Inverse modelling techniques for determining hydraulic properties of coarse-textured porous media by transient outflow methods

A series of multi-step outflow experiments was carried out to identify the unsaturated hydraulic properties of two homogeneous coarse-textured porous media (glass beads and sand). Because of the measured sharp fronts of water content decrease during these experiments the hydraulic functions are assumed to be represented by the complete van Genuchten–Mualem closed-form expressions with variable coefficients α, n, m and θ_r. The values of θ_s and K_s were measured directly. A sensitivity analysis with respect to α, n, and m shows that conditions of local identifiability are satisfied if measurements of water content at some inner points inside the column are considered. The inverse modelling technique consists of two steps: first, computation of objective function values based on water content data responses to obtain initial parameter estimations, and second, a more detailed parameter determination using a Levenberg–Marquardt scheme. In both steps a numerical model incorporating the hydraulic functions is utilized to simulate theoretical pressure head and water content distributions along the column. For both porous media unique solutions of the inverse problem could be obtained, and afterwards, the corresponding hydraulic functions were verified from additional drainage experiments.©1998 Elsevier Science Limited. All rights reserved     

Boundary integral formulation and two-dimensional fundamental solutions for dynamic behavior analysis of unsaturated soils

In this paper the coupled equations governing the dynamic behavior of unsaturated soils are derived based on the poromechanics theory within the framework of the suction-based mathematical model presented by Gatmiri (1997) [Gatmiri B. Analysis of fully coupled behavior of unsaturated porous medium under stress, suction and temperature gradient. Final report of CERMES-EDF, 1997] and Gatmiri et al. (1998) [Gatmiri B, Delage P, Cerrolaza M, UDAM: a powerful finite element software for the analysis of unsaturated porous media. Adv Eng Software 1998; 29(1): 29–43]. In this formulation, the solid skeleton displacements, water pressure and air pressure are presumed to be independent variables. The Boundary Integral formulations as well as fundamental solutions for such a dynamic u–p_w–p_a theory are presented in this paper for the first time. The boundary integral equations are derived via the use of the weighted residuals method in a way that permits an easy discretization and implementation in a Boundary Element code. Also, the associated two dimensional (2D) fundamental solutions for such deformable porous medium with linear elastic behavior are derived in Laplace transform domain using the method of Hörmander. Finally, some numerical results are presented to show the accuracy of the proposed solutions. The derived results are verified analytically by comparison with the previously introduced corresponding fundamental solutions in elastodynamic limiting case.     

A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain

An unstructured Godunov-type finite volume model is developed for the numerical simulation of geometrically challenging two-dimensional shallow water flows with wetting and drying over convoluted topography. In the framework of sloping bottom model, a modified formulation of shallow water equations is used to preserve mass conservation during flooding and recession. The key ingredient of the model is the use of this combination of the sloping bottom model and the modified shallow water equations to provide a robust technique for wet/dry fronts tracking and, together with centered discretization of the bed slope source term, to exactly preserve the static flow on irregular topographies. The variable reconstruction technique ensures nonnegative reconstructed water depth and reasonable reconstructed velocity, and the friction terms are solved by semi-implicit scheme that does not invert the direction of velocity components. The robustness and accuracy of the proposed model are assessed by comparing numerical and reference results of extensive test cases. Moreover, the results of a dam-break flooding over real topography are presented to show the capability of the model on field-scale application.     

Numerical simulation of a dam break for an actual river terrain environment

A two‐dimensional (2D) finite‐difference shallow water model based on a second‐order hybrid type of total variation diminishing (TVD) approximate solver with a MUSCL limiter function was developed to model flooding and inundation problems where the evolution of the drying and wetting interface is numerically challenging. Both a minimum positive depth (MPD) scheme and a non‐MPD scheme were employed to handle the advancement of drying and wetting fronts. We used several model problems to verify the model, including a dam break in a slope channel, a dam break flooding over a triangular obstacle, an idealized circular dam‐break, and a tide flow over a mound. Computed results agreed well with the experiment data and other numerical results available. The model was then applied to simulate the dam breaking and flooding of Hsindien Creek, Taiwan, with the detailed river basin topography. Computed flooding scenarios show reasonable flow characteristics. Though the average speed of flooding is 6–7 m s^?1, which corresponds to the subcritical flow condition (Fr < 1), the local maximum speed of flooding is 14·12 m s^?1, which corresponds to the supercritical flow condition (Fr ≈ 1·31). It is necessary to conduct some kind of comparison of the numerical results with measurements/experiments in further studies. Nevertheless, the model exhibits its capability to capture the essential features of dam‐break flows with drying and wetting fronts. It also exhibits the potential to provide the basis for computationally efficient flood routing and warning information. Copyright © 2006 John Wiley & Sons, Ltd.     

