When modelling contamination transport in the subsurface
and aquifers, it is crucial to assess the heterogeneities of the porous
medium, including the vertical distribution of the aquifer parameter. This
issue is generally addressed thanks to geophysical investigations.
As an alternative, a method is proposed using estimated hydraulic parameters
from a 2D calibrated flow model (solely reliant on piezometric series) as
parametrization constraints for a 3D hydrogeological model. The methodology
is tested via a synthetic model, ensuring full knowledge and control of its
structure. The synthetic aquifer is composed of five lithofacies,
distributed according to a sedimentary pattern, and functions in an
unconfined regime. The level of heterogeneity for hydraulic conductivity
spans 3 orders of magnitude. It provides the piezometric chronicles used
to inverse 2D flow parameter fields and the lithological logs used to
interpolate the 3D lithological model. Finally, the parameters of each
facies (hydraulic conductivity and porosity) are obtained through an
optimization loop, which minimizes the difference between the 2D calibrated
transmissivity and the transmissivity computed with the estimated 3D facies
parameters.
The method estimates values close to the known parameters, even with sparse
piezometric and lithological data sampling. The maximal discrepancy is
45 % of the known value for the hydraulic conductivity and 6 % for the
porosity (mean error 26 % and 3 %, respectively). Although the
methodology does not prevent interpolation errors, it succeeds in
reconstructing flow and transport dynamics close to the control data. Due to
the inherent limitations of the 2D inversion approach, the method only
applies to the saturated zone at this point.
Introduction
To simulate contamination transport in the subsurface and aquifers, it is
crucial to assess and reliably describe the heterogeneities of the porous
medium. The development of inverse methods in recent decades is mainly based
on two-dimensional flow models and focused on the horizontal structure of
heterogeneities with the collection of piezometric data as a cornerstone
(Poeter and Hill, 1997; Carrera et al., 2005; Hendricks Franssen et al.,
2009). But the latter is less sensitive to the vertical structure of the
aquifer, leaving its estimation dependent on complex and expensive field
methods, for example, pumping tests (De Caro et al., 2020), tracer tests (Linde et
al., 2006), electrical resistivity (Coscia et al., 2011; Priyanka and Mohan
Kumar, 2019), radar tomography (Boni et al., 2020), self-potential methods
(Eppelbaum, 2021), crosshole testing (Klotzsche et al., 2013; Doetsch et
al., 2010), hydraulic tomography (Sanchez-León et al., 2015; Luo et al.,
2020; Fischer et al., 2020), and/or laboratory analysis, for example grain-size
analysis from core samples (Marini et al., 2018) and ex situ permeability
tests (Zhang and Brusseau, 2005). The collection of this information,
describing the vertical heterogeneity of the aquifer, allows for the development
of 3D inversion techniques. For example, some successful methods combine
direct parameter quantification and stochastic geological modelling
(Guadagnini et al., 2004; Fu and Gómez-Hernández, 2008; Cardiff
and Kitanidis, 2009), and others incorporate water head data and more advanced
geophysical measurements to the (joint) inversion procedure (Straface et
al., 2011; Lee and Kitanidis, 2014).
Hydrogeological models are usually two-dimensional, and transmissivities are
estimated through model calibration. Two-dimensional models are easier to handle
considering fieldwork, parameter measurements, data manipulation, and
calibration than 3D models. We question here the possibility of transferring
data from a 2D calibrated hydrogeological model to a 3D configuration.
Viaroli et al. (2019) recently employed a simplified 2D model to specify the
boundary conditions and recharge of a 3D model already designed by other
means. Incidentally, the design and parametrization of the 3D model itself
are even more seldom independent of geophysical methods. In this line, we
propose an original method using data from a 2D calibrated flow model
(solely reliant on piezometric time series) as parametrization constraints
for a 3D hydrogeological model resulting from interpolation of borehole
data. The use of 2D calibrated transmissivities allows our technique to be
completely unrelated to geophysical methods and less heavily computational
than 3D joint inversion approaches. Moreover, the articulation of the method
also allows a pre-existing 2D calibrated model to be taken advantage of, if any.
The method is tested on a synthetic test case constituted by five
hydrofacies, distributed according to a sedimentary pattern, with a level of
heterogeneity for hydraulic conductivity spanning 3 orders of magnitude.
This work can be considered an improvement of the method proposed by Harp
et al. (2008), who also tested the combination of 2D inversion and an
interpolation method but on a two-dimensional transect model composed of
only two facies.
In order to evaluate our methodology's robustness, it is first carried out
with a very profuse data sampling (piezometric for the 2D inversion and
lithological for the 3D model interpolation), assessing the consistency
between the different numerical codes. Second, a sparser sampling is tested
to approximate more realistic field conditions.
The detail of the methodology is described in Sect. 2, including the
synthetic data framework, the mathematical background of the tools used, and
the link between them. The results for both samplings concerning the
inversions, the facies interpolation, and the final model outputs (in terms
of parameter, piezometric series, and contamination plumes) are discussed in
Sect. 3.
Material and methods
The methodology we propose and analyse in this paper is the following:
We propose estimates of transmissivity and porosity from a 2D calibrated flow model
based on piezometric heads. These transmissivities exist for each
element/cell of the 2D mesh. It is a huge dataset constrained by the
piezometric heads.
