The document proposes using joint inversion of real and presumed data to analyze the predictive certainty of groundwater flow models. This combines historical aquifer data with presumed data about alternative conceptual models or potential changes. Parameter values are estimated that balance fitting both data types. If the presumed data is also well-fit, the alternative model or changes are considered compatible with available information. A case study applies this to test if a presumed window in a confining layer is consistent with historical head and drawdown measurements from pumping tests and river stages. While the presumed window improved the model fit, drawdown data suggested the model was biased, so the existence of the window was not conclusively refuted.

1. Groundwater Flow Model Predictive Analysis
David J. Dahlstroml
and John Doherty2
lBarr Engineering Company
Minneapolis, MN, USA
2Watermark Numerical Computing
Corinda, Australia
ABSTRACT
Joint inversion of real and presumed information is proposed as a direct approach to analyzing
the predictive certainty of groundwater flow models. This approach combines historical
information about an aquifer system with presumed information regarding an alternative
conceptual model or with presumed effects of potential modifications to the aquifer system.
Automated inverse methods are used to estimate parameter values that balance the historical and
presumed information. Results of the inversion indicate whether the alternative conceptual
model or the presumed effects are likely based on their compatibility with available information
regarding the aquifer system.
Data utilized in the inversion include at least two independent historical observation groups plus
an observation group consisting of presumed prior information regarding the alternative
conceptual model or presumed responses of the modified aquifer system.
At least some of the historical observations must be sensitive to the parameters associated with
the alternative conceptual model to yield a meaningful analysis of predictive certainty. Likewise,
those parameters to which the presumed observations are most sensitive must be
well-constrained by the historical information.
INTRODUCTION
Analysis of flow model predictive certainty typically boils down to one of the following
questions: 1) "Could a different conceptual model yield a comparable match to my historical
data?" or 2) "Can models based on a single set of parameters conform to my historical
information and yield a presumed set of outcomes in a predictive simulation?" Automated
inverse modeling methods can be used to address these questions in a straightforward manner:
through a joint inversion based on historical and presumed data.
For questions of the first type, the presumed data might be fictitious prior information regarding
an aquifer parameter. An example is presented in which a zone within a confining layer
separating two aquifers is presumed to be a window of high hydraulic conductivity. The effect of
Dahlstrom, D.J. and J. Doherty, 1998. Groundwater Flow Model Predictive Analysis,
in MODFLOW'98 Proceedings, Vol. II, E. Poeter, C. Zheng, and M. Hill, eds.,
Colorado School of Mines, Golden, CO, pp. 767-774.
Printed with permission of the publisher.

2. this presumed window on the model's ability to match historical data is used to assess the
probability that the window is real.
For questions of the second type, the presumed effects might represent a worst-case scenario
resulting from natural or man-made modifications to an aquifer system. Examples of worst-case
effects include unacceptable well interference, excessive reduction of groundwater discharge to a
river due to pumping, and extreme water table declines during a drought.
PREDICTIVE ANALYSIS PROCEDURE
Predictive analysis of a flow model begins with an automated parameter optimization in which
the most robust flow model is produced given the available data. Two or more independent
historical observation groups are recommended for the initial optimization, because the
additional information can markedly improve parameter estimates (ASTM, 1997, p. 322).
Revise the flow model to accommodate the alternative conceptual model or predictive
simulation; and expand the objective function for the inverse model to include the presumed
prior information or worst-case observations. The contribution by each independent data set and
the prior information or possible outcomes should be approximately equal.
Repeat the optimization using the modified objective function; and assess whether the alternative
conceptual model or worst-case observations are consistent with the historical information.
CASE STUDY
Background and Modeling Methods
The case study concerns a site located adjacent to the Mississippi River (Figure 1). A sequence
of fluvial deposits up to 83 meters thick underlies the site. The fluvial stratigraphy beneath the
site consists of a continuous confining unit overlying a thin sand layer (upper aquifer layer), a
discontinuous clay layer (lower confining unit), and a thick sequence of sand and gravel (lower
aquifer layer) (Figure 2). The fluvial deposits are incised into bedrock aquifers ranging in age
from Devonian to Ordovician. Regionally, the bedrock aquifers and the fluvial deposits
discharge to the river. Recharge is locally induced from the river to the fluvial deposits by
domestic and industrial groundwater withdrawal.
Discharge from the margins of the bedrock valley was modeled using the general head boundary
package (McDonald and Harbaugh, 1988). Flux from the base of the bedrock valley was
modeled using the well package with injection rates assigned from an analytic element method
model (Strack, 1989) prepared previously for the site. The river and storm basin were modeled
using the river package. The lower confining unit was modeled as a zone of low horizontal and
vertical hydraulic conductivity.

