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• A high-speed camera observed gravity-driven unstable finger formation. • At the fingertip, water advances intermittently one pore at a time. • Pore invasions occur at high velocities. • High velocities increase dynamic contact angles. • Greater contact angles reduce the matric potential at the front and cause instability. The imbibition of fluids in porous media has been studied widely. Still, processes of preferential flow under gravity due to instability at the wetting front, crucial in groundwater contamination, have yet to be fully understood. Recent theories using dynamic contact angles could describe unstable flow phenomena but have not been proven experimentally. Therefore, infiltration experiments in small sand-filled chambers were conducted to explore the effect of dynamic contact angles. A high-speed camera recorded pore invasion at the unstable imbibition liquid front. A tensiometer recorded the matric potential. The results show that water moved in milliseconds through a small pore at the wetting front, followed by a stationary period. The maximum observed pore water velocity was 25 cm/s, exceeding the saturated hydraulic conductivity by three orders of magnitude. The relationship between dynamic contact angle and water velocity found by Hoffman in glass tubes could describe which was observed in soils. [Display omitted] [ABSTRACT FROM AUTHOR]

• Preferential pathways at the Darcy scale can be characterized by their bifurcations, and USR formation formed by their tortuous path. • The bifurcation fraction, USR, tortuosity, and fractal dimension scale with the conductivity heterogeneity. • The same power law exponent captures the scaling of bifurcation fraction, USR, tortuosity, and fractal dimension. Transport in porous media on the Darcy scale can be both Fickian and non-Fickian, an outcome dependent on the degree of homogeneity of the hydraulic conductivity pattern, as well as the boundary conditions and flow rate. The non-Fickian manifestation is generally associated with heterogeneous media, which promotes the formation of preferential pathways that funnel the transport. Yet this funneling occurs in both weakly and strongly heterogeneous domains, through a mechanism that is yet to be fully characterized. We model Darcy-scale transport using a particle tracking code (PT) that samples a 2D, lognormally distributed conductivity field with a variance representing a range of homogeneous to heterogeneous domains. We find that the resulting preferential pathways tend to split into more pathways (bifurcations), leaving regions into which particles do not invade, which we refer to as "under sampled regions" (USR), while forming a tortuous path. The fraction of bifurcations decreases downstream, reaching an asymptotic value, with a trend that can be fitted as a power-law of the variance. We show that the same power-law exponent relating the bifurcations to the variance holds true for the USR fraction, tortuosity, and fractal dimension with the same variance. An extension of our work is also presented for varying correlation length of the conductivity spatial distribution. We further expand our analysis to a case of impermeable fraction in a uniform conductivity field and show that the power-law fit still holds. Key Points Preferential pathways at the Darcy scale can be characterized by their bifurcations, and USR formation formed by their tortuous path. The bifurcation fraction, USR, tortuosity, and fractal dimension scale with the conductivity heterogeneity. The same power law exponent captures the scaling of bifurcation fraction, USR, tortuosity, and fractal dimension. [ABSTRACT FROM AUTHOR]

• Physics Informed Neural Networks (PINNs) is a promising technique. • Current implementation of PINNs face convergence issues in heterogeneous domains. • A new implementation is proposed based on the mixed formulation of the equations. • The new implementation simulate flow, regardless the level and type of heterogeneity. • It provide high accurate results without additional learning time. Current implementations of Physics Informed Neural Networks (PINNs) can experience convergence problems in simulating fluid flow in porous media with highly heterogeneous domains. The objective of this work is to develop an accurate implementation of PINNs to model fluid flow in heterogeneous porous media that avoids alternative approaches in the literature that tend to impose unphysical continuity of the hydraulic conductivity field. The proposed implementation is inspired by the mixed formulation of the governing equations, which separates the mass continuity equation and Darcy's law. This methodology has been used with the mixed finite element method and is known to provide accurate pressure and velocity fields in highly heterogeneous domains. In this work, a similar methodology is applied to PINNs, where the training loss function is based on the decoupled continuity equation and Darcy's law. The separation of the continuity equation and Darcy's law allows for calculating the residual term in the learning loss function without evaluating the spatial derivatives of the discontinuous hydraulic conductivity and associated non-differentiable functions. This approach provides more accurate automatic differentiation of the neural networks for pressure and velocity fields. The new implementation of PINNs can be applied to simulate flow in porous media, regardless of the type and level of heterogeneity with a discontinuous hydraulic conductivity distribution. A variety of structures of the new implementation are investigated. Numerical experiments show that the structure based on different neural networks for the pressure and each component of the velocity field provides the highest accuracy with equivalent training time as other structures. The proposed methodology presents a novel approach to broaden the scope of PINNs in modeling fluid flow in porous media, while preserving the precise representation of the domain's discontinuous features. [ABSTRACT FROM AUTHOR]

