Stepwise inversion of a groundwater flow model with multi-scale observation data Zhenxue Dai & Elizabeth Keating & Carl Gable & Daniel Levitt & Jeff Heikoop & Ardyth Simmons Abstract Based on the regional hydrogeology and the stratigraphy beneath the Los Alamos National Laboratory (LANL) site, New Mexico (USA), a site-scale groundwater model has been built with more than 20 stratified hydrofacies. A stepwise inverse method was developed to estimate permeabilities for these hydrofacies by coupling observation data from different sources and at various spatial scales including single-well test, multiple-well pumping test and regional aquifer monitoring data. Statistical analyses of outcrop permeability measurements and single-well test results were used to define the prior distributions of the parameters. These distributions were used to define the parameter initial values and the lower and upper bounds for inverse modeling. A number of inverse modeling steps were performed including the use of drawdown data from the pump tests at two wells (PM-2 and PM-4) separately, and a joint inversion coupling PM-2 and PM-4 pump test data and head data from regional aquifer monitoring. Parameter sensitivity coefficients for different data sets were computed to analyze if the model parameters can be estimated accurately with the data provided at different steps. The joint inversion offers a reasonable fit to all data sets. The uncertainty of estimated parameters for the hydrofacies is addressed with the parameter confidence intervals. Keywords Stepwise inversion . Hydrofacies . Multi-scale data . Parameter sensitivity . Joint inversion . USA
Introduction Understanding and predicting groundwater flow and contaminant transport at large spatial scales in subsurface
Received: 23 October 2008 / Accepted: 5 October 2009 Published online: 10 November 2009 © Springer-Verlag 2009 Z. Dai ()) : E. Keating : C. Gable : D. Levitt : J. Heikoop : A. Simmons Earth and Environmental Sciences Division, Los Alamos National Laboratory, EES-16, T003, Los Alamos, NM 87545, USA e-mail:
[email protected] Hydrogeology Journal (2010) 18: 607–624
systems entails developing and then integrating knowledge of aquifer heterogeneity and field-scale parameterizations. Inverse modeling combined with aquifer pumping tests has been widely applied to regional or site-scale aquifer characterization and parameter estimation (e.g. Carrera and Neuman 1986; Sun and Yeh 1990; Wagner 1992; Hill 1992; Poeter and Hill 1997; Doherty 2005; Robinson et al. 2005; Dai and Samper 2006; Samper et al. 2006; Kwicklis et al. 2006 and Zhu and Yeh 2006). In general, the longer the duration of the pumping test and the larger the pumping rate, the more valuable the test. However, limited by the budgets of each test, very few pumping tests can be conducted long enough or with high enough pumping rates to fully interpret aquifer heterogeneity. There is a need to develop a new inverse method for aquifer parameter estimation by taking advantage of multi-scale observation data, which are collected from site-scale monitoring and small-scale pumping tests conducted at different locations with different influence areas. This paper uses the calibration of the Los Alamos National Laboratory (LANL) site-scale model as an example to illustrate the newly developed stepwise inversion methodology. Los Alamos National Laboratory and adjacent communities are situated on the Pajarito Plateau, New Mexico, USA (Fig. 1). The Plateau occupies the south-western part of the Española Basin, which is bounded on the east by the Rio Grande and on the west by the eastern Jemez Mountains. The major aquifer hydrostratigraphic unit beneath the Plateau is the Santa Fe group, which is the primary source of water supply for the communities of Española, Los Alamos, and numerous pueblos (Griggs and Hem 1964; Nylander et al. 2003; Newman and Robinson 2005; Collins et al. 2005). For simulating groundwater flow in the Española Basin, the US Geological Survey (USGS) developed two numerical models. The first was developed by Hearne (1985). The second was developed by McAda and Wasiolek (1988) (and later refined by Frenzel 1995), using the MODFLOW code (McDonald and Harbaugh 1988). These models have made important contributions to the understanding of the basin hydrology and regional water-balance issues, particularly concerning impacts of pumping on stream-flow in the Rio Grande. At present, local and state agencies continue to refine and apply both these models to address water supply issues in this basin. However, both the DOI 10.1007/s10040-009-0543-y
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N NM Mexico
Study area
Gr a
Ri o
LANL
nd
Española
0
10
Santa Fe
20
30
40
50
60
70 Kilometers
Fig. 1 Location of the study area in New Mexico (NM). Basin-scale model domain (pink line) and current site-scale model domain (black line) and domains of USGS models of Hearne (1985) in red and McAda and Wasiolek (1988) in green (modified from Keating 2005). LANL Los Alamos National Laboratory. Towns are shaded yellow
USGS models of this aquifer system place a model boundary along the western edge of LANL, and do not include the recharge area in their model domains; therefore, use of these models for the LANL site would be compromised by boundary effects. To ensure that all model boundaries were far from the area of interest (e.g. LANL) and to incorporate the possible influences of regional flow on local conditions, Keating et al. (2003) developed an Española basin-scale model with a coarse grid and a LANL site-scale model with fine grid for modeling groundwater flow and solute transport under the Pajarito Plateau and Española basin, based on the hydrogeological work conducted by LANL, the USGS and other organizations. The boundaries of the Hearne (1985), McAda and Wasiolek (1988), and the LANL basin-scale and site-scale models are shown in Fig. 1. The initial conceptual model of groundwater flow in the regional aquifer under the Pajarito Plateau was outlined in the Hydrogeologic Workplan (LANL 1998), and the original Española basin-scale model was developed to provide a regional context for flow and transport simulations at the scale of the LANL site (Carey et al. 1999; Keating et al. 2003). Researchers at the USGS and other organizations contributed to the LANL effort of defining the initial conceptual model and provided the supporting datasets. The basin-scale model also provides the water balance computation and boundary calibration for the site-scale model. In 2000, a site-scale model for the Pajarito Plateau (or LANL site) was developed with much higher grid resolution than could be achieved with the basin-scale model, and was coupled to the basin-scale model (Keating Hydrogeology Journal (2010) 18: 607–624
et al. 2003). The site-scale model, run on the LANL’s flow and transport simulator FEHM (Zyvoloski et al. 1997), has been used to make contaminant transport calculations, to conduct capture zone analyses, to support monitoringwell siting decisions, and to estimate groundwater velocities. Keating et al. (2003, 2005) described the geologic and hydrologic settings and framework model, FEHM simulator, material properties, model boundaries, and initial calibration strategies. The site-scale model has been periodically updated and improved when new data become available for model calibration. Recently, two pumping tests were conducted at Los Alamos County municipal water supply wells PM-2 and PM-4, both using multiple observation wells. Analytical methods were used by McLin (2005, 2006a) to interpret the pumping test data. However, because the analytical solutions are based on homogeneous assumptions and infinite boundary conditions, they cannot simulate the complex heterogeneity of the aquifer system under the Pajarito Plateau. In this paper, using observation data from PM-2 and PM-4 multiple-well pumping tests and regional aquifer monitoring data, the stepwise inversion method has been coupled with the LANL site-scale numerical model to estimate the flow parameters for the hydrofacies beneath the Pajarito Plateau. Statistical analyses of outcrop permeability measurements and single-well slug or pumping test results are used to define the prior distributions of the parameters. This prior information was used to define the parameter initial values and the lower and upper bounds for the stepwise inverse modeling. At each step, the inverse problem is solved with PEST (Doherty 2005). The uncerDOI 10.1007/s10040-009-0543-y
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tainty of the final estimated parameters for the hydrofacies is addressed with the parameter confidence intervals.
