Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue1.1999:

Advances in Water Resources 23 (1999) 1±13

Joint simulation of transmissivity and storativity ®elds conditional to
steady-state and transient hydraulic head data
mez-Hern
Harrie-Jan Hendricks Franssen, J. Jaime Go
andez *, Jose E. Capilla,
Andres Sahuquillo
Departamento de Ingenierõa Hidr
aulica y Medio Ambiente, Universidad Polit
ecnica 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 transmissivity 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 di€erences 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 e€ort to incorporate di€erent types of information to better characterize the spatial variability of transmissivity. Much
e€ort has gone into the inverse modeling of groundwater
¯ow and the incorporation of hydraulic head data into
the generation of transmissivity ®elds that are conditioned 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 methods. 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

*

Corresponding author. Tel.: +349-7638-79614; fax: +349-763877618; e-mail: jaime@dihma.upv.es

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 transmissivity ®elds honoring transmissivity 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 purpose of generating transmissivity ®elds with realistic
spatial variability patterns, conditional to the transmissivity 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

0309-1708/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 0 6 - 8

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H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

for the estimation of single best estimates, into conditional simulation approaches. The self-calibrated method (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 involves geostatistics, the second one, non-linear optimization using a parameterization of the perturbation ®eld
with a small number of independent parameters. Because 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 measurement error variance. The extension of the pilot point

method by Ramarao et al. [7] into a conditional simulation 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 dependence 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 transmissivity; furthermore, standard multivariate geostastistical techniques could be used for the generation
of transmissivity and head ®elds conditional to transmissivity and head data. However, the ®elds so generated do not respect the ¯ow equation but its linearized
approximation. The three methods then enter an iterative procedure to modify the linearly related transmissivity and head ®elds so that they satisfy the ¯ow
equation. The methods di€er in the way this modi®cation is obtained. Mention 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 measurements together with an estimate of the joint conditional 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, storativity, steady-state heads and transient head measurements. None of the previously referred to methods for
inverse conditional simulation have considered the joint

generation of transmissivity and storativity. And, although 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 likelihood 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 involving storativity values are taken from a synthetic
storativity ®eld generated on the basis of real measurements ± but to prove the capability of SCM to generate
jointly conditional realizations of transmissivity and
storativity, and also to analyze the impact that considering 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
G
omez-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 (Y ˆ log T) and a seed
log-storativity (Z ˆ log S) ®eld are jointly generated
conditional to Y and Z data, using, for instance, sequential 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 available, the covariances of Y and Z are estimated from the
Y and Z data. When few data are available, covariances
should be postulated on the basis of prior experience on
similar formations. Measurement errors can be accounted for in this ®rst step. The simplest way is to estimate which proportion of the nugget e€ect 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.
(2) The steady-state 2-D groundwater ¯ow equation
(if steady-state head data are available) and the transient
2-D groundwater ¯ow equation are solved for the current-iteration Y±Z ®eld couple, with given external
stresses, boundary and initial conditions. The resulting
steady-state head solution h0 …x; y† and transient head
solution h…x; y; t† (with x and y being the spatial coordinates and t being time) at measurement locations are
then compared to the measured values. Generally, the

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

steady-state head solution is used as the initial condition
for the transient simulation. If this is not deemed appropriate, 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 G
omez-Hern
andez et al. [3].
(3) An objective function is de®ned that measures the
mismatch between simulated and observed heads:
Jˆ

Nm
X

2

MEAS
ni0 …hSIM
† ‡
i;0 ÿ hi;0

iˆ1

Nt X
Nm
X

2

nij …hSIM
ÿ hMEAS
†;
i;t
i;t

tˆ1 iˆ1

…1†
where the ®rst term corresponds to the discrepancies at
the steady state and the second term to the discrepancies
at the transient state. Nm is the number of measurement
locations, Nt the number of time steps with measurements, 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 weights ni0 and nit are chosen inverse-proportional
to the estimated measurement errors. We say that the Y
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 corresponding T and S ®elds are obtained by taking the antilog of the resulting Y and Z ®elds.
If J is not small enough, a perturbation ®eld DY and a
perturbation ®eld DZ have to be calculated to be added
to the current Y and Z ®elds. These perturbation ®elds
are parameterized as functions of the individual perturbations 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 ˆ
DZij ˆ

