This work is the fourth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. The general TCAT framework and the mathematical foundation presented in previous works are built upon by formulating macroscale models for conservation of mass, momentum, and energy, and the balance of entropy for a species in a phase volume, interface, and common curve. In addition, classical irreversible thermodynamic relations for species in entities are averaged from the microscale to the macroscale. Finally, we comment on alternative approaches that can be used to connect species and entity conservation equations to a constrained system entropy inequality, which is a key component of the TCAT approach. The formulations detailed in this work can be built upon to develop models for species transport and reactions in a variety of multiphase systems.
This work is the seventh in a series that introduces and employs the thermodynamically constrained averaging theory (TCAT) for modeling flow and transport in multiscale porous medium systems. This paper expands the previous analyses in the series by developing models at a scale where spatial variations within the system are not considered. Thus the time variation of variables averaged over the entire system is modeled in relation to fluxes at the boundary of the system. This implementation of TCAT makes use of conservation equations for mass, momentum, and energy as well as an entropy balance. Additionally, classical irreversible thermodynamics is assumed to hold at the microscale and is averaged to the megascale, or system scale. The fact that the local equilibrium assumption does not apply at the megascale points to the importance of obtaining closure relations that account for the large-scale manifestation of small-scale variations. Example applications built on this foundation are suggested to stimulate future work.
This work is the eighth in a series that develops the fundamental aspects of the thermodynamically constrained averaging theory (TCAT) that allows for a systematic increase in the scale at which multiphase transport phenomena is modeled in porous medium systems. In these systems, the explicit locations of interfaces between phases and common curves, where three or more interfaces meet, are not considered at scales above the microscale. Rather, the densities of these quantities arise as areas per volume or length per volume. Modeling of the dynamics of these measures is an important challenge for robust models of flow and transport phenomena in porous medium systems, as the extent of these regions can have important implications for mass, momentum, and energy transport between and among phases, and formulation of a capillary pressure relation with minimal hysteresis. These densities do not exist at the microscale, where the interfaces and common curves correspond to particular locations. Therefore, it is necessary for a well-developed macroscale theory to provide evolution equations that describe the dynamics of interface and common curve densities. Here we point out the challenges and pitfalls in producing such evolution equations, develop a set of such equations based on averaging theorems, and identify the terms that require particular attention in experimental and computational efforts to parameterize the equations. We use the evolution equations developed to specify a closed two-fluid-phase flow model.
This work is the fifth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. The general TCAT framework and the mathematical foundation presented in previous works are used to develop models that describe species transport and single-fluid-phase flow through a porous medium system in varying physical regimes. Classical irreversible thermodynamics formulations for species in fluids, solids, and interfaces are developed. Two different approaches are presented, one that makes use of a momentum equation for each entity along with constitutive relations for species diffusion and dispersion, and a second approach that makes use of a momentum equation for each species in an entity. The alternative models are developed by relying upon different approaches to constrain an entropy inequality using mass, momentum, and energy conservation equations. The resultant constrained entropy inequality is simplified and used to guide the development of closed models. Specific instances of dilute and non-dilute systems are examined and compared to alternative formulation approaches.
Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.
Standard models of flow of two immiscible fluids in a porous medium make use of an expression for the dependence of capillary pressure on the saturation of a fluid phase. Data to support the mathematical expression is most often obtained through a sequence of equilibrium experiments. In addition to such expressions being hysteretic, recent experimental and theoretical studies have suggested that the equilibrium functional forms obtained may be inadequate for modeling dynamic systems. This situation has led to efforts to express relaxation of a system to an equilibrium capillary pressure in relation to the rate of change of saturation. Here, based on insights gained from the thermodynamically constrained averaging theory (TCAT) we propose that dynamic processes are related to changes in interfacial area between phases as well as saturation. A more complete formulation of capillary pressure dynamics is presented leading to an equation that is suitable for experimental study.
The National Airborne Field Experiment 2006 (NAFE’06) was conducted during a three week period of November 2006 in the Murrumbidgee River catchment, located in southeastern Australia. One objective of NAFE’06 was to explore the suitability of the area for SMOS (Soil Moisture and Ocean Salinity) calibration/validation and develop downscaling and assimilation techniques for when SMOS does come on line. Airborne L-band brightness temperature was mapped at 1 km resolution 11 times (every 1–3 days) over a 40 by 55 km area in the Yanco region and 3 times over a 40 by 50 km area that includes Kyeamba Creek catchment. Moreover, multi-resolution, multi-angle and multi-spectral airborne data including surface temperature, surface reflectance (green, read and near infrared), lidar data and aerial photos were acquired over selected areas to develop downscaling algorithms and test multi-angle and multi-spectral retrieval approaches. The near-surface soil moisture was measured extensively on the ground in eight sampling areas concurrently with aircraft flights, and the soil moisture profile was continuously monitored at 41 sites. Preliminary analyses indicate that (i) the uncertainty of a single ground measurement was typically less than 5% vol. (ii) the spatial variability of ground measurements at 1 km resolution was up to 10% vol. and (iii) the validation of 1 km resolution L-band data is facilitated by selecting pixels with a spatial soil moisture variability lower than the point-scale uncertainty. The sensitivity of passive microwave and thermal data is also compared at 1 km resolution to illustrate the multi-spectral synergy for soil moisture monitoring at improved accuracy and resolution. The data described in this paper are available at www.nafe.unimelb.edu.au.