We analyse the aquifer lithology at boreholes. Lithology is usually
described at each borehole. It provides a qualitative description of aquifer
heterogeneity. This qualitative description can be interpreted in terms of
facies. This description is used here to define an optimal number of facies
that have been identified within the aquifer.
The 3D discretization of the aquifer is a vertical extension of the 2D model,
which avoids interpolation of the 2D parameters. Facies are interpolated
over the 3D domain based on the borehole local data.
The hydraulic conductivity and porosity for each facies are estimated
through optimization using the 2D data, which are considered to be vertical
integrations of the 3D data. Optimization is required because the number of
unknowns is quite small (twice the number of facies) compared to the number
of constraints (twice the number of elements of the 2D flow model at the
most). Of course, the constraints are correlated through the flow model and
cannot be considered independent.
To evaluate this approach, we built a synthetic test case (Fig. 1) generated
by the following:
a 3D aquifer design;
computation of the 3D flow using the software TRACES (Hoteit and Ackerer,
2004);
selection of representative head data, used as constraints for the 2D
inversion after vertical averaging;
estimation of transmissivities and mean vertical porosity by a 2D flow model
calibration based on the selected head data using PINOGRI (Rambourg et al.,
2020);
selection of representative boreholes for lithological data and facies
definition;
design of the 3D facies distribution using a geostatistical interpolator
(GemPy; de la Varga et al., 2019) or a deterministic interpolator (splines; Lee et al., 1997);
estimation of each facies' hydrodynamic parameters (hydraulic conductivity,
porosity) using an optimization procedure constrained by the 2D calibrated
values;
comparison of local hydrodynamic parameters, simulated water heads, and
concentrations between the “true” aquifer and the reconstructed
(estimated) aquifer.
Methodology flowchart – TRACES: Transport of Radioactive Elements
in Subsurface (Hoteit and Ackerer, 2004); PINOGRI: Parameter Inversion
Numerically Optimized for Groundwater Issues (Rambourg et al., 2020); GEMPY:
Open-source 3D geological modelling (de la Varga et al., 2019); SPLINES:
QGIS/SAGA multilevel B-spline interpolation (Lee et al., 1997).
The computations are run on a PC with Intel(R) Core(TM) i7-6700 CPU @
3.40 GHz processor and 16 GB RAM.
Synthetic three-dimensional datasetThe aquifer model
The synthetic aquifer consists of five hydrogeological facies (also referred
to as hydrofacies) distributed along a sedimentary pattern over a 10×10 km
area and 20 m depth (Fig. 2).
Each hydrofacies is characterized by a hydraulic conductivity 5 times
higher than the underlying facies (Table 1). Their porosity is less
heterogeneous as it is defined in the range of permeable sedimentary
materials (10 %–30 %). In practice, a hydrofacies is defined by
clustering lithofacies with comparable hydrodynamic properties. The
limitations and pitfalls inherent in this step are not addressed in this
study, which is assumed to be flawless.
Three-dimensional facies distribution of the synthetic aquifer and
aquifer discretization – the black dot is the location of the pumping well and
the red dots the location of the contaminant sources.
Hydraulic boundary conditions are of null-Neumann type (no flux) except the
northwest corner where a 21 m head is imposed (Dirichlet boundary), acting
as the outlet of the aquifer. A constant pumping well (18 m3 h-1)
is positioned in the southeast part of the model, intercepting the whole
thickness. For solute transport, a zero solute flux is prescribed at the
boundary, except at the aquifer outlet, where the outflow is considered
purely advective.
1 Proportion of facies at the scale of the whole domain. 2 Proportion of facies within the surface elements.
The aquifer is exclusively fed by rainfall (720 mm yr-1 on average),
which is assumed homogeneous over the whole area. However, five different
recharge patterns are imposed according to the most superficial facies,
whose hydrodynamic parameters greatly influence the amount and dynamics of
water infiltration. Thus, recharge zone 5 (formed by the most permeable
surface facies) is subject to major and fast infiltration, in contrast to
zone 1 (least permeable), where the seepage signal is very attenuated and
spread over time (see Sect. 3).
Piezometric and contamination reference data generation
The behaviour of groundwater and dissolved contamination is computed using
TRACES (Transport of Radioactive Elements in Subsurface) software (Hoteit
and Ackerer, 2004), a numerical code written in FORTRAN 90 for the
simulation of flow and reactive transport in saturated/unsaturated porous
media.
The three-dimensional flow model is the combination of the conservation of
mass and Darcy's laws, generalized to also apply to the unsaturated zone
(Darcy–Buckingham law), resulting in the Jacob–Richards equation (Eq. 1):
∂θ∂t+sθϕ∂h∂t-∇⋅(K∇h)=f,
where θ and ϕ are the water content [–] and porosity [–],
respectively, necessary to deal with the unsaturated zone. s and
K are the specific storage coefficient [m-1] and
hydraulic conductivity tensor [m s-1], respectively. h is the water
head [m], and f is the sink–source term [s-1].
To limit inconsistencies with the 2D inversion (where unsaturated flow is
not addressed via a physical model), the 3D model is reduced to a fully
saturated approach. Therefore, the flow equation (Eq. 1) is simplified and shown
with the adequate initial and boundary conditions as Eq. (2).
