Identification of Manning’s Roughness Coefficients in Shallow Water Flows

A numerical method based on optimal control theories for identifying Manning’s roughness coefficients (Manning’s n) in modeling of shallow water flows is presented. The coefficients are difficult to be determined especially when the spatial variation is significant, and are usually estimated empirically. The present methodology is applied to determine the optimal values of the spatially distributed parameters, which give least overall discrepancies between simulations and measurements. Through a series of systematic studies to identify the n values in both a hypothetical open channel and a natural stream stretch, several identification procedures based on unconstrained and constrained minimizations are analyzed. It is found that the limited-memory quasi-Newton method has the advantages of higher rate of convergence, numerical stability and computational efficiency. Although the identification of Manning’s n is chosen as an example, the identification methods can be applied to numerical simulations of various flow problems.     

Design Aid for Grass-Lined Channels

The Manning roughness factor varies with the hydraulic radius in grass-lined channels. The functional relationship between the roughness factor and the hydraulic radius is highly nonlinear. This necessitates a lengthy and tedious trial-and-error method to solve the Manning formula in the different stages of the traditional design procedures for grass-lined channels. Various charts are presented in this technical note that can be used in designing grass-lined channels without a need for trial-and-error. The charts were developed using predetermined solutions to the Manning formula. They are presented in terms of dimensionless parameters and can be employed with any consistent unit system. The use of the charts is demonstrated through an example.     

Design of Circular Urban Storm Sewer Systems Using Multilinear Muskingum Flow Routing Method

In this study, a multilinear Muskingum method is presented for hydrologic routing through circular conduits. In order to increase accuracy, the reference discharge is assumed to be a nonlinear function of conduit diameter, Manning coefficient, bed slope, and peak discharge of the inflow hydrograph. The reference discharge function has been determined using a nonlinear regression technique. Flow depths are computed at every time step by solving the continuity equation using an implicit finite difference scheme. Many storm hydrographs were routed through circular conduits of various sizes by the proposed model. The calculated routed hydrographs and water surface profiles indicate close agreement with those obtained by solving Saint Venant equations. Using this method, a branched urban sewer system was designed. This indicates that the method can be easily implemented for design purposes because of its simplicity, accuracy, and computational efficiency.     

Rational Coefficients for Steeply Sloped Watersheds

When computing peak discharges for the design of drainage systems using the rational method, it is important to have an accurate value for the rational coefficient (C). For steeply sloped watersheds the origin of values of the rational coefficient are unknown and lack even modeling verification. A model that shows the relationship between the rational coefficient and watershed slope was developed for steeply sloped watersheds. Using Horton’s infiltration equation, Manning’s equation, the velocity method for computing times of concentration, and generalized intensity-duration-frequency curves, a model was developed to test the effect of variation of several watershed characteristics on the relationship between slope and the rational coefficient. Analyses with the model showed that both Manning’s coefficient and land use had the greatest effect on the relationship between C and slope. A mathematical function was then developed from data generated from the Horton–Manning model. This model allows C to be estimated for a given slope and a value of Manning’s coefficient for the land cover. A rational coefficient at a 6% slope is also required input. The model was tested using several watersheds with moderate to steep slopes. This relationship should be used to better estimate values of C on steep slopes, and thereby, lead to more accurately hydrologic designs.     

Optimal Design of Open Channel Section Incorporating Critical Flow Condition

The flow at critical condition of an open channel is unstable. At critical condition, a small change in specific energy will cause abrupt fluctuation in water depth of the channel. This is because the specific energy curve is almost vertical at critical state. Therefore, if the design depth of the channel is near or equal to critical depth of the channel, the shape of the channel must be altered to avoid a large fluctuation in water depth. In the present study, a nonlinear optimization model is presented for designing an optimal channel section incorporating the critical flow condition of the channel. The optimization model derives the optimal channel section at a desirable difference from the critical condition of the channel so that a small change in the specific energy of the channel will not cause an abrupt change in flow depth. The objective of the optimization model is to minimize the total construction costs of the channel. Manning’s equation is used to specify the uniform flow condition in the channel. The developed optimization model is solved by sequential quadratic programming using MATLAB. Applicability of the model is demonstrated for a trapezoidal channel section with composite roughness. However, it also can be extended to other shapes of channel.     