Bioaugmentation of microbial cultures is a potential method to enhance the performance of in situ bioremediation. In this study we evaluated the bioaugmentation of aerobic microorganisms that grow on butane that can transform chlorinated aliphatic hydrocarbon (CAH) mixtures, such as 1,1,1-trichloroethane (1,1,1-TCA), 1,1-dichhloroethane (1,1-DCA) and 1,1-dichloroethene (I,1-DCE). This mixture of contaminants is of interest, since 1, 1,1-TCA was a frequently used solvent at Department of Defense (DoD) facilities in the United States, and 1,1-DCE and 1,1-DCA are abiotic and biotic transformation products of 1,1,1-TCA. Kinetic studies with butane grown enrichment cultures and pure cultures isolated from the enrichment culture showed effective transformation of mixtures of these contaminants, with 1,1-DCE most rapidly transformed, followed by 1,1-DCA, and 1,1,1-TCA. In laboratory microcosm batch experiments, with aquifer material and groundwater from the field site, microcosms bioaugmented with mixed and pure cultures outperformed microcosms where indigenous butane-uti I i zing microorganisms were stimulated. The microcosm tests were consistent with the kinetics from mixed and pure cultures. Field studies were conducted in the saturated zone at the Moffett Field In Situ Test Facility in California. Tests were performed in an indigenous test leg along with a bioaugmented test leg, and the bioaugmented test leg outperformed the indigenous test leg. In the bioaugmented leg, 1,1-DCE was more effectively transformed, followed by 1. 1-DCA, and 1,1,1-TCA, consistent with the results from laboratory kinetic studies and microcosm studies. (c) 2007 Elsevier Masson SAS. All fights reserved.
The main processes affecting the migration of a solute in a fissured aquifer will be advection and dispersion in the fissures, diffusion into the porous matrix; and adsorption. This paper considers solute transport in an idealized fissured aquifer consisting of slabs of saturated rock-matrix separated by equally spaced, planar fissures. The solution of the transport equations is developed as far as Laplace transforms of the solute concentrations in the fissure and matrix water. Numerical inversion of the transforms is used to investigate characteristic behaviour of the model for a number of special cases.
The immense diversity of microbial life found in the vadose zone reflects the extremely heterogeneous and highly dynamic aquatic and chemical environments formed within soil pore spaces. The notion of planktonian free swimming microbes is unrealistic under most unsaturated conditions. Experimental and theoretical evidence suggests that surface attachment is the prevailing lifestyle, where bacterial colonies are embedded in biosynthesized extracellular polymeric substances (EPS). This strategy represents a successful adaptation to the variable and unpredictable hydration conditions near the earth surface. The EPS matrix serves as the interface with the environment; it enhances hydration and transport properties in the immediate vicinity of microbial cells, and dampens effects of highly transient fluctuations in water and nutrient fluxes. The primary effect of soil pore geometry and hydration status is on diffusion pathways to and away from stationary microbial colonies. Microbial dependency on diffusion processes occurs at all scales, but is particularly important at the colony scale. We illustrate the critical role of diffusion pathways with their complex spatial and temporal patterns in promoting coexistence and diversity. We review specific features and adaptations of microbial life to the particular conditions of terrestrial soil environments. The physical and related chemical conditions that shape microbial habitats and govern key processes in unsaturated soils are reviewed in a quantitative framework. Key physiological adaptations and biological responses to challenges presented by unsaturated conditions are discussed. Finally, we discuss potential impacts of microbial activity on properties and characteristics of the host porous medium. This review is an attempt to establish an interdisciplinary dialogue between hydrologists and microbiologists towards a quantitative integration of the role of hydrologic conditions on microbial activity and the role of microbiology in controlling macroscopic fluxes within this important compartment of the biosphere.
The assumption of local mass equilibrium for describing the transport of a contaminant in an aquifer containing a trapped non-aqueous-phase liquid has been under investigation for the past decade. Laboratory and field studies have shown that this assumption fails under certain circumstances, and heuristic macroscopic models have been introduced to describe the dispersion process when the assumption of local mass equilibrium is removed from the analysis. In this paper, a macroscopic model is developed for homogeneous porous media using the method of local volume averaging. The resulting macroscopic equation involves a dispersion tensor that is influenced by the mass transfer process, additional convective transport terms, and a linear form for the interfacial mass flux. Two local closure problems are provided that allow one to compute the effective transport coefficients so that theory and experiment can be compared in the absence of adjustable parameters. Numerical methods are used to solve the two local closure problems, and preliminary results are presented for two-dimensional, spatially periodic models of porous media.
A robust numerical method for saturated-unsaturated flow is developed which uses a monotone discretization and variable substitution. This method is compared to a conventional formulation and to a two phase (active air phase) model. On some published test examples of infiltration into dry media, the variable substitution method shows an order of magnitude improvement (in terms of nonlinear iterations) compared to the conventional pressure based method. One, two and three dimensional computations using both finite element and finite volume discretizations are presented.
Transport processes in heterogeneous porous media are often treated in terms of one-equation models. Such treatment assumes that the velocity, pressure, temperature, and concentration can be represented in terms of a single large-scale averaged quantity in regions having significantly different mechanical, thermal, and chemical properties. In this paper we explore the process of single-phase flow in a two-region model of heterogeneous porous media. The region-averaged equations are developed for the case of a slightly compressible flow which is an accurate representation for a certain class of liquid-phase flows. The analysis leads to a pair of transport equations for the region averaged pressures that are coupled through a classic exchange term, in addition to being coupled by a diffusive cross effect. The domain of validity of the theory has been identified in terms of a series of length and timescale constraints.In Part II the theory is tested, in the absence of adjustable parameters, by comparison with numerical experiments for transient, slightly compressible flow in both stratified and nodular models of heterogeneous porous media. Good agreement between theory and experiment is obtained for nodular and stratified systems, and effective transport coefficients for a wide range of conditions are presented on the basis of solutions of the three closure problems that appear in the theory. Part III of this paper deals with the principle of large-scale mechanical equilibrium and the region-averaged form of Darcy's law. This form is necessary for the development and solution of the region-averaged solute transport equations that are presented in Part IV. Finally, in Part V we present results for the dispersion tensors and the exchange coefficient associated with the two-region model of solute transport with adsorption.
