Henry Problem Research Articles

A Generalized Semi-Analytical Solution for the Dispersive Henry Problem: Effect of Stratification and Anisotropy on Seawater Intrusion

The Henry problem (HP) continues to play a useful role in theoretical and practical studies related to seawater intrusion (SWI) into coastal aquifers. The popularity of this problem is attributed to its simplicity and precision to the existence of semi-analytical (SA) solutions. The first SA solution has been developed for a high uniform diffusion coefficient. Several further studies have contributed more realistic solutions with lower diffusion coefficients or velocity-dependent dispersion. All the existing SA solutions are limited to homogenous and isotropic domains. This work attempts to improve the realism of the SA solution of the dispersive HP by extending it to heterogeneous and anisotropic coastal aquifers. The solution is obtained using the Fourier series method. A special hydraulic conductivity–depth model describing stratified heterogeneity is used for mathematical convenience. An efficient technique is developed to solve the flow and transport equations in the spectral space. With this technique, we show that the HP can be solved in the spectral space with the salt concentration as primary unknown. Several examples are generated, and the SA solutions are compared against an in-house finite element code. The results provide high-quality data assessed by quantitative indicators that can be effectively used for code verification in realistic configurations of heterogeneity and anisotropy. The SA solution is used to explain contradictory results stated in the previous works about the effect of anisotropy on the saltwater wedge. It is also used to investigate the combined influence of stratification and anisotropy on relevant metrics characterizing SWI. At a constant gravity number, anisotropy leads to landward migration of the saltwater wedge, more intense saltwater flux, a wider mixing zone and shallower groundwater discharge zone to the sea. The influence of stratified heterogeneity is more pronounced in highly anisotropic aquifers. The stratification rate and anisotropy have complementary effects on all SWI metrics, except for the depth of the discharge zone.

Fourier series solution for an anisotropic and layered configuration of the dispersive Henry Problem

Henry Problem (HP) still plays an important role in benchmarking numerical models of seawater intrusion (SWI) as well as being applied to practical and managerial purposes. The popularity of this problem is due to having a closed-form semi-analytical (SA) solution. The early SA solutions obtained for HP were limited to extensive assumptions that restrict its application in practical works. Several further studies expended the generality of the solution by assuming lower diffusion coefficients or including velocity-dependent dispersion in the results. However, all these studies are limited to homogeneous and isotropic domains. The present work made an attempt to improve the reality of the SA solution obtained for dispersive HP by considering anisotropic and stratified heterogeneous coastal aquifers. The solution is obtained by defining Fourier series for both stream function and salt concentration, applying a Galerkin treatment using the Fourier modes as trial functions and solving the flow and the salt transport equations simultaneously in the spectral space. In order to include stratified heterogeneity, a special depth-hydraulic conductivity model is applied that can be solved analytically without significant mathematical complexity. Several examples are proposed and studied. The results show excellent agreement between the SA and numerical solutions obtained with an in-house advanced finite element code.

On the use of COMSOL Multiphysics for seawater intrusion in fractured coastal aquifers

COMSOL Multiphysics is a comprehensive simulation software environment for a wide range of applications. COMSOL has an interactive interface that facilitates the modeling procedure and allows an easy coupling of different physical processes. The Subsurface Flow module extends the COMSOL modeling environment to applications related to fluid flow in saturated and variably saturated porous media. COMSOL is increasingly used in the investigation of geophysical, hydrogeological and environmental phenomena. The main goal of this work is to explore the ability of COMSOL for simulating seawater intrusion (SWI) in fractured coastal aquifers. Numerical modeling of such a problem is of high interest as fractured/karstic coastal aquifers are widespread and processes of SWI in the presence of fractures remains poorly understood. We set up a COSMOL model for the popular Henry problem. The accuracy of COMSOL is highlighted by comparison against the semianalytical solutions for simple homogeneous aquifers. For fractured aquifers, the performance of COSMOL is evaluated by comparison against an in-house finite element model based on the discrete fracture model and against the results of existing works. Given its versatility and flexibility, COMSOL shows promise as a tool for SWI in coastal aquifers.