Optimal Design of Channel Having Horizontal Bottom and Parabolic Sides

The cost of open channels can be minimized by using (1) the optimal design concept; (2) a new geometric shape to substitute for the trapezoidal channels, and/or (3) a composite channel. The channels in which the roughness along the wetted perimeter become distinctly different from part to part of the perimeter are called composite channels. The feasibility of a new cross-sectional shape that has a horizontal bed and two parabolic sides and lined as a composite channel is investigated to substitute for the trapezoidal cross section. The optimal design concept is used to establish the efficacy of the proposed new cross-sectional shape, because it gives the best and unique design of open channels. In optimal design concept, the geometric dimensions of a channel cross section are determined in a manner to minimize the total construction costs. The constraints are the given channel capacity and other imposed restrictions on geometric dimensions. The Lagrange multiplier technique is used to solve the resulting channel optimization models. The developed optimization models are applied to design the proposed and trapezoidal channels to convey a given design flow considering various design scenarios which include unrestricted, flow depth constrained, side slopes constrained, and top width constrained design. Each of these design scenarios again takes into account fixed freeboard, and depth-dependent freeboard cases of design. An analysis of the optimization results establishes the cost-saving capability of the proposed cross-sectional shape in comparison to a trapezoidal cross section.     

Automatic Calibration of the U.S. EPA SWMM Model for a Large Urban Catchment

The Storm Water Management Model was adapted and calibrated to the Ballona Creek Watershed, a large urban catchment in Southern California. A geographic information system (GIS) was used to process the input data and generate the spatial distribution of precipitation. An optimization procedure using the complex method was incorporated to estimate runoff parameters, and ten storms were used for calibration and validation. The calibrated model predicted the observed outputs with reasonable accuracy. A sensitivity analysis showed the impact of the model parameters, and results were most sensitive to imperviousness and impervious depression storage and least sensitive to Manning roughness for surface flow. Optimized imperviousness was greater than imperviousness predicted from land-use information. The results demonstrate that this methodology of integrating GIS and stormwater model with a constrained optimization technique can be applied to large watersheds.     

Spatial and Temporal Variation of Manning’s Roughness Coefficient in Furrow Irrigation

Manning’s roughness coefficient is one of the input parameters in many surface irrigation simulation models. It affects the velocity of flow and thereby its variation with time and distance along the field length influence water application. In this study, variation of Manning’s roughness coefficient was studied for a furrow plot consisting of three 40 m long free drained furrows of parabolic shape and having a top width of 0.30 m, a depth of 0.15 m and a slope of 0.5%. The irrigation experiments were carried out with the inflow rates of 0.2, 0.3, 0.4, and 0.5?L?s1; and 0.3, 0.4, 0.5, 0.6, and 0.7?L?s?1 under bare; and cropped field conditions, respectively. Furrow cross-section data were collected before each irrigation event at 0.5, 13, 26 and 39.5 m from the head end along the center furrow using a profilometer. During the irrigation event, water depth and velocity of flow were measured at these locations at an interval of 15 min using point gauge and color dye, respectively. The furrow cross-section data were fitted into a second-degree polynomial equation to determine the furrow shape parameters that were used along with the flow depth data for determining the wetted area and wetted perimeter. The wetted area, wetted perimeter, and the velocity data were used to estimate Manning’s roughness coefficient spatially and temporally. It is found that for both bare and cropped field conditions, Manning’s roughness coefficient was more at second and last quarter of the furrow due to soil erosion at these locations. Manning’s roughness coefficient at these locations varied from 0.019 to 0.022 and 0.015 to 0.018 for bare field whereas from 0.02 to 0.024, and 0.019 to 0.022 for cropped field, respectively. The temporal variation of Manning’s roughness coefficient for both bare and cropped furrow conditions decreased with the elapsed time. However, these decreasing trends were observed more for lower inflow rates. Further, the average Manning’s roughness coefficient for the subsequent irrigations was varied from 0.018 to 0.02 and from 0.019 to 0.0245 for bare and cropped conditions, respectively. Thus, the values of Manning’s roughness coefficients were more for cropped furrow conditions than for bare furrow.     

Optimization-Based Simulation and Design of Tile Drainage Systems

Optimization-based models for simulation of tile drainage systems are formulated and solved with a finite-difference approximated Richard's equation as an embedded constraint. The solution methodology minimizes the sum of absolute violations of the discretized variably saturated porous media flow equations. The optimal design minimizes the sum of the pressure heads to obtain optimal drain locations and discharges for a given lateral spacing. The projected augmented Lagrangian method of nonlinear programming is used to solve the optimization models. The performance of the simulation is verified using the steady-state square embankment drainage problem. Thereafter, the simulation and design models are solved for a hypothetical tile drainage system. The results demonstrate the capability of the models.     

Dynamic Routing Model with Real-Time Roughness Updating for Flood Forecasting

A flood routing model with time-varying roughness updating was developed to simulate flows through natural channels based on the dynamic wave theory. Taking observed stages as the targets, a roughness updating technique was developed using the Gauss-Newton method to update the Manning n in each time step of the routing processes. The technique provides a reasonable roughness coefficient estimate and reliable initialization of stage profile for the forecast. The examinations including the initialization of stage profile, conservation of mass, iteration convergence, effectiveness evaluation, and convergence with different initial values were conducted to verify the predictive capability of the roughness updating technique. The forecasting results show that the stage recalculated by updating the Manning n in current time has a good agreement with the observed stage. The presented model can improve the forecast for a lead time up to 6?h in the Tanshui River of northern Taiwan.     