On 8–9 September 2002, an extreme rainfall event caused by a stationary mesoscale convective system (MCS) occurred in the Gard region, France. Distributed hydrologic and hydraulic modelling has been carried out to assess and compare the various sources of data collected operationally and during the post-event field surveys. Distributed hydrological modelling was performed with n-TOPMODELs and assessed for ungauged basins with the discharge estimates of the post-event surveys. A careful examination of the occurrence in time and space of the flash floods over the head watersheds indicates that flooding was controlled by the trajectory of the convective part of the MCS. Stationarity of the MCS over the Gardon watershed (1858 km2 at Remoulins) for 28 h was responsible for the exceptional magnitude of the flood at this scale. The flood dynamics were characterized by an extensive inundation of the Gardonnenque plain upstream of the Gardon Gorges resulting in a significant peak flow reduction downstream. One-dimensional unsteady-flow hydraulic modelling was found to be required to reproduce these dynamics. Hydraulic modelling also proved to be potentially useful for the critical analysis and extrapolation of operational discharge rating curves.
Secondary motions are commonly present in open channel flows. This study aims to investigate, both experimentally and analytically, time-mean characteristics of cellular secondary flows generated by longitudinal bedforms. Experiments were conducted in a tilting, rectangular flume with six different longitudinal bedforms, including alternate bed strips with different roughness heights and bed ridges of wavy and rectangular shapes. Various flows were sampled using a two-dimensional laser Doppler anemometer (LDA) and a one-dimensional ultrasonic Doppler velocimeter (UDV). Experimental results demonstrate secondary flows appearing basically in cellular fashion over the modeled longitudinal bedforms. It is also shown that the cellular structures can be described analytically with kinematic considerations. The discrepancies between theoretical and measurement results are discussed. An empirical relationship between maximum vertical velocity and bed configuration is finally proposed based on the experimental data.
In this article we consider the transport of an adsorbing solute in a two-region model of a chemically and mechanically heterogeneous porous medium when the condition of large-scale mechanical equilibrium is valid. Under these circumstances, a one-equation model can be used to predict the large-scale averaged velocity, but a two-equation model may be required to predict the regional velocities that are needed to accurately describe the solute transport process. If the condition of large-scale mass equilibrium is valid, the solute transport process can be represented in terms of a one-equation model and the analysis is simplified greatly. The constraints associated with the condition of large-scale mass equilibrium are developed, and when these constraints are satisfied the mass transport process can be described in terms of the large-scale average velocity, an average adsorption isotherm, and a single large-scale dispersion tensor. When the condition of large-scale mass equilibrium is not valid, two equations are required to describe the mass transfer process, and these two equations contain two adsorption isotherms, two dispersion tensors, and an exchange coefficient. The extension of the analysis to multi-region models is straight forward but tedious.
A two-dimensional (2D) unsteady simulation model is applied to the problem of a submerged warm water discharge into a stratified lake or reservoir with an ice cover. Numerical simulations and analyses are conducted to gain insight into large-scale convective recirculation and flow processes in a cold waterbody induced by a buoyant jet. Jet behaviors under various discharge temperatures are captured by directly modeling flow and thermal fields. Flow structures and processes are described by the simulated spatial and temporal distributions of velocity and temperature in various regions: deflection, recirculation, attachment, and impingement. Some peculiar hydrothermal and dynamic features, e.g. reversal of buoyancy due to the dilution of a warm jet by entraining cold ambient water, are identified and examined. Simulation results show that buoyancy is the most important factor controlling jet behavior and mixing processes. The inflow boundary is treated as a liquid wall from which the jet is offset. Similarity and difference in effects of boundaries perpendicular and parallel to flow, and of buoyancy on jet attachment and impingement, are discussed. Symmetric flow configuration is used to de-emphasize the Coanda effect caused by offset.
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.
Five physically based models for predicting liquid saturation from light transmission in 2D laboratory systems containing translucent porous media were developed and tested (Models A–E). The models were based upon various simplifying assumptions concerning pore geometry, wettability, and drainage. Models A–D assumed uniform-sized pores, and liquid saturation was an explicit function of light transmission. Model E considered a distribution of pore sizes whose drainage characteristics were inferred from the Laplace equation. Mass balances were calculated using data from drainage and infiltration experiments, in four textures of silica sand with water as the fluid. Model E performed the best overall, with systematic errors of less than 2.3% saturation. Model E represents a robust easily applied new method for the determination of liquid saturation by light transmission. The other four models are presented, and compared, for reasons of historical interest and to investigate the impact of the various simplifying assumptions.
Two analytical solution methods are presented for regional steady-state groundwater flow in a two-dimensional stratified aquifer cross section where the water table is approximated by the topographic surface. For the first solution, the surficial aquifer is represented as a set of dipping parallel layers with different, but piecewise constant, anisotropic hydraulic conductivities, where the anisotropy is aligned with the dip of the layered formation. The model may be viewed as a generalization of the solutions developed by [Tóth JA. A theoretical analysis of groundwater flows in small drainage basins. J Geophys Res 1963;68(16):4795–812; Freeze R, Witherspoon P. Theoretical analysis of regional groundwater flow 1) analytical and numerical solution to the mathematical model, water resources research. Water Resour Res 1966;2(4):641–56; Selim HM. Water flow through multilayered stratified hillside. Water Resour Res 1975;11:949–57] to an multi-layer aquifer with general anisotropy, layer orientation, and a topographic surface that may intersect multiple layers. The second solution presumes curved (syncline) layer stratification with layer-dependent anisotropy aligned with the polar coordinate system. Both solutions are exact everywhere in the domain except at the topographic surface, where a Dirichlet condition is met in a least-squared sense at a set of control points; the governing equation and no-flow/continuity conditions are met exactly. The solutions are derived and demonstrated on multiple test cases. The error incurred at the location where the layer boundaries intersect the surface is assessed.