The case of the Biscayne Bay and aquifer near Miami, Florida: density-driven flow of seawater or gravitationally driven discharge of deep saline groundwater?

Coastal groundwater flow investigations at the Cutler site of the Biscayne Bay south of Miami, Florida, gave rise to the dominating concept of density-driven flow of seawater into coastal aquifers indicated as a saltwater wedge. Within that wedge, convection-type return flow of seawater and a dispersion zone were concluded to be the cause of the Biscayne aquifer ‘seawater wedge.’ This conclusion was merely based on the chloride distribution within the aquifer and on an analytical model concept assuming convection flow within a confined aquifer without taking non-chemical field data into consideration. This concept was later labeled the ‘Henry problem,’ which any numerical variable-density flow program has to be able to simulate to be considered acceptable. Revisiting the summarizing publication with its record of piezometric field data (heads) showed that the so-called seawater wedge was actually caused by discharging deep saline groundwater driven by regional gravitational groundwater flow systems. Density-driven flow of seawater into the aquifer was not found reflected in the head measurements for low and high tide which had been taken contemporaneously with the chloride measurements. These head measurements had not been included in the assumption of a seawater wedge and associated dispersion zone and convection cell. The Biscayne situation emphasizes the need for any chemical interpretation of flow pattern to be backed up by head data as energy indicators of flow fields. At the Biscayne site density-driven flow of seawater did not and does not exist. This conclusion was confirmed by five independent methods. The hydrostatic use of vertical buoyancy forces needs, under hydrodynamic boundary conditions, to be replaced with buoyancy forces along the direction of the pressure potential forces [(grad p)/density] which, in the subsurface, can be pointed in any direction in space.

Numerical analysis of the effects of changing hydraulic parameters on saltwater intrusion in coastal aquifers

PurposeThe purpose of this paper is to develop and validate a numerical model to study the effect of changing hydraulic parameters on saltwater intrusion in coastal aquifers.Design/methodology/approachThe numerical model SEAWAT is validated and applied to a hypothetical case (Henry problem) and a real case study (Biscayne aquifer, Florida, USA) for different values of hydraulic parameters including; hydraulic conductivity, porosity, dispersion, diffusion, fluid density and solute concentration. The dimensional analysis technique is used to correlate these parameters with the intrusion length.FindingsThe results show that the hydraulic parameters have a clear effect on saltwater intrusion as they increase the intrusion in some cases and decrease it in some other cases. The results indicate that changing hydraulic parameters may be used as a control method to protect coastal aquifers from saltwater intrusion.Practical implicationsThe results of the application of the model to the Biscayne aquifer in Florida showed that the intrusion can be reduced to 50 percent when the hydraulic conductivity is reduced to 50 percent. Decreasing hydraulic conductivity by injecting some relatively cheap materials such as bentonite can help to reduce the intrusion of saltwater. So the saltwater intrusion can be reduced with relatively low cost through changing some hydraulic parameters.Originality/valueA relationship to calculate intrusion length in coastal aquifer is developed and the impact of different hydraulic parameters on saltwater intrusion is highlighted. Control of saltwater intrusion using relatively cheap method is presented.