S∂h∂t-∇⋅T∇h=Fhx,0=h0xxϵΩhx,t=hDx,txϵΓDtϵ0,TT∇hx,t⋅n=qNx,txϵΓNtϵ0,T,
where S is the storativity [–], equivalent to the
effective porosity in an unconfined context. T is the
transmissivity [m2 s-1], the integration of the hydraulic
conductivity over the vertical of the model. F [m s-1] is the
sink–source term, x is a position in Ω,
the model domain, and h0x represents
the initial conditions. ΓD and ΓN
are partitions of the domain boundaries that correspond to Dirichlet and
Neumann conditions, respectively, and n is the unit
vector normal to the boundary, counted positive outward. hDx,t is the prescribed head value at the Dirichlet
boundaries, and qNx,t is the
prescribed flux at the Neumann boundaries, both defined at each time t of
the simulated period T.
The assumption of a locally constant transmissivity is satisfied, with water
head variations of a maximum of 5.2 % (and 3.5 % on average) of the
local mean water head.
TRACES addresses the migration of contaminants via an
dispersion–diffusion equation, supporting adsorption,
precipitation, and degradation (reactive transport) phenomena. However, the
study considers one inert species, giving form to Eq. (3).
∂θC∂t-∇⋅θD∇C+qC=Qq=-K∇hCx,0=C0xxϵΩ-D∇C⋅nAt+BtC=qttϵ0,T,
where C is the solute concentration [kg m-3], D
is the dispersion–diffusion tensor [m2 s-1], q
is Darcy's velocity [m s-1], and Q is the solute sink–source term
[kg m-3 s-1]. C0x is the
initial concentration; and At, Bt, and
qt are the parameters to define the boundary conditions
(see Hoteit and Ackerer, 2004).
The dispersion and diffusion parameters are set identically for all the
facies (Table 2).
These parameters are transferred into the dispersion–diffusion tensor as
shown in Eq. (4).
θD=θDm⋅τ+DTqδij+DL-DTqiqjqwithτ=θ7/3/ϕ2,
where Dm is the molecular diffusion coefficient [m2 s-1], and
τ is the tortuosity factor of the porous medium [–] (according to
Millington and Quirk, 1961). DT and DL are the transversal and
longitudinal dispersivities [m], respectively, while δ is the
Kronecker function, with i and j the position indexes in the tensors.
Flow and transport equations implemented in the code TRACES are solved under
transient or steady-state computation in 2D or 3D heterogeneous domains.
Mixed hybrid finite elements are used to solve the flow equation and the
diffusive–dispersive components of the transport. A mass lumping formulation
is used to limit the occurrence of numerical oscillations. The advective
part of the transport is solved using a discontinuous Galerkin finite element method,
which also prevents numerical oscillations in the simulations and strongly
limits numerical diffusion. These numerical schemes ensure an exact mass
balance at the element level and are very flexible in space discretization
(triangular or quadrangular element in 2D, tetrahedral, prismatic of
hexahedral elements in 3D).
As shown in Fig. 2, the model consists of 10 440 triangular prisms, 6193 nodes, and 27 544 facies. The horizontal edges have a characteristic length
of 500 m, while the vertical edges are 2 m long.
The initial state of the water table is derived from a preliminary
steady-state calculation involving averaged recharge. The aquifer is
initially uncontaminated (C0=0 over the whole domain) and undergoes a
pollution episode from three surface sources (Fig. 2), each discharging 0.1 g s-1 for the first 24 h of the simulated time.
Structural and piezometric reference data sampling strategies
Structural (hydrofacies) and piezometric data are sampled following two
subsequent strategies in order to validate the method (Fig. 3).
First, the methodology is conducted with a very dense dataset (400 control
points) to assess its potential under ideal conditions and verify the
numerical approaches' compatibility. The piezometric chronicles used for the
inversion cover the 9 years of simulation (see Sect. 3).
Second, a sparser dataset is extracted to evaluate the method in more
realistic conditions. The control points are reduced to 40, and the
piezometric chronicles are shortened randomly (down to 2 years). Meanwhile,
the hydrofacies logs extracted at the location of the control points are
kept in their integrity.
Transmissivity map of the aquifer and location of the wells for
sparse (black dots) and dense (white dots) samplings.
The sparse sampling is pseudo-random so that each recharge area is covered.
The points at each corner of the model are included in both sampling
strategies to avoid extrapolation issues. The sparse sampling accounts for
10 % of the lithological information of the dense dataset and only 6 %
of the water head data.
Two-dimensional flow model inversion
The sampled piezometric data are used as inversion constraints for the
PINOGRI (Parameter Inversion Numerically Optimized for Groundwater Issues;
see Rambourg et al., 2020) software, developed at ITES (Strasbourg). As water
heads are less sensitive to the vertical heterogeneities of the porous
media, the inversion approach is restricted to a two-dimensional scale,
where heads are vertically averaged. This step results in the estimation of
transmissivity and average porosity fields at the scale of each mesh of the
model. The inversion procedure consists of minimizing an objective function
(the quadratic difference between measured and computed piezometric heads)
with parameter optimization guided by a gradient descent method.
Although piezometric data are subject to uncertainty in a field context, we
do not address this aspect in the present study, and the water heads'
measurement errors are considered negligible.
The flow model
Two-dimensional groundwater flow in the aquifer is described by a
diffusion-type equation, akin to the TRACES approach, but with a constant
head over depth assumption (Dupuit–Forchheimer's hypothesis), reducing the
problem's dimension.
