Modeling variations in salt composition components of river water

The balance of a component contained in river water is considered taking into account its input with lateral inflow and decay in the aquatic environment. Random changes in lateral inflow causes fluctuations in the parameters of component input and decay. A stochastic equation of component balance is derived and used as the basis for the construction of an equation for the probability density of component concentration. The solution of this equation shows that the probability density follows lognormal law. This theoretical result is applicable to the analysis of time series of water salt composition components, including pH, alkalinity, chlorides, ammonia, iron, and aluminum. The applicability of the lognormal law is proved and distribution parameters are evaluated. The distributions of three components (pH, alkalinity, and chlorides) are found to split into two lognormal branches, describing high and low component concentrations. In the case of pH and alkalinity, this splitting is due to seasonal effects, while in the case of chlorides, it is caused by the difference between concentrations in the surface runoff at the early and final stages of snow melting and rainfalls. The application of the statistical distributions for probabilistic forecasting of extreme component concentrations is considered. The exceedance probability of standard limits of the components is considered. The use of exceedance probability in hydrochemical standardization is demonstrated.     

Statistical assessment of relationships between water flow in a river and water turbidity in water intakes

Probability distributions of water flow and its turbidity after passing through water intake structures are considered. Heavy tails with a power distribution are shown to exist. These distributions are used to find the dependence between water turbidity at the inlet to the water station and water flow in the river. Quantiles of these distributions are estimated to characterize the levels of water flow and turbidity with a given exceedance probability.Translated from Vodnye Resursy, Vol. 32, No. 2, 2005, pp. 196–204.Original Russian Text Copyright © 2005 by Dolgonosov, Korchagin.     

Generalized flux law,with an application

A generalized flux law, which can specialize into power‐ and diffusive‐type flux laws, is proposed. When coupled with the continuity equation, a generalized flow equation is the result. The generalized equation specializes into several cases, one of which is the kinematic wave equation. By applying the generalized flow equation and its variants to flow routing in two rivers, the usefulness of the generalized flux law is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

Development and testing of a physically based,three-dimensional model of surface and subsurface hydrology

We present a numerical, catchment-scale model that solves flow equations of surface and subsurface flow in a three-dimensional domain. Surface flow is described by the two-dimensional parabolic approximation of the St. Venant equation, using Manning’s equation of motion; subsurface flow is described by the three-dimensional Richards’ equation for the unsaturated zone and by three-dimensional Darcy’s law for the saturated zone, using an integrated finite difference formulation. The hydrological component is a dynamic link library implemented within a comprehensive model which simulates surface energy, radiation budget, snow melt, potential evapotranspiration, plant development and plant water uptake. We tested the model by comparing distributed and integrated three-dimensional simulated and observed perched water depth (PWD), stream flow data, and soil water contents for a small catchment. Additional tests were performed for the snow melting algorithm as well as the different hydrological processes involved. The model successfully described the water balance and its components as evidenced by good agreement between measured and modelled data.     

A universal field equation for dispersive processes in heterogeneous media

When formulated properly, most geophysical transport-type process involving passive scalars or motile particles may be described by the same space–time nonlocal field equation which consists of a classical mass balance coupled with a space–time nonlocal convective/dispersive flux. Specific examples employed here include stretched and compressed Brownian motion, diffusion in slit-nanopores, subdiffusive continuous-time random walks (CTRW), super diffusion in the turbulent atmosphere and dispersion of motile and passive particles in fractal porous media. Stretched and compressed Brownian motion, which may be thought of as Brownian motions run with nonlinear clocks, are defined as the limit processes of a special class of random walks possessing nonstationary increments. The limit process has a mean square displacement that increases as t^α+1 where α > −1 is a constant. If α = 0 the process is classical Brownian, if α < 0 we say the process is compressed Brownian while if α > 0 it is stretched. The Fokker–Planck equations for these processes are classical ade’s with dispersion coefficient proportional to t^α. The Brownian-type walks have fixed time step, but nonstationary spatial increments that are Gaussian with power law variance. With the CTRW, both the time increment and the spatial increment are random. The subdiffusive Fokker–Planck equation is fractional in time for the CTRW’s considered in this article. The second moments for a Levy spatial trajectory are infinite while the Fokker–Planck equation is an advective–dispersive equation, ade, with constant diffusion coefficient and fractional spatial derivatives. If the Lagrangian velocity is assumed Levy rather than the position, then a similar Fokker–Planck equation is obtained, but the diffusion coefficient is a power law in time. All these Fokker–Planck equations are special cases of the general non-local balance law.     

