AD-Ai59 921 ONE-DIMENSIONRL MODEL FOR MUD FLONS(U) HYDROLOGIC 1/ENGINEERING CENTER DAYIS CR D R SCHAMBER ET RL. OCT 85HEC-TP-169

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US Army Corpsof EngineersThe HydrologicEngineering Center

" One-Dimensional ModelMFor Mud FlowsCDIn

byD. R. Schamber DTIC

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R. C. MacArthur O0CT09ED

Technical Paper No. 109 =IT eat ha. bees hasr-be-

st 7 October 1985

85 10 08 001

Papers in this series have resulted from technical activities of theHydrologic Engineering Center. Versions of some of these have beenpublished in technical journals or in conference proceedings. Thepurpose of this series is to make the information available for use in theCenter's training program and for distribution within the Corps ofEngineers.

The findings in this report are not to be construed as an officialDepartment of the Army position unless so designated by otherauthorized documents.

The contents of this report are not tc be used for advertising,publication, or promotional purposes. Citation of trade names does notconstitute an official endorsement or approval of the use of suchcommercial products.

ONE-DIMENSIONAL MODEL FOR MUD FLOMS1

David R. Schamber2 , A.M. ASCE and Robert C. MacArthur3, A.N. ASCE

ABSTRACT

In this paper transient, one-dimensional model for dynamic floodrouting of mud flows is presented. The governing equations of mass andmomentum conservation incorporate laminar flow resistance effects andutilize a power law expression to represent the cross-sectional geometryof the channel. The equations are solved by the method ofcharacteristics on fixed time lines and program execution is performedon a micro-computer. Numerical results are compared with publishedexperimental data for a laminar flow, dambreak problem of a viscous oil

" INTRODUCTION

During the spring of 1983, widespread landslides and debris flowscaused an estimated 250 million dollars in damage in the state of Utah.Along a thirty-mile length of the Wasatch Front Mountains, over ninetysignificant landslides and debris flows sent torrents of mud, debris andwater down steep canyons onto residential areas located on alluvial fansat the base of the mountains.

The ability to model these types of events is clearly needed andwill be useful in preparing maps which delineate potential flood damageareas. The purpose of this paper is to present a one-dimensionalmathematical model which can be used to route a mudflow down a confiningchannel. Equations of mass and momentum conservation are presented,with frictional resistance terms, which account for the laminar flow ofa Bingham plastic fluid. The equations are solved by the method ofcharacteristics on fixed time lines. To verify the model, comparison ismade with experimental results of a laminar flow dambreak prnhlem.

GOVERNING EQUATIONS

The flow is governed by the equations of mass and momentumconservation which are given respectively by [6]

1Presented at the ASCE Hydraulic Division Specialty Conference,Hydraulics and Hydrology in the Small Computer Age, Orlando, Florida,

2Associate Professor, Civil Engineering Department, University of Utah,Salt Lake City, Utah 84112

3Research Hydraulic Engineer, U. S. Amy Corps of Engineers, TheHydrologic Engineering Center, 609 Second Street, Davis, California95616

A + VB +B ' + VAY= 0 (1)

aV +V A g(S -sf) (2)

in which x = coordinate along the channel; t = time; A = cross-sectionalarea of flow; V = average velocity; B = channel top width; y = flowdepth; AY = rate of change of area with x for a constant depth(nonprisiatic term); g - gravitational constant; So = slope of thechannel bottom; and Sf - resistance slope.

In most hydraulic applications, the flow is turbulent and Sf isgenerally given by Manning's equation. The flow of mud presents anentirely different situation. DeLeon and Jeppson [1] summarize the datafrom a number of debris flows, mud flows and pipe sludge flows andconclude that the flow is usually laminar. By fitting a line through anumber of data points, these authors postulate a power law relationbetween the Chezy coefficient and the flow Reynolds number. Jeyapalanet al. [2], in their analysis of mine tailing dam failures, develop anexpression for Sf by analyzing the laminar flow of materials withBingham plastic fluid characteristics. Other researchers, [1,5] havenoted a similar behavior for mud flows, which often exhibit plug likeflow with a critical yield stress. In this work, the resistance termfor a Bingham plastic fluid is adopted [2]. Mathematically,

2npVh 2 t hSf :+_3 (3

y y2R2 yyR (3)

in which np= plastic viscosity; Y = unit weight of the fluid; Ty =

yield stress of the fluid; h = hydraulic depth; and R - hydraulicradius. The first term on the right hand side of Eq. 3 is similar inform to the expression postulated by DeLeon and Jeppson [1].