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4. Parameters optimized in the inverse model included conductance of the river bed, conductance of
the general head boundaries, infiltration rates on two zones, horizontal hydraulic conductivity of
the upper and lower aquifer layers, and vertical hydraulic conductivity of the lower confining
unit. Automated inverse modeling was performed using PEST98 (Watermark Computing, 1998,
1994) in conjunction with ModIME (Zhang, et aI, 1995).
Historical observations included water level measurements at near-record high and low river
stages and steady-state drawdowns measured in response to two aquifer tests: one with a
pumping well screened in the upper aquifer layer and one with a pumping well screened in the
upper 12 meters of the lower aquifer layer. Drawdowns were used as observations rather than
hydraulic heads to prevent any bias in the head distribution of the river stage simulations from
being propagated into the observations based on the aquifer test simulations (Hill, 1992, p. 174).
Dense non-aqueous phase liquids released at the site in the past appear to have been trapped by
the lower confining unit. A drilling program defined the lateral, erosional limits of this unit. No
windows through the lower confining unit were identified within it's erosional limits. The
possibility of the existence of a window located between the soil borings was tested by adding a
new zone to the vertical hydraulic conductivity array for the second model layer. An additional
parameter was added to the optimization: Kvwin, the vertical hydraulic conductivity of the zone
representing the presumed window.
Starting values for the aquifer parameters were set based on analysis of the aquifer tests. Two
optimizations were performed using only the historical information: one with the initial value of
Kvwin set equal to the value for the zone representing the lower confining unit; and one with the
initial value set equal to the vertical hydraulic conductivity of the upper aquifer layer
(optimizations 1 and 2 of Table 1, respectively).
Table 1
Summary of Optimization Results
Optimi- Vertical hydraulic conduc- Sensitivity of all Simulated discharge
zation tivity of the presumed observations to the given through the window with
window (Kvwin) (m/day) parameters a well pumping in the
upper aquifer layer
Starting Optimized Kvwin All para- Discharge % of dis-
Value Value meters (m3/day) charge from
the well
1 2.44E-04 2.48E-04 1.70E-03 248 7.02E-02 4.3E-02
2 1.54 4. 18E-02 2.6 252 17.7 10.8
3 1.54 1.48 6.4 236 30.4 18.6

5. A joint inversion was then performed in which presumed prior information regardingthe value of
Kvwin was included in the objective function (optimization 3 of Table 1). The weight on the
prior information was set so its contribution would be one-third of the total sum-of-squared-
errors if the parameter value declined to its final value from the second optimization.
Results of the Predictive Analysis
As expected, the distribution of sensitivity of modeled drawdowns caused by pumping in the
lower aquifer layer shows that those observations in the upper aquifer layer closest to the window
are the most sensitive to Kvwin (Figure 3). However, the insert on Figure 3 shows that even at
the well with the highest sensitivity to Kvwin, the sensitivities to many of the other model
parameters are markedly higher. The contribution from Kvwin should dominate the sensitivity
distribution of at least some of the historical data to conclude that the alternative conceptual
model is inconsistent with the available information.
Presumed window
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Site Outline
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• 0.003 - 0.004
• 0.004 - 0.005
Lower Aquifer Layer
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+
Figure 3. Distribution ofsensitivities ofmodeled drawdown caused by pumping in the
lower aquifer layer to vertical hydraulic conductivity ofthe window. Pie chart shows the
breakdown ofsensitivity of the modeled drawdown at one well to all model parameters.