A novel methodology for the solution of the 2D shallow water equations is proposed. The algorithm is based on a fractional step decomposition of the original system in (1) a convective prediction, (2) a convective correction, and (3) a diffusive correction step. The convective components are solved using a Marching in Space and Time (MAST) procedure, that solves a sequence of small ODEs systems, one for each computational cell, ordered according to the cell value of a scalar approximated potential. The scalar potential is sought after computing first the minimum of a functional via the solution of a large linear system and then refining locally the optimum search. Model results are compared with the experimental data of two laboratory tests and with the results of other simulations carried out for the same tests by different authors. A comparison with the analytical solution of the oblique jump test has been also considered. Numerical results of the proposed scheme are in good agreement with measured data, as well as with analytical and higher order approximation methods results. The growth of the CPU time versus the cell number is investigated successively refining the elements of an initially coarse mesh. The CPU specific time, per element and per time step, is found out to be almost constant and no evidence of Courant–Friedrichs–Levi (CFL) number limitation has been detected in all the numerical experiments.
Four finite-volume component-wise total variation diminishing (TVD) schemes are proposed for solving the two-dimensional shallow water equations. In the framework of the finite volume method, a proposed algorithm using the flux-splitting technique is established by modifying the MacCormack scheme to preserve second-order accuracy in both space and time. Based on this algorithm, four component-wise TVD schemes, including the Liou–Steffen splitting (LSS), van Leer splitting, Steger–Warming splitting and local Lax–Friedrichs splitting schemes, are developed. These schemes are verified through the simulations of the 1D dam-break, the oblique hydraulic jump, the partial dam-break and circular dam-break problems. It is demonstrated that the proposed schemes are accurate, efficient and robust to capture the discontinuous shock waves without any spurious oscillations in the complex flow domains with dry-bed situation, bottom slope or friction. The simulated results also show that the LSS scheme has the best numerical accuracy among the schemes tested.
A finite volume MUSCL scheme for the numerical integration of 2D shallow water equations is presented. In the framework of the SLIC scheme, the proposed weighted surface-depth gradient method (WSDGM) computes intercell water depths through a weighted average of DGM and SGM reconstructions, in which the weight function depends on the local Froude number. This combination makes the scheme capable of performing a robust tracking of wet/dry fronts and, together with an unsplit centered discretization of the bed slope source term, of maintaining the static condition on non-flat topographies (C-property). A correction of the numerical fluxes in the computational cells with water depth smaller than a fixed tolerance enables a drastic reduction of the mass error in the presence of wetting and drying fronts. The effectiveness and robustness of the proposed scheme are assessed by comparing numerical results with analytical and reference solutions of a set of test cases. Moreover, to show the capability of the numerical model on field-scale applications, the results of a dam-break scenario are presented.
An approach to represent drying and wetting processes in a three-dimensional finite element sigma coordinate model is described. This approach makes use of capillaries in dry areas, which can connect to the nearby wet areas. The time marching of the mass conservation equation is modified by introducing a “size factor” coefficient and a water level diffusion term. Therefore, the fictitious water level of the dry nodes can fluctuate with the adjacent wet nodes. This eliminates the artificial pressure gradient appearing in some drying and wetting approaches in the partially wet (transition) elements. This approach results in a null momentum computation at the dry areas, which can guarantee numerical stability and satisfy the mass and momentum conservation. The approach has been applied in a hypothetical case and a real case in Xiamen Estuary, China, with satisfactory results.
Many popular groundwater modeling codes are based on the finite differences or finite volume method for orthogonal grids. In cases of complex subsurface geometries this type of grid either leads to coarse geometric representations or to extremely fine meshes. We use a coordinate transformation method (CTM) to circumvent this shortcoming. In computational fluid dynamics (CFD), this method has been applied successfully to the general Navier–Stokes equation. The method is based on tensor analysis and performs a transformation of a curvilinear into a rectangular unit grid, on which a modified formulation of the differential equations is applied. Therefore, it is not necessary to reformulate the code in total. We applied the CTM to an existing three-dimensional code (SHEMAT), a simulator for heat conduction and advection in porous media. The finite volume discretization scheme for the non-orthogonal, structured, hexahedral grid leads to a 19-point stencil and a correspondingly banded system matrix. The implementation is straightforward and it is possible to use some existing routines without modification. The accuracy of the modified code is demonstrated for single phase flow on a two-dimensional analytical solution for flow and heat transport. Additionally, a simple case of potential flow is shown for a two-dimensional grid which is increasingly deformed. The result reveals that the corresponding error increases only slightly. Finally, a thermal free-convection benchmark is discussed. The result shows, that the solution obtained with the new code is in good agreement with the ones obtained by other codes.
