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Characterizing groundwater flow parameters is crucial for understanding complex aquifer systems, and inverse techniques play a fundamental role in modeling hydrogeological parameters and assessing their uncertainties. Nonetheless, the use of a forward model in these methods can be highly time-consuming, especially with an increasing number of model parameters. To address this issue, we propose a surrogate model based on a U-Net architecture that replaces the transient groundwater flow model, reducing runtime and enabling a fast quantification of uncertainties related to key parameters, including heterogeneous hydraulic conductivity, boundary conditions, specific storage, and pumping rate. The surrogate is trained using limited evaluations of the forward model to learn the physical relationship between hydraulic conductivity fields and transient hydraulic heads measured on-site. The physical principles of the studied problem, including boundary conditions, specific storage, and source terms, are also mapped and introduced as inputs to the model to enhance its understanding of the governing equation of transient groundwater flow. To speed up learning using image–image regression, the previously predicted transient hydraulic heads also serve as an input to predict the transient heads at the current time step. Once the model is trained, we use a spectral geostatistical method to solve the inverse problem, a pumping test of 12 h, using the surrogate model in place of the forward model. Our study demonstrates that the trained U-Net accurately reproduces the state variables corresponding to a specific parameter field, and in terms of computational demand, using U-Net as a surrogate model reduces the required computational time by approximately an order of magnitude for the defined problem. The proposed approach offers an efficient and accurate method for groundwater flow parameter characterization and uncertainty quantification in complex aquifer systems. [Display omitted] • A U-Net surrogate approach is developed for transient groundwater inverse problems. • Uncertainties are rapidly evaluated using a geostatistical spectral method. • Uncertain parameters include log(K), Q, Ss, and boundary conditions • Integrate time into image–image regression for capturing time-varying processes. • The surrogate model significantly reduces computation times for inverse problems. [ABSTRACT FROM AUTHOR]

• Cross-gradients constraint inversion is fused into hydraulic tomography. • The method permits deterministic optimization of structural similarities. • Limited improvements are obtained from the cross-gradients constraint. The inclusion of a priori information is helpful to obtain high resolution parameter identifications. A common practice is to exploit structural information in the search for detailed hydrogeological characterizations of subsurface. In this study, we proposed to incorporate cross-gradients as a constraint in nonlinear inverse problems. The resultant inversion scheme is successfully applied to three synthetic experiments and one laboratory rock block experiment with structural similarities in hydraulic parameters and also compared to estimated results with a traditional method. Our results show similar estimated hydraulic parameters whether the objective function has the cross-gradients operator or not. The improvements are only enhanced when assuming one type of parameter is completely known. The limited improvement is proved by the small contribution from the cross-gradients term in the objective function. Therefore, the cross-gradients constraint is a weak structural constraint. The main reason is that drawdown data has already contained correlated hydrogeological parameter information. Estimated results substantiate the rationalization for a simplified assumption that hydraulic conductivity and specific storage are treated as mutual independent variables. [ABSTRACT FROM AUTHOR]

