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]     

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     

Optimization-simulation models for the design of an Irrigation project

Optimization-simulation models were used for the systems analysis of a water resources system. The Karjan Irrigation reservoir project in India was taken as the system. Two types of optimization models, i.e., linear programming, and dynamic programming (continuous and discontinuous) were used for preliminary design purposes. The simulation technique was used for further screening. The linear programming model is most suitable for finding reservoir capacity. Dynamic programming (continuous and discontinuous models) may be used for further refining the output targets and finding the possible reservoir carry-over storages, respectively. The simulation should then be used to obtain the near optimum values of the design variables.Notations a ₁ Unit irrigation benefit [Rs.10⁵ L^–3] - B ₁ Gross annual irrigation benefit [Rs.10⁵] - B _1,t Gross irrigation benefit in periodt [Rs.10⁵] - C ₁ Annual capital cost of irrigation [Rs.10⁵] - C ₁ Annual capital cost function for irrigation [Rs.10⁵ L^–3] - C _1,t Fraction of annual capital cost for irrigation in periodt [Rs.10⁵] - C ₂ Annual capital cost of reservoir [Rs.10⁵] - C ₂ Annual capital cost function for reservoir [Rs.10⁵ L^–3] - C _2,t Fraction of annual capital cost for reservoir in periodt [Rs.10⁵] - El _t Reservoir evaporation in timet [L³] - f _t Optimal return from staget [Rs.10⁵] - g _t The return function for periodt [Rs.10⁵] - I _t Catchment inflow into the reservoir in periodt [L³] - I _t Water that joins the main stem just above the irrigation diversion canal in timet [L³] - _t Local inflow to the reservoir from the surrounding area in timet [L³] - Ir Annual irrigation target [L³] - K _t Proportion of annual irrigation targetIr to be diverted for irrigation in timet - K _t Amount by whichK _t exceeds unity is the fraction of the end storage which is assigned to reservoir evaporation losses - L Loss in irrigation benefits per unit deficit in the supply [Rs.10⁵ L^–3] - L ₁ Lower bound on annual irrigation target,Ir [L³] - L ₂ Lower bound on reservoir capacity,Y [L³] - N Number of time periods in the planning horizon - O _t Total water release from the reservoir in periodt [L³] - O _t ^* The optimal total water release from the reservoir in timet [L³] - _t Secondary water release from the reservoir in timet [L³] - O _t Reservoir release to the natural channel in timet [L³] - Od _t Irrigation demand in timet [L³] - Om ₁ Annual OM cost of irrigation [Rs.10⁵] - Om ₁ Annual OM cost function for irrigation [Rs.10⁵ L^–3] - Om _1,t Fraction of annual OM cost for irrigation in periodt [Rs.10⁵] - Om ₂ Annual OM cost of reservoir [Rs.10⁵] - Om ₂ Annual OM cost function for reservoir [Rs.10⁵ L^–3] - Om _2,t Fraction of annual OM cost for reservoir in periodt [L³] - Omin_t Lower bound onO _t in timet [L³] - Omax_t Upper bound onO _t in timet [L³] - P _t Precipitation directly upon reservoir in timet [L³] - S _t Gross/live reservoir storage at the end of timet (gross storage in the linear program and live storage in the dynamic program) [L³] - S _t–1 Gross/live reservoir storage at the beginning of timet [L³] - t Any time period - Tr_t Transformation function - U ₁ Upper bound onIr [L³] - U ₂ Upper bound onY [L³] - Y Total capacity of reservoir at maximum pool level [L³] - Ya Fixed active (live) capacity of the reservoir (Y-Yd) [L³] - Ya _t Active (live) capacity (Ymax_t –Ymin_t) of the reservoir in timet [L³] - Yd Dead storage of the reservoir [L³] - Ymax_t Capacity up to the normal pool level of the reservoir in timet [L³] - Ymax_t Live capacity up to the normal pool level of the reservoir in timet [L³] - Ymin_t Capacity up to the minimum pool level of the reservoir in timet [L³] - Ymin_t Live capacity up to the minimum pool level of the reservoir in timet [L³]     

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]     

