Anisotropic dispersive Henry problem

The Henry problem has played a key role in our understanding of seawater intrusion into coastal aquifers and in benchmarking density dependent flow codes. This paper seeks to modify Henry’s problem to ensure sensitivity to density variations and vertical salinity profiles that resemble field observations. In the proposed problem, the “dispersive Henry problem”, mixing is represented by means of the traditional Scheidegger dispersion tensor (dispersivity times water flux). Anisotropy in the hydraulic conductivity is acknowledged and Henry’s seaside boundary condition of prescribed salt concentration is replaced by a flux dependent boundary condition, which represents more realistically salt transport across the seaside boundary. This problem turns out to be very sensitive to density variations and its solution gets closer to reality. However, an improvement in the traditional Henry problem (gain in sensitivity and realism) can be also achieved if the value of the Peclet number is significantly reduced.Although the dispersive problem lacks an analytical solution, it can shed light on flow in coastal aquifers. It provides significant information about the factors controlling seawater penetration, width of the mixing zone and influx of seawater. The width of the mixing zone depends basically on dispersion with longitudinal and transverse dispersion controlling different parts of the mixing zone but displaying similar overall effects. Toe penetration is mainly controlled by the horizontal permeability and by the geometric mean of the dispersivities. Finally, transverse dispersivity and the geometric mean of the hydraulic conductivity are the leading parameters controlling the amount of saltwater that enters the aquifer.     

Brine transport in porous media: self-similar solutions

In this paper we analyze a model for brine transport in porous media, which includes a mass balance for the fluid, a mass balance for salt, Darcy's law and an equation of state, which relates the fluid density to the salt mass fraction. This model incorporates the effect of local volume changes due to variations in the salt concentration. Density variations affect the compressibility of the fluid, which in turn cause additional fluid flow. Two specific situations are investigated that lead to self similarity. We study the relative importance of the compressibility effect in terms of the relative density difference. Semi-analytical solutions are obtained as well as asymptotic expressions in terms of the relative density difference. It is found that the volume changes have a small but noticeable effect on the mass transport only when the salt concentration gradients are large. Some results on the simultaneous transport of brine and dissolved (radioactive) tracers are presented.     

Coupled groundwater flow and transport: 1. Verification of variable density flow and transport models

This work examines variable density flow and corresponding solute transport in groundwater systems. Fluid dynamics of salty solutions with significant density variations are of increasing interest in many problems of subsurface hydrology. The mathematical model comprises a set of non-linear, coupled, partial differential equations to be solved for pressure/hydraulic head and mass fraction/concentration of the solute component. The governing equations and underlying assumptions are developed and discussed. The equation of solute mass conservation is formulated in terms of mass fraction and mass concentration. Different levels of the approximation of density variations in the mass balance equations are used for convection problems (e.g. the Boussinesq approximation and its extension, fully density approximation). The impact of these simplifications is studied by use of numerical modelling.Numerical models for nonlinear problems, such as density-driven convection, must be carefully verified in a particular series of tests. Standard benchmarks for proving variable density flow models are the Henry, Elder, and salt dome (HYDROCOIN level 1 case 5) problems. We studied these benchmarks using two finite element simulators - ROCKFLOW, which was developed at the Institute of Fluid Mechanics and Computer Applications in Civil Engineering and FEFLOW, which was developed at the Institute for Water Resources Planning and Systems Research Ltd. Although both simulators are based on the Galerkin finite element method, they differ in many approximation details such as temporal discretization (Crank-Nicolson vs predictor-corrector schemes), spatial discretization (triangular and quadrilateral elements), finite element basis functions (linear, bilinear, biquadratic), iteration schemes (Newton, Picard) and solvers (direct, iterative). The numerical analysis illustrates discretization effects and defects arising from the different levels of the density of approximation. We contribute new results for the salt dome problem, for which inconsistent findings exist in literature. Applications of the verified numerical models to more complex problems, such as thermohaline and three-dimensional convection systems, will be presented in the second part of this paper.     

