Incorporating groundwater flow direction and gradient into a flow model calibration

Transcription

1 Calibration and Reliability in Groundwater Modelling (Proceedings of the ModelCARE 96 Conference held at Golden, Colorado, September 1996). IAHS Publ. no. 237, Incorporating groundwater flow direction and gradient into a flow model calibration GREGORY J. RUSKAUFF INTERA Inc., 1650 University Blvd. NE, Ste. 300, Albuquerque, New Mexico 87102, USA JAMES O. RUMBAUGH, III Environmental Simulations Inc., 2997 Emerald Chase Drive, Herndon, Virginia 22091, USA Abstract Groundwater flow models most commonly use observed hydraulic-head data as the sole calibration data, however it is possible to achieve a reasonable head calibration and have incorrectflowdirections. Bounds on model gradient and direction errors are set using a previously derived analytic stochastic solution for evaluating hydraulic gradient and direction in a three-well monitoring network. Analysis of gradient errors propagated by Monte Carlo simulation in a syntheticflowmodel showed that the proposed approach works well, and that even a well-calibrated model can have substantial errors in gradient and direction. Solute transport analysis on the flow fields showed substantial differences in concentration histories with even a relatively small amount of flow-field variation. INTRODUCTION It is well known in well testing that fitting to both the drawdown data and its derivative produces superior results to fitting on drawdown data alone (Ehlig-Economides, 1988). This is due to the fact that the derivative amplifies subtle differences and trends. Likewise in groundwater flow modelling it would be superior to match both hydraulic head and its gradient. Hydraulic head is a scalar field, and its first derivative (the hydraulic gradient) is a vector field having both direction and magnitude. The gradient and direction of groundwater flow can be computed from any three head observations using the three-point method and does not require use of any other data. In the method described here the gradient and direction of the groundwater flow field is computed using the head calibration data. This is compared to the gradient and direction computed from the simulated head at the observation locations. Van Rooy & Rosbjerg (1988) showed how the use of various types of data (transmissivity, head, and concentration) contributes to constraining a groundwater flow model. In simulations conditioned on head data the uncertainty in head decreased markedly, but the transport uncertainty remained unaffected. Duffield et al. (1990) used observed plume migration data to explicitly constrain flow simulations. They calibrated a groundwater flow model with an inverse procedure and then performed particle tracking to see how well the movement of a conservative plume was matched. Heterogeneity was added until plume velocity was duplicated. Significantly more

2 72 Gregory J. Ruskauff & James O. Rumbaugh, III parameterization was required to match travel times and directions with only a minor increase in the goodness of the flow model fit. Anderman et al. (1994) used advective particle tracking in an approach similar to that of Duffield et al. The advective front of a conservative plume was defined as a function of time and advective particle tracking incorporated into MODFLOWP to give an inverse solution. Increased parameter sensitivity or ability to discriminate heterogeneity was not observed, but parameter correlation was decreased. Guo & Zhang (1994) presented an automated inverse approach that considers hydraulic head and the hydraulic gradient. TECHNIQUE DEVELOPMENT Computation of hydraulic gradient The hydraulic gradient, i, is defined as the change in potentiometric level over a specified distance along a streamline as i = dh/dl. The direction of groundwater flow can be computed using the three-point method of plane analytic geometry as described by Fetter (1993). Vacher (1989) presents a BASIC program to do this computation. A more complex treatment using basis functions was presented by Pinder etal. (1981). It is assumed that the potentiometric surface is a plane but, because of heterogeneity in aquifer properties, this assumption is only approximate. For calibrating a groundwater flow model the set of observation wells is first divided into groups of three (called a network in this paper), and a computer program then used to compute true and simulated gradient and direction. Assessing acceptable gradient and direction error bounds We propose using the stochastic analysis of Mizell (1980) to set acceptable error bounds in numerical models for gradient magnitude and direction. Mizell's solutions were developed to address the problem of estimating the magnitude and direction of groundwater flow from three observation wells for two-dimensional steady-state flow (a fairly common numerical model configuration). Two solutions are used in this analysis: the first gives the expected value of the direction error, and the second gives the variance of the hydraulic gradient. The errors are a function of several variables, including orientation of the network relative to the flow direction, independent measurement error, and ratio of the network size to head-field correlation scale. Some of these variables, such as the variance of ln, are not easily determined. A key facet of Mizell's analysis is that for values of the ratio of the network scale to head correlation of less than 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. The bounding characteristics of the solution for direction error can be readily used. However, the gradient magnitude variance is dependent on the square of the head-correlation scale that may be difficult to estimate from limited data. Limitations on the solutions include the fact that they were obtained by perturbation analysis, and may not be accurate for extremely heterogeneous media,