Overtopping Probability Constrained Optimal Design of Composite Channels Using Swarm Intelligence Technique

In this paper swarm intelligence based methodology is proposed for optimal and reliable design of irrigation channels. The input parameters involved in channel design are prone to uncertainty and the solution of deterministic model may result in flooding risk and affect the stability of the channel. To provide reliability in the design, an overtopping probability constrained design is presented in this study. The deterministic equivalent of the probabilistic constraint is derived by following the principle of first order uncertainty analysis. In order to account for the uncertainty of design parameters in the objective function, a modified cost function is proposed. A methodology is propounded to solve it in a metaheuristic environment and solved it using elitist-mutated particle swarm optimization (EMPSO) method. The EMPSO based solutions are found to be quite successful and better than the classical optimization methods. Finally, it is concluded that the proposed methodology has a good potential for reliable design of composite channels for designer specified reliability values.     

Physical Modeling of Sheet Flow on Rough Impervious Surfaces

This paper presents results from an extensive experimental study of sheet flow on rough impervious surfaces that are used to represent highway pavement. Experiments were performed on three surfaces under no-rainfall and simulated rainfall conditions, and with slopes of 1, 2, and 3%. Measurements include flow depth and unit discharge. Turbulent boundary layer theory for a rough surface is used to describe the depth-discharge relationship, resulting in a model with a single parameter directly related to the surface roughness. Comparisons are made with Manning’s equation, and the variability of the Manning coefficient is assessed. Hydraulic effects of rainfall are generally found to be small compared to other factors.     

Optimization of Multiple Reservoir Networks for Sedimentation Control

An optimal control methodology is developed to evaluate reservoir management policies that minimize sediment scour and deposition in multiple-reservoir river networks. Consideration is given to adverse effects occurring in both reservoirs and rivers of a hydraulic network. The sedimentation problem is formulated as a discrete-time optimal control problem in which a successive approximation linear quadratic regulator optimization scheme is coupled with the U.S. Army Corps of Engineers HEC-6 sediment transport simulation model. Operational constraints imposed on reservoir storage levels and releases are accommodated using either a bracket or hyperbolic penalty function method. The resulting model also allows the user to evaluate policies that alternatively maximize sedimentation or consider adverse effects only at specified locations. Comparisons of the computational efficiencies between the successive approximation linear quadratic regulator and differential dynamic programming methodology and between different penalty functions are performed. Capabilities of the model are demonstrated through applications to a hypothetical three-reservoir network and the Yazoo Basin river-reservoir network in Mississippi.     

Distributed Sensitivity Analysis of Flood Inundation Model Calibration

Uncertainties in hydrodynamic model calibration and boundary conditions can have a significant influence on flood inundation predictions. Uncertainty analysis involves quantification of these uncertainties and their propagation through to inundation predictions. In this paper the inverse problem of sensitivity analysis is tackled, in order to diagnose the influence that model input variables, together and in combination, have on the uncertainty in the inundation model prediction. Variance-based global sensitivity analysis is applied to simulation of a flood on a reach of the River Thames (United Kingdom) for which a synthetic aperture radar image of the extent of flooding was available for model validation. The sensitivity analysis using the method of Sobol’ quantifies the significant influence of variance in the Manning channel roughness coefficient in raster-based flood inundation model predictions of flood outline and flood depth. The spatial influence of the Manning channel roughness coefficient is analyzed by dividing the channel into subreaches and calculating variance-based sensitivity indices for each subreach. Replicated Latin hypercube sampling is used for sensitivity analysis with correlated input variables. The methodology identifies subreaches of channel that have the most influence on variance in the model predictions, demonstrating how far boundary effects propagate into the model and indicating where further data acquisition and nested higher-resolution model studies should be targeted.     

Optimal Control of Open-Channel Flow Using Adjoint Sensitivity Analysis

An optimal flow control methodology based on adjoint sensitivity analysis for controlling nonlinear open channel flows with complex geometries is presented. The adjoint equations, derived from the nonlinear Saint-Venant equations, are generally capable of evaluating the time-dependent sensitivities with respect to a variety of control variables under complex flow conditions and cross-section shapes. The internal boundary conditions of the adjoint equations at a confluence (junction) derived by the variational approach make the flow control model applicable to solve optimal flow control problems in a channel network over a watershed. As a result, an optimal flow control software package has been developed, in which two basic modules, i.e., a hydrodynamic module and a bound constrained optimization module using the limited-memory quasi-Newton algorithm, are integrated. The effectiveness and applicability of this integrated optimal control tool are demonstrated thoroughly by implementing flood diversion controls in rivers, from one reach with a single or multiple floodgates (with or without constraints), to a channel network with multiple floodgates. This new optimal flow control model can be generally applied to make optimal decisions in real-time flood control and water resource management in a watershed.     

