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The incorporation of hard data in geostatistical modeling is crucial for enhancing the accuracy of interpolating or stochastically estimating subsurface spatial features. The hard data at specified points in the model domain serve as a guide in optimizing the unknown parameters to follow the patterns of the hard data. Recently, a novel approach to solving hydrogeologic/reservoir modeling problems has emerged by using deep generative models, specifically generative adversarial networks (GANs), to generate realistic and diverse images of channelized aquifers. This subsequently can be coupled with inverse models to solve parameter estimation problems. This study focused on using an improved GAN, called a progressive growing generative adversarial network (PGGAN), conditioned with hard data to perform parameter estimation of complex facies models by coupling an ensemble smoother with multiple data assimilation (ES-MDA). First, the PGGAN was trained to an image with 128 × 128 resolution. The trained PGGAN was used to generate hydraulic conductivity fields when fed an ensemble of latent variables and hard data. The ES-MDA then was used to update the latent variable with the help of hydraulic head data obtained from the groundwater model. The approach was tested on synthetic hydraulic conductivity data. Results show that this approach was able to perform efficient estimation of an unknown facies model domain. Additionally, the proposed method was applied to a different test case of a facies model exhibiting different statistical characteristics. The results were satisfactory, demonstrating that the method is not constrained to the particular hydraulic conductivity fields introduced in the generator's training. • Coupling deep learning with data assimilation for groundwater modeling. • Both hard data and soft data are conditioned. • Synthetic examples are used to demonstrate the effectiveness. [ABSTRACT FROM AUTHOR]

Highlights • The continuous adjoint state associated with the transport equation is derived. • Hydraulic conductivity and porosity fields are estimated by inverse modeling. • Concentration data complementing heads slightly improve inverse solutions to flow. Abstract A parameter estimation methodology has been developed on the basis of model inversion using a Quasi–Newton method and adaptive parameterization. The continuous adjoint state equations for both flow and transport in porous media are employed as the tool calculating the gradient components of the objective function with respect to parameters. Solving the continuous form of the adjoint equations can be implemented independently of the code used to solve the forward problem, which renders the technique non-intrusive. The developed methodology is applied to the identification of hydraulic conductivity and porosity fields conditioned by piezometric head data associated with steady-state flow and transient solute concentrations. Synthetic numerical experiments have been undertaken for test cases of increasing complexity, from an almost uniform flow sweeping the modeled domain with a prescribed uniform continuous injection of solute at the inflow boundary, to spatially highly variable flow conditions obtained through source/sink terms within the flow domain and a local stepwise solute injection. The results of inversions are analyzed using criteria based on the comparisons between estimated concentration and reference concentration values as well as comparisons between estimated hydraulic conductivity (and porosity for one test case) and reference hydraulic conductivity fields. The results show that employing a continuous adjoint state technique computed independently of the direct problem is an efficient option for parameter estimation relying jointly upon flow and transport data. In the reported numerical examples that are characterized by the identifiability of the flow problem on the basis of hydraulic head observations, concentration data from solute transport scenarios bring few added value to sought solutions of hydraulic conductivities. The spatial structure of the conductivity fields is slightly improved compared with the reference, but the overall system in terms of head distribution, identification of main flow paths, and solute transit times, only inherits cosmetics. [ABSTRACT FROM AUTHOR]

Accurate characterization of subsurface hydraulic conductivity is vital for modeling of subsurface flow and transport. The iterative ensemble smoother (IES) has been proposed to estimate the heterogeneous parameter field. As a Monte Carlo-based method, IES requires a relatively large ensemble size to guarantee its performance. To improve the computational efficiency, we propose an adaptive Gaussian process (GP)-based iterative ensemble smoother (GPIES) in this study. At each iteration, the GP surrogate is adaptively refined by adding a few new base points chosen from the updated parameter realizations. Then the sensitivity information between model parameters and measurements is calculated from a large number of realizations generated by the GP surrogate with virtually no computational cost. Since the original model evaluations are only required for base points, whose number is much smaller than the ensemble size, the computational cost is significantly reduced. The applicability of GPIES in estimating heterogeneous conductivity is evaluated by the saturated and unsaturated flow problems, respectively. Without sacrificing estimation accuracy, GPIES achieves about an order of magnitude of speed-up compared with the standard IES. Although subsurface flow problems are considered in this study, the proposed method can be equally applied to other hydrological models. [ABSTRACT FROM AUTHOR]

Estimating the statistical parameters (mean, variance, and integral scale) that define the spatial structure of the transmissivity or hydraulic conductivity fields is a fundamental step for the accurate prediction of subsurface flow and contaminant transport. In practice, the determination of the spatial structure is a challenge because of spatial heterogeneity and data scarcity. In this paper, we describe a novel approach that uses time drawdown data from multiple pumping tests to determine the transmissivity statistical spatial structure. The method builds on the pumping test interpretation procedure of Copty et al. (2011) (Continuous Derivation method, CD), which uses the time-drawdown data and its time derivative to estimate apparent transmissivity values as a function of radial distance from the pumping well. A Bayesian approach is then used to infer the statistical parameters of the transmissivity field by combining prior information about the parameters and the likelihood function expressed in terms of radially-dependent apparent transmissivities determined from pumping tests. A major advantage of the proposed Bayesian approach is that the likelihood function is readily determined from randomly generated multiple realizations of the transmissivity field, without the need to solve the groundwater flow equation. Applying the method to synthetically-generated pumping test data, we demonstrate that, through a relatively simple procedure, information on the spatial structure of the transmissivity may be inferred from pumping tests data. It is also shown that the prior parameter distribution has a significant influence on the estimation procedure, given the non-uniqueness of the estimation procedure. Results also indicate that the reliability of the estimated transmissivity statistical parameters increases with the number of available pumping tests. [ABSTRACT FROM AUTHOR]