The mathematical model is solved by a two-dimensional nonconforming finite
element method (Crouzeix and Raviart, 1973), ensuring flux continuity, mass
balance (like the finite volume method), flexibility in geometry, and
rigorous computation of full tensor transmissivity (like conforming finite
elements, as stated by Ackerer et al., 2014). The time discretization scheme
is implicit, giving the direct problem the form of Eq. (5).
Aht=Ft-1,
where h is the water head vector at the calculation time t, and F is
the sink–source term vector produced at the previous time step.
A is the flow coefficient matrix, depending on the mesh
geometry and the parameter vector.
The inverse problem
The groundwater flow parameters are estimated through the minimization of an
objective function (Eq. 6) based on weighted least squares (Carrera and
Neuman, 1986; Tarantola, 2005).
JP=hP-h∗TW-1hP-h∗,
where J is the objective function, P represents the
vector of the parameters to be estimated, h∗ is the measured
piezometric head (obtained from vertical averaging of 3D sampled data), and
h is the corresponding simulated values. T is the transpose
operator, and W is the weighting matrix, depending on
measurement errors, able to prioritize optimization effort on specific
locations. Therefore, in this study, all data are weighted equally.
Moreover, because no a priori hydraulic parameters information is added in
the study, the objective function does not include the plausibility
criterion of the maximum likelihood approach.
Due to the great number of parameters and measurements, the minimization of
the objective function is led by a quasi-Newton method, which is less
time-consuming compared to Gauss–Newton and other Jacobian-based approaches
(Kitanidis and Lane, 1985). The gradient and an approximate Hessian of the
objective function are calculated using the discrete adjoint state method
(Carter et al., 1974) and the limited-memory BFGS
(Broyden–Fletcher–Goldfarb–Shanno) algorithm (Byrd et al., 1995),
respectively. In our case, the adjoint state method is used to compute the
gradient of the objective function (required by the L-BFGS algorithm) as an
optimization problem of a Lagrangian, constrained by the head values
obtained from the direct calculation. On the other hand, instead of
calculating the sensitivity coefficients for each parameter at each
iteration required by Newton methods, the L-BFGS algorithm (quasi-Newton)
approaches a Hessian approximation by converging an initial matrix (e.g. the
identity matrix) according to the results from a limited number of previous
iterations. As the parametrization of the inversion can lead to a high
number of degrees of freedom, this set of techniques has been found more
efficient than standard sensitivity approaches (Townley and Wilson, 1985).
Finally, three stopping criteria are set to end the algorithm: (i) the
objective function J or its gradient is sufficiently low, (ii) the
adjustment of the parameters P or the decrease of J
between two iterations is too small, or (iii) the number of iterations has
reached a user-set maximum value.
Incidentally, the inverse problems generally suffer from being ill-posed;
i.e. the number of data (locally known piezometry) is too low compared to
the number of unknowns (hydraulic conductivity and porosity at the scale of
each mesh element). This leads to issues of non-uniqueness and instability
of solutions. One way to limit these inconveniences is to reduce the number
of unknowns via a parametrization technique. In PINOGRI, the parameter
spatial pattern is inferred using an adaptive multiscale triangulation (AMT; Majdalani and Ackerer, 2011; Hassane and Ackerer, 2017). The parameters,
borne by the vertices of the AMT mesh, are interpolated into each element of
the calculus mesh (see Fig. 4). If during the inversion process, the
minimization criteria are not met at the scale of a parameter cell, the
latter is divided into four, increasing the optimization's degrees of
freedom. Refinement is halted either when the objective function at the
element level drops below a user-defined threshold, when the number of
iterations reaches a user-defined maximum, or when the last iteration fails
to produce a better optimization than the previous one. This adaptive
approach allows for more flexibility and needs fewer preconceptions about the
model structure than fixed parametrizations, such as zonation or
interpolations. A detailed description of the mathematical developments and
the algorithm can be found in Ackerer et al. (2014) and Hassane and Ackerer (2017).
Inversion algorithm (adapted from Rambourg et al., 2020) – on the
left, the general algorithm, and on the right, the adaptive parametrization and
its first refinement pattern.
A maximum of three adaptive multiscale iterations is set to ensure a
satisfactory calibration while preventing over-parametrization.
The boundary conditions of the 2D approach are exactly the same as for the
3D synthetic model. In contrast, the initial conditions cannot be integrally
transposed, the knowledge of the water head being limited by data sampling.
Thus, the initial water head for the 2D approach is derived from preliminary
steady-state inversions constrained by time-averaged water head data.
The parameter bounds of the inversion are 5×10-6-5×10-2 for the hydraulic conductivity
[m s-1] and 6 %–30 % for the porosity.
Facies interpolation
For comparison purposes, two interpolation methods are used to reconstruct
the 3D facies distribution, both based on sampled lithological logs. The
interpolation estimates the distribution between control points, using
statistical dependency of measures in the case of geostatistical methods or
drawing geometric surfaces independently of the spatial statistical
repartition in the case of deterministic methods. For both approaches, the
resulting 3D models contain only qualitative (indicator) data. The
interpolation methods selected are not exclusive, as the general methodology
presented can accommodate any interpolation tool that produces a 3D facies
model.
