# Fixed and Variable Area Discharges Update

## How pressure influences N1, with updated calculations

Management of distribution system pressures to reduce leakage and bursts, and extend asset life, is resulting in increasingly widespread reduction of excess pressures, and lowered pressures at times of low consumption. Recent research has enabled a more thorough application of FAVAD concepts in the practical methods used to analyse and predict pressure:leak flow relationships.

This webpage is longer, and contains more information, that most Concept webpages in LEAKSSuite, as it describes the background to the updates and provides links to additional webpages and software which are relevant to the updates.

The Fixed and Variable Area Discharges (FAVAD) concept, proposed by John May in 1994 reconciled Japanese (1979), UK (1980) and other international research data. The FAVAD concept considers the area of some leakage paths to be fixed, whilst the area of other types of leaks varies linearly with pressure. This explains why leak flow rates (volume/second) from most leaking pipes and distribution systems are more responsive to changes in pressure than leak velocity (distance/second), which only varies with the square root of pressure.Since 1994 the N1 Power Law – a simplified approximate version of the FAVAD concept – has been widely used internationally for practical assessment of pressure-dependent leakage in water distribution systems. The Power Law assumes that leak flow rate varies with pressure to the power N1, where N1 is evaluated by field testing or assessed for a particular system. If the range of pressures is small, N1 is assumed for simplicity to be constant, but that would only be true if all leaks are fixed area (N1 = 0.5) or all are variable area (N1 = 1.5).

Zonal field tests to calculate N1 involve waiting until minimum night flow rate and average zone pressure (AZP) have stabilized, then reducing the pressure, observing the reduction in night flow, and deducting an allowance for customer night use to obtain the night leakage rates (L_{o}, L_{1}, L_{2}) and the corresponding average zone pressures (AZP_{o}, AZP_{1}, AZP_{2}). The values for N1 can then be calculated from the equations shown below.

Once N1 is known, it can be used for predictions of how leak flow rates change with pressure. If AZP_{o} and L_{o} are the initial average zone pressure and leakage rate, and N1 is known or can be predicted, then at a different average zone pressure AZP_{1}, leakage rate L_{1} is predicted from the equation:

By 2005, N1 values calculated from Zonal tests at night in the UK, Japan, Brazil, Cyprus, USA, Australia, New Zealand and Malaysia had shown that N1 usually lay between 0.50 and 1.50, with occasional values outside the 0.5 to 1.5 range (which may or may not be due to data or testing error). The limited published N1 data that exists is usually used by others without reference to, or knowledge of, the circumstances of the original tests. For example, one of the most frequently quoted average N1 values (1.15) was originally based on 8 field tests on around 2km of metal mains and 300 services, almost 50 years ago. In two other series of tests, all detectable bursts were repaired before testing, and in one of these the N1 values include customer night use.

For practical purposes, a linear relationship (N1 = 1.0) is often assumed to apply for large zones (as in the UARL formula), or where no specific evidence exists; i.e., a 10% reduction in average zone pressure will be assumed to reduce leak flow rate by 10%. Despite the approximations inherent in the Power Law equation, it has proved successful in introducing, to practitioners, calculations relating leak flow rates to average zone pressure in most distribution systems.

## How is more extensive use of pressure management likely to influence values of N1?

Improved understanding of pressure:burst frequency relationships and advances in valve control technology and data transfer have resulted in increasingly complex forms of advanced pressure management, with generally lower and more varied pressures at times of low consumption, particularly at night. Because variable leakage area reduces with pressure, when the average zone pressure for any particular system is reduced, the N1 value will also reduce, to a greater or lesser extent.

Research by Professor Kobus van Zyl and colleagues at University of Cape Town from 2012 to 2017 on cracks in different pipe materials has confirmed the validity of the assumptions in the FAVAD equation relating leak flow rate L to pressure P, which can be written as:-Where Af is area of Fixed Area leaks, and Av is area of Variable Area leaks which increases linearly (slope m) with pressure. Cd √(2gP) is the established equation for velocity of flow through an orifice.

Although the N1 parameter is now widely recognized and used in the international water industry, the more complex FAVAD equation (2) is not, and can appear intimidating to some practitioners.

## Fast track Analysis of N1 tests shows how N1 varies with Average Zone Pressure

Lambert et al (Paper 2017L) show how basic data from N1 tests can be rapidly and more thoroughly analysed using the principles of the FAVAD equation. A spreadsheet in the free PaLN1Calcs software calculates equations and graphs relating N1 and AZP, for practitioners to use. In the following excerpt from PaLN1Calcs software, data and preferred units are entered in the yellow cells, and pink cells show calculated outputs.

