Designation: D5388 − 93 (Reapproved 2013)Standard Test Method forIndirect Measurements of Discharge by Step-BackwaterMethod1This standard is issued under the fixed designation D5388; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This test method covers the computation of discharge ofwater in open channels or streams using representative cross-sectional characteristics, the water-surface elevation of theupstream-most cross section, and coefficients of channelroughness as input to gradually-varied flow computations.21.2 This test method produces an indirect measurement ofthe discharge for one flow event, usually a specific flood. Thecomputed discharge may be used to define a point on thestage-discharge relation.1.3 The values stated in inch-pound units are to be regardedas the standard. The SI units given in parentheses are forinformation only.1.4 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:3D1129 Terminology Relating to WaterD2777 Practice for Determination of Precision and Bias ofApplicable Test Methods of Committee D19 on WaterD3858 Test Method for Open-Channel Flow Measurementof Water by Velocity-Area Method3. Terminology3.1 Definitions:3.1.1 For definitions of terms used in this test method, referto Terminology D1129.3.2 Definitions of Terms Specific to This Standard:NOTE—Several of the following terms are illustrated in Fig.1.3.2.1 alpha (α)—a dimensionless velocity-head coefficientthat represents the ratio of the true velocity head to the velocityhead computed on the basis of the mean velocity. It is assumedequal to unity if the cross section is not subdivided. Forsubdivided sections, α is computed as follows:α 5(ki3ai2KT3AT2(1)where:k and a = the conveyance and area of the subsection indi-cated by the subscript i , andK and A = the conveyance and area of the total crosssection indicated by the subscript T.3.2.2 conveyance (K)—a measure of the carrying capacity ofa channel without regard to slope and has dimensions of cubicfeet per second. Conveyance is computed as follows:K 51.49nAR2/3(2)3.2.3 cross-section area (A)—the area at the water below thewater-surface elevation that it computed. The area is computedas the summation of the products of mean depth multiplied bythe width between stations of the cross section.3.2.4 cross sections (numbered consecutively in downstreamorder)—representative of a reach and channel and are posi-tioned as nearly as possible at right angles to the direction offlow. They must be defined by coordinates of horizontaldistance and ground elevation. Sufficient ground points mustbe obtained so that straight-line connection of the coordinateswill adequately describe the cross-section geometry.3.2.5 expansion or contraction loss (ho)—in the reach iscomputed by multiplying the change in velocity head throughthe reach by a coefficient. For an expanding reach:1This test method is under the jurisdiction of ASTM Committee D19 on Waterand is the direct responsibility of Subcommittee D19.07 on Sediments,Geomorphology, and Open-Channel Flow.Current edition approved Jan. 1, 2013. Published January 2013. Originallyapproved in 1993. Last previous edition approved in 2007 as D5388 – 93 (2007).DOI: 10.1520/D5388-93R13.2Barnes, H. H., Jr., “Roughness Characteristics of Natural Streams,” U.S.Geological Survey Water Supply Paper 1849, 1967.3For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1ho 5 Ke~hv12 hv2!(3)and for a contracting reach:ho 5 Kc~hv22 hv1!(4)where:hv= velocity head at the respective section, andKe and Kc = coefficients.3.2.5.1 Discussion—The values of the coefficients can rangefrom zero for ideal transitions to 1.0 for Ke and 0.5 for Kc forabrupt changes.3.2.6 fall (∆h)—the drop in the water surface, in ft (m),computed as the difference in the water-surface elevation atadjacent cross sections (see Fig. 1):∆h 5 h12 h2(5)3.2.7 friction loss (hf)—the loss due to boundary friction inthe reach and is computed as follows:hf5LQ2K1K2(6)where:L = length of reach, feet (metres), andK = conveyance at the respective section.3.2.8 Froude number (F)—an index to the state of flow inthe channel. In a prismatic channel, the flow is tranquil orsubcritical if the Froude number is less than unity and a rapidor supercritical if it is greater than unity. The Froude number iscomputed as follows:F 5V=gdm(7)where:V = the mean velocity, ft/s (m/s),dm = the mean depth in the cross section, feet, andg = the acceleration of gravity, ft/s/s (m/s/s).3.2.9 hydraulic radius (R)—defined as the area of a crosssection or subsection divided by the corresponding wettedperimeter. The wetted perimeter is the distance along theground surface of a cross section or subsection.3.2.10 Manning’s equation—Manning’s equation for