Improving the robustness of the Comparison Model Method for the identification of hydraulic transmissivities [post]

Alessandro Comunian, Mauro Giudici
2021 unpublished
Approaches to improve the robustness of a direct inversion method are explored • Guidelines are provided to deal with the numerical issues related to small gradients • A tomographic approach allows to improve the robustness of the inversion procedure • An open source implementation of the comparison model method is provided Improving the robustness of the Comparison Model Method for the identification of hydraulic transmissivities ⋆ A B S T R A C T The Comparison Model Method (CMM) is a
more » ... (CMM) is a relatively simple and computationally efficient direct method for the identification of the transmissivity of a confined aquifer by solution of an inverse problem. However, it suffers some of the classical drawbacks related to ill-posedness and ill-conditioning of inverse methods. Effectiveness of the CMM can be improved by some approaches. First of all, the introduction of a factor which permits to limit the negative effects of small hydraulic gradients by selection of a single parameter of the algorithm. Moreover, the CMM can be cast in a tomographic framework, i.e., by profiting of multiple sets of data, corresponding to different flow situations, produced by different boundary conditions or sources terms. Numerical tests are performed on a synthetic aquifer, by means of an open-source implementation based on the use of flopy for the solution of the forward problems. The tests show that the above mentioned approaches improve the robustness of the CMM with respect to error on the input head data. CMM (model evaluation) runs. Despite the growing capabilities of parallel computing, dealing with the high number of forward problem runs remains an open research question. A possible solution are hybrid optimization methods, that blend in a smart way global and local optimization methods, as proposed for example by Vesselinov and Harp (2012) . Another possibility is to find a proxy or a surrogate for the forward model, as proposed for example by Dagasan et al. (2020) with the use of Generative Adversarial Networks. Alternatively, one can also look for improving the robustness of direct approaches, as it is done in this work. In this paper, the Comparison Model Method (CMM) is considered. The CMM is a direct inversion method first proposed by Scarascia and Ponzini (1972) . Its basic properties and further improvements are described by Ponzini and Lozej (1982) ; Ponzini and Crosta (1988); Ponzini et al. (1989) . In real cases applications, the CMM was used to identify the parameters of coastal aquifers De Filippis et al., 2016 , of regional models to support groundwater-surface water interactions (Baratelli et al., 2016) , of hydrogeological models of highly irrigated alluvial plains Vassena et al. (2008 Vassena et al. ( , 2012 . Moreover, its applications are not limited to hydrology, as the method was successfully applied by Lesnic (2010) to determine the flexural rigidity of a beam and by Ponzini et al. (1989) to identify thermal conductivities. In addition, the CMM was also included in the groundwater modeling software YAGMod (Cattaneo et al., 2015) . Some recent developments of the CMM include an hybrid approach that blends this direct inversion method with multiple-point statistics (Comunian and Giudici, 2018) . The CMM is a direct method of inversion based on an auxiliary model, hereinafter called Comparison Model (CM), and on the knowledge of the head field and of the source terms throughout the whole domain. The CM shares the same boundary conditions and source/sink terms with the predictive model to be developed, but the transmissivity field corresponds to a first guess, possibly informed by the results of field tests, geological or geophysical information. Basically, the CMM estimates the transmissivity field by correcting the initial guess with the ratio of the hydraulic gradients computed from the solution to the forward problem for the CM and the reference hydraulic gradients. The basic physical idea behind such an approach is that, if the initial guess is not far from the real transmissivity field, the flow rates computed with the CM should be a reasonable approximation of the real ones, obtained as the product of the real transmissivity times the reference hydraulic gradient. Similarly to any other inverse method, the CMM suffers from some drawbacks. Some of the most important problems are related to the good knowledge of the head field and to the non-boundedness of derivative operators. The inverse problem is very prone to errors on heads and, in particular, even very small errors at short wavelength might induce huge errors in the estimate of the hydraulic gradient and finally of the transmissivity field. Like other direct methods of inversion, e.g., the Differential System Method Giudici et al., 1995; Giudici and Vassena, 2006) , the head field has to be known everywhere. Since measurements are available at possibly few, sparse points, an interpolation is necessary. Often, interpolation can be a critical step: for example, if one uses kriging, the available data might not be sufficient to build up a reliable variogram model, and the errors made in the model selection will affect the interpolated head field. In order to reduce the uncertainty intrinsic in this procedure, it is good practice to use any other additional information, e.g., no flow conditions related to the outcropping of impermeable rocks. Furthermore, interpolation errors add themselves to measurement errors at the monitoring points. This might be critical, above all, in areas of low gradients, because the relative error in those areas could be particularly great, and in extreme cases the estimated hydraulic gradient could show a direction opposite to the real one. Moreover, it should be recalled that the balance equation acts as a low pass filter for wave number components of the head field . In other words, the high wave number (short wavelength) components of the transmissivity field are filtered out by the balance equation, so that the head field does not carry correct information about the variability of the transmissivity field at short distances. This aspect was also investigated by Comunian and Giudici (2018) , where the CMM was used in conjunction with the Direct Sampling multiple-point statistics simulation method (Mariethoz et al., 2010) in a hybrid approach to improve the reproduction of small scale details. In order to overcome some of these difficulties, the approaches considered in this work are twofold. First, the correction initially proposed by Ponzini and Lozej (1982) and lately improved by Vassena et al. (2012) to limit the heavy effects of low hydraulic gradients is considered. In particular, the factor which was introduced in the aforementioned studies is defined in a more precise and general way in this paper. This factor depends on a coefficient , which permits to tune the amount of cells where low hydraulic gradients should be corrected. Besides studying the implications of different choices of , in this work a straightforward criterion to select this parameter based on the percentage of cells where the hydraulic gradient has to be corrected is proposed. Second, the CMM can be cast in a "tomographic" framework, by profiting of the availability of multiple data sets related to different flow conditions. If this is the case, the forcing of the natural system in such a way as to An implementation of the CMM based on flopy and MODFLOW 6, called cmmpy (version 0.1.3) is provided, together with the accompanying scripts and parameter files, at the following link: https://bitbucket.org/alecomunian/cmmpy with a GNU General Public License. The corresponding Python package is also included in the Python Package Index (PyPI), and it can be easily installed with the command pip install cmmpy. The documentation is available at the link https://cmmpy.readthedocs.io.
doi:10.31223/x5231f fatcat:rwkrg2okfncvfaimcefhp7einu