ML20155D585
| ML20155D585 | |
| Person / Time | |
|---|---|
| Issue date: | 09/30/1988 |
| From: | Codell R NRC OFFICE OF NUCLEAR MATERIAL SAFETY & SAFEGUARDS (NMSS) |
| To: | |
| References | |
| NUREG-1263, NUDOCS 8810110307 | |
| Download: ML20155D585 (102) | |
Text
NU REG-1263 l
M Hydrologic Design for Riprap On Embankment Slopes E
a 5
5 o.
ae g
U.S. Nuclear Regulatory Commission f
Office of Nuclear Material Safety and Safeguards Technical Review Branch a
a f
R. B. Codell
?t i
e' "' G v, c x.
4
- r s
3 e
S 0b10110307 000930 NUREO PDR
]R6'J
NOTICE Availability of Reference Materials Cited in NRC Publications Most documents cited in NRC publications will be available from one of the following sources:
- 1. The NRC Public Document Room,1717 H Street, N.W.
Washington, DC 20555
- 2. The Superintendent of Documents, U.S. Government Printing Office, Post Office Box 37082, Washington, DC 20013 7082
- 3. The National Technical information Service, Springfield, VA 22161 Although the listing that follows represents the majority of documents cited in NRC publications, it is at intended to be exhaustive.
Referenced documents available for inspection and copying for a fee from the NRC Public Docu-ment Room include NRC correspondence and internal NRC memoranda; NRC Office of Inspection and Enforcement bulletins, circulars, information notices, bspection and investigation notices; Licensee Event Reports; vendor reports and correspondence; Commission papers; and applicant and licensee documents and correspondence.
The following documents in the NUREG series are available for purchase from the GPO Sales Program: formal NRC staff and contractor reports, NRC sporsored conference proceedings, and NRC booklets and brochures. Also available are Regt.fatory Guides, NRC regulations in the Code of Federal Regul1tions, and Nuclear Regulatory Commission Issuance.t Documents available from the National Technical information Service include NUREG series reports and technical reports prepared by other federal agencies and reports prepared by the Atomic Energy Commission, forerunner agency to the Nuclear Rcgulatory Commission.
Documents available from public and special technical libraries include all open literature items, such as books, journal and periodical articles, and transactions. Federal Register notices, federal and state legislation, and congressional reports can usually be obtained from these libraries.
{
Documents such as theses, dissertations, foreign reports and translations, and non NRC conference proceedings are available for purchase from the organization sponsoring the publication cited.
Single copies of NRC draft reports are available free, to the extent of supply, upon written request l
to the Division of Information Support Services, Distribution Section, U.S. Nuclear Regulatory i
Conimission, Washington, DC 20555.
Copies of industry codes and standards used in a substantive manner in the NRC regulatory process are maintained at the NRC Library, 7920 Norfolk Avenue, Bethesda, Maryland, and are available there for reference use by the public. Codes and standards are usually copyrighted and may be purchased from the originating organization or, if they are American National Standards, from the American National Standards institute,1430 Broadway, New York, NY 10018.
I
i NUREG-1263 i
Hydrologic Design for Riprap On Embankment Slopes Manuscript Completed: May 1988 Date Published: September 1988 R. B. Codell Office of Nuclear Material Safety and Safeguards U.S. Nuclear Regulatory Commission Washington, DC 20555 a
s....,..-
,. - -,,. -, -,. ~.. - - -. --- -. -. - -. -, - - - - - -,
., - - - ~.,. _, -
ABSTRACT Waste impoundments for uranium tailings and other hatardous substances are often protected by compacted earth and clay, covered with a layer of loose rock (rip-rap).
The report outlines procedures that could be followed to design riprap to withstand forces caused by runoff resulting from extreme rainfall directly on the embankment.
The Probable Maximum Precipitation for very small areas is developed from considerations of severe storus of short duration at mid-latitudes.
A two-dimensional finite difference model is then used to calculate the runoff from severe rainfall events.
The procedure takes into account flow both beneath and above the rock layer and approximates the concentration in flow which could be caused by a non-level or slumped embankment.
