Coniract Report HL-93-4December 1993

US Army Corps A2 6of Engineers AD-A274 611Waterways Experiment I l 11Station

Flood Control Channels Research Program

Modification of the Ackers-WhiteProcedure to Calculate SedimentTransport by Size Fractions

by Alan L. PrasuhnSouth Dakota State University

DTICELECTEJAN 111994

Approved For Public Release; Distribution Is Unlimited

94-01180.94 1 10 149

Prepared for Headquarters, U.S. Army Corps of Engineers

The contents of this report are not to be used for advertising.publication, or promotional purposes. Citation of trade namedoes not constitute an official endorsemet or approval of the useof such commercial products.

O ftNfrND 3CTOZDA~m

Flood Control Channels Contract Report HL-93-4Remarch Program December 1993

Modification of the Ackers-WhiteProcedure to Calculate SedimentTransport by Size Fractions

by Aln L PrasuhnSouth Dakot State UniversityBrookings, SD

DI'JST QUJATY TINMT M

Accesion For

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Prepared for U.S. Army Corps of EngineersWashington, DC 20314-1000

Monitored by Hydraulics LaboratoryU.S. Army Engineer Waterways Experiment Station3909 Hagls Ferry Road, Vicksburg, MS 39180-6199

Under Work Unit 32552

US Army Corpsof EngineersWaterays Expeimnt

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ratoiy USAmyEgneWarways Experiment Station.hk~nhdc~ Dt

65 p. :I1L ; 28 cm. - (Contract report ; HL-93-4)

indudes bibiarphical reerences.1. Sediment transport - Data, processing. 2. Bled load - Measurement - Mathematics. I.I

United States. Army. Corps of Enginer. 11. U.S. Army Engineer Waterways Experimient Sta-tion. Ill. Flood Control Channels Research Program. IV. Title. V. Series: Contrac repor.(U.S. Army Ek wv Waterways Experkiment Station) ; HL-93-4.TA7 W34c rt. .-A-4

Contents

Prreface ................................................ iv

Conversion Factors, Non-SI to SI Units of Measure .................. v

1-Introduction and Procedures ................................ 1

2-Expansion of HEC-6 to Include Five Additional Sediment Sizes ....... 2

3-Original Ackers-White Procedure (Summary) .................... 4

4-Modifications to the Ackers-White Procedure (Prasuhn Procedure) ...... 7

5-Verification of the Prasuhn Procedure .......................... 9

6-Incorporation of the Prasuhn Procedure into HEC-6 ............... 11

Problems with Incorporation.. ............................ 11Adjustments to m and C .................................. 13

7-Conclusions and Recommendations.......................... 16

References ............................................. 18

Tables 1-9

Figre 1-47

SF298

IU

Preface

The investigation reported herein was conducted for the US Army EngineerWaterways Experiment Station (WES) under Contract No. DACW39-89-K-0018 by Dr. Alan L Prasuha, South Dakota State University, Brokings, SD,under Work Unit 32552, "Sediment Transport in Small Channels," of the FloodControl Channels Research Program of the US Army Corps of Engineers CivilWorks Research and Development Program. It documents modifications to theoriginal Ackers-White sediment transport function to allow for multiple grainsizes, to include the new routine in HEC-6, and to increase the number ofgrain size classes into the cobble-boulder range.

The study, conducted during the period 1989 to 1990, was under the gen-eral supervision of Messrs. F. A. Herrmann, Jr., Director of the HydraulicsLaboratory; Mr. R. A. Sager, Assistant Director of the Hydraulics Laboratory;Mr. M. B. Boyd, Chief of the Waterways Division, Hydraulics Laboratory; andunder the direct supervision of Mr. W. A. Thomas, Research Hydraulic Engi-neer, Waterways Division. This report was prepared by Dr. Prasuln as part ofthe contract, and was reviewed by Mr. Thomas, who was the ContractingOfficer's Representative.

At the time of publication of this report, Director of WES wasDr. Robert W. Whalin. Commander was COL Bruce IL Howard, EN.

Iv

Conversion Factors,Non-Sl to Si Units of Measure

Non-SI units of measurement used in this report can be converted to SI(metric) units as follows:

m*IpI __________to Obidn

cubicm let 0.OI65 cubic mWtes

dgrsa ClMul0 s digras o kskis1

bet0.304 nmi~

UM CLOW pon& mass 9"0.1847 dloTwm

1 To babin Ciluu (C) IsmpifMuu readings -1 ro Fdmrenhwe (F) rdng umf ie kdlowingC- (F )-32). Toohn Kalvin Q() readng u: K. (5/)(F-- 32) 27.1.

V

1 Introduction andProcedures

The purpose of this report is to indicate what has been accomplishedconcernig the modification of the Ackers-White sediment ranqmot procedurand its incorporation into HEC-6. The original proposed changes to theAckers-White procedure will be considered. lhese will be followed be a dis-cussion of the initial problems that arose when the modified Ackers-Whiteprocedure was incorporated into HEC-6. Finally, the procedure that evolved asa result of the above difficulties and recommendations conceming their use inHEC-6 will be presented.

The original Ackers-White pocedure (1973) has been thboougily discussedin the literature, e.g., Prasuhn, Lewandowaki and Bagherzadeh (1987). Thedevelopment, other than that necessary to explain the modificatons, will not berepeated here.

In addition, the HEC-6 code was expanded to include five additional sedi-ment sizes (from small cobbles up to and including large boulder). T willbe covered first and in a fairly brief fashion.

chmpbr I&, and satmo s

2 Expansion of HEC-6 toInclude Five AdditionalSediment Sizes

71h existing HEC-6 code will handle 15 size fractions of sediment(U.-S Army Engineer Hydrologic Engineering Center 1977). The researchproposal included the expansion of that code to include small and largecobbles. Consistent with the format which addressed the sand and gravel sizesin groupings of five sizes, the expansion actually includes three additionalsizes: small boulders, medium boulders, and large boulders. Subsequent tothat report a complete listing of all the required changes and where they occurin HEC-6, subroutine by subroutine, has been furnished. They were also high-lighted in a copy of the computer program.

