Joint simulation of transmissivity and storativity ®elds conditional to
steady-state and transient hydraulic head data
Harrie-Jan Hendricks Franssen, J. Jaime G
omez-Hern
andez
*, Jose E. Capilla,
Andres Sahuquillo
Departamento de Ingenierõa Hidraulica y Medio Ambiente, Universidad Politecnica de Valencia, Apartado de Correos 22012, 46080 Valencia, Spain
Received 15 September 1998; accepted 15 February 1999
Abstract
The self-calibrated method has been extended for the generation of equally likely realizations of transmissivity and storativity conditional to transmissivity and storativity data and to steady-state and transient hydraulic head data. Conditioning to trans-missivity and storativity data is achieved by means of standard geostatistical co-simulation algorithms, whereas conditioning to hydraulic head data, given its non-linear relation to transmissivity and storativity, is achieved through non-linear optimization, similar to standard inverse algorithms. The algorithm is demonstrated in a synthetic study based on data from the WIPP site in New Mexico. Seven alternative scenarios are investigated, generating 100 realizations for each of them. The dierences among the scenarios range from the number of conditioning data, to their spatial con®guration, to the pumping strategies at the pumping wells. In all scenarios, the self-calibrated algorithm is able to generate transmissivity±storativity realization couples conditional to all the sample data. For the speci®c case studied here the results are not surprising. Of the piezometric head data, the steady-state values are the most consequential for transmissivity characterization. Conditioning to transient head data only introduces local adjustments on the transmissivity ®elds and serves to improve the characterization of the storativity ®elds. Ó 1999 Elsevier Science Ltd. All rights
reserved.
Keywords:Heterogeneity; Geostatistics; Conditioning; Inverse modeling; Self-calibrated algorithm; Network design
1. Introduction
In recent years, there has been a large eort to in-corporate dierent types of information to better char-acterize the spatial variability of transmissivity. Much eort has gone into the inverse modeling of groundwater ¯ow and the incorporation of hydraulic head data into the generation of transmissivity ®elds that are condi-tioned to both transmissivity and head data. The reader is referred to Ref. [1] for a review of the state-of-the-art of conditional simulation in the context of the inverse problem along with an intercomparison of seven meth-ods. Here, we will refer only to the latest methods able to generate realizations of transmissivity conditional to transmissivity and head data without limitation on transmissivity variance. Traditional inverse methods seeking the determination of a single best estimate will
not be reviewed: smooth estimates of the transmissivity spatial distribution, even after calibration to head data, would yield bias predictions of ¯ow and transport variables at unsampled locations. Only conditional transmissivity realizations displaying the same patterns of spatial variability as observed in the ®eld should be used for ¯ow and transport predictions.
To the best of our knowledge, the ®rst conditional simulation algorithm capable of producing trans-missivity ®elds honoring transtrans-missivity and piezometric head data without restrictions on transmissivity variance is the self-calibrated method [2±6]. This method was conceived and developed from the outset with the pur-pose of generating transmissivity ®elds with realistic spatial variability patterns, conditional to the trans-missivity measurements on which the solution of the groundwater ¯ow equation reproduces, as close as possible, the observed piezometric heads. In opposition were the methods later presented by Ramarao et al. [7], Gutjahr et al. [8], Kitanidis [9] and Hanna and Yeh [10], which are evolutions of techniques, originally developed
*Corresponding author. Tel.: 79614; fax:
+349-7638-77618; e-mail: jaime@dihma.upv.es
for the estimation of single best estimates, into condi-tional simulation approaches. The self-calibrated meth-od (SCM), which is later described in more detail, consists of two steps, in the ®rst step a seed realization of transmissivity conditional to the transmissivity data is generated, then, in the second step, a perturbation is added to the seed realization to modify it into one conditional to the piezometric heads. The ®rst step in-volves geostatistics, the second one, non-linear optimi-zation using a parameterioptimi-zation of the perturbation ®eld with a small number of independent parameters. Be-cause the conditioning to piezometric head is carried out by the minimization of an objective function, an exact reproduction of the piezometric head is not pursued, the ®eld is said to be conditional if the heads are reproduced within a tolerance value, proportional to the measure-ment error variance. The extension of the pilot point method by Ramarao et al. [7] into a conditional simu-lation algorithm is similar to the SCM in that it consists of the same two steps; however, the perturbation ®eld is computed by small patches in an iterative fashion. The methods by Gutjahr et al. [8], Kitanidis [9] and Hanna and Yeh [10], are all based on the fact that if the de-pendence of heads on transmissivities is linearized, then the cross-spectral or cross-covariance structure of the bivariate transmissivity-head random function is fully de®ned from the spectrum or covariance of trans-missivity; furthermore, standard multivariate ge-ostastistical techniques could be used for the generation of transmissivity and head ®elds conditional to trans-missivity and head data. However, the ®elds so gener-ated do not respect the ¯ow equation but its linearized approximation. The three methods then enter an itera-tive procedure to modify the linearly related trans-missivity and head ®elds so that they satisfy the ¯ow equation. The methods dier in the way this modi®ca-tion is obtained. Menmodi®ca-tion should also be made of the maximum likelihood method of Carrera and Neuman [11,12], which was used as a conditional simulation technique in the context of the intercomparison exercise carried out by the US SANDIA National Laboratories [1]. In their pioneering inverse method, Carrera and Neuman compute a maximum likelihood estimate of transmissivity given the transmissivity and head mea-surements together with an estimate of the joint condi-tional covariance matrix. This joint conditional covariance matrix could then be used in the context of an LU decomposition stochastic simulation approach for the generation of the conditional transmissivity ®elds.