3 Incorporating groundwater flow direction and gradient into a flow model calibration 73 an infinite domain (no boundary effects), and the wells were assumed to be arranged in a right triangle. The potential error in gradient magnitude and direction can also be addressed with a simple deterministic analysis. The calculation of hydraulic gradient involves use of three potentiometric-level data points. Error in any point will create error in the flow direction calculated from the model heads. The calculation is done by raising the upgradient and lowering the downgradient heads in the observation network, which results in the maximum gradient change relative to the target value. EXAMPLES Synthetic test problem The technique is illustrated with a perfectly-calibrated synthetic problem modified from Rumbaugh (1993) with noise introduced by geostatistical simulation. The acceptable head error was estimated at 0.23 m (0.75 ft) using the technique of Anderson & Woessner (1992). Alternatively, the total head drop in the observation wells is 3 m (10 ft), and using a criteria that any individual error should not exceed 10% of the drop TECHNIQUE DEVELOPMENT Computation of hydraulic gradient The hydraulic gradient, i, is defined as the change in potentiometric level over a specified distance along a streamline as / = dh/dl. The direction of groundwater flow can be computed using the three-point method of plane analytic geometry as described by Fetter (1993). Vacher (1989) presents a BASIC program to do this computation. A more complex treatment using basis functions was presented by Pinder et al. (1981). It is assumed that the potentiometric surface is a plane but, because of heterogeneity in aquifer properties, this assumption is only approximate. For calibrating a groundwater flow model the set of observation wells is first divided into groups of three (called a network in this paper), and a computer program then used to compute true and simulated gradient and direction. Assessing acceptable gradient and direction error bounds We propose using the stochastic analysis of Mizell (1980) to set acceptable error bounds in numerical models for gradient magnitude and direction. Mizell's solutions were developed to address the problem of estimating the magnitude and direction of groundwater flow from three observation wells for two-dimensional steady-state flow (a fairly common numerical model configuration). Two solutions are used in this analysis: the first gives the expected value of the direction error, and the second gives the variance of the hydraulic gradient. The errors are a function of several variables, including orientation of the network relative to the flow direction, independent

4 74 Gregory J. Ruskauff & James O. Rumbaugh, III measurement error, and ratio of the network size to head-field correlation scale. Some of these variables, such as the variance of InK, are not easily determined. A key facet of Mizell's analysis is that for values of the ratio of the network scale to head correlation of less than 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. The bounding characteristics of the solution for direction error can be readily used. However, the gradient magnitude variance is dependent on the square of the head-correlation scale that may be difficult to estimate from limited data. Limitations on the solutions include the fact that they were obtained by perturbation analysis, and may not be accurate for extremely heterogeneous media, an infinite domain (no boundary effects), and the wells were assumed to be arranged in a right triangle. The potential error in gradient magnitude and direction can also be addressed with a simple deterministic analysis. The calculation of hydraulic gradient involves use of three potentiometric-level data points. Error in any point will create error in the flow direction calculated from the model heads. The calculation is done by raising the upgradient and lowering the downgradient heads in the observation network, which results in the maximum gradient change relative to the target value. EXAMPLES Synthetic test problem The technique is illustrated with a perfectly-calibrated synthetic problem modified from Rumbaugh (1993) with noise introduced by geostatistical simulation. The acceptable head error was estimated at 0.23 m (0.75 ft) using the technique of Anderson & Woessner (1992). Alternatively, the total head drop in the observation wells is 3 m (10 ft), and using a criteria that any individual error should not exceed 10% of the drop gives a target error of 0.3 m (1 ft). Gradient direction and magnitude errors in the synthetic problem were investigated in detail for three differently sized observation networks. The network sizes were 100 m (330 ft), 365 m (1200 ft), and 670 m (2200 ft). Table 1 summarizes the gradient and direction computed for the networks. Table 1 Summary gradient magnitude and direction statistics. Network True V, 8 V Mean, stand. 0 Mean, stand, dev., (ave. %, SD % V error) size dev., CV a CV [ave., SD" D error] Small Medium Large , , , , , , , , , 38% 17% 15% ' Coefficient of variation = mean/standard deviation. 282, 14.8, 5.2% 181.5, 8, 4.4% 202, 4.4, 2.2% (29, 27) [4, 3.3 ] (14, 11) [6.2, 4.9 ] (8.5,5.5) [1.8, 1.2 ]