Stochastic sketching of infiltration-advance isochrones

A technique has been developed for predicting the irregular advance pattern often observed as water spreads on the surface of the ground. The technique is a combination of stochastic sketching, potential theory, probability theory, and a mass balance equation in the form of an advance equation. The technique can be used on flat as well as sloping terrain and addresses any form of obstructions or constraints to the flow of the water. The stochastic sketching portion of the technique uses cellular automata with transition probability movement rules to sketch the dynamics of small volume water elements in the defined environment. Randomly selected small volume flow path segments are computed and plotted. The envelope of these segments defines the wetted area and the advance front. Several examples are presented showing the patterns produced for various situations.     

Measuring hydraulic permeability in a streambed using the packer test

Permeability of a streambed is an important factor regulating nutrient and oxygen availability for aquatic biota. In order to investigate the relationship, an accurate permeability should be measured. However, it is difficult to measure permeability in a coarse gravel bed using a conventional permeability test. Moreover, turbulent flow may occur in coarse bed material, and then deviations from Darcy's law do occur. Thus, permeability calculated following Darcy's law may be overestimated under turbulent flow conditions and should be corrected. The packer test can be used in highly permeable gravel beds. We developed a field method applicable to a gravel bed using the packer test and derived an equation adopting a law of turbulent flow to study the problems under any type of flow condition. The accuracy of the equation was examined using a laboratory flume with a gravel bed. The results suggested that permeability calculated from Hvorslev's equation is overestimated for turbulent flow. In contrast, our equation, developed here, could evaluate permeability accurately under any type of flow condition. Copyright © 2005 John Wiley & Sons, Ltd.     

Solution of the nonlinear transport equation using modified Picard iteration

The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard iteration algorithm of Celia et al.⁵ for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard iteration. In general, the new solver resulted in negligible mass balance errors (< ∥10⁻⁸∥%) and required less computational time than the conventional iteration scheme for the test examples, including transport involving highly nonlinear adsorption under steady-state as well as transient flow conditions. In contrast, mass balance errors resulting from the conventional Picard iteration method were higher than 10% for some highly nonlinear problems. Application of the modified Picard iteration scheme to solve the nonlinear transport equation may greatly reduce the mass balance errors and increase computational efficiency.     

Brine transport in porous media: self-similar solutions

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

Suspended load in flows on erodible bed

Steady state suspended-load of sediment transported in flow over erodible beds usually is treated by the advection-diffusion approach, though in recent years, it is being treated as a two-phase flow phenomenon incorporating kinetics of sediment particles. Among the advection-diffusion approaches, Rouse＇s equation is the well-known, although a number of researchers in later periods have attempted to improve it by modifying the mixing length concept taking into account other aspects. In this paper, the advection-diffusion approach and associated logarithmic law of flow velocity are revisited. It is concluded from the logarithmic law that the Reynolds shear stress is a linear function of height above the bed, which reduces to bed shear stress in the case of a long horizontal channel. As a consequence, it is shown that the volumetric concentration of sediment is best approximated by the sum of two power laws of height above the bed. An equation is derived for the suspended-load transport rate in terms of elementary functions.     