Parallel iterative solution for h and p approximations of the shallow water equations

A p finite element scheme and parallel iterative solver are introduced for a modified form of the shallow water equations. The governing equations are the three-dimensional shallow water equations. After a harmonic decomposition in time and rearrangement, the resulting equations are a complex Helmholz problem for surface elevation, and a complex momentum equation for the horizontal velocity. Both equations are nonlinear and the resulting system is solved using the Picard iteration combined with a preconditioned biconjugate gradient (PBCG) method for the linearized subproblems. A subdomain-based parallel preconditioner is developed which uses incomplete LU factorization with thresholding (ILUT) methods within subdomains, overlapping ILUT factorizations for subdomain boundaries and under-relaxed iteration for the resulting block system. The method builds on techniques successfully applied to linear elements by introducing ordering and condensation techniques to handle uniform p refinement. The combined methods show good performance for a range of p (element order), h (element size), and N (number of processors). Performance and scalability results are presented for a field scale problem where up to 512 processors are used.     

A multi-envelope vertical coordinate system for numerical ocean modelling

A multi-envelope generalised coordinate system for numerical ocean modelling is introduced. In this system, computational levels are curved and adjusted to multiple ‘virtual bottoms’ (aka envelopes) rather than following geopotential levels or the actual bathymetry. This allows defining computational levels which are optimised to best represent different physical processes in different sub-domains of the model. In particular, we show how it can be used to improve the representation of tracer advection in the ocean interior. The new vertical system is compared with a widely used z-partial step scheme. The modelling skill of the models is assessed by comparison with the analytical solutions or results produced by a model with a very high-resolution z-level grid. Three idealised process-oriented numerical experiments are carried out. Experiments show that numerical errors produced by the new scheme are much smaller than those produced by the standard z-partial step scheme at a comparable vertical resolution. In particular, the new scheme shows superiority in simulating the formation of a cold intermediate layer in the ocean interior and in representing dense water cascading down a steep topography.     

High-order balanced CWENO scheme for movable bed shallow water equations

In this work the numerical integration of 1D shallow water equations (SWE) over movable bed is performed using a well-balanced central weighted essentially non-oscillatory (CWENO) scheme, fourth-order accurate in space and in time. Time accuracy is obtained following a Runge–Kutta (RK) procedure, coupled with its natural continuous extension (NCE). Spatial accuracy is obtained using WENO reconstructions of conservative variables and of flux and bed derivatives. An original treatment for bed slope source term, which maintains the established order of accuracy and satisfies the property of exactly preserving the quiescent flow (C-property), is introduced in the scheme. This treatment consists of two procedures. The former involves the evaluation of the point-values of the flux derivative, considered as a whole with the bed slope source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the expected regularity of the free surface elevation. The high accuracy of the scheme allows to obtain good results using coarse grids, with consequent gain in terms of computational effort. The well-balancing of the scheme allows to reproduce small perturbations of the free surface and of the bottom otherwise of the same order of magnitude of the numerical errors induced by the non-balancing. The accuracy, the well-balancing and the good resolution of the model in reproducing free surface flow over movable bed are tested over analytical solutions and over numerical results available in literature.     