GemPy (geostatistical interpolation)
GemPy (de la Varga et al., 2019) is an open-source 3D geomodelling package
written in Python. It specializes in the reconstruction of stratigraphic
series, with the possibility of modelling complex environments by adding
faults, folds, plutonic intrusions, and other anomalies. GemPy's
mathematical background is a development of the work of Lajaunie et al. (1997; Calcagno et al., 2008), using universal
cokriging methods to interpolate potential fields (scalar fields).
Kriging covers a set of exact (unbiased) linear estimation techniques that
minimize the estimation variance using the variogram, a function
representing the correlation level of a random variable as a function of
distance. Initially limited to stationary variables (simple and ordinary
kriging), universal kriging has extended the use of this type of
geostatistical methods to non-stationary variables. Eventually, cokriging
not only uses the spatial correlation of a variable with itself, but also
incorporates the cross-correlations between two or more random variables.
In the software, the interpolation concerns two types of data: isosurfaces
(including the interface between stacked lithologies and the boundaries of
fault planes or unconformities) on the one hand and surface orientation (the
gradient of the said isosurfaces) on the other hand. This last source of
data allows for the computation of a very smooth and continuous sedimentary
structure, which is rarely the case in other freeware geostatistical tools
(dell'Arciprete et al., 2012; Langousis et al., 2017). In our case, the
orientations (geological poles) are obtained by calculating the normal of
the planes, defined by triangulation between the hydrofacies interfaces at
the sampled data points.
Being specialized in geological modelling, GemPy handles the second-order
(weak) stationarity of universal kriging by assuming a linear trend in the
mean value of the scalar field. In addition, the random function defined for
universal cokriging does not bear any physical meaning as it only aims at
ensuring equality at every point of the isosurface (no matter the value).
As a result, the cross-variogram, inherent to cokriging, cannot be
empirically determined. The shape of the surfaces mainly depends on the
orientations provided and on an arbitrary spherical covariance function that
only balances the relative weight of the surfaces and their orientation in
the cokriging. Hence, the variogram parameters do not bear any physical
meaning as well and are arbitrarily chosen to ensure stability to the
computation according to the GemPy's developers' guidelines (De la Varga et
al., 2019): the nugget effect should be small (set to 10 in our case) and
the range equal to the domain's extension (10 000 m in our case). As the
variogram is not differentiated according to the search direction, the
vertical component of the model must be exaggerated (×500 in our case) so
that its dimension is compatible with the previously quoted values.
Finally, GemPy produces a 3D facies model made of 50×50×10 hexahedron
elements. After rescaling on the z direction, it has the same extension as
the mesh used for the other models and a finer resolution. Henceforth, the
facies in the flow/transport mesh for TRACES are determined according to the
majority facies of the GemPy elements intersecting each 3D prismatic
element.
The calculation of both the scalar fields and their derivatives is handled
by the Theano Python library, which also allows for developments toward
stochastic modelling. A more precise description of the software is
available in De la Varga et al. (2019).
B-splines (deterministic interpolation)
Spline methods are also suitable for the construction of sedimentary models
characterized by smooth surfaces. By definition, their interpolation adjusts
continuous polynomial equations to the data, ensuring no discontinuities and
exact fitting (Prautzsch et al., 2002). Splines can be assimilated to
flexible surfaces constrained to fit the observation values while minimizing
their bending energy. Contrary to a simpler deterministic method (e.g. trend
surface) that operates via a single polynomial equation, splines represent
the surface in pieces and therefore require the computation of a large
number of equations.
However, this method is chosen for its ability to reproduce smooth surfaces,
compatible with a sedimentary morphology, and can be easily carried out through a
GIS procedure (QGIS/SAGA multilevel B-spline interpolation; Lee et al.,
1997). To avoid anomalies in the stacking of the facies, the interpolation
is carried on their thickness instead of their boundaries' z coordinates. In
addition, the first underlying facies is not interpolated but considered to be
the background (filling) lithology. The thicknesses of the four remaining
facies are delivered in raster format, with integer values between 0 and 10
(i.e. the number of layers in the final 3D model) and a resolution of 200 m. Eventually, the facies stacking is transcribed for each column of
prismatic elements in the 3D flow/transport mesh for TRACES according to the
same majority analysis as for the GemPy procedure.
Hydrofacies parametrization and 3D simulations
The lithological models resulting from the interpolations do not have any
assigned hydrodynamic parameters. Thus, an optimization procedure is
implemented to find the hydraulic parameters of the five facies by
minimizing the quadratic difference between 2D and 3D estimated
transmissivities and porosities (Fig. 1, Eq. 7). Both previous steps of the
methodology draw continuous data over the modelled domain (2D averaged
parameters on the one hand and lithological structure on the other).
Conceptually, the optimization could be performed with as many constraints
as the number of elements of the mesh. However, the inversion and
interpolation errors are expected to be minimal at the sampled data
location. Therefore, the algorithm is carried out only with the parameter
values and the lithological successions in these locations, minimizing
uncertainties related to lack of sensitivity for transmissivity values or
related to interpolation for lithological data.
The optimization is handled thanks to a Levenberg–Marquardt algorithm, whose
unknowns are the hydraulic parameters (porosity and hydraulic conductivity)
for each facies, i.e. 2×5 unknowns over the all domain. The
constraints are the 2D mean values (transmissivity and porosity) at the
sampled locations. In order to integrate the least number of preconceptions
in the method, the bounds of values within which the algorithm can pick
during the optimization are not differentiated by facies (the bounds are
10-6 and 10-2 m s-1 for the hydraulic conductivity, 2 % and
50 % for the porosity). The objective function of the optimization
problem takes the form of Eq. (7).