Np
X

kkij DYk ‡

Np
X

kˆ1

kˆ1

Np
X

Np
X

kˆ1

tkij DYk ‡

3

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 locations, 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 another. 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 transmissivity 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 of DYk and DZk at the master locations
are obtained by minimizing the objective function J. The
minimization of the objective function is achieved by
non-linear optimization as outlined below.
(4) The gradient vector g containing the derivatives of
J with respect to the 2Np perturbations of Y and Z at the
master locations {oJ/oDYk , oJ/oDZk , k ˆ 1, Np } is determined using the adjoint-state formulation. The adjoint-state approach allows computation of the gradient
vector eciently ± especially when modeling transient
¯ow as compared with the computation using sensitivity
coecients [15].
(5) The updating direction d is 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 ˆ ÿHl gl ‡ alÿ1 dlÿ1 ;

lkij DZk ;
…2†

xkij DZk ;

kˆ1

where Np is the number of master locations, DYk and
DZk the perturbations at the master locations, and the
coecients kkij , lkij , tkij and xkij 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 simpli®es since the weights lkij , tkij will be zero. The master
blocks form an essential part of the methodology because 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

where d is the updating vector (of dimension 2Np ), H the
Hessian matrix containing the second order derivatives
of the objective function with respect to the perturbations, a a step parameter, the value of which is determined di€erently depending on the optimization
algorithm used and l the iteration counter. (Due to the
complexity of the Hessian matrix computation, it is always replaced by an approximate estimation, i.e., for the
conjugate gradient and the steepest descent algorithms,
it is simply set equal to the identity matrix.) As recommended by Carrera and Neuman [15], a more stable and
faster convergence is achieved when the optimization
algorithm rotates among the algorithms mentioned after
a small number of iterations. Once the updating direction is determined, a linear search in that direction is
carried out to determine the step size bl that yields the

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H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

perturbation vector bl dl minimizing the objective function in the updating direction. It is convenient to constrain 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 Y and S ®elds resulting in the updated ®elds for the current iteration:
Yijl ˆ Yijlÿ1 ‡ DYij ;
Zijl ˆ Zijlÿ1 ‡ DZij :
The iteration counter l is increased by one and the algorithm 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 the Y, Z and
h data given in the reports by LaVenue et al. [16] and
Cau€man et al. [17].
A total of seven di€erent scenarios have been analyzed. For each scenario 100 equally likely Y±Z ®eld
couples are generated conditional to Y, Z and h data. 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 interested in analyzing the impact of the spatial variability of
both transmissivity and storativity, and did not want
the results to be also a€ected by error measurements.
For all seven scenarios the same reference Z ®eld has
been used. The seven scenarios di€er with respect to the
reference Y ®eld, the sampling density of the reference
Z ®eld, the sampling density of the reference h ®eld and
the amount, location and pumping rates of the pumping
wells.

3.2. Reference transmissivity ®elds
Two di€erent reference Y ˆ log T ®elds have been
generated. Field 1 is generated by sequential simulation
conditioned to 36 Y measurements. Field 2 is generated
by self-calibration conditioned to 36 Y measurements
and 34 steady-state head measurements. For the generation of ®eld 1, and the seed ®eld necessary by the selfcalibration 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 ®elds Y1 and Y2 are
shown in Fig. 1.
The reference ®elds are sampled at the same 36 locations at which logtransmissivity is reported by LaVenue et al. [16] (See Fig. 2).
3.3. Reference storativity ®eld
Thirteen S data from the WIPP area have been used
to estimate a variogram of Z ˆ log S. An isotropic
spherical variogram with nugget 0.02 (log(m/m))2 , sill of
1.78 (log(m/m))2 and range of 11.8 km has been adopted
for Z. The Z variogram and the Z data have been used
as input to the sequential simulation program GCOSIM3D 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 between measured logtransmissivity and measured logstorativity at the locations in which both parameters were
available. Therefore, the logstorativity and logtransmissivity ®elds are generated independent of each
other.
Three samples are taken from the reference storativity
®eld. Sample 1 contains 13 values at the WIPP measurement locations. Sample 2 contains 30 samples regularly spaced over the site. Sample 3 contains 9 samples,
also regularly spaced. (See Fig. 2.)

3.1. Spatial domain

3.4. Reference hydraulic heads

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 prescribed 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.

Three pumping test scenarios have been considered
di€ering on the number of wells and rates being
pumped. In all cases, the reference transmissivity and
storativity ®elds are used to determine the reference
hydraulic heads that are later sampled to produce hydraulic head measurement data sets. The initial heads
for the transient simulations correspond to the steadystate heads prior to pumping.

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

5

Fig. 1. Reference ®elds. Logtransmissivity reference ®eld Y1 is generated by sequential simulation just conditional to 35 logtransmissivity measurements from the WIPP site; logtransmissivity reference ®eld Y2 is 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.

Fig. 2. Location of the logtransmissivity and logstorativity sample data sets.

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

6

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

Fig. 3. Location of pumping wells (black dots) and head monitoring locations (white dots).

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 a€ect 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 randomly chosen within the site (see Fig. 3). The sampling
frequency is the same as above.
Pumping test Scenario 3. In order to analyze the e€ect
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.

