Usually, subsoil data for groundwater models are generated from borehole data, using upscaling techniques. Since the assumed hydraulic properties for litho-classes in boreholes are uncertain, and upscaling may add inaccuracies, the groundwater model has to be calibrated. In this paper, a method is presented that uses a calibrated groundwater model to improve the quality of a hydrogeological model (layer thickness and hydraulic properties) as obtained from borehole data. To achieve this, all borehole data are defined by random variables and related to aquifer and aquitard properties at the same support as the groundwater model, using complete probability density functions. Subsequently, the calibrated parameter values of the groundwater model are assumed to be the truth and are used to find the most likely combination of layer thicknesses and hydraulic conductivities for the lithological layers making up the aquifer or aquitard. The presented example is an application of the proposed method to aquitards. Nevertheless, the method can be applied to aquifers as well. The analysis of the results gives rise to the discussion about the correctness of the hydrogeological interpretation of the borehole data as well as the correctness of the calibration results of the groundwater flow model. In order to make the problem tractable, computationally feasible, and avoid assumptions about the distribution form, piecewise linear probability density functions are used, instead of parametrized functions.
A clear understanding of the subsoil is important when building
hydrogeological and groundwater models. Because the subsoil is
generally sparsely observed, only limited information is available
about its properties. Building a hydrogeological model of the subsoil
is therefore a challenging task with many uncertainties.
Identification of deposits with distinct hydraulic properties, at
observed and unobserved locations, is vital for the creation of such
a model
The aim of this study is to find the most likely (hydro)geological representations of the subsoil by using results from a calibrated groundwater model. These improvements include the litho-layer thickness and conductivity for each model grid cell.
In this study, a readily calibrated groundwater model, namely the
Azure groundwater model
With the proposed method, effects of prior assumptions concerning the litho-layers can be analyzed, and discrepancies between the geological and hydrological interpretation of the subsoil become clear. These discrepancies show up as very unlikely values of the litho-class thickness or conductivities, or unlikely combinations of these parameter values. Herewith, our method not only yields the most likely litho-layer properties, it may also serve as a communication tool between the geologist and the groundwater modeler as well.
This paper is organized as follows. In Sect.
The proposed method is applicable to the vertical resistance of an aquitard as well as to the transmissivity of an aquifer. In this paper, we focus on the vertical resistance of aquitards.
The vertical resistance of a litho-layer can be derived from
observations of the layer thickness and the vertical conductivity of
the deposits. These observations always yield uncertain parameter
values and they might not be representative for the required model
scale. In the REGIS information system
Since the resistance depends on the vertical conductivity and the
thickness of the litho-layer, we need the PDF of both. The PDFs of
the vertical conductivity are available from earlier studies and, in
the Netherlands, supplied by the REGIS information system. The full
PDF of the thickness can not be obtained from existing data bases. In
practice, often only the mean or mode of the distributions is used to
represent the layer thickness. In our method we need the full PDF. In
Sect.
Let the value of the vertical resistance of an aquitard at grid block
The vertical resistance of a litho-layer is calculated as
Because of this piecewise linearity, no analytical solution is
available to find the most likely marginal layer thicknesses
(
With this method, a multidimensional joint PDF can successively be evaluated to find the ML values of all marginal distributions.
The update algorithm, as described in the former section, needs for
each litho-class the PDFs of the layer thickness and the conductivity
at each grid cell. In the information system REGIS, a PDF of the
hydraulic conductivity is assigned to each litho-class, independent of
the spatial coordinates. So everywhere in the subsurface where
a particular litho-class exists, the probability distribution of the
hydraulic conductivity is assumed to be known. Therefore, only the
PDFs of the layer thickness for each litho-class have to be spatially
predicted. This spatial prediction is performed by using ordinary
kriging (OK). For every litho-class in the study area,
a semi-variogram model for the litho-layer thickness is estimated.
Since layer thicknesses are greater than or equal to zero, the
interpolation method must account for this
Using kriging interpolation, the estimation of the interpolated
thickness
Ordinary kriging (OK) tends to generate negative weight factors,
beside the positive ones, when the spatial distribution of the
observations is somehow unbalanced around the estimation location,
known as the screen effect. Apart from the physical meaning of
negative weight factors, this influences the interpolated result
Subsequently, the PDF of the vertical hydraulic resistance is obtained
by division of the PDF of the interpolated litho-layer thickness by
the PDF of the vertical hydraulic conductivity (Eq.
The described method needs complete borehole descriptions for a model-layer (aquifer or aquitard) at all borehole locations. Therefore, when a description is incomplete for a borehole, this borehole is neglected for that layer.
Within the extent of the study area and within the considered
model-layer, a limited set of litho-classes is found. Because of
heterogeneity of the subsurface, not every litho-class is present at
every borehole location. However, the absence of a litho-class in
a certain borehole is an observation as well. Therefore, when
a litho-class is absent in a borehole, it is added with a zero layer
thickness. The assignment of the variance to the layer thickness is
described in Sect.