Site hydrogeology and pumping tests Site hydrogeology The regional aquifer beneath the Pajarito Plateau consists of the sedimentary rocks of the Puye Formation and the Santa Fe Group, the fractured volcanic rocks of the Tschicoma Formation, Cerros del Rio basalts, and older basalt flows. These units are described in detail in Broxton and Vaniman (2005) and Cole et al. (2006). The sources of recharge to the regional aquifer are diffuse recharge in the Sierra de los Valles and focused recharge from wet canyons on the Pajarito Plateau. Natural discharge from the regional aquifer is primarily into the Rio Grande directly, to springs that flow into the Rio Grande, and to basins to the south (Keating et al. 2003). The aquifer is under water-table conditions across much of the Plateau, but exhibits more confined aquifer behavior near the Rio
Grande. Hydraulic properties are highly anisotropic, with vertical hydraulic conductivities much smaller than horizontal hydraulic conductivities (Collins et al. 2005). Over the past 60 years numerous aquifer tests at different scales were conducted in the regional aquifer below the Pajarito Plateau. Figure 2 shows the spatial distributions of the historical pumping wells and observation wells on the Pajarito Plateau. The test data were used to characterize hydraulic properties of the saturated geological units. Analyses of these test results have revealed a complex regional aquifer that is a highly stratified, heterogeneous system. McLin (2006b) summarized the major aquifer test data and interpreted the aquifer flow parameters individually for each test. Although those interpretations were based on various analytical and numerical models, the estimated aquifer parameters give an indication of the spatial variation and uncertainty in this heterogeneous aquifer system. Using the interpretation results of McLin (2006b), as well as outcrop measurements (Gaud et al. 2004; Dai et al. 2005), a statistical analysis was performed to bound the horizontal
Fig. 2 Locations of Pajarito Plateau wells and the influence areas of PM-2 pumping test (dashed line) and PM-4 pumping test (dotted line) (modified from McLin 2006b). BLM Bureau of Land Management; DOE Department of Energy Hydrogeology Journal (2010) 18: 607–624
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permeability distributions, and compute the standard deviation, lower and upper bounds (minimum and maximum) of the permeability of each of the hydrofacies listed in Table 1. For the hydrofacies where there were not enough data to compute the lower and upper bounds, their prior information as parameter values in the site-scale numerical model was assigned because these hydrofacies are distributed either at the deep part of the model (e.g. dPC, dPM and Ts-deep) or at the top of the model (e.g. Qbt and sTk) and their parameters are not very sensitive to the pumping tests. PM-2 pumping test A 25-day aquifer pumping test was conducted at municipal water supply well PM-2 at a constant pump rate of 4.73 m3/min. The PM-2 well was completed in Pajarito Canyon in 1965 (Purtymun 1995) to a depth of 701 m. This well is on the south side of Pajarito Road, approximately 4 km northwest of White Rock. Immediately before the pumping test there was an initial non-pumping recovery period of 3 months beginning in 5 November 2002. The pump test began on 3 February and ended 28 February 2003. Surrounding observation wells were used to record drawdown data. These data revealed horizontal propagation of drawdown in the regional aquifer beyond 2,682 m from well PM-2. In Fig. 2 the dashed line roughly defines the pumping influence area. A total of 12 observation wells were used for this pump test, including: PM-1, PM-3, PM-4, PM-5, R-12, R-13, R-14, R-15, R-20, R-21, R-22, and R-32. Drawdown was observed in four observation wells: R-20, R-32, PM-4, and PM-5 (see Table 2). No drawdown was observed in the remaining nine wells (McLin 2005).
conducted at PM-4 at a constant pump rate of 5.66 m3/min. This pumping test was preceded by an initial non-pumping recovery period of 6 months beginning in August 2004. The pump test began on 8 February 2005, and ended on 1 March 2005. The observed drawdown data revealed horizontal propagation of the area of influence in the regional aquifer beyond 2652 m from well PM-4. In Fig. 2 the approximate area of influence of this test is shown with the dotted line. A total of 11 observation wells were used for this pump test, including: PM-2, PM-5, R-13, R-14, R-15, R-19, R-20, R22, R-32, TW-8, and CdV-R37-2. Drawdown was observed in nine of the 11 observation wells (see Table 2). No drawdown was observed at TW-8 and CdV-R37-2 because TW-8 is not deep enough to penetrate the major part of the Santa Fe group and the well CdV-R37-2 is located too far from PM-4. McLin (2005, 2006a) used a variety of analytical methods to derive aquifer parameters from the two pump tests. McLin concluded that although both confined and leaky-confined conceptual models were approximately consistent with the data, leaky-confined models matched the data much better and thus were the most appropriate conceptual models. More importantly, McLin (2005, 2006a) reported that the two pump tests demonstrated a remarkably complex aquifer response over space and time that is not easily interpreted with the analytical methods, and the regional aquifer materials in the area of the pumping tests are strongly heterogeneous and exhibit pronounced horizontal and vertical anisotropy in hydraulic transmitting properties.