Table 1
The seven scenarios
Scenario

Reference
transmissivity ®eld

Pumping
test scenario

Storativity
data set

1
2
3
4
5
6
7

2
2
1
1
1
1
1

1
1
2
2
2
3
3

1
2
1
2
3
1
2

Reference transmissivity ®eld: 1 ˆ conditioned to transmissivity measurements from WIPP, 2 ˆ conditioned to transmissivity and steadystate head measurements from WIPP. Pumping test scenario: 1 ˆ single
well pumping test with 34 head monitoring locations corresponding to
WIPP sampling locations, 2 ˆ multiple well pumping test with 35
randomly selected head monitoring locations, 3 ˆ multiple well pumping test with pumping rates varying from well to well and 35 head
monitoring locations evenly distributed over the aquifer. Storativity
data set: 1 ˆ 13 sampling locations corresponding to the WIPP sampling locations, 2 ˆ 30 sampling locations on a regular grid, 3 ˆ 9
sampling locations on a regular grid.

3.5. Seven scenarios analyzed
A number of scenarios has been studied to analyze
how the di€erent types of data in¯uence the characterization of the spatial heterogeneity of the logtransmissivity and logstorativity reference ®elds, as well
as the reproduction of the hydraulic head reference
®elds. Seven combinations of the sample data sets described 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 generation 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 (comparison between Scenarios 3 and 6, and between Scenarios 4 and 7),

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

3. the position of the head monitoring points (comparison between Scenarios 3 and 6, and between Scenarios 4 and 7),
4. the number of storativity samples (comparison between 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.

7

4. Results
For each scenario, the self-calibration method is used
to generate 100 Y±Z ®eld couples. No statistical correlation was considered between the two parameters, resulting in a simpler updating of the seed ®elds than if
such a correlation had been considered: the weights lkij
and tkij 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 to Y, Z
and h data are shown together with the steady-state head solution and the head ®eld at the end of the pumping period (the latter two ®elds are the
solution of the ¯ow equation in the Y±Z couple and are also conditional to the h measurements).

8

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

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 Y and Z data is automatic in the

seed ®eld generation step, and posterior updating;
however, conditioning to the h data (in the sense that the
solution of the transient ¯ow equation using the updated
Y±Z couple reproduces, within a preset tolerance limit,
the measured heads) is carried out through an optimization 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.

Fig. 6. Ensemble average of the logtransmissivity ®elds conditional to Y, Z and h data for Scenarios 1 and 2. The reference ®eld Y2 is also shown.

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

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 dicult 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 reference ones is made through the use of the average absolute
error (AAE), and the average ensemble variance (AEV):

AAE…X † ˆ

NNODES
X 

1
X SIM;i ÿ XREF;i ;
NNODES iˆ1

AEV…X † ˆ

NNODES
X
1
r2 ;
NNODES iˆ1 Xi

9

where NNODES is the number of discretization grid
cells, and i is a grid cell index, X represents either logtransmissivity (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,
r2Xi is the ensemble variance of X at a given node. The
smaller these averages are, the better characterized, the
Y, Z and h ®elds are. (Notice that if the above averages
were computed over the conditioning points only, they
would be zero in the Y and Z ®elds, and close to it for
the h ®eld.)

Fig. 7. Ensemble average of the logtransmissivity ®elds conditional to Y, Z and h data for Scenarios 3±7. The reference ®eld Y1 is also shown.

10

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

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 logtransmissivity or logstorativity data, then after conditioning to steady-state head data, and ®nally after
conditioning to transient head data.
Fig. 5 shows the ensemble average Y and Z seed ®elds
for the di€erent scenarios. These ensembles averages are
conditioned only to the Y or Z data, respectively, and
therefore, do not carry any information about the ¯ow
behavior of the formation.
Figs. 6 and 7 show the ensemble average Y ®eld for
all seven scenarios alongside their respective reference
®elds. Recall that the di€erence between the two Y reference ®elds is that Scenarios 1 and 2 use a logtransmissivity conditional to the 35 logtransmissivity
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 delineation of the main patterns of spatial variability existing in
the reference ®elds. It is dicult to distinguish large
di€erences between the ensemble averages in Fig. 6, or
among those in Fig. 7, which indicates that the aspects
di€erentiating 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 conditioning to steady and transient state heads shown in
Figs. 6 and 7. The AAE(Y) displays a noticeable decrease 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 interpreted as an average measure of local uncertainty. All
scenarios that use Y1 as the reference logtransmissivity
®eld display a reduction on AEV(Y) from the seed ®elds
to the updated ®elds after self-calibration to the transient head data. This indicates a reduction in the uncertainty on the prediction of the reference ®eld by the
ensemble of conditional realizations. However, Scenarios 1 and 2 that use Y2 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 between the variogram used for the generation of the seed
®elds (the same for all scenarios and for the generation
of Y1 ) 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
Steady h
Transient h
Y, Z
Steady h
Transient h
Y, Z
Steady h
Transient h
Y, Z
steady h
transient h
Y, Z
steady h
transient h
Y, Z
steady h
transient h
Y, Z
steady h
transient h