Heat is transported in aquifers by advection and conduction. Spatial variability of hydraulic conductivity causes fluctuations in small scale advection, whose effect can be represented by a dispersion term. However, the use of this term is still subject to controversy among modelers. The effect of heterogeneity on the heat plume generated by a groundwater heat exchanger (GHE) in a three-dimensional aquifer under steady state conditions is examined. Transverse dispersion is estimated using a stochastic approach in which a distinction between effective and ensemble dispersion coefficients is made. The former quantifies the typical width of the heat plume and the latter takes into account the uncertainty of the lateral plume position. Simulations show that transverse dispersion is proportional to the variance and correlation length of the log-conductivity field. On the one hand, the ensemble transverse dispersion coefficient, which can be used for risk analysis to find the mean temperature and the potential plume spread, is high near the heat source and then decreases. On the other hand, the effective transverse dispersion coefficient, the one required to simulate actual temperature values and plume width, displays a less marked dependence on the distance from the source. For modeling purposes it can be approximated as , where is the variance of the log-conductivity field and Lx its correlation length in the mean flow direction. However, a zero dispersion should be used to compute the energy dissipated by the GHE.
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.
Two equivalent permeability tensors are defined for 3D heterogeneous media, Kp and Kq, valid respectively for linear pressure and constant flux conditions at the block boundary. Both tensors are symmetric and positive-definite and the second one produces lower magnitude of directional permeability than the first one. These tensors only depends upon the internal block structure and 3D distribution of the local permeability values. On this basis, we develop first a straightforward method for evaluating the coefficients of the 2D tensor for the problem of flow through fracture traces in a cross-section, subject to linear pressure conditions. A quartzite rock mass is used as an example to illustrate this method. Then, an approximated method is proposed to build up the 3D permeability tensor of the fractured block from the ellipses within cross-sections in varied orientations.
This paper presents a 3D model in sigma coordinates. Although the principles it is based on have been established for some time, some original aspects for this type of 3D mode splitting model are presented here. The model was designed to simulate flows in various coastal areas from the regional scale down to the inshore scale of small bays or estuaries where circulation is generally driven by a mix of processes. The processes to be modeled enable simplifications of the Navier–Stokes equations on the classic Boussinesq and hydrostatic hypotheses. These equations are transformed within a sigma framework to make free surface processing easier. The main point of our demonstration focuses on the original aspect of the coupling between barotropic and baroclinic modes especially designed for ADI. It explains how full consistency of the transport calculated within the 2D and 3D equation sets was obtained. Lastly, we describe the physical processes simulated on a realistic configuration at a regional scale in the Bay of Biscay.
A new method is presented to construct a simple and general site bond correlated 3D HYdraulic POre Network model (HYPON) of hydraulic behavior of porous media for a wide range of permeability and porosity. Pore scale microstructure in this model is captured through simple power functions of Beti's influence lines that fix both the location and the size of throat (the narrowest section of bond) by relating the important elements of microstructure such as coordination number, porebody sizes and pore wall curvature. The new element in pore-network architecture is thus, the location of throat, which is important for smooth hydraulic transitions during steady state flow conditions. Despite the reduced number of parameters in comparison with other pore-network models, the morphological characteristics of HYPON compare well to those of the process-based predictive models in literature, and these characteristics are sensitive to the variance of porebody sizes rather than to the used type of the porebody size distributions. Processes such as diagenesis and dissolution are captured implicitly through the pore wall curvature parameter. Different combinations of porosity and permeability relations are obtained if the bond curvature and porebody sizes are varied. These relations reveal that effects of diagenesis and dissolution on the permeability may be ignored as they are secondary to effects on porosity.
Improved network flow models require the incorporation of increasingly accurate geometrical characterization of the microscale pore structure as well as greater information on fluid–fluid interaction (interfaces) at pore scales. We report on three dimensional (3D) pore scale medium characterization, absolute permeability computations for throat structures, and pore scale residual fluid distribution in a Berea core. X-ray computed microtomography combined with X-ray attenuating dopants is used to obtain 3D images of the pore network and to resolve phase distributions in the pore space.We present results on pore characterization, including distributions for pore volume, pore surface area, throat surface area, and principal direction diameters for pores and throats. Lattice Boltzmann computations are used to predict absolute permeabilities for individual throats reconstructed from the images. We present results on oil and water distribution in the pore space at residual conditions. We also consider the effects on residual fluid distribution due to the injection and gelation of a water-based gel. In extensive studies of Berea cores it has been observed that introducing water-based gels in the displacement process reduces permeability to water more than to oil. Our results provide supporting evidence for the involvement of gel compaction (dehydration) and oil trapping, while discounting gel blockage in throats, as mechanisms contributing to this effect.
The objective of this work is to develop a new numerical approach for the three-dimensional modelling of flow and transient solute transport in fractured porous media which would provide an accurate and efficient treatment of 3D complex geometries and inhomogeneities. For this reason, and in order to eliminate as much as possible the number of degrees of freedom, the fracture network, fractures and their intersections, are solved with a coupled 2D–1D model while the porous matrix is solved independently with a 3D model. The interaction between both models is accounted for by a coupling iterative technique. In this way it is possible to improve efficiency and reduce CPU usage by avoiding 3D mesh refinements of the fractures. The approach is based on the discrete-fracture model in which the exact geometry and location of each fracture in the network must be provided as an input. The formulation is based on a multidimensional coupling of the boundary element method-multidomain (BEM-MD) scheme for the flow and boundary element dual reciprocity method-multidomain (BE-DRM-MD) scheme for the transport. Accurate results and high efficiency have been obtained and are reported in this paper.
