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Gradually varied flow (GVF) in open channel hydraulics occurs when free water surface level gradually varies in a steady flow due to geometry alteration in the channel. Investigation of GVF profile is important because water depth along the channel should be determined when designing the channel. Mathematically, GVF profile is obtained by solution of a nonlinear ordinary differential equation. However, due to extreme nonlinearity of the equation, providing an analytical solution is practically unattainable unless under highly simplified conditions. The current study presents semi-analytical solutions for the equation in prismatic ordinary and wide rectangular channels by perturbation method (PM). At the first step, a perturbation solution is derived for wide rectangular channels using a constant Chezy coefficient. At the second step, the wide rectangular channel is investigated using Manning equation as the resistance equation. Finally, the perturbation solution is presented for an ordinary rectangular channel using Manning equation. Obtained GVF profiles using the perturbation solutions are compared with finite difference method (FDM) profiles. Results show that PM profiles are in excellent agreements with FDM ones, and therefore, PM solutions may be used to obtain GVF profiles in mentioned cases with high levels of accuracy. It was concluded that developed semi-analytical solutions may be used as reference solutions for efficiency assessment in various numerical techniques that provide solution for GVF profile in open channels.

Modeling and simulation of water infiltration from the ground surface into unsaturated–saturated soil is very important in the study of the hydrological cycle process. Transient fluid flow through unsaturated–saturated soil is generally described by Richards’ equation. The equation is a nonlinear parabolic partial differential equation for which many numerical and limited analytical solutions exist. In this paper, exponential models for water content and hydraulic conductivity are selected so the nonlinear equation becomes linear and then separation of variables and Fourier series expansion techniques are used to derive analytical solutions to 2D water infiltration into unsaturated–semi-saturated soils with different distributions of water content on a part and/or parts of the top boundary. 2D water infiltrations are investigated in a rectangular soil domain, where high water content zone is considered as semi-saturated zone and residual water content zone is considered as unsaturated zone at initial time. A total of three cases are investigated in this paper. In case 1, a uniform distribution of water content on a part at the top boundary and no-flow boundaries at side edges are considered. Also, horizontal water table in unsaturated–semi-saturated domain is assumed as initial condition. In case 2, an inclined water table initial condition and a uniform distribution of water content on a part at the top boundary are investigated. Finally, in case 3, an analytical solution for ramp distributions of water content on two parts at the top boundary and an inclined water table initial condition is obtained. To illustrate the use of the derived equations, water content values are obtained from a numerical solution and compared to those from analytical solutions for two cases, showing less than 2% errors. These analytical solutions may be used as a benchmark for verification and efficiency assessment of numerical approaches.

In this paper, two modified pressure gradient models based on Taylor series expansion are proposed to enhance the higher order source term MPS (MPS-HS) method. The modified models consist of gradient correction matrices applied to the existing (base) pressure gradient models. To validate the modified pressure gradient models first hydrostatic pressure test is simulated and compared to both the base and modified MPS methods. Using the modified models are shown to reduce unphysical pressure oscillations observed in the base models. Second, an evolution of an elliptical drop in a 2D flow field is examined and shown to verify the models. Third, the proposed models illustrated appropriate stability and consistency properties against analytical solutions when an altered gravitational acceleration was superimposed to the hydrostatic pressure test. In addition, an improved performance is observed when Higher order Laplacian (HL) and Error-Compensating Source (ECS) of the Poisson Pressure Equation (PPE) schemes are coupled with the modified pressure gradient models compared to coupling them with the base gradient models. Finally, the modified MPS methods enhanced performances are validated in a free-surface flow simulation for a dam break problem with impact pressure, and a violent sloshing flow in a rectangular tank when compared to the base MPS methods against an existing experimental data.

Fluid infiltration and imbibition into unsaturated soil are of vital significance from many perspectives. Mathematically, such transient flows are described by Richards’ equation, a nonlinear parabolic partial differential equation with limited analytical solutions in the literature. However, the choice of exponential model for water content and hydraulic conductivity linearizes the nonlinear Richards’ equation, making it possible to obtain an analytical solution via classical approaches. In this study, separation of variables and Fourier series expansion techniques are used to derive new analytical solutions to 2-D vertical and horizontal infiltration and imbibition into unsaturated soils for nonsymmetrical boundary and nonuniform initial conditions. A total of 11 cases are considered, where high water content is imposed on the top, side, or bottom edges of the sample and water is infiltrated (from the top and/or side boundaries) and imbibed (from the bottom boundary) into the sample. Residual water content and/or no-flow boundary condition are assumed on other edges of the sample. Initial conditions include 7 cases of constant residual water content, 2 cases of sinusoidal, and 2 cases of exponential water content functions over the sample. Presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed-form solution. Two-dimensional and 3-D plots of water content are presented for the transient as well as steady-state conditions. To illustrate the use of the derived equations, water content values from numerical solutions are compared to those from analytical solutions for four cases, showing a maximum error of

In this study, the critical (or maximum) discharge rates before saline water enters a well were determined for vertical and horizontal wells in a freshwater aquifer which is separated from a static saline aquifer by a sharp interface. Flow around the well was solved by integration of a point sink solution along the well axis, and both the critical discharge rate and critical interface rise were determined through a comparison of the heads and vertical gradients at the saline–fresh water interface. The rates were determined for vertical and horizontal wells with various lengths and depths for different aquifer salinities. Results were generalized by drawing dimensionless type curves. The results showed that the dimensionless total critical discharge rates are higher for the longer horizontal wells and longer vertical wells with a certain bottom depth, and they almost linearly decrease with well depth at rates of 0.7–0.9. For the dimensionless well length of 0.2, the dimensionless total discharge rate of a horizontal well is about 0.1 more than that of a vertical well with the same length and well-top depth. Also, the critical discharge rates per unit length of well are inversely proportional to well length and remarkably higher for shallower wells. Additionally, the critical pumping rate is proportional to the salinity difference of the aquifers. These results were confirmed by comparison to existing solutions for vertical wells with dimensionless lengths of 0.2, 0.5 and 0.6, and for critical interface rises in the range of 0.75–0.9.

