Random Hydraulic Conductivity Research Articles

Comment on “Modeling groundwater flow with random hydraulic conductivity using radial basis function partition of unity method” by Shile et al. (2024)

The goal of this article is to explain some striking discrepancies between the Monte Carlo inferences for flow in heterogeneous aquifers presented in (Shile et al., 2024) and reliable and verifiable results previously published. Comparisons with statistical inferences done within the same numerical setup and using the same benchmark codes demonstrate that quantifiable aspects, such as a too small departure from the linear theory and the decrease of the standard deviations with the increase of the aquifer heterogeneity, cannot be produced with the tools used by the authors.

Modeling groundwater flow with random hydraulic conductivity using radial basis function partition of unity method

Simulating groundwater flows in heterogeneous aquifers is one of the most widely studied problems. The heterogeneity is modeled through random hydraulic conductivity fields log-normally distributed. In this paper, we aim to generate the realization of the log-normal hydraulic conductivity by summing up a finite number of random periodic modes with the Kraichnan algorithm. To address Neumann conditions, we propose a change of variables that results in Dirichlet boundary conditions along all boundaries of the computational domain. Then, we suggest using the radial basis function partition of unity method (RBFPUM) and RBFPUM with QR factorization (RBFPUM-QR) to analyze groundwater flow in the heterogeneous porous medium. We evaluate the performance of the proposed methods by increasing the number of random modes and the variance of the log-hydraulic conductivity fields using Gaussian and exponential correlation. In addition, we verify the effectiveness of the presented approaches in one and two dimensions using the method of manufactured solutions. Furthermore, we provide statistical inferences through Monte Carlo simulations. Numerical results demonstrate that RBFPUM-QR performs well in handling higher heterogeneity in both single realizations and Monte Carlo simulations.

Conditional generative adversarial networks for groundwater contamination characterization and source identification

Accurate characterization of sources and spatial distribution of contamination remains crucial in groundwater survey and remediation efforts. Nevertheless, conventional geo-statistical methods, like kriging, usually produce overly smooth concentration fields that violate the transport physics. Furthermore, the identification of contaminant sources based on kriging methods may be misleading. As a deep learning approach, Generative adversarial networks (GANs) is being increasingly employed to extract patterns and insights from intricate geospatial data. The conditional variant of GANs (CGANs), can reconstruct complete spatial data from sparse point observations. In this study, we propose a deep learning framework with CGANs for the purpose of groundwater contamination characterization and source identification. In ensure compliance with the principles of transport physics, we use a numerical transport model as the underlying prior model for training the CGANs. Through the adaptation of network architecture, the generator transforms low-dimensional random variables and observation data into concentration images, while the discriminator is responsible for extracting source information from these images. The developed framework is tested through two numerical cases with channelized random hydraulic conductivity fields, which pose a challenge due to the complexity of the concentration fields. The results demonstrate that the proposed method can give accurate results for both contaminant characterization and source identification in groundwater. CGANs excel in reconstructing concentration fields compared to traditional interpolation methods such as Kriging and inverse distance weighting (IDW). Additionally, CGANs offer the capability to estimate uncertainty and exhibit a certain level of error robustness.

A space-time multigrid method for poroelasticity equations with random hydraulic conductivity

In this work, we propose a space-time multigrid method for solving a finite difference discretization of the linear Biot’s model in two space dimensions considering random hydraulic conductivity. This method is an extension of the one developed by Franco et al. [1] and which was applied to the heat equation. Particularly for the poroelasticity problem, we need to use the Tri-Diagonal Matrix Algorithm (TDMA) for systems of equations with multiple variables (block TDMA) solver on the planes xt and yt with zebra-type wise-manner (the domain is divided into planes and solve each odd plane first and then each even plane independently [2]). On the planes xy, the F-cycle multigrid with fixed-stress smoother is improved by red-black relaxation. A fixed-stress smoother is compound solver, where the mechanical and flow parts are addressed by two distinctive solvers and symmetric Gauss-Seidel iteration. This is the basis of the proposed method which does not converge only with the standard line-in-time red-black solver [1]. This combination allows the code parallelization. The use of standard coarsening associated with line or plane smoothers in the strong coupling direction allows an application in real world problems and, in particular, the random hydraulic conductivity case. This would not be possible using the standard space-time method with semicoarsening in the strong coupling direction. The spatial and temporal approximations of the problem are performed, respectively, by using Central Difference Scheme (CDS) and implicit Euler methods. The robustness and excellent performance of the multigrid method, even with random hydraulic conductivity, are illustrated with numerical experiments through the average convergence factor.