0.892
0.745
0.741
0.892
0.745
0.744
0.652
0.586
0.574
0.652
0.586
0.583
0.652
0.586
0.593
0.652
0.588
0.578
0.652
0.588
0.580

1.05
0.85
1.06
1.05
0.85
1.04
0.90
0.60
0.65
0.90
0.60
0.59
0.90
0.60
0.67
0.90
0.59
0.88
0.90
0.59
0.78

0.856
±
0.826
0.563
±
0.541
0.856
±
0.808
0.563
±
0.525
0.752
±
0.759
0.856
±
0.762
0.563
±
0.527

1.19
±
0.79
0.55
±
0.44
1.19
±
0.84
0.55
±
0.45
2.68
±
1.05
1.19
±
0.96
0.55
±
0.57

2

3

4

5

6

7

AAE is the average absolute error and AEV is the average ensemble variance, both are computed cell by cell through the ensemble of 100 realizations.
To appreciate the impact that conditioning to heads has in the characterization process, both AAE and AEV are given for the seed ®elds (just
conditioned to Y and Z data), for the updated ®elds conditioned to steady-state head data and for the ®nal updated ®elds conditional also to
transient heads.

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

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 average Z ®eld for all seven
scenarios alongside the reference ®eld. These ®elds
should be compared to those in Fig. 5 to notice the impact 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-

11

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
a€ect 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 km2 a€ected by the test.

Fig. 8. Ensemble average of the logstorativity ®elds conditional to Y, Z and h data for the seven scenarios. The reference logstorativity ®eld is also
shown.

12

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

Table 3
Hydraulic head characterization measures
Scenario

Time (days)

AAE initial (m)

AAE updated steady (m)

AAE updated transient (m)

1

0
17
35
0
17
35
0
17
35
0
17
35
0
17
35
0
17
35
0
17
35

2.22
3.60
4.52
1.98
2.14
2.26
1.04
3.81
5.56
1.03
2.58
3.35
1.03
3.94
5.29
1.03
183.0
321.2
1.03
104.5
193.9

0.60
1.61
2.12
0.60
0.73
0.83
0.55
3.61
5.40
0.54
2.30
3.10
0.55
3.58
4.98
0.55
202.9
360.0
0.55
103.1
193.0

0.61
0.69
0.76
0.61
0.68
0.75
0.59
3.39
5.20
0.62
2.15
2.90
0.68
2.90
3.75
0.58
11.9
22.8
0.57
7.46
13.0

2

3

4

5

6

7

The average absolute error computed cell by cell through the ensemble of 100 realizations is determined for the seven scenarios: at steady-state prior
to pumping (time 0), at 17 days after pumping started and at the end of the pumping period (35 days). To appreciate the e€ect of conditioning to the
head data, the AAE is computed in the seed ®elds just conditioned to Y and Z data (AAE initial), for the updated ®elds conditioned to steady-state
head data (AAE updated steady) and for the ®nal updated ®elds conditional also to transient heads (AAE updated transient).

The sampling pattern and frequency of Z data play
important roles in the reduction of the AAE(Z). The
reduction is larger for the scenarios with 30 logstorativity 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 e€ect of the conditioning to transient head than the reduction of AAE(Z) is the reduction of the AEV(Z). The AEV(Z), interpreted as a
measure of the degree of local uncertainty in the estimates 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 di€erent scenarios. It appears as if using multiple
pumping wells does not help in improving the characterization of the main patterns of spatial variability of Y
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 characterization of the main patterns of Y spatial variability. Ad-

ditional conditioning to transient head data only
produces local updates of the Y and 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 reference ®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 di€erent types of head data helps in
reproducing the heads everywhere within the simulation
domain.
In all cases, as expected, the AAE(h) is always reduced 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 prediction of the transient-state reference ®elds. Further
conditioning to transient data drastically reduces the
departure between measured and predicted transient
head values. This e€ect is particularly noticeable for
Scenarios 6 and 7 in which the head drawdowns are very

H.-J.H. Franssen et al. / Advances in Water Resources 23 (1999) 1±13

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 logtransmissivity and logstorativity are generated using cosimulation 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 different 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 measurement errors or uncertainty on the boundary conditions
or the variogram in order to focus on the worth of the
di€erent 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 di€erent types
of data with regard to the characterization of the different parameters and variables involved. This paper
does not present a systematic analysis of these trade-o€s,
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 variables of interest.

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