com-puting discharge for gradually-varied flow is:Q 51.49nAR2/3Sf1/2(8)where:Q = discharge, ft3/s (m3/s),n = Manning’s roughness coefficient,A = cross-section area, ft2(m2),R = hydraulic radius, ft, (m), andSf= friction slope, ft/ft (m/m).3.2.11 roughness coeffıcient (n)—or Manning’s n is used inthe Manning equation. Roughness coefficient or Manning’s n isa measure of the resistance to flow in a channel. The factorsthat influence the magnitude of the resistance to flow includethe character of the bed material, cross-section irregularities,depth of flow, vegetation, and channel alignment. A reasonableevaluation of the resistance to flow in a channel depends on theexperience of the person selecting the coefficient and referenceto texts and reports that contain values for similar stream andflow conditions (see 10.3).3.2.12 velocity head (hv)—in ft(m), compute velocity headas follows:hv5αV22g(9)where:α = velocity-head coefficient,V = the mean velocity in the cross section, ft/s (m/s), andg = the acceleration of gravity, ft/s/s (m/s/s).4. Summary of Test Method4.1 The step-backwater test method is used to indirectlydetermine the discharge through a reach of channel. Thestep-backwater test method needs only one high-water eleva-tion and that being at the upstream most cross section. A fieldsurvey is made to define cross sections of the stream anddetermine distances between them. These data are used tocompute selected properties of the section. The information isused along with Manning’s n to compute the change inwater-surface elevation between cross sections. For one-dimensional and steady flow the following equation is writtenfor the sketch shown in Fig. 1:h15 h21hv21hf1ho 2 hv1(10)where:h = elevation of the water surface above a common datumat the respective sections,hf = the loss due to boundary friction in the reach, andho = the energy loss due to deceleration or acceleration ofthe flow (in the downstream direction) in an expand-ing or contracting reach.FIG. 1 Definition Sketch of Step-Backwater ReachD5388 − 93 (2013)25. Significance and Use5.1 This test method is particularly useful for determiningthe discharge when it cannot be measured directly (such asduring high flow conditions) by some type of current meter toobtain velocities and with sounding weights to determine thecross section (refer to Test Method D3858). This test methodrequires only one high-water elevation, unlike the slope-areatest method that requires numerous high-water marks to definethe fall in the reach. It can be used to determine a stage-discharge relation without needing data from several high-water events.5.1.1 The user is encouraged to verify the theoreticalstage-discharge relation with direct current-meter measure-ments when possible.5.1.2 To develop a rating curve, plot stage versus dischargefor several discharges and their computed stages on a ratingcurve together with direct current-meter measurements.6. Interferences6.1 The cross sections selected should be typical and rep-resentative of the reach half way to each adjacent cross section.If there are abrupt changes between adjacent cross sections, theresults could be suspect. The ratio of the conveyance to theconveyance at an adjacent cross section should stay within 0.7and 1.4.6.2 Care must be taken in selecting the water-surfaceelevation for the downstream cross section. It should not be sohigh that it would reflect backwater at the upstream crosssection or so low that it would be in super-critical flow. A goodrule of thumb is to select a stage so that the conveyance of thedownstream cross section is approximately equal to the con-veyance of the upstream-most cross section.6.3 The only way to be certain that the water-surfaceelevation is not too high or too low or that the reach issufficiently long enough or that enough cross sections are used,or all of the above, is to use the converging profile method. Inthis method, several profiles are developed using a range ofstarting water-surface elevations. The slope of the profiles fromthe higher starting elevations should increase as you move inan upstream direction. The slope of the profiles from the lowerstarting elevations should decrease as you move in an upstreamdirection. At some distance upstream, the profiles will con-verge.6.4 A minimum of about ten cross sections are needed todevelop a smooth backwater curve.7. Apparatus7.1 The equipment generally used for a “transit-stadia”survey is recommended. An engineer’s transit, a self-levelinglevel with azimuth circle, newer equipment using electroniccircuitry, or other advanced surveying instruments may beused. Standard level rods, a telescoping 25-ft (7.62-m) levelrod, rod levels, head levels, steel and metallic