Numerical models and stepwise inverse method Site-scale numerical model
PM-4 pumping test The PM-4 well is a municipal water supply well completed in 1981 (Purtymun 1995) to a depth of 876 m. This well is on the north side of Pajarito Road, approximately 5 km northwest of White Rock. A 21-day aquifer pumping test was
Key aspects of the site-scale numerical model (Keating et al. 2003 and 2005) are briefly summarized here. The model extent, shown in Fig. 1, extends from Santa Clara Canyon (to the north), the eastern margin of the Valles Caldera (to the west), the Rio Grande (to the east), and the
Table 1 Hydrofacies and their volumetric proportion (Vol), permeability (K), lower and upper bounds used in the numerical model Hydrofacies name
Symbol
Vol (%)
K (log m2)
Lower bound
Upper bound
PreCambrian Palezoic/Mesozoic Tf Galisteo Fanglomeratic Santa Fe Sandy Santa Fe-deep Sandy Santa Fe-shallow Keres Group-deep Keres Group-shallow Oldest basalt Bayo Canyon basalt Basalts-Cerros del Rio Tschicoma flows Totavi Lentil Pumiceous Puye Puye Fanglomerate Bandelier Tuff
dPC dPM gal Tf Ts-deep Ts dTk sTk Tb1 Tb2 Tb4 Tt Tpt Tpp Tpf Qbt
0.007 0.261 0.240 0.044 0.004 0.294 0.018 0.036 0.004 0.007 0.013 0.035 0.002 0.003 0.027 0.006
–18.0 –16.66 –13.5 –14.21 –16.07 –13.33 –13.73 –12.78 –13.63 –11.61 –12.36 –12.98 –12.22 –12.51 –13.09 –15.33
N/A N/A –15.0 –15.0 N/A –14.72 N/A N/A –15.0 –13.83 –15.83 –14.5 –14.06 –14.54 –14.06 N/A
N/A N/A –12.0 –10.73 N/A –10.42 N/A N/A –11.0 –10.65 –9.87 –11.5 –10.85 –10.07 –9.39 N/A
N/A There are not enough data to define the bounds
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611 Table 2 Drawdown measured during PM-2 and PM-4 pump tests Pumping and observation wells
PM-2 pumping test PM-4 pumping test Drawdown at 25 days (m) Distance from PM-2 (m) Drawdown at 21 days (m) Distance from PM-4 (m)
PM-2 PM-4 PM-5 R-20-1 R-20-2 R-20-3 R-32-1 R-32-2 R-32-3 R-12 R-13 R-14-2 R-15 R-19-4 R-22 PM-1 PM-3 TW-8 CdV-R37-2
25.69 2.97 0.73 0.33 1.03 4.57 0.36 0.46 0.53 0a 0a 0a 0a 0a 0a 0a 0a N/A N/A
a
0 1,365 2,685 373 373 373 1,457 1,457 1,457 3,949 2,350 3,452 2,543 2,128 2,693 4,101 3,308 2,907 5,388
4.41 21.31 2.46 0.06 0.61 4.96 0.0a 0.28 0.34 0a 0.11 0.01 0.52 0.09 0a N/A N/A 0a 0a
1,365 0 1,418 1,679 1,679 1,679 2,656 2,656 2,656 3,851 1,740 2,177 1,170 2,211 3,651 4,025 2,601 1,725 5,257
No response or less than detectable
Rio Frijoles (to the south). Analysis of regional precipitation, streamflow data, and hydraulic head data, integrated using the basin-scale groundwater flow model, enabled the quantification of the range of plausible fluxes that might cross these four lateral boundaries (Keating et al. 2003). The top of the model is set to be coincident with the measured water table surface at the beginning of the PM-2 pumping test. The water table is modeled as a free surface using FEHM (Keating and Zyvoloski 2009). The site-scale model includes a total of 397,574 nodes, which represent a model volume of 1,098 km3. The hydrofacies (or units) and their volumetric proportions in the numerical model are listed in Table 1. The grid resolutions are varied in the x–y plane from 125 to 500 m. In the vicinity of PM-2 and PM-4, the resolution is 125 m while towards the north it increases to 500 m. The vertical resolution is 12.5 m at elevations greater than 1,000 m; at lower elevations the grid resolution increases to 500 m.
Stepwise inverse method Comprehensive reviews of the general inverse problem of aquifer flow and solute transport can be found in Yeh (1986), Carrera and Neuman (1986), Sun (1994), and Dai and Samper (2004). Here, a stepwise inverse approach is introduced which provides a strategy to determine unknown model parameters by fitting the model output with multiplescale or different-source observation data step by step. The formulation of the stepwise inverse problem is based on a generalized least squares criterion for parameter estimation. Let p = (p1, p2, p3, ..., pM) be the vector of M unknown parameters. The generalized least squares criterion, E(p), can be expressed as, E ð pÞ ¼
NE X
Wi Ei ð pÞ;
i¼1
Hydrogeology Journal (2010) 18: 607–624
ð1Þ
where i denotes different scales or different sources of data, i=1 for PM-2 pumping test data; i=2 for PM-4 pumping tests; i=3 for site-wide water head monitoring data (NE=3), Wi is the weighting coefficient of the i-th generalized leastsquares criterion Ei(p) which is defined as: Ei ð pÞ ¼
Li P l¼1
w2li rli2 ð pÞ;
ð2Þ