Experiments with a numerical model of coastal currents and tidal mixing fronts

A three-dimensional numerical model is used to simulate the development of disturbances on shelf-sea coastal currents and fronts. The model, which has a free surface, uses a finite difference grid ☐ scheme based on sigma coordinates. It has a semi-implicit scheme for the barotropic flow and a hydrid advection scheme to retain sharp fronts. The results demonstrate that (i) eddy formation follows changes at the inflow of a coastal current, (ii) a simple radiation boundary condition at the outflow produces nearly identical results for different outflow boundary positions, (iii) eddy growth, with matching behaviour of surface and bottom fronts, follows a small displacement on a tidal mixing front and (iv) effects of friction and mixing can significantly alter the behaviour of the front and the relative strength of the cyclonic and anticyclonic eddies formed.     

Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed

In this paper, we study the numerical approximation of the two-dimensional morphodynamic model governed by the shallow water equations and bed-load transport following a coupled solution strategy. The resulting system of governing equations contains non-conservative products and it is solved simultaneously within each time step. The numerical solution is obtained using a new high-order accurate centered scheme of the finite volume type on unstructured meshes, which is an extension of the one-dimensional PRICE-C scheme recently proposed in Canestrelli et al. (2009) [5]. The resulting first-order accurate centered method is then extended to high order of accuracy in space via a high order WENO reconstruction technique and in time via a local continuous space–time Galerkin predictor method. The scheme is applied to the shallow water equations and the well-balanced properties of the method are investigated. Finally, we apply the new scheme to different test cases with both fixed and movable bed. An attractive future of the proposed method is that it is particularly suitable for engineering applications since it allows practitioners to adopt the most suitable sediment transport formula which better fits the field data.     

Predicted positions of tidal fronts in the Gulf of Maine region

The distribution ofSimpson andHunter's (1974, Nature, 250, 404–406) tidal mixing parameter in the Gulf of Maine region, calculated fromGreenberg's (1983, Journal of Physical Oceanography, 13, 886–904) numerical model of the M₂ tide, is presented and used in a discussion of the summertime positions of tidal fronts and their variability. The role of wind mixing is assessed using a version of Simpson and Hunter's criterion reformulated to include a fixed fraction of the climatological wind-derived energy in the Gulf. Two alternative criteria for the extent of tidally well-mixed areas, one based on a near-surface energy partition and the other on the bottom Ekman layer thickness, are also examined. It is suggested that all of these factors should be given further consideration in criteria for the extent of vertically well-mixed areas on continental shelves.     

Hydrodynamic accumulation of Karenia off the west coast of Florida

Blooms of the toxic dinoflagellates, Karenia spp. occur nearly annually in the eastern Gulf of Mexico with cell abundances typically >10⁵ cells L⁻¹. Thermal and ocean color satellite imagery shows sea surface temperature patterns indicative of upwelling events and the concentration of chlorophyll at fronts along the west Florida continental shelf. Daily cell counts of Karenia show greater increases in cell concentrations at fronts than can be explained by Karenia's maximum specific growth rate. This is observed in satellite images as up to a 10-fold greater increase in chlorophyll biomass over 1–2 d periods than can be explained by in situ growth. In this study, we propose a model that explains why surface blooms of Karenia may develop even when nutrients on the west Florida shelf are low. In the summer, northward winds produce a net flow east and southeast bringing water and nutrients from the Mississippi River plume onto the west Florida shelf at depths of 20–50 m. This water mass supplies utilizable inorganic and organic forms of nitrogen that promote the growth of Karenia to pre-bloom concentrations in sub-surface waters in the mid-shelf region. In the fall, a change to upwelling favorable winds produces onshore transport. This transport, coupled with the swimming behavior of Karenia, leads to physical accumulation at frontal regions near the coast, resulting in fall blooms. Strong thermal fronts during the winter provide a mechanism for re-intensification of the blooms, if Karenia cells are located north of the fronts. This conceptual model leads to testable hypotheses on bloom development throughout the Gulf of Mexico.     