5 Incorporating groundwater flow direction and gradient into a flow model calibration 75 Application of Mizell's solutions suggested that a reasonable value of direction error would be ±21 with a InK variance of 1 (a representative value). If a normal distribution of errors is assumed the likelihood of the directional error can be quantified. There is about a 67 % (1 standard deviation) chance that the error will lie within ±21, and a 95% (two standard deviations) chance that it will be within about ±42. The solutions for gradient variance involve the square of the head-field correlation scale making their use sensitive to this parameter. The head correlation scale (X) could not be estimated for this problem. Instead, the effect of a range of Xs on gradient error is illustrated. With X of 152 m (300 ft), 305 m (1000 ft), 609 m (2000 ft), and 1524 m (5000 ft) the gradient standard deviation was , , , and , respectively. At the two smallest correlation scales the standard deviation is larger than the actual gradients observed in all or some of the networks in the synthetic model, which illustrates the difficulty in using this solution. If the smallest standard deviation is used (X = 1524 m) allowable gradient errors are 29%, 13%, and 11 % for the smallest to largest networks, respectively. When the perturbation method was applied for the three networks the acceptable gradient error was 85%, 44%, and 33% for the smallest to largest networks, respectively. Thus even a relatively accurate flow model may have substantial errors in hydraulic gradient. Inspection revealed that the gradient was low in the first case, and the 0.23 m change represented a much larger change there than at the other locations. A similar problem arises when using the standard deviation of the gradient; the amount of acceptable error expressed as a percentage of the true gradient will be larger where the gradient is lower, which seems counterintuitive (larger errors would be expected in areas of high gradient). From all these considerations a reasonable gradient magnitude error is about 25 %. Direction errors determined from simple sensitivity analysis showed a range depending on how the head at the downgradient wells was altered. If both downgradient heads were depressed by the error value (0.23 m) the gradient direction errors were 36, 41, and 13 for the smallest to largest networks, respectively. If one downgradient well was raised by the error amount and the other depressed by the same amount the gradient direction errors were 12, 1, and 8 for the smallest to largest networks, respectively. As with the gradient magnitude this illustrates that even a well-calibrated model with respect to head can have significant gradient direction errors. From the previous results it seems questionable practice to set an acceptable absolute gradient error to be applied globally. It would be better to set allowable error as a percentage of the local gradient accounting for the fact that what may be acceptable error in an area of high gradient may be unacceptable in an area of low gradient. Furthermore, it may not be reasonable to set a single acceptable level of head error as proposed by Anderson & Woessner (1992) in a nonuniform flow system. For instance, in a model with both a gentle regionalflowsystem and intense pumping centres the acceptable error will be different in each area. To further investigate model gradient error 50 steady-state Monte Carlo realizations were generated using Stochastic MODFLOW (Ruskauff, 1994). A InK" variance of 0.5 was used to generate the geostatistical fields for input into MODFLOW. From an allowable error of 0.23 m (0.75 ft) a well-calibrated model would have a residual sum of squares (RSS) of 0.79 m 2 (8.5 ft 2 ) for the 15 targets. Summary calibration statistics

6 Gregory J. Ruskauff & James O. Rumbaugh, III CM o o 00 ooooo Realization #12 (best) ***** Realization #31 (median) o o oo o Realization #50 (worst) Observed Head (m) 1 Plot of observed vs simulated head for realizations 12, 31, and Smallest network * Medium network x Largest network 1 Error bounds Realization Number 2 Simulated hydraulic gradient and error bounds for all realizations.