rli ðpÞ ¼ uil ð pÞ e uil ; where uil ð pÞ is the model simulated value of the i-th scale or source of data at the l-th observation point; e uil are measured values; Li is the number of observations, both in space and in time, for the i-th scale of data; and rli is the residual or difference between model outcome and measurement; wli is the weighting coefficient for the l-th measurement. The values of wli depend on the accuracy of observations. If some data are judged to be unreliable, they are assigned small weights in order to prevent their deleterious effect on the optimization process. Weighting of different scales or sources of data Equation 1 is a weighted multi-objective optimization criterion. It is difficult to designate optimal weighting coefficients. Several methods were proposed by Neuman and Yakowitz (1979) and Carrera and Neuman (1986). Generally, the determination of optimal weights should follow an iterative procedure. Carrera and Neuman (1986), Sun (1994) and Dai and Samper (2004) use the following equation for generalization of different types of data:
Wi ¼
W0i Ei ðpÞ Li
;
ð3Þ
DOI 10.1007/s10040-009-0543-y
612 Table 3 Summary of model results and estimated parameters by stepwise inversion Step Model description No. of estimated parameters Total objective value Sub-objective values
Estimated parameters
ΦPM-2 ΦPM-4 ΦPM-5 ΦR-20 ΦR-32 ΦR-13 ΦR-14 ΦR-15 ΦR-19 Φ0Da ΦFluxb ΦSSMc Sy Ss Ktb2 Ktpt Ktpf Ktf Kts Ktt Ktpp Ktb1 Ktb4 Zftb2 Zftpt Zftpf Zftf Zfts Zftt Zftpp
1 PM-2 Forward N/A 1068.6
PM-2 inverse 15 52.20
2 PM-4 forward N/A 1118
PM-4 inverse 15 96.5
3 Joint inversion 18 445.38 PM-2 PM-4
388.3 0.96 118.7 485.1 75.6 N/A N/A N/A N/A N/A N/A N/A 0.1 –3.16 –11.61 –12.22 –13.09 –14.21 –13.33 –12.98 –12.51 –13.63 –12.56 1.00 0.06 0.11 1.58 0.45 1.00 0.04
1.22 10.06 7.52 10.66 9.26 N/A N/A N/A N/A 13.42 0.035 N/A 0.1 –5.773 –12.99 –12.89 –13.10 –11.62 –12.43 –12.11 –12.29 –13.63d –12.56d 0.0013 0.3539 1.00d 0.0174 0.001 0.0142 0.0428
0.50 100.3 2.12 32.96 36.28 350.4 0.15 3.74 572.2 18.99 0.15 N/A 0.08 –5.4798 –12.99 –12.89 –13.10 –11.62 –12.43 –12.11 –12.29 –13.63 –12.56 0.0013 0.3539 1.00 0.0174 0.001 0.0142 0.0428
2.76 2.72 2.35 39.91 9.16 8.44 0.02 5.12 13.06 12.97 0.0001 N/A 0.08 –5.4798 –12.99 –13.59 –12.20 –11.54 –12.39 –12.33 –12.80 –13.63d –12.56d 0.0013d 0.0537 0.3374 0.0966 0.0008 0.6097 0.1217
14.20 14.64 9.79 11.29 8.72 N/A N/A N/A N/A 12.06 39.65 177.57 0.111 –6.241 –12.99 –11.99 –12.33 –11.64 –12.40 –12.90 –13.62 –13.60 –13.20 0.0001 0.001 0.2465 0.0583 0.0042 0.2791 0.6891
1.24 14.66 2.81 37.29 32.84 10.97 0.03 35.18 15.82 6.63
a
Refer to zero drawdown data Discharge to Rio Grande c Refer to site-wide aquifer monitoring data d The parameter is fixed to prior information because it is insensitive b
where W0i are user-defined dimensionless initial weighting coefficients for different scales or different sources of observation data. Generally, they are taken as being equal to 1. Wi have dimensions which are reciprocal of those of Ei(p). Generally Ei(p) has units equal to the square of the unit used for each type of data (i.e., m2 for heads or drawdown and (kg/s)2 for water-flow rate data (Stauffer 2006). Weights Wi are updated automatically during the iterative optimization process according to Eq. (3). However, in this stepwise strategy, three steps are used to perform the model inversion: Step 1: W01 =1, W02 =W03 =0 Step 2: W01 =0, W02 =1, W03 =0 Step 3: W01 =W02 =W03 =1
coefficients and correlation structures. There are many risks in trying to estimate a large number of parameters in one step. Some steps of the stepwise procedure should be devoted to analyze parameter identifiability (which mea-
Kwicklis et al., 2005 Keating et al., 2005 Keating, 2005 Keating et al., 2003 McAda and Wasiolek, 1988 Hearne, 1985
The main advantages and reasons to adopt a stepwise strategy are, first, that stepwise inversion helps to avoid over-parameterization. One must carefully analyze which parameters are identifiable and can be subsequently estimated accurately through inverse modeling. This process is performed by comparing parameter sensitivity Hydrogeology Journal (2010) 18: 607–624
0
100
200 300 Discharge (kg/s)
400
500
Fig. 3 Summary of previous estimates of discharge (unit: kg/s) beneath the Pajarito Plateau to the Rio Grande (adapted from Keating et al. 2005). Keating (2005) was calculated using streamflow analysis while Keating et al. (2005) was calculated in the uncertainty analysis DOI 10.1007/s10040-009-0543-y
613 Table 4 Calculated parameter sensitivity coefficients Parameter name
Initial value
Sensitivity coefficient
Relative sensitivity
Kdpc Kdpm Kts-deep Kgal Kts Ktf Kdtk Kstk Ktb4 Ktb1 Ktb2 Ktt Ktpt Ktpp Ktpf Kqbt Sy Ss Zftb2 Zftpt Zftpf Zftf Zftpp Zftt Zfts
–18.0 –16.66 –16.07 –13.50 –13.33 –14.21 –13.73 –12.78 –12.36 –13.63 –11.61 –12.98 –12.22 –12.51 –13.09 –15.33 0.1 –3.16 1.0 0.06 0.11 1.58 0.04 1.0 0.45
0.0000 0.0005 0.0000 0.0005 0.1648 0.1737 0.0002 0.0003 0.0092 0.0003 0.0013 0.0538 0.0281 0.0861 0.0789 0.0002 0.4848 0.2935 1.3239 0.2099 0.0043 0.3922 0.2880 0.0643 0.0325
0.0000 0.0008 0.0008 0.0066 2.0061 2.0844 0.0023 0.0046 0.0041 0.0032 0.0224 0.6601 0.2906 1.0781 1.0490 0.0020 0.0800 1.4756 0.0153 0.0059 0.0030 0.0404 0.1020 0.0643 0.4266
Sensitive rank
problem requires not only obtaining optimum parameter values, but also identifying relevant boundary conditions and parameter structures, known as conceptual model identification in the stepwise inversion (Dai et al. 2006).