The expanded program now accommodates the five additional sizes and theprintout should be consistent with all previous procedures. The dredgingroutines, $DREDOE, SSED, SVOL, and perhaps some of the other specialoptions, however, were not thoroughly tested. There should be no operationalproblems. The print format should be checked, however. The program willnow handle 20 size fractions of sedimentL one size of clay, four sizes of silt,five sizes sand, riwe sizes of gravel, small and large cobbles, and small,medium and large boulders. It still remains, of course, to find a transportfunction for the large sediments. It is felt that transport of at least the cobblesizes are acceptable using this procedure, the Scboklitsch procedure, andmaybe the Meyer-Peter and Muller procedure.

The program has been tested with many combinations of sizes up to themaximum. This included tests with both four and less than four silt sizes.Changes to the format statements have been avoided with few exceptions. TheE-level printout which gives all the Ackers-White parameters did not provideenough room for large values of Dvr This column has been changed, but theheadings (VF, F, etc.) need adjustment too as they awe inconsistent with thecobble and boulder designations.

Whereas there are several sediment transport functions that may be able togive a reasonable estimate of the cobble transport, there is almost no verifica-tion of boulder transport. If EC-6 is used for the transport of boulders, care

2 chopsw 2 Ep HECE4 lo kwkdo F M iAnd Seimmi Slams

should be taken to consider the reasonableness of the remsts. "his expansioof the code will not be discussed futher in this report.

APSW 2 Em; on of .EC t R.e A hndd F. Mied SsinlI Sk,

3 Original Ackers-WhiteProcedure (Summary)

According to Ackers and White (1973), the sediment transport dependsupon a mobility factor F given by the following expression

_ds-l) 632 log(Vy/ds) (1)

Here s is the specific gravity of sediment particles, U. = the shear velocity, gis the acceleration of gravity, V is the average velocity, y is the depth, ds is therepresentative grain diameter (assumed by Ackers and White to be the d35sizes), a is the rough turbulent flow coefficient (assumed by Ackers and Whiteto equal 10), and n is a factor reflecting the sediment size. For D8 >60,n = 0, otherwise.

n - 1.0 - 0.56 log D8 r (2)

The dimensionless grain size, D8? is given by

D . dfI(sYl)/V2]V/W (3)

where v is the kinematic viscosity.

ITe dimensionless transport G8r is then calculated from the mobility factoraccording to

4ct 0., S AdlWh4 s Pocmdu, (SMn.mra

G f - C(FgylA - 1)' (4)

where the coefficient C, the exponent in, and the initiation of motion parameterA are all determined by regression analysis as follows:

Transition range (I < Dgr < 60)

log C . 2.86 log Dg - (log Dg")2 -3.53 (5)

m = 9.66/D.r +.1.34 (6)

A = 0.23/ID + 0.14 (7)

Coarse range (Dgr > 60)

C = 0.025 (8)

M -1.50 (9)

A = 0.17 (10)

The resulting dimensionless transport G r is related to the concentration X(in Ib sediment/lb water, or N sediment/N *ater), by

GO = (Xls)(yldX)(u 4J) (11)

The actual sediment transport may then be determined by either

Ggr((43.2)ioX (wns/day) (12)

or

5

G - (NIS) (13)

The resulting sediment transport is consequently the total bed material loadbased on the representative size. It was not suggested by Ackers and Whitethat their procedure could be used to calculate the transport by size fractions.

6 3 Ornk Ackn,.ihb Pmood O

4 Modifications to theAckers-White Procedure(Prasuhn Procedure)

The original proposal to modify the Acke-White proposal has been pro-sited and discussed on several occasions (Prasuhn, L.ewandowsui, andBahezadeh 1967; Heng 1969, Pruubn and Heng 1990). In its final form itmay be summarized as follows.

In their procedure, Ackems and White Inuoduced an initiation of motionparameter A which they expressed as a function of the dimensionless grainsize, D Using rmay tranport data collected by White and Day (1962)at the rallingford Research Station (designated as HRS Series A) in a wideflume with a broadly graded sample a similar initiation of motion parameter A'has been determined. The A' values are the best A values based on the actualWhite and Day measured sediment concentrations followed by back calcula-ions for A us•ng the.Ackes-White equations. Tlhese curves are plotted in

Figure 1 along with the White and Day dats. Th bed distribution of thesdata can be expressed by the gradation coeficient o = (dad 16 )12 . 4.198.The odgial Ackers and White data are considered to work best for a singlesediment size; thus the A versus Dcurve is associated with a value a a=1.For values of D < 56, A' is grat;er than A, reflecting a *hiding factor'similar to that or insein and thereby reducing th transport of the fimr sizes.When PF > 56, A' is laes than A, resulting in an "exposure factlo for thelage sites and resulting in increased transpotL

"The proposed procedure uses the a value of a given bed distribution as aninterpolating factor to get a best A' from Figure 1, for each size of bedmaterial, and hence of D For each grain size the best As is calculated asfollows:- If I < c< 4.1(then A' is interpolated from the respective valuesof A and A', baed on the given D If cy = 1,then A' a A, if = 4.19,thenA' = A'. If a > 4.198, then A' is ixtrolated based on the reqpective valuesof A and A'. Thus anA' is determined for each sediment size. One this isaccomp FlIshed, the transport of each size is calculated according to the

conventional Ades-White pr . individual transport is theaadjusted by the pi factor to reflect the actual per cent of that size found in thebed.

ChIpg 4 MogdEemlar to AdwmWN. Pwmdm (Prim Pmemdiar.

A second procedure, to be referred to as the Heng procedure, involved theuse of two graduation coefficients, o1 and q2, defined by oa = do/d16 and02 = d94ldio. These were used as interpolation factors in the same way as oabove. They were used to reflect the bimodal nature of many bed distributionsand in tests of river data, did a slightly better job than the Prasuhn Procedure.However, this procedure raised additional problems when incorporated intoHEC-6 and further, was difficult to justify theoretically. Therefore, althoughthe Heng procedure will be included in many of the figures, it will not berecommended nor discussed further.

8 CapW 4 Modiftals to t Ak-Wtb ProFokdure (P•asuhn Pfoodu.)

5 Verification of the PrasuhnProcedure

Verification of the modified Ackers-White procedure has been reportedpreviously (Prasuhn, Uwandowski, and Bagherzadeh 1987; Heng 1989; andPrasuhn and Heng 1990). Additional results have been included here forselected flume and river data. The concentrations shown in Figures 2-45represent either the concentration by size fractions or the concentration of thetotal bed material load as indicated. There is dearly a variation in theaccuracy of the results, but they are generally quite acceptable and ae as goodas can be achieved by other transport functions. As a general comment theoverestimation of the very fine sand and the tendency for gravel sizes toincrease as the particle size increases, major points that will be considered indetail later, are not obvious here.