**Step 1:** Calculate the N1 value from the ‘before’ and ‘after’ AZP and Leak Flow Rates data

**Step 2:** Calculate %s of Fixed and Variable Area leaks at average AZP pressure in the N1 test Image Blur?

% of Variable Area Leakage = N1 – 0.5 = 79%; % of Fixed Area Leakage = 1.5 – N1 = 21% {as corrected 7 December 2017}

N1 is plotted against the Average AZNP, and variable area leakage % is added to the graph

Because N1 varies with Average Zone Pressure, it is not sufficient to quote N1 (1.29) on its own; the Average Zone Night Pressure (AZNP) corresponding to the stated N1 (40.5m) must also be quoted at the same time. This leads to the calculation of AZP_{N=1} when N1 = 1.0, which is used in the equation relating N1 to AZP for this N1 test, calculated in Step 3.

**Step 3:** Calculate the equation relating N1 to AZP pressure, to predict how the zonal N1 from the test would vary with changes in Average Zone Pressures, and show this as a graph.

For this N1 test, it can be seen that:

• at the average AZP pressure during the N1 test (40.5 metres), the N1 was 1.29 and variable area leaks accounted for 79% (= N1 – 0.5) of the leakage paths

• if the AZP pressure was to be reduced to say, 20 metres, the predicted N1 would reduce to 1.15, with variable area leaks accounting for 65% (= N1 – 0.5) of the leakage paths.

• when AZP = 10.8 metres, N1 = 1.0, and areas of fixed and variable paths are equal

• At AZP = 0, N1 = 0.5 and all leakage paths are fixed area, with no variable area component

**CAUTION 1:** If the N1 vs AZP equation is extrapolated to higher AZP pressures than the maximum AZP in the N1 test, there is the risk of creating new leaks which may change the N1 vs AZP relationship. However, if pressure is reduced (without causing significant pressure transients), the predicted reduced N1 values at lower AZP pressures should provide best estimates of N1 without causing additional leaks and bursts.

**CAUTION 2:** The N1 calculations and predictions in Step 1 to Step 3 above require that an N1 Test Protocol is followed. This includes systematic definition of an Average Zone Point (AZP) and measurements of pressure at the AZP during the N1 test. Pressures should always be reduced (not increased) during an N1 test. If you are not prepared to measure AZP pressures during N1 tests, or you do not follow the N1 Test Protocol, then don’t bother with the tests; you will not only be wasting your time and resources, but the results could not be relied upon.

Individual N1 vs AZP equations from individual N1 tests should always lie somewhere within the general relationships between N1 and AZP pressure, as shown in the graph below for AZP pressures up to 100 metres.

It is expected that the above graph will be used for many different purpose, but perhaps the first and simplest is a quick check to assess if it is reasonable to assume that the value of N1 is almost constant for the range of AZP pressures for any particular zone. This is most likely in Zones where:

a) the pressures at the AZP point do not vary greatly around the average daily value, and/or

b) the N1 vs AZP pressures lines have flatter slopes, which mainly occurs at higher pressures.

Just like a blood pressure test by a medical practitioner, even the most reliable N1 test by a leakage practitioner only gives a ‘snapshot’ value of N1, as (particularly in small Zones) the relative %s of fixed and variable area leakage paths in zones with mixed pipe materials can vary over time, as new leaks occur and existing leaks are repaired.

## Fast track calculation of leak flow rate versus Average Zone Pressure

N1 values are useful for generally assessing how sensitive leak flow rates in a Zone are to changes in pressure, but practitioners also need to know the relationship between leak flow rate versus AZP. Because the studies by van Zyl and colleagues have confirmed the assumptions in the FAVAD equation, the FAVAD equation (2) can be rewritten simply as:-

where A and B are Zonal coefficients derived from an N1 test using any preferred combination of units for leak flow rate and AZP. The equations to calculate coefficients B and A are shown in Lambert et al (Leak flow using fast track FAVAD), and used to calculate the L vs AZP equation directly from the N1 test.

## Continuous calculations of leak flow rates using recorded Average Zone Pressures

Utilities which establish continuous pressure measurement at Average Zone Points (which is best practice for most types of leakage calculations) will also be able to use reliable N1 test data to calculate and update equations (based on FAVAD principles) relating leak flow rate to AZP pressure, in their preferred choice of measuring units.