Characterization of flow topology is essential to understand the effects of the heterogeneity and dimensionality of geological formations on the mixing of inert solute clouds in these same geological formations. In this work, we numerically study two indicators of flow topology, the averaged vorticity magnitude | ω | av and the averaged positive second invariant Q av of the deformation tensor ∇ u , in steady Darcy flows through exponentially correlated lognormal hydraulic conductivity fields K. Our numerical results allow us to establish the relationships between the two indicators considered here and the hydraulic conductivity variance σ 2 in 2D and 3D, highlighting the role played by the spatial structure of these porous media on flow topology, and indirectly on mixing. This work leads us to assess the maximum dilution index E max , indicator of mixing, theoretically known to increase monotonically in steady Darcy flows through isotropic heterogeneous porous media. Our numerical results allow us to test this hypothesis by establishing the relationship between the slope a of maximum dilution index E max and the averaged positive second invariant Q av of deformation tensor ∇ u. The parameters of this relationship depend on molecular diffusion and dimensionality of problem considered. [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]

To study subsurface microbial processes, a coupled model which has been developed within a Thermal-Hydraulic-Chemical-Mechanical (THCM) framework is presented. The work presented here, focuses on microbial transport, growth and decay mechanisms under the influence of multiphase flow and bio-geochemical reactions. In this paper, theoretical formulations and numerical implementations of the microbial model are presented. The model has been verified and also evaluated against relevant experimental results. Simulated results show that the microbial processes have been accurately implemented and their impacts on porous media properties can be predicted either qualitatively or quantitatively or both. The model has been applied to investigate biofilm growth in a sandstone core that is subjected to a two-phase flow and variable pH conditions. The results indicate that biofilm growth (if not limited by substrates) in a multiphase system largely depends on the hydraulic properties of the medium. When the change in porewater pH which occurred due to dissolution of carbon dioxide gas is considered, growth processes are affected. For the given parameter regime, it has been shown that the net biofilm growth is favoured by higher pH; whilst the processes are considerably retarded at lower pH values. The capabilities of the model to predict microbial respiration in a fully coupled multiphase flow condition and microbial fermentation leading to production of a gas phase are also demonstrated. [ABSTRACT FROM AUTHOR]

The porous medium equation (PME) is a generalization of the traditional Boussinesq equation for hydraulic conductivity as a power law function of height. We analyze the horizontal recharge of an initially dry unconfined aquifer of semi-infinite extent, as would be found in an aquifer adjacent a rising river. If the water level can be modeled as a power law function of time, similarity variables can be introduced and the original problem can be reduced to a boundary value problem for a nonlinear ordinary differential equation. The position of the advancing front is not known ahead of time and must be found in the process of solution. We present an analytical solution in the form of a power series, with the coefficients of the series given by a recurrence relation. The analytical solution compares favorably with a highly accurate numerical solution, and only a small number of terms of the series are needed to achieve high accuracy in the scenarios considered here. We also conduct a series of physical experiments in an initially dry wedged Hele-Shaw cell, where flow is modeled by a special form of the PME. Our analytical solution closely matches the hydraulic head profiles in the Hele-Shaw cell experiment. [ABSTRACT FROM AUTHOR]