Summary In this study, fractal and multifractal self-affinity in two datasets (d1 and d2) of groundwater level fluctuations time series, each having more than 2000 data points, were evaluated. Variation method, power spectrum, and Detrended Fluctuation Analysis (DFA) were used to estimate fluctuation exponent of the datasets. Multifractal detrended fluctuation analysis (MF-DFA) was used to evaluate multifractal behavior of d1 and d2. Results obtained in this study were compared to two other natural self-affine time series; turbulent open channel flow and river runoff. It was shown that a relatively high (and almost identical) level of multifractality exists in both datasets. The multifractality was postulated to be due to different long-range correlations for small and large fluctuations in the series. Minimum number of data points required to get a stable fractal dimension for groundwater level fluctuation time series (∼1500) was found to be much smaller than that for velocity variations in an open channel (∼2500). Comparing λ exponents of several river runoff time series (0.55

Waste burial in uncontrolled landfills can cause serious environmental damages and unpleasant consequences. Leachates produced in landfills have the potential to contaminate soil and groundwater resources. Leachate management is one of the major issues with respect to landfills environmental impacts. Improper design of landfills can lead to leachate spread in the environment, and hence, engineered landfills are required to have leachate monitoring programs. The high cost of such programs may be greatly reduced and cost efficiency of the program may be optimized if one can predict leachate contamination level and foresee management and treatment strategies. The aim of this study is to develop two expert systems consisting of Artificial Neural Network (ANN) and Principal Component Analysis-M5P (PCA-M5P) models to predict Chemical Oxygen Demand (COD) load in leachates produced in lab-scale landfills. Measured data from three landfill lysimeters, including rainfall depth, number of days after waste deposition, thickness of top and bottom Compacted Clay Liners (CCLs), and thickness of top cover over the lysimeter, were utilized to develop, train, validate, and test the expert systems and predict the leachate COD load. Statistical analysis of the prediction results showed that both models possess good prediction ability with a slight superiority for ANN over PCA-M5P. Based on test datasets, the mean absolute percentage error for ANN and PCA-M5P models were 4% and 12%, respectively, and the correlation coefficient for both models was greater than 0.98. Developed models may be used as a rough estimate for leachate COD load prediction in primary landfill designs, where the effect of a top and/or bottom liner is disputed.

Fractals are objects which have similar appearances when viewed at different scales. Such objects have details at arbitrarily small scales, making them too complex to be represented by Euclidian space; hence, they are assigned a non-integer dimension. Some natural phenomena have been modeled as fractals with success; examples include geologic deposits, topographic surfaces and seismic activities. In particular, time series have been represented as a curve with fractal dimensions between one and two. There are different ways to define fractal dimension, most being equivalent in the continuous domain. However, when applied in practice to discrete data sets, different ways lead to different results. In this study, three methods for estimating fractal dimension are described and two standard algorithms, Hurst’s rescaled range analysis and box-counting method (BC), are compared with the recently introduced variation method (VM). It was confirmed that the last method offers a superior efficiency and accuracy, and hence may be recommended for fractal dimension calculations for time series data. All methods were applied to the measured temporal variation of velocity components in turbulent flows in an open channel in Shiraz University laboratory. The analyses were applied to 2500 measurements at different Reynold’s numbers and it was concluded that a certain degree of randomness may be associated with the velocity in all directions which is a unique character of the flow independent of the Reynold’s number. Results also suggest that the rigid lateral confinement of flow to the fixed channel width allows for designation of a more-or-less constant fractal dimension for the spanwise velocity component. On the contrary, in vertical and streamwise directions more freedom of movements for fluid particles sets more room for variation in fractal dimension at different Reynold’s numbers.

Surfactant-enhanced sorbent zones have potential utility for contaminant plume management in the subsurface. Changes in the hydraulic properties associated with the formation of such zones were studied by measuring hydraulic conductivities of sandy-loam soil mixtures with different fine contents (6, 12, 18, and 24%) that were either untreated, or treated in batch systems with hexadecyltrimethylammonium (HDTMA). Experimental conditions included treatment to 0.1, 0.7, 1.0, and 2.3 times the cation exchange capacities (CEC), and effective stresses from 0 to 766 kPa. Achieved treatment levels and the kinetics of HDTMA adsorption were evaluated using 14 C-labeled HDTMA. Soil suspension turbidity was used to infer the degree of clay dispersion or flocculation; clays were dispersed in 0, 0.1, and 2.3 CEC treated soils and flocculated in 0.7 and 1.0 CEC treated soils. Consistent with their flocculated clay structures, the 0.7 and 1.0 CEC treated soils showed equal or higher conductivities compared to 0, 0.1, and 2.3 CEC treated soils. The 0.7, 1.0, and 2.3 CEC treated soils also showed higher compressibility. HDTMA adsorption occurred rapidly at levels up to 1.0 CEC but was kinetically limited at the high treatment level (2.3 CEC). Creation of sorbent zones by ex situ batch treatment with cationic surfactants was deemed hydraulically feasible as evidenced by equal or higher conductivity of HDTMA-treated soils to 0.7 and 1.0 CEC treatment levels, as compared to untreated soil, under all loads.