Multi-resolution adaptive modeling of groundwater flow and transport problems

Many groundwater flow and transport problems, especially those with sharp fronts, narrow transition zones, layers and fingers, require extensive computational resources. In this paper, we present a novel multi-resolution adaptive Fup approach to solve the above mentioned problems. Our numerical procedure is the Adaptive Fup Collocation Method (AFCM), based on Fup basis functions and designed through a method of lines (MOL). Fup basis functions are localized and infinitely differentiable functions with compact support and are related to more standard choices such as splines or wavelets. This method enables the adaptive multi-resolution approach to solve problems with different spatial and temporal scales with a desired level of accuracy using the entire family of Fup basis functions. In addition, the utilized collocation algorithm enables the mesh free approach with consistent velocity approximation and flux continuity due to properties of the Fup basis functions. The introduced numerical procedure was tested and verified by a few characteristic groundwater flow and transport problems, the Buckley–Leverett multiphase flow problem, the 1-D vertical density driven problem and the standard 2-D seawater intrusion benchmark–Henry problem. The results demonstrate that the method is robust and efficient particularly when describing sharp fronts and narrow transition zones changing in space and time.     

A three-dimensional numerical shelf-sea front model with variable eddy viscosity and diffusivity

A three-dimensional numerical model of circulation and eddy development in shelf-sea fronts is applied to three frontal structures, with two parameterization schemes for vertical eddy viscosity and diffusivity. The three fronts resemble those in the German Bight (a front between relatively fresh coastal water and saltier water offshore, with an interface extending from surface to bottom), the Norwegian Coastal Current (also formed by fresh coastal water but with a thermocline on one side), and the Celtic Sea (a front between water which is stratified in summer and water which is well mixed throughout the year). The two mixing assumptions, modelling the reduction of turbulence in stratified zones, are based on the Munk-Anderson scheme and the turbulent energy equation. Many features of frontal dynamics are common to all the results: strong surface currents along the front, cross-frontal circulation cells, a considerable enhancement of vertical velocities when eddies are formed, and development of eddies into cyclonic-anticyclonic vortex pairs. Cross-frontal circulation and frontal sharpening are the variables most sensitive to the different mixing assumptions. The German Bight front is the one most affected by changing these assumptions. The comparisons suggest that realistic results may be obtained from models despite the present uncertainty about vertical mixing in stratified shelf seas.     

Error analysis of finite difference methods for two-dimensional advection–dispersion–reaction equation

In this paper, the numerical errors associated with the finite difference solutions of two-dimensional advection–dispersion equation with linear sorption are obtained from a Taylor analysis and are removed from numerical solution. The error expressions are based on a general form of the corresponding difference equation. The variation of these numerical truncation errors is presented as a function of Peclet and Courant numbers in X and Y direction, a Sink/Source dimensionless number and new form of Peclet and Courant numbers in X–Y plane. It is shown that the Crank–Nicolson method is the most accurate scheme based on the truncation error analysis. The effects of these truncation errors on the numerical solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution for predicting contaminant plume distribution in uniform flow field. Considering computational efficiency, an alternating direction implicit method is used for the numerical solution of governing equation. The results show that removing these errors improves numerical result and reduces differences between numerical and analytical solution.     

Temperature distribution related to shelf-sea fronts on the Patagonian Shelf

Hydrographic data taken from the fishery research ship Dr. Holmberg and an infrared image from the NOAA-6 satellite show the location of shelf-sea fronts over the Patagonian Shelf off the coast of Argentina.High tidal dissipation rates in this area [Simpson and Bowers (1981) Deep-Sea Research, 28, 727–738], intensified by headlands such as the Peninsula of Valdes, generate zones of enhanced vertical mixing in the near-shore waters, which lowers the sea-surface temperature. Shelf fronts separate this water from the seasonally stratified water offshore. The irregular sea-surface temperature pattern suggests the transfer of some of the cold mixed water into the upper layer of stratified water farther offshore, leading to the inference of a northeastward mean flow.     

A three dimensional k-ε-A_p model for water-sediment movement

A three-dimensional k-ε-Ap two-fluid turbulence model is proposed to study liquid-particle two-phase flow and bed deformation. By solving coupled liquid-phase and solid-phase governing equations in a finite-volume method, the model can calculate the movement of both water and sediment. The model was validated by water-sediment transport in a 180° channel bend with a movable bed. The validation concerns two-phase time-averaged velocities, bed deformation, water depth, depth-averaged streamwise velocity, cross-stream bed profiles, and two-phase secondary flow velocity vectors. The agreement between numerical results and experimental results was generally good. The comparisons of the numerical results of different models show that the three-dimensional k-ε-Ap two-fluid turbulence model has a relatively higher accuracy than one-fluid model.     