A test case for the simulation of three-dimensional variable-density flow and solute transport in discretely-fractured porous media

A test case has been developed for three-dimensional simulations of variable-density flow and solute transport in discretely-fractured porous media. The simulation domain is a low-permeability porous matrix cube containing a single non-planar fracture. The initial solute concentration is zero everywhere. A constant solute concentration is assigned to the top of the domain, which increases near-top fluid density and induces downward density-driven flow. The test case is therefore comparable to downwelling of a dense brine below a saline disposal basin or a waste repository. Numerous fingers and distinct convection cells develop early in the fracture but the fingers later coalesce and convection becomes less apparent. To help test other variable-density flow and transport models, results of the test case are presented both qualitatively (concentration contours and velocity fields) and quantitatively (penetration depth, mass flux, total mass stored, maximum fracture and matrix velocity).     

Coupled groundwater flow and transport: 2. Thermohaline and 3D convection systems

This work continues the analysis of variable density flow in groundwater systems. It focuses on both thermohaline (double-diffusive) and three-dimensional (3D) buoyancy-driven convection processes. The finite-element method is utilized to tackle these complex non-linear problems in two and three dimensions. The preferred numerical approaches are discussed regarding appropriate basic formulations, balance-consistent discretization techniques for derivative quantitites, extension of the Boussinesq approximation, proper constraint conditions, time marching schemes, and computational strategies for solving large systems. Applications are presented for the thermohaline Elder and salt dome problem as well as for the 3D extension of the Elder problem with and without thermohaline effects and a 3D Bénard convection process. The simulations are performed by using the package FEFLOW. Conclusions are drawn with respect to numerical efforts and the appropriateness for practical needs.     

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.     

Influence of Boundary Condition Types on Unstable Density‐Dependent Flow

Boundary conditions are required to close the mathematical formulation of unstable density‐dependent flow systems. Proper implementation of boundary conditions, for both flow and transport equations, in numerical simulation are critical. In this paper, numerical simulations using the FEFLOW model are employed to study the influence of the different boundary conditions for unstable density‐dependent flow systems. A similar set up to the Elder problem is studied. It is well known that the numerical simulation results of the standard Elder problem are strongly dependent on spatial discretization. This work shows that for the cases where a solute mass flux boundary condition is employed instead of a specified concentration boundary condition at the solute source, the numerical simulation results do not vary between different convective solution modes (i.e., plume configurations) due to the spatial discretization. Also, the influence of various boundary condition types for nonsource boundaries was studied. It is shown that in addition to other factors such as spatial and temporal discretization, the forms of the solute transport equation such as divergent and convective forms as well as the type of boundary condition employed in the nonsource boundary conditions influence the convective solution mode in coarser meshes. On basis of the numerical experiments performed here, higher sensitivities regarding the numerical solution stability are observed for the Adams‐Bashford/Backward Trapezoidal time integration approach in comparison to the Euler‐Backward/Euler‐Forward time marching approach. The results of this study emphasize the significant consequences of boundary condition choice in the numerical modeling of unstable density‐dependent flow.     

Theoretical analysis of the worthiness of Henry and Elder problems as benchmarks of density-dependent groundwater flow models

Computer models must be tested to ensure that the mathematical statements and solution schemes accurately represent the physical processes of interest. Because the availability of benchmark problems for testing density-dependent groundwater models is limited, one should be careful in using these problems appropriately. Details of a Galerkin finite-element model for the simulation of density-dependent, variably saturated flow processes are presented here. The model is tested using the Henry salt-water intrusion problem and Elder salt convection problem. The quality of these benchmark problems is then evaluated by solving the problems in the standard density-coupled mode and in a new density-uncoupled mode. The differences between the solutions indicate that the Henry salt-water intrusion problem has limited usefulness in benchmarking density-dependent flow models because the internal flow dynamics are largely determined by the boundary forcing. Alternatively, the Elder salt-convection problem is more suited to the model testing process because the flow patterns are completely determined by the internal balance of pressure and gravity forces.     