As the AZP pressure determines the instantaneous leak flow rate, measured AZP pressures can be used to calculate 15 minute, hourly and daily leak flow rates, allowing inflow rates to be split into leakage and consumption on a continuous basis. The example below is from a Mexican low pressure system with roof tanks, pressure reduction at night, occasional intermittent supply, and reliable N1 tests, analysed using the LfromAZPCalcs software.

## Influence of variable N1 on Night-Day Factors

Night leakage rate (volume/hour) can be converted to daily leakage using Night-Day Factor NDF (Hour-Day Factor HDF in the UK), which depends on variation of AZP pressure and assumed N1 value. Until now, a constant N1 value (either estimated or from N1 tests) has been assumed, but as N1 varies with Average Zone Pressure, current methods of calculating NDF need to be reviewed. Several options to do these checks are shown and discussed in detail, with examples, in the Night-Day Factor Webpage.

NDF calculations can become quite detailed, and not many practitioners understand the complexities, so Water Loss Research & Analysis Ltd has applied the corrections for variable leakage paths that expand with pressure, and developed a simple approach for quick calculations of NDF. The method initially calculates NDF using the common (but approximate) assumption that N1 is constant at 1.0, then applies a correction for variable area leakage using a Correction Factor CF. CF depends on the ratio of average AZP/AZP at the hour of minimum night flow (AZPave/AZPmnf) and N1. The equation for NDF is

### NDF (hours/day) = CF x 24 x AZPave/AZPmnf

Step 1: for a particular Zone, calculate AZPave/AZPmnf : e.g. AZPave/AZPmnf = 1.50

Step 2: Approximate estimate of NDF, if N1 is constant at 1.0, = 24 x 1.50 = 36.0 hours/day

Step 3: get values of CF at AZPave/AZPmnf = 1.50 from the graph below.

CF for N1 = 1.0 is 0.98, so true NDF at N1 = 1.0 = 0.98 x 24 x1.50 = 35.3 hours/day

Step 4: CF max at AZPave/AZPmnf = 1.22, so NDFmax = 1.22 x 24 x 1.50 = 43.9 hours/day;

CF min at AZPave/AZPmnf = 0.82, so NDFmin = 0.82 x 24 x 1.50 = 29.5 hours/day

Step 4: carry out N1 test, define N1 at the Average AZP for the Zone, finalise NDF estimate Image Blur?

The above approach has the advantage that it starts from a method which is already being widely used (calculate NDF assuming N1 = 1.0), and is readily adaptable to automatic data processing of recorded AZP pressures, with periodic N1 tests.

For a Research Report which derives the equations used, further information on the methodology, equations, look-up tables, training etc, Contact Us will put you in touch with Water Loss Research and Analysis Ltd, the originators of this approach based on Correction Factor.

Alternatively, the NDFs can be read directly (but less accurately) off the graph of NDF vs AZPave/AZPmnf below, for different values of N1 at the average zone pressure. This graph has also been adjusted for fixed and variable leakage areas.

**Summary:** this webpage has provided an outline of the some of the basic additional calculations that can be carried out by leakage practitioners when the FAVAD concepts are applied to the data from a reliable N1 test.

**Acknowledgements:** Kobus van Zyl, Mark Shepherd, Marco Fantozzi, Julian Thornton

**Author:** Allan Lambert, Water Loss Research and Analysis Ltd

**Additional webpages and software **needed to carry out these analyses are:

**N1 Test Protocol:** free Protocol for carrying out reliable N1 tests at night, also part of PaLNICalcs software.

**Average Zone Point:** webpage describing importance, systematic identification, establishment, benefits and use of Average Zone Points in distribution Zones’

**PaLNICalcs:** low cost software to calculate N1 and %s of fixed and variable area leakage at average AZNP during N1 test; AZP pressure when N1 = 1.0; equation relating N1 to AZP, and equation relating leak flow rate to AZP showing fixed and variable area components.

**LfromAZPCalcs** pro software: uses reliable N1 tests to split zonal inflows into leakage and consumption components on a continuous basis.

All of the above softwares are being rewritten and updated, and will be issued by, Water Loss Research and Analysis Ltd. It is recommended that all recipients/purchasers receive certified training in the concepts and use of the above software from WLR&A Ltd or WLR&A Ltd registered trainers.

6th November 2017