A benchmark for variably saturated variable-density variable-viscosity flow and solute transport in porous media

In natural environments, fluid density and viscosity can be affected by spatial and temporal variations of solute concentration, for example, due to saltwater intrusion in coastal aquifers, leachate infiltration from waste disposal sites, and upconing of saline water from deep aquifers. Potentially unstable situations may arise in which a dense fluid overlies a less dense fluid. This situation can produce instabilities manifested by dense plume fingers moving vertically downwards counterbalanced by vertical upward flow of the less dense fluid. The resulting free convection increases solute transport rates over large distances and times relative to constant-density flow. Unstable brine flow is further complicated if the porous medium is variably saturated. The results from a laboratory experiment of variably saturated variable-density flow and solute transport from Simmons et al. (2002) are used as the physical basis to define a new mathematical benchmark. This benchmark aims at realistically reproducing the experimental fingering patterns. Random hydraulic conductivity fields were used in the simulations as a numerical perturbation method to realistically mimic the observed dense plume fingering. The HydroGeoSphere code coupled with PEST are used to calibrate the parameter set that defines the benchmark. A grid convergence analysis is performed to obtain the adequate spatial and temporal discretizations. The new mathematical benchmark is useful for model comparison and testing of variably saturated variable-density flow in porous media. Simmons CT, Pierini ML, Hutson JL (2002) Laboratory investigation of variable-density flow and solute transport in unsaturated–saturated porous media. Transp Porous Media. 47(2): 215–244, 10.1023/A:1015568724369.

A Multigrid Multilevel Monte Carlo Method for Stokes–Darcy Model with Random Hydraulic Conductivity and Beavers–Joseph Condition

A Multigrid Multilevel Monte Carlo Method for Stokes–Darcy Model with Random Hydraulic Conductivity and Beavers–Joseph Condition

A stochastic collocation method based on sparse grids for a stochastic Stokes-Darcy model

<p style='text-indent:20px;'>In this paper, we develop a sparse grid stochastic collocation method to improve the computational efficiency in handling the steady Stokes-Darcy model with random hydraulic conductivity. To represent the random hydraulic conductivity, the truncated Karhunen-Loève expansion is used. For the discrete form in probability space, we adopt the stochastic collocation method and then use the Smolyak sparse grid method to improve the efficiency. For the uncoupled deterministic subproblems at collocation nodes, we apply the general coupled finite element method. Numerical experiment results are presented to illustrate the features of this method, such as the sample size, convergence, and randomness transmission through the interface.</p>

Numerical analysis of a second order ensemble algorithm for numerical approximation of stochastic Stokes–Darcy equations

Numerical approximation of stochastic Stokes–Darcy equations usually requires repeated sampling of the random hydraulic conductivity tensor and then simulating flow ensembles. In this setting, we propose an efficient, second order, ensemble algorithm for fast computation of the whole set of realizations of the stochastic Stokes–Darcy model corresponding to different random hydraulic conductivity tensor samples. The ensemble algorithm only requires the solution of two linear systems that have the same constant coefficient matrices for all realizations. We give a complete long time stability and convergence analysis for the method. Numerical experiments are presented to support theoretical results and demonstrate the application of the method.

Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media

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.