a: PM-2 and PM-4
2 1
PM-2 computed Corrected data Raw data PM-4 computed Measured PM-5 computed Measured
25
13 6 8 4 5 10 3 14 15
Drawdown (m)
20 15 10 5 0 0
12 9 11 7
5
10
Drawdown (m)
sures if the model parameters can be estimated accurately with the data provided). Secondly, stepwise inversion provides a procedure to upscale the flow parameters that were obtained from small-scale pumping test inversions to the site-scale model by coupled inversion of multi-scale observation data. Thirdly, stepwise inversion allows the use of inverse models of increasing complexity. Therefore, until sufficient insight is gained on the nature of the inverse problem of the individual PM-2 and PM-4 pumping test, the scientific principle of analyzing complex problems was followed by using models of increasing complexity. Fourthly, stepwise inversion can be useful for conceptual model identification. Solving the inverse
6
12 9
20
25
2
0 0
5
10
15
30
Time (days) 1.0
Drawdown (m)
Measured drawdown (m)
15
25
4
c: R-32 Computed 1 Observed 1 Computed 2 Observed 2 Computed 3 Observed 3
0.8 18
20
b: R-20 Computed 1 Observed 1 Computed 2 Observed 2 Computed 3 Observed 3
8
2
K is the log permeability (logm ); Sy is the specific yield; Ss is the elastic storage coefficient and Zf is the ratio of the vertical and horizontal permeability
15
Time (days)
0.6
0.4
0.2
6 3
0.0 -5
0 0
3
6
9
12
15
18
Computed drawdown (m)
Fig. 4 Measured versus calculated drawdown for the inversion of PM-2 pumping test (the solid line is 1:1 line) Hydrogeology Journal (2010) 18: 607–624
0
5
10
15
20
25
30
Time (days) Fig. 5 Measured and calculated drawdown versus time at the main observation wells for the inversion of the PM-2 pumping test (Drawdown correction in the pumping well was based on Peaceman 1978) DOI 10.1007/s10040-009-0543-y
614 0.25
the well R-32). This procedure differs from that used for different types of data such as drawdowns, water flow rate, and prior information. The weight for the l-th measurement of the k-th observation well wlk, is computed as:
a Computed R-12 Computed R-13 Computed R-14-1 Computed R-14-2
Drawdown (m)
0.20
0.15
wkl ¼
0.10
w0l ; sk
ð4Þ
where w0l is an initial weight for the l-th measured value which should be equal to the reciprocal of its analytical error, and σk is the standard deviation of measured drawdown of the k-th well,
0.05
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u X 2 s k ¼ tðN 1Þ1 Fi k Fk ;
0.00 -5
0
5
10
15
20
25
30
Time (days)
Fk ¼ N 1
i¼1
0.25
N X
Fi k :
i¼1
ð5Þ
b Computed R-15 Computed R-23 Computed PM-1 Computed PM-3
Drawdown (m)
0.20
0.15
Fik is the i-th measured value of the k-th well; N is the number of drawdown data of the k-th well; F k is the average of measured drawdown of the k-th well. This weighting system ensures that all observation wells have similar contributions to the global objective function and that drawdown data in one well are neither “drowned” by other wells nor dominate in the estimation process.
0.10
0.05
a
0.00 -5
0
5
10
15
20
25
30
Z
Time (days)
PM-2
0.25
R-19
0.20
Drawdown (m)
PM-4 Qbt
R-20 R-32
Y
X
Tpf
c
Tpt
Computed R-22-1 Computed R-22-2
0.15
Tb4
Ts
Tpp
Tf
Tf
Tb4 Tpp
Tb2 Ts
0.10
b
Z
0.05
PM-4 PM-2 R-19
0.00 -5
0
5
10
15
20
25
30
Time (days) Fig. 6 Calculated drawdown versus time for the inversion of the PM-2 pumping test at the wells a where there are no available observation data or b and c no apparent drawdown was observed
Weighting of different observation wells A special procedure is used for weighting drawdown data from different observation wells because they usually vary within different logarithmic ranges. For example, in the PM-2 pumping test the drawdown in PM-2 is more than 25 m, but it is about 0.08 m in R-32-1 (the first screen of Hydrogeology Journal (2010) 18: 607–624
Y
X
R-20 R-22
5 4 2 1 0.8 0.5 0.2 0.1 0
Fig. 7 a Outline of geological framework around PM-2 pumping well and b the drawdown distribution of the PM-2 pumping test at 25 days (PM-2 well is located at the center of two cross sections, drawdown unit is meters, and refer to Table 1 for hydrofacies units) DOI 10.1007/s10040-009-0543-y
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Fig. 8 Outline of hydrofacies distribution in XY plane (a Z=1,700 m and b Z=1,600 m), and simulated drawdown contour of the PM-2 pumping test at 25 days in XY plane (c Z=1,700 m and d Z=1,600 m). Drawdown unit is meters
Grid-size correction factor for drawdown in pumping Rejection of the outlier points in the observation data wells
Hydrogeology Journal (2010) 18: 607–624
During pumping tests, a measurement may be read, recorded, or transcribed wrongly, or a mistake may be made in the way in which a treatment was applied for this 15
Measured drawdown (m)
With the numerical models described here, the simulated head in any grid block represents an average head over the entire control volume. This average head will be a poor approximation to water levels in a pumping well if the size of the control volume is much larger than the diameter of the pumping well. For this reason, a method has been developed to calculate the expected relationship between measured water level in a pumping well and the corresponding simulated head in a large control volume representing the pumping well. By using analytical well functions, Peaceman (1978) defined the equivalent radius as the radius at which the steady-state flowing pressure for the actual well is equal to the numerically calculated pressure for the well control volume. Based on the approach derived by Peaceman (1978), the observed drawdowns in pumping wells PM-2 and PM-4 were corrected to eliminate the computing errors due to the large grid size of the numerical model, but the drawdown data in the observation wells were not corrected.
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Fig. 9 Measured and calculated drawdown at all observation wells for inverse modeling of PM-4 pumping test DOI 10.1007/s10040-009-0543-y
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measurement. A major error greatly distorts the mean and the standard deviation, and affects conclusions of inverse modeling. The principal safeguards are vigilance in carrying out the operating instructions in the measuring and recording process, and visual inspection of the observation data. If a value in the data to be analyzed looks suspicious, an inquiry about this observation sometimes shows that there was a gross error. If no explanation of an extreme
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observation is discovered, one may consider rejecting it. The rules for the rejection of observations have been based on some type of significance test (Snedecor and Cochran 1976). The probability that a residual as large as the suspect value would occur by chance is computed. If this probability is sufficiently small, the suspect is rejected. Anscombe and Tukey (1963) present a rule that rejects an observation whose residual has the value of d if |d| > Cs, where C is a constant to be determined from
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Fig. 10 Measured and calculated drawdown versus time at observation wells for PM-4 pumping test Hydrogeology Journal (2010) 18: 607–624
DOI 10.1007/s10040-009-0543-y
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the conceptual model or the model parameters should be re-calibrated accordingly. By using the site-scale model, a forward run was conducted to simulate the PM-2 pumping test. The predicted drawdown did not match the observed drawdown well. The major fitting errors include under-predicting response at the PM-2 pumping well and over-predicting very early response at all observation wells (including R-20, R-32, PM-4 and PM-5). The computed objective function is 1068.6 for this forward run (see Table 3). Overall, these results indicate that predictions of drawdown from the PM-2 pumping test are not satisfied using the previous site-scale model. The forward modeling results indicate that there is a need to re-calibrate the model with PM-2 pumping test data.
