Analytical solutions of unsteady drainage problems for soils subjected to variable recharge from a semi-confined aquifer

In drainage of agricultural lands, the upward vertical recharge from a semi-confined aquifer depends on the difference of the piezometric heads on the two sides of the semi-impermeable layer through which this recharge takes place. This means that the recharge through the semi-impermeable base depends on the unknown height of the unsteady water table. In the nonhomogeneous Boussinesq equation, which describes the drainage problems, the downward uniform rate of the recharge from rain or irrigation and the recharge from the semiconfined aquifer are expressed by two terms. By solving the Boussinesq equation expressions for the nondimensional height of the water table and the nondimensional discharge of the drains per unit drained area are obtained for three different initial conditions. Some known solutions are shown as special cases of the present solutions. Variation of nondimensional water table heights at half distance of the drain spacing and the nondimensional discharge of the drains with nondimensional time have been graphically illustrated with the help of synthetic examples.Notation B _s thickness of the semi-impervious layer [L] - c hydraulic resistance of the semi-impervious layer [T] - D depth of the drains from the base [L] - d _e equivalent depth [L] - h=h(x, t) height of the water table [L] - h ₀ initial height of the water table [L] - h _t water table height at mid-distance of drains att [L] - h _j ,h _k water table height at mid-distance of drains at timej andfk, respectively [L] - H ₀ piezometric head in the semi-confined aquifer [L] - K hydraulic conductivity of the soil [LT^–1] - K _s hydraulic conductivity of the semi-impervious layer [LT^–1] - k ₀,k ₁,k ₂ nondimensional constants - L distance between the drains [L] - q ₀ upward recharge per unit surface area through the semi-impervious layer [LT^–1] - q _t discharge per unit drainable area of drains at timet [LT^–1] - R,R ₀ recharge per unit surface area from rain or irrigation during the unsteady and steady-state, respectively, [LT^–1] - S specific yield of the soil - t time of observation [T] - x distance measured from the drain [L] - leakage factor [L] - nondimensional distance - nondimensional time     

Combined analytical solution of overland flow and sediment transport

With reference to the kinematic wave theory coupled with the hypothesis of constant linear velocity for the rating curve, rising limb analytical solutions have been calculated for overland flow, over an Hortonian-infiltrating surface, and sediment discharge. These analytical solutions are certainly easier to use than the numerical integration of the basic equations and they may be used to obtain an initial evaluation of the parameters of more complex models generally devised for complicated cases.Notation a exponent of the Horton law [T^–1] - b exponent of the rill erosion equation - B inter-rill erosion coefficient [ML^–m–2T ^m–1] - c sediment concentration [ML^–3] - c _o reference sediment concentration [ML^–3] - E _I inter-rill erosion [ML^–2T^–1] - E _R rill erosion [ML^–2T^–1] - f _c final infiltration rate of the soil [LT^–1] - f _o initial infiltration rate of the soil [LT^–1] - h flow depth [L] - h _o reference flow depth [L] - i infiltration rate [LT^–1] - k rill erosion coefficient [ML^–1–b T^–1] - K integration constant - L() Laplace transformation - m exponent of the inter-rill erosion equation - n Manning's coefficient [L^–1/3T] - p rainfall intensity [LT^–1] - q water discharge per unit width [L²T^–1] - q _s sediment discharge per unit width [ML^–1T^–1] - t time [T] - t _p ponding time [T] - x distance along the flow direction [L] Greek Letters coefficient of the stage-discharge equation [L^2–T^–1] - exponent of the stage-discharge equation - rill erosion coefficient [L^–1]     