Groundwater Throughflow and Seawater Intrusion in High Quality Coastal Aquifers

High quality coastal aquifer systems provide vast quantities of potable groundwater for millions of people worldwide. Managing this setting has economic and environmental consequences. Specific knowledge of the dynamic relationship between fresh terrestrial groundwater discharging to the ocean and seawater intrusion is necessary. We present multi- disciplinary research that assesses the relationships between groundwater throughflow and seawater intrusion. This combines numerical simulation, geophysics, and analysis of more than 30 years of data from a seawater intrusion monitoring site. The monitoring wells are set in a shallow karstic aquifer system located along the southwest coast of Western Australia, where hundreds of gigalitres of fresh groundwater flow into the ocean annually. There is clear evidence for seawater intrusion along this coastal margin. We demonstrate how hydraulic anisotropy will impact on the landward extent of seawater for a given groundwater throughflow. Our examples show how the distance between the ocean and the seawater interface toe can shrink by over 100% after increasing the rotation angle of hydraulic conductivity anisotropy when compared to a homogeneous aquifer. We observe extreme variability in the properties of the shallow aquifer from ground penetrating radar, hand samples, and hydraulic parameters estimated from field measurements. This motived us to complete numerical experiments with sets of spatially correlated random hydraulic conductivity fields, representative of karstic aquifers. The hydraulic conductivity proximal to the zone of submarine groundwater discharge is shown to be significant in determining the overall geometry and landward extent of the seawater interface. Electrical resistivity imaging (ERI) data was acquired and assessed for its ability to recover the seawater interface. Imaging outcomes from field ERI data are compared with simulated ERI outcomes derived from transport modelling with a range of hydraulic conductivity distributions. This process allows for interpretation of the approximate geometry of the seawater interface, however recovery of an accurate resistivity distribution across the wedge and mixing zone remains challenging. We reveal extremes in groundwater velocity, particularly where fresh terrestrial groundwater discharges to the ocean, and across the seawater recirculation cell. An overarching conclusion is that conventional seawater intrusion monitoring wells may not be suitable to constrain numerical simulation of the seawater intrusion. Based on these lessons, we present future options for groundwater monitoring that are specifically designed to quantify the distribution of; (i) high vertical and horizontal pressure gradients, (ii) sharp variations in subsurface flow velocity, (iii) extremes in hydraulic properties, and (iv) rapid changes in groundwater chemistry. These extremes in parameter distribution are common in karstic aquifer systems at the transition from land to ocean. Our research provides new insights into the behaviour of groundwater in dynamic, densely populated, and ecologically sensitive coastal environments found worldwide.

Seepage Analysis of Railway Earthen Embankment in Mokama Using RFEM Considering Random Hydraulic Conductivity Field

In this study, seepage through the body of an earthen embankment for construction of railway lines in Mokama in Bihar is estimated using finite element method. Seepage of water through the body of earthen embankment adversely affects its stability. In order to check the reliability of the constructed railway embankment stretching for a long distance, Random Finite Element Method (RFEM) originally developed by Fenton and Griffiths (Fenton and Griffiths, 1996) is used. The influence of variability in spatially distributed hydraulic conductivity/permeability field of the embankment is investigated by carrying out seepage analysis. Local Averaging Subdivision (LAS) technique is used to build the random field of hydraulic conductivity of soil domain under consideration. Afterwards, Monte Carlo simulation is performed for 1000 such random fields. The mean of total flow rate $$\left( {m_{Q} } \right)$$ , mean of logarithmic of total flow rate $$\left( {m_{\ln Q} } \right)$$ and variance of logarithmic of total flow rate $$\left( {s_{\ln Q}^{2} } \right)$$ from seepage analyses for 1000 realizations of random field are reported. The results show that as the correlation length decreases, the output parameters $$m_{Q}$$ , $$m_{\ln Q}$$ and $$s_{\ln Q}^{2}$$ follow a decreasing trend and vice versa. These variations are studied for the embankment with and without drain. The effect of drain on the mean hydraulic conductivity and hydraulic gradient standard deviation of the embankment are investigated in which the embankment with drains is found safer than embankment without drain.

An artificial compressibility ensemble algorithm for a stochastic Stokes‐Darcy model with random hydraulic conductivity and interface conditions

SummaryWe propose and analyze an efficient ensemble algorithm with artificial compressibility (AC) for fast decoupled computation of multiple realizations of the stochastic Stokes‐Darcy model with random hydraulic conductivity (including the one in the interface conditions), source terms, and initial conditions. The solutions are found by solving three smaller decoupled subproblems with two common time‐independent coefficient matrices for all realizations, which significantly improves the efficiency for both assembling and solving the matrix systems. The fully coupled Stokes‐Darcy system can be first decoupled into two smaller subphysics problems by the idea of the partitioned time stepping, which reduces the size of the linear systems and allows parallel computing for each subphysics problem. The AC further decouples the velocity and pressure which further reduces storage requirements and improves computational efficiency. We prove the long time stability and the convergence for this new ensemble method. Three numerical examples are presented to support the theoretical results and illustrate the features of the algorithm, including the convergence, stability, efficiency, and applicability.