The Henry problem: New semianalytical solution for velocity‐dependent dispersion

AbstractA new semianalytical solution is developed for the velocity‐dependent dispersion Henry problem using the Fourier‐Galerkin method (FG). The integral arising from the velocity‐dependent dispersion term is evaluated numerically using an accurate technique based on an adaptive scheme. Numerical integration and nonlinear dependence of the dispersion on the velocity render the semianalytical solution impractical. To alleviate this issue and to obtain the solution at affordable computational cost, a robust implementation for solving the nonlinear system arising from the FG method is developed. It allows for reducing the number of attempts of the iterative procedure and the computational cost by iteration. The accuracy of the semianalytical solution is assessed in terms of the truncation orders of the Fourier series. An appropriate algorithm based on the sensitivity of the solution to the number of Fourier modes is used to obtain the required truncation levels. The resulting Fourier series are used to analytically evaluate the position of the principal isochlors and metrics characterizing the saltwater wedge. They are also used to calculate longitudinal and transverse dispersive fluxes and to provide physical insight into the dispersion mechanisms within the mixing zone. The developed semianalytical solutions are compared against numerical solutions obtained using an in house code based on variant techniques for both space and time discretization. The comparison provides better confidence on the accuracy of both numerical and semianalytical results. It shows that the new solutions are highly sensitive to the approximation techniques used in the numerical code which highlights their benefits for code benchmarking.

Reactive Henry problem: effect of calcite dissolution on seawater intrusion

Freshwater stored in a coastal aquifer is extensively extracted through pumping wells due to high water demand in coastal area (touristic, industrial, and public use). To enhance freshwater security and to avoid contamination of water reserves, seawater intrusion becomes a topic of great interest for hydrogeologists. During the last few decades, hydrogeologists have provided a deeper understanding of the prediction, processes, investigative tools, and management of such systems. The majority of the studies quantifies these hydrogeological systems using traditional density-dependent flow and transport models and does not consider the effect of the chemical reactions. Interdependence of density-dependent flow and chemical reactions and their effects on the porosity and permeability is the most important key toward a reliable modeling of these complex systems. Seawater intrusion can increase by the solid matrix dissolution processes in the saltwater–freshwater mixing zone in the case of a coastal carbonate aquifer. The dissolution of such rocks can easily induce a development of porosity and permeability as a result of the mixing processes. The increase of permeability would enhance further seawater flux to the freshwater side. In this work, a relatively complete modeling scheme is presented to quantify and predict this risk. The modeling of such a problem requires a set of highly nonlinearly coupled equations. In this regard, GEODENS code used in this work can solve these equations by a finite element procedure; it can handle density-dependent flow, transport, and geochemical reactions in porous media. Its main purpose is to represent the physicochemical processes in the subsurface system. The code is used to simulate the effect of calcite dissolution during seawater intrusion in a coastal carbonate homogeneous aquifer.

Extension of the Henry semi-analytical solution for saltwater intrusion in stratified domains

The Henry semi-analytical solution is developed for stratified aquifers with exponential and Gaussian permeability–depth relationships and small dispersion. The semi-analytical solution is sought by expanding the stream function and the concentration in an infinite Fourier series truncated at given orders. Due to the heterogeneity, the semi-analytical solution contains additional terms and quickly becomes impractical because of the high truncation orders required to yield stable results. An appropriate evaluation of the most expensive term, involving fairly complex summations, is proposed to render the computation of the semi-analytical solution affordable. Three configurations with the same transmissivity as that for the homogeneous Henry problem are investigated. The first configuration considers an exponential decay of the permeability (k) with depth. The second and third configurations consider an exponential increase and a Gaussian high-low-high distribution of k with depth, respectively. The three configurations are also investigated numerically using an accurate numerical code based on the method of lines and advanced spatial discretization schemes. An excellent agreement is obtained between the semi-analytical and the numerical solutions for all test cases. The results show that the amount of saltwater intrusion is strongly dependent on the heterogeneity. Further, the heterogeneous Henry problem is found to be more suitable for benchmarking density-driven numerical codes because the numerical solution is more sensitive to the grid size than for the homogeneous problem.