Figures 2-14 represent selected Wallingford HRS Series A data (White andDay, 1982). Since the A"I curve was based on these data, it does not representa true verification. Figure 2 is the total concentration for each run and theremaining figures are by size fractions.- On these figures, and those that fol-low, the L-B results can be ignored as representing early worL The A&Wdata refer to the original Ackers-White procedure, and the Proposed refers tothe Heng procedure. Sediment sizes ranged from 0.063, the beginning of sizefraction 1, up to 15.7 mm, the end of size fraction 8. Th'ose results labeledeither Prasuhn or A&W will usually be of the greatest interest. Some of thelower numbered runs refer to very low discharges, but generally a selection ofboth good and poor results will be included. Note that eact pair of figuresrefers to a specific run, in one case referencing the calculated to the measureddata by a line of perfect agreement, and in the other comparing the calculatedto the measured data on a size by size basis. The dashed lines in the first casegive the range of variation within a factor of two. The size by size compari-son is a much tougher comparison since it is on an arithmetic plot.

Figures 15-20 represent selected HRS Series B (White and Day, 1982)results for which the sediment sizes ranged from 0.063 to 635 mm. These

data were not utilized in the developmental process. Figure 15 is once againthe total concentration, but based on the calculation by size fractions, exceptfor the original Ackers-White procedure (A&W). Note that the originalAckers-White procedure overestimated the sediment transport in both cases.

9COusr 5 Vwmcasm d #w Prauhn PRocedwe

Thie remainder of the figures are for transport by size fractions, however, all ofthe runs are combined in Figure 16 which pertains to just the Prasuhn proce-dure. In some cases there is considerable deviation from the measured data,but not in any consistent fashion.

Thie remaining flume data which will be considered here were the data setscollected at the St. Anthony Falls Labortory and reported by Hubbel et al.(1987). The generally coarse material was distributed from 0.5 up to 32 mm.These results, all based on the individual size fractions, ae given in Faig-ures 21-32. These would appear to give an excellent veriation based oanindependent flume data.

Figures 33-42 are based on sand and gravel data from the Platte River(Kircher 1963). With the exception of the first figure, all results are shown bysize fractions on arithmetic plots. Both the Prasulh procedure and the originalAckers-White procedure give fair results.

Total concentration results are given for two sites on the Rio Grande Riverin Figures 43 and 44. Although the bed material was mostly medium tocoarse sand, the sediment ranged from very fine sand to medium gravel(Nordin 1964).

1The final river shown (Figue 45) is the Snake River data (Jones and Sietz1980). Since the measured sediment sizes ranged up to 181 mm, it is unfortu-nate that this is one of the poorer sets of total concentration results. As itturns out, the original Ackers-White procedure gives the best results which isinconsistent with ow usual observation concerning the application of theAckers-White procedure for broadly graded sediments.

10 1 op- s Vwmamon d to Piamuf proafu

6 Incorporation of thePrasuhn Procedure IntoHEC-6

Problems with Incorporation

The original process of icorporx g the modified Acker-Wbht procedureinto HEC-6 involved little more than setting up a separate subroutine whichwas essentially parallel to dte exsting Ackera-White subroutine. However the

inoporation of the above proposed procedure into HEC-6 created a numberof problems which will be enumerated here.

The problems arose during eithe the coding or testing stagm They will betreated separately in no particular order below, although they may beinterrelated. Tle resolution of the problems, or recommendations therefrom,will be considered as they occur as well as summarized at the end of thesection.

a. toe gradation coefficient, a = (d0 /d 1692 wa ued to ex s thespread of the bed sediment distribution and as an interpolating andextrapolating parameter for the incipient motion prameter KA. Previ-ously only sand and larger siza were considered, whereas in HE4signlificant quantities of silt and clay we frequently encountered. It wasnot reasonable to include the clay and silt sizes in the omputation of oso for purposes of this Computation only, the bed wasp prOpronallyreompoMed tr include only the sand and larger sizes. This can not befully justified on the basis of hiding and expmo factors, but no alter-native could be found. It does not appear to be a serious problem

b. In the computation of a within HEC4, evolving bed distributionswhich may not always have been reasonable, led to values of a wellbeyond what had been previously experienced or tested. (The HRSSeries A data has a value o a 4.198.) It was felt that this wasunrlistic and a was set equal to 6 when values of a exceeded 6. Thiagin seemed to work wC4 but the value of 6 was somewhat aibitrary.

11O~u'O hsm~purdno le~t Prmiui Ptoosiar bt MO4S

c. During the testing with HEC-6 it was concluded that the hiding andexposure effects could not be pronounced when only sand sizes werepresent so the modification procedure for A* was bypassed when DS <50 for all sizes. This increased the very find sand transport, makingmore apparent the concerns relative to the overestimation of the finermaterial by the Ackers-White procedure. This apparent problem will bediscussed further, below.

d. Some considerable evidence is now available concerning the overestima-tion of the fine sediments by the original Ackers-White procedure. Incomparisons based on the HEC-6 test runs, both the original Ackers-White procedure and the modified Ackes-White procedure tended tooverestimate the transport of the finer material and very fine sand inparticular. It has been concluded that this is a valid criticism of thebasic Ackers-White procedure requiring additional analysis. It is sug-gested that the best remedy lies in the adjustment of the exponent m.The rationale and method of adjustment is discussed along with thecoefficient C, below.

e. A second problem coming out of the original Ackers-White procedure isthe behavior of the coarser material, primarily gravel and larger sizes.This may have escaped notice previously because the transport wasusually limited to only the smaller gravel sizes. However, if the trans-port is normalized so that each size is considered to cover 100% of thebed, transport will increase as the sediment size increases.

"The problem which is presented in paragraph e is demonstrated in Fig-are 46 where a Froude number range from 0.3 to 12. is considered. (Theeffect becomes mjich more pronounced at still higher Froude numbers.) Ahypothetical discharge, and width were picked for a wide rectangular channel."Tle required depth was then calculated to match the selected Froude numbers.Finally, the slope was determined so as to satisfy the Manning equation atconstant Manning a. TIhe calculations for sediment transport oncentration arebased on the assumption that each sediment size (very fine sand up to largecobbles) completely covers the bed.