The sensitivity to various assumptions, such as the shape and size of the rock, the thickness of the layer, and the shape of the embankment, suggests that peak runoff from an armored slope could be attenuated with proper design.
Frictional relationships for complex flow regimes are developed on the basis of flow through rock-filled dams and in mountain streams.
These relationships are tected against experimental data collected in laboratory flumes; the tests provide excellent results.
The re-sulting runoff is then 'Jsed in either the Stephenson or safety factor method to find the stable rock diameter.
The rock sizes determined by this procedure for a given flow have been compared with data on the failure of rock layers in ex-perimental fiumes, again with excellent results.
Comouter programs are included for implementing the method.
NUREG-1263 iii
TABLE OF CONTENTS Page ABSTRACT...............................................................
iii NOTATIONS.............................................................
ix 1 INTRODUCTION........................................................
1-1 1.1 Need for Protection of Mill Tailings Embankments...............
1-1 1.2 Hydrologic Design..............................................
1-1 2 RUN0FF MODEL........................................................
2-1 2.1 Flow Equations.................................................
2-1 2.2 Resistance to Flow.............................................
2-2 2.2.1 Flow Confined to Riprap Layer...........................
2-2 2.2.2 Flow Over Top of Rock...................................
2-5 2.2.3 Definition of the Effective Top-of-Rock Datum...........
2-5 2.2.4 Rating Curve............................................
2-6 2.2.5 Comparison of Model and Data............................
2-7 2.2.5.1 Flow Below the Top of Rock.....................
2-8 2.2.5.2 Combined Flow Relationships....................
2-15 2.3 Numerical Solution.............................................
2-15 2.4 Precipitation Model............................................
2-23 2.5 Model Results and Sensitivity Experiments......................
2-25 2.5.1 Benchmark Case..........................................
2-25 2.5.2 Flow Concentration......................................
2-28 2.5.2.1 Embankment Slumping............................
2-29 2.5.2.2 Reduced Conveyance.............................
2-30 2.5.2.3 Peak Intensity of Precipitation................
2-31 2.5.2.4 Infilling of Ro:k..............................
2-32 2.6 Conclusions....................................................
2-33 3 OETERMINING THE SIZE OF THE RIPRAP..................................
3-1 3.1 Introduction...................................................
3-1 3.2 Safety Factor Method...........................................
3-1 3.3 Stephenson's Method............................................
3-2 3.4 Discussion.....................................................
3-3 3.5 Example Calculations for Rock Armor...........................
3-5 NUREG-1263 v
J TABLE OF CONTENTS (Continued)
.P,,a gg 4 CONCLUSIONS.........................................................
4-1 5 REFERENCES..........................................................
5-1 APPENDIX A USER'S GUIDE FOR SLOPE 2D APPENDIX B PROGRAM ROCKSIZE FIGURES 2-1 Tailings embankment in profile....................................
2-2 2-2 Cross section of embankment.......................................
2-3 2-3 Rating curve example..............................................
2-8 2-4 Diagram of outdoor f1ume..........................................
2-9 2-5 Diagram of indoor f1ume...........................................
2-10 2-6 Correlation of calculated and measured discharge for flow below surface of riprap........................................
2-14 2-7 Velocity in 56-mm riprap layer vs.
stage..........................
2-16 2-8 Stage vs. discharge for 26-mm riprap..............................
2-17 2-9 Stage vs. discharge for 56-mm riprap..............................
2-18 2-10 Stage vs. discharge for 104-mm riprap.............................
2-19 2-11 Stage vs. discharge for 130-mm riprap.............................
2-20 2-12 Stage vs. discharge for 157-mm riprap.............................
2-21 2-13 Finite-difference grid............................................
2-22 2-14 Emba n kme n t f a il u re s cena ri o s......................................
2-26 2-15 Transient runoff for benchmark case...............................
2-27 2-16 Flow concentration for steady rate of precipitation...............
2-29 2-17 Transient runoff for 1/2% inward s1 ump............................
2-30 2-18 Runof f reduction on thick armored embankment......................