Equations I and 2 are hyperbolic in nature and have the propertythat, through linear combination, they can be reduced to equationsinvolving differentiation in one less direction than the originalequations [6]. This characteristic form for Eqs. I and 2 is given by

w c vAY (V +c)Jy-2 - Bdd (V + ) = g(S o Sf) T -E .A ; V c y a2 -c dn (4)

0 f t A xr 0 ~2 (4)

dx V v+ c (5)dt

in which c = celerity of an elementary gravity wave is given by

c ((6)B

2

and w = Escoffier stage variable is given by

W = .1dn (7)

Eqs. 4 comprise a forward (+) and a backward (-) characteristic equationvalid on the curves in the x-t plane defined by Eqs. 5, respectively.

A power law expression is used to represent the top width and areain Eqs. 4 and 5. Mathematically,

B = (kL + kR) ym (8)

A = ( ) y+l (9)m+1

Here kL and kR define the left and right width at any depth y and theexponent m defines the shape of the cross-section. The parameters kL,kR and m can be specified functions of distance x to capture thenonprismatic nature of the channel. Using the definitions of Eqs. 8 and9, Eqs. 6 and 7 reduce to

C gy (10)m + 1)

w 2[g(m + 1)y]" (11)

NUMERICAL SOLUTION

The numerical solution of Eqs. 4 and 5 is developed with referenceto Fig. 1. At a sequence of points xk, k = 1,2,. ,n, at some time ti,the solution is known. It Is desired to find the solution for thepoints xk on time line t 1 , an interval 6t later. The characteristic

t

PXo°I.tiIX In

cession For

k-I L kI IiIS GRA&I- L W R WIC TAB

announced 0*X stif lcat on _

Fig. 1 - Characteristics Computational Scheme. stributiton/

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low

a cr, . a' a s.aT p s: _ . at. l.in--: . r --a : ,-A r - . . .a 4 - J. , - -- , - --- -.. -...

curves in Fig. 1, i.e., L-P and R-P are approximated by parabolas in thex-t plane. With this approximation, the finite difference form of Eqs.5 is given by

XP XL A (Vt + cL ) + Xfp(Vp + Cp) (12)6t L L r r

P R AR (VR" CR) + P(VP - cp) (13)

6t

in which XL = ;p = AR= The forward and backward version of Eqs. 4are also written in finite difference form. Mathematically,

(Vp w ) - (VL +w1 ) = +1FL APe (4____________________(15)

*(VP -WP) -(VR- R) X= F + X F -(56t

in which

F- g(So Sf( cL dn (16)+c0 Sf) A VA 0 (V 2 C)

' The set of four nonlinear equations, Eqs. 12-15, determines the,* locations of points L and R as well as VP and yp. The variation of YL,

V, yR and VR is determined by parabolic interpolation along time line't- simple search procedure assures that the interpolation nodes

" (Xk_1 , xk, xk,1) always straddle the points in question, so thatextrapolation s avoided.

Eqs. 12-15 are solved iteratively by Newton's method [4]. A first, guess to the solution is found by solving a linear version of Eqs. 12-15

in which - XR = 1 and X = 0. The equations are solved at a numberetween x1 and xn to define the wave profile. At the

*boundaries of the flow domain, if only the velocity or depth isspecified, the remaining unknown is determined by application of theappropriate backward or forward characteristic equation. For the caseof advance on a dry bed, Whitham's assumption is used, i.e., Vn = Vn.

During the early stages of flooding, the effects of boundaryroughness and channel slope are small. A solution which ignoresfriction and slope is therefore used as the initial condition from whichto start the numerical solution.

