Water Flow Equations Research Articles

A comprehensive mathematical model for transport of soil-dissolved chemicals by overland flow

The model developed in this study simulates the contamination of overland flow by soil chemicals that reside near its surface during a surface runoff event. The model includes mass-balance equations for both water flow and chemical transport in the soil profile and surface runoff. A rate-limited mass transfer through an overland-flow boundary layer at the soil overland flow interface controls the dissolved chemical transfer from soil solution to overland flow, once formed. The model predicts water flow and chemical transport in the soil profile prior to the rainfall ponding (when overland flow starts) and during the surface runoff event. The predictions of these variables, together with the total load to the surface runoff, were successfully compared with the measured data of Hubbard et al. [Trans. ASAE, 32(4) (1989) 1239]. Being physically based, the model was used to investigate the dependence of surface runoff pollution and its extent on the system hydrological parameters. A key factor on the availability of soil chemicals to pollute the overland flow is their displacement by infiltrating water prior to runoff initiation. Being dependent on soil moisture prior to rainfall initiation and on rainfall intensity, a lower chemical concentration and a lower load in surface runoff are obtained for longer ponding times, ones that are associated with lower rainfall rates and initially drier soil profiles. During the surface runoff flow, the chemical concentration in overland flow at the slope outlet is affected by the contact time of an overland flow parcel with the soil surface. Thus, it increases for higher values of equilibrium time — t E, lower rainfall rates, slope gradients, and higher soil-surface roughness coefficients. These parameters have an inverse effect on the surface runoff concentration by affecting the transfer coefficient of soil chemical to overland flow. A different insight into the relationship between the relevant dynamic processes throughout the storm event is achieved by studying the transient variation of soil chemical flux to overland flow, the chemical flux at the slope outlet, and the change of chemical mass in the overland flow.

Numerical simulation of sea water intrusion near Beihai, China

A leaky aquifer system occurs in the coastal plain near Beihai, China. Seawater intrusion into the confined aquifer took place along the northern coast. Chloride concentrations at some observation wells increased steadily from 1988 and were at their peak in 1993. A quasi-three-dimensional element model has been developed to simulate the spatial and temporal evolution of hydraulic heads and chloride concentrations of the groundwater near the northern coast. The simulation model was based on the transition zone approach, which requires simultaneous solution of the governing water flow and solute transport equations. An irregular grid of a quadrangle was used to discretize the flow domain. Various aquifer parameters were verified with the numerical model in order to obtain satisfactory matches between computed values and observed data from an investigation. Three pumpage schemes were designed to use the calibrated model for prediction of future changes in water levels and chloride concentrations in groundwater in the study area. Results show that seawater intrusion would worsen in the confined aquifer if the current rates of groundwater pumpage continue. The alternative, to eliminate pumpage in the intruded area and to moderate pumpage rates from water supply wells far from the seashore, may limit seawater intrusion significantly and is considered attractive in the area.

Modeling flow and transport in a two-dimensional dual-permeability system with spatially variable hydraulic properties

Most field soils exhibit soil spatial variability as well as soil structure. The challenge is to account adequately for both types of spatial heterogeneity in simulation models. A numerical finite element code was used to compare single- and dual-permeability approaches for modeling variably saturated flow and transport in two-dimensional heterogeneous soil systems. The code was based on the Richards' equation for water flow and the advection-dispersion equation for solute transport. Spatial variability in the soil hydraulic properties was accounted for by randomly generating a hydraulic conductivity field using a one-dimensional first-order Markov process. Soil structural effects were modeled with a two-domain concept in which a first-order kinetic expression is used to describe the transfer of water and solute between the two domains. Numerical experiments were carried out for the case of furrow irrigation, including the breakthrough of a conservative solute to the groundwater table. We compared five different scenarios: a single domain having uniform hydraulic properties (SU), a single domain with a randomly distributed hydraulic conductivity (SR), a dual-permeability system with uniform hydraulic properties (DU), a dual-permeability system with a randomly distributed fracture hydraulic conductivity (DRF), and a dual-permeability system having a randomly distributed matrix hydraulic conductivity (DRM). All scenarios started with pressure heads in equilibrium with a constant groundwater table 150 cm below the soil surface and zero initial solute concentrations. The simulated two-dimensional (2D) vertical concentration profiles showed preferential pathways resulting from both the spatial variability (SR) and soil structure (DRF) scenarios. As expected, drainage of water from the bottom of the profile occurred significantly earlier for dual-than for single-permeability scenarios. The combination of having spatial variability in the hydraulic properties and invoking the dual-permeability approach yielded the quickest and largest leaching of solute. The 2D dual-permeability approach should considerably improve the simulation of water and solute movement in naturally heterogeneous field soils.