Y
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Fig. 11 a Hydrofacies distribution in YZ plane and b drawdown distribution in YZ plane (X=19,294) of the PM-4 pumping test at 21 days. Drawdown is in meters
10 5 2 1 0.5 0.2 0.1 0
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Eq. 6, which is obtained from Snedecor and Cochran (1976), and s is the standard deviation of the observations. qffiffiffiffiffiffiffiffi 2 N 1 C ¼ K 1 4KðN 2 1Þ N ; ð6Þ K ¼ 1:4 þ 0:85z; where N is the number of the observations. z corresponds to the one-tailed probability value of N1 N P in the normal distribution, where P is the small premium which is involved in a rejection rule, say 2.5%. This rejection rule has been used to test and reject some of the observation data for PM-2 and PM-4 pumping tests.
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Step 1: forward and inverse modeling of PM-2 pumping test Forward modeling Previous to this study, PM-2 pump test data were not used to calibrate or otherwise develop the site-scale numerical models (Keating et al. 2005). Therefore, this dataset provides a useful benchmark to test the degree to which the site-scale models can predict the pump test behavior. It is reasonable to expect that the previous versions of the site-scale model developed to address pump tests should predict an envelope of behavior within which a good match between measured and predicted pump test response could be identified. If this were not the case, Hydrogeology Journal (2010) 18: 607–624
R-13
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Fig. 12 Simulated drawdown contour of the PM-4 pumping test at 21 days in XY plane (a Z=1,700 m and b 1,600 m). Drawdown is in meters DOI 10.1007/s10040-009-0543-y
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ð7Þ
where Si is the sensitivity coefficient of i-th parameter, m is the number of observations with non-zero weights, J is the Jacobian matrix and W is the weighting matrix. The relative sensitivity (Rsi) of the i-th parameter is obtained by multiplying its composite sensitivity by the magnitude of the value of the parameter pi as
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are sensitive enough to the smaller scale PM-2 pumping test data. Only the sensitive parameters are identifiable. Therefore, before conducting the inverse modeling, a sensitivity analysis of the inverse model to the involved parameters was performed. The sensitivity coefficient (or composite sensitivity) of the parameters are computed according to Doherty (2005) as
4 Kilometers
Fig. 13 Locations of all observation wells used in joint inversion. The circles represent the observation wells for the PM-2 and PM-4 pumping tests. PM-2 and PM-4 themselves are green circles. The triangles represent the site-wide aquifer monitoring wells
Inverse modeling In addition to PM-2 pumping test data, the previously estimated discharge rate to the Rio Grande was also included as one of the fitting targets or a sub-objective function in the inverse model. Previous investigations to estimate steady-state discharge through the regional aquifer beneath the Pajarito Plateau were conducted by Keating (2005), Keating et al. (2003, 2005), Kwicklis et al. (2005), McAda and Wasiolek (1988), and Hearne (1985). Their estimates of discharge beneath the Pajarito Plateau are summarized in Fig. 3 The mean of the above cited estimates of discharge (which is 332.75 kg/s with a standard deviation of 68.72 kg/s) was used as the observed steady-state discharge data for model fitting. The site-scale numerical model includes more than 20 parameters. One first needs to know how many parameters
Rsi ¼
jpi j T 12 J WJ ii : m
ð8Þ
The relative sensitivity is thus a measure of the composite changes in model outputs that are incurred by a fractional change in the value of the parameter. The sensitivity coefficient computed in Table 4 presents the sensitivity of each parameter with respect to all observations of the PM-2 pumping test. The use of relative sensitivities in addition to composite sensitivities assists in comparing the effects that different parameters have on the parameter estimation process when these parameters are of different types, and of very different magnitudes. By ranking the relative sensitivity coefficients (see Table 4), the 14 most sensitive parameters were selected for parameter estimation with the PM-2 pumping test data and an improved fit to the observation data was obtained compared to the forward modeling results. The objective function decreased to 52.2 from 1068.6. The final estimated parameter set from the inverse modeling of the PM-2 pump test is given in Table 3, and
Table 5 Comparison of parameter sensitivities from different data sets Parameters
Step 1 (PM-2) Sensitivity Relative sensitivity
Step 2 (PM-4) Sensitivity Relative sensitivity
Step 3 (joint inversion) Sensitivity Relative sensitivity
Ss Sy Ktb1 Ktb2 Ktb4 Ktf Ktpf Ktpp Ktpt Kts Ktt Zfstf Zftb2 Zftpp Zftpt Zftpf Zfts Zftt
0.065 0.219 < 0.001 0.117 < 0.001 0.554 0.200 0.282 0.020 1.202 0.003 10.006 7.341 10.288 0.030 < 0.001 209.145 0.280
0.039 0.224 < 0.001 0.032 < 0.001 0.113 0.065 0.026 0.020 0.470 0.004 0.076 < 0.001 0.942 0.263 0.080 65.187 0.025
0.006 0.154 0.010 0.035 0.073 0.600 0.383 0.052 0.035 0.547 0.280 0.342 11.227 0.060 20.981 0.171 39.416 0.042
0.355 0.054 < 0.001 1.520 < 0.001 6.415 2.635 3.475 0.250 14.947 0.034 0.068 0.022 0.102 0.019 < 0.001 0.210 0.014
Hydrogeology Journal (2010) 18: 607–624
0.231 0.018 < 0.001 0.419 < 0.001 1.276 0.803 0.328 0.271 5.831 0.055 0.030 < 0.001 0.093 0.005 0.027 0.053 0.013
0.034 0.054 0.114 0.450 0.968 6.984 4.741 0.713 0.420 6.755 3.608 0.023 0.011 0.040 0.023 0.042 0.127 0.012
DOI 10.1007/s10040-009-0543-y
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Step 1 (PM-2) Step 2 (PM-4) Step 3 (Joint)
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Fig. 14 Comparison of parameter relative sensitivities from different data sets. Apparent missing data (e.g. Ktb1, Ktb4 and Zftpf) have relative sensitivity < 0.001
the hydrograph fits are presented in Figs. 4 and 5. Figure 6 shows the calculated drawdowns for the observation wells where no apparent drawdown was observed. Most of the calculated values are less than 0.05 m and close to zero drawdowns. The simulated drawdowns in the pumping well and observation wells match reasonably well with the observation data. The calculated discharge (334.62 kg/s) is very close to the observed value (332.75 kg/s). Therefore, the uncertainty of estimated parameters is limited. This set of parameters represents the final results of inverse modeling of the PM-2 pump test. Figure 7 shows two orthogonal cross sections extracted from the geological framework around the PM-2 well and the simulated drawdown distributions after 25 days. Figure 8 presents the plane view of hydrofacies and drawdowns (also at 25 days) at elevations 1,700 and 1,600 m. At elevations between 1,700 and 1,600 m, the drawdown increases and the influence ranges of the pumping test also increase.