The motivation of this paper is the extension of the SCM to the joint generation of transmissivity and storativity ®elds conditional to transmissivity, stor-ativity, steady-state heads and transient head measure-ments. None of the previously referred to methods for inverse conditional simulation have considered the joint
generation of transmissivity and storativity. And, al-though the extensions of any of those methods to handle transient head data may appear as straightforward, only the pilot point method [7,13] and the maximum likeli-hood method [11,12] have been demonstrated with transient data. The new implementation of the SCM method is applied to a synthetic case built on real data from the Waste Isolation Pilot Plan in New Mexico. This site is characterized by its strong spatial variability. The objective of the paper is not to draw any conclusion about the WIPP site ± mostly because the data sets in-volving storativity values are taken from a synthetic storativity ®eld generated on the basis of real measure-ments ± but to prove the capability of SCM to generate jointly conditional realizations of transmissivity and storativity, and also to analyze the impact that consid-ering the spatial variability of storativity may have into the characterization of the formation.
2. The self-calibrated method
A detailed description of the SCM can be found in Gomez-Hernandez et al. [3]. The main steps of the method are summarized next, along with its extension to the joint simulation of transmissivity and storativity ®elds conditional to transient head data. They are the following.
(1) A seed log-transmissivity (YlogT) and a seed log-storativity (ZlogS) ®eld are jointly generated conditional to Y and Z data, using, for instance, se-quential co-simulation [14]. The ®elds reproduce the spatial variability observed in the ®eld as modeled by their respective auto-covariance and cross-covariance functions. (In principle, the method is presented for the most general case of non-negligible cross-correlation between the two attributes.) If enough data are avail-able, the covariances ofYandZare estimated from the YandZdata. When few data are available, covariances should be postulated on the basis of prior experience on similar formations. Measurement errors can be ac-counted for in this ®rst step. The simplest way is to es-timate which proportion of the nugget eect appearing in the experimental covariance is due to measurement error ± not to short scale spatial variability ± and to assume the measurement errors as spatially uncorrelated.
The iteration counter is set to zero.
steady-state head solution is used as the initial condition for the transient simulation. If this is not deemed ap-propriate, an initial head ®eld must be supplied. In this case, the initial head may be known, at most, at a few locations from which an estimate of the initial head over the entire aquifer will have to be inferred. This estimate could also be subject to calibration by the SCM method in a manner similar to the calibration of the boundary conditions as described by Gomez-Hernandez et al. [3]. (3) An objective function is de®ned that measures the mismatch between simulated and observed heads:
JX where the ®rst term corresponds to the discrepancies at the steady state and the second term to the discrepancies at the transient state.Nmis the number of measurement
locations, Nt the number of time steps with
measure-ments, hi;0 represents the steady-state heads, hi;t the
transient heads and the superscripts SIM and MEAS refer to `simulated' and `measured' values, respectively. The weightsni0 and nit are chosen inverse-proportional to the estimated measurement errors. We say that theY and Z ®elds are conditional to the h data when J is smaller than a prede®ned tolerance value. No ®eld will be accepted unless this condition is met. The corre-spondingTandS®elds are obtained by taking the an-tilog of the resultingYandZ®elds.