A three-dimensional numerical method of moments has been developed for solute flux through nonstationary flows in porous media. The solute flux is described as a space–time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. Flow nonstationarity may stem from various sources, such as the medium’s conductivity nonstationarity and complex hydraulic boundary conditions. The first two statistics of solute flux are derived using a Lagrangian framework and are expressed in terms of the probability density functions (PDFs). These PDFs are given in terms of one- and two-parcel moments of travel time and transverse locations, and these moments are related to the Eulerian velocity moments. The moment equations obtained analytically for flow and transport are so complex that numerical techniques are used to obtain solutions. In this study, we investigate the influence of various factors, such as the grid resolution relative to correlation length and the number of solute parcels comprising a source, on the accuracy of the calculation results. It has been found that for the computation of means and variances using the developed moment equations, hydraulic head requires at least one numerical grid element per correlation length scale. At least two grid elements are required for velocity, and 1–2 grid elements for the solute flux variance. Five parcels are required per correlation length scale to approximate the initial solute source distribution. The effects of boundary and hydraulic conductivity nonstationarity on flow and transport are also considered. Flow nonstationarity caused by either hydraulic boundary condition or conductivity nonstationarity significantly influences the transport process. The calculation results of numerical method of moments are compared with Monte Carlo simulations. The comparison indicates that the two methods are consistent with each other for head variance, velocity covariance in longitudinal direction, and mean and variance of total solute flux, but numerical method of moment underestimates the velocity variance in transverse direction. The method is applied to an environmental project for predicting the solute flux in the saturated zone below the Yucca Mountain project area, demonstrating the applicability of the method to complex subsurface environments.
We consider 3D steady flow of fresh water over a salt water body in a confined aquifer of constant thickness D, with application to a pumping well in a coastal aquifer. With neglect of mixing, a sharp interface separates the two fluid bodies and an existing analytical solution, based on the Dupuit assumption, is adopted. The aim is to solve for the mixing between the fresh and salt waters for αT/D ≪ 1 (αT transverse dispersivity), as field studies indicate that αT = O(10−3 − 10−2 m). The mixing zone around the interface is narrow and solutions by existing codes experience numerical difficulties. The problem is solved by the boundary layer (BL) approximation, extending a method, applied previously to two-dimensional flows. The BL equations of variable-density flow are solved by using the Von Karman integral method, to determine the BL thickness and the rate of entrainment of salt water along the interface. Application to the pumping well problem yields the salinity of the pumped water, as function of the parameters of the problem (well discharge, seaward discharge, well distance from the coast and density difference).
A spatially distributed snow model procedure for estimating snow melt, snow water equivalent and snow cover area is formulated and tested with data from the American River basin in California’s Sierra Nevada. An adaptation of the operational National Weather Service snow accumulation and ablation model is used for each model grid cell forced by spatially distributed precipitation and temperature data. The model was implemented with 6-hourly time steps on 1 km2 grid cells for the snow season of 1999–2003. Temperature is spatially interpolated using the prevailing lapse rate and digital terrain elevation data. Precipitation is spatially interpolated using regional climatological analyses obtained from PRISM. Parameters that control snow melt are distributed using ground surface aspect. The model simulations are compared with data from 12 snow-sensors located in the basin and the daily 500-m snow cover extent product from the MODIS/Terra satellite mission. The results show that the distribution of snow pack over the area is generally captured. The snow pack quantity compared to snow gauges is well estimated in high elevations with increasing uncertainty in the snow pack at lower elevations. Sensitivity and uncertainty analyses indicate that the significant input uncertainty for precipitation and temperature is primarily responsible for model errors in lower elevations and near the snow line. The model is suitable for producing spatially resolved realistic snow pack simulations when forced with operationally available observed or predicted data.
Constant head borehole infiltration tests are widely used for the in situ evaluation of saturated hydraulic conductivities of unsaturated soils above the water table. The formulae employed in analysing the results of such tests disregard the fact that some of the infiltrating water may flow under unsaturated conditions. Instead, these formulae are based on various approximations of the classical free surface theory which treats the flow region as if it were fully saturated and enclosed within a distinct envelope, the so-called ‘free surface’. A finite element model capable of solving free surface problems is used to examine the mathematical accuracy of the borehole infiltration formulae. The results show that in the hypothetical case where unsaturated flow does not exist, the approximate formulae are reasonably accurate within·a practical range of borehole conditions. To see what happens under conditions closer to those actually encountered in the field, the effect of unsaturated flow on borehole infiltration is investigated by means of two different numerical models: a mixed explicit-implicit finite element model, and a mixed explicit-implicit integrated finite difference model. Both of these models give nearly identical results; however, the integrated finite difference model is considerably faster than the finite element model. The relatively low computational efficiency of the finite element scheme is attributed to the large number of operations required in order to re-evaluate the conductivity (stiffness) matrix at each iteration in this highly non-linear saturated-unsaturated flow problem. The saturated-unsaturated analysis demonstrates that the classical free surface approach provides a distorted picture of the flow pattern in the soil. Contrary to what one would expect on the basis of this theory, only a finite region of the soil in the immediate vicinity of the borehole is saturated, whereas a significant percentage of the flow takes place under unsaturated conditions. As a consequence of disregarding unsaturated flow, the available formulae may underestimate the saturated hydraulic conductivity of fine grained soils by a factor of two, three, or more. Our saturated-unsaturated analysis leads to an improved design of borehole infiltration tests and a more accurate method for interpreting the results of such tests. The analysis also shows how one can predict the steady state rate of infiltration from data collected during the early transient period of the test.