Fully Coupled Model of Ice-Jam Dynamics

This paper presents a numerical model that simulates the dynamic failure and re-formation of an ice jam. The model comprises a simultaneous solution of the 1D, unsteady flow equations (conservation of mass and momentum) for water flow and ice motion. Its use is demonstrated in the routing of flow hydrographs through a channel containing an initial jam or accumulation of broken ice. The model shows the extent to which changes in flow discharge may affect profiles of jam thickness and flow depth for these situations. It also shows how ice momentum may affect the jam-thickness profile. Dimensionless parameters, characterizing initial jam stability and the shape of the hydrograph, are identified to indicate conditions when changes in flow discharge and ice momentum should be taken into account when estimating jam thickness.

Modeling the Vacuum Filtration of Fine Coal. III. Comparison of Models for Predicting Desaturation Kinetics

Three mathematical models devised by Wakeman, by Baluais, and by Nicolaou have been compared in their ability to predict the desaturation kinetics of fine (−0.5+0 mm) coal using vacuum filtration. Although similar in their basic approach and many of their underlying assumptions, the models diverge considerably in the techniques used to solve the equations of air and water flow through a porous filter cake. The models were assessed as a function of vacuum level and cake height, with a deviation of below 1 wt% moisture from the experimental data being deemed a success. All models predicted cake moisture after 300 seconds desaturation time to within 1 wt% moisture, except Baluais, which deviated significantly (by 2.3–3.2 wt% moisture) at the lowest vacuum (40 kPa) and lowest cake height (8–9 mm) tested. At shorter times (up to 50 seconds), the Wakeman model showed the best all-round performance, never deviating more than 1.5 wt% from experiment, and frequently being much closer. The Nicolaou model was particularly good at predicting desaturation kinetics for the thinnest cakes. Taking the data set as a whole, there were no obvious trends for any of the models in systematically under- or overestimating product moisture.

NUMERICAL STUDY ON A SHALLOW WATER MODEL WITH LIGNOSA FROM LARGE-EDDY SIMULATION

In a river, diffusion which is caused by horizontal shear has a great effbct on the environmental problem and in previous studies we made a shallow water flow equation including turbulent term on the basis of horizontal shear.In this study we propose a shallow water flow equation taking into accout Iignosa by using large eddy simulation theory and carried out experiments which types are high density, low density and middle density of lignosa. As results, in case of high density the water depth obtained by a equation including horizontal shear is more similar to the experimental results than a equation not including horizontal shear and in case of low density the water depth is consistent with the experimental results greatly by including horizontal shear.

Modeling of soil deformation and water flow in a swelling soil

Soil deformation and unsaturated transient water flow in swelling soil on a laboratory scale is predicted using a one-dimensional numerical model. The model is based on a soil water flow equation and extended to soil deformation using Lagrangian description (LD). The specific features of the model are inclusion of an overburden component in the total potential of the flow equation, introduction of a shrinkage–swelling characteristic (SSC) known as a third soil hydraulic function, and two-dimensional analysis of soil deformation using a geometry factor. This paper describes an evaluation of the model, which was previously verified with a case of shrinkage of marine clay soil, with a data set from an infiltration experiment performed under swelling conditions. Soil hydraulic properties such as moisture retention characteristic (MRC), hydraulic conductivity function (K(h)) and SSC were derived from simultaneous measurement of volumetric moisture content (VMC), bulk density and pressure heads in a framework of the Eulerian description (ED). Two different K(h) relationships, with respect to moving solid particles, were obtained from both the LD and ED. Simulation was performed using two different hydraulic conductivities, i.e., Eulerian descriptive hydraulic conductivity (EDHC) and Lagrangian descriptive hydraulic conductivity (LDHC). Model performance was evaluated using four different statistical parameters representing the goodness of model predictions on the transient variations of soil moisture contents. Results of the evaluation reveal that the model predicts reasonably well the moisture content changes, although the use of LDHC gives slightly better results than the EDHC. Overprediction of moisture contents by the simulation model especially near the bottom of the sample is attributed to the incomplete dissipation of swelling pressure during expansion of the soil material. As a result, the soil shows anisotropic swelling although it was forced to deform in the one-dimensional vertical direction by experimental design. Unlike other one-dimensional consolidation models, the model can be used for two-dimensional analysis of soil deformation in porous media with either self-weight or external load application, at either shrinking or swelling phase and under both saturated and unsaturated flow conditions.

