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     

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]     

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]     

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     

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     

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]     

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     

Water-table variation in a sloping aquifer due to random recharge

Variation in the level of the water table is closely linked with recharge. Therefore, any uncertainty associated with the recharge rate is bound to affect the nature of the water-table fluctuation. In this note, a ditch-drainage problem of a sloping aquifer is considered to investigate the effect of uncertainty in the recharge rate on water-table fluctuation. The rate of recharge is taken as an exponentially decaying function with its decay constant as a Gaussian random variable. Expressions for the first two moments of the water-table height, i.e. mean and standard deviation, are presented. By using these expressions, the effect of uncertainty in the recharge rate on the water-table fluctuation has been analyzed with the help of a numerical example.     

Nitrate Accumulation and Other Components of the Groundwater in Relation to Cropping System in an Aquifer in Southwestern Spain

Inappropriate farming techniques and 4 years of drought (1992–1995) produced conditions for high nitrate concentration in an aquifer located in the Province of Seville (Southwest Spain). The objectives of this study were to study groundwater components and to determine the relationship between the cropping system and nitrate concentrations in groundwater. From October 1994 to May 1996, 16 groundwater samples were obtained from a network of 35 sample points. Groundwater concentrations were analysed by multivariate statistical techniques (factor and cluster analysis). Two factors were found: firstly, agricultural pollution and secondly, salinization (from natural mineralization of groundwater and from anthropogenic effects). High levels of NO_3– were closely associated with intensive cotton (Gossypium herbaceum L.) and potato production (Solanum tuberosum L.). The salinization factor includes electrical conductivity, Cl_–, Na₊, Ca_{2 +}, Mg_{2 +}, SO_{42 –} and HCO_3–. Our study permits the impact of crops on groundwater to be determined. In comparison to the other crops, cotton (Gossypium herbaceum L.) and potato (Solanum tuberosum L.) made the greatest contribution to nitrate pollution and salinization. Use of the groundwater for human consumption in drought periods could represent a public health risk. Alternative farming techniques for these crops such as monitored fractional fertilization and controlled irrigation are recommended.     

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     

Impact of Treated Wastewater Reuse on Agriculture and Aquifer Recharge in a Coastal Area: Korba Case Study

Treated wastewater (TWW) reuse has increasingly been integrated in the planning and development of water resources in Tunisia. The present study aimed the evaluation of the environmental and health impact that would have the reuse of TWW for crops direct irrigation or for the recharge of the local aquifer in Korba (Tunisia). For this purpose water analyses were carried on the TWW intended for the aquifer recharge and on underground water of this area. As for underground water before recharge, no contamination by organic matter or heavy metals is shown but high salinity, nitrate, potassium and chloride concentrations are detected. The bacteriological analyses show the occurrence of faecal streptococcus, thermo-tolerant coliforms, total coliforms and E coli, but absence of salmonella. These results indicate that this water is not suitable for irrigation worse still for drinking purpose. The monitoring of TWW pollutants has demonstrated that oxygen demands (COD and BOD) do not exceed the Tunisian standards for TWW used in agriculture (NT 106.03) except for August when samples reach high values (COD = 139 mg O₂ L^− 1, BOD = 34). It is also the case for temperature, electrical conductivity (EC), salinity and pH. Heavy metal concentrations are under the detection limit. The determination of nutrients shows relatively low concentrations of nitrates, nitrites and orthophosphate (the maxima in mg L^− 1 are respectively 6.6, 5.6 and 0.92) whereas the potassium levels are high (up to 48.8 mg L^− 1) and the ammonia levels very high, reaching 60.6 mg L^− 1. As for bacteriological pollution, while no salmonella and intestinal nematods are detected, high concentrations of total coliforms, thermo-tolerant coliforms, faecal streptococci and E. coli are analysed. Consequently, the better use of TWW in this region would be the use of infiltration basins for the recharge of the deteriorated aquifer by TWW. It would give the opportunity to better the quality of the TWW reaching the groundwater by an additional treatment for bacteriological and suspended solid (TSS) contaminants while being an alternative water for the aquifer recharge and a coastal barrier against seawater intrusion.     

Observations on Vertical Variability in Groundwater Quality: Implications for Aquifer Management

This work demonstrates significant vertical variability in the chemical composition of groundwater (Cl⁻ from 150 to 550 mg/L, NO₃^-{\rm{NO}}_{3}^{-} from 3 to 70 mg/L, trichloroethene from 350 to 55,000 μg/L and Cr_Total from 3 to 2,900 μg/L) along a 130 m thick vertical section passing through two subaquifers of the Costal Plain aquifer of Israel. Water samples were obtained by multilevel sampler under natural gradient flow conditions from a monitoring well which penetrates the entire aquifer. The vertical chemical variability detected at a single point in time in this well was found to be similar to the range of concentrations detected for Cl⁻ in pumping wells located over an area of about 1,000 km² in the same aquifer. Similarly, vertical variations in NO₃^-{\rm{NO}}_{3}^{-} concentration in the single monitoring well represent more than 50% of the total variation in nitrate levels in pumping wells sampled across the entire aquifer. These results graphically illustrate that data from pumping wells, which extract unknown mixtures of groundwater from different depths, cannot provide adequate information about groundwater quality for management purposes.     