Analytical models of contaminant transport in coastal aquifers

The Henry formulation, which couples subsurface flow and salt transport via a variable-density flow formulation, can be used to evaluate the extent of sea water intrusion into coastal aquifers. The coupling gives rise to nontrivial flow patterns that are very different from those observed in inland aquifers. We investigate the influence of these flow patterns on the transport of conservative contaminants in a coastal aquifer. The flow is characterized by two dimensionless parameters: the Péclet number, which compares the relative effects of advective and dispersive transport mechanisms, and a coupling parameter, which describes the importance of the salt water boundary on the flow. We focus our attention on two regimes – low and intermediate Péclet number flows. Two transport scenarios are solved analytically by means of a perturbation analysis. The first, a natural attenuation scenario, describes the flushing of a contaminant from a coastal aquifer by clean fresh water, while the second, a contaminant spill scenario, considers an isolated point source.     

Multi-functional heat pulse probe measurements of coupled vadose zone flow and transport

Simultaneous measurement of coupled water, heat, and solute transport in unsaturated porous media is made possible with the multi-functional heat pulse probe (MFHPP). The probe combines a heat pulse technique for estimating soil heat properties, water flux, and water content with a Wenner array measurement of bulk soil electrical conductivity (EC_bulk). To evaluate the MFHPP, we conducted controlled steady-state flow experiments in a sand column for a wide range of water saturations, flow velocities, and solute concentrations. Flow and transport processes were monitored continuously using the MFHPP. Experimental data were analyzed by inverse modeling of simultaneous water, heat, and solute transport using an adapted HYDRUS-2D model. Various optimization scenarios yielded simultaneous estimation of thermal, solute, and hydraulic parameters and variables, including thermal conductivity, volumetric water content, water flux, and thermal and solute dispersivities. We conclude that the MFHPP holds great promise as an excellent instrument for the continuous monitoring and characterization of the vadose zone.     

Validation of classical density-dependent solute transport theory for stable,high-concentration-gradient brine displacements in coarse and medium sands

A series of careful laboratory soil column experiments was carried out for the purpose of providing data for testing recently presented theories of miscible density-dependent flow and transport. In particular, modifications to the standard theory involve extensions to both Darcy's (for flow) and Fick's laws (for diffusive/dispersive solute flux). Both coarse- and medium-grained sands were used in the experiments. All experiments concerned upward (i.e., stable) displacement of fresh water within the soil by a brine solution, under either constant head or constant volume flux conditions. The experimental data were analysed using accurate numerical solutions of the standard governing flow and transport model, as well as models with modified Darcy's and Fick's laws. Model parameters were determined by a step-wise fitting procedure based on the least-squares criterion. The results show clearly that, for large density contrasts, an extended Darcy's law was not necessary. On the other hand, an extension to Fick's Law was needed to model the experimental data accurately.     

Analysis of the thickness of a fresh water lens and of the transition zone between this lens and upwelling saline water

In regions with saline groundwater, fresh water lenses may develop due to rainwater infiltration. The amount of fresh water that is available for e.g. agricultural crops depends on the thickness of the lens and the extent of mixing between fresh and saline water. In this paper, we consider the mixing of fresh water and upward moving saline ground water in low-lying deltaic areas. The parameters that dominate the flow and transport problem are investigated using dimensionless groups and scaled sensitivities. We characterize the numerically simulated thicknesses of the lens and of the mixing zone by spatial moments. Rayleigh number and mass flux ratio, which is the ratio of the salt water seepage and the precipitation, determine the thickness of the fresh water lens. The local thickness of the mixing zone is mainly influenced by the dispersive/diffusive groups and the mass flux ratio. In addition, convergence of streamlines towards an outflow boundary affects the thickness, particularly in the vicinity of this boundary.Analytical and numerical steady state solutions for lens thickness are compared, taking into account upward seepage, for the two cases with and without a density difference between lens and underlying groundwater. Agreement between the numerical and analytical solutions for the lens thickness is good except when the mass flux ratio becomes small. For zero mass flux ratio, it is implicitly assumed in the analytical solution that salt water is stagnant, and that is unrealistic. Relative contributions of longitudinal and transversal hydrodynamic dispersion and diffusion to the thickness of the mixing zone are quantified numerically for different phases of lens formation. Longitudinal dispersion dominates in the early stages of lens formation, while diffusion and transversal dispersion dominate at steady state.     

