This note suggests a technique for including pressure dependent demand and leakage terms in simulation models for water distribution systems. Empirical functions are used to relate consumer outflows and leakage losses to the network pressures and the inclusion of these functions in the mathematical formulation of the network analysis problem is described. An application to an existing distribution system is presented where it is shown that the extended network model proposed leads to more realistic simulation results when the network pressures are too low to provide specified consumer demands, or high enough to cause significant leakage losses. It is also found that the computational requirements of network analysis and simulation are not significantly affected by the inclusion of the additional terms.

A technical note on the inclusion of pressure dependent demand and leakage terms in water supply network models

The paper concerns the problems of minimization of leakage in water-distribution networks. It has been reported that leakage from some networks may account for a significant amount of the water put into supply. For some aging urban networks, rates of up to 50% have been quoted, with average rates of 25% being quite typical. These high rates of leakage represent a significant economic loss. An algorithm for the determination of flow control valve settings to minimize leakage is presented. The nonlinear network equations describing nodal heads and pipeflows are augmented by terms that explicitly account for pressure-dependent leakage and by terms that model the effect of valve actions. Successive linearization of these equations using the linear-theory method allows a linear program that minimizes leakage to be formulated and solved. The performance of the method is demonstrated by application to an example network.

Optimal Valve Control in Water‐Distribution Networks

The reduction of leakage is an important objective of the United Kingdom water industry. The inclusion of pressure-dependent leakage terms in network analysis allows the application of formal optimization techniques to identify the most effective means of reducing water losses in distribution systems. The development of an optimization method to minimize leakage in water distribution systems through the most effective settings of flow reduction valves is described. The method shows a significant advantage compared with previously published techniques in terms of robustness and computational efficiency. A particular feature of the approach is the use of an objective function that allows minor violations in the targeted pressure requirements. This allows a much greater improvement in the exceedance of minor pressure requirements than would otherwise be achieved.

Leakage Reduction in Water Distribution Systems: Optimal Valve Control

This paper presents a numerical model for simulating the transfer of suspended sediment from a channel to an adjacent flood plain under conditions of steady, longitudinally uniform flow. The model accounts for transverse transfer of sediment by convection and turbulence. Turbulent transfer is modelled by the diffusion analogy and empirical relationships are used to evaluate the distributions of diffusivity and bed shear stress. The elliptic partial differential equation describing the distribution of suspended sediment concentration under the influence of these transfer mechanisms is solved numerically by the method of Successive Over-Relaxation. The model can be applied to sediments of different sizes and densities and can be used to predict the extent and relative distributions of deposits over the flood plain as well as the size and density characteristics of deposited material at any position on the flood plain. The model is verified by comparing predicted distributions of fine sand deposits with thos...