Seawater intrusion in heterogeneous coastal aquifers under flooding events

Coastal flooding due to storm surge and sea-level rise (SLR) affects seawater intrusion (SWI) in aquifers. Under coastal flooding events, SWI from both sea boundary and land-surface is expected to increase. SWI in coastal aquifers is also strongly influenced by geological heterogeneity characteristics. In this study, the combined impacts of fully geological heterogeneity and coastal flooding on SWI is quantified in a two-dimensional unconfined coastal aquifer. Three indices including the seawater toe location (xtoe), the mixing volume (vmv) and the SWI volume (vswiv) are used to assess the vulnerability to the intrusion of the seawater. A series of 50 log-normally distributed random hydraulic conductivity with a spherical correlation function is generated using the GCOSIM3D code. Then, numerical simulations of transient, saturated-unsaturated variable-density flow and solute transport, SUTRA, are performed in a stochastic Monte Carlo framework to understand how coastal flooding and geological heterogeneity effects can change the groundwater salinity.During storm surge events, seawater penetrates into the aquifer and leads to reduce temporary of the freshwater quality. The results indicate that the storm surge has a short-term impact on fresh groundwater. The aquifer is significantly influenced by SLR as a long-term effect while it causes a larger xtoe in comparison with pre-flooding events’ conditions. Also, the results obtained under gradual and instantaneous assumption of SLR show that the gradual assumption leads to a smaller xtoe which is more realistic. Comparison of the homogeneous and heterogeneous scenarios indicates the heterogeneity affects SWI status both in the presence and the absence of coastal flooding events. It causes reduction in xtoe and vswiv as well as increase in vmv. This study shows the significant value of considering a wide range of variations due to heterogeneity, in order to better manage the coastal fresh groundwater aquifers.

An efficient ensemble algorithm for numerical approximation of stochastic Stokes–Darcy equations

We propose and analyze an efficient ensemble algorithm for fast computation of multiple realizations of the stochastic Stokes–Darcy model with a random hydraulic conductivity tensor. The algorithm results in a common coefficient matrix for all realizations at each time step making solving the linear systems much less expensive while maintaining comparable accuracy to traditional methods that compute each realization separately. Moreover, it decouples the Stokes–Darcy system into two smaller sub-physics problems, which reduces the size of the linear systems and allows parallel computation of the two sub-physics problems. We prove the ensemble method is long time stable and first-order in time convergent under a time-step condition and two parameter conditions. Numerical examples are presented to support the theoretical results and illustrate the application of the algorithm.

An Effective Kalman Filter-Based Method for Groundwater Pollution Source Identification and Plume Morphology Characterization

The identification of unknown groundwater pollution sources and the characterization of pollution plume remains a challenging problem. In this study, we addressed this problem by a linked simulation-optimization approach. This approach couples a contaminant transport simulation model with a Kalman filter-based method to identify groundwater pollution source and characterize plume morphology. In the proposed methodology, the concentration field library, the covariance reduction with a Kalman filter, an alpha-cut technique of fuzzy set, and a linear programming model are integrated for solving this inverse problem. The performance of this methodology is evaluated on an illustrative groundwater pollution source identification problem. The evaluation considered the random hydraulic conductivity filed, erroneous monitoring data, a prior information shortage of potential pollution sources, and an unexpected and unknown pumping well. The identified results indicate that, under these conditions, the proposed Kalman filter-based optimization model can give satisfactory estimations to pollution sources and plume morphology for domains with small and moderate heterogeneity but cannot validate the transport in the relatively high heterogeneous field.

Efficient Uncertainty Quantification for Unconfined Flow in Heterogeneous Media with the Sparse Polynomial Chaos Expansion

In this study, we explore an efficient stochastic approach for uncertainty quantification of unconfined groundwater flow in heterogeneous media, where a sparse polynomial chaos expansion (PCE) surrogate model is constructed with the aid of the feature selection method. The feature selection method is introduced to construct a sparse PCE surrogate model with a reduced number of basis functions, which is accomplished by the least absolute shrinkage and selection operator-modified least angle regression and cross-validation. The training samples are enriched sequentially with the quasi-optimal samples until the results are satisfactory. In this study, we test the performance of the sparse PCE method for unconfined flow with the presence of random hydraulic conductivity and recharge, as well as pumping well. Numerical experiments reveal that, even with large spatial variability and high random dimensionality, the sparse PCE approach is able to accurately estimate the flow statistics with greatly reduced computational efforts compared to Monte Carlo simulations.