A litho-class may appear multiple times within one model-layer in one borehole. The thicknesses and variances of all these occurrences are added to one thickness and variance before further processing. Consequently, the horizontal connectivity of individual litho-layers of a litho-class between boreholes is neglected.
The method presented in the former sections needs a quantification of the uncertainty of litho-layer thicknesses. However, no quantitative data about this uncertainty are available. In this section, a method is described to provide all litho-layers of the borehole descriptions with an appropriate uncertainty.
During drilling of a borehole, the measured layer thicknesses are always rounded off. This causes uncertainty in the layer thickness. The magnitude of the round-off error depends, for instance, on the drilling method and the way the borehole descriptions are made. Therefore, it is likely that drilling methods which can distinguish the layers more accurately have a smaller round-off error than drilling methods with a lower accuracy. Reversing this reasoning it may be concluded that small round-off values give a more accurate layer thickness than large round-off values. The question is how to recognize the order of magnitude of round-off error in the borehole description data.
From the REGIS data base, about 475 000 litho-layer thicknesses of
about 16 000 borehole descriptions are available. The remainder of
all these thicknesses, when dividing by one meter, is calculated and
shown as a cumulative distribution in Fig.
Since no quantitative information is available about the uncertainty
of the litho-layer thicknesses, an arbitrary choice has to be made. To
justify the choice, a sensitivity analysis is carried out to test the
performance of different options. These options include two types of
distributions, and the magnitude of the variance. When the type of
distribution is unknown, the normal distribution is usually a safe
choice, because the round-off errors may be assumed symmetric around
zero. Since the normal distribution may yield negative layer
thicknesses, the log-normal distribution is tested as well. The
standard deviation for each litho-layer is linearly related to its
truncation class value using a fixed factor for each standard
deviation class (Table
When a litho-class is observed absent in a borehole, it should get
a thickness of 0
The location of the study area is shown in
Fig.
The calibrated layer properties (transmissivity and resistance values)
originate from the Azure groundwater model, developed by Deltares, the
Netherlands
In this paper, we focus on the fourth aquitard in the Azure
groundwater model. This aquitard is a high vertical resistance clay
patch, surrounded by an area where the clay layer is thin or
absent. This aquitard is found between 20 and 85
The aquitard consists of seven different litho-classes. The hydraulic
properties of these litho-classes, as defined in REGIS, are shown in
Table
Table
The calibrated vertical resistance from the groundwater model is
divided over the seven litho-classes, according to the method
described in Sect.
Figure
The ML method yields for each litho-class for each grid cell the most
likely resistance, thickness and conductivity values. The position of
these ML values in their prior probability distribution can be
indicated by the corresponding cumulative probability value. These
cumulative probabilities of the litho-layer thickness, the vertical
resistance, and the hydraulic conductivity of litho-class EE-k are
depicted in Fig.
The uncertainties of the observations of the litho-layer thickness and
the vertical conductance are all represented by PDFs. In
Fig.
The prior distributions of the litho-class conductivities, i.e. the
distributions obtained from the REGIS system, represent the best
estimates given the available hydrogeological knowledge. One goal of
the ML method is to improve these distributions. The CDFs of the
conductivity values of litho-class EE-k and EE-kz, as discussed in the
former section, are shown in Fig.
In the study area, the majority of the calibrated
The results are based on a small study area and can currently not be extrapolated to the whole REGIS database.
The observed litho-layer thicknesses of the available borehole data
are expected to be uncertain, but the variance and distribution type
are unknown. However, the proposed method needs probability
distributions of these observations. As stated in
Sect.
For a given litho-layer the observations of the layer thickness fall
into two groups: one with the observed litho-classes and one group
with the litho-classes observed absent. These groups are denoted as
observed-thickness and zero-thickness, respectively. Two
characteristics of the PDFs are important when judging the usability,
the probability of negative values and the width of the distribution.
We defined the latter as the width of the 95 % probability
interval, which is the distance between the 2.5 and the 97.5 %
quantiles. In Table
Interpolation of the litho-layer thicknesses are performed using the
settings as described above. The normal distribution often yields
negative ML thicknesses (column percentile
The lower-left and the upper-right corner of the study area are dominated by zero-thickness observations. The variances shown in these areas of the low (upper row) and medium variance (middle row) are low compared to areas dominated by the observed-thickness locations. Therefore, the high variance settings (lower row) for the zero-thickness observations are used for further ML analysis. Corollary, for further calculations the medium variance for the observed-thickness locations is used and the high variance for the zero-thickness locations.
Depending on the nature of the observed data, and the associated assumptions of the underlying random field model, the appropriate form of data-transformation and kriging is chosen before performing kriging interpolation. Herewith, a decision has to be made whether or not to perform a data transformation. Hereafter, the decisions made are justified.