Except for the smaller Froude numbers, there is a reversal of the curve forsediment sizes in excess of Dr equal to approximately 60. Beyond this point,the sediment transport increaies as the sediment size and D increase. This,of course, is entirely unrealistic. The effect is more pronounced for Fr > 0.8(the recommended upper limit of validity as recommended by Ackers andWhite).

Here, there is little to go on except the illogical behavior of the currentAckers-White procedure with regard to the larger sizes. One possible fix forthis problem is to replace the existing functions for C with alternative equa-tions. The proposed changes to C will also be discussed below.

12 Chaper on~m (• pa ~ -o t nuh row kt• ME0,4

Adjustments to m and C

It has been established that there are problems with both the Ackers-Whiteprocedure and the modifications thereto. The recommended adjustment for mis to replace the existing functions with an alternative expression whichreduces the magnitude of m for very fine sand and, to a lesser degree, finesand without affecting its magnitude for the larger sediments. The followingset of equations were tested at the U.S. Army Engineer Waterways Experiment(WES) to determine a best equation:

m , 8.419/DSr 0.9 + 1.289 (14a)

m = 8.027/Dr 0.9 + 1.298 (14b)

mn - 7.6351Dg 0.9 + 1.308 (14c)

S- 7.2441D 0.9 . 1.3 18 (14d)

S- 6.85 grD 0.9 + 1328 (14e)

"These equations all achieve the goal of reducing the trahhport of the finersediments without materially affecting the trnslout of the lage sizes. Equa-tion 14e has the most effect. In each case the extpomt of 0.9 was chosen tominimize the effect on the larger sizes.

Typical values of m for the original equation (Equation 6) as well as Equa-tions 14a-e are given for various values of Di in Table 1.

The coefficient C was likewise to be tested by a set of equations. For Dovalues up to 166.7, the original equation (Equation 5) was used, but nowextended over the greater range.

log C - 2.86 log Dgr - (log gr)2 - 3.53 (5)

Above D8 = 166.7 the alternative equations were

13Chaptrw *I hmpaoafl of te PraMhn Proedu k"rn HEC4

c - 0.00772 (111)

C - 03166/Dsr * 0.00582 (15b)

C - 0.6296/Dr + 0.00394 (15C)

C - 0.9426/Dp. + 0.00207 (15d)

C - 1.235/Di + 0.000316 (15.)

Typical values of C fornD 166.7 baed on Equations a- we Sgivm forvarious values of D in Tal 2. Unfortunately, adequate time was notavailable at WES t6 satisfactorily test all of the above equation as originallyintended. 71e tat sets that were completed only involved the sand sizes sothe range of equations for C was not amessed. The only tub that ran success-fully were to pertaing to the HEC-6 Example 1 involving the O*rkReservoir. Example 1 ran with most of the proposed equations for m, theoriginal Ackers-White procedure (MTC - 7), Test Example 1 using theToffalet pmcedre (MTC a 1), and Test Example 3 whIch is identical to TetExample I except that the Laursen procedure (MTC a 3) is used for sedimenttrnspor Regretay, there was omne confusion in these rms and remlts foronly four of the five m equations survive.

The HIC-6 computer results vs.tabulated in Tables 3-9. TAbls 3,4, and5 ve the Toffaleti, Laurmen, and original AckemWhite resut, respectively.Each of the tables contai two parts reflecting runs of I and 64 days. Thoriginal Ackers-White procedue gives hie tumrport of the finer material aexpected, by as much as a factor of 2. The very fine sand ia the sediment out-flow perhaps best demonstrates this tendency. The effect is reduced during thesecond time interval, but this involves the HBC-6 computations and iterationsto a greater extent. TabMes 6, 7, 8, and 9 rdw to successive equations for m.The increased reduction in very fine sand is apparent in eah table. Apin theeffect is reduced for the second time period. It is felt that the reductionafforded by Equation 14b ab 7) gives dte best reults, and this becomes dthreommended equation forms, • ltough additional study Is also rommen

Although -the tea run at WES did not provide any insight into the behaviorof differet equations for C on HEC-6 output, the equations had beoo tested tosame exent previously. Equation Le wceved the greaten atteation, and onthis smewhat limited bais Equation Ld is recommended at presea T1is isalso subject to the reommmendation that additional testing be undertmken.

14 Ohopw 6 pmmn ft U wm. P wan hft HkU4

The above changes must be reasonable and consistent with the originalAckers-White procedure. Refer to Figure 47, which is a copy of Ackers andWhite's Figure 3 (1973). Here the proposed changes to m and C have beenadded to the Ackers and White graphs, illustrating that the proposed changesare consistent with their original data. In addition, they must not contradict thesoundness or logic of the original Ackers-White procedure. The r•st pointwas somewhat satisfied by testing. The second can be justified as follows,based on the development by Ackers and White (1973). The numerical valuesin the expressions for the parameters A, n, m and C were ultimately determinedby regression analysis. They (Ackers and White) explain that the regressionfirst led to the expression for n followed by the incipient motion parameter A.At this point, the expressions for m and C were determined. Since the pro-posed changes in m and C are in mutually exclusive ranges of D thereshould be no violation of the original logic, other than the changes in m and Cthemselves.

15Chqftr 6 h-no pn son-of t Pumftm Proom m ki. HE-C

7 Conclusions andRecommendations

No mater which version or form of Aakes-White is used, the argument ofthe log function in the expmion for the mobility number or factor (Eqution 1) mumst remain geter than zero. This requires that ylds : 0.1. Thisshould be included in any code involving Ackers-White. For example, in thepresent subroutine ACKER, the following IF statement should be added shortlyafter statement 131 and immediately after the comment line 'FOR=SEDIMENT MOBIFXrY FACTOR":

IF (EFD LE 0- 'SD(Q))THENFOiROGOTO (the write statement immediately ahead of existing statementnumber statement 141)

ENDIF

Sy, under the above conditions, the calculation for 0P(I) needs to bebypsased so that GP(1) will remain at the initialized value, Sl. (A standardfix up in the existing log funtion may already accomplish the se- gool, butif it does not, this should avoid problem.)

The expansion of the aode to cover 15 and and larger sediment sizs is inoperational order. It should be useul for at least cobble Led streams and willicrease the range of HEC-6 when inorporated into the prgrm.