IfJis not small enough, a perturbation ®eldDYand a perturbation ®eldDZhave to be calculated to be added to the currentYandZ®elds. These perturbation ®elds are parameterized as functions of the individual per-turbations at a number of selected locations, referred to as master blocks. The values away from the master blocks are obtained by ordinary co-kriging interpolation of the master blocks perturbations:
DYij
where Np is the number of master locations, DYk and DZk the perturbations at the master locations, and the coecientskk
ij are the co-kriging coe-cients weights for the interpolation of the perturbation at location ij from the master location perturbations. Obviously, if no cross-correlation is observed between transmissivity and storativity, the above equation sim-pli®es since the weightslk
ij, tkij will be zero. The master blocks form an essential part of the methodology be-cause they reduce the dimensionality of the optimization problem. The number of master blocks should be as small as possible to minimize the number of independent
parameters describing the perturbation ®elds, but large enough so that there are enough degrees of freedom for the optimization process to determine the perturbation ®elds that result in a match to the piezometric heads. For each case study, it is necessary to carry out a small exercise to determine the optimal number of master lo-cations, which, as a rule of the thumb, should be in the order of two per correlation range. The master blocks are normally laid out on a regular grid with a random starting point that changes from one realization to an-other. They could be selected by random sampling or random strati®ed sampling, but our experience shows that, in those cases, the convergence of the optimization process is slower. In addition, the transmissivity and storativity measurement locations are always included in the set of master blocks. In case of error-free trans-missivity and storativity measurements, the perturbation at the data locations is forced to be null, otherwise the perturbation at these locations is allowed to vary within an interval proportional to the magnitude of the error measurement variance.
The values ofDYk and DZk at the master locations are obtained by minimizing the objective functionJ. The minimization of the objective function is achieved by non-linear optimization as outlined below.
(4) The gradient vectorgcontaining the derivatives of Jwith respect to the 2Npperturbations ofYandZat the
master locations {oJ/oDYk, oJ/oDZk, k1,Np} is
de-termined using the adjoint-state formulation. The ad-joint-state approach allows computation of the gradient vector eciently ± especially when modeling transient ¯ow as compared with the computation using sensitivity coecients [15].
(5) The updating directiondis computed using one of the following algorithms: steepest descent, Fletcher± Reeves conjugate gradient, Hestenes±Stiefel conjugate gradient or quasi-Newton. The updating direction is given by
dl ÿHl
glalÿ1
dlÿ1;
wheredis the updating vector (of dimension 2Np),Hthe
perturbation vector bldl minimizing the objective func-tion in the updating direcfunc-tion. It is convenient to con-strain the magnitude of the perturbations at the master locations to prevent instabilities in the optimization process. We generally apply a constrain so that the ®nal logtransmissivity and logstorativity ®elds are within plus/minus three co-kriging standard deviations of the co-kriging estimates at the master locations obtained by ordinary co-kriging of the data.
(6) The resulting perturbations DYk and DSk at master locations are interpolated by ordinary co-kriging to the rest of the blocks, using Eq. (2). The perturbation ®elds are added to the last iteration Yand S ®elds re-sulting in the updated ®elds for the current iteration:
YijlYijlÿ1DYij;
Zijl Zijlÿ1DZij:
The iteration counter l is increased by one and the al-gorithm returns to Step 2.
3. A synthetic study based on real data
We have tested the performance of the method using a synthetic aquifer resembling the Culebra formation at the Waste Isolation Pilot Plant in New Mexico (USA). The reference ®elds are built conditional to theY,Zand h data given in the reports by LaVenue et al. [16] and Cauman et al. [17].
A total of seven dierent scenarios have been ana-lyzed. For each scenario 100 equally likely Y±Z ®eld couples are generated conditional toY,Zandhdata. In this synthetic study we have not considered errors in the measurements or in the covariance estimate. However, these errors could be easily accounted for. The reason for assuming error-free data is because we were inter-ested in analyzing the impact of the spatial variability of both transmissivity and storativity, and did not want the results to be also aected by error measurements. For all seven scenarios the same reference Z ®eld has been used. The seven scenarios dier with respect to the reference Y®eld, the sampling density of the reference Z®eld, the sampling density of the referenceh®eld and the amount, location and pumping rates of the pumping wells.