Optimal Spacing in an Array of Fully Penetrating Ditches for Subsurface Drainage

A methodology for optimal spacing in an array of ditches fully penetrating into homogeneous and isotropic porous medium of finite depth over an impervious layer is presented. The cost function includes the depth-dependent earthwork cost and the capitalized cost of pumping of drain discharge. Essentially, it is a problem of minimization of a nonlinear objective function of single variable. The input variables consist of rainfall intensity, hydraulic conductivity of the porous medium, width and depth of ditches, earthwork cost, cost of pumps and pumping energy cost, efficiency of pumping unit, and rate of interest. Using nonlinear data fitting method an explicit equation has been proposed for computing the optimal spacing between the ditches.     

Aspect Ratio to Maximize Sediment Transport in Rigid Bank Channels

The continuity equation, Manning’s equation, Einstein’s wall correction procedure and sediment transport equations are combined to indicate channel aspect ratios which maximize sediment transport for a given water discharge in rigid-bank trapezoidal and rectangular channels with fixed slope. Higher aspect ratios are required to maximize sediment transport for channels conveying bed load than for those with a dominant suspended load. A total load equation predicts optimum aspect ratios lying in between those for bed load and suspended load channels. The equations imply that the optimum aspect ratio increases markedly as the channel bank to channel bed roughness ratio increases. The resulting optimum ratios are smaller than the aspect ratios of many natural rivers.     

Optimal Design with Probabilistic Objective and Constraints

Significant challenges are associated with solving optimal structural design problems involving the failure probability in the objective and constraint functions. In this paper, we develop gradient-based optimization algorithms for estimating the solution of three classes of such problems in the case of continuous design variables. Our approach is based on a sequence of approximating design problems, which is constructed and then solved by a semiinfinite optimization algorithm. The construction consists of two steps: First, the failure probability terms in the objective function are replaced by auxiliary variables resulting in a simplified objective function. The auxiliary variables are determined automatically by the optimization algorithm. Second, the failure probability constraints are replaced by a parametrized first-order approximation. The parameter values are determined in an adaptive manner based on separate estimations of the failure probability. Any computational reliability method, including first-order reliability and second-order reliability methods and Monte Carlo simulation, can be used for this purpose. After repeatedly solving the approximating problem, an approximate solution of the original design problem is found, which satisfies the failure probability constraints at a precision level corresponding to the selected reliability method. The approach is illustrated by a series of examples involving optimal design and maintenance planning of a reinforced concrete bridge girder.     

Optimal Design of a Stable Trapezoidal Channel Section Using Hybrid Optimization Techniques

A cost effective channel section for a specified flow rate, roughness coefficients, longitudinal slope, and various cost parameters can be determined using an optimization technique. However, the derived optimal channel section may not be feasible for construction because of in situ conditions. The local soil conditions may not support the optimal side slope of the channel and if constructed, the slope may fail. It is therefore necessary to also incorporate the criteria for side slope stability in designing an optimal open channel section. In this paper, a new methodology has been developed to design a stable and optimal channel section using hybrid optimization techniques. A genetic algorithm based optimization model is developed initially to determine the factor of safety of a channel slope for given soil parameters. This optimization model is then externally linked with a separate sequential quadratic programming based optimization model to evaluate the parameters of the stable and optimal channel section. Solution for various example problems incorporating different soil parameters are illustrated to demonstrate the applicability of the developed methodology.     

New Strategy for Optimizing Water Application under Trickle Irrigation

The determination of water application parameters for creating an optimal soil moisture profile represents a complex nonlinear optimization problem which renders traditional optimization into a cumbersome procedure. For this reason, an alternative methodology is proposed which combines a numerical subsurface flow model and artificial neural networks (ANN) for solving the problem in two, fully separate steps. The first step employs the flow model for calculating a large number of wetting profiles (output), obtained from a systematic variation of both water application and initial soil moisture (input). The resulting matrix of corresponding input/output values is used for training the ANN. The second step, the application of the fully trained ANN, then provides the irrigation parameters which range from a specified initial soil moisture to a desired crop-specific soil moisture profile. In order to avoid substantial disadvantages associated with the common feedforward backpropagation approach, a self-organizing topological feature map is implemented to perform this task. After a comprehensive sensitivity analysis, the new methodology is applied to the outcome of an irrigation experiment. The convincing results recommend the new methodology as a positive contribution towards an improved irrigation efficiency.