Predicting sediment transport by interrill overland flow on rough surfaces

Modelling soil erosion requires an equation for predicting the sediment transport capacity by interrill overland flow on rough surfaces. The conventional practice of partitioning total shear stress into grain and form shear stress and predicting transport capacity using grain shear stress lacks rigour and is prone to underestimation. This study therefore explores the possibility that inasmuch as surface roughness affects flow hydraulic variables which, in turn, determine transport capacity, there may be one or more hydraulic variables which capture the effect of surface roughness on transport capacity suffciently well for good predictions of transport capacity to be achieved from data on these variables alone. To investigate this possibility, regression analyses were performed on data from 1506 flume experiments in which discharge, slope, water temperature, rainfall intensity, and roughness size, shape and concentration were varied. The analyses reveal that 89·8 per cent of the variance in transport capacity can be accounted for by excess flow power and flow depth. Including roughness size and concentration in the regression improves that explained variance by only 3·5 per cent. Evidently, flow depth, when used in combination with excess flow power, largely captures the effect of surface roughness on transport capacity. This finding promises to simplify greatly the task of developing a general sediment equation for interrill overland flow on rough surfaces. Copyright © 1998 John Wiley & Sons, Ltd.     

Backward probabilistic model of groundwater contamination in non-uniform and transient flow

Backward location and travel time probabilities, which provide information about the former location of contamination in an aquifer, can be used to identify unknown contamination sources. Backward location probability describes the possible upgradient positions of contamination at a known time in the past, and backward travel time probability describes the time required for contamination to travel from a known upgradient location to an observation point. These probabilities are related to adjoint states of resident concentration, and their governing equation is the adjoint of a forward contaminant transport model. Using adjoint theory to obtain the appropriate governing equation, we extend the backward probability model for conservative solutes to more general non-uniform and transient flow fields. In particular, we address three important extensions, spatially-varying porosity, transient flow and temporally-varying porosity, and internal distributed sources and sinks of solute and water. For the first time we learn that forward and backward location and travel time probabilities are not necessarily equivalent to adjoint states, but are related to them. The extensions are illustrated using a vertically-integrated groundwater model, creating transient flow by a step change in pumping and using areal recharge as an internal distributed source. Both the movement and spread of probabilities are affected. With internal sources of water, there are two interpretations of backward probability, depending on whether or not the source of water is also a source of solute. The results demonstrate how the backward probability model can be applied to other, perhaps more important, non-uniform and transient flow conditions, with time- and space-varying water storage, such as time-varying pumping or unsaturated (or saturated–unsaturated) flow and transport with spatially- and temporally-varying moisture content.     

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

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

Testing a theoretical resistance law for overland flow on a stony hillslope

Overland flow, sediments, and nutrients transported in runoff are important processes involved in soil erosion and water pollution. Modelling transport of sediments and chemicals requires accurate estimates of hydraulic resistance, which is one of the key variables characterizing runoff water depth and velocity. In this paper, a new theoretical power–velocity profile, originally deduced neglecting the impact effect of rainfall, was initially modified for taking into account the effect of rainfall intensity. Then a theoretical flow resistance law was obtained by integration of the new flow velocity distribution. This flow resistance law was tested using field measurements by Nearing for the condition of overland flow under simulated rainfall. Measurements of the Darcy–Weisbach friction factor, corresponding to flow Reynolds number ranging from 48 to 194, were obtained for simulated rainfall with two different rainfall intensity values (59 and 178 mm hr⁻¹). The database, including measurements of flow velocity, water depth, cross-sectional area, wetted perimeter, and bed slope, allowed for calibration of the relationship between the velocity profile parameter Γ, the slope steepness s, and the flow Froude number F, taking also into account the influence of rainfall intensity i. Results yielded the following conclusions: (a) The proposed theoretical flow resistance equation accurately estimated the Darcy–Weisbach friction factor for overland flow under simulated rainfall, (b) the flow resistance increased with rainfall intensity for laminar overland flow, and (c) the mean flow velocity was quasi-independent of the slope gradient.     

Statistical distributions of phytoplankton concentration in river water

Daily time series of phytoplankton concentration are theoretically analyzed with the use of population dynamics equation with fluctuations in the growth factor and environmental carrying capacity taken into account. The statistical distributions of phytoplankton cell concentration are shown to obey different laws, depending on the season of the year: lognormal in the winter and logarithmic in the vegetation period. The probability of extremely high concentration numbers is described by the normal law. The verification of the obtained relationships against a body of empirical data confirmed the theoretical forecasts. The obtained results make it possible to predict the probabilities of various phytoplankton concentration values within a wide range, including the domain of large values, which are of greatest hazard in terms of water quality, water treatment processes, and aquatic ecosystem well-being.     