Determination of optimal unit hydrographs by linear programming

A unit hydrograph (UH) obtained from past storms can be used to predict a direct runoff hydrograph (DRH) based on the effective rainfall hyetograph (ERH) of a new storm. The objective functions in commonly used linear programming (LP) formulations for obtaining an optimal UH are (1) minimizing the sum of absolute deviations (MSAD) and (2) minimizing the largest absolute deviation (MLAD). This paper proposes two alternative LP formulations for obtaining an optimal UH, namely, (1) minimizing the weighted sum of absolute deviations (MWSAD) and (2) minimizing the range of deviations (MRNG). In this paper the predicted DRHs as well as the regenerated DRHs by using the UHs obtained from different LP formulations were compared using a statistical cross-validation technique. The golden section search method was used to determine the optimal weights for the model of MWSAD. The numerical results show that the UH by MRNG is better than that by MLAD in regenerating and predicting DRHs. It is also found that the model MWSAD with a properly selected weighing function would produce a UH that is better in predicting the DRHs than the commonly used MSAD.Notations M number of effective rainfall increments - N number of direct runoff hydrograph ordinates - R number of storms - MSAD minimize sum of absolute deviation - MWSAD minimize weighted sum of absolute deviation - MLAD minimize the largest absolute deviation - MRNG minimize the range of deviation - RMSE root mean square error - P _m effective rainfall in time interval [(m–1)t,mt] - Q _n direct runoff at discrete timent - U _k unit hydrograph ordinate at discrete timekt - W _n weight assigned to error associated with estimatingQ _n - _n ⁺ error associated with over-estimation ofQ _n - _n ^– error associated with under-estimation ofQ _n - _max ⁺ maximum positive error in fitting direct runoff hydrograph - _max ^– maximum negative error in fitting direct runoff hydrograph - _max largest absolute error in fitting obtained direct runoff - E _r,1 thelth error criterion measuring the fit between the observed DRHs and the predicted (or reproduced) DRHs for therth storm - E ₁ averaged value of error criterion overR storms     

Optimal flood control in multireservoir cascade systems with deterministic inflow forecasts

This article presents the formal analysis of a problem of the optimal flood control in systems of serially connected multiple water reservoirs. It is assumed, that the basic goal is minimization of the peak flow measured at a point (cross-section) located downstream from all reservoirs and that inflows to the system are deterministic. A theorem expressing sufficient conditions of optimality for combinations of releases from the reservoirs is presented together with the relevant proof. The main features of the optimal combinations of controls are thoroughly explained. Afterwards, two methods of determining the optimal releases are presented. Finally, the results of the application of the proposed methodology to a small, four reservoir system are presented.Notations c _i contribution of theith,i=1, ...,m, reservoir to the total storage capacity of the multireservoir system - d _i (t) one of the uncontrolled inflows to the cascade at timet (fori=1 main inflow to the cascade, fori=2, ...,m, side inflow to theith reservoir, fori=m+1 side inflow at pointP) - total inflow to theith reservoir,i=2, ...,m, at timet (i.e., inflowd _i augmented with properly delayed releaser _i–1 from the previous reservoir) (used only in figures) - d(t),d _S (t) (the first term is used in text, the second one in figures) aggregated inflow to the cascade (natural flow at pointP) at timet - time derivative of the aggregated inflow at timet - i reservoir index - m number of reservoirs in cascade - P control point, flood damage center - minimal peak of the flow at pointP (cutting level) - Q _p (t) flow measured at pointP at timet - flow measured at pointP at timet, corresponding to the optimal control of the cascade - r _i (t) release from theith reservoir at timet, i=1, ...,m - optimal release from theith reservoir at timet, i=1, ...,m - r ₁ ^* (t) a certain release from theith reservoir at timet, different than ,i=1, ...,m, (used only in the proof of Theorem 1) - a piece of the optimal release from themth reservoir outside period at timet - assumed storage of theith reservoir at time (used only in the proof of Theorem 1) - s _i (t) storage of theith reservoir at timet, i=1, ...,m - time derivative of the storage of theith reservoir at timet, i=1, ...,m - storage capacity of theith reservoir,i=1, ...,m - (the first term is used in text, the second one in figures) total storage capacity of the cascade of reservoirs - S* sum of storages, caused by implementingr _i ^* ,i=1, ...,m, of all reservoirs measured at (used only in the proof of Theorem 1) - t time variable (continuous) - t ₀ initial time of the control horizon - t _a initial time of the period of constant flow equal at pointP - initial time of the period of the essential filling of theith reservoir,i=1, ...,m (used only in the proof of Theorem 1) - t _b final time of the period of constant flow equal at pointP - final time of the period of the essential filling of theith reservoir,i=1, ...,m (used only in the proof of Theorem 1) - time of filling up of theith reservoir while applying method with switching of the active reservoir - t _f final time of the control horizon - fori=1, ...,m–1, time lag betweenith andi+1th reservoir; fori=m time lag between the lowest reservoir of the cascade and the control pointP     

Water table fluctuation in response to transient recharge from a rectangular basin