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RESULTS

The model is compared with several dambreak experiments performedby Jeyapalan et al. [3]. In these experiments, oil is used to simulatea laminar flo Tr-a viscous fluid. The experiments are conducted in a 6foot long glass flume which has a constant width of 1 foot. The dam islocated 4 feet from the downstream edge of the flume giving a reservoirlength of 2 feet. Table I gives the parameters characterizing the

Table I Flume Test Parameters

Test No. H (ft) 0(degrees) y(lb/fts) r(lb sec/ft2)2[0.50 0 56 0.078

I.2 0.50 0 56 0.078

6 0.75 0 56 0.078

7 0.50 0 56 0.156

examples presented herein. The test numbers listed in Table Icorrespond to several of the flood examples presented in [3]. In Table1, HO = depth of oil immediately behind the dam before failure; 1 =bottom slope of the flume; and for all cases Ty - 0.

Results of the numerical simulation are presented in Figs. 2-3 andcompared with the available experimental data. The agreement betweentheory and experiment is generally good. The numerical algorithm isprogrammed in Fortran and executed on an Apple Macintosh micro-computer.Computation times are on the order of 0.25-0.34 seconds per computationalnode.

5i

Test 6 Test 2 Test 744

E3

A2 Experimental Results0 Jeyapalan at al. (3) -

C6- Numerical Results

4 8 12 i6Time (Sec.)

Fig. 2 - Wave Advance.

5m

4 * . . . .

* Experimental eutjoyapalon et of. (3)

Numerical 0.5

Results (F.

IL* 2- 0 - - - 2

Fig. 3 -Wave Profile at tl.95 sec., Test 2.

ACKNWLEDGWNT

Thsresearch is Supported by The Hydrologic Ennerg Cnt,Army Corps of Engineers.Cntr

REFERENCES

1. DeLeon, A. A. , and Jeppson, R. W.. Hdalc n ueiaSoltios f Sead.-State but Spatially Varied Debris Flow," Report,Utah Water Research Laboratory, Logan, Utah, July 1982, 95 pp.2. Jeyapalan, J. K., Duncan, J. M., and Seed, H. B., "Analyses of FlowFailures of Mine Tailings Dams,"1 Journal f Geotechnica nineering,ASCE, Vol. 109, No. 2, Feb. 1983, pp.1011

3. Jeyapalan. J. K., Duncan, J. m., and Seed, H. B.. "Investigation ofFlow Failures of Tailings Dams," Journal f GeotechniaEnagineering, ASCE, Vol. 109 No. 2, Feb.*93 p.12194. Katopodes, N. 0.,1 and Strelkoff, T.0 "Hydrodynamics of BorderIrrigation - Complete Model," Journal of the Irriatio and Dr n g2Lviin SE, Vol. 103, No. 1R3, ep. 97, p.39245. Pierson,, T. C., "Composition and Dynamics of Rudd Canyon Mudf lows,'presentation, Specialty Conference on the Delineation of Landslide,Flash Flood and Debris Flow Hazards in Utah, June 14-15, 1984, UtahState University, Logan, Utah.

6. Strelkoff, T., "Numerical Solution of Saint-Venant Equations,"Journal of the

Di-v i on ASCE Vol -CS

No. v~l

Jan.

1970,-pp. ."ydraulics iii ACo.96 N. Y Jn

6

TECHNICAL PAPERS (TP)

Technical papers are written by the staff of the HEC, sometimes incollaboration with persons from other organizations, for presentationat various conferences, meetings, seminars and other professionalgatherings.

This listing includes publications starting in 1978.

HEC HEC NTISNUMBER TITLE PRICE NUMBER

82.00 Each

TP-52 Potential Use of Digital Computer Ground AD-A106 251Water Models, D. L. Gundlach,Apr 78, 38 pp.

TP-53 Development of Generalized Free Surface AD-A106 252Flow Models Using Finite ElementTechniques, D. M. Gee andR. C. MacArthur, Jul 78, 21 pp.

TP-54 Adjustment of Peak Discharge Rates for AD-A106 253Urbanization, D. L. Gundlach,Sep 78, 7 pp.