wells R-13 (350.4 or 31.3%) and R-19 (572.2 or 51.2%), totaling 82.5%, because the calculated drawdowns are much larger than the observed drawdowns for these two wells, as is the case for the pumping well PM-4. Note that the weighting coefficients for these two observation wells are also very large, because the weights are larger at points with lower observation drawdowns and a very small standard deviation (see Eq. 4). The model parameters developed to match the PM-2 pump test in step 1 do not provide a good fit compared to the observation data for the PM-4 pump test. The poor prediction may be because the PM-2 pump test influences only a small fraction of the site-scale model domain. Subsequently the estimated parameters can represent the heterogeneity of the aquifer in the area close to well PM2, but cannot represent the parameter distributions for the entire aquifer or even the area surrounding PM-4. The conclusion of this forward simulation is that only using PM-2 data to develop a site-scale model for predicting drawdown at PM-4 is an inadequate approach. The inclusion of PM-4 pumping test data for parameter calibration may be necessary for developing a better predictive model. Inverse modeling For inverse modeling of PM-4 pumping test data, the results from the PM-2 pumping test (step 1) were used as 15
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Forward modeling As a first step, the final estimated parameter set from inverse modeling of the PM-2 pumping test (estimated parameters in Table 3) was used to simulate the PM-4 pumping test. The quantitative evaluation is listed in Table 3 with a total objective value of 1118. The model can approximately fit the drawdowns of the observation wells PM-2, PM-5, R-20 and R-15, and the calculated discharge (336.66 kg/s) is close to the observed discharge (332.75 kg/s). However, the model cannot fit the drawdowns in the pumping well PM-4 and the observation wells R-13 and R-19. In fact, the major contributions to the total objective function (1118) come from observation
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Fig. 15 The overall fitting results from the joint inverse model: a PM-2 and PM-4 pumping test data; b site-wide water head data. The green lines are 1:1 lines DOI 10.1007/s10040-009-0543-y
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the initial parameters and the PM-4 pumping test data were used as the fitting target, which includes the drawdown data from PM-4, PM-2, PM-5, R-13, R-14-2, R-15, R-20-3 and R-32, zero drawdown data (a zero drawdown was assigned to the observation wells where no apparent drawdown was observed) from R-14-1 (the first screen of well R-14), and R-22, and discharge data. The observed drawdown data in pumping well PM-4 were also corrected with a factor of 0.52 as discussed in the previous section. The final estimated parameter set from the inverse
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modeling of PM-4 pumping test is listed in Table 3, and the hydrograph fits are presented in Figs. 9 and 10. The objective function value is greatly reduced to 96.5 from 1118 of the forward simulation. The fitting results to the pumping well and observation wells are improved. Most of the calculated drawdowns for the observation wells where no apparent drawdown was observed (R-14-1, R22-1 and R-22-2) are less than 0.05 m and close to zero drawdowns. The observation well R-20-3 (the deepest screen of R-20) is the only well that did not closely match
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Fig. 16 The fitting results for PM-2 pumping test using the joint inverse model (a–f) (Drawdown correction in the pumping well was based on Peaceman 1978) Hydrogeology Journal (2010) 18: 607–624
DOI 10.1007/s10040-009-0543-y
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the drawdown development in the middle and lower part of the pumping well. Figure 12 presents the XY plane views of drawdowns at different elevations (Z=1,700 and 1600 m) after 21 days. From elevation 1,700 to 1600 m, the drawdowns increase and the influence ranges of the pumping test also increase. The corresponding hydrofacies distributions at these elevations can be found in Fig. 8. The calculated discharge (332.86 kg/s) is well matched to the observed value (332.75 kg/s).
the data. This may be due to a high conductivity fault or fracture between R-20-3 and PM-4, and this numerical model did not capture it. So, the computed drawdowns at R-20-3 are much lower than the observed drawdowns. Figure 11 shows the hydrofacies cross-section and the corresponding drawdown distribution in YZ plane (X= 19,294 m) around the PM-4 pumping well. The largest drawdown occurred at about an elevation of 1,600 m. The relatively smaller permeabilities in Tb1 and Tb4 limited
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Fig. 17 The fitting results for PM-4 pumping test using the joint inverse model (a–f) (Drawdown correction in the pumping well was based on Peaceman 1978) Hydrogeology Journal (2010) 18: 607–624
DOI 10.1007/s10040-009-0543-y
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From step 1 to step 2, the biggest changes in the estimated parameters are changes in log permeabilities of Tpt (Ktpt) and Tpf (Ktpf). The former is reduced from – 12.89 to –13.59 log(m2) (see Table 3), and the latter increases from –13.1 to –12.2 log(m2). This is due to heterogeneity within hydrofacies that were not considered in the site-scale model. It means that Tpt has a high permeability in the local area around PM-2 but a relatively lower permeability around PM-4, while Tpf is the opposite. This inverse model only estimates the mean permeability in each hydrofacies based on the individual pumping data. Therefore, different estimates for these parameters were obtained from different pumping test data. In order to obtain a set of parameters suitable to the entire aquifer system of the site model, the next step is to couple the PM-2 and PM-4 pumping test data, discharge data, and the site-wide aquifer monitoring data for a joint inversion of the model parameters.