Use of two‐step split‐operator approach for 2D shallow water flow computation

The objective of this paper is to present a methodology of using a two-step split-operator approach for solving the shallow water flow equations in terms of an orthogonal curvilinear co-ordinate system. This approach is in fact one kind of the so-called fractional step method that has been popularly used for computations of dynamic flow. By following that the momentum equations are decomposed into two portions, the computation procedure involves two steps. The first step (dispersion step) is to compute the provisional velocity in the momentum equation without the pressure gradient. The second step (propagation step) is to correct the provisional velocity by considering a divergence-free velocity field, including the effect of the pressure gradient. This newly proposed method, other than the conventional split-operator methods, such as the projection method, considers the effects of pressure gradient and bed friction in the second step. The advantage of this treatment is that it increases flexibility, efficiency and applicability of numerical simulation for various hydraulic problems. Four cases, including back-water flow, reverse flow, circular basin flow and unsteady flow, have been demonstrated to show the accuracy and practical application of the method. Copyright © 1999 John Wiley & Sons, Ltd.

Use of two-step split-operator approach for 2D shallow water flow computation

The objective of this paper is to present a methodology of using a two-step split-operator approach for solving the shallow water flow equations in terms of an orthogonal curvilinear co-ordinate system. This approach is in fact one kind of the so-called fractional step method that has been popularly used for computations of dynamic flow. By following that the momentum equations are decomposed into two portions, the computation procedure involves two steps. The first step (dispersion step) is to compute the provisional velocity in the momentum equation without the pressure gradient. The second step (propagation step) is to correct the provisional velocity by considering a divergence-free velocity field, including the effect of the pressure gradient. This newly proposed method, other than the conventional split-operator methods, such as the projection method, considers the effects of pressure gradient and bed friction in the second step. The advantage of this treatment is that it increases flexibility, efficiency and applicability of numerical simulation for various hydraulic problems. Four cases, including back-water flow, reverse flow, circular basin flow and unsteady flow, have been demonstrated to show the accuracy and practical application of the method. Copyright © 1999 John Wiley & Sons, Ltd.

A numerical model for density-and-viscosity-dependent flows in two-dimensional variably saturated porous media

We present a formulation for water flow and solute transport in two-dimensional variably saturated media that accounts for the effects of the solute on water density and viscosity. The governing equations are cast in a dimensionless form that depends on six dimensionless groups of parameters. These equations are discretized in space using the Galerkin finite element formulation and integrated in time using the backward Euler scheme with mass lumping. The modified Picard method is used to linearize the water flow equation. The resulting numerical model, the MARUN model, is verified by comparison to published numerical results. It is then used to investigate beach hydraulics at seawater concentration (about 30 g l −1) in the context of nutrients delivery for bioremediation of oil spills on beaches. Numerical simulations that we conducted in a rectangular section of a hypothetical beach revealed that buoyancy in the unsaturated zone is significant in soils that are fine textured, with low anisotropy ratio, and/or exhibiting low physical dispersion. In such situations, application of dissolved nutrients to a contaminated beach in a freshwater solution is superior to their application in a seawater solution. Concentration-engendered viscosity effects were negligible with respect to concentration-engendered density effects for the cases that we considered.

FLOW AROUND A CIRCULAR CYLINDER IN A SUPERCRITICAL FLOW

Supercritical flows sometimes occur during floods and shock waves can be easily found on the both sides of the pier. In order to know the influences of this type flow toward a pier, it is necessary to study on a flow around a circular cylinder in a supercritical flow. In this study we carried out experiments on a flow around a circular cylinder with the Froudc number of 1.98 and 2.06. Numerical simulations also were performed by using a shallow water flow equation in which the SGS effect is taken into account. As a result, the results of the numerical simulations agreed with those of the experiments, for example, Mach angle of shock waves and the intensified velocities behind shock waves.

A regional-scale land surface parameterization based on areally-averaged hydrological conservation equations

In order to account for subgrid-scale spatial variability (heterogeneity) of land surface characteristics in regional-scale hydrological-atmospheric models, a land surface parameterization of areally-averaged sensible heat and evapotranspiration fluxes which is based upon areally-averaged hydrological soil water flow and soil heat flow equations, was developed. This land surface parameterization is fully coupled in a two-way interaction with the atmospheric boundary layer and the regional atmospheric model's first layer. The Monin-Obukhov similarity theory which is utilized in modelling the atmospheric boundary layer, and the areallyaveraged hydrological conservation equations are strictly valid only over stationary-heterogeneous areas where the fluctuations of the hydrological and boundary layer state variable values, parameter values, and of boundary conditions have spatially invariant means, spatially invariant higher moments and spatially invariant probability distributions. Therefore, the coupled two-way interactive model was run first for the computation of land surface fluxes over stationary-heterogeneous land patches, each corresponding to a single soil texture-vegetation class. Then, utilizing a mosaic scheme, the land surface fluxes over a regional model grid cell were obtained by numerical probabilistic averaging. The application of the regional model with this land surface parameterization to California during April 1989 has produced promising results.

