My presentation from ModelCARE96

1. Incorporating Groundwater Flow
Direction and Gradient into Flow
Model Calibration
Gregory J. Ruskauff
INTERA Inc., Albuquerque, NM
James O. Rumbaugh, III
Environmental Systems Inc., Herndon, VA

2. Brief Review
 Van Rooy & Rosbjerg (1988) used plume to
constrain flow model.
 Duffield et al. (1990) used advective particle
tracking of plume to constrain flow model.
 Anderman et al. (1994) used advective
particle tracking and inverse estimation for
flow model.
 Guo & Zhang (1994) presented an inverse
method that considers head and gradient .

3. What if you don’t have a plume?
 No plume, no analysis.
 Any surrogate we could use instead?
 More thorough use of head data?
 Assume that, as a first approximation,
the shape of the potentiometric surface
can be used.

4. Potentiometric Surface Shape
 Gradient gives steepness of surface.
 Direction computed from plane geometry.
 Pinder et al. (1981), Vacher (1989), and Cole
and Silliman (1996) describe various
approaches and computer programs.
 Not currently quantitatively included in
calibration.
 Can be added to the calibration methodology.

5. Establishing Gradient/Direction
Error Bounds
 Any measurement incorporates error:
– field measurement error
– scale effects
– interpolation effects
– error from unmodeled small-scale heterogeneity
 Unmodeled small-scale heterogeneity is
probably the most difficult to quantify.
 Analytic solutions of Mizell (1980) address
the error in direction and gradient in a 3-well
monitoring network.

6. Approach to Setting Error Bounds
 Gelhar (1986) presented an approach that used
analytic stochastic solutions to help set reasonable
model error.
 In an analogous approach, Mizell’s solutions are
used to help set reasonable gradient and direction
errors for using this information in calibration.
Solutions consider:
– network size and orientation relative to flow
– correlation scale
– measurement error
 Assume all error from heterogeneity.

7. Analytic Solution Assumptions
 Networks form isosceles triangles.
 Infinite domain.
 Solutions obtained by pertubation analysis,
may not be accurate for extremely
heterogeneous media.

8. Characteristics of Mizell’s Analytic
Solutions
 2 solutions; first gives the expected value of the
direction error, and the second gives the variance of the
hydraulic gradient.
 Gradient solution difficult to use since it involves
knowledge of variance of ln K.
 For values of the ratio of network scale to head
correlation of <1 the error in direction and variance in
gradient are bounded at a maximum. This reflects the
fact that the head field is strongly correlated over short
distances, and implies that less error will result in
measuring the mean gradient from a larger network
(Gelhar, 1991).

9. Bounding Direction Error-
Stochastic Solution
 Practically,L/λ cannot be known.
 Upper bound is given
10
by assuming L/λ < 1.
1
 Gives error in θ ± 21°
0.1
0.01
0.001
0.1 1 10 100
L/λ

10. Synthetic Problem
 2D version of ModelCad sample problem.
 Initially run with homogenous properties to
establish “truth”.
 50 realizations of K generated and MODFLOW
flow simulations done.
 Head calibration goals set, and each realization
analyzed for heads and direction/gradient.
 Acceptable error estimated at 0.75 ft; giving a
residual sum of squares of 8.5 ft2 or less.

11. Synthetic Problem Domain
3 sizes of
networks
 Constant heads
and drains
 Recharge
 120x120 (every
2nd shown)
 Steady-state

12. Reference Conditions
26
22
18
14
10
6
feet

13. Bounding Direction/Gradient
Error- Sensitivity Analysis Approach
 Using calibration goal head (0.75 ft) perform
sensitivity analysis on direction/gradient errors.
 85%, 44%, and 33% gradient errors for smallest to
largest networks from perturbing by target error.
 29%, 13%, and 11% from theory.
 36°, 41°, and 13° for smallest to largest networks from
perturbing by target error.

14. Head Calibration Goals
 50 realizations run, 24
all simulations below Best
Median
22 Worst
RSS goal. Best, median,
Computed Head (ft)
and worst simulations 20
further analyzed. 18
16
14
14 16 18 20 22 24
Observed Head (ft)

15. Monte Carlo Results-Gradient
 25% error bound

16. Monte Carlo Results-Direction
 Smallest network
has largest θ
variability
 Largest network
has lowest
variability
 Error in θ ± 21°

17. Ensemble Direction Errors
16
14
12
10
8 Smallest
6 Medium
4 Largest
2
0
Standard CV% Ave. Error SD Error
Deviation

18. Solute Transport Analysis
 MT3D, constant
mass loading
 Small network
shows most
variability
(shown)

19. Real-World Test
 Truckee Meadows, Nevada, USA
– 7 networks along plume path well
matched by advective particle tracking
– 3 networks had θ errors < 5°
– 3 networks near upper bound (22°, 27°,
31°)
– 1 error of 46° (furthest away from path)

20. Visualizing Gradient and
Direction Errors
 Polar plot.
 True (yellow) and computed (red)
directions plotted and connected with
an arc.
 Percent error in gradient plotted as
displacement away from origin.

21. Gradient/Direction Error
Diagnostic Plot

22. Discussion
 Currently use post plots or contour maps of residuals to
assess bias.
 Can get swamped by data.
 Doesn’t directly consider relations between wells.
 Is bias necessarily bad? If all heads off by 1 foot then
direction and gradient will be correct. Worst head
calibration does not have worst θ error or breakthrough.

23. Discussion- cont.
 Can
be difficult to find meaningful
combinations of wells.
6 5 5 6 7
Stream

24. Conclusions - 1
 Significant direction and gradient errors
can exist even in reasonably calibrated
models.
 An upper bound on direction error from theory
is about ± 21°.
 Sensitivity and Monte Carlo analysis suggests
~ ± 10° is readily achievable.
 Gradient errors of ± 25% common. Harder to
set bounds on gradient error. Should be relative
error.

25. Conclusions-2
 Real world case show reasonable
correspondance with theory and synthetic
analysis in terms of achievable accuracy.
 Presented another tool for assessing calibration.
 Anisotropy not relevant, shape would still be the
same.
 Use of head data alone still leaves room for
substantial transport uncertainty

26. Incorporating Groundwater Flow
Direction and Gradient into Flow
Model Calibration
Gregory J. Ruskauff
INTERA Inc., Albuquerque, NM
James O. Rumbaugh, III
Environmental Systems Inc., Herndon, VA