Seawater intrusion in fractured coastal aquifers: A preliminary numerical investigation using a fractured Henry problem

Despite that fractured coastal aquifers are widespread, the influence of fracture characteristics on seawater intrusion (SWI) has not been explored in previous studies. This research uses numerical modelling in a first step towards understanding the influence of fracture orientation, location and density on the extent of seawater and accompanying patterns of groundwater discharge in an idealised coastal aquifer. Specifically, aquifers containing single fractures or networks of regularly spaced fractures are studied using modified forms of the Henry SWI benchmark problem. The applicability of equivalent porous media (EPM) models for representing simple fracture networks in steady-state simulations of SWI is tested. The results indicate that the influence of fractures on SWI is likely to be mixed, ranging from enhancement to reduction in seawater extent and the width of the mixing zone. For the conceptual models considered here, vertical fractures in contact with the seawater wedge increase the width of the mixing zone, whereas vertical fractures inland of the wedge have minimal impact on the seawater distribution. Horizontal fractures in the lower part of the aquifer force the wedge seaward, whereas horizontal fractures located within the zone of freshwater discharge enhance the wedge. Inclined fractures roughly parallel to the seawater-freshwater interface increase the landward extent of seawater and fractures perpendicular to the interface inhibit the wedge. The results show that EPM models are likely inadequate for inferring salinity distributions in most of the fractured cases, although the EPM approach may be suitable for orthogonal fracture networks if fracture density is high and appropriate dispersivity values can be determined.

A new benchmark semi-analytical solution for density-driven flow in porous media

A new benchmark semi-analytical solution is proposed for the verification of density-driven flow codes. The problem deals with a synthetic square porous cavity subject to different salt concentrations at its vertical walls. A steady state semi-analytical solution is investigated using the Fourier–Galerkin method. Contrarily to the standard Henry problem, the cavity benchmark allows high truncation orders in the Fourier series and provides semi-analytical solutions for very small diffusion cases. The problem is also investigated numerically to validate the semi-analytical solution. The obtained results represent a set of new test case high quality data that can be effectively used for benchmarking density-driven flow codes.

Sampling efficiency in Monte Carlo based uncertainty propagation strategies: Application in seawater intrusion simulations

The implementation of Monte Carlo simulations (MCSs) for the propagation of uncertainty in real-world seawater intrusion (SWI) numerical models often becomes computationally prohibitive due to the large number of deterministic solves needed to achieve an acceptable level of accuracy. Previous studies have mostly relied on parallelization and grid computing to decrease the computational time of MCSs. However, another approach which has received less attention in the literature is to decrease the number of deterministic simulations by using more efficient sampling strategies. Sampling efficiency is a measure of the optimality of a sampling strategy. A more efficient sampling strategy requires fewer simulations and less computational time to reach a certain level of accuracy. The efficiency of a sampling strategy is highly related to its space-filling characteristics.This paper illustrates that the use of optimized Latin hypercube sampling (OLHS) strategies instead of the widely employed simple random sampling (SRS) and Latin hypercube sampling (LHS) strategies, can significantly improve sampling efficiency and hence decrease the simulation time of MCSs. Nine OLHS strategies are evaluated including: improved Latin hypercube sampling (IHS); optimum Latin hypercube (OLH) sampling; genetic optimum Latin hypercube (GOLH) sampling; three sampling strategies based on the enhanced stochastic evolutionary (ESE) algorithm namely φp-ESE which employs the φp space-filling criterion, CLD-ESE which utilizes the centered L2-discrepancy (CLD) space-filling criterion, and SLD-ESE which uses the star L2-discrepancy (SLD) space-filling criterion; and three sampling strategies based on the simulated annealing (SA) algorithm namely φp-SA which employs the φp criterion, CLD-SA which uses the CLD criterion, and SLD-SA which utilizes the SLD criterion. The study applies SRS, LHS and the nine OLHS strategies to MCSs of two synthetic test cases of SWI. The two test cases are the Henry problem and a two-dimensional radial representation of SWI in a circular island. The comparison demonstrates that the CLD-ESE strategy is the most efficient among the evaluated strategies. This paper also demonstrates how the space-filling characteristics of different OLHS designs change with variations in the input arguments of their optimization algorithms.