Management Model for Conjunctive Use of Coastal Surface Water and Ground Water

A nonlinear multiobjective management model has been developed for managing water use within a coastal region. Two conflicting objectives are considered: cost-effective allocation of surface water and ground water supplies, and minimization of saltwater intrusion. Optimal control of the system is examined by studying the response of these objectives to changes in ground water pumping rates and transfer of surface water between sources and users. System constraints include economic, operational, and institutional requirements. The objectives, some of the constraints, and the ground water flow equations are formulated as nonlinear functions of the decision variables. The flow model simulates the flow of ground water using a quasi-three-dimensional finite-difference model based on the sharp interface assumption. The nonlinear optimization problem is solved with MINOS. The flow simulation and the optimization model are considered simultaneously by linking the simulation directly into the optimization procedure. The multiobjective aspect of the problem is solved using the constraint method. To test the methodology, a hypothetical case study based on the Waialae aquifer of Hawaii was developed and examined.

Suppression of the Karman Vortex Street due to Positive Buoyancy. Effect of Fluid Type on Wake Oscillation and Surface Value.

Effect of fluid type on wake oscillation and surface value in suppression of the Karman vortex street due to positive buoyancy from a heated circular cylinder submerged in an upward mainstream is elucidated by numerical calculation of the Navier-Stokes equations with buoyant term and the energy equation for mercury, air and water flow. Numerical results are validated. The Strouhal number decreases abruptly at the critical Richardson number Ric, which can be regarded as the exact boundary between the regions of the transformed vortex street and the plume flow. The largest Ric is for water, the smallest is for air. The mean Nusselt number of water flow decreases abruptly at Ric. This is because that the local Nusselt number near the rear stagnant point decreases at Ric due to a formation of a twin vortex.

A NUMERICAL STUDY ON A SHALLOW WATER FLOW USING CIP METHOD

In a mountainous region, run-offs are generally produced by infiltration flows and surfacerun-offs. Surface run-offs include streaming flows and rapid flows. Therefore, in order to reproducea flow in a mountainous region, we must calculate the flow in which a streaming flow and a rapid floware intermingled. For this calculation, we applied CIP to a shallow water flow equation. The purposeof this study is to reproduce two types, of flows. In one of them, the resistance coefficient isconstant, and in another it is given by Hey equation. As a result, the experiment of a mountainousflow and the simulated one agreed considerably.

Water flow modeling in the unsaturated zone with imprecise parameters using a fuzzy approach

Abstract An alternative approach, based on fuzzy set theory, is presented to express imprecision of parameters in a non-probabilistic sense. Imprecision may originate from indirect measurements, estimation routines, subjective interpretation and expert judgement of available information. One dimensional, steady state water flow in the unsaturated zone of homogeneous soils, which is described by the Darcy-Buckingham equation, was chosen to evaluate and to incorporate fuzzy soil hydraulic properties and boundary conditions in the modeling procedure. It is here used to describe soil water pressures with depth, as well as to calculate maximum evapotranspiration rates under steady state conditions. Solving the fuzzy equation for steady state water flow results in minimizing/maximizing procedures, from where resulting membership functions of the dependent variable are calculated. A comparison to a more classical stochastic approach points out the main differences between fuzzy and stochastic concepts to account for uncertainties. Finally, a sensitivity analysis shows the strong impact of different shapes of membership functions of the input parameters on the resulting membership functions of maximum evapotranspiration rates and soil water pressures.