Most important an the conclusions and P P n - concerning theAckers-White ptocedure. It is still felt that there is some significant pin inthe modified procedure as developed at South Daotma State University. How-eva, it is now felt that the imprvements are strictly valid only when the beddistribution is broadly graded. It may still be desirable to hnorporate theprogram into HEC-6 under these conditions. Although reasonably verifiedagainst a broad rangp of rver data, this study does not demnmtrate a signif-icantly improved funcion when tse in HEC-6. The sting did identifyproblems and point the way to simple, yet direct, improvements to the Acku,.-White procedure. By replacing the functions for m and Ci i the originalAcker.-White procedure, much of th criticism of the procedure is eiminaed.It is renommended that the eti subroutine ACKER be used in HEC-6 with

16 Chinp- 7 onm•wWa and -ý.w.

the changes made to the equations for m and C as given in Equations 14b and15d. If the modified procedure using the bed coefficient S is still of interest toWES, that code can be provided.

By the end of this study the direction of work had changed abruptly, butfor the better. The end product is simpler than envisioned at the start, but therecommended changes are significant. Time was not available at the end tofully evaluate these changes. Consequently, while the changes will improvethe performance of the subroutine ACKER in HEC-6, they may not yet beoptimized. It is recommended that the equations for m and C be tested further.

17chap•er 7 Ca.ndhuons and RmNm-n ra

References

Ackers, P., and Whie W. K. 1973. 'Sediment Transport: New Approachand AnalyhiW Jowxs HyS Div., ASCE, Vol 95, No. HYll, pp 2041-2060.

Heng, H. H. 1989. 'Sediment Transport Procedure for Nornniform Bed Sedi-nient by Modification of the Ackers and White Procedure,- 7hesis, SouthDakota State University, Brookings, SD.

Hubbel, D. W., et al. 1987. 'Laboratory Data on Coars Sediment Transportfor Bed LoAd Sampler Calibrations,' USGS Water Supply Paper No. 2299,U.S. Geological Survey, Denver, CO.

Jones, M. 1, and Sielz IL RL 1960. -Sediment Transport in the Snake andClearwater River in the Viciity of Lewisto, Idaho,* USGS Open FileReport 80-690.

Kircher, J. F. 1983. -Interpretaion of Sediment Data for South Platte Riverin Colorado and Nebraska, and North Plantt and Platte Rivers in Nebraska,*USGS Professional Paper No. 1277-P.

Nordin, C. F. 1964. -Aspect of Flow Residstace and Sediment Thnsport,Rio Grande Near Bernalillo, New Mexico,' USGS Water Supply PaperNo. 1498-H, US Geological Survey, Waitinglon, Dr-

Prasuba, A. L 1967. Paudamenta of Hydraulc Engineering, Bolt, Rinehartand Winston, New York.

Prasuha, A. ", and Heng, IL IL 1990. 'Application of the Ackers-WhitePtocedure for Transport by Sizn Fraction and Testing in HECA6'Hydraulic Enguaterang; Proceeding of die 1990 Nainal Conference onHydraualc Engineering, Howard IL Chang and Joseph C. Hill, eda., July30-August 3, 1990, San Diego, CA, Hydraulis Divsion, American Societyof Civil Engineers New York, pp 481-486&

Prasuin, A. L, Lewandowaki J. A., and BagherAdleb, M. IL 1967. 'Ackers-White Equation: Transport by Size Fractions,' Hydrau&i Engineerixg;Proceedxng of iMe 1987 Na"iWa Confere nc Hydrauli Engineering,

is -u

R. M. Ragan, ed., August 3-7, 1987, Williamsburg, VA, HydraulicsDivision, American Society of Civil Engineers, New York, pp. 936-941.

U.S. Army Engineer Hydrologic Engineering Center. 1977. "HEC-6: Scourand Deposition in River and Reservoirs, Users Manual,* Davis, CA.

White, W. R., and Day, T. J. 1982. "Transport of Graded Bed Material,"Sedimet Transport in Gravel Bed Rivers, R. D. Hey, ed., Wiley, NewYork, pp 181-223.

19Rlmnmem

Table 1Values of m versus D, for Equation 14

E number1 2 10 20 40 s0

5 11.00 6.17 2.31 1.82 1.58 1.50

14a 9.71 5.80 2.35 1.86 1.58 1.50

14b 9.33 5.60 2.31 1.84 1.56 1.50

14c &.94 5.40 L22 1.62 1.56 1.50

14d L.56 5.20 2.23 1.81 1.56 1.50

14. 6.18 5.00 2.19 1.79 1.56 1.50

Table 2Values of C versus D,, for Equation 15

Eq.number 166.7 600 1000 2000 4000

15. 0.00772 0.00772 0.00772 0.00772 0.00772

15b 0.00772 0.00645 0.00614 0.00596 0.00590

15C 0.00772 0.00520 0.00457 0.00425 0.00410

15d 0.00772 0.00396 0.00301 0.00254 0.00231

15. 0.00772 0.00279 0.00155 0.00003 0.00062

-MJ-

00

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94o4...94*.

0: . .

.4 9

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Lo N.449 49IVIV 10*3 .p "0* 444

:o t t3 %aJJJJw 94 9 S .* 4.4044999

4b 94 49 .4 q 44

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0I IiIl e

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a4~

* ~0S* 4 C

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C ..* ~0-

* bC 0

a; GDa

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3.16 Ina f lotP.PI.A.W. 0

ds~599 ftwt f W11 ow It Ow

WO4 0 W, I.- 444*4 40

* 09%0 C roHi .9 $a 11

I . . S b

es io 0C10

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1.0S

I - Modified

0.52

$ I aml

0C0.2 S a-

0.1I I I I I I

1 2 5 10 20 50 100 200 500 1000

D om Gin Sim dgr

I u. l IFigumi . Curvefor Avs dg

' /0

HRS SERIES A 0

,%1000 0 - /O / /0

E 0.000

C O.,,y ,/0 100' 0

C -, 00000 PRSUHN(0) -

- - sssa* PROPOSED'a /g

0 0 CpO 100. 0 I

(-)S/

I,,0

1 10 100000Actual Concentration (mg/)°°

Figure 2. HRIS Sedas A, Total Cociraln

HRS SERIES A RLUI 5

¶ - X= L-B- 's 00000 PRASUHN/ -•-9. **•**PROPOSED

0.1

C -

00.01 I

0IC) 1

0.001 0.01 0.1Actual concentration (mg/'I)