A problem of water-table fluctuation in a finite two-dimensional aquifer system in response to transient recharge from an overlying rectangular area is studied. An analytical solution is obtained by using the method of finite Fourier transform to predict the transient position of the water-table. The solution for constant rate of recharge is shown as a special case of the present solution. Effects of variation in the rate of recharge on the growth of two-dimensional groundwater mound is illustrated with the help of a numerical example.Notation A half width of the aquifer [L] - B half length of the aquifer [L] - D half width of the recharge basin [L] - e specific yield - h varying water-table height [L] - h ₀ initial water-table height [L] - h weighted mean of the depth of saturation [L] - K hydraulic conductivity [LT^–1] - L half length of the recharge basin [L] - P(t) time varying rate of recharge [LT^–1] - P ₁ +P ₀ initial rate of time varying recharge [LT^–1] - P ₁ final rate of time varying recharge [LT^–1] - t time of observation [T] - x, y coordinate axes - decay constant [T^–1]     

An analytical solution for water-table fluctuation in a finite aquifer due to transient recharge from a strip basin

Two cases of water-table fluctuation in a finite aquifer in response to transient recharge from a strip basin are investigated. In the first case the aquifer is bounded by open water-bodies, whereas in second one the aquifer is bounded by impermeable boundaries on both sides. Analytical solutions are presented to predict the transient position of the water-table. The solutions are obtained by using finite Fourier sine and cosine transforms.Notations A width of the aquifer [L] - e specific yield - h variable water-table height [L] - h ₀ initial water-table height [L] - weighted mean of the depth of saturation [L] - K hydraulic conductivity [LT^–1] - m,n integers - P ₁ +P ₀ initial rate of transient recharge [LT^–1] - P ₁ final rate of transient recharge [LT^–1] - P constant rate of recharge [LT^–1] - x ₁ distance of left boundary of the strip basin [L] - x ₂ distance of right boundary of the strip basin [L] - t time of observation [T] - decay constant [T^–1]     

Water table fluctuation due to transient recharge in a 2-D aquifer system with inclined base

Recharging of aquifers due to irrigation, seepage from canal beds and other sources leads to the growth of water table near to the ground surface causing problems like water logging and increase of salinity in top soils in many regions of the world. This problem can be alleviated if proper knowledge of the spatio — temporal variation of the water table is available. In this paper an analytical solution for the water table fluctuation is presented for a 2-D aquifer system having inclined impervious base with a small slope in one — direction and receiving time varying vertical recharge. Application of the solution in estimation of water table fluctuation is demonstrated with the help of an example problem.Notations A length of the aquifer [L] - B width of the aquifer [L] - D mean depth of saturation [L] - e specific yields - h variable water table height [L] - K hydraulic conductivity [LT ^–1] - P(t) transient recharge rate [LT ^–1] - P ₁+P _o initial rate of transient recharge [LT ^–1] - P ₁ final rate of transient recharge [LT ^–1] - q slope of the aquifer base in percentage - r decay constant [T ^–1] - t time of observation [T] - x, y coordinate axes - x ₂–x ₁ length of the recharge basin [L] - y ₂–y ₁ width of the recharge basin [L]     

Forecast model of water consumption for Naples

The data refer to the monthly water consumption in the Neapolitan area over more than a 30 year period. The model proposed makes it possible to separate the trend in the water consumption time series from the seasonal fluctuation characterized by monthly peak coefficients with residual component. An ARMA (1,1) model has been used to fit the residual component process. Furthermore, the availability of daily water consumption data for a three-year period allows the calculation of the daily peak coefficients for each month, and makes it possible to determine future water demand on the day of peak water consumption.Notation j numerical order of the month in the year - i numerical order of the year in the time series - t numerical order of the month in the time series - h numerical order of the month in the sequence of measured and predicted consumption values after the final stage t of the observation period - Z _ji effective monthly water consumption in the month j in the year i (expressed as m³/day) - T _ji predicted monthly water consumption in the month j in the year i minus the seasonal and stochastic component (expressed as m³/day) - C _ji monthly peak coefficient - E _ji stochastic component of the monthly water consumption in the month of j in the year i - Z _i water consumption in the year i (expressed as m³/year) - Z _j (t) water consumption in the month j during the observation period (expressed as m³/day) - evaluation of the correlation coefficient - Z _j (t) water consumption in the month j during the observation period minus the trend - Y _t transformed stochastic component from E _t : Y _t =ln Et - Y _t+h measured value of stochastic component for t+h period after the final stage t of the observation period - Y _t (h) predicted value of stochastic component for t+h period after the final stage t of the observation period - _j transformation coefficients from the ARMA process (m, n) to the MA () process     