Solute and Heat Transport Model of the Henry and Hilleke Laboratory Experiment

SEAWAT is a coupled version of MODFLOW and MT3DMS designed to simulate variable-density ground water flow and solute transport. The most recent version of SEAWAT, called SEAWAT Version 4, includes new capabilities to represent simultaneous multispecies solute and heat transport. To test the new features in SEAWAT, the laboratory experiment of Henry and Hilleke (1972) was simulated. Henry and Hilleke used warm fresh water to recharge a large sand-filled glass tank. A cold salt water boundary was represented on one side. Adjustable heating pads were used to heat the bottom and left sides of the tank. In the laboratory experiment, Henry and Hilleke observed both salt water and fresh water flow systems separated by a narrow transition zone. After minor tuning of several input parameters with a parameter estimation program, results from the SEAWAT simulation show good agreement with the experiment. SEAWAT results suggest that heat loss to the room was more than expected by Henry and Hilleke, and that multiple thermal convection cells are the likely cause of the widened transition zone near the hot end of the tank. Other computer programs with similar capabilities may benefit from benchmark testing with the Henry and Hilleke laboratory experiment.     

Variable-density groundwater flow and solute transport in irregular 2D fracture networks

Numerical simulations of variable-density flow and solute transport have been conducted to investigate dense plume migration for various configurations of 2D fracture networks. For orthogonal fractures, simulations demonstrate that dispersive mixing in fractures with small aperture does not stabilize vertical plume migration in fractures with large aperture. Simulations in non-orthogonal 2D fracture networks indicate that convection cells form and that they overlap both the porous matrix and fractures. Thus, transport rates in convection cells depend on matrix and fracture flow properties. A series of simulations in statistically equivalent networks of fractures with irregular orientation show that the migration of a dense plume is highly sensitive to the geometry of the network. If fractures in a random network are connected equidistantly to the solute source, few equidistantly distributed fractures favor density-driven transport. On the other hand, numerous fractures have a stabilizing effect, especially if diffusive transport rates are high. A sensitivity analysis for a network with few equidistantly distributed fractures shows that low fracture aperture, low matrix permeability and high matrix porosity impede density-driven transport because these parameters reduce groundwater flow velocities in both the matrix and the fractures. Enhanced molecular diffusion slows down density-driven transport because it favors solute diffusion from the fractures into the low-permeability porous matrix where groundwater velocities are smaller. For the configurations tested, variable-density flow and solute transport are most sensitive to the permeability and porosity of the matrix, which are properties that can be determined more accurately than the geometry and hydraulic properties of the fracture network, which have a smaller impact on density-driven transport.     

Transport and fate of nonaqueous phase liquid (NAPL) in variably saturated porous media with evolving scales of heterogeneity

 A stochastic simulation is performed to study multiphase flow and contaminant transport in fractal porous media with evolving scales of heterogeneity. Numerical simulations of residual NAPL mass transfer and subsequent transport of dissolved and/or volatilized NAPL mass in variably saturated media are carried out in conjunction with Monte Carlo techniques. The impact of fractal dimension, plume scale and anisotropy (stratification) of fractal media on relative dispersivities is investigated and discussed. The results indicate the significance of evolving scale of porous media heterogeneity to the NAPL transport in the subsurface. In general, the fractal porous media enhance the dispersivities of NAPL mass plume transport in both the water phase and the gas phase while the influence on the water phase is more significant. The porous media with larger fractal dimension have larger relative dispersivities. The aqueous horizontal dispersivity exhibits a most significant increase against the plume scale.     