The knowledge of subsurface heterogeneity is a prerequisite to describe flow and transport in porous media. Of particular interest are the variance and the correlation scale of hydraulic conductivity. In this study, we present how these aquifer parameters can be inferred using empirical steady state pumping test data. We refer to a previously developed analytical solution of “effective well flow” and examine its applicability to pumping test data as under field conditions. It is examined how the accuracy and confidence of parameter estimates of variance and correlation length depend on the number and location of head measurements. Simulations of steady state pumping tests in a confined virtual aquifer are used to systematically reduce sampling size while determining the rating of the estimates at each level of data density. The method was then applied to estimate the statistical parameters of a fluvial heterogeneous aquifer at the test site Horkheimer Insel, Germany. We conclude that the “effective well flow” solution is a simple alternative to laboratory investigations to estimate the statistical heterogeneity parameter using steady state pumping tests. However, the accuracy and uncertainty of the estimates depend on the design of the field study. In this regard, our results can help to improve the conceptual design of pumping tests with regard to the parameter of interest. [ABSTRACT FROM AUTHOR]

Nitrate contamination of aquifers is threatening the sustainability of water resource production around the globe. Denitrification is an ubiquitous process in groundwater systems and needs to be represented in managerial and predictive efforts. Yet, fully understanding the temporal and spatial dynamics of all biogeochemical and physical properties controlling nitrate reduction at the basin scale remains a challenging task. Conceptual simplifications such as homogenization are used, but their implications on model performance are often not well assessed. This paper (1) analyzes the importance of denitrification spatial variability on the fate of non-point source nitrate contamination, and (2) assesses the performance of upscaled denitrification rate coefficient values. Heterogeneity in the hydraulic conductivity of a typical unconsolidated alluvial aquifer system and its associated uncertainty is systematically represented in our 3D numerical simulations of flow and transport. The spatial variability in the nitrate reduction capacities of the aquifer is accounted for by 3 different conceptual models: a positive and a negative correlation with the aquifer hydraulic properties and a uniform random distribution. Our results indicate that heterogeneity in the denitrification rate coefficient – as well as the model describing spatial variability – has a significant impact on concentration statistics observed at a series of extraction wells. However, we also show that, under the modeling setting used in this study (e.g., stationarity of reactivity), upscaled values of denitrification rate coefficients allow a virtually perfect reproduction of nitrate concentration statistics, a promising outcome for employing parameter estimation (calibration) of effective rate coefficients of denitrification: A detailed description of the spatial variability of the nitrate reduction capacity of the alluvial aquifer is not necessary. Yet, these homogeneous, upscaled denitrification rate coefficients result from a complex interaction between the biochemistry and the physics of the aquifer, which makes their direct estimation a challenging task. [Display omitted] • Spatial variability structure of denitrification rate coefficients influence nitrate dynamics. • Upscaled, effective denitrification rate coefficients reproduce well concentration statistics. • Upscaling denitrification rates is complex due to interplays between key processes. [ABSTRACT FROM AUTHOR]

Highlights: [•] E. coli transport below freshly-excavated pond receiving latrine effluent modeled. [•] Anisotropy in hydraulic conductivity greatly limited E. coli depth penetration. [•] Reversible and irreversible attachment of E. coli required to match field data. [•] Laboratory-predicted E. coli transport matches field data within factor of two. [•] Kinetic interaction parameters relatively insensitive to scale of measurement. [ABSTRACT FROM AUTHOR]

Abstract: Parameters of functions to describe soil hydraulic properties are derived from measurements by means of parameter estimation. Of crucial importance here is the choice of correct constraints in the parameter space. Often, the parameters are mere shape parameters without physical meaning, giving flexibility to the model. A fundamental requirement is that the hydraulic functions are monotonic: the retention function and the conductivity function can only decrease as the capillary suction increases. A stricter physical requirement for the conductivity function is that its decrease with respect to saturation is at least linear. This linear decrease would occur if all pores of a capillary bundle had an equal radius. In the first part of this contribution, we derive constraints for the so-called tortuosity parameter of the Mualem conductivity model, which allow highest possible flexibility on one hand and guarantee physical consistency on the other hand. In combination with the retention functions of Brooks and Corey, van Genuchten, or Durner, such a constraint can be expressed as a function of the pore-size distribution parameters. In the second part, we show that a common modification of retention models, which is applied to reach zero water content at finite suction, can lead to the physically unrealistic case of increasing water content with increasing suction. We propose a solution for this problem by slightly modifying these models and introducing a correct parameter constraint. [Copyright &y& Elsevier]

• 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]