Extreme dry weather intervals of the growing season in Bačka,Yugoslavia

A method of describing and analyzing the stochastic process of droughts, which are defined here as the upper extremes of intervals of no rainfall, is recommended. All important components of extreme dry weather intervals such as their duration, time of occurrence, their total number in a given time interval [0, t], the longest drought duration in a given time interval [0, t], and time T(t) of occurrence of the longest drought are taken into consideration. Application of the method is performed using the records of nine meteorological stations in Baka, Yugoslavia and a good agreement is found between the theoretical and empirical distribution functions for all analyzed components of the process. On the basis of the performed computations, a set of maps showing the contours of extreme dry weather intervals, having return periods from 2, 5... up to 100 years, related to the growing season (1 April –30 September), for the region of Baka in Yugoslavia is obtained. If the period of exploitation of an irrigation system is 60 years, then it could be expected to appear as 20 dry weather intervals having 3 year return periods. The duration of dry weather intervals is given on the set of maps. The obtained results give a prognosis of an average state of droughts during long time intervals (60, 100, 200... years).     

A real-time flood forecasting model based on maximum-entropy spectral analysis: II. Application

The MESA-based model, developed in the first paper, for real-time flood forecasting was verified on five watersheds from different regions of the world. The sampling time interval and forecast lead time varied from several minutes to one day. The model was found to be superior to a state-space model for all events where it was difficult to obtain prior information about model parameters. The mathematical form of the model was found to be similar to a bivariate autoregressive (AR) model, and under certain conditions, these two models became equivalent.Notation A _k parameter matrix of the bivariate AR model - B backshift operator in time series analysis - eT forecast error (vector) at timet = T - _t uncorrelated random series (white noise) - F _k forward extension matrix of the entropy model forkth lag - I identity matrix - m order of the entropy model - N number of observations - P order of the AR model - Q _p peak of the direct runoff hydrograph - R correlation matrix - t _p time to peak of the direct runoff hydrograph - ₁ coefficient of variation - ₂ ratio of absolute error to the mean - forecasted runoff - x _i observed runoff - mean of the observed runoff - X ^–1 inverse ofX matrix - X* transpose of theX matrix Abbreviations AIC Akaike information criterion - AR autoregressive (model) - AR(p) autoregressive process of thepth order - ARIMA autoregressive integrated moving average (model) - acf autocorrelation function - ccf cross-correlation function - FLT forecast lead time - MESA maximum entropy spectral analysis - MSE mean square error - STI sampling time interval     

The artificial recharge of groundwater: A solution by successive variations of steady states

The artificial recharge of groundwater aims at the modification of water quality, an increase of groundwater resources, and the optimization of the exploitation and recovery of contaminated aquifers. The purpose of this work is to develop a new mathematical model for the problem of an artificial recharge well, using the method of successive variations of steady states. Applying this method, one arrives at an expression of time as a double integral. This integral contains the time-dependent radius of the recharge boundary and the piezometric head of the well, calculated with the finite-element method. The new model is simple and useful, and can be applied to many practical problems, using the designed dimensionless graphs.Notations A area of the finite element (m²) - c the Euler constant (0.5772156649...) - e index of the finite element - E _i the exponential integral function - F _j nodal values of the functionF - h piezometric head, (m) - h ₀ piezometric head at timet=0 (m) - h _w piezometric head on the well contour (m) - i, j, k nodal indices of the finite element - K hydraulic contactivity (ms^–1) - N _i interpolation function - Q discharge (m³ s^–1) - r cylindrical coordinate (m) - r ₀ the action radius of the well (m) - r _w the radius of the well (m) - S the effective porosity - t the time (s) - T the transmissivity of the aquifer (m²s^–1) - V the stored water volume (m³) - x, y, dummy variables     