7 Incorporating groundwater flow direction and gradient into a flow model calibration 77 for each realization were computed, and all the simulations were at or below the 0.79 m 2 goal. However global measures of model calibration can be misleading, and more detailed analysis of the worst, median, and best simulations was done. Figure 1 shows a plot of observed vs computed heads with calibration statistic information for realizations 12 (best), 31 (median), and 50 (worst). Realizations 12 and 31 had no targets with error greater than that allowable, and realization 50 had only 1. Using the results from Fig. 1 as being representative of all the simulations the calibrations of all the realizations would be judged good. The variation in simulated gradient magnitude over all the realizations is shown in Fig. 2 along with a 25% error bound for each network. As noted previously it was not possible to estimate the correlation scale for head for this problem in order to use MizelPs gradient solution. However, the semi-empirical 25% error bound bounds the simulation results quite well. The simulated gradient direction for each realization for the three networks is shown in Fig. 3. The error bounds on flow direction are also shown. Note that the bounds are rarely exceeded. The gradient errors were consistently larger than the direction errors as shown in Table 1. Variability in gradient and direction also decreased with increasing network size as Mizell's analysis says it should. Solute transport analysis The effects of flow field variation on solute transport analysis was investigated by using MT3D (Zheng, 1990) with all 50 flow fields. Constant mass loading was specified at o- m KD 0 Smallest network * Medium network x Largest network 1 Error bounds C O ' * o O o- o o 0 o oo o Q O CD- CD "** vf Realization Number Fig. 3 Simulated flow direction and error bounds for all realizations.

8 78 Gregory J. Ruskauff & James O. Rumbaugh, III 'mo Time (days) Fig. 4 Ensemble of breakthrough curves, smallest network. / >00 ( i ' <") Time (days) Fig. 5 Ensemble of breakthrough curves, medium network.

9 Incorporating groundwater flow direction and gradient into a flow model calibration 79 C a-* V. "c CD O c o (J" Time (days) 7300 Fig. 6 Ensemble of breakthrough curves, largest network 9125 three locations in order to generate plumes that pass through the area near each network. Figures 4, 5, and 6 show the simulated concentration histories for observation points located near the small, medium, and large networks, respectively. Note that even with the relatively mild variation in the flow fields a large variation in peak concentrations and shape of the breakthrough curves results. This shows that, while a plume can constrain a flow model, adequate plume characterization is crucial. Real-world application A model in the Truckee Meadows, Nevada USA was assessed with the method discussed here. The model (Geraghty & Miller, 1993) replicated observed advective plume migration direction well, and also had a good head calibration. Summary statistics for the 105 calibration targets are as follows: RSS of head 34.3 m 2 (369 ft 2 ), mean error of m (-0.32 ft), error standard deviation of 0.57 m (1.86 ft). Seven networks along the flow path of the plume were examined. Three of the networks had direction errors of 5 or less, three had direction errors near the limiting bound (±21 ) of 22, 27, and 31, and one had a direction error of 46. This shows that the direction calibration goals proposed in this paper are achievable in real life, and that a good direction fit can be correlated to an accurate depiction of plume migration.