For strongly non-linear and high-dimensional inverse problems, Markov chain Monte Carlo (MCMC) methods may fail to properly explore the posterior probability density function (PDF) given a realistic computational budget and are generally poorly amenable to parallelization. Particle methods approximate the posterior PDF using the states and weights of a population of evolving particles and they are very well suited to parallelization. We focus on adaptive sequential Monte Carlo (ASMC), an extension of annealed importance sampling (AIS). In AIS and ASMC, importance sampling is performed over a sequence of intermediate distributions, known as power posteriors, linking the prior to the posterior PDF. The AIS and ASMC algorithms also provide estimates of the evidence (marginal likelihood) as needed for Bayesian model selection, at basically no additional cost. ASMC performs better than AIS as it adaptively tunes the tempering schedule and performs resampling of particles when the variance of the particle weights becomes too large. We consider a challenging synthetic groundwater transport inverse problem with a categorical channelized 2D hydraulic conductivity field defined such that the posterior facies distribution includes two distinct modes. The model proposals are obtained by iteratively re-simulating a fraction of the current model using conditional multiple-point statistics (MPS) simulations. We examine how ASMC explores the posterior PDF and compare with results obtained with parallel tempering (PT), a state-of-the-art MCMC inversion approach that runs multiple interacting chains targeting different power posteriors. For a similar computational budget, ASMC outperforms PT as the ASMC-derived models fit the data better and recover the reference likelihood. Moreover, we show that ASMC partly retrieves both posterior modes, while none of them is recovered by PT. Lastly, we demonstrate how the power posteriors obtained by ASMC can be used to assess the influence of the assumed data errors on the posterior means and variances, as well as on the evidence. We suggest that ASMC can advantageously replace MCMC for solving many challenging inverse problems arising in the field of water resources. • Adaptive sequential Monte Carlo (ASMC) approximates the posterior and the evidence. • ASMC is combined with sequential geostatistical resampling to handle complex priors. • ASMC outperforms parallel tempering considering the same computational budget. • Intermediate ASMC results are interpreted as results for larger assumed data errors. [ABSTRACT FROM AUTHOR]

Karst aquifers are characterized by extreme heterogeneity due to the presence of karst conduits embedded in a fractured matrix having a much lower hydraulic conductivity. The resulting contrast in the physical properties of the system implies that the system reacts very rapidly to some changes in the boundary conditions and that numerical models are extremely sensitive to small modifications in properties or positions of the conduits. Furthermore, one major issue in all those models is that the location and size of the conduits is generally unknown. For all those reasons, estimating karst network geometry and their properties by solving an inverse problem is a particularly difficult problem. In this paper, two numerical experiments are described. In the first one, 18,000 flow and transport simulations have been computed and used in a systematic manner to assess statistically if one can retrieve the parameters of a model (geometry and radius of the conduits, hydraulic conductivity of the conduits) from head and tracer data. When two tracer test data sets are available, the solution of the inverse problems indicate with high certainty that there are indeed two conduits and not more. The radius of the conduits are usually well identified but not the properties of the matrix. If more conduits are present in the system, but only two tracer test data sets are available, the inverse problem is still able to identify the true solution as the most probable but it also indicates that the data are insufficient to conclude with high certainty. In the second experiment, a more complex model (including non linear flow equations in conduits) is considered. In this example, gradient-based optimization techniques are proved to be efficient for estimating the radius of the conduits and the hydraulic conductivity of the matrix in a promising and efficient manner. These results suggest that, despite the numerical difficulties, inverse methods should be used to constrain numerical models of karstic systems using flow and transport data. They also suggest that a pragmatic approach for these complex systems could be to generate a large set of karst conduit network realizations using a pseudo-genetic approach such as SKS, and for each karst realization, flow and transport parameters could be optimized using a gradient-based search such as the one implemented in PEST. [ABSTRACT FROM AUTHOR]

• We propose a deep neural network to predict 2D subsurface property fields from 1D measurement data (e.g., time series) thereby offering a new way to perform inversion. • The method is illustrated using inversion of synthetic (1) first-arrival GPR travel times and (2) time series of transient hydraulic heads from a multi-pumping experiment. • The method generalizes by producing more appropriate models when fed with data than those found in the training set. • Uncertainty is assessed using a deep ensemble. We present vec2pix , a deep neural network designed to predict categorical or continuous 2D subsurface property fields from one-dimensional measurement data (e.g., time series), thereby offering an alternative approach to solve inverse problems. The performance of the method is investigated through two types of synthetic inverse problems: (a) a crosshole ground penetrating radar (GPR) tomography experiment with GPR travel times being used to infer a 2D velocity field, and (2) a multi-well pumping experiment within an unconfined aquifer with time series of transient hydraulic heads being used to retrieve a 2D hydraulic conductivity field. For each type of problem, both a multi-Gaussian and a binary channelized subsurface domain with long-range connectivity are considered. Using a training set of 20,000 examples (implying as many forward model evaluations), the method is found to recover a 2D model that is in much closer agreement with the true model than the closest training model in the forward-simulated data space. Further testing with smaller training sample sizes shows only a moderate reduction in performance when using 5000 training examples only. Although these 5000 to 20,000 forward runs can be performed offline in parallel, the associated computational expense may still be prohibitive for very demanding forward solvers. In addition, re-training is required for each new measurement configuration. Even if the recovered models are visually close to the true ones, the data misfits associated with their forward responses are generally larger than the noise level used to contaminate the true data. Uncertainty of the inverse solution is partially assessed using deep ensembles, in which the network is trained repeatedly with random initialization. Overall, this study advances understanding of how to use deep learning to infer subsurface models from indirect measurement data. More work is needed to evaluate the suitability of our proposed approach for real data, for which model errors and uncertainty in the geologic scenario will bring additional complexities. [ABSTRACT FROM AUTHOR]