In this study, a numerical manifold method (NMM) model is developed to analyze flow in porous media with discrete fractures in a non-conforming mesh. This new model is based on a two-cover-mesh system with a uniform triangular mathematical mesh and boundary/fracture-divided physical covers, where local independent cover functions are defined. The overlapping parts of the physical covers are elements where the global approximation is defined by the weighted average of the physical cover functions. The mesh is generated by a tree-cutting algorithm. A new model that does not introduce additional degrees of freedom (DOF) for fractures was developed for fluid flow in fractures. The fracture surfaces that belong to different physical covers are used to represent fracture flow in the direction of the fractures. In the direction normal to the fractures, the fracture surfaces are regarded as Dirichlet boundaries to exchange fluxes with the rock matrix. Furthermore, fractures that intersect with Dirichlet or Neumann boundaries are considered. Simulation examples are designed to verify the efficiency of the tree-cutting algorithm, the calculation's independency from the mesh orientation, and accuracy when modeling porous media that contain fractures with multiple intersections and different orientations. The simulation results show good agreement with available analytical solutions. Finally, the model is applied to cases that involve nine intersecting fractures and a complex network of 100 fractures, both of which achieve reasonable results. The new model is very practical for modeling flow in fractured porous media, even for a geometrically complex fracture network with large hydraulic conductivity contrasts between fractures and the matrix. [ABSTRACT FROM AUTHOR]

Modeling flow and solute transport in large-scale (e.g.) on the order of 10 3 m heterogeneous porous media involves substantial computational burden. A common approach to alleviate the problem is to utilize an upscaling method that generates models that require less intensive computations. The method must also preserve the important properties of the spatial distribution of the hydraulic conductivity ( K ) field. We use an upscaling method based on the wavelet transformations (WTs) that coarsens the computational grid based on the spatial distribution of K . The technique is applied to a porous formation with broadly distributed and correlated K values, and the governing equation for solute transport in the formation is solved numerically. The WT upscaling preserves the resolution of the initial highly-resolved computational grid in the high K zones, as well as that of the zones with sharp contrasts between the neighboring K , whereas the low- K zones are averaged out. To demonstrate the accuracy of the method, we simulate fluid flow and nonreactive solute transport in both the high-resolution and upscaled grids, and compare the concentration profiles and the breakthrough times. The results indicate that the WT upscaling of a K field generates non-uniform upscaled grids with a number of grid blocks that on average is about two percent of the number of the blocks in the original high-resolution computational grids, while the concentration profiles, the breakthrough times and the second moment of the concentration distribution, computed for both models, are virtually identical. A systematic parametric study is also carried out in order to investigate the sensitivity of the method to the broadness of the K field, the nature of the correlations in the field (positive versus negative), and the size of the computational grid. As the broadness of the K field and the size of the computational domain increase, better agreement between the results for the high-resolution and upscaled models is obtained. [ABSTRACT FROM AUTHOR]

A common way to simulate fluid flow in porous media is to use Lattice Boltzmann (LB) methods. Permeability predictions from such flow simulations are controlled by parameters whose settings must be calibrated in order to produce realistic modelling results. Herein we focus on the simplest and most commonly used implementation of the LB method: the single-relaxation-time BGK model. A key parameter in the BGK model is the relaxation time τ which controls flow velocity and has a substantial influence on the permeability calculation. Currently there is no rigorous scheme to calibrate its value for models of real media. We show that the standard method of calibration, by matching the flow profile of the analytic Hagen-Poiseuille pipe-flow model, results in a BGK-LB model that is unable to accurately predict permeability even in simple realistic porous media (herein, Fontainebleau sandstone). In order to reconcile the differences between predicted permeability and experimental data, we propose a method to calibrate τ using an enhanced Transitional Markov Chain Monte Carlo method, which is suitable for parallel computer architectures. We also propose a porosity-dependent τ calibration that provides an excellent fit to experimental data and which creates an empirical model that can be used to choose τ for new samples of known porosity. Our Bayesian framework thus provides robust predictions of permeability of realistic porous media, herein demonstrated on the BGK-LB model, and should therefore replace the standard pipe-flow based methods of calibration for more complex media. The calibration methodology can also be extended to more advanced LB methods. [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]