The hydrodynamics of a system where there is a coupled flow above and below a sediment–water interface (SWI) are not completely understood. We numerically simulate mean two-dimensional, unidirectional, steady, viscous flow in these systems using a sequentially coupled formulation. Simulations were conducted to determine fundamental relationships between bedform geometry, Reynolds number for the water-column flow (Re), interfacial exchange zone depth (dz) in the sediments, and flux through the SWI (qint); the latter two parameters play a significant role in biogeochemical and aquatic-life processes across the SWI. dz and Re are functionally related through an asymptotic growth-curve model while qint and Re follow a power function. These relationships are dynamically explained by the manner in which pressure gradients along the SWI develop due to current–bedform interactions at different Res and by Darcy’s Law. We found that the coupling between water column and exchange zone flow is controlled by the behavior of the water-column eddy. The eddy detaches at or near the point of minimum pressure along the interface, and reattaches near the point of maximum pressure. These two critical points determine the pressure gradient along the bed surface that controls the exchange zone flow field. Moreover, the reattachment point corresponds to flow divides within the sediments. Lastly, pore-water velocities drop with depth below the SWI, and are larger below the bedform crests than below the troughs.
The spontaneous expansion and mobilization of discontinuous gas above dense non-aqueous-phase liquid (DNAPL) pools can affect the aqueous-phase concentrations of the DNAPL constituents above the pool. The results of an intermediate-scale, two-dimensional flow cell experiment showed that the discontinuous gas flow produced by spontaneous expansion, driven by the partitioning of 1,1,1-TCA from the surface of a DNAPL pool, resulted in detectable aqueous-phase concentrations of 1,1,1-TCA well above the pool surface. In comparison to a conventional model for DNAPL pool dissolution in the absence of a discontinuous gas phase, these concentrations were greater than expected, and were present at greater than expected elevations. Additionally, this study showed that the discontinuous gas flow produced transient behavior in the aqueous-phase concentrations, where the elevated concentrations occurred as short-term, pulse-like events. These results suggest that the spontaneous expansion and mobilization of discontinuous gas in DNAPL source zones could lead to the misdiagnosis of source zone architecture using aqueous concentration data, and that the transient nature of the elevated concentrations could further complicate the difficult task of source zone characterization.
A new experimental method is introduced for determining the relative magnitudes of liquid and vapor diffusion by using a small amount of soluble chemical as a tracer. The theoretical justification of the method is presented for the case where ice is absent. The feasibility of the method is demonstrated by an experiment using marine-deposited clay.
Connectivity of high/low-permeability areas has been recognized to significantly impact groundwater flow and solute transport. The task of defining a rigorous quantitative measure of connectivity for continuous variables has failed so far, and thus there exist a suite of connectivity indicators which are dependent on the specific hydrodynamic processes and the interpretation method. Amongst the many existing indicators, we concentrate on those characterizing connectivity between the points involved in a hydraulic or tracer test. The flow connectivity indicator used here is based on the time elapsed for hydraulic response in a pumping test (e.g., the storage coefficient estimated by the Cooper–Jacob method, Sest). Regarding transport, we select the estimated porosity from the breakthrough curve (ϕest). According to Knudby and Carrera [Knudby C, Carrera J. On the relationship between indicators of geostatistical, flow and transport connectivity. Adv Water Resour 2005;28(4):405–21] these two indicators measure connectivity differently, and are poorly correlated. Here, we use perturbation theory to analytically investigate the intrinsic relationship between Sest and ϕest. We find that ϕest can be expressed as a weighted line integral along the particle trajectory involving two parameters: the transmissivity point values, T, and the estimated values of Sest along the particle path. The weighting function is linear with the distance from the pumping well, thus the influence of the weighting function is maximum at the injection area, whereas the hydraulic information close to the pumping well becomes redundant (null weight). The relative importance of these two factors is explored using numerical simulations in a given synthetic aquifer and tested against intermediate-scale laboratory tracer experiments. We conclude that the degree of connectivity between two points of an aquifer (point-to-point connectivity) is a key issue for risk assessment studies aimed at predicting the travel time of a potential contaminant.
In this article, we propose a new model, called LBLR for Linear Backwater Lag-and-Route, which approximates the Saint-Venant equations linearized around a non-uniform flow in a finite channel (with a downstream boundary condition). A classical frequency approach is used to build the distributed Saint-Venant transfer function providing the discharge at any point in the channel in the Laplace domain with respect to the upstream discharge. The moment matching method is used to match a second-order-with-delay model on the theoretical distributed Saint-Venant transfer function. Model parameters are then expressed analytically as functions of the pool characteristics. The proposed model efficiently accounts for the effects of downstream boundary condition on the channel dynamics.
Two recently developed approaches to quantification of model (conceptual) error in a single groundwater model, a per-datum calibration methodology and a Bayesian model error analysis, were applied to a problem of 90Sr migration to water wells at Chernobyl, Ukraine. The intent of this composition is to demonstrate their utility to accounting for the uncertainty due to model error in estimating risks (or costs) in decision models. Bayesian model error analysis resulted in a more conservative estimate of the probability of the Pripyat Town well field contamination than did the per-datum calibration approach. This difference in risk estimates is a result of the conceptual differences between the two methods. Per-datum calibration relies primarily on information on model error contained in the measurements of the dependent variables to quantify its effect on model predictions. The Bayesian model error analysis assigns equal importance to prior information on the parameters and measurements of the dependent variable, thus allowing the incorporation of a more informative description of parameter distributions, as well as subjective judgement into a risk analysis. The suitability of either of the two methods, when applied to a specific problem, may be determined based on the nature and quantity of available data.