2-31 3-1 Angle of repose for typical rock armor...........................
3-3 3-2 Sample problem - interactive session for rock size on side slope with program ROCKSIZE..........................................
3-6 3-3 Sample problem - interactive session for rock size on top slope with program ROCKSIZE..........................................
3-7 A-1 Four quadrants of armored embankment.............................
A-2 A-2 Finite-difference grid for example problem.......................
A-4 A-3 Gradation of rock armor for example problem......................
A-5 A-4 Inputs to computer program SLOPE 2D...............................
A-7 (a) Benchmark embankment.........................................
A-7 (b) Changes for 1/2% slump.......................................
A-8 (c) Changes for trench case......................................
A-9 A-5 Outputs from computer program SLOPE 20............................
A-10 (a) Benchmark case...............................................
A-10 (b) 1/2% inward slump case.......................................
A-12 (c) 200-ft-wide, 1% inward slump trench..........................
A-13 NUREG-1263 vi
TABLE OF CONTENTS (Continued)
FIGURES (Continued)
Page A-6 Listing of program SLOPE 20.......................................
A-17 B-1 Listing of program ROCKSIZE......................................
B-2 TABLES l
2-1 Properties of riprap and filter rock.............................
2-4 j
2-2 Da ta s umma ry fo r outdoo r f i ume...................................
2-11 j
2-3 Data summary for indoor flume....................................
2-12 1
2-4 Effective porosities back-calculated from measured interstitial velocities.....................................................
2-14 2-5 Spline curve for rainfall intensity vs. duration.................
2-24 2-6 Rainfall rate for Probable Maximum Precipitation.................
2-24 2-7 Inputs for sample design of stable rock..........................
2-25 2-8 Summary of model experiments.....................................
2-26 3-1 Modeled and measured flowrates for riprap failure................
3-4 A-1 Calculation of harmonic mean rock diameters.......................
A-6 l
l l
l l
l NUREG-1263 vii
l HYDROLOGIC DESIGN FOR RIPRAP ON EMBANKMENT SLOPES 1
INTRODUCTION 1.1 Need for Protection of Mill Tailings Embankments The long-term storage of hazardous waste, radioactive waste, and uranium mill tailings presents a unique challenge to engineers.
Unprotected impoundments can pose a significant long-term risk to nearby inhabitants and the environment.
The engineering designs should provide overall site stability with little or no maintenance, and should not place an undue burden on future generations (EPA, 1985).
One means of providing long-term stabilization of a waste impoundment is to place a protective filter blanket and a layer of loose rock (riprap) over a thick earth cover.
Typical embankments for uranium mill tailings have surface areas of a few acres, with gentle top slopes (0-2% grade) and steep side slopos (10-20% grade).
The tailings are generally covered with a 6-to 8-foot layer of silt and clay, topped with about 1 to 2 feet of rock armor.
The riprap de-sign must be conservative enough to ensure cover stabilization, yet be economi-cally attractive.
Rock armor is of ten the most suitable protection for large embankments, especially in arid climates where other means of slope stabiliza-tion such as vegetation may be impractical.
1.2 Hydrologic Design The rock armor must, among other things, withstand the runoff caused by intense rainfall directly on the embankments.
Design procedures haea been establishn for stabilizing embankment toes and side slopes of channels fron, arosive fuces; little attention, however, has been devoted to situations in which the~ direct runoff from intense rainfall flows through and/or overtops the riprap.
Models exist for calculating overland flow on hillsides (e.g., Morris 1980), but no models have been found that deal explicitly with the routing of runoff water on armored embankments.
Overland flow models for runoff would not be very useful for computing velocity and depth on armored embankments, because the rock layer has a large capacity to store rainfall temporarily in its void space.
In addi-tion, runoff on armored embankments differs from typical overland flow, because substantial turbulent flow can occur beneath the surface of the rock layer at low runoff, and can also occur both above and below the surface at high runoff.
Furthermore, other than work on rubble and rock-filled dams (e.g., Stephenson 1979, Olivier, 1967), relatively little attention has been paid to the stability of rock armor for overtopping conditions with flow down an embankment.
