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We introduce an approach for the stochastic characterization of the geometric and hydraulic parameters of a three‐dimensional (3D) discrete fracture network (DFN) and for estimating their uncertainty based on data from hydraulic tomography experiments. The inversion approach relies on a Bayesian framework and the resulting posterior distribution is characterized by generating samples by Markov chain Monte Carlo (MCMC) methods. The inversion method is evaluated for four synthetic test cases related to the Grimsel test site in Switzerland. Comparison of original and reconstructed DFN models shows that the presented approach is suitable for identifying variable fracture locations and orientations. This is especially the case for those fractures that represent the preferential flow paths in the simulated experiments. It is also revealed that the Bayesian framework is useful to discriminate fractures based on the reliability of the inversion, which is illustrated by fracture probability maps taken as sections through the studied rock mass. Moreover, it is demonstrated that the hydraulic apertures can be calibrated together with the fracture geometries. A premise for applicability in practice, however, is that the hydraulic measurements are complemented by additional information to sufficiently constrain the value ranges of the geometric and hydraulic parameters to be inverted together. The presented work expands the applicability of a previously presented promising two‐dimensional procedure based on transdimensional inversion to field‐based 3D problems. The theoretical findings here open the door for highly flexible structural characterization in practice based on hydraulic tomography, as well as alternative or complementary tomographic methods. Key Points: We infer the structural properties of a three‐dimensional discrete fracture network given data from hydraulic tomography experimentsThe method is demonstrated for synthetic test cases based on the Grimsel site in SwitzerlandThe inversion yields major geometric parameters and their uncertainties [ABSTRACT FROM AUTHOR]

Fluctuating stream stages and peak‐flow events can significantly influence the interactions between streams and aquifers and modify the hydraulic gradient, the flux exchange and the subsurface flow paths. As a result, stagnation zones and reverse flow may appear in different parts of an aquifer and at different times. These features of the flow field play a relevant role in the transport, transformation, and residence time of solutes, pollutants, and nutrients in the subsurface. However, their identification using numerical models is complex not only because of highly nonlinear dynamics, but also due to significant uncertainties in the model input data which propagate into the quantities of interest. In this work, we use an approach based on polynomial chaos expansions to map the probability of occurrence of stagnation zones and reverse flow during a flood event. We quantify the propagation of uncertainty into the groundwater flow field due to the applied river boundary conditions. Then, we evaluate the responses of the posterior probabilities in an element‐wise fashion using a set of flow classification criteria and kernel density estimations. The proposed methodology is flexible because it employs a nonintrusive pseudo‐spectral technique and, consequently, it can be applied straightforwardly in preexisting models. The regions near the confluence of two streams in the studied area are prone to present transient stagnation and reverse flow. Key Points: The probability of occurrence of stagnation zones and reverse flow is computed based on the uncertainty caused by river boundary conditionsPolynomial chaos expansions are used in an element‐wise fashion for quantifying the propagation of uncertaintyThe regions near the confluence of two streams are very dynamic and prone to the presence of stagnation points and reverse flow [ABSTRACT FROM AUTHOR]

Uncertainty quantification of groundwater (GW) aquifer parameters is critical for efficient management and sustainable extraction of GW resources. These uncertainties are introduced by the data, model, and prior information on the parameters. Here, we develop a Bayesian inversion framework that uses Interferometric Synthetic Aperture Radar (InSAR) surface deformation data to infer the laterally heterogeneous permeability of a transient linear poroelastic model of a confined GW aquifer. The Bayesian solution of this inverse problem takes the form of a posterior probability density of the permeability. Exploring this posterior using classical Markov chain Monte Carlo (MCMC) methods is computationally prohibitive due to the large dimension of the discretized permeability field and the expense of solving the poroelastic forward problem. However, in many partial differential equation (PDE)‐based Bayesian inversion problems, the data are only informative in a few directions in parameter space. For the poroelasticity problem, we prove this property theoretically for a one‐dimensional problem and demonstrate it numerically for a three‐dimensional aquifer model. We design a generalized preconditioned Crank‐Nicolson (gpCN) MCMC method that exploits this intrinsic low dimensionality by using a low‐rank‐based Laplace approximation of the posterior as a proposal, which we build scalably. The feasibility of our approach is demonstrated through a real GW aquifer test in Nevada. The inherently two‐dimensional nature of InSAR surface deformation data informs a sufficient number of modes of the permeability field to allow detection of major structures within the aquifer, significantly reducing the uncertainty in the pressure and the displacement quantities of interest. Key Points: Using Interferometric Synthetic Aperture Radar data reduces the uncertainty in selected quantities of interest compared to using prior knowledge onlyThe preconditioned Crank‐Nicolson (pCN) Markov chain Monte Carlo (MCMC) method is extended to exploit posterior curvature and allow better chain mixingWe demonstrate the intrinsic low dimensionality of the poroelastic inverse problem that is, critical for the success of the MCMC method [ABSTRACT FROM AUTHOR]