O=∑i∑jli,jpj-PiTσi-1∑jli,jpj-Pi,
where O is the objective function, i is the index for the constraint
(i.e. the sampled location retained for the optimization), j is the index
for each facies, and li,j is the thickness [m] of facies j at location
i. p represents the parameters to be optimized (the hydraulic
conductivity or the effective porosity of each facies) and P the 2D mean
values calibrated during the inversion stage, weighted by the matrix
σ representing this calibration uncertainty. We
consider only the diagonal of the matrix, containing the inverse of the
variance given at location i by the 2D calibration.
The final uncertainty of the optimized parameters is given by Eq. (8).
ϵp=φO^m1/2Cp1/2,
where ϵp is the uncertainty of the parameter p, O^ is
the objective function at end of the optimization, and m is the number of
data. The coefficient φ is determined through Fisher's
distribution, assuming a normal distribution of the uncertainty (for an
estimation at 95 % of confidence, φ=1.96).
Cp is the variance of the parameter p, derived from
the Jacobian (sensitivity matrix) of the model.
Once the optimization estimated each facies' hydraulic parameters, the 3D
model is parametrized. Flow and contamination simulations are carried out
with TRACES, as described previously, with the new facies distribution and
the new parameter set. The boundary conditions and the recharge distribution
are kept unchanged from the reference synthetic model. However, the initial
state data are directly taken from the 2D simulations.
Parameter variability between solutions in each batch.
Transmissivity Porosity relative standard deviation relative standard deviation MeanMaxMeanMaxDense sampling0.5 %8.7 %1.0 %22.1 %Sparse sampling0.2 %2.1 %1.5 %23.4 %
Facies distribution of the models – the different facies are in
greyscale; each model is shown with the ratio of misplaced facies at the
scale of the entire domain on the one hand and of the conditional data on the
other hand.
Results and discussionCalibrated 2D models
The inversion algorithm is run 80 times for each sampling case to gather a
set of possible solutions to the inverse problem, a model inversion lasting
approximately 1 h on average in these conditions. Each set of solutions is
called a batch.
To avoid over-parametrization, the adaptive refinement of the parameter grid,
initially composed of 21 nodes (i.e. degrees of freedom for the
minimization), is limited to three refinements. The number of new parameter
located at the parameter grid vertices is also constrained by the number and
location of the piezometric control points. Therefore, the final average
number of parameter nodes is 497 for the dense sampling (400 control points)
and 74 for the sparser sampling (40 control points).
Each solution batch produces very stable parameter fields (Table 3) and
piezometer chronicles with a mean absolute discrepancy of less than 5 mm
compared to the sampled reference data. For the sparse sampling, the mean
absolute error increases to 37 cm when all control points from the dense
sampling are included in the evaluation.
The mean relative standard deviation of the parameter at the element mesh
scale stays at a very low level in both cases. The variability of
transmissivity is even lower for the sparse sampling (0.2 % vs.
0.5 %), as it has fewer degrees of freedom.
The consistency between the estimated and the reference parameters is
described in Sect. 3.3.
Three-dimensional interpolations
As the reference model is shaped according to a simple sedimentary pattern
(absence of faults), both interpolation methods (geostatistical and
deterministic) produce results of similar quality.
Differences in the facies composition of the models are marginal, even with
a sparse distribution of the conditional data (Fig. 5).
The two percentages accompanying each model in Fig. 5 represent the
proportion of elements incorrectly parameterized for the whole dataset and
the conditional data, respectively. With a dense conditional data sampling,
the deterministic approach yields slightly better results than the
geostatistical one (2.6 % vs. 4.8 % of elements parametrized with the
wrong facies). The GemPy algorithm handles sparser constraints slightly
better (9.5 % vs 11.0 % of errors). However, the differences between
the two interpolation methods can be considered small and assumed to be
dependent on the case study.
Parameter comparison
The results of the inversions (mean 2D values and their associated variance)
and the known facies distributions at the sampling location are used to
optimize 3D hydrodynamic parameters as explained in Sect. 2.4. These
optimized values and their uncertainties are shown in Fig. 6.
In all cases, the hydraulic conductivity of each facies stays in the same
order of magnitude as the reference data (the mean errors account for
26 % of the known hydraulic conductivity values). The largest discrepancy
(0.26 log units, 45.5 % of the reference value in a linear scale) affects
the facies 4 in the sparse-sampling-based estimation. The gaps between the
data and the estimated porosity are also very low, with mean and maximal
absolute errors of 1.3 % and 2.9 % (i.e. 2.9 % and 6.4 % of the
data values). These consistent results are due to the fact that the
parametrization optimization is carried out with input at the sampled data
location, where the 3D facies vertical succession is known and where the
errors on the 2D mean parameters are low.
Calibrated facies parameter values and uncertainties – on the
x axes, the reference parameter values, on the y axes, the estimated values. The facies are differentiated by colours, and the types of parameter are
differentiated by markers (hollow markers are for sparse sampling). The 95 %
confidence intervals are only reported for the sparse models.