TP-55 The Development and Servicing of Spatial AD-A106 254Data Management Techniques in theCorps of Engineers, R. P. Webb andD. W. Davis, Jul 78, 26 pp.

TP-56 Experiences of the Hydrologic Engineering AD-A106 255Center in Maintaining Widely UsedHydrologic and Water ResourceComputer Models, B. S. Eichert,Nov 78, 16 pp.

TP-57 Flood Damage Assessments Using Spatial AD-A106 256Data Management Techniques, D. W. Davisand R. P. Webb, May 78, 27 pp.

TP-58 A Model for Evaluating Runoff-Quality in AD-A106 257Metropolitan Master Planning,L. A. Roesner, H. M. Nichandros,R. P. Shubinski, A. D. Feldman,J. W. Abbott, and A. 0. Friedland,Apr 72, 81 pp.

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TECHNICAL PAPERS (TP)(Continued)

HC HEC NTISNUMBR TITLE PRICK NUMBER

$2.00 Each

TP-59 Testing of Several Runoff Models on an AD-A106 258

Urban Watershed, J. Abbott,Oct 78, 53 pp.

* TP-60 Operational Simulation of a Reservoir AD-A106 259System with Pumped Storage,G. F. Mclahon, V. R. Bonner andB. S. Eichert, Feb 79, 32 pp.

TP-61 Technical Factors in Small Hydropower AD-A109 757Planning, D. W. Davis, Feb 79,35 pp.

TP-62 Flood Hydrograph and Peak Flow Frequency AD-A109 758Analysis, A. D. Feldman, Mar 79 21 pp.

TP-63 HEC Contribution to Reservoir System AD-A109 759Operation, B. S. Kichert andV. R. Bonner, Aug 79, 28 pp.

TP-64 Determining Peak-Discharge Frequencies in AD-A109 760an Urbanizing Watershed: A Case Study,S. F. Daly and J. C. Peters, Jul 79, 15 pp.

TP-65 Feasibility Analysis in Small Hydropower AD-A109 761Planning, D. W. Davis and B. W. Smith,Aug 79, 20 pp.

TP-66 Reservoir Storage Determination by Computer AD-AI09 762Simulation of Flood Control andConservation Systems, B. S. Kichert,Oct 79, 10 pp.

TP-67 Hydrologic Land Use Classification Using AD-AI09 763LANDSAT, R. J. Cermak, A. D. Feldmanand R. P. Webb, Oct 79, 26 pp.

TP-68 Interactive Nonstructural Flood-Control AD-A109 764Plannng, D. T. Ford, Jun 80, 12 pp.

9

' 2*" , € " € ' , " . . """""€ ,: '""" . .' "". ''"" " ' """"' .. , . . ' ""

TECHNICAL PAPERS (TP)(Continued)

HEC HEC NTIS5 M ER TITLE PRICE NUMBER

$2.00 Each

TP-69 Critical Water Surface by Minimum Specific AD-A951 599

Energy Using the Parabolic Method,B. S. Eichert, 1969, 15 pp.

TP-70 Corps of Engineers Experience with AD-Al09 765Automatic Calibration of aPrecipitation-Runoff Model, D. T. Ford,E. C. Morris, and A. D. Feldman,May 80, 12 pp.

TP-71 Determination of Land Use from Satellite AD-Al09 766Imagery for Input to Hydrologic Models,R. P. Webb, R. Cermak, and A. D. Feldman,Apr 80, 18 pp.

TP-72 Application of the Finite Element Method to AD-A109 767Vertically Stratified Hydrodynamic Flowand Water Quality, R. C. MacArthur andW. R. Norton, May 80, 12 pp.

TP-73 Flood Mitigation Planning Using HEC-SAM, AD-A109 756D. W. Davis, Jun 80, 17 pp.

TP-74 Hydrographs by Single Linear Reservoir AD-Al09 768Model, J. T. Pederson, J. C. Peters,and 0. J. Helweg, May 80, 17 pp.

TP-75 HEC Activities in Reservoir Analysis, AD-A109 769V. R. Bonner, Jun 80, 10 pp.