Step 3: joint inverse modeling The joint inversion model couples all the previously described observation data together to form a composite inverse model. The generalized objective function of this model takes into account the head data from the site-wide aquifer monitoring, drawdown data from the PM-2 and PM-4 pump tests, and discharge data. For forward simulations, the site model is run to steady-state to fit the site-wide water head data and the discharge data. Then, using the obtained steady-state flow field as the initial conditions, two transient simulations are run for the 25-day PM-2 pumping test and the 21-day PM-4 pumping test, respectively. Finally, the joint inverse model calculates the generalized objective function from the forward modeling results. Figure 13 shows the locations of all the observation wells used in joint inversion. In order to test the sensitivity of the joint inverse model to the parameters, the computed sensitivity coefficients from these three steps are summarized in Table 5 (also see Fig. 14). The sensitivity (or composite sensitivity) in Table 5 presents the sensitivity of each parameter with respect to different scales and different sources of observations; while the relative sensitivity of a parameter is obtained by multiplying its composite sensitivity by the magnitude of the value of the parameter. For all these three steps, the permeability of Ts (Sandy Santa Fe) is the most sensitive parameter because Ts has the largest fraction of the total volume (0.294, see Table 1) and both PM-2 and PM-4 pumping wells intersect this unit. For the same reason, permeabilities of Tf (Fanglomeratic Santa Fe), Tpf (Puye fanglomerate) and Tt (Tschicoma flows) are the second most sensitive parameters in these three simulations. Since the joint model incorporates much more observation data than the individual inverse model of PM-2 or PM-4, the joint model is sensitive to more parameters. For example, in step 1 and 2, the permeabilities of Tb1 and Tb4 are not sensitive enough to be estimated because they are distributed fairly far away from the PM-2 and PM-4 pumping wells. When the Hydrogeology Journal (2010) 18: 607–624
site-wide aquifer monitoring data are included in the joint inverse model, these two parameters are identifiable in the joint inverse model. The Z factors (which is the ratio of the vertical and horizontal permeability) of zftpf and zftb2 are not sensitive in step 1 and 2, respectively, while they are sensitive enough to be estimated in the joint inverse model. In the joint model a total of 18 parameters were estimated. The joint inverse model yielded good results in the form of a reasonably low overall objective function (see the last column in Table 3), although the sub-objective values for the PM-2 and PM-4 pumping tests (70.7 and 157.5, respectively) are slightly larger than those obtained from step 1 (52.2 for PM-2) and step 2 (96.5 for PM-4). These results indicate that the joint inverse model provides good predictions of water heads or drawdowns at most of the observation wells. Figure 15 shows the overall fitting results. Figure 15a compares the calculated versus measured drawdowns of PM-2 and PM-4 pumping tests. With the exception of drawdown data from R-20-3 in the PM-4 pumping test, one can fit all the other data. The reason for the poor fitting of R-20-3 was discussed previously. Figure 15b shows a very good match of the calculated and measured site-wide water head data. Figures 16 and 17 show the fitting results of the measured versus calculated time-series drawdowns for PM-2 and PM-4 pumping tests. Simulated drawdowns at the two pumping wells match the corrected drawdown data quite well. The calculated drawdowns for observation wells where zero drawdown was observed are less than 0.05 m for both pumping tests. Table 6 illustrates the new estimated parameters and their 95% confidence intervals. Because more observation data were used for the joint inverse model, all of the other 18 parameters are well-constrained and have reasonable lower and upper bounds. Therefore, the uncertainty of estimated parameters is well limited in the joint inversion. This set of parameters represents the final results of inverse modeling of the site-scale model. Table 6 The estimated parameters and 95% confidence intervals from the joint inversion Parameters
Estimated values
95% confidence intervals Lower limit Upper limit
Kts Ktf Ktb4 Ktb1 Ktb2 Ktt Ktpt Ktpp Ktpf Ss Sy Zftb2 Zftpt Zftpf Zfstf Zfts Zftpp Zftt
–12.40 –11.64 –13.20 –13.60 –12.99 –12.90 –11.99 –13.62 –12.33 –5.75 0.35 0.0001 0.001 0.247 0.058 0.004 0.689 0.279
–12.45 –11.73 –14.11 –15.55 –13.57 –13.22 –12.49 –14.22 –12.52 –9.98 0.165 0.0001 0.0001 0.078 0.004 0.003 0.0001 0.0001
–12.35 –11.54 –12.29 –11.64 –12.41 –12.58 –11.50 –13.03 –12.14 –1.63 0.535 0.001 0.002 0.415 0.113 0.006 1.386 1.091
DOI 10.1007/s10040-009-0543-y
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Summary and conclusions Inverse modeling provides a method whereby measurements of state are used to determine unknown model structures and parameters by fitting model output with observation data. This technique combined with a stepwise strategy has been successfully applied to the LANL site-scale model to estimate model parameters by coupling observation data from different sources and at various spatial scales including single-well tests, multiple-well pumping tests to regional aquifer monitoring data. To determine the flow parameters for the site-scale model, statistical analyses of single-well slug and pumping test results were first conducted to define the prior distributions of the key parameters. This prior information was used to define the parameter initial values and the lower and upper bounds for inverse modeling. From the stepwise progression of inverse modeling, a number of steps were used by inversions of drawdown data from the PM-2 and PM-4 pump tests separately, and a joint inversion was undertaken by coupling PM-2 and PM-4 pump test data with head data from the site-wide aquifer monitoring network. Parameter sensitivity coefficients for different data sets were calculated to analyze the parameter identifiability in different steps. Finally, the joint inverse model was derived with a generalized composite objective function. Solving the joint inverse problem leads to a well-constrained estimate of model parameters and reduces the uncertainty of estimated parameters. The joint inversion results provide a reasonable fit to all three data sets. The uncertainty of estimated parameters for the hydrofacies is addressed with the estimated parameter confidence intervals. The parameters estimated from the joint inverse model should provide appropriate overall predictions of drawdown at future pumping tests in LANL site. The previous versions of the site-scale model and the smaller scale flow and transport models (e.g. individual watershed-scale models), using different calibration criteria and observation data (Keating et al. 2003 and 2005), may incorporate different parameter values from those presented here and are still valuable for flow and transport modeling in this site. Acknowledgements We wish to extend our appreciation to K. Birdsell, Z. Lu and P. Stauffer for their thoughtful reviews of this paper, and to D. Broxton and S. McLin for the use of figures from their reports. We are grateful to the editor and to the two reviewers for their comments and suggestions and the time they invested in our manuscript.
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DOI 10.1007/s10040-009-0543-y