On the predictability of the water-table variation in a ditch-drainage system

The irrigation in regions of brackish groundwater in many parts of the world results in the rise of the water-table very close to the groundsurface. The salinity of the productive soils is therefore increased. A proper layout of the ditch-drainage system and the prediction of the spatio-temporal variation of the water table under such conditions are of crucial importance in order to control the undesirable growth of the water-table. In this paper, an approximate solution of the nonlinear Boussinesq equation has been derived to describe the water-table variations in a ditch-drainage system with a random initial condition and transient recharge. The applications of the solution is discussed with the help of a synthetic example.Notations a lower value of the random variable representing the initial water-table height at the groundwater divide - a+b upper value of the random variable representing the initial water-table height at the groundwater divide - h variable water-table height measured from the base of the aquifer - K hydraulic conductivity - L half width between ditches - m ₀ initial water-table height at the groundwater divide - N(t) rate of transient recharge at time t - N ₀ initial rate of transient recharge - P N ₀/K - S Specific yield - t time of observation - t ₀ logarithmic decrement of the recharge function - T Kt/SL - x distance measured from the ditch boundary - X x/L - Y h/L - Y mean of Y - Y Variance of Y     

A real-time flood forecasting model based on maximum-entropy spectral analysis: I. Development

This paper, the first of two, develops a real-time flood forecasting model using Burg's maximum-entropy spectral analysis (MESA). Fundamental to MESA is the extension of autocovariance and cross-covariance matrices describing the correlations within and between rainfall and runoff series. These matrices are used to derive the model forecasting equations (with and without feedback). The model may be potentially applicable to any pair of correlated hydrologic processes.Notation a _k extension coefficient of the model atkth step - B _k backward extension matrix forkth step - B _ijk element of the matrixB _k (i,j=1, 2) - c _k coefficient of the entropy model atkth step in the LB algorithm - e _k (e _x ,e _y )k = forecast error vector atkth step - E _k error matrix atkth step - E _ijk element of theE _k (i,j=1, 2) - f frequency - F _k forward extension matrix atkth step - F _ijk element of theF _k matrix (i,j=1, 2) - H(f) entropy expressed in terms of frequency - H _X entropy of the rainfall process (X) - H _Y entropy of the runoff process (Y) - H _XY entropy of the rainfall-runoff process - I identity matrix - forecast lead time - m model order, number of autocorrelations - R correlation matrix - S _x standard deviation of the rainfall data - S _y standard deviation of the runoff data - t time - T ₁ rainfall record - T ₂ runoff record - T rainfall-runoff record (T=T ₁ T ₂) - x _t rainfall data (depth) - X X() = rainfall process - mean of the rainfall data - y _t direct runoff data (discharge) - Y Y() = runoff process - mean of the runoff data - (x, y) _t rainfall-runoff data (att T) - (x, y, z) _t rainfall-runoff-sediment yield data (att T) - z complex number (in spectral analysis) - _k coefficient of the LB algorithm atkth step - _nj Lagrange multiplier atjth location in the _n matrix - _{n n} = matrix of the Lagrange multiplier atkth step - _X (k), _Y (k) autocorrelation function of rainfall and runoff processes atkth lag - _XY (k) cross-correlation function of rainfall and runoff processes atkth lag - W ₁(f) power spectrum of rainfall or runoff - W ₂(f) cross-spectrum of rainfall or runoff Abbreviations acf autocorrelation function - ARMA autoregressive moving average (model) - ARMAX ARMA with exogenous input - ccf cross-correlation function - det() determinant of the (...) matrix - E[...] expectation of [...] - FLT forecast lead time - KF Kalman filter - LB Levinson-Burg (algorithm) - MESA maximum entropy spectral analysis - MSE mean square error - SS state-space (model) - STI sampling time interval - forecast ofx - forecast ofx -step ahead - x _F feedback ofx-value (real value) - |x| module (absolute value) ofx - X ^–1 inverse of the matrixX - X* transpose of the matrixX     

An application of linear programming to determine optimal systems for rainwater management: An economic view

Two decision models, one for determining optimal systems for rainwater management and the other for allocating additional water supplies from managed rainfall in conjunction with irrigation water, are formulated. The application of a rainwater management model to the command and to a watercourse, decides the minimum cost activities to manage rainwater. The output from the first model is used as the input in the second model which optimally allocates water to competing crops. It has been shown that 80% of rainwater could be managed economically in rice fields and in storage underground through artificial recharge. Optimal allocation of managed rainwater in conjunction with irrigation water increases the income of the project area to the extent of 14%.List of symbols AER Total available energy kWh - B _max Maximized value of the objective function, Rs - C _W Cost of canal water, Rs/10³ m³ - C _i Cost of managing rainwater through activityi, Rs/10³/m³ - C _min Minimized cost of managing surplus rainwater, Rs - C _RF Average cost of managed rainwater through activityi, Rs/10³ m³ - E _i Energy consumption in rainwater management activityi, kWh/10³ m³ (only energy required for pumping water is considered) - FLS Available capacity for fallow land storage, 10³ m³ - FPS Total storage in lined and unlined farm ponds, 10³ m³ - GWR Runoff diversion for artificial recharge through inverted tubewells, 10³ m³ - i A suffix for management activities having values 1,2,3,..., - j Crop index having values 1,2,3,..., - k Index for crop season, 1=kharif (summer) and 2=rabi(winter) - MRF Maximum rainfall surplus (runoff) available for management. (Runoff value at a 5-year return period was adopted) - P _j Income from crop activityj, Rs/ha - RFL Storage in fallow alkali land, 10³ m³ - RFS Storage in rice fields up to various depths, 10³ m³ - RWM _i Volume of rainwater managed through activityi, 10³ m³ - VCW Volume of canal water, 10³ m³ - VGW Volume of ground water, 10³ m³ - X _j Area under cropj, ha.     