Solution of a stochastic Darcy equation by polynomial chaos expansion

This paper deals with solving a boundary value problem for the Darcy equation with a random hydraulic conductivity field.We use an approach based on polynomial chaos expansion in a probability space of input data.We use a probabilistic collocation method to calculate the coefficients of the polynomial chaos expansion. The computational complexity of this algorithm is determined by the order of the polynomial chaos expansion and the number of terms in the Karhunen–Loeve expansion. We calculate various Eulerian and Lagrangian statistical characteristics of the flow by the conventional Monte Carlo and probabilistic collocation methods. Our calculations show a significant advantage of the probabilistic collocation method over the directMonte Carlo algorithm.

Stochastic analysis of seepage under water-retaining structures

This paper investigated the problem of confined flow under dams and water-retaining structures using stochastic modelling. The approach advocated in the study combined a finite-element method based on the equation governing the dynamics of incompressible fluid flow through a porous medium with a random field generator that generates random hydraulic conductivity based on log normal probability distribution. The resulting model was then used to analyse the confined flow under a hydraulic structure. Cases for a structure provided with cut-off wall and when the wall did not exist were both tested. Various statistical parameters that reflected different degrees of heterogeneity were examined and the changes in the mean seepage flow, the mean uplift force and the mean exit gradient observed under the structure were analysed. Results reveal that under heterogeneous conditions, the reduction made by the sheet pile in the uplift force and exit hydraulic gradient may be underestimated when deterministic solutions are used.

A computationally efficient parallel Levenberg‐Marquardt algorithm for highly parameterized inverse model analyses

AbstractInverse modeling seeks model parameters given a set of observations. However, for practical problems because the number of measurements is often large and the model parameters are also numerous, conventional methods for inverse modeling can be computationally expensive. We have developed a new, computationally efficient parallel Levenberg‐Marquardt method for solving inverse modeling problems with a highly parameterized model space. Levenberg‐Marquardt methods require the solution of a linear system of equations which can be prohibitively expensive to compute for moderate to large‐scale problems. Our novel method projects the original linear problem down to a Krylov subspace such that the dimensionality of the problem can be significantly reduced. Furthermore, we store the Krylov subspace computed when using the first damping parameter and recycle the subspace for the subsequent damping parameters. The efficiency of our new inverse modeling algorithm is significantly improved using these computational techniques. We apply this new inverse modeling method to invert for random transmissivity fields in 2‐D and a random hydraulic conductivity field in 3‐D. Our algorithm is fast enough to solve for the distributed model parameters (transmissivity) in the model domain. The algorithm is coded in Julia and implemented in the MADS computational framework (http://mads.lanl.gov). By comparing with Levenberg‐Marquardt methods using standard linear inversion techniques such as QR or SVD methods, our Levenberg‐Marquardt method yields a speed‐up ratio on the order of to in a multicore computational environment. Therefore, our new inverse modeling method is a powerful tool for characterizing subsurface heterogeneity for moderate to large‐scale problems.

Heat-based hyporheic flux calculations in heterogeneous salmon spawning gravels

Groundwater-surface water interactions in rivers are a critically important factor for fish spawning, as streamwater downwelling or upwelling of low-oxygen groundwater can affect egg survival. Assessing such dynamics at the reach scale using distributed temperature measurements as a tracer proved reliable in determining flux rates and directions in the hyporheic zone in a number of studies. Here, we report heat-based vertical flux rates from a heterogeneous gravel-bed stream reach used by spawning Atlantic salmon in the Scottish Highlands. Results showed mostly small downwelling fluxes (~0.3 m d−1), which were largely independent of discharge. Contrasting, and at times unusual flux-depth profiles (e.g., increasing flux with depth) were detected, consistent with the heterogeneous streambed material causing diverse hyporheic flow paths. This was tested in a numerical 2-D model setup attempting to reproduce such behavior with variable random hydraulic conductivity (K) fields. The 2-D model clearly demonstrated that strong deviations from the expected decrease of fluxes with depth can be explained by high heterogeneity coupled with relatively low K fields. This showed that using simple 1-D heat-based flux estimates in combination with 2-D models is a useful approach to testing hypotheses about the influence of variable streambed materials on groundwater–surface water exchange in an ecological context.