Characterizing the properties of groundwater aquifers is essential for predicting aquifer response and managing groundwater resources. In this work, we develop a high‐dimensional scalable Bayesian inversion framework governed by a three‐dimensional quasi‐static linear poroelastic model to characterize lateral permeability variations in groundwater aquifers. We determine the maximum a posteriori (MAP) point of the posterior permeability distribution from centimeter‐level surface deformation measurements obtained from Interferometric Synthetic Aperture Radar (InSAR). The scalability of our method to high parameter dimension is achieved through the use of adjoint‐based derivatives, inexact Newton methods to determine the MAP point, and a Matérn class sparse prior precision operator. Together, these guarantee that the MAP point is found at a cost, measured in number of forward/adjoint poroelasticity solves, that is independent of the parameter dimension. We apply our methodology to a test case for a municipal well in Mesquite, Nevada, in which InSAR and GPS surface deformation data are available. We solve problems with up to 320,824 state variable degrees of freedom (DOFs) and 16,896 parameter DOFs. A consistent treatment of noise level is employed so that the aquifer characterization result does not depend on the pixel spacing of surface deformation data. Our results show that the use of InSAR data significantly improves characterization of lateral aquifer heterogeneity, and the InSAR‐based aquifer characterization recovers complex lateral displacement trends observed by independent daily GPS measurements. Key Points: A Bayesian framework was developed for characterizing heterogeneous aquifer properties from surface deformation dataScalable algorithms allow the inference of high‐dimensional discretized parameters in a transient fully coupled 3‐D poroelastic aquifer modelInSAR data significantly improve characterization of lateral aquifer heterogeneity at a test site in Nevada [ABSTRACT FROM AUTHOR]

The locations of wetting front and water table are key variables in an integrated surface‐groundwater modeling. In current land surface models, they are either diagnosed from pressure head profiles or predicted using certain parameterization schemes. Both approaches need to make assumptions on how pressure head changes vertically and numerical results may have large errors when these assumptions are violated. In this work, we propose a numerical framework, which can explicitly track the locations of wetting front and water table. Existing numerical solvers modeling soil water movement usually use water content, pressure head, or both of them as prognostic variables, while in this framework, two additional prognostic variables representing the locations of wetting front and water table are introduced. Besides, another two improvements are explored to make the framework stable and efficient. First, a new equivalent hydraulic conductivity formula accommodating nonlinear hydraulic curves is proposed. The formula is unconditionally stable and can yield oscillation‐free solution of Richards' equation in unsaturated homogeneous soils. Second, a mixed implicit‐explicit temporal discretization is adopted. It uses an implicit scheme initially at every time step to improve the stability and accuracy, while in case of failure in convergence it replaces the iterations with an explicit scheme to guarantee mass conservation. Based on all these improvements, the framework is potentially used to simulate variably saturated flow in stratified soils in land surface models. Seven numerical tests covering various flow situations, boundary conditions, and soil geometries are carried out and illustrate the characteristics of the proposed framework. Key Points: Locations of water table and wetting front are tracked explicitly in solving Richards' equationAn equivalent hydraulic conductivity formula, which can yield oscillation‐free solution of Richards' equation, is proposedA mixed implicit‐explicit temporal discretization is adopted to improve the stability and guarantee mass conservation of water [ABSTRACT FROM AUTHOR]

A closure relation for capillary pressure plays an important role in the formulation of both traditional and evolving models of two‐fluid‐phase flow in porous medium systems. We review the traditional approaches to define capillary pressure, to describe it mathematically, to determine parameters for this relation, and to constrain the domain of applicability of this relation. In contrast to the traditional approach, we provide a rigorous, multiscale definition of capillary pressure, define the state domain of interest in practice, summarize computational and experimental approaches to investigate the system state, and apply the methods for two‐fluid states in a model ink bottle system, the classical Finney pack of spheres, and a synthetic sphere pack system. The results of these applications show that a state equation exists that describes capillary pressure without hysteresis. This state equation parameterizes a function that describes the nonwetting phase volume fraction in terms of the capillary pressure, the interfacial area, and the specific Euler characteristic of the nonwetting phase. Furthermore, this state equation applies over the complete range of conditions encountered in practice, and it applies under both equilibrium and dynamic conditions. This state equation involving capillary pressure forms an important foundation for the development of the next generation of macroscale two‐fluid‐phase flow models in porous medium systems. Key Points: A hysteretic‐free capillary pressure state equation is developedThe state equation is validated using both analytical and numerical simulation approachesThis state equation applies under both equilibrium and dynamic conditions [ABSTRACT FROM AUTHOR]

Existing analytical and semianalytical solutions for density‐driven flow (DDF) in porous media are limited to 2‐D domains. In this work, we develop a semianalytical solution using the Fourier Galerkin method to describe DDF induced by salinity gradients in a 3‐D porous enclosure. The solution is constructed by deriving the vector potential form of the governing equations and changing variables to obtain periodic boundary conditions. Solving the 3‐D spectral system of equations can be computationally challenging. To alleviate computations, we develop an efficient approach, based on reducing the number of primary unknowns and simplifying the nonlinear terms, which allows us to simplify and solve the problem using only salt concentration as primary unknown. Test cases dealing with different Rayleigh numbers are solved to analyze the solution and gain physical insight into 3‐D DDF processes. In fact, the solution displays a 3‐D convective cell (actually a vortex) that resembles the quarter of a torus, which would not be possible in 2‐D. Results also show that 3‐D effects become more important at high Rayleigh number. We compare the semianalytical solution to research (Transport of RadioACtive Elements in Subsurface) and industrial (COMSOL Multiphysics®) codes. We show cases (high Raleigh number) where the numerical solution suffers from numerical artifacts, which highlight the worthiness of our semianalytical solution for code verification and benchmarking. In this context, we propose quantitative indicators based on several metrics characterizing the fluid flow and mass transfer processes and we provide open access to the source code of the semianalytical solution and to the corresponding numerical models. Key Points: A 3‐D semianalytical solution for the density‐driven flow model is developed, for the first time, using Fourier series methodThe solution is used to gain physical insight into 3‐D density‐driven flow processes in the case of horizontal crossed density gradientsNumerical simulations, using COMSOL and advanced research code, show the worthiness of the semianalytical solution for benchmarking 3‐D codes [ABSTRACT FROM AUTHOR]