On the prediction of groundwater mound formation due to transient recharge from a rectangular area

Recharge to the aquifer leads to the growth of a groundwater mound. Therefore, for the proper management of an aquifer system, an accurate prediction of the spatio-temporal variation of the water table is very essential. In this paper, a problem of groundwater mound formation in response to a transient recharge from a rectangular area is investigated. An approximate analytical solution has been developed to predict the transient evolution of the water table. Application of the solution and its sensitivity to the variation of the recharge rate have been illustrated with the help of a numerical example.Notations a = Kh/e [L²/T] - A = aquifer's extent in the x-direction [L] - B = aquifer's extent in the y-direction [L] - e = effective porosity - h = variable water table height [L] - h ₀= initial water table height [L] - h = weighted mean of the depth of saturation [L] - K = hydraulic conductivity [L] - m, n = integers - P = constant rate of recharge [L/T] - P ₁+P₀= initial rate of transient recharge [L/T] - P ₁= final rate of transient recharge [L/T] - s = h ²–h ₀ ² [L²] - t = time of observation [T] - x,y = space coordinates - x ₂–x₁= length of recharge area in x-direction [L] - y ₂–y₁= width of recharge area in y-direction [L] - z = decay constant [T^-1]     

Use of parameter estimation in determining soil hydraulic properties from unsaturated transient flow

This paper proposes a model for determining the parameters given by the closed-form equations of van Genuchten. An objective function is made by the observed data from vertical drainage, and the solutions of optimization show that less computation and more accurate estimates are made as head profiles are taken into account rather than cumulative drainage. Sensitivity analysis of the error vector to parameters interprets this reason. The convergence and stability of solutions are evaluated with different magnitudes of measured errors in the head, and the results show good estimates will be obtained if a sufficient pressure head at the soil bottom is applied. A variable _k is introduced to avoid the estimations of and n being affected by the uncertainties of Ks and _s .     

Water table fluctuation with a random initial condition

The nonlinear Boussinesq equation is used to understand water table fluctuations in various ditch drainage problems. An approximate solution of this equation with a random initial condition and deterministic boundary conditions, recharge rate and aquifer parameters has been developed to predict a transient water table in a ditch-drainage system. The effects of uncertainty in the initial condition on the water table are illustrated with the help of a synthetic example. These results would find applications in ditch-drainage design.Notation A / tanh t - a lower value of the random variable representing the initial water table height at the mid point - a+b Upper value of the random variable representing the initial water table height at the midpoint - B tanh t - C 4/ - h variable water table height - h mean of the variable water table height - h _m variable water table height at the mid point - h _m mean of the variable water table height at the mid point - K hydraulic conductivity - L half spacing between the ditches - m ₀ initial water table height at the mid point - N Uniform rate of recharge - S specific yield - t time of observation - x distance measured from the ditch boundary - (4/SL)(NK)^1/2 - (L/4)(N/K)^1/2 - dummy integral variable     