A semi-analytical solution for saltwater intrusion with a very narrow transition zone

The Henry saltwater intrusion problem provides a semi-analytical solution that is largely used for benchmarking density-dependent groundwater flow models. The major drawback of this problem arises from the high dispersion value used by Henry (represented by the dimensionless parameter b = 0.1). Finding a stable semi-analytical solution for small values of b is challenging due to the low convergence of the corresponding nonlinear system. In this work, an accurate semi-analytical solution is developed in the case of a very narrow transition zone corresponding to b = 0.005. About 6,330 terms are used in the Fourier series to accurately represent the solution. The resolution of the corresponding highly nonlinear system is made possible by the modified Powell hybrid algorithm due to the analytical evaluation of the Jacobian, which drastically reduces the computational time. The new test problem is also investigated numerically using different numerical methods and different mesh sizes to show its high worthiness, compared to the standard Henry problem, for benchmarking density driven flow codes.

Model reduction of a coupled numerical model using proper orthogonal decomposition

Summary Numerical models for variable-density flow and solute transport (VDFST) are widely used to simulate seawater intrusion and related problems. The mathematical model for VDFST is a coupled nonlinear dynamical system, so the numerical discretizations in time and space are usually required to be as fine as possible. As a result, fine-scale transient models require large computational time, which is a disadvantage for state estimation, forward prediction or model inversion. The purpose of this research is to develop mathematical and numerical methods to simulate VDFST via a model order reduction technique called Proper Orthogonal Decomposition (POD) designed for nonlinear dynamical systems. POD was applied to extract leading “model features” (basis functions) through singular value decomposition (SVD) from observational data or simulations (snapshots) of high-dimensional systems. These basis functions were then used in the Galerkin projection procedure that yielded low-dimensional (reduced-order) models. The original full numerical models were also discretized by the Galerkin Finite-Element Method (GFEM). The implementation of the POD reduced-order method was straightforward when applied to the full order model to the complex model. The developed GFEM-POD model was applied to solve two classic VDFST cases, the Henry problem and the Elder problem, in order to investigate the accuracy and efficiency of the POD model reduction method. Once the snapshots from full model results are obtained, the reduced-order model can reproduce the full model results with acceptable accuracy but with less computational cost in comparison with the full model, which is useful for model calibration and data assimilation problems. We found that the accuracy and efficiency of the POD reduced-order model is mainly determined by the optimal selection of snapshots and POD bases. Validation and verification experiments confirmed our POD model reduction procedure.

Fluid dispersion effects on density-driven thermohaline flow and transport in porous media

This study introduces the dispersive fluid flux of total fluid mass to the density-driven flow equation to improve thermohaline modeling of salt and heat transports in porous media. The dispersive fluid flux in the flow equation is derived to account for an additional fluid flux driven by the density gradient and mechanical dispersion. The coupled flow, salt transport and heat transport governing equations are numerically solved by a fully implicit finite difference method to investigate solution changes due to the dispersive fluid flux. The numerical solutions are verified by the Henry problem and the thermal Elder problem under a moderate density effect and by the brine Elder problem under a strong density effect. It is found that increment of the maximum ratio of the dispersive fluid flux to the advective fluid flux results in increasing dispersivity for the Henry problem and the brine Elder problem. The effects of the dispersive fluid flux on salt and heat transports under high density differences and high dispersivities are more noticeable than under low density differences and low dispersivities. Values of quantitative indicators such as the Nusselt number, mass flux, salt mass stored and maximum penetration depth in the brine Elder problem show noticeable changes by the dispersive fluid flux. In the thermohaline Elder problem, the dispersive fluid flux shows a considerable effect on the shape and the number of developed fingers and makes either an upwelling or a downwelling flow in the center of the domain. In conclusion, for the general case that involves strong density-driven flow and transport modeling in porous media, the dispersive fluid flux should be considered in the flow equation.