Figure 3. HRS Swim A, Run 5, Conoerolron by Si, Fracdons

7-

HPS SEPIES A PUll 56-

E GGa9 ACTUAL068 L-B

4- 0+4-EC PRASUH1JSPROPOSED

0

4-3-

01

03

1 2 34 567 8 9 1011121314151617181920

Size Fractions

Figure 4. HRS Seie A Run 5, Concenbation by Size Fractons

HPS SERIES A RflI 7 '0

E 10' D

C. Of

AV-E o~ ., 00000 L-B10 *4**=* PPCFOSEO

C /

C 0.1 1 Q

C c m 1' -

'•) 0.01 , d

25 -

094) ACTP,/'

0.001

0.® oPo, o.,, :o1D

Actual concentration (m /)

Figure 5. HRS Series A, Run 7, Concnsrutton by Size Fractions

• HRS SERIES A PUrl 7

. 25.o I;

O9G• ACTULLEL. • PASUHI 1

v • \ PROPOSED

CC

.o9•L°15., •

10

U 5-

0'12 3 45 6 78 91011113141516171S1920

Size Fractions

Figure 6. HRS Series A, Run 7, Concerirgon by Size Fraction

HRS SERIES A RUII 8

10-

__ , /

/b_, 1 * /

C• 0 , , 'C, U, -

C'

L.oo -

" " 0.1 0 0 S0e000 L-BIH44,00 PRASUHII& * * * * PROPOSED

/p

I, C

- 0.01 *,

C)

0.0011 -

0.001 0.01 7. 1 ,1'Actual concentration (mg/I)

Figure 7. HRS Swerie A, Run 8, Cancwtion by Siz Fractieo

HPS SERIES A P11ll 8

15 GeBC ACI"UAL0990L-B

* e* PRASUHI I"•" *' * PROP')SED

Ex

10-

c 'A0 . h

4-•

0 . / " \A .6 7 8 91 1 1 1 1 1 1 1 1 1 2SizeFra\ion

Flgue O HF• •mlu , Rn 8, • b 81e F

S. .. .,,,i = == = m = ,, l l l 1 11 I III

HPS SERIES A RL'N 9

E 10 0 0

C%C0 0 /

-'- /

C 0.1.2) i

0 / /

S/ / a= L-B"": •;PP51~t

0.0 1 v Y/ .H../ PPOPOSED

0.001 -0.001 0.01 0.1 1

Actual concentration (rmg//1)

Figure 9. HRS Series A, Run 9, Concsntrgion by Size Fracion

HRS SERIES A RULJ 9

40. O A--CTUAL:3BB L-BS(ec PRASUHI I

PROPOSZ ED

E30.

.o

o20- ' .

C I010•

0 ' I I\ 1 1 11 .e

C) \

1 2 3 4 5 6 7 8 9 1011121314151617181920Size Froctions

Figure 10. HRS Series A, Run 9, Conentration by Size Fractiors

HPS SERIES A RUN 10

1 fj --0- /.00 L-

*1. U /N .

U. I.

U /

0.001 0.01 Q.1 1 10Actual concentration (mg/I')

Figure 11. HRS Series A, Run 10, Cono•erlmn by Size Fracims

70-HF'S SEPIES A RUN 10

60_ 0999e ACTUAL

9 -.• 6') PRASUHN,50 , PPOPOSEDE

1030-

20-'

0

1 2 3 4 5 6 7 8 9 1011121.314151617181920

Size Fractions

Figure 12. -RS Series A, Run 10, Concertralln by Size Fractions

100"HFS SEPEI3-, A RUIN 11

A

10 Ile

00

C // //

//

0 .00' /

- .;,,PAUI

r,/ / ** ** *PPOPOSED

00 0.01 0- 1

0.00)1

Actual Concentration (rmg/I)

Figure 13. HRS Series A, Run 11, Concnralion by Size Fractions

HPS SEREIS A PLIN I ,I 0

100\

C0eeo ACTUAL"--" IL-B /p".)-() 'PRASUHI*' 4 PPCO)POSED

E/

0.oE1 50-• •

CU -C0

C'k

1 3 4', 6 78 910 11 1213 141'516 1`718192.0

Si~ze Fractions

Figure 14. HRS Series A, Run 11, Concentration Size Fractions

HRS SERIES 81000,

E'100 0

C i-i-0 0 - /

-6-0

0 0 /

10 C0000A&WC . 0DDIO: L-Ba) .. 00000 PRASUHNIo / *• PROPOSEDCo / 0'

0 D/100

0.10.1 1 10 100 1Q0.

Actual Concentration (mg/I)

Figure 15. HRS Serie B, Tota Ccs-rallkon

HRS SERIES B

100

E10 /

C 00

" 0 00000 PRASUHNC 00 0/C ' / 00

0.1 - t 96

0.01 . . p ..

0.01 0.1 1 10 100Actual Concentration (mg/I)

Figure 16. HRS Seris B, AN Rum, Cmnwlon by Size Fracionm

3.00 HRS SERIES B RUN 3GI • L-B

O PRASUHNPROPOSED

Ges ACTUAL

E2.00

o0,.'

c- 1.001(1)04.'

oC)

1 2 45 7891 110 11l 12 1,3 1,4 115 1F6 17

Size Fractions

Figre 17. HRS Sews B., Run 3, Conceuation by Size Fracdons

HRS SERIES B RUN 514.0E0 L-B

14.00 9)• PRASUHNPROPOSED

oe998 ACTUALo'lZ12.00

E io.oo

o 8.000

_6.00

4-.0cC) 4;00

0

2.00

0.00•

1 2 .3 4 561 7 8 9 10 11 12 1'31'4A11617

Size Fractions

Figure 18. HRS Smde B, Run S. Conceaon by SIz Fain

120- HRS SERIES 8 RUN 6

em. L-Be0 PRASUHN4 '** PROPOSED

S100- 0699 ACTUAL

E ao-

160S

.9 RPRSUH

140P

0

20-•

0 1 23 ,5 6 7 a 9 O10 1 12 13 14 15 1617Size Fractions

Figure 19. HRS Seria B, Run 8, Conoi'•n by Size Fracio

160- MRS SERIES B RUN 8

SPRASUHN140 - PROPOSED

08 ACTUAL

.2- 80"

0-i

C 40-0

20

2 45 6 7 a 9 10 1112 1314 1516 17Size Fractions

Figure 20. HRS Sorias B. Run 8, Cc -0 bodni by Sim Fmdblm

100 SAINT ANTHONY FALL LAODTU RUN1M DAY7 7

S/0

11-

210 /0

.0-I O@O@PRASWINC -00. PROPOSED

0)

1Actual Concentration (mg/-)

Figure 21. SL Anrd" Fab, Run 1, Concwmdion by Siz Frcd

o SAINT ANTHONY FALLS LABMIXTURE RUN 1 DAY 7

m ##W0 PReSrUHNEm PROPOSED

3-

C)

70

040

Size Fractions

Figure 22. SL. An•mw Fab, P=r 1, Commen&*k by Size Fraction=

...... .. ... ......a. ..