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     

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     

A water policy intercomparison model for a region with three seasons

A water policy model is proposed as a solution to the problem of obtaining maximum net benefit from providing irrigation and urban water in regions where the major source of supply is groundwater. In essence, the model introduces an innovative scheme based on two types of penalties. These intervene when either watertable elevation falls below a critical value during the operation of a system of wells or a remote source is used to partially cover the needed amount of water expected from the basic groundwater source. Another specific idea of the model is the consideration, for southern regions, of a three-season division in the climatic character of a year. The algorithm is illustrated by a numerical example in which five possible alternatives are compared. The conclusion of the study (although a function of regional economics, natural conditions, as well as specific zonal water policy constraints) reveals a compromise between limiting the amount provided from remote sources and confining the aquifer operation to critical values of the water-table elevation.Notation B benefits from water use, 10³ US$ - C sum of total costs, 10³ US$ - CMR total maintenance-repair cost, 10³ US$ - CO total cost of operation, 10³ US$ - C _HO unit cost of operation per hour, US$ hr^–1 - C _y unit average cost of a repaired pump, US$/(year × well) - d screen diameter for each well, m - H _G average groundwater elevation, m - H _{W cr} critical water elevation value in wells, m - H _{W dj} hydraulic head deficit in wells (belowH _{W cr} ), m - H _{W j} seasonal water elevation in each well, m - j current season - K average hydraulic conductivity of the aquifer, m s^–1 - NB net benefit, 10³ US$ - Ns maximum number of seasons - N _HO number of hours of operation per well and month - N _{W j} number of wells in operation over a seasonj - n _j number of months over each seasonj - P sum of total penalties, 10³ US$ - PH total penalty for pumping whenH _{W dj} >0, 10³ US$ - PQ total penalty for remote source use, 10³ US$ - P _RSj unit penalty for remote source use, 10³ US$ month^–1 - P _{W dj} unit penalty for pumping whenH _{W dj} >0, 10³ US$ month^–1 - pRM percentage of repaired and maintained pumps yearly, % - Q _iRj discharge needed for irrigation use, m³ s^–1 - Q _Nj total discharge needed by users, m³ s^–1 - Q _Pj total seasonal yield capacity of the battery, m³ s^–1 - Q _RSj discharge covered from remote sources, m³ s^–1 - Q _UWj discharge needed for urban water use, m³ s^–1 - Q _j seasonal operated pumping rate in each well, m³ s^–1 - Q _waj weighted average of pumping rate at timet _j , m³ s^–1 - S _Y average specific yield of the aquifer - S _cr critical drawdown value in wells, m - S _j seasonal drawdown in each well - T average transmissivity of the aquifer, m² s^–1 - t _OPj current duration of system operation - t _j –t _j–1 duration of each seasonj     

Errors of kinematic-wave and diffusion-wave approximations for time-independent flows

Time-independent (or steady-state) cases of planar (overland) flow were treated. Errors of the kinematic-wave and diffusion-wave approximations were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth-gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with error in the range of 1–2% for values ofKF ₀ ² (7.5). Even for small values ofKF ₂ ⁰ (e.g.,KF ₂ ⁰ =0.75), the errors were typically in the range of 11–15%. The accuracy of the diffusion wave approximation was greatly influenced by the downstream boundary condition. The error of the kinematic wave approximation was found to vary from 7 to 13% in the regions 0.05x0.95 forKF ₀ ² =0.75 and was greater than 30% forKF ₀ ² =0.75.     

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     

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     

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     

A variable sorptivity infiltration equation

Horizontal and vertical one-dimensional infiltration are compared when they both occur in a homogeneous isotropic porous body initially at a uniform low water content _n under constant concentration (₀) or constant pressure head (H ₀) conditions. From a consideration of the physics governing infiltration under such conditions, the conclusion is reached that the magnitude of the pressure head gradient atx=0, wherex=0 denotes the infiltration surface in the horizontal case, must be larger than the magnitude of the pressure head gradient atz=0, wherez=0 denotes the infiltration surface in the vertical case, for all finitet>0, so that for the hydraulic head gradient atz=0 to be greater than (1/2K ₀)S _x t ^–1/2 but smaller than [(1/2K ₀)S _x t ^–1/2+1],K ₀ being the hydraulic conductivity at ₀ andS _x the sorptivity during horizontal infiltration. On these grounds, it is further argued that if the sorptivityS _z is introduced for the case of vertical infiltration, then it must be equal toS _x fort=0 only and that it must decrease with time. Results obtained by solving soil-water flow equations for the infiltration conditions defined above, and from experiment, support the above conclusions. An equation for the relationship between cumulative infiltration and time during vertical infiltration is developed after assuming thatS _z decreases with time in an exponential manner. Cumulative infiltration versus time relationships given by this equation are compared with those obtained from the numerical solution of the soil-water flow equation and from experiment.     

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