Groundwater ages in coastal aquifers

The interpretation of groundwater ages in coastal aquifers requires an improved understanding of relationships between age distributions and the processes accompanying dispersive, density-dependent flow and transport. This study uses numerical modeling to examine the influence of mixing and a selection of other hydrogeological factors on steady-state age distributions in coastal aquifers. Three methods of age estimation are compared: the piston flow age, the direct age, and the tracer-based age. These are applied to various forms of the Henry problem, as well as to three variants of a larger, hypothetical coastal aquifer. Circulation of water within the seawater wedge results in markedly higher ages in the transition zone than in the underlying saltwater or overlying freshwater. Piston flow ages show a sharp increase where the freshwater and saltwater systems meet, whereas direct- and tracer-based simulations result in a smoother age distribution, as expected. Greater degrees of mixing result in larger differences between piston flow and direct or tracer-based ages, and bring about lower ages in the saltwater wedge. Tracer-based ages are preferred over direct- and piston flow ages for comparison with field data, especially in cases with wide transition zones. Despite the relatively simple conditions used for the simulations, complex age distributions with depth were obtained. Hence, the assessment of ages in field cases will be difficult, particularly where accurate ages in the transition zone are sought.

Reactive dispersive contaminant transport in coastal aquifers: Numerical simulation of a reactive Henry problem

The reactive mixing between seawater and terrestrial water in coastal aquifers influences the water quality of submarine groundwater discharge. While these waters come into contact at the seawater groundwater interface by density driven flow, their chemical components dilute and react through dispersion. A larger interface and wider mixing zone may provide favorable conditions for the natural attenuation of contaminant plumes. It has been claimed that the extent of this mixing is controlled by both, porous media properties and flow conditions. In this study, the interplay between dispersion and reactive processes in coastal aquifers is investigated by means of numerical experiments. Particularly, the impact of dispersion coefficients, the velocity field induced by density driven flow and chemical component reactivities on reactive transport in such aquifers is studied. To do this, a hybrid finite-element finite-volume method and a reactive simulator are coupled, and model accuracy and applicability are assessed. A simple redox reaction is considered to describe the degradation of a contaminant which requires mixing of the contaminated groundwater and the seawater containing the terminal electron acceptor. The resulting degradation is observed for different scenarios considering different magnitudes of dispersion and chemical reactivity. Three reactive transport regimes are found: reaction controlled, reaction-dispersion controlled and dispersion controlled. Computational results suggest that the chemical components' reactivity as well as dispersion coefficients play a significant role on controlling reactive mixing zones and extent of contaminant removal in coastal aquifers. Further, our results confirm that the dilution index is a better alternative to the second central spatial moment of a plume to describe the mixing of reactive solutes in coastal aquifers.

The Henry semianalytical solution for saltwater intrusion with reduced dispersion

The Henry semianalytical solution for salt water intrusion is widely used for benchmarking density dependent flow codes. The method consists of replacing the stream function and the concentration by a double set of Fourier series. These series are truncated at a given order and the remaining coefficients are calculated by solving a highly nonlinear system of algebraic equations. The solution of this system is often subject to substantial numerical difficulties. Previous works succeeded to provide semianalytical solutions only for saltwater intrusion problems with unrealistic large amount of dispersion. In this work, different truncations for the Fourier series are tested and the Levenberg‐Marquardt algorithm, which has a quadratic rate of convergence, is applied to calculate their coefficients. The obtained results provide semianalytical solutions for the Henry problem in the case of reduced dispersion coefficients and for two freshwater recharge values: the initial value suggested by Henry (1964) and the reduced one suggested by Simpson and Clement (2004). The developed semianalytical solutions are compared against numerical results obtained by using the method of lines and advanced spatial discretization schemes. The obtained semianalytical solutions improve considerably the worthiness of the Henry problem and therefore, they are more suitable for testing density dependent flow codes.