On the Prediction of Ground-Water Mound Formation in Response to Transient Recharge from a Circular Basin

An analytical solution of the linearized Boussinesq equation is developed to predict the formation of a ground-water mound in an aquifer system in response to localized time-varying recharge. The recharge is applied from a centrally located circular basin. The solution is obtained using an eigenvalue-eigenfunction method. The solution for a constant recharge rate is shown as a special case of the solution for a time-varying recharge rate. Application of the solution to predict ground-water mound formation is demonstrated by a numerical example. Effects of variation in the rate of reachrge, size of recharge basin and the saturated hydraulic conductivity on the growth of the water-table are also investigated.     

Characteristics of pressure fluctuations beyond spillways with a ledge

Conclusions When calculating the intensity of pressure fluctuation on a revetment for the surface regime beyond a dam with a ledge by the appropriate relations [3] it is necessary to take into account that the dispersion of the turbulent component given by this calculation is commensurate with the dispersion of the wave component of presssure fluctuation, which presently can be determined from the results of hydraulic investigations. The data given in [5] can be used for estimating the intensity of the total value of pressure fluctuation on a revetment for a surface regime beyond a dam with a ledge under conditions of the two-dimensional problem and in the range of relative heights of the ledge d/h_c=3–6.5 and Froude numbers in the vena contracta on the ledge Fr_c=8–50.Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 3, pp. 26–28, March, 1986.     

Study of Rejim Maatoug groundwater in southern Tunisia using isotope methods

The Rejim Maatoug aquifer, which located in south-western Tunisia, constitutes a good example of overexploited resources. Hydrogeological and geochemical data were examined to determine the main factors and mechanisms controlling the groundwater chemistry and the hydrodynamic functioning of this aquifer based on a multi-tracer approach. Thirty-one groundwater samples from the CT Miocene aquifer of Rejim Maatoug and Djerid regions were collected in 2010 for major chemical element analysis. Physical parameters such as temperature, pH and electrical conductivity (EC) were measured in the field. Three groundwater samples were selected for ¹⁴C measurements. Four groundwater samples were chosen for ³H measurements. The chemical data indicates that dissolution of evaporate minerals, evaporation and cation exchange processes seem to be the main factors controlling groundwater mineralization. The geological and hydrogeological studies show the important role of the deep existing faults in controlling the hydrodynamic functioning of the aquifer systems in southern Tunisia. The isotopic data shows that groundwater in this area is a mixture of recent shallow waters located in the Mio-Plio-Quaternary sands and paleo-deep waters of the carbonate Senonian aquifer. Low ³H and ¹⁴C contents, observed in some sampled wells (³H = 0.15 TU in C1N1, ¹⁴C = 8.8 pmc in Matr2), indicates that the recharge of the Miocene aquifer occurs mainly through upflow leakage of Senonian deep aquifer while there is no evidence of significant recharge in major part of the plain and mountains piedmonts that can be observed.     

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     

First estimation of the diffusive methane flux and concentrations from Lake Winnipeg,a large,shallow and eutrophic lake

Freshwater lakes are increasingly recognized as significant sources of atmospheric methane (CH₄), potentially offsetting the terrestrial carbon sink. We present the first study of dissolved CH₄ distributions and lake-air flux from Lake Winnipeg, based on two-years of observations collected during all seasons. Methane concentrations across two years had a median of value of 24.6 nmol L^-1 (mean: 41.6 ± 68.2 nmol L^-1) and ranged between 5.0 and 733.8 nmol L^-1, with a 2018 annual median of 24.4 nmol L^-1 (mean: 46.8 ± 99.3 nmol L^-1) and 25.1 nmol L^-1 (mean: 38.8 ± 45.2 nmol L^-1) in 2019. The median lake-air flux was 1.1 µmol m^?2 h^?1 (range: 0.46–70.1 µmol m^?2h^?1, mean: 2.9 ± 10.2 µmol m^?2 h^?1) in 2018, and 5.5 µmol m^?2h^?1 (range: 0.0–78.4 µmol m^?2 h^?1, mean: 2.7 ± 8.5 µmol m^?2 h^?1) in 2019, for a total diffusive emission of 0.001 Tg of CH₄-C yr^?1. We found evidence of consistent spatial variability, with higher concentrations near river inflows. Significant seasonal trends in CH₄ concentrations were not observed, though fluxes were highest during the fall season due to strong winds. Our findings suggest Lake Winnipeg is a CH₄ source of similar mean magnitude to Lake Erie, with lower concentrations and fluxes per unit area than smaller mid- to high-latitude lakes. Additional work is needed to understand the factors underlying observed spatial variability in dissolved gas concentration, including estimations of production and consumption rates in the water column and sediments.