Extreme rainfall events are of particular importance due to their severe impacts on the economy, the environment and the society. Characterization and quantification of extremes and their spatial dependence structure may lead to a better understanding of extreme events. An important concept in statistical modeling is the tail dependence coefficient (TDC) that describes the degree of association between concurrent rainfall extremes at different locations. Accurate knowledge of the spatial characteristics of the TDC can help improve on the existing models of the occurrence probability of extreme storms. In this study, efficient estimation of the TDC in rainfall is investigated using a dense network of rain gauges located in south Louisiana, USA. The inter-gauge distances in this network range from about 1 km to 9 km. Four different nonparametric TDC estimators are implemented on samples of the rain gauge data and their advantages and disadvantages are discussed. Three averaging time-scales are considered: 1 h, 2 h and 3 h. The results indicate that a significant tail dependency may exist that cannot be ignored for realistic modeling of multivariate rainfall fields. Presence of a strong dependence among extremes contradicts with the assumption of joint normality, commonly used in hydrologic applications.
On the basis that hydrological users need to know the forecast uncertainty at the time that the forecast is issued, we computed distributions of radar rainfall forecast uncertainty as a function of forecast lead time, basin size, and forecasted rainfall intensity using data from the US 3-D National Mosaic of radar data. We document how exceptional forecasts such as those of heavy rainfall are generally biased. Since forecast uncertainty is also weather dependent, we tried to find good predictors to help either reduce the forecast uncertainty or better define it. These predictors were based either on characteristics of the current precipitation field or on the performance of the nowcast in the immediate past. The value of some predictors, especially those based on the properties of large-scale rainfall patterns, was significant though modest, the predictors being generally more skillful at characterizing forecast uncertainty than at improving forecast accuracy.
The main objective of this paper is to estimate the error in the rainfall derived from a polarimetric X-band radar, by comparison with the corresponding estimate of a rain gauge network. However the present analysis also considers the errors inherent to rain gauge, in particular instrumental and representativeness errors. A special emphasis is addressed to the spatial variability of the rainfall in order to appreciate the representativeness error of the rain gauge with respect to the 1 km square average, typical of the radar derived estimate. For this purpose the spatial correlation function of the rainfall is analyzed.The data set consists of 1-year radar data collected by the X-band polarimetric radar HYDRIX®, located in Beauce region (80 km south of Paris). All data were processed in real time using the ZPHI® algorithm. A dense 25 rain gauge network provided ground comparison data.The various sources of uncertainties (instrumental and representativeness) are then analyzed and quantified for each sensor.
Solving the kinematic wave equations for overland flow using the conventional consistent Galerkin finite element scheme is known to result in numerical oscillations due to the non-symmetric first spatial derivative terms in the kinematic wave equations. In this paper the lumped and the upwind finite element schemes are evaluated as alternatives to the consistent Galerkin finite element scheme. Stability analysis of the upwind scheme shows that the damping effect, that could reduce the oscillations, is small for the high Courant numbers encountered in overland flow problems. The upwind scheme, using upwind factors of 0.1 and 1.0, did not provide any improvement to the stability of the lumped and the consistent schemes. The lumped scheme considerably reduces oscillations without significant reduction in the overall solution accuracy. No analytical guidelines for time-step criteria that will insure stability and accuracy were provided by the stability analysis performed for the three schemes. Problem specific accuracy-based dynamic time-step criteria was developed and evaluated for the lumped scheme. These time-steps were found to be on average, double the size of the dynamic time-steps for the consistent scheme.
We propose an algorithm for particle tracking based on Cheng's method [Int. J. Numer. Meth. 39 (1996) 1111–1136]. Velocities in a flow field are known at a series of points and interpolated between them by finite element local functions. Tracking is performed in local coordinates, element by element, using any standard ODE solution method. The exit from an element is found using the polynomials to interpolate between the tracking points. The algorithm was tested and compared to Pollock's and Cheng's method in a series of numerical experiments, in which the Euler, Runge–Kutta 2, Runge–Kutta 5(4) and Runge–Kutta 6(4) ODE solution methods were combined with first-, second-, third- and fifth-order exit polynomials. The known velocities had a random error with standard deviation of 0%, 0.1% and 1% of the velocity. Meaningful results were obtained only when the spatial interpolation error and the error of the tracking method were calculated separately, otherwise some results were misleading. The numerical experiments confirmed that the accuracy of the exit polynomial has to be consistent with the ODE solution method. Quadratic interpolation of velocities on a coarser mesh often gives more accurate path lines and requires less computational time than linear interpolation. Pollock's method for particle tracking is viable only if input data are rather inaccurate and path lines nearly straight. Cheng's method is appropriate for moderately accurate input data, while the proposed algorithm with Runge–Kutta 5(4) or Runge–Kutta 6(4) method and fifth-order exit polynomial has excellent accuracy. Computational time is about 10 times longer than for Cheng's method while the accuracy is increased by several orders of magnitude.
Soil moisture satellite mission accuracy, repeat time and spatial resolution requirements are addressed through a numerical twin data assimilation study. Simulated soil moisture profile retrievals were made by assimilating near-surface soil moisture observations with various accuracy (0, 1, 2, 3, 4, 5 and 10%v/v standard deviation) repeat time (1, 2, 3, 5, 10, 15, 20 and 30 days), and spatial resolution (0.5, 6, 12 18, 30, 60 and 120 arc-min). This study found that near-surface soil moisture observation error must be less than the model forecast error required for a specific application when used as data assimilation input, else slight model forecast degradation may result. It also found that near-surface soil moisture observations must have an accuracy better than 5%v/v to positively impact soil moisture forecasts, and that daily near-surface soil moisture observations achieved the best soil moisture and evapotranspiration forecasts for the repeat times assessed, with 1–5 day repeat times having the greatest impact. Near-surface soil moisture observations with a spatial resolution finer than the land surface model resolution (∼30 arc-min) produced the best results, with spatial resolutions coarser than the model resolution yielding only a slight degradation. Observations at half the land surface model spatial resolution were found to be appropriate for our application. Moreover, it was found that satisfying the spatial resolution and accuracy requirements was much more important than repeat time.