100 SAWN ANTHONY FALLS LAO0 MM 2RUN2 DAY5 5

C1 0 .

o 9 C/,

0-0

6) 4'*****PROPOSW

0.1 ,1 ,"

o Actual e tio (mg/l) 100

Figure 23. SL ,Arhn Fab, Run 2, - 011 bW 1b Size Fract

0SAW ANT FALLS LABMIXTU RUN 2 DAY 3

471te~ PRASUHNE~o PtlromOE

(4 ) 4

0.10

SAze Fractions

Fgum 24. ft Ano Fab, Run 2, Centeo by Stze Fmrctim

100" SAINT ANTHONY FALLS LAB

MIXIURE RUN 3 DAY 3 /

E /

/4-'/0-

PPSU,..*"4-'/ ,,•, - 00000C• *****PROPOSED

C Io ,,

*/ //

0.10.1 1 10

Actual Concentration (mg//I)

Figure 25. SL Anthony Fails, Run 3, Day 3, Concetation by Size Fractions

60 SAINT ANTHONY FALLS LABMIXTURE RUN 3 DAY 3

01E4o ***** PROPOSEDC

20

.o3.

0L.

4-'

0010-

Size Fractions

Figure 26. SL Anhony Fidls, Run 3, Day 3, Concerradon by Size Fractions

SAINT ANTHONY FALLS LAD

100 23.50 mm RUN3 DAY 9

0 ,

C 10 -

o 0 -* - -• S•0 / 0

I.. /

/ / OO000 PRASMI4N

0) ' 4

-0,

0.1•110 100lActual Concentration (mgI

Figure 27. St. An"hn Falls, Run 3, Day 9, Conremlation by Size Fractonm

SAINT ANTHONY FALLS LAB50o 23.50 mm RUN 3 DAY 9

4o0

30-

r-)

C0

0.10

10 1"10

Size Fractions

Figure 28. SL Anthon Fabl, Run 3, Day 9, Concntaon by Size Fracmn

100 SAINT ANi iNY FALLS LABMIXTURE RUN 4 DAY 3

I O z•,/ ,"

C1 0 -. ,i

I--, 000o0 PRASUHN0***** PROPOSED

C) / i

I -. ' / ifI o / ,"

0 . 1 • s 5 . u S I* I U I i |

0.1 1 10 100I Actual Concentration (mg/I)

Figure 29. SL An" Fals, Run 4, Day 3. Concentration by Size Fractions

SAINT ANTHONY FALLS LABMIXTURE RUN 4 DAY 3

120

""100 Aws CTUALIm4~ PRASUHN

E ***PROPOSED

so-0

L..

40-

40C-)20-

01Size Fractions

Figure 30. St. Anthony Fabs, Run 4, Day 3, Concenratdon by Size FractoS

SAINT ANTHONY FALLS LAM1000 23.50 mm RUN 4 DAY 6

/

E

C -.00' - -

C.100 /.~1o0-# -PRASUHN

- - - tV PROPOSM0) - -

(/)/

o ,0 4 -1

10 f10 c00 lowO

Actual Concentration (mg/I)

Figure 31. SL Anrh Fals, Run 4, Day 6, C=4onrMion by Sin Fracfto

SAINT ANTHONY FALLS LAB200- 23.50 mm RUN 4 DAY 6

180-

E120,

.2 lOO- QMMA •IML

0.C

0 4

011

Size Fractions

FIgure 32. SL AMn y Fals, Run 4, Day 6, Concsrtlond by Size Fractions

PLATTE PIVER NEBRASKA & COLOPADO13 SAMPLE

(71000-Er

"/

0.0 ,0

L.

OOC00OA&W/ 1***0 PROPOSED4) 100 //, 0 000 L-BU ",t 0000•0 PPASUHHJC0

6

10 100 100Actual Concentration (mg/I)

Figure 33. Plt Rer, TdaI Cor ano

160-4 PLATTE RIVER SAMPLE 1

140-Boo L-8OO0 PRASUHN

",1 20- **.*" PROPOSEDaGSSS ACTUAL

100

C

0 aCO-)

20

I-

1 40.

0

-80 '

20.

1 2 3 4 6 7 8

Size FractionsFigure 34. Pke River, SBmnl 1, Cwoetation by Size Fraclon

100-

PLATTE RIVER SAMPLE 4UL-B

'Z 8 4HC PRASUHNN ~d~ib*PROPOSED

osese ACTUALE

Go0

40

0

Size FractionsFigure 35. PI. River, Sample 4, Cocnrlnby Size Fracftio

100- PLATTE RIVER SAMPLE 6

mieL-B"WO PRASUI4N

~ 80 - PROPOSEDso- 388190 ACTUAL

E60

40

0 20-

01 4 5

Sie rotin

Fiuewpo lCr mb6 1 o yrwFct

50 PLATTE RIVER SAMPLE 8

G88 L-B"W PRASUHN

"*'* PROPOSED0e90 ACTUAL

--'•40-

E30

0

-20-C

0010

1 2 3 4 5 6 7 8 9

Size Fractions

Figure 37. Platte River, Sample 6, Conrcenraton by Size Fractlion

180 PLATTE RIVER SAMPLE 10

160-G•BBW L-B0940-) PRASUHN

Z. 140 . PROPOSED0) - Geeee ACTUAL

E 120-

C 100-

0

6 80-

'V 60

U0

0 40

20.