3.1. Spatial domain
The study is carried out in a rectangular domain of 21.5 ´30.5 km2 discretized into 43´ 61 square cells of
500 ´500 m2 in size. This domain corresponds to the
WIPP model area. The boundary conditions are pre-scribed heads along the perimeter of the modeling site, the values used are the same reported by LaVenue et al. [16] in their model, implying an average gradient of 32 m across the formation forcing ¯ow from north to south.
3.2. Reference transmissivity ®elds
Two dierent reference YlogT ®elds have been generated. Field 1 is generated by sequential simulation conditioned to 36Ymeasurements. Field 2 is generated by self-calibration conditioned to 36 Y measurements and 34 steady-state head measurements. For the gener-ation of ®eld 1, and the seed ®eld necessary by the self-calibration of Field 2, a modi®ed version of the sequential simulation program GCOSIM3D [14] has been used. This modi®cation accounts for a linear trend in the logtransmissivity ®eld [18]. The variogram of the logtransmissivity residuals is spherical with nugget 0.195 (log(m2/s))2, sill of 1.33 (log(m2/s))2 and an isotropic
range of 11.3 km. The two reference ®eldsY1andY2are
shown in Fig. 1.
The reference ®elds are sampled at the same 36 lo-cations at which logtransmissivity is reported by LaVe-nue et al. [16] (See Fig. 2).
3.3. Reference storativity ®eld
Thirteen Sdata from the WIPP area have been used to estimate a variogram of ZlogS. An isotropic spherical variogram with nugget 0.02 (log(m/m))2, sill of
1.78 (log(m/m))2and range of 11.8 km has been adopted
forZ. TheZvariogram and theZdata have been used
as input to the sequential simulation program GCO-SIM3D for the generation of a conditional simulation of
Z, which is used as the reference Z ®eld for all seven
scenarios. (The realization is conditional to the 13 S values measured at WIPP.) The reference Z ®eld is shown in Fig. 1.
There is no apparent statistical cross-correlation be-tween measured logtransmissivity and measured logs-torativity at the locations in which both parameters were available. Therefore, the logstorativity and log-transmissivity ®elds are generated independent of each other.
Three samples are taken from the reference storativity ®eld. Sample 1 contains 13 values at the WIPP mea-surement locations. Sample 2 contains 30 samples reg-ularly spaced over the site. Sample 3 contains 9 samples, also regularly spaced. (See Fig. 2.)
3.4. Reference hydraulic heads
Pumping test Scenario 1. It mimics one of the long-term pumping tests performed at the WIPP site. A single well pumps at a constant rate of 1.93 l/s
during 35 days. Hydraulic head is monitored at 34 locations (the same ones monitored at the WIPP site), (see Fig. 3). At each location, head is reported
Fig. 1. Reference ®elds. Logtransmissivity reference ®eldY1is generated by sequential simulation just conditional to 35 logtransmissivity mea-surements from the WIPP site; logtransmissivity reference ®eldY2is generated by self-calibration conditional to 35 logtransmissivity measurements and 34 steady-state head data from the WIPP site; logstorativity reference ®eld is generated by sequential simulation conditional to 13 measurements from the WIPP site.
at 21 time steps, the step size follows a geometric progression of ratio 1.2. This pumping test, although realistic, only acts over a limited area of the modeling domain.
Pumping test Scenario 2. In order to aect the entire formation with a transient event, 24 wells, regularly distributed, are pumped at a rate of 1.93 l/s during 35 days. Hydraulic head is monitored at 35 locations ran-domly chosen within the site (see Fig. 3). The sampling frequency is the same as above.
Pumping test Scenario 3. In order to analyze the eect of varying pumping rates, and correspondingly, a large heterogeneity on head drawdowns, 24 wells, regularly distributed, are pumped at rates varying between 0.00193 and 193 l/s. Hydraulic head is monitored at 35 locations distributed regularly so that each pumping well has a monitoring location nearby (see Fig. 3). The sampling frequency is the same as above.
3.5. Seven scenarios analyzed
A number of scenarios has been studied to analyze how the dierent types of data in¯uence the character-ization of the spatial heterogeneity of the log-transmissivity and logstorativity reference ®elds, as well as the reproduction of the hydraulic head reference ®elds. Seven combinations of the sample data sets de-scribed above have been chosen to de®ne the seven scenarios studied. These scenarios are summarized in Table 1.