tapes, tag lines(small wires with markers fixed at known spacings), vividlycolored flagging, survey stakes, a camera (preferably stereo)with built-in light meter with color film, and ample note paperare necessary items.7.2 Additional equipment that may expedite a survey in-cludes axes, machetes, a boat with oars and motor, hip boots,waders, rain gear, sounding equipment, and two-way radios.7.3 Safety equipment should include life jackets, first aidkit, drinking water, and pocket knives.8. Sampling8.1 Sampling as defined in Terminology D1129 is notapplicable in this test method.9. Calibration9.1 Check the surveying instruments, levels, transits, etc.adjustments before each use, and possibly daily when incontinuous use, or after some occurrence that may haveaffected the adjustment.9.2 The standard check is the two-peg or double-peg test. Ifthe error is over 0.03 ft in 100 ft (0.009 m in 30.4 m), adjustinstrument. The two-peg test and how to adjust the instrumentare described in many surveying textbooks and in instructionsprovided by the manufacturer. Refer to manufacturer’s manualfor the electronic instruments.9.3 If the reciprocal leveling technique is used in the survey,it is the equivalent of the two-peg test between each of the twosuccessive hubs.9.4 Check sectional and telescoping level rods visually atfrequent intervals to be sure sections are not separated. Aproper fit at each joint can be quickly checked by measure-ments across the joint with a steel tape.9.5 Check all field notes of the transit-stadia survey beforeproceeding with the computations.10. Procedure10.1 Selection of a reach of channel is the first and probablythe most important step to obtain reliable results. Ideal reachesrarely exist; thus the various elements in a reach must beevaluated and compromises made so that the best reachavailable is selected. This test method requires that the reach bemuch longer than a reach using the slope-area test method.10.2 The reach of the channel should be as uniform aspossible. Changes in channel conveyance should be fairlyuniform from section to section. Avoid abrupt changes inchannel shape because of uncertainties regarding the value ofthe expansion/contraction loss coefficient and the frictionlosses in the reach.10.3 A reach with flow confined to a roughly trapezoidalchannel is desirable because roughness coefficients have beendetermined for such shapes. However, compound channels,those with overbank flow, for example, can be used if they areproperly subdivided into sub-areas that are approximatelytrapezoidal.10.4 The reach should be long enough to develop a fall thatis approximately equal to half of the average depth.10.5 Cross sections represent the geometry of a reach ofchannel. For example, a section should be typical of the reachfrom halfway to the next section upstream to halfway to theD5388 − 93 (2013)3next section downstream. A minimum of about ten crosssections is recommended.10.6 The roughness coefficient, n , is assigned to a crosssection or to subdivisions of a section, but the n selected shouldrepresent conditions in the reach for which the section istypical. Most texts on hydraulics give techniques of determin-ing values of n. One particularly helpful reference usesphotographs and descriptive stream-channel data to describevalues of n3. Cowan developed a procedure for estimating theeffects of these factors to determine the value of n for achannel.411. Interpretation of Results11.1 Compute the discharge by trial and error. The dischargeand a water-surface elevation at the downstream most crosssection are assumed. A good water-surface elevation for thedownstream most cross section is the given water-surfaceelevation at the upstream most cross section and to adjust it forthe natural slope of the stream. Compute a backwater profile bystarting at the downstream-most cross section and progressingupstream to the upstream-most cross section.5Compute awater-surface elevation for each cross section.11.2 Compute the water-surface elevation for the first crosssection upstream from the downstream-most cross section.Compute this water-surface elevation using the equations in4.1. This computation is done by trial and error. A water-surface elevation is first assumed for this section. With theassumed elevation, compute the area, conveyance, and othersection properties. Use these values in the equations in 4.1 tocompute the change in water-surface elevation between thissection and the downstream-most cross section. Using thischange in water-surface elevation, compute an elevation forthis cross section. The computed elevation should be the sameas the assumed elevation for the section properties to becorrect. When the water-surface ele