Percolation losses in paddy fields with a dynamic soil structure: model development and applications

The hydraulic characteristics of the plough pan of paddy fields provide continuous ponding conditions during the growing season and control the water use efficiency in wet rice production. Its saturated hydraulic conductivity K_s, however, exhibits a large spatiotemporal variability as a consequence of a highly dynamic soil structure involving temporary shrinkage cracks. Water flow through the earthen bunds surrounding the fields further contributes to the uncertainty in water flux calculations. The objective of this study was to develop a simple deterministic model with stochastic elements (‘PADDY‐FLUX’) for depiction of deep percolation, and to assess the effect of different water management scenarios on percolation in two channel command areas. Darcy's law is used as the fundamental equation for water flow calculations with the ponding water depth h as a time‐dependent variable. Flux uncertainty is estimated by a Monte‐Carlo‐type implementation. K_s is treated as a random variable of a bimodal probability density function (PDF), which is the weighted sum of two Gaussian PDFs (accounting for a matrix and a preferential flow domain). The weighing factor α is a function of h, reflecting an increasing risk for preferential flow situations after desiccation and the development of shrinkage cracks. Under‐bund percolation is calculated using transfer functions. The results demonstrate that percolation losses increase in the following order: continuous soil saturation < continuous flooding (CF) < mid‐season drainage and intermittent irrigation (MD + II) < mid‐season drainage and continuous flooding. The bunds contribute up to 54 and 17% to total fluxes under CF and MD + II, respectively. Preferential water fluxes are responsible for the major part of water losses as soon as desiccation causes the formation of shrinkage cracks. As a conclusion, continuous soil saturation should be promoted as the least water‐intensive irrigation regime, while intermittent irrigation is recommended only in case that irreversible shrinkage cracks have already developed. Copyright © 2010 John Wiley & Sons, Ltd.     

Assessment of transmission losses and groundwater recharge from runoff events in a wadi under shortage of data on lateral inflow,Negev, Israel

A hydrological–lithostratigraphical model was developed for assessment of transmission losses and groundwater recharge from runoff events in arid water courses where hydrological and meteorological records are incomplete. Water balance equations were established for reaches between hydrometric stations. Because rainfall and tributary flow data are scarce, lateral inflow, which is an essential component of the water balance equation, could not be estimated directly. The solution was obtained by developing a method which includes a hydrological–lithostratigraphical analogy. This is based on the following assumptions: (a) runoff resulting from a given rainfall event is related to the watershed surface lithology; (b) for a given event, the spatial distribution of runoff reflects the distribution of rainfall: and (c) transmission losses are uniquely related to the total inflow to the reach. The latter relationship, called the loss function, and the water balance equation comprise a model which simultaneously assesses lateral inflow and transmission losses for runoff events recorded at the terminal stations. The model was applied to three reaches of the arid Nahal Tsin in Israel. In this case study, the transmission losses were of the same order of magnitude as the flow at the major hydrometric stations. The losses were subdivided into channel moistening, which subsequently evaporates, and deep percolation, which recharges groundwater. For large runoff events, evaporation was substantially smaller than the losses. The mean annual recharge of groundwater from runoff events in the Tsin watershed was 4·1×10⁶ m³, while the mean annual flow volume at the major stations ranged from 0·6 to 1·5×10⁶ m³. Once in 100 years, the annual recharge may be seven times higher than the mean annual value, but the recharge during most years is very small. Copyright © 1999 John Wiley & Sons, Ltd.     

Estimating the energy of an ice-covered flow

Energy balance of an ice-covered flow in a constant-slope reach of a regulated river is considered. The ratio of energy losses for viscous friction to the total flow energy is evaluated. The turbulence intensities of an open and ice-covered flows are compared.Translated from Vodnye Resursy, Vol. 32, No. 1, 2005, pp. 48–54.Original Russian Text Copyright © 2005 by Dolgopolova.