A methodology to design riprap for embankments depends on the relationship of the forces exerted by the water on the rock (e.g., traction, uplift, overturn-ing, buoyancy) and the resistance to movement of the rock (e.g., gravity).
Fluce tests can measure directly the stability of a particular rock armor layer to a range of flows for relatively simple geometries.
The fundamental relation-NUREG-1263 1-1
ships of +he forces at play must be understood, however, to extrapolate labora-tory measure >nents to other more complicated configurations of armor with dif-ferent rock sizes and properties.
An understanding of the frictional and con-veyance relationships for flow through and over the riprap is a prerequisite of a complete routing model capable of simulating the velocity and depth of water on armored embankments of complex geometry resulting from intense rainfall.
This report outlines procedures that could be followed to design riprap for the protection of uranium mill tailings.
The principles of runoff calculations for armored embankments will be derived.
Methods will then be presented that can solve the equations for flow on the armored embankments.
These principles are formalized into a computer program that solves the finite difference equa-tions for flow.
The proper choice of the Probable Maximum Precipitation onto the embankment will be covered.
These techniques will allow the computation of the maximum credible flow over the embankment.
As the last step, procedures will be described that allow the characteristics of the riprap to be chosen on the basis of rates calculated.
Procedures are supported with experimental results wherever possible.
Relationships for flow resistance are based on previously published studies for flow through rock layers and in gravel beds and mountain rivers, and are compared with data collected in experimental fiumes at the Colorado State University (CSU).
Similarly, the CSU fiume data on the stability of rock armor are compared with published methods used for designing riprap.
s i
NUREG-1263 1-2
2 RUN0FF MODEL This chapter details the basic relationships for runoff from armored embank-ments. The constitutive relationships for flow will be developed first, fol-lowed by a development for flow resistance on armored embankments and a model for severe precipitation.
The flow resistance model is compared with data col-1ected in experimental flumes at Colorado State University (Abt et al., 1987).
2.1 Flow Equations Rain falling on an armored embankment will flow downhill, except for the frac-tion infiltrating the ground, which, for the present case, can be neglected.
Referring to Figure 2-1, the flow of water on the embankment may be described for a two-dimensional case by a macroscopic mass balance and the kinematic ap-proximation of the energy balance (Overton, 1976).
The kinematic approximation neglects acceleration, which can be shown to be small, and balances friction versus hydraulic gradiant only.
The kinematic equations for runoff are stated:
a((U),a((V), btaj = R (2-1) 8x By Oh +
gn3<d
~b
=0 (2-2) x 0[y+KVU 0
+ V4 -S
=0 (2-3) y where ( = water depth above an impermeable layer V = flux of water across the embankment V = flux of water down the embankment n = rock void porosity t = time R = rainfall rate 0 = a factor used to adjust the surface gradient (see Section 2.2.3) g = acceleration of gravity K = friction factor
<d> = representative rock diameter S = the slopes across the embankment x
S = the slope down the embankment y
The upper end of the top slope is assummed to be a no-flow boundary:
0 V = 0 where
= 0 (2-4)
NUREG-1263 2-1
p cot I Figure 2-1 Tailings embankment in profile The water level is continuous across the break between the top and side slopes.
Free slip and no flow are assumed at the lateral boundaries.
The flow boundary condition at the base of the lower slope considers that the depth of the water layer is determined only by the balance between friction and gravity:
b _ bl K I h 9
g<d>S (2-5) y where qb = the discharge across the downstream boundary 2.2 Resistance to Flow Host armored embankments employ a filter layer beneath the riprap layer.
Run-off from the slope could be conveyed in the filter layer, riprap layer, and over the top of the rock.
The filter generally will be a shallow layer of rela-tively small, well graded rock.
Criteria for filter design are covered in Sherard (1963).
Flow through the filter will be significantly smaller than flow in the riprap layer, but not necessarily negligible.
Increased conveyance in the rock layers is a generally favorable condition as far as the stability of the riprap layer to overtopping flow is concerned, so neglecting the convey-ance in the filter layer will be a conservative assumption.