Sediment transfer to overbank sections

1 Basic Principles and Mathematical Models of Engineering Mechanics Problems.- 1.0 Introduction.- 1.1 Ideal-Fluid Flow.- 1.2 Porous Media Flow.- 1.3 Fickian Diffusion.- 1.4 Heat Flow.- 1.5 Elasticity Problems.- 1.6 Use of Laplace Equation.- 1.7 Mathematical Modeling.- 1.8 Computational Mechanics and Hydraulics.- 1.9 Modeling of Potential Problems.- 2 A Review of Complex Variable Theory.- 2.0 Introduction.- 2.1 Preliminary Definitions.- 2.2 Polar Forms of Complex Numbers.- 2.3 Limits and Continuity.- 2.4 Derivatives: Analytic Functions.- 2.5 The Cauchy-Riemann Equations and Harmonic Functions.- 2.6 Complex Line Integration.- 2.7 Cauchy's Integral Theorem.- 2.8 The Cauchy Integral Formula.- 2.9 Taylor Series.- 2.10 Program I: A Complex Polynomial Approximation Method.- 2.11 Complex Variables and Two-Dimensional Fluid Flow.- 3 Mathematical Development of The Complex Variable Boundary Element Method (CVBEM).- 3.0 Introduction.- 3.1 Basic Definitions.- 3.2 Characteristics of the Linear Global Trial Function.- 3.3 The H1 Approximation Function.- 3.4 Higher Order HK Approximation Functions.- 3.5 Using ? or ? Functions as Boundary Conditions.- 4 A Computer Algorithm for the Complex Variable Boundary Element Method.- 4.0 Introduction.- 4.1 A Complex Variable Boundary Element Approximation Model.- 4.2 The Analytic Function Defined by the Approximation $$\hat w\left( z \right)$$.- 4.3 Program 2: A Linear Basis Function Approximation $$\hat w\left( z \right)$$.- 4.4 A Constant Boundary Element Method.- 4.5 Summary of Basic CVBEM Modeling Algorithm.- 5 Reducing CVBEM Approximation Error.- 5.0 Introduction.- 5.1 Application of the CVBEM to the Unit Circle.- 5.2 Approximation Error from the CVBEM.- 5.3 A CVBEM Modeling Strategy to Reduce Approximation Error.- 5.4 A Modified CVBEM Numerical Model.- 5.5 Program 3: A Modified CVBEM Numerical Model.- 5.6 Determining Some Useful Error Bounds for the CVBEM.- 6 The Approximative Boundary.- 6.0 Introduction.- 6.1 Expansion of the HK Approximation Function.- 6.2 Upper Half Plane Boundary Value Problems.- 6.3 The Approximative Boundary for Error Analysis.- 6.4 Program 4: Analytic Continuation Model.- 6.5 Locating Additional Nodal Points on ?.- 7 CVBEM Modeling Techniques.- 7.0 Introduction.- 7.1 Sources and Sinks.- 7.2 Program 5: Source and Sink Model.- 7.3 Regional Inhomogeneity.- 7.4 Program 6: Nonhomogeneous Domain.- 7.5 The Poisson Equation.- 7.6 Computer-Aided-Analysis and the CVBEM.- 8 CVBEM Applications.- 8.0 Introduction.- 8.1 Modeling Nonuniform St. Venant Torsion.- 8.2 Numerical Calibration of Domain Models.- 8.3 Modeling Steady State, Advective Contaminant Transport.- 8.4 A Simple Model of Soil Water Phase Change.- 8.5 Modeling Two-Dimensional Steady State Soil Freezing Fronts.- References.- List of Symbols.- Mathematical Notations.

The Complex Variable Boundary Element Method in Engineering Analysis

A previously unpublished numerical method for the steady-state solution of the flow distribution in a pipe network is presented. It relies on an iterative linearization procedure to solve for junction heads, and is highly suited for use on small computers. Its relationship to other solution methods is explicitly developed. This new method is extremely simple to formulate, and requires minimum data preorganization for computation.

Linear Theory Methods for Pipe Network Analysis

Advantages of boundary integral equation methods (BIEM) for the analysis of two-dimensional flows through porous media are numerous. The data required and the number of unknowns are significantly fewer than for other general, numerical methods such as finite differences of finite element methods. A BIEM based on the Cauchy integral equation is developed. This formulation has the advantage that the equation used for any unknown is a Fredholm equation of the second kind which results in well-conditioned equations and allows nodes to be closely spaced in the region of singularities. The method is extended to the analysis of multizone, anisotropic flows. Computational algorithms for solution of the equations by a simple iterative method and by direct solution of the full set of simultaneous linear equations are developed.

Integral Equation Formulation for Ground-Water Flow

The forces experienced by models of three different types of bridge piers have been measured for angles of inclination between the stream flow and the pier axis in the range 0○ to 50○, and the results are presented in the form of drag coefficients and lift coefficients. The types of pier tested were a plate pier, dumb-bell piers, and twin cylinder piers. The values obtained for the lift coefficient for the first two types were large, their magnitudes generally falling between one and two for angles of inclination in excess of 10○, whereas those obtained for the third type were relatively small. The forces measured for all pier types possessed fluctuating components the characteristics of which are described. The fluctuating components experienced by the dumb-bell and twin cylinder piers were generally quite large and complex. The relevance of the model results to prototype conditions is discussed and it is concluded that the force coefficients obtained can be used for preliminary design calculations.

Bridge Piers—Hydrodynamic Force Coefficients

A curved cubic triangular element which has as nodal parameters the value of the function and its two derivatives is derived by use of a transformation similar to that used for quadrilateral isoparametric elements. A special form of the element which may be used to satisfy the condition that the function is constant along a boundary is also presented.

Lewis T. Isaacs

Papers

Linear Theory Methods for Pipe Network Analysis

Integral Equation Formulation for Ground-Water Flow

Bridge Piers—Hydrodynamic Force Coefficients

A curved cubic triangular finite element for potential flow problems

On the estimation of the optimum accelerator for sor applied to finite element methods