The uncertainty attached to the optimization varies with the density of the
data. The values related to the dense sampling are negligible (the highest
uncertainties are at 0.02 log units for the hydraulic conductivity and
0.5 % for the porosity); therefore only the ones of the sparse sampling
are shown. At the most, the uncertainty extends over 0.44 log units for the
hydraulic conductivity (facies 2) and 3.3 % for the porosity (facies 5).
Consequently, the confidence intervals almost always include the
corresponding reference value.
Map of transmissivity discrepancies – the values represent the
difference between the models output and the initial synthetic data; the
samplings (dots) are underlined on the 2D calibrated (inverse) models.
The conjunction of the parameter discrepancies and facies misplacements
results in transmissivity errors, as shown in Fig. 7. The transmissivity
discrepancies are always below 1 order of magnitude and mostly below 0.2
log units. The large-scale heterogeneities are very well reproduced in every
case, notably in the models based on sparse data.
Comparisons between the 2D parameters and the 3D parameters show that the
estimation errors in the former do not propagate entirely in the latter.
Indeed, the errors of the inverse models are not only located at the
interfaces between the large-scale heterogeneities, but also at smaller
scales, within the heterogeneities, when the density of the constraint data
is reduced (in the absence of local piezometric data, the hydrodynamic
parameters are less constrained; see bottom left of Fig. 7). The 3D
interpolation techniques are not subject to these small-scale errors as they
generate very smooth and continuous facies distributions (Fig. 5).
Therefore, the final discrepancies are mainly at the transitions between the
large-scale horizontal heterogeneities where the facies interpolation
generates errors (in particular, the overestimation of transmissivity is
visible where the interpolations have extended facies 1 in excess).
Piezometric head comparison
In view of the few parametrization errors produced with the dense sampling,
the results with respect to piezometry are shown only for the sparse
sampling. In addition to the simulations incorporating the optimized
hydrodynamic parameters per facies, four additional runs are conducted for each
interpolation method in order to study the propagation of parameter
uncertainties. For each additional run, the parameter values are set as follows: (i) at the upper bound of the confidence interval, (ii) at the lower bound,
(iii) alternately at the lower (for the facies 1, 3, 5) and upper (facies 2, 4)
bounds, and (iv) alternately at the lower (for the facies 2, 4) and upper (facies
1, 3, 5) bounds of the parameter confidence interval. These simulations
generate chronicles and maps whose extreme values are retained to construct
“envelope curves”, showing the final uncertainty of the piezometry
(Fig. 8).
The final piezometric state for each model (water head averaged on the
vertical) is shown in the central map of Fig. 8. The piezometric contours
are well reproduced, especially for the 40 and 30 m isolines. Differences
between the interpolation strategies are marginal. The main discrepancy
between the models and the reference data is visible in the middle of the
right border of the domain, where the interpolations underestimated the
presence of facies 1, in favour of facies 2. However, the narrow confidence
interval on the parameters of facies 1 is reflected by a low uncertainty on
the piezometry in the lower right corner of the domain, where this facies
predominates. In contrast, the uncertainty in piezometry is significant
where facies 2, 3, and 4 predominate.
The water head fluctuations are also consistent with the reference
chronicles over the entire period of simulation and for every recharge zone.
Amongst the chosen piezometers, those identified with an asterisk were not
used as constraint data in the sparse sampling. The mean absolute errors of
each model are 44.6 cm for the GemPy model and 47.6 cm for the B-splines.
The deviations mainly take the form of a shift (by excess or by default) in
the base level when the fluctuations are well reproduced. Overall, the
parametrization discrepancies (Fig. 6) are too small to significantly modify
the flow dynamics. Combined with the small differences in the model's
composition (Fig. 5), the water head equilibrium is slightly shifted, as
shown in the charts. Therefore, a more significant deviation would occur in
a permanent flow simulation (with averaged recharge). As with the contour
map, the highest levels of uncertainty are for piezometers intercepting
significant thicknesses of facies 2 (P2, P3, P6, and P7), this one having the
widest confidence intervals.
Comparisons of the 3D piezometric head variations. Piezometric
head values (chronicles and map) are differentiated by colours (black:
reference model, purple: GemPy, green: B-splines). The dashed and dotted
lines give the uncertainty intervals.
Contamination comparison
The study of contaminant transport is another prime use of hydrogeological
models. The outputs of the transport simulations are shown in Fig. 9, in the
form of breakthrough curves and iso-concentration maps (10 mg L-1) at
different times of the simulation. As for the piezometric data, envelope
curves of uncertainty are deduced from the four simulations involving the
parameters at the bounds of the confidence intervals.
The models' output errors are attributable to parametrization discrepancy
on individual facies, interpolation misplacements, and the accumulation of
these same errors upstream of each surveyed location. The results confirm
that contaminant transport is much more sensitive to parametrization errors
than piezometry. Indeed, the latter is governed by the transmissivity, where
individual facies misparametrization can be buffered by the vertical
integration. For contaminant transport, each voxel parametrization may
influence the outcome.
For instance, source 1 is located on facies 1 in the reference model and on
facies 2 in the interpolated models. Therefore, the dynamics of the
breakthrough curve and the pollutant plume are significantly different (i.e.
a lower spike due to a higher porosity and a faster depletion due to a
higher conductivity). Conversely, the facies distribution is preserved
in the location of source S2 (in facies 4), where the discrepancies are
mainly due to hydrodynamic parametrization errors on this very same facies.