Simulation of flow and solute coupled 2‐D problems with velocity‐dependent dispersion coefficient based on the network method

AbstractThe governing equations of coupled density‐driven flow and solute transport problems in porous media, with velocity‐dependent dispersion coefficient, are strongly nonlinear and must be solved numerically. This contribution presents a network model, based on the network simulation method, capable of simulating the transient solution to this kind of problem efficiently and with a relatively low computational time. The mathematical model is formulated using the stream function and concentration variables. Simulation of the network model is carried out in the standard electric circuit simulation code, Pspice. The present model is first applied to simulate the original benchmark Henry problem, and the solution is compared with those obtained by other authors. A study of the grid size is also carried out. In addition, the modified version of Simpson and Clement of the Henry problem, as well as the groundwater flow in the closed desert basin of Pilot Valley, is studied using the proposed model. Copyright © 2011 John Wiley & Sons, Ltd.

Predictive uncertainty analysis of a saltwater intrusion model using null‐space Monte Carlo

Because of the extensive computational burden and perhaps a lack of awareness of existing methods, rigorous uncertainty analyses are rarely conducted for variable‐density flow and transport models. For this reason, a recently developed null‐space Monte Carlo (NSMC) method for quantifying prediction uncertainty was tested for a synthetic saltwater intrusion model patterned after the Henry problem. Saltwater intrusion caused by a reduction in fresh groundwater discharge was simulated for 1000 randomly generated hydraulic conductivity distributions, representing a mildly heterogeneous aquifer. From these 1000 simulations, the hydraulic conductivity distribution giving rise to the most extreme case of saltwater intrusion was selected and was assumed to represent the “true” system. Head and salinity values from this true model were then extracted and used as observations for subsequent model calibration. Random noise was added to the observations to approximate realistic field conditions. The NSMC method was used to calculate 1000 calibration‐constrained parameter fields. If the dimensionality of the solution space was set appropriately, the estimated uncertainty range from the NSMC analysis encompassed the truth. Several variants of the method were implemented to investigate their effect on the efficiency of the NSMC method. Reducing the dimensionality of the null‐space for the processing of the random parameter sets did not result in any significant gains in efficiency and compromised the ability of the NSMC method to encompass the true prediction value. The addition of intrapilot point heterogeneity to the NSMC process was also tested. According to a variogram comparison, this provided the same scale of heterogeneity that was used to generate the truth. However, incorporation of intrapilot point variability did not make a noticeable difference to the uncertainty of the prediction. With this higher level of heterogeneity, however, the computational burden of generating calibration‐constrained parameter fields approximately doubled. Predictive uncertainty variance computed through the NSMC method was compared with that computed through linear analysis. The results were in good agreement, with the NSMC method estimate showing a slightly smaller range of prediction uncertainty than was calculated by the linear method.

Two‐relaxation‐time lattice Boltzmann method for the anisotropic dispersive Henry problem

This study develops a lattice Boltzmann method (LBM) with a two‐relaxation‐time collision operator (TRT) to cope with anisotropic heterogeneous hydraulic conductivity and anisotropic velocity‐dependent hydrodynamic dispersion in the saltwater intrusion problem. The directional‐speed‐of‐sound technique is further developed to address anisotropic hydraulic conductivity and dispersion tensors. Forcing terms are introduced in the LBM to correct numerical errors that arise during the recovery procedure and to describe the sink/source terms in the flow and transport equations. In order to facilitate the LBM implementation, the forcing terms are combined with the equilibrium distribution functions (EDFs) to create pseudo‐EDFs. This study performs linear stability analysis and derives LBM stability domains to solve the anisotropic advection‐dispersion equation. The stability domains are used to select the time step at which the lattice Boltzmann method provides stable solutions to the numerical examples. The LBM was implemented for the anisotropic dispersive Henry problem with high ratios of longitudinal to transverse dispersivities, and the results compared well to the solutions in the work of Abarca et al. (2007).