6. In the initial optimization, Kvwin remained essentially unchanged, a reflection of the model's
extremely low sensitivity to that parameter at that initial value. In the second optimization,
Kvwin was reduced substantially from its starting value, but was not well constrained by the
historical data (Table 1). None of the other parameters changed radically in the joint inversion.
Even at the higher starting value, heads and drawdowns calculated by the flow model have a
relatively low sensitivity to the parameter Kvwin, whereas simulated discharge through the
window is highly sensitive to Kvwin (Table 1).
Drawdowns measured in the upper aquifer layer in response to pumping in the lower aquifer
layer showed no correlation to distance from the pumping well (Figure 4). "Short-circuiting"
around the limits of the lower confining unit was inferred as the primary cause of the observed
drawdown rather than leakage through the lower confining unit. The lack of outliers of
excessively large drawdown was originally taken as support for the conceptual model of a
continuous lower confining unit within its erosional limits. However, increasing Kvwin
improved the model's ability to match this data subset without causing the simulated drawdown
to exceed the measured value at any well (Figure 4).
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Figure 4. Distance-drawdown plot of the measured and calculated water-level responses in the
upper aquifer to pumping in the lower aquifer.
Distance-drawdown data from the aquifer test in the upper aquifer layer qualitatively suggest that
Kvwin was artificially high in the joint inversion (Figure 5). The simulated drawdown at the well

7. located closest to the presumed window drops further from its observed value as Kvwin rises, due
primarily to leakage from the lower aquifer layer through the window reducing the drawdown in
the upper aquifer layer (Table 1).
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Figure 5. Distance-drawdown plot of the measured and calculated water-level responses in the
upper aquifer to pumping in the upper aquifer.
DISCUSSION
Joint inversion of historical and presumed information directly addresses groundwater flow
model predictive certainty. Key parameters and observations are identified through analysis of
the spatial distribution of sensitivities and comparison of the relative magnitudes of the absolute
values of the sensitivities. The parameters to which presumed prior information is applied must
be well-constrained by the historical data for the predictive analysis of an alternative conceptual
model to be conclusive. The absolute values of the sensitivities of the worst-case observations to
at least some parameters must be relatively large and those parameters must be well-constrained
by the historical data for the predictive analysis of a worst-case scenario to be conclusive.
As more sets of independent information are combined in the objective function of an inverse
model, the sum-of-squared errors between observed and simulated values for any given subset of
observations will generally rise. However, a more robust model is produced because the total
sum-of-squared errors over all observations is lower. The result can be a "biased" model with

8. respect to a given subset of data. For example, the simulated drawdownin the upper aquifer
layer in response to pumping in the lower aquifer layer was underestimated at all wells in all of
the optimizations (Figure 4). If this data subset had been the only element of the objection
function, the residuals would undoubtedly have been more randomly distributed, but the resulting
model would have poorly predicted the system's responses to pumping in the upper layer and to
changes in river stage.
A similar data subset bias is likely when the objective function is expanded to include fictitious
prior information or presumed observations. The challenge in applying the proposed approach
will be deciding when the model has strayed too far from the original, calibrated parameter set,
i.e., when the bias introduced is excessive.
The case study suggests the limitation of using only hydraulic head data: in spite of a large and
varied historical database (including pumping of wells located directly above and below the
confining layer, near the presumed window) the results of the predictive analysis do not
unequivocally refute the existence of the presumed window in the lower confining unit. Adding
discharge measurements and independent types of information to the historical database (such as
stable isotope data or temperature data) would likely make the joint inversion more conclusive.
REFERENCES
American Society for Testing and Materials, 1997. Standard Guide for Comparing Ground-water
Flow Model Simulations to Site-Specific Information. ASTM Standard D 5490-93. 1997
Annual Book of ASTM Standards, Section 4, Construction. Volume 04.09, p. 323-329.
Hill, M.e., 1992. A Computer Program (MODFLOWP) for Estimating Parameters of a
Transient, Three-Dimensional, Ground-Water Flow Model Using Non-Linear Regression.
U.S.G.S. Open-File Report 91-484. U.S. Geological Survey, Denver, Colorado. 358 p.
McDonald, M.G. and A.W. Harbaugh, 1988. A Modular Three-Dimensional Finite-Difference
Ground-Water Flow Model. Techniques of the Water-Resources Investigations of the
U.S.G.S., Book 6, Chapter AI. Scientific Software Group, Washington, D.e.
Strack, O.D.L., 1989. Groundwater Mechanics. Prentice Hall, Englewood Cliffs, NJ. 732 p.
Watermark Computing, 1994. PEST, Model-Independent Parameter Estimation. Computer
software manual.
Watermark Computing, 1998. PEST98 Upgrade Notes. Computer software manual.
Zhang, Y., C. Zheng, C.J. Neville, and e.B. Andrews, 1995. ModIME User's Guide. An
Integrated Modeling Environment for MODFLOW. PATH3D, and MT3D. Version 1.0. S.S.
Papadopulos & Associates, Inc.