TP-76 Institutional Support of Water Resource AD-A109 770Models, J. C. Peters, May 80, 23 pp.

TP-77 Investigation of Soil Conservation Service AD-A109 771Urban Hydrology Techniques,

D. G. Altman, W. H. Espey, Jr. and

A. D. Feldman, May 80, 14 pp.

TP-78 Potential for Increasing the Output of AD-A109 772Existing Hydroelectric Plants,

D. W. Davis and J. J. Buckley,

Jun 81, 20 pp.

'. o.. " "" ' ".-.'. • "." " .'. . .. .2 '. ..-.'." "- " . .. C.-"- -,' " . - . v-. "- -.-' .. -. , ..

TECHNICAL PAPERS (TP)(Continued)

HEC HEC NTIS

NUMBER TITLE PRICE NUMBER

$2.00 Each

TP-79 Potential Energy and Capacity Gains from AD-A109 787Flood Control Storage Reallocationat Existing U. S. HydropowerReservoirs, B. S. Eichert andV. R. Bonner, Jun 81, 18 pp.

TP-80 Use of Non-Sequential Techniques in the AD-A109 788Analysis of Power Potential at StorageProjects, G. M. Franc, Jun 81, 18 pp.

TP-81 Data Management Systems for Water Resources AD-All4 650Planning, D. W. Davis, Aug 81, 12 pp.

TP-82 The New HEC-1 Flood Hydrograph Package, A. D. AD-A114 360Feldman, P. B. Ely and D. M. Goldman,May 81, 28 pp.

TP-83 River and Reservoir Systems Water Quality AD-AI14 192Modeling Capability, R. G. Willey,Apr 82, 15 pp.

TP-84 Generalized Real-Time Flood Control System AD-A114 359Model, B. S. Eichert and A. F. Pabst,Apr 82, 18 pp.

TP-85 Operation Policy Analysis: Sam Rayburn AD-A123 526Reservoir, D. T. Ford, R. Garlandand C. Sullivan, Oct 81, 16 pp.

TP-86 Training the Practitioner: The Hydrologic AD-A123 568Engineering Center Program,W. K. Johnson, Oct 81, 20 pp.

TP-87 Documentation Needs for Water Resources AD-A123 558Models, W. K. Johnson, Aug 82, 16 pp.

TP-88 Reservoir System Regulation for Water AD-A130 829Quality Control, R.G. Willey,Mar 83, 18 pp.

I_ TP-89 A Software System to Aid in Making Real-Time AD-A138 616Water Control Decisions, A. F. Pabstand J. C. Peters, Sep 83, 17 pp.

" q

TECHNICAL PAPERS (TP)(Continued)

HEC HEC NTISNUMBER TITLE PRICE NUMBER

$2.00 Each

TP-90 Calibration, Verification and Application AD-A135 668of a Two-Dimensional Flow Model,D. N. Gee, Sep 83, 6 pp.

TP-91 HEC Software Development and Support, AD-A139 009B. S. Eichert, Nov 83, 12 pp.

TP-92 Hydrologic Engineering Center AD-A139 010Planning ModelsD. T. Ford and D. W. Davis,Dec 83, 17 pp.

TP-93 Flood Routing Through a Flat, Complex AD-A139 011Floodplain Using A One-DimensionalUnsteady Flow Computer Program,J. C. Peters, Dec 83, 8 pp.

TP-94 Dredged-Material Disposal Management AD-A139 008Model, D. T. Ford, Jan 84, 18 pp.

TP-95 Inflitration and Soil Moisture Redistribution AD-A141 626in HEC-1, A. D. Feldman, Jan 84,

TP-96 The Hydrologic Engineering Center Experience AD-A141 860in Nonstructural Planning, W. K. Johnsonand D. W. Davis, Feb 84, 7 pp.

TP-97 Prediction of the Effects of a Flood Control AD-A141 951Project on a Meandering Stream,D. M. Gee, Mar 84, 12 pp.

TP-98 Evolution in Computer Programs Causes Evolution AD-A145 601in Training Needs: The HydrologicEngineering Center Experience, V. R. Bonner,Jul 84, 20 pp.