This paper presents a nonlocal, derivative free model for transient flow in unsaturated, heterogeneous, and anisotropic soils. The formulation is based on the peridynamic model for solid mechanics. In the proposed model, flow and changes in moisture content are driven by pairwise interactions with other points across finite distances, and are expressed as functional integrals of the hydraulic potential field. Peridynamic expressions of the rate of change in moisture content, moisture flux, and flow power are derived, as are relationships between the peridynamic and the classic hydraulic conductivities; in addition, the model is validated. The absence of spacial derivatives makes the model a good candidate for flow simulations in fractured soils and lends itself to coupling with peridynamic mechanical models for simulating crack formation triggered by shrinkage and swelling, and assessing their potential impact on a wide range of processes, such as infiltration, contaminant transport, and slope stability. [ABSTRACT FROM AUTHOR]

Flows showing complex topology are ubiquitous in natural systems. However, contrasting evidence exists on the helical nature of flow in porous media and on the occurrence of groundwater whirls. We analyze the topology of steady-state flow fields in porous media, highlighting the importance of considering the three-dimensionality of the flow field to properly capture the complexity of the system dynamics controlling the deformation of material surfaces, which is widely recognized as the main driver of mixing. We use the helicity density as topological measure and investigate the necessary and sufficient conditions to obtain non-zero helicity density for Darcy flow. We show that helical groundwater flow can develop in both homogeneous and heterogeneous porous media, provided that the hydraulic conductivity is anisotropic. In the homogeneous case, the additional condition of non-vanishing mixed second spatial derivatives of hydraulic head is required, while in heterogeneous media, helical flow may occur even when the hydraulic gradient is uniform. We present illustrative examples of complex flow topology in three-dimensional porous media and discuss the computed streamline patterns and their potential implications for mixing processes. [ABSTRACT FROM AUTHOR]

Abstract: The hydraulic conductivity of heterogeneous porous media depends on the distribution function and the geometry of local conductivities at the smaller scale. There are various approaches to estimate the effective conductivity K eff at the larger scale based on information about the small scale heterogeneity. A critical geometric property in this ‘upscaling’ procedure is the spatial connectivity of the small-scale conductivities. We present an approach based on the Euler-number to quantify the topological properties of heterogeneous conductivity fields, and we derive two key parameters which are used to estimate K eff. The required coefficients for the upscaling formula are obtained by regression based on numerical simulations of various heterogeneous fields. They are found to be generally valid for various different isotropic structures. The effective unsaturated conductivity function K eff (ψ m) could be predicted satisfactorily. We compare our approach with an alternative based on percolation theory and critical path analysis which yield the same type of topological parameters. An advantage of using the Euler-number in comparison to percolation theory is the fact that it can be obtained from local measurements without the need to analyze the entire structure. We found that for the heterogeneous field used in this study both methods are equivalent. [Copyright &y& Elsevier]