Overall, the results at the outlet E and the plume maps show that, in our
case, the parametrization errors have a more significant impact on the fate
of sources S2 and S3 than for source 1.
The confidence intervals on the facies parameters (especially on facies 2, 3,
and 4) generate scenarios where the plumes disappear early on the one hand or
extend in excess on the other hand.
Contamination breakthrough curves. GemPy (green) and B-spline
(yellow) parametrized models (dashed for sparse data) are compared to the
initial synthetic dataset (black). The two columns on the left show depletion
or breakthrough curves, and the last column on the right depicts contamination
plumes at four different times.
Perspectives regarding uncertainties
In this study, only the uncertainty related to the calibration of the 2D
parameters is calculated and propagated to the 3D models. However, it must
be emphasized that in the context of distributed hydrogeological models,
many other sources of uncertainty occur (Pechlivanidis et al., 2011).
First, there are data uncertainties: piezometric measurements, rainfall and
radiative data (leading to recharge estimates), and lithological descriptions
(both in terms of their categorization and altimetry) are all subject to
error. Our approach via a synthetic case encouraged us to postpone this
aspect and to concentrate on the analysis of the methodology. These
uncertainties will be taken into account in future work dealing with real
cases.
Second, all uncertainties related to the parameters have not been addressed
in this study. Focused on the determination of hydrodynamic parameters, it
did not integrate the uncertainty related to the dispersivity and molecular
diffusion parameters inherent to the contaminant transport phenomena.
Third, the structure of the models and the way they are defined also involve
uncertainties. In particular, interpolation methods produce uncertainty and
propagate those inherent in the lithological data (Phillips and Mark, 1996;
Lloyd and Atkison, 2001; White, 2017). Our data sampling strategy allowed this pitfall to be circumvented, but it becomes unavoidable when there are
significant gaps in the data.
Monte Carlo simulations (Wagener and Kollat, 2007; Beven, 2009) allow for the
joint estimation of these uncertainties, at great computational cost (the
simulations must cover a range of value for each input or parameter at
stake). Several random (Olsson and Sandberg, 2002) or Bayesian (Vrugt et
al., 2009) sampling strategies are at hand in order to reduce the number of
iterations needed to obtain a representative view of the uncertainties.
Integrating this type of analysis into the framework of our method is one the
important improvements planned.
Conclusions
A method has been developed to assess 3D aquifer parameters by combining hydrodynamic parameters estimated by a 2D model calibration and 3D facies
interpolation. While direct 3D parameter estimation is generally based on a
heavy geophysical survey, the proposed methodology is based solely on
piezometric series and geological logs (commonly available at the same
locations). Both 2D flow model calibration and 3D interpolation parts of the
algorithm are independent. Therefore, the approach is not restricted to the
tools described in the article (i.e. other interpolation methods than GemPy
and B-splines can be used) and can potentially incorporate a pre-existing 2D
model.
The synthetic test carried out with a relatively sparse dataset yields a
consistent hydrodynamic parametrization (highest discrepancy: 45.5 % of
the initial value for hydraulic conductivity, 6.4 % for effective
porosity) and quite low errors in facies distribution (11 % of misplaced
facies at the most). Subsequently, the reconstructed piezometric series show
very consistent dynamics, with a maximal mean difference of 47.6 cm, mainly
due to shifting the base level, while the fluctuations and the hydraulic
gradients are generally unaltered. The discrepancies concerning the
transport simulations are more significant, the phenomenon being more
sensitive to parametrization errors at the individual voxel scale.
Comparatively to joint inversion methods, the need of data acquisition and
the computation efforts are lower. However, in a field context, the method
is very dependent on the characterization of the hydrofacies and the quality
of the piezometric survey. Indeed, if uncertainties related to the 2D flow
calibration were propagated to the 3D parameter optimization and,
thereafter, into the piezometry and contaminant transport simulation, other
sources of uncertainty hindering the hydrogeological modelling process were
not accounted for at the time of publication. Among them, we can cite the
uncertainties related to input data, transport parameters, or the structure
of the model itself (especially the lithology interpolation step).
Another pitfall of the method, inherent to the 2D step, lies in the fact
that low-hydraulic-conductivity facies may be masked by more permeable
facies in the transmissivity term, making their parametrization somehow
difficult. Also, some other important modelling points are not addressed in
the study, for example, the vadose zone dynamics and the transport parameters, which
require separate estimates. Following this synthetic case, we plan to test
the method on a real case, in order to confirm its operational potential,
completed by comprehensive sensitivity and uncertainty analyses.
Code and data availability
TRACES, PINOGRI, and the synthetic case dataset can be provided on request
to the corresponding author.
GemPy is an open-source 3D geomodelling package (https://www.gempy.org, last access: 5 December 2022).
The B-spline interpolation method used is an add-in tool of the free and
open-source geographic information system QGIS/SAGA.
Author contributions
DR and PA designed the method workflow, DR
designed the synthetic case and carried out the simulations. RDC
adapted the GemPy Python package, and PA developed TRACES (with
co-authors cited in the references) and co-developed PINOGRI with DR
(and previous contributors to the code development, as cited in the references).
DR prepared the manuscript with contributions from both
co-authors.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Review statement
This paper was edited by Alberto Guadagnini and reviewed by two anonymous referees.
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