2.2.1 Flow Confined to Riprap Layer Typical embankments are covered by one or more rock layers as illustrated in Figure 2-2, e.g., a filter layer and a riprap layer.
Stephenson (1979) has proposed an empirical formula for flux V through each rock layer:
NUREG-1263 2-2
\\
RAINFALL BATE R t
i [
I
'I OytRTOP FLOW TO P '
40 cP g
C 4
I,JD*%
/ / /// //
09 0
/ Do g el
y
~f
\\
V = l SQ"'< d')I g
(2-6)
\\
/
where S = slope g = 9.8 m/sec2
-The dimensionless friction factor K' for the riprap or filter layer is defined (Stephenson,1979):
800 K' = k + g3-(2-7) where k = 1 for smooth marbles k = 2 for rounded gravel k = 4 for crushed rock Re=Reynoldsnumber=#d[Y
~
y v = kinematic viscosity The estimated values of k for rock used in the present study are given in Table 2-1.
t 4
64tl L i
1 l
t 4
1 Figure 2-2 Cross section of embankment l
NUREG-1263 2-3
Y Su!face a
s, 1
1
-.;}'
ggptS9
~~
y
, pi\\t9I 3.
1
,,,,,es se"
(
l l
8
\\
h t
Table 2-1 Properties of riprap and filter rock C4) d o,
- ds4, d
H(3)
H 3
mS' C (1) C (2) n mm mm Shape k
mm mm u
z 26(5) 30 23 1.7 1.3 0.44 76 152 Sub-angular 56(6) 74 50 2.1 1.3 0.45 152 Angular 4
104 135 89 2.2 1.1 0.44 305 Angular 4
130 178 125 1.6 1
0.46 305 Angular 4
157 203 144 1.7 1.1 0.46 305 Angular 4
3.4(7) 5.8 2.7 2.9 1.2 0.3(0) 152 Rounded 2
6.1(9) 14.5 4.5 3.8 0.9 0.3(8) 152 Rounded 2
(1) Coefficient of uniformity = deo/d o.
i d2 (2) Coefficientofgradation=doh*
i (3) Thickness of layer when used as riprap.
(4) Thickness of layer when used as filter.
(5) This rock is also used for filter in 157-mm riprap experiments.
(6) This rock is also used for filter in 104-and 130-mm riprap experiments.
(7) Filter for 26-mm riprap.
(8) Estimated.
(9) Filter layer for 56-mm riprap.
Stephenson suggests that the representative rock diameter <d> should be the harmonic mean diameter:
<d> = dh* N pq (2~0) 1 1-1 d 4
where pg = the fraction of rocks of diameter d4 by mass N = the number of size classifications Stephenson (1979) based his correlations of friction almost exclusively on the median rock diameter dso, however, because the true gradations of the rock were usually not available.
In,the present study, the harmonic mean diameters deter-mined from Equation 2-8 with N = 10 and pg = 0.1 will be used.
It is important to note that the flux predicted by Stephensor,*3 formula is independent of stage.
The veracity of this assumption will be questioned later in Section 2.2.5.1 by comparison to the data collected for the present study.
NUREG-1263 2-4
2.2.2 Flow Over Top of Rock The Darcy-Weisbach friction factor f is used to express the flux for flows that overtop the rock layer:
/ S.1)h 8g V3=l (2-9)
\\')
where Y is the water depth normal to the flow above the effective top-of rock datum as shown in Figure 2-2.
The flux Va is the average flow per unit cross-sectional area (i.e., (m3/sec)/m2].
It is useful to regard V3 as a flux rather than the average velocity in the layer, in order to be consistent with the fluxes through the rock layers.
The conveyance of the embankment increases sharply as the stage exceeds the top of the rock layer.
Flow resistance is a function of depth above the rock sur-face.
Individual rocks protrude above the water surface at low flows, and the relative roughness of the rock layer surface is large.
At greater flow rates, the rocks become increasingly submerged, and the relative roughness decreases.