• Finite particle method (FPM) simulates groundwater solute transport. • FPM outperforms smoothed particle hydrodynamics (SPH) with irregular particles. • FPM improves SPH by using a modified kernel gradient for concentration gradient. When a smoothed particle hydrodynamics (SPH) method, a Lagrangian and meshfree numerical scheme, is used to solve the advection-dispersion equation (ADE), SPH solutions may not be accurate when particles are irregularly distributed, i.e., the distance between two neighbor particles varies irregularly in a simulation domain. Particle irregularity may be caused by nonuniform groundwater flow in a heterogeneous field of hydraulic conductivity. This study explores for the first time whether the Finite Particle Method (FPM) can provide more accurate ADE solutions than SPH does for irregularly distributed particles. FPM is similar to SPH in theory, but uses a modified kernel gradient to construct a SPH approximation of solute concentration gradients. Performance of SPH and FPM with irregularly distributed particles is evaluated by using two numerical cases. The first case considers only diffusive transport, and has an analytical solution for the evaluation. The second case considers both advection and dispersion, and uses a numerical solution as a reference for the evaluation. For each of the two cases, several numerical experiments are conducted using multiple sets of irregularly distributed particles with different levels of particle irregularity due to different levels of heterogeneity of hydraulic conductivity. Numerical results indicate that, for the numerical experiments of this study, FPM outperforms SPH to yield more accurate ADE solutions. However, FPM solutions are still subject to numerical errors, and the errors increase when the level of heterogeneity of hydraulic conductivity increases. Further improvement of FPM is warranted in a future study. [ABSTRACT FROM AUTHOR]

Three-dimensional (3D) natural convection (NC) processes in heterogeneous porous media and associated energy losses and mixing processes are still poorly understood. Studies are limited to two-dimensional domains because of computational burden, worsened by heterogeneity, which may demand grid refinement at high permeability zones for accurate evaluation of buoyancy forces. We develop a meshless Fourier series (FS) solution of the natural convection problem in a porous enclosure driven by thermal or compositional variations. We derive the vector potential formulation of the governing equations for vertical and horizontal heterogeneity of hydraulic conductivity and implement an efficient method to solve the spectral system with an optimized number of Fourier modes. 3D effects are induced either by heterogeneity or variable boundary conditions. The developed FS solution is verified against a finite element solution obtained using COMSOL Multiphysics. We evaluate entropy generation (viscous dissipation and mixing) indicators using FS expansions and assess how they are affected by heterogeneity. We define a large-scale Rayleigh number to account for heterogeneity by adopting an arithmetic average effective permeability. The FS solution is used to investigate the effect of the large-scale Rayleigh number and level of heterogeneity on NC processes and energy losses. Results show that increasing the Rayleigh number intensifies fluid flow, thus enhancing convective transfer, which causes a dramatic increase in total entropy generation. Both viscous dissipation and mixing (and thus chemical reactions in the solute transport case) increase. The third dimension effect, which also enhances flow and entropy indicators, is more pronounced at high Rayleigh numbers. Surprisingly, entropy variation indicators remain virtually unchanged in response to changes in heterogeneity, for fixed Rayleigh number, which we attribute to the arithmetic average permeability being indeed appropriate for NC in 3D. This study not only explores the effect of Rayleigh number and heterogeneity on natural convection processes and the associated entropy generation and mixing processes, but also provides a highly accurate solution that can be used for codes benchmarking. [ABSTRACT FROM AUTHOR]

• Flow and transport equations for porous media are solved by new random walk schemes. • Global random walk (GRW) algorithms are designed as L-schemes for Richards equation. • The GRW L-schemes solve nonlinearities and model unsaturated/saturated transitions. • The coupled flow-transport solvers are robust and second-order convergent in space. • GRW schemes for both flow and transport are practically free of numerical diffusion. In this article, we present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the L -scheme developed in finite element/volume approaches. The resulting GRW L -schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards' equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes. [ABSTRACT FROM AUTHOR]