Layer thicknesses are, obviously, required to be greater than or equal to zero. Therefore, not every PDF is appropriate to describe the uncertain thickness. In the kriging interpolation with uncertain observations, two parameters need to be assigned a probability density function: the observations of the layer thickness (former section), and the interpolation error. Usually, the interpolation error is assumed to be normal distributed. No accurate information is available about the true shape of these PDFs. Therefore, the performance of the use of normal and log-normal distributions was tested. Both distributions have their own deficiency, especially when the standard deviation is large compared to the mean value. In that case, the normal distributions may yield negative thicknesses with too high probability, and the log-normal distribution may become very skewed. The latter is a disadvantage in finding representative ML values because of the difference between the mode and the mean of the distribution. Because of the potential negative values of the normal distribution, the log-normal distribution is tested for the interpolation error as well.
One way to avoid negative interpolated values is to transform the
observations to their log values before interpolation, and
back-transform them afterwards. Applied to block-kriging, the
different way the block average is calculated has to be
considered. When kriging the log-transformed values, the block average
is the geometric mean, kriging the non-transformed values yields the
arithmetic mean. In Fig.
The implementation of the maximum likelihood (ML) method, in conjunction with kriging interpolation, appears useful in updating hydrogeological information from borehole data, with information derived from calibrated groundwater models. From the uncertain hydrogeological data, described by a multidimensional probability density function, the most likely parameter values are derived given the information available from calibrated parameter values in groundwater models. The ML method is applied to layer thicknesses and vertical conductivities at litho-class support. Herewith, the most likely litho-layer thickness and vertical conductivity values are obtained for the studied aquitard.
In the REGIS database the a priori probability distributions of the vertical conductivity, for a given litho-class, are assumed location independent. The posterior distributions of the two most important litho-classes show much less variability than the corresponding prior distributions do. This can be expected as additional knowledge is added using results from a calibrated groundwater model. However, at this point this is yet not a reason to update the prior distribution in the REGIS database, because the posterior distribution is based on a small study area, while the prior distribution is based on data from the whole data base. Moreover, it is not unlikely that the a priori distribution of the vertical conductivity of a litho-class is spatially varying. Subsequent application of this method to a larger area will give more certainty about this.
The values of some parameters, obtained by the ML method, show a strong systematic deviation from the prior distribution, with the majority of the values either lower or higher than the median of the prior distribution. In case of a data update the posterior distribution should of course divert from the prior distribution, but a strong systematic deviation may indicate errors, either caused by data errors or a wrong perception about the hydrological system. The proposed method can thus serve as a tool to guide the discussion between experts from different domains.
With the described method, the ML values of the PDFs are derived for each layer separately, neglecting the thickness of adjacent layers. Obviously, it is not possible to change the thickness of a layer without affecting the thickness of the adjacent layers. The present study does not take this into account and only aims to describe a method to find the most likely combination of layer thickness and conductivity. A future study should account for all layers of the hydrogeological model, where the sum of all model-layer thicknesses is constrained and, preferably, described by RVs.
The proposed method assigns a PDF to the thickness of every single
litho-layer from the borehole descriptions. In
Sect.
As with the assignment of litho-classes, also the calibrated vertical resistance of the groundwater model is regarded as perfectly known. A valuable extension to the presented method is to account for uncertainty of the calibration results.
The use of piecewise linear PDFs, instead of parametrized PDFs, makes it possible to perform the necessary calculations without the burden of deriving intractable analytical solutions or resort to time-consuming Monte Carlo analysis. Herewith, many different calculations can be tested with relatively little effort.
This section contains the derivations of calculating the maximum likelihood (ML) of the result of an elementary operation. All random variables (RV) are described by piecewise linear probability density functions (PDF).
Hereafter,
Applying elementary operations,
In the next sections this method is applied to four elementary operations.
Let
Let
Let
Let
Three classes of standard deviation related to the truncation classes.
Probability data of the vertical hydraulic conductivity values of each litho-class. The distributions are defined by the 2.5 and 97.5 % percentile values and are assumed to be log-normal. The presented mean and SD are derived from the PDFs.
Variogram model for thickness of each litho-class.
Effect of SD of the observed-thicknesses distributions. The mean value of 1
Effect of the SD of the zero-thickness distributions.
Connection of
Cumulative distribution of the remainder of about 475 000 litho-layer thicknesses after division by one meter. The vertical lines show the position of the round-off values at every ten centimeters.
Study area. The gray area is the extent of the Azure
groundwater model. The small rectangle denotes the study area with
the vertical resistance, as shown in Fig.
The
Most likely vertical resistance. The squares denote observations where the litho-class is present, the plus signs where it is absent.
Most likely thickness of each litho-class. The circles denote observations where the litho-class is present, the plus signs where it is absent. The circles are colored with the observed thickness.
Thickness of litho-class EE-k (top) and EE-kz (bottom). Mean kriging thickness (left) compared to ML thickness (right).
Cumulative probability of the ML values of the uncalibrated
(top) and calibrated (bottom) parameters of litho-class EE-k. The
data is clipped at 0.05
Cumulative probability of the ML values of the uncalibrated
(top) and calibrated (bottom) parameters of litho-class EE-kz. The
data is clipped at 0.05
Comparison of conductivity distributions of litho-class
Maps of interpolation variances of litho-class EE-kz for
different settings of the observation variances. The used settings
for the variances are shown in Tables
Mean value of interpolated layer thickness of litho-class
EE-k. Shown is