TP-99 Reservoir System Analysis for Water Quality, AD-A145 680J. H. Duke, D. J. Smith and R. C. Willey,Aug 84, 27 pp.

) 5

- r. - ... . - . . -. . . " - " . - - - - 7

TECHNICAL PAPERS (TP)(Continued)

, KEC HEC NTIS

NUMBER TITLE PRICE NUEBER

$2.00 Each

TP-100 Probable Maxinum Flood Estimation - Eastern AD-AI46 536United States, P. B. Ely and J. C. Peters,Jun 84, 5 pp.

TP-101 Use of Computer Program HEC-5 For Water AD-A146 535Supply Analysis, R. J. Hayes andBill S. Kichert, Aug 84, 7 pp.

TP-102 Role of Calibration in the Application AD-A149 269of HEC-6, D. Michael Gee, Dec 84,19 pp.

TP-103 Engineering and Economic Considerations A150 154in Formulating Nonstructural Plans,M. W. Burnham, Jan 85, 16 pp.

TP-104 Modeling Water Resources Systems for AD-A154 288Water Quality, R. G. Willey,D. J. Smith and J. H. Duke,Feb 85, 10 pp.

TP-105 Use of a Two-Dimensional Flow Model to AD-A154 287Quantify Aquatic Habitat, D. M. Geeand D. B. Wilcox, Apr 85, 10 pp.

TF-106 Flood-Runoff Forecasting with HECIF, AD-A154 286J. C. Peters and P. B. Ely,May 85, 7 pp.

TP-107 Dredged-Material Disposal System CapacityExpansion, D. T. Ford, Aug 85, 23 pp.

TP-108 Role of Small Computers in Two-DimensionalFlow Modeling, D. M. Gee, Oct 85, 6 pp.

TP-109 One-Dimensional Model For Mud Flows,D. R. Schamber and R. C. MacArthur,Oct 85, 6 pp.

... :. .% . . .-;*% ; .- .: % . . . . .-'' '.G,

UNCLASSIFIEDS;ECURITY CLASSIFICATION OF THIS PAGE (IWhen Data Entered)

REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM

I. REPORT NUMBER 2. GOVT ACCESSION NO. HECIPIENT'S CATALOG NUMBER

Technical Paper No. 109

4. TITLE (and Subtitle) -. TYPE OF REPORT & PERIOD COVERED

One-Dimensional Model For Mud Flows

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(eQ S. CONTRACT OR GRANT NUMBER(s)

Schamber, David R. andMacArthur, Robert C.

9..PRFORMING DRGANIZAJOM NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASKU. . Army ,orps of Engineers AREA & WORK UNIT NUMBERS

Hydrologic Engineering Center609 Second StreetDavie, CA

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October, 198513. NUMBER OF PAGES

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ISI. SUPPLEMENTARY NOTES

Presented at the American Society of Civil Engineers Hydraulic DivisionSpecialty Conference on Hydraulics and Hydrology in the Small Computer

Age, Orlando, Florida, 12-17 August 1985.

19. KEY WORDS (Continue on revere aide if necessary and identify by block number)

Mud and Debris Flows, One-Dimensional Unsteady Flows, Numerical Modeling,Non-Newtonian Fluid Properties, Bingham Fluids, Laminar Flows, High Viscosity -High Solids Concentrations, Model Verification, Method of Characteristics,

v Micro-Computers.

120. ADSINAC (Cinthw - ,rwmers N neeoaim tdewtfy by block number)

In this paper a transient, one-dimensional model for dynamic flood routing of* mud flows is presented. The governing equations of mass and momentum

conservation incorporate laminar flow resistance effects and utilize a powerlaw expression to represent the cross-sectional geometry of the channel. Theequations are solved by the method of characteristics on fixed time lines andprogram execution is performed on a micro-computer. Numerical results arecompared with published experimental data for a laminar flow, dambreak problem"of a viscous ol.

AO I 1473 gD9ro oP, or SImO.ETE UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAr.E (Whon Dte Fnteted)

FILMED

11-85

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