• Log-conductivity (Y) is modeled as a generalized sub-Gaussian (GSG) field. • Average longitudinal spreading is smaller for GSG than for Gaussian Y fields. • Plume stretching in GSG fields is enhanced with respect to Gaussian fields. • Local dispersion masks identification of impacts of non-Gaussianity on transport. • The GSG nature of Y can be difficult to identify relying only on transport metrics. There are increasing evidences that probability distributions and associated statistical moments of a variety of hydrogeological and soil science variables and their spatial increments display distinctive scale-dependent features that are not captured by a typical Gaussian model. A Generalized Sub-Gaussian (GSG) model is able to capture key aspects of this pattern. We present the results of a suite of computational analyses set in a Monte Carlo framework and aimed at assessing the impact of a GSG structure of log hydraulic conductivity (Y) on transport of a conservative solute through a three-dimensional bounded porous medium under steady-state saturated Darcy flow. Our results indicate that the longitudinal spreading of a plume is on average significantly smaller for Sub-Gaussian than for Gaussian Y fields. Otherwise, the velocity field arising from a Sub-Gaussian Y field induces enhanced plume stretching with respect to what can be observed in a Gaussian Y setting, this aspect potentially influencing the strength of solute mixing within these two types of conductivity domains. We also find that, in some cases, it may be difficult to identify the nature of the underlying conductivity field relying solely on observations of solute concentrations migrating within the system. In this regard, we show that the action of local dispersion tends to mask the influence of Sub-Gaussianity on major transport metrics. [ABSTRACT FROM AUTHOR]

• Represents local heterogeneity at micro-scale level with an average error variance less than 0.02. • The FSM shows consistent results at well node locations with changing grid dimensions. • ROM is computationally efficient and replicates FSM with desired accuracy. • Both FSM and ROM shows good mesh convergence. Groundwater, being a vital component of the natural water resource system, needs continuous monitoring and dynamic management strategies. That said, we require computationally inexpensive groundwater flow models for repetitive solutions with desirable accuracy under budgetary limitation(s). Natural aquifer systems inherit strong heterogeneity at local scales. In this work, we have proposed ordinary kriging-based sequential algorithm for generating replicates of randomly distributed heterogeneous hydraulic conductivity field (Monte Carlo method-based algorithm) conditioned by field values from sampled locations in an irregular-unstructured grid system. Finite Volume method-based groundwater models often encounter difficulties with the representation of point source/sink terms operating within the domain. In this paper, we have proposed an irregular-unstructured grid Finite Volume discretization technique for overcoming the singularity of point source/sink term to yield a consistent output with different grid dimensions. Furthermore, full-system groundwater models often come with a substantial computational burden. Hence, reduction in model order cuts down the computational expenses (in terms of CPU time and usage) to a significant level. We have also put forth a model order reduction methodology for three different illustrative pumping tests. The proposed framework for the model order reduction projects the governing groundwater flow equation onto a set of identified patterns or orthonormal basis functions, applying the Galerkin Projection method to compute a vector of time-dependent coefficients. We have performed pattern identification by Singular Value Decomposition (SVD) of snapshots of full-system model simulation data at selected time instants within the pumping test time domain. The numerical results of the proposed reduced-order models show a good approximation of the full-system models at a comparatively lesser computational time. The accuracy and efficiency of the models attempt to ensure their potential applicability for identifying groundwater dynamics. [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]

• A convolutional neural network (CNN) is built to estimate macro-dispersivity from hydraulic conductivity field directly. • The validation of the CNN shows that most estimations fall within ± 20% error range for conductivity fields with moderate heterogeneity. • For a given size of training datasets, the performance of estimating drops with an increase in the heterogeneity of conductivity field. • The CNN trained by conductivity fields with high heterogeneity can achieve superior performance in estimations for fields with less heterogeneity. Building a quantitative relation between the spatial heterogeneity of the hydraulic conductivity fields and the macroscale behavior of solute transport is fundamental for groundwater environment problem. In this work, the deep learning technique is explored to build the functional mapping between the hydraulic conductivity field and the longitudinal macro-dispersivity. We examine the capability of the deep neural network in estimating macro-dispersivities of conductivity fields with different variances. The universality of the trained deep neural network is investigated. Comparisons of the neural network results and the reference values (macro-dispersivities from transport simulation) suggest the promising potential of deep learning technique in porous media with moderate heterogeneity. For a given size of training datasets, the deep neural network produces better macro-dispersivity estimation for the conductivity field with smaller variance. The trained neural network by conductivity fields with larger variance has stronger universality for macro-dispersivity estimation. This study demonstrates that deep neural network can be an effective alternative for estimating macroscale behavior of solute transport by directly interpreting hydraulic conductivity fields. [ABSTRACT FROM AUTHOR]