There does not appear to be a unified theoretical approach to quantifying flow resistance at sites over wide ranges of relative roughness caused by changing discharges (Bathurst, 1985).
Hey (1979) developed an expression for the Darcy-Weisbach friction factor f in gravel bed streams:
1
= 2.03 log (2-10) 3' 3
\\
- *)
Where ds4 is the 84th percentile finer diameter of the riprap determined from a grid and number sampling procedure.
The factor a ranges from about 11.08 to 13.46, and depends on channel geometry.
A value of a = 11.08 is appropriate for wide, flat channels and is used in the present study.
This equation, although not meant to represent flow resistance where rocks are only slightly submerged, has been shown to perform well in thi; situation (Thorne, 1985).
Bathurst (1985) developed an empirical relationship for the Darcy-Weisbach friction factor from data on mountain rivers of 0.4% to 4% slope, and for a wide range of relative roughness:
IO9
+4 d
(2-11) 1 0
8 The Bathurst form of the Darcy-Weisbach friction factor has been adopted for the present model, although the Hey relat onship, equation 2-10, gives nearly identical results.
2.2.3 Definition of the Effective Top-of-Rock Datum In the present series of fiume experiments, the rock layers were dump-placed and leveled to give the appearance of a uniform surface.
The physical top-of-rock NUREG-1263 2-5
datum, z = H i + H, where z is the distance above the bottom of the filter layer 2
normal to the slope, was measured to the approximate height of the top of the rock deterra ced by a flat plate parallel to the slope.
The transition between flow through the rock layer and overtopping flow is indistinct.
It is clear, however, that the relationa ps for flux above the rock layer implicitly assume some flow within the rock layers.
The effective datum for overtopping flow therefore must be beneath the physical top-of-rock datum, as illustrated in Figure 2-2.
For the present model, the effective top-of-rock datum, Y = 0, is defined at a point below the physical top-of-rock datum so that the flux in the riprap layer V2 defined by equation 2-6 equals the flux in the overtopping layer V defined by equation 2-9 at the physical top-of-rock datum z = Hg + H, where 3
2 H and H2 are the thicknesses of the filter and riprap layers, respectively.
iThe intent of this definition is to assure that the runoff is 6 monotonically increasing function of stage.
The stage Y = AH of the physical top-of-rock datum is calculated by setting V2 from equations 2-6 and 2-7 equal to V3 from equations 2-9 and 2-11 and solving iteratively for Y:
(AH \\
fn d
\\l3 1
h2 log
= 5.62 AHK
~4 (2-l2}
2 where K2 = the dimensionless friction factor of the riprap layer dh2 = the harmonic mean diameter of the riprap layer The stage Y used in equations 2-9 and 2-11 is defined therefore in terms of the measured stage 2:
Y = z - (H3 +H2 - AH)
(2-13) 2.2.4 Rating Curve A depth-discharge rating curve can be expressed by following the algorithm:
q = Vgz where z 1 H3 (2-14) q=VH3
+ V (z - H ) where H31 z < (H + Hz)
(2-15) t 2
1 q=VH +VH2 + Va(z - Hi - H ) where z > (H + Hz)
(2-16) t 2
2 where q = flow rate per unit width V
flux in filter layer from equation 2-6
=
V2 = flux in riprap layer from equation 2-6 3
flux in flow in the overtopping layer from equation 2-9 V u Equations 2-14 through 2-16 (hereafter referred to as "Model 1") are a rigorous representation for conveyance on the embankment slopes.
The simulation model, however, employs a slightly different arrangement for flow, in terms of a modified friction factor Ka, hereafter called "Model 2 " and conservatively neglects the conveyance of the filter layer.
For flow over the top of the rock layer, the depth (, becomes a virtual depth; that is, the depth that the water would have to assume if the riprap layer ware NUREG-1263 2-6
i f
water surface for flows that overtop the rock layer.
It is equal to unity if flow is below the level of the rock layer and equal to the porosity n if flow is above the rock layer surface.
Consider, for the time being, only the flow down a slope that is covered by a
~
uniform layer of rock.
The total flow q past