10 80 Gregory J. Ruskauff & James O. Rumbaugh, III DISCUSSION It is possible that, using typical techniques for setting acceptable model head calibration error, there will be significant deviation between true and observed gradient and direction. Obviously as the goodness of fit to the head field increases the fit to the direction and gradient should also improve. However, as shown by the synthetic model results, the calibration stopping point for flow model calibration can be much greater than the goodness of fit required to produce a good gradient and direction match. While the method discussed here sets an objective framework for evaluating the goodness-of-fit of groundwater flow directions it cannot constrain the inverse procedure as well as the use of plume data can. It can be seen from inspection of Darcy's law Q = KA (d/z/d/) that the use of head data alone cannot define all the necessary parameters. Current practice for checking gradient and direction errors has the analyst scanning residual plots looking for areas where bias may be present and semi-quantitatively checking gradient and direction errors with contour maps and particle tracking analysis. In the author's experience this can be inadequate. The method proposed here gives a systematic framework for more completely examining the quality of the head-field calibration by explicitly quantifying the relationship between groups of calibration points by checking the shape of the potentiometric surface, rather than just single points. As with any technique there are limitations with the flow-direction calibration method. In areas of consistent bias the heads could be significantly in error, but the flow direction and gradient would be correct. The appropriateness of the wells selected for the gradient and direction computation can be an issue, particularly when there are rapid fluctuations in the head field and the wells are far apart. Finally, the gradient and direction could be correct, but anisotropy can cause the true groundwater travel path to deviate significantly. REFERENCES Anderman, E. R., Hill, M. C. & Poeter, E. P. (1994) Two-dimensional advective transport in nonlinear regression: Sensitivities and uncertainty of plume-front observations. In: Proc. of 1994 Groundwater Modelling Conference (ed. by J. W. Warner & P. K. M. Van der Heijde) (Fort Collins, Colorado), Anderson, M. P. &Woessner, W. W. (1992)AppliedGroundwaterModelling:Simulationof'FlowandAdvectiveTransport. Academic Press, San Diego, California, USA. Duffield, G. M., Stephenson, D. E. & Buss, D. R. (1990) Velocity prediction errors related to flow model calibration uncertainty. In: Calibration and Reliability in Groundwater Modelling (ed. by K. Kovar) (Proc. Int. Conf. ModelCARE'90, The Hague, September 1990), IAHS Publ. No Ehlig-Economides, C. (1988) Use of the pressure derivative for diagnosing pressure-transientbehavior. /. Petrol. Technol. 40(10), Fetter, C. W. (1993) AppliedHydrogeology, 3rd edn. MacMillan, New York, New York, USA. Geraghty & Miller Inc. (1993) Modelling/conceptualengineering design for the Sparks solvent fuel site, Sparks, Nevada. Consultant's report. Guo, X. & Zhang, C. M. (1994) Use of the physical feature of groundwater flow system to reduce the mathematical complexity in parameter identification - a practical and efficient automated procedure. In: Proc. of the 1994 Groundwater Modelling Conference (ed. by J. W. Warner & P. K. M. Van der Heijde) (Fort Collins, Colorado), Mizell, S. A. (1980) Stochastic analysis of spatial variability in two-dimensional groundwater flow with implications for observation well network design. PhD Thesis, New Mexico Institute of Mining and Technology, Socorro, New Mexico, USA. Pinder, G. F., Celia, M. & Gray, W. G. (1981) Velocity calculation from randomly located hydraulicheads. Groundwater 19(3), Rumbaugh, J. O., Ill (1993)ModelCad 386, vol. 2, Tutorial. Geraghty & Miller Inc., Reston, Virginia, USA.

11 Incorporating groundwater flow direction and gradient into a flow model calibration 81 Ruskauff, G. J. (1994) A methodology for performing Monte Carlo analysis with MODFLOW and MODPATH. In: Proc. of 1994 Groundwater Modelling Conference (ed. by J. W. Warner & P. K. M. Van der Heijde) (Fort Collins, Colorado), Vacher, H. L. (1989) The three-pointproblem in the contextof elementary vector analysis./. Geol. Education 37, Van Rooy, D. & Rosbjerg, D. (1988) The effect of conditioning transport simulations on transmissivity, head and concentrationdata.in: ConsequencesofSpatialVariability in Aquifer Properties anddatalimitationsfor Groundwater Modelling Practice (ed. by A. Peck, S. M. Gorelick, G. De Marsily, S. Foster & V. Kovalevsky), IAHS Publication No Zheng, C. (1990) MT3D, a modular three-dimensional transport model. S. S. Papadopolus& Assoc, Rockville, Maryland, USA.