration based on system structure avoids
identification issues with ARMAX approach