Water Flux below the Root Zone vs. Drip‐Line Spacing in Drip‐Irrigated Corn

AbstractUse of microirrigation (drip) is increasing, prompted by factors such as a greater ability to control losses of water and nutrients from the root zone. The cost of drip can be reduced by using wider drip‐line spacings. Our objective was to evaluate water flux below the root zone (1.5‐m soil depth) with a subsurface drip irrigation system having drip‐line spacings of 1.5, 2.3, and 3.1 m near Colby, KS. The soil is a deep silt loam that formed in loess. The crop was corn (Zea mays L.), planted in rows spaced 0.76 m apart. Water flux at the 1.5‐m soil depth was determined in five treatments (Trt.): 1.5‐m spacing, full irrigation; 2.3‐m spacing, 67% of full irrigation; 2.3‐m spacing, full irrigation; 3.1‐m spacing, 50% of full irrigation; and 3.1‐m spacing, full irrigation. With full irrigation, irrigation plus effective rain equaled calculated corn evapotranspiration (ET). Tensiometers were placed below the drip line and at increments of 0.4 m from the drip line at soil depths of 1.4 and 1.7 m. Water flux was calculated by using a hydraulic conductivity (K) vs. matric potential (ψm) relationship, ψm data from tensiometers within the corn plots, and Darcy's equation of water flow. In‐season water fluxes below the root zone (1.5 m) were 51, 118, and 124 mm at drip‐line spacings of 1.5, 2.3, and 3.1 m, respectively. In 1990, corn grain yield was not significantly affected by spacing. In 1991, with drier initial soil‐water conditions, corn yielded significantly less grain at spacings of 2.3 m (11.5 Mg ha‐1) and 3.1 m (10.7 Mg ha‐1) than at a drip‐line spacing of 1.5 m (13.1 Mg ha‐1). If spacing between drip lines is increased beyond 1.5 m in the silt loam soils of western Kansas, there would be an associated increase in internal drainage from the root zone and decrease in corn yields.

NUMERICAL MODEL OF 3-DIMENSIONAL ANISOTROPIC DEFORMATION AND 1-DIMENSIONAL WATER FLOW IN SWELLING SOILS

Current models of water flow in deforming soils generally involve a transformation from spatial to material coordinates. Existing forms of this coordinate transformation either assume that soil deformation is one-dimensional, or that it is isotropic. In the present article, we propose a new expression of the transformation gradient tensor, that allows different extents of deformation in the vertical and horizontal directions. The resulting generalized water flow equation is calibrated with experimental data obtained for one-dimensional vertical infiltration in a bentonite sample. The hydraulic characteristics obtained from this calibration are then used to analyze, via simulations, the sensitivity of water flow to anisotropy in soil deformation. The results indicate that the extent of the lateral deformation strongly influences not only the height of the soil surface, as expected, but also the distribution of water and the total volume of water in a swelling/shrinking soil undergoing infiltration or drainage. Consequently, this lateral deformation should be taken into account explicitly in modeling efforts or in the determination of the hydraulic characteristics of soils that deform anisotropically.

Water Flux Below the Root Zone vs. Irrigation Amount in Drip‐Irrigated Corn

AbstractWith increasing demands on water resources, greater efficiency is needed in irrigated agriculture. Internal drainage from the root zone is a loss that can be reduced or managed to improve irrigation efficiency. Our objective was to determine the relationship between seasonal water flux below the root zone (1.5 m) and irrigation amount with corn (Zea mays L.). Subsurface drip irrigation systems near Colby and Holcomb, KS, were used to supply water. Driplines were buried at a soil depth of 0.40 to 0.45 m, with a spacing of 1.5 m. The two soils are deep silt Ioams that formed in loess. Water flux at the 1.5‐m soil depth was determined in four irrigation treatments during 1990 and 1991. Tensiometers were placed at soil depths of 1.4 and 1.7 m and at distances from the dripline of 0, 0.4, and 0.8 m. Water flux was calculated using predetermined hydraulic conductivity vs. matric potential (Ψm) relationships, Ψm data from tensiometers within the corn plots, and Darcy's equation of water flow. Irrigation was applied to four treatment such that irrigation plus rain equaled 125, 100, 75, and 50% of calculated corn evapotranspiration (ET). From a regression analysis relating integrated water flux below the root zone (1.5 m) and in‐season irrigation, net upward water flux occurred with in‐season irrigation < 296 mm (RMSE = 71 mm), whereas net downward water flux occurred with irrigation > 296 mm. Compared with the 100% ET treatment (full irrigation), the 75% ET treatment had 76% of the in‐season irrigation, 25% of the in‐season water flux (net downward) below the root zone (1.5 m), and 93% of the corn grain yield. Near‐maximum corn grain yields can be obtained with significant decreases in irrigation amount and internal drainage from the root zone compared with full irrigation.

A finite volume method for steady state 2D shallow water flows

Describes the solution of the shallow water flow equations in strongly conservative form using a finite volume method. A SIMPLE‐like scheme is developed to treat the velocity depth coupling. The method is applied to flow in a sharply curved channel and the results compared with published data. An error analysis is included which indicates that the method proposed is suitable for solving two‐dimensional steady state problems in open channel flow.