1 3 4 5 6 7 8

Size Fractions

Figure 38. Platte River, Sample 10, Concenation by Siz Fractiorn

PLATTE RIVER SAMPLE 11160-

•,140 - L-B000 PRASUHN..*PROPOSED

0120. -• ACTUALE/

100-0

o• 801

S40.0

20

0

S 2 3 4 5 6 7 8Size Fractions

um~rs 39. Pkft WIver, Sunpb 11. CQnouktgl I by Sie Friemon

200- PLATTE RIVER SAMPLE 12ISO-180-

U-m L-B160 -040W PRASUHN*.." PROPOSED

0)140- ACTUAL

120O

C ..2loo-

ao-I

V

) 60-

o 40.

20

0* 2 3. 4 5 6 7 a

Size Fractions

F•,r 40. Plft Rvr, SWI*p 12, C by Siz FhkCton

PLATTE RIVER SAMPLE 13

140-

88uu L-BS1 20. 0)44 PRASUHN

N�*.'* PROPOSEDoeeeo ACTUAL

E iooc S-o

- 60'

c

40-c0

S20"

0

1 2 3 4 5 6 7 8Size Fractions

FigUre 41. PIU R~v, Sonpis 13, Can ~asi tion by SIM FRUdNs

PLAflE RIVER SAMPLE 14

100,

U-L-B")0O PRASUHN*.*. PROPOSED

o-80 GS88O ACTUAL

EC 6 0 '

0

40-

020

0

2 3 4 5 6 8

Size Froctions

ft"ur 42. p1MW p~ve, Swnpl 14, Canceerilo by SIM FracOns

RIO GRA14DE RIVERNEW MEXICO SECTION A-2

.1000- 0 //0 /

- 0 PROPOSED

( = 0 0A&W

0)

U 1 ,- o .~O ~ ORSH

o o'10ER•

( ) - /

/ 7 O"

C.

100 //////100 1000

Actual Concentration (mg/I)

mgr 43. Ro Ckrad. Rie. Seeda A-2. TOW= Cancemnic i

RIO GRAJIIDE RIVERN4EW MEXICO SECTION F

ci

c PRO~POSED

U) 0 COOOC PRASUH14

ci

100100 '1000

Actual Concentration (mg//l)

Figur 44. Rio Grwtde River. Secdan F, Totld Cancmitrion

1000- SNAKE RIV'ER IDAHO

0 -'

0 0 01

0 0 C,*

100-b ,

W ***** PROPOSEDo /0 00000OQC PRASUH1l

C..

101010 1001000

Actual Concentration (mg,/l')Figure 45. Snake Rime, Total Cioncmnirstion

TRANSPORT AT VARIOUS FROUDE NUMBER

1000000000

100000000 r04

10000000 r06

m~ 1000000 r08

'- 100000

0

0 1000V-'

C 100a)U 10C0

0.1

0.01 W

Dgr

Figure 46. Ccncmntrdon Distrbuton as a Function of the Froude Number

Le -OMO AWAASI.. Sa-lo SWlm

a-,0

1. - I 0o

I I I

.ma

Figur4. EQ15ds an hte(93 Fgr-

REPORT DOCUMENTATION PAGE om ow o, ,ga4Jeuf rond mamunenhg the dmn1 nuu• ad axndoeleung an evueweng the Clctonm, of nf-aco Sn comm € regWardg thu burden estmate at any, other aggact.•_ oftm ~of/meu an ,ncudng lugre fo r reducing thu _ .den, to was Hington Ieadquarten e•,&.v Ovectow ate for information operatond m pRe •, 1 II$

ae Hg.way Suite 5Q.. ArlIngt=o0n. 2202-4N2. and to the Offce of Management f dt. PPeW*ork ftadci Pvoet (070-100). Wahiwtan. DC 2r03•

1. AGENCY USE ONLY (Leave Wank) 2. REPORT DATE t3. REPORT TYPE AND DATES COVERED

December 1993 Final report

4. TITLE AND SUBTITLE S. FUNDING NUMBERS

Modification of the Ackers-White Procedure to Calculate Sediment WU 32552Transport by Size Fractions6. AUTHOR(S)

Alan L Prasuha

7. PERFORMING ORGANIZATION NAME(S) AND ADORESS(ES) 8. PERFORMING ORGANIZATIONREPORT NUMBER

South Dakota State University,Brookings, SD

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADORESS(ES) 10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

US Army Corps of Engineers, Washington, DC 20314-1000

USAE Waterways Experiment Station, Hydraulics Laboratory, Contract Report3909 Halls Ferry Road, Vicksburg, MS 39180-6199 HL-93-4

11. SUPPLEMENTARY NOTES

Available from National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22161.

12a. DISTRIBUTION / AVAILABULTY STATEMENT 121L DISTIBUTION CODE

Approved for public release; distribution is unlimited.

13. ABSTRACT (Maximum 200 worctsThis report documents modifications to the original Ackers-White sediment transport function to calculate

sediment transport by grain size classes, to include the new routine in HEC-6, and to increase the number of grainsize classes into the cobble-boulder range.

The Ackers and White sediment transport procedure gives a total bed material load for a sediment mixturebased on the representative particle size of the mixture. The dimensionless transport Gv is calculated from themobility factor F8r according to

G, - C(Fgr4 - 1)"

Ackers and White did not suggest that their procedure could be used to calculate transport by size fractions."This research extended the calculations to particle size classes by modifying the m and C coefficients.

(Continued)14. SUBJECT TERMS 15. NUMBER OF PAGESAckers and White Cobbles Sediment mixtures 65Bed material load HEC-6 16. PmCE CODEBoulders Multiple grain sizes

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 13. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT

UNCLASSEFME UNCIASIFIEDIIANSN 7S401-280-500 Standard Form 298 (Rev. 2-89)

Pvewnoed by ANSI S•d. Z39-16296102

14. (Cohclmkd

The existing suboutine ACKER should continue to be used in HEC-6 with changes made to the equations for mand C as give in Equations 14b and 15d in this report. In the cas tested, this study demonstrs•tes a significantlyixproved function when the bed distribution is broadly graded. However, it should be tested in HEC-6 usingadditiceal, mor diverse data sets.

This modified function can be applied to particle sizes up through cobbles. It still remains, of course, tofind a transport function for boulder size sediments. If HEC-6 is used for the transport of boulders, care should betaken to consider the reasonableness of the results.

No matter which version of Ackers-White is used, the argument of the log function in the expressio forthe mobility factor (Equation 1) must remain greater than zero, i.., y/d, > 0.1. This should be included in anycode involving AckeM-White.