The effect of dispersion on the stability of density-driven flows in saturated homogeneous porous media

This work is a continuation of Musuuza et al. [37] in which a stability criterion for density-driven flow in a saturated homogeneous medium was derived. The criterion predicted the stability of a system as a function of the density and viscosity contrasts, the magnitude of the flow velocity and the concentration gradients for flow aligned orthogonal to gravity. It could not accurately predict stability transition with increasing velocity, a failure we attributed to dispersion effects that were not included. Small-scale dispersion and molecular diffusion, the main stabilising mechanisms in homogeneous media can stabilise flow if the instability wavelengths are below a certain cutoff. The width of the mixing zone is also central in controlling the range of wavelengths that persist into fingers. We propose a method of quantifying the cutoff wavelength and the width of the mixing zone, which are incorporated into the earlier criterion as constituents of the dispersive part. The earlier criterion is reformulated in terms of the Rayleigh number and with the dispersive part added, we attempt to predict the number of fingers formed which is directly related to the physical stability of the system. The inclusion of the Rayleigh number and dispersion into a single stability criterion provides new insight in the way dispersion affects vertical flow systems. Stability numbers computed with the new criterion are in reasonable agreement with numerical simulations for a range of physical variables. The numerical computations are performed with the software package d³f, which uses the cell-centred finite volume and the implicit Euler methods for the spatial and temporal discretisations, respectively. The admission of the density and dispersivities as inputs into the criterion makes it usable in practical problems.     

Perspective on theories of non-Fickian transport in heterogeneous media

Subsurface fluid flow and solute transport take place in a multiscale heterogeneous environment. Neither these phenomena nor their host environment can be observed or described with certainty at all scales and locations of relevance. The resulting ambiguity has led to alternative conceptualizations of flow and transport and multiple ways of addressing their scale and space–time dependencies. We focus our attention on four approaches that give rise to nonlocal representations of advective and dispersive transport of nonreactive tracers in randomly heterogeneous porous or fractured continua. We compare these approaches theoretically on the basis of their underlying premises and the mathematical forms of the corresponding nonlocal advective–dispersive terms. One of the four approaches describes transport at some reference support scale by a classical (Fickian) advection–dispersion equation (ADE) in which velocity is a spatially (and possibly temporally) correlated random field. The randomness of the velocity, which is given by Darcy’s law, stems from random fluctuations in hydraulic conductivity (and advective porosity though this is often disregarded). Averaging the stochastic ADE over an ensemble of velocity fields results in a space–time-nonlocal representation of mean advective–dispersive flux, an approach we designate as stnADE. A closely related space–time-nonlocal representation of ensemble mean transport is obtained upon averaging the motion of solute particles through a random velocity field within a Lagrangian framework, an approach we designate stnL. The concept of continuous time random walk (CTRW) yields a representation of advective–dispersive flux that is nonlocal in time but local in space. Closely related to the latter are forms of ADE entailing fractional derivatives (fADE) which leads to representations of advective–dispersive flux that are nonlocal in space but local in time; nonlocality in time arises in the context of multirate mass transfer models, which we exclude from consideration in this paper. We describe briefly each of these four nonlocal approaches and offer a perspective on their differences, commonalities, and relative merits as analytical and predictive tools.     