Numerical Modeling of Flow Kinematics in a Segment of Inerodable Channel

A numerical model of a riffle segment of a large river is applied to examine the kinematic flow structure at different phases of water regime. A finite-difference scheme on an irregular triangular grid is proposed for solving shallow-water equations. The results of field measurements and calculations are presented.     

Horizontal patterns of water temperature and salinity in an estuarine tidal channel: Ria de Aveiro

This work presents results from two complementary and interconnected approaches to study water temperature and salinity patterns in an estuarine tidal channel. This channel is one of the four main branches of the Ria de Aveiro, a shallow lagoon located in the Northwest coast of the Iberian Peninsula. Longitudinal and cross-sectional fields of water temperature and salinity were determined by spatial interpolation of field measurements. A numerical model (Mohid) was used in a 2D depth-integrated mode in order to compute water temperature and salinity patterns. The main purpose of this work was to determine the horizontal patterns of water temperature and salinity in the study area, evaluating the effects of the main forcing factors. The field results were depth-integrated and compared to numerical model results. These results obtained using extreme tidal and river runoff forcing, are also presented. The field results reveal that, when the river flow is weak, the tidal intrusion is the main forcing mechanism, generating saline and thermal fronts which migrate with the neap/spring tidal cycle. When the river flow increases, the influence of the freshwater extends almost as far as the mouth of the lagoon and vertical stratification is established. Results of numerical modelling reveal that the implemented model reproduces quite well the observed horizontal patterns. The model was also used to study the hydrology of the study area under extreme forcing conditions. When the model is forced with a low river flow (1 m³ s⁻¹) the results confirm that the hydrology is tidally dominated. When the model is forced with a high river flow (1,000 m³ s⁻¹) the hydrology is dominated by freshwater, as would be expected in such an area.     

A marching in space and time (MAST) solver of the shallow water equations. Part I: The 1D model

A new approach is presented for the numerical solution of the complete 1D Saint-Venant equations. At each time step, the governing system of partial differential equations (PDEs) is split, using a fractional time step methodology, into a convective prediction system and a diffusive correction system. Convective prediction system is further split into a convective prediction and a convective correction system, according to a specified approximated potential. If a scalar exact potential of the flow field exists, correction vanishes and the solution of the convective correction system is the same solution of the prediction system. Both convective prediction and correction systems are shown to have at each x − t point a single characteristic line, and a corresponding eigenvalue equal to the local velocity. A marching in space and time (MAST) technique is used for the solution of the two systems. MAST solves a system of two ordinary differential equations (ODEs) in each computational cell, using for the time discretization a self-adjusting fraction of the original time step. The computational cells are ordered and solved according to the decreasing value of the potential in the convective prediction step and to the increasing value of the same potential in the convective correction step. The diffusive correction system is solved using an implicit scheme, that leads to the solution of a large linear system, with the same order of the cell number, but sparse, symmetric and well conditioned. The numerical model shows unconditional stability with regard of the Courant–Friedrichs–Levi (CFL) number, requires no special treatment of the source terms and a computational effort almost proportional to the cell number. Several tests have been carried out and results of the proposed scheme are in good agreement with analytical solutions, as well as with experimental data.     

High-order finite volume WENO schemes for the shallow water equations with dry states

The shallow water equations are used to model flows in rivers and coastal areas, and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. These equations have still water steady state solutions in which the flux gradients are balanced by the source term. It is desirable to develop numerical methods which preserve exactly these steady state solutions. Another main difficulty usually arising from the simulation of dam breaks and flood waves flows is the appearance of dry areas where no water is present. If no special attention is paid, standard numerical methods may fail near dry/wet front and produce non-physical negative water height. A high-order accurate finite volume weighted essentially non-oscillatory (WENO) scheme is proposed in this paper to address these difficulties and to provide an efficient and robust method for solving the shallow water equations. A simple, easy-to-implement positivity-preserving limiter is introduced. One- and two-dimensional numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.