• Standard DNN methods might lead to highly uncertain and inaccurate predictions. • Multiphysics-informed neural network (MPINN) method is proposed. • Multiphisics constraints improve predictive ability of DNNs. • MPINN allows assimilating multiphysics data. • MPINN reduces data requirements for parameter estimation. Data assimilation for parameter and state estimation in subsurface transport problems remains a significant challenge because of the sparsity of measurements, the heterogeneity of porous media, and the high computational cost of forward numerical models. We present a multiphysics-informed deep neural network machine learning method for estimating space-dependent hydraulic conductivity, hydraulic head, and concentration fields from sparse measurements. In this approach, we employ individual deep neural networks (DNNs) to approximate the unknown parameters (e.g., hydraulic conductivity) and states (e.g., hydraulic head and concentration) of a physical system. Next, we jointly train these DNNs by minimizing the loss function that consists of the governing equations residuals in addition to the error with respect to measurement data. We apply this approach to assimilate conductivity, hydraulic head, and concentration measurements for the joint inversion of these parameter and states in a steady-state advection–dispersion problem. We study the accuracy of the proposed data assimilation approach with respect to the data size (i.e., the number of measured variables and the number of measurements of each variable), DNN size, and the complexity of the parameter field. We demonstrate that the physics-informed DNNs are significantly more accurate than the standard data-driven DNNs, especially when the training set consists of sparse data. We also show that the accuracy of parameter estimation increases as more different multiphysics variables are inverted jointly. [ABSTRACT FROM AUTHOR]

• This paper introduces the concept of Traveling Pilot Points (TRIPS) for solving the categorical inverse problem • The TRIPS methodology combined with subspace methods is used to identify channel structures in a categorical aquifer system while honoring calibration and prior constraints • Tests on a synthetic example suggest that the TRIPS methodology has promise and could be tested on more complicated problems Categorical parameter distributions are common-place in hydrogeological systems consisting of geologic facies/categories with distinct properties, e.g., high-permeability channels embedded in a low-permeability matrix. Parameter estimation is difficult in such systems because the discontinuities in the parameter space hinder the inverse problem. Previous research in this area has been focused on the use of stochastic methods. In this paper, we present a novel approach based on Traveling Pilot points (TRIPS) combined with subspace parameter estimation methods to generate realistic categorical parameter distributions that honor calibration constraints (e.g., - measured water levels). In traditional implementations, aquifer properties (e.g., hydraulic conductivity) are estimated at fixed pilot point locations. In the TRIPS implementation, both the properties associated with the pilot points and their locations are estimated. Tikhonov regularization constraints are incorporated in the parameter estimation process to produce realistic parameter depictions. For a synthetic aquifer system, we solved the categorical inverse problem by combining the TRIPS methodology with two subspace methods: Null Space Monte Carlo (NSMC) and Posterior Covariance (PC). A posterior ensemble developed with the rejection sampling (RS) method is compared against the TRIPS ensembles. The comparisons indicated similarities between the various ensembles and to the reference parameter distribution. Between the two subspace methods, the NSMC method produced an ensemble with more variability than the PC method. These preliminary results suggest that the TRIPS methodology has promise and could be tested on more complicated problems. [ABSTRACT FROM AUTHOR]