Transport and structure of the Weddell Gyre

A cyclonic gyre controls the advection of source waters into the formation areas of bottom water in the southern and western parts of the Weddell Sea and the subsequent transport of modified water masses to the north. Determination of the structure of the Weddell Gyre and of the associated transports was one of the objectives of the “Weddell Gyre Study” which began in September 1989 and ended in January 1993. The collected data set comprises records of moored current meters and profiles of temperature and salinity distributed along a transect between the northern tip of the Antarctic Peninsula and Kapp Norvegia. The circulation pattern on the transect is dominated by stable boundary currents of several hundred kilometers width at the eastern and western sides of the basin. They are of comparable size on both sides and provide nearly 90% of the volume transport of the gyre which amounts to 29.5 Sv. In the interior, a weak anticyclonic cell of 800 km diameter transports less than 4 Sv. Apart from the continental slopes, the near-bottom currents flow at some locations in an opposite direction to those in the water column above, indicating a significant baroclinic component of the current field. The intensity of the boundary currents is subject to seasonal fluctuations, whereas in the interior, time scales from days to weeks dominate. The large-scale circulation pattern is persistent during the years 1989 to 1991. The heat transport into the southern Weddell Sea is estimated to be 3.48×10¹³ W. This implies an equivalent heat loss through the sea surface of 19 W m⁻², as an average value for the area south of the transect. The derived salt transport is not significantly different from zero; consequently, the salt gain by sea ice formation has to compensate almost entirely the fresh water gain from the melting ice shelves and from precipitation. Estimation of water mass formation rates from the thermohaline differences of the inflow and outflow through the transect indicates that 6.0 Sv of Warm Deep Water are transformed into 2.6 Sv of Weddell Sea Bottom Water, into 1.2 Sv of Weddell Sea Deep Water, and into 2.2 Sv of surface water.     

A quasi three-dimensional model for flow and transport in unsaturated and saturated zones: 1. Implementation of the quasi two-dimensional case

A quasi three-dimensional (QUASI 3-D) model is presented for simulating the subsurface water flow and solute transport in the unsaturated and in the saturated zones of soil. The model is based on the assumptions of vertical flow in the unsaturated zone and essentially horizontal groundwater flow. The 1-D Richards equation for the unsaturated zone is coupled at the phreatic surface with the 2-D flow equation for the saturated zone. The latter was obtained by averaging 3-D flow equation in the saturated zone over the aquifer thickness. Unlike the Boussinesq equation for a leaky-phreatic aquifer, the developed model does not contain a storage term with specific yield and a source term for natural replenishment. Instead it includes a water flux term at the phreatic surface through which the Richards equation is linked with the groundwater flow equation. The vertical water flux in the saturated zone is evaluated on the basis of the fluid mass balance equation while the horizontal fluxes, in that equation, are prescribed by Darcy law. A 3-D transport equation is used to simulate the solute migration. A numerical algorithm to solve the problem for the general quasi 3-D case was developed. The developed methodology was exemplified for the quasi 2-D cross-sectional case (QUASI2D). Simulations for three synthetic problems demonstrate good agreement between the results obtained by QUASI2D and two fully 2-D flow and transport codes (SUTRA and 2DSOIL). Yet, simulations with the QUASI2D code were several times faster than those by the SUTRA and the 2DSOIL codes.     

Variability of the meridional overturning circulation of the North Atlantic: sensitivity to overflows of dense water masses

A numerical model of the Atlantic Ocean was used to study the low-frequency variability of meridional transports in the North Atlantic. The model shows a behaviour similar to those used in previous studies, and the temporal variability of certain variables compares favourably to observed time series. By changing the depth and width of the sills between the subpolar North Atlantic and the Nordic Seas, the mean horizontal and overturning circulation as well as some water mass properties are modified significantly. The reaction of meridional oceanic transports to atmospheric forcing fluctuations remains, however, unchanged. The critical role of the surface heat flux retroaction term for the meridional heat transport in stand-alone ocean models is discussed. The experiments underline the role of atmospheric variability for fluctuations of the large-scale ocean circulation on time scales from years to decades, and they support the hypothesis that the mean overturning strength is controlled by the model representation of the density of the overflow water masses.Responsible Editor: Dirk Olbers