• Model for effective conductivity of highly heterogeneous random media. • Computational studies for conductivity of random fields. • Effects of spatial variability of flow fields. • Upscaling the effects of percolation on the macro scale conductivity of porous media. We develop a phenomenological model for the effective conductivity of porous media that consist of two distinct materials characterized by different values of hydraulic conductivity. Our focus is two-fold: first, to relate the terms and parameters of macroscale conductivity to the spatial variability of local flow and conductivity observed in computational experiments, and second to develop a reduced order model for effective conductivity based on those observations. Darcys law is assumed to hold at local (mesoscopic) and large (macroscopic) scales. At the mesoscale a composite medium is a configuration of irregular subdomains consisting of two different materials. The resulting heterogeneities force fluid to flow along irregular paths, and this produces spatial variability in the magnitude and the orientation of the Darcy flux. This variability, along with the macroscopic effective conductivity, depends on the proportion of the total volume allocated to each material, the ratio of the two conductivity values, and the spatial connectivity of material subvolumes. Computational experiments indicate that the effects of heterogeneity are most pronounced when the two conductivity values are very different, and when the volume fraction of the more conductive material is near a percolation threshold. The percolation threshold is the critical volume fraction of the more conductive material at which the medium is no longer traversed by connected paths through that material. The percolation threshold determines the existence of three regimes in effective conductivity, two where the effective conductivity obeys power laws and is dominated by one of the materials, and a third intermediate regime that interpolates between the power laws. [ABSTRACT FROM AUTHOR]

• Comparison of the micro-CT images obtained under the spreading and nonspreading conditions allowed us to gain a significantly improved understanding of the governing displacement physics and the matrix-fracture interactions in fractured rocks. • During secondary gas injection under the spreading condition, the oil spread and formed thick and stable layers between brine and gas. These spreading layers were present not only in the matrix pores but also in the fracture conduit. • The oil phase in the matrix established a hydraulic connectivity through the connected and stable spreading layers with the oil in the fracture, thereby allowing the oil in the matrix to drain into the fracture, which resulted in higher oil recovery. • However, infrequent oil layers observed under nonspreading condition were not stable and collapsed easily as capillary pressure was increased during gas injection. This meant that oil had poor connectivity that hindered oil movement from the matrix to the fracture, leading to lower oil recovery compared to that of the spreading system. This paper presents the results of a detailed experimental study performed to examine fluid flow in a water-wet fractured sandstone rock. Using high-resolution X-ray micro computed tomography technique, we systematically investigate the pore-scale displacement mechanisms and governing interactions between the matrix and fracture during gas injection. We perform two sets of flow experiments using brine/Soltrol 170 (spreading oil)/nitrogen and brine/decalin (nonspreading oil)/nitrogen fluid systems to probe the possible beneficial role of spreading phenomena in transferring oil from the matrix to the fracture during gas injection. Gas injection after primary oil drainage was used to generate a wide range of oil saturations and pore fluid arrangements with both fluid systems. Direct visualization of the pore fluid occupancies at different oil saturations in the medium reveals the significant role that the spreading oil layers play in maintaining the hydraulic conductivity of the oil phase between the matrix and the fracture. This mainly takes place at low remaining oil saturations under the spreading condition where layer drainage, in the presence of stable and connected spreading oil layers, is the dominant displacement mechanism. In the case of high remaining oil saturations, it is observed that the oil movement is primarily governed by piston-like displacements as well as the ability of the gas phase to access the pore elements adjacent to the fracture. Under the nonspreading condition, oil cannot maintain its connectivity due to the absence of the spreading layers, leading to higher residual oil saturations in the matrix and lower ultimate oil recovery. [ABSTRACT FROM AUTHOR]