On the parametric approach to unit hydrograph identification

Unit hydrograph identification by the parametric approach is based on the assumption of a proper analytical form for its shape, using a limited number of parameters. This paper presents various suitable analytical forms for the instantaneous unit hydrograph, originated from known probability density functions or transformations of them. Analytical expressions for the moments of area of these form versus their definition parameters are theoretically derived. The relation between moments and specific shape characteristics are also examined. Two different methods of parameter estimation are studied, the first being the well-known method of moments, while the second is based on the minimization of the integral error between derived and recorded flood hydrographs. The above tasks are illustrated with application examples originated from case studies of catchments in Greece.Notations A catchment area - a,b,c definition parameters (generallya is a scale parameter, whileb andc are shape parameters) - C _v coefficient of variation - C _s skewness coefficient - D net rainfall duration - f( ) probability density function (PDF) - F( ) cumulative (probability) distribution function (CDF) - g( ) objective function - H net rainfall depth - H ₀ unit (net) rainfall depth (=10 mm) - I(t) net hyetograph - i(t) standardized net hyetograph (SNH) - I _n n ^th central moment of the standardized net hyetograph - Q(t) surface runoff hydrograph - q(t) standardized surface runoff hyrograph (SSRH) - Q _n n ^th central moment of the standardized surface runoff hydrograph - S _D (t) S-curve derived from a unit hydrograph of durationD - s(t) standardizedS-curve (SSC) - t time - T _D flood duration of the unit hydrographU _D (t) - T ₀ flood duration of the instantaneous unit hydrographU ₀(t) (= right bound of the functionU ₀(t)) - t _U IUH lag time (defined as the time from the origin to the center of area of IUH or SIUH) - t _I time from the origin to the center of the area of the net hyetograph - t _Q time from the origin to the center of the area of the surface runoff hydrograph - t _p time from the origin to the peak of IUH (or SIUH) - U _D (t) unit hydrograph for rainfall of durationD (DUH) - U _o (t) instantaneous unit hydrograph (IUH) - u(t) standardized instantaneous unit hydrograph (SIUH) - U _n nth central moment of area of IUH - U _n nth moment of IUH about the origin - U _n nth moment of IUH about the right bound (when exists) - V surface runoff volume - V ₀ volume corresponding to the unit hydrograph     

Time-dependent soil-water distribution under a circular trickle source

Soil-water distribution in homogeneous soil profiles of Yolo clay loam and Yolo sand (Typic xerorthents) irrigated from a circular source of water, was measured several times after the initiation of irrigation. The effect of trickle discharge rates and soil type on the locations of the wetting front and soil-water distribution was considered. Soil-water tension and hydraulic conductivity, as functions of soil-water content, were also measured. The theories of time-dependent, linearized infiltration from a circular source and a finite-element solution of the two-dimensional transient soil-water equation were compared with the experimental results. In general, for both soils the computer horizontal and vertical advances of the wetting front were closely related to those observed. With both theories, a better prediction of the wetting front position for the clay loam soil than for the sandy soil is shown. The calculated and measured horizontal vertical advances did not agree over long periods of time. With the linearized solution, overestimated and underestimated vertical advances for the clay and sandy soils, respectively, were shown. The finite-element model approximate in a better way the vertical advances than the linearized solution, while an opposite tendency for the horizontal advances indicated, especially in sandy soil.Notation k constant (dK/d) - K hydraulic conductivity - K ₀ saturated hydraulic conductivity - J ₀,J ₁ Bessel functions of the first kind - h soil water tension - q Q/r ₀ ² - Q discharge rate - r cylindrical coordinate; also horizontal distance in soil surface - R dimensionless quantity forr - r ₀ constant pond radius - R ₀ dimensionless quantity forr ₀ - t time - T dimensionless quantity fort - x, y Cartesian coordinates - z vertical coordinate; also vertical distance along thez axis chosen positively downward - Z dimensionless quantity forz - empirical soil characteristic constant - dummy variable of integration - volumetric soil water content - matrix flux potential - dimensionless quantity for      

Chemical Evaluation of Ma’an Sewage Effluents and its Reuse in Irrigation Purposes

The present study investigates the chemical composition of Ma’an Wastewater Treatment Plant in south Jordan. Samples of effluent of this plant were collected over 1 year period. All samples were analyzed for pH, conductivity, major ions (Cl⁻, , , , , Na⁺, K⁺, Ca^2 + and Mg^2 +) and trace metals B, Fe, Cu, Zn, Cd and Pb. The pH value ranges from 6.79 to 8.15 with a median value of 7.39 ± 0.32. The water quality was characterized by its high salinity hazard (C3) and low sodium hazard (S1) which can be considered as marginal for human consumption. Moreover, concentrations of trace metals in treated wastewater were found to be low and within guidelines for irrigation water due to low level of industrialization activities in the study area. Generally, the result of this study suggests that the treated wastewater is suitable for irrigational purposes, while these effluents can be considered as possible additional resources for irrigation in Jordan.