For each scenario, the challenge is the joint genera-tion of couples of logtransmissivity and logstorativity ®elds conditional to the given sample data sets. The de®nition of the scenarios aims to analyze the in¯uence of:
1. the number of pumping wells (comparison between Scenarios 1 and 3, and between Scenarios 2 and 4), 2. the spatial heterogeneity of head drawdowns
(com-parison between Scenarios 3 and 6, and between Sce-narios 4 and 7),
Fig. 3. Location of pumping wells (black dots) and head monitoring locations (white dots).
Table 1
The seven scenarios
Scenario Reference
transmissivity ®eld
Pumping test scenario
Storativity data set
1 2 1 1
2 2 1 2
3 1 2 1
4 1 2 2
5 1 2 3
6 1 3 1
7 1 3 2
Reference transmissivity ®eld: 1conditioned to transmissivity mea-surements from WIPP, 2conditioned to transmissivity and steady-state head measurements from WIPP. Pumping test scenario: 1single well pumping test with 34 head monitoring locations corresponding to
WIPP sampling locations, 2multiple well pumping test with 35
3. the position of the head monitoring points (compari-son between Scenarios 3 and 6, and between Scenar-ios 4 and 7),
4. the number of storativity samples (comparison be-tween Scenarios 1 and 2, among Scenarios 3, 4 and 5, and between Scenarios 6 and 7).
For illustration purposes, Fig. 4 shows a realization from Scenarios 1 and 4.
4. Results
For each scenario, the self-calibration method is used to generate 100 Y±Z®eld couples. No statistical corlation was considered between the two parameters, re-sulting in a simpler updating of the seed ®elds than if such a correlation had been considered: the weights lk
ij andtk
ij in Eq. (2) are zero. However, there is an implicit
Fig. 4. A realization from Scenarios 1 (left column) and 4 (right column). The logtransmissivity and logstorativity ®eld couples conditional toY,Z
correlation, through the transient groundwater ¯ow equation, since each couple is conditional to the same set of transient head data.
The sequential self-calibration generates Y±Z ®eld couples conditional to the measured, Y,Z and h data. Conditioning to the Yand Z data is automatic in the
seed ®eld generation step, and posterior updating; however, conditioning to thehdata (in the sense that the solution of the transient ¯ow equation using the updated Y±Zcouple reproduces, within a preset tolerance limit, the measured heads) is carried out through an optimi-zation algorithm that does, in principle, not ensure exact
Fig. 5. Ensemble average of the seed ®elds. Since the seed ®elds do not incorporate any hydraulic head information, these ensemble averages correspond to the kriging estimates.
reproduction of the measured heads. In this respect, in Scenarios 1±5, the ®nal values of the objective function (1) (not reported) are close to zero, indicative of a good conditioning in all realizations. However, for Scenarios 6 and 7, which correspond to the pumping scenario that produces very heterogeneous drawdowns (as large as a hundred meters in some wells) to achieve conditioning to the measured transient heads was dicult and CPU time consuming: some realizations required more than one hundred iterations in the non-linear optimization step, as opposed to less than 10 iterations on an average for the rest of the scenarios.
The next issue addressed is how conditioning to the input data sets helps in improving the characterization of the logtransmissivity, logstorativity and hydraulic head ®elds over the entire model area. For this purpose, the comparison between the generated ®elds and the refer-ence ones is made through the use of the average absolute error (AAE), and the average ensemble variance (AEV):
AAE X 1 NNODES
X
NNODES
i1
XSIM;i
ÿXREF;i ;
AEV X 1 NNODES
X
NNODES
i1 r2
Xi;
where NNODES is the number of discretization grid cells, andi is a grid cell index,X represents either log-transmissivity (Y), logstorativity (Z), or head (h) for a given time step, the overbar indicates ensemble average, that is, the average, at a given grid cell, through the 100 realizations, the subscript SIM refers to the realizations, and the subscript REF to the reference values; ®nally,
r2
Xi is the ensemble variance ofX at a given node. The smaller these averages are, the better characterized, the Y,Zandh ®elds are. (Notice that if the above averages were computed over the conditioning points only, they would be zero in theYandZ®elds, and close to it for theh®eld.)
To evaluate the impact of the successive conditioning to steady-state and to transient heads, the AAE is computed in the ®elds only conditioned to log-transmissivity or logstorativity data, then after condi-tioning to steady-state head data, and ®nally after conditioning to transient head data.
Fig. 5 shows the ensemble averageYandZseed ®elds for the dierent scenarios. These ensembles averages are conditioned only to the Y or Zdata, respectively, and therefore, do not carry any information about the ¯ow behavior of the formation.
Figs. 6 and 7 show the ensemble averageY®eld for all seven scenarios alongside their respective reference ®elds. Recall that the dierence between the two Y ref-erence ®elds is that Scenarios 1 and 2 use a log-transmissivity conditional to the 35 loglog-transmissivity data and the 34 steady-state head data provided in the report by LaVenue et al. [16], whereas Scenarios 3±7 use a logtransmissivity ®eld that is only conditional to the same 35 logtransmissivity data. These ensemble averages should be compared to those in Fig. 5 to appreciate how conditioning to the head data helps in a better delinea-tion of the main patterns of spatial variability existing in the reference ®elds. It is dicult to distinguish large dierences between the ensemble averages in Fig. 6, or among those in Fig. 7, which indicates that the aspects dierentiating the scenarios are not very important for the characterization of the logtransmissivity ®eld, in this speci®c case. This is corroborated when analyzing the AAE(Y) in Table 2. This table shows the AAE(Y)
computed in the seed ®elds of Fig. 5, in the ensemble average of the updated ®elds after conditioning to steady-state heads (not displayed in any ®gure) and in the ensemble averages of the updated ®elds after con-ditioning to steady and transient state heads shown in Figs. 6 and 7. The AAE(Y) displays a noticeable de-crease after conditioning to steady-state head data and a much smaller decrease after conditioning to transient heads. It appears that conditioning to between 30 and 40 logtransmissivity and steady-state head data is enough for a good characterization of the logtransmissivity spatial variability.
Table 2 also shows the AEV(Y) that can be inter-preted as an average measure of local uncertainty. All scenarios that useY1 as the reference logtransmissivity
®eld display a reduction on AEV(Y) from the seed ®elds to the updated ®elds after self-calibration to the tran-sient head data. This indicates a reduction in the un-certainty on the prediction of the reference ®eld by the ensemble of conditional realizations. However, Scenar-ios 1 and 2 that useY2 as the reference ®eld display a
small increase on AEV(Y) from the seed ®elds to the updated ®elds self-calibrated to the transient head data. This behavior is most likely due to the mismatch be-tween the variogram used for the generation of the seed ®elds (the same for all scenarios and for the generation ofY1) and the variogram of Y2. The calibration to the
steady-state head data, carried out for the generation of Y2, modi®ed the seed ®eld producing a reference ®eld
with a larger intrinsic variability, and therefore a
Table 2
Logtransmissivity and logstorativity characterization measures
Scenario Cond. Stage AAE(Y) (log2(m/s))2 AEV(Y) log2(m/s) AAE(Z) log2(m/m) AEV(Z) log(m/m)
1 Y,Z 0.892 1.05 0.856 1.19
Steadyh 0.745 0.85 ± ±
Transienth 0.741 1.06 0.826 0.79
2 Y,Z 0.892 1.05 0.563 0.55
Steadyh 0.745 0.85 ± ±
Transienth 0.744 1.04 0.541 0.44
3 Y,Z 0.652 0.90 0.856 1.19
Steadyh 0.586 0.60 ± ±
Transienth 0.574 0.65 0.808 0.84
4 Y,Z 0.652 0.90 0.563 0.55
steadyh 0.586 0.60 ± ±
transienth 0.583 0.59 0.525 0.45
5 Y,Z 0.652 0.90 0.752 2.68
steadyh 0.586 0.60 ± ±
transienth 0.593 0.67 0.759 1.05
6 Y,Z 0.652 0.90 0.856 1.19
steadyh 0.588 0.59 ± ±
transienth 0.578 0.88 0.762 0.96
7 Y,Z 0.652 0.90 0.563 0.55
steadyh 0.588 0.59 ± ±
transienth 0.580 0.78 0.527 0.57
variogram with a larger sill than the one used for the generation of the seed ®eld, as it can be noticed in Fig. 1. Fig. 8 shows the ensemble averageZ®eld for all seven scenarios alongside the reference ®eld. These ®elds should be compared to those in Fig. 5 to notice the im-pact that conditioning to transient heads has in a better characterization of the logstorativity ®eld. In six out of the seven scenarios, conditioning to transient head
re-duces the value of AAE(Z) by a factor between 4% and 11% (see Table 3). The reduction is more important in those scenarios with multiple pumping tests since they aect a larger part of the aquifer. An illustrative case is Scenario 2 with a single pumping test, whereas the reduction of the AAE(Z) is 4% when computed over all the formation, it is as large as 45% if the computation is limited to the 1.5´1.5 km2aected by the test.
The sampling pattern and frequency ofZ data play important roles in the reduction of the AAE(Z). The reduction is larger for the scenarios with 30 logs-torativity data than for those with only 13 data, but it is also larger for the scenarios of nine regularly spaced data than for the 13 irregularly sampled values.
A more important eect of the conditioning to tran-sient head than the reduction of AAE(Z) is the reduc-tion of the AEV(Z). The AEV(Z), interpreted as a measure of the degree of local uncertainty in the esti-mates of logstorativity, is substantially reduced in most scenarios, indicating that the ensemble of conditional realizations ¯uctuate more closely about its mean value than the ensemble of realizations not conditioned to head data. The lesser logstorativity data there are, the larger the reduction of the ensemble variance is. The largest reduction occurs for Scenario 5 (61%) with only nine Z data. The smaller reduction occurs when the drawdown heterogeneity is large (Scenarios 6 and 7).
The ensemble averages of both logtransmissivity and logstorativity realizations do not change much among the dierent scenarios. It appears as if using multiple pumping wells does not help in improving the charac-terization of the main patterns of spatial variability ofY
and Z. For this particular case study, the main reason
explaining this uniformity is that all realizations and all scenarios with the same logtransmissivity reference ®eld are conditional to the same steady-state heads. This conditioning seems to be enough for the characteriza-tion of the main patterns of Yspatial variability.
Ad-ditional conditioning to transient head data only produces local updates of theYand Z®elds which are enough to match the transient head data but hardly noticeable in the average ®elds. (It must be stressed here that this behavior is speci®c to this data set. Experiments carried out with other data sets, in which the number of steady-state head data was smaller, showed precisely the opposite behavior, the transient head data were the most relevant to the characterization of the logtransmissivity ®eld.)
To determine the degree of reproduction of the ref-erence ®elds by the realizations, the discrepancy between realizations and reference is measured by the AAE(h) at times 0 (steady-state), 17 days and 35 days (end of the pumping period). These values are also computed in the seed ®elds, and in the ®elds conditional only to steady-state heads, with the objective to analyze how conditioning to dierent types of head data helps in reproducing the heads everywhere within the simulation domain.
In all cases, as expected, the AAE(h) is always re-duced after conditioning to head data. It is interesting to notice that conditioning to steady-state head data helps improving the overall prediction of the steady-state reference ®eld but has little impact in the overall pre-diction of the transient-state reference ®elds. Further conditioning to transient data drastically reduces the departure between measured and predicted transient head values. This eect is particularly noticeable for Scenarios 6 and 7 in which the head drawdowns are very
Table 3
Hydraulic head characterization measures
Scenario Time (days) AAE initial (m) AAE updated steady (m) AAE updated transient (m)
1 0 2.22 0.60 0.61
heterogeneous and the predictions in the seed ®elds or the steady-state conditioning data depart more than 300 m on an average at the end of the pumping period.
5. Summary and conclusions
The formulation of the self-calibration algorithm has been extended to the conditioning to transient head data and to the joint generation of logtransmissivity and logstorativity ®elds. The main changes with respect to the original formulation are that the seed ®elds of log-transmissivity and logstorativity are generated using co-simulation and then updated using co-kriging, and that the gradient of the objective function in the updating step is computed using the adjoint-state equations.
The self-calibration algorithm has been tested in dif-ferent situations regarding the number and type of conditioning data, and in all cases, was capable of generating conditional realizations of logtransmissivity and logstorativity.
In this study we have not considered data measure-ment errors or uncertainty on the boundary conditions or the variogram in order to focus on the worth of the dierent sources of data. However, SCM is able to handle these sources of uncertainty. In particular, the method has been shown to be very robust to variogram mismatch [4].
There is a clear trade-o between the dierent types of data with regard to the characterization of the dif-ferent parameters and variables involved. This paper does not present a systematic analysis of these trade-os, although points to the potential that the self-calibration algorithm has for the design of the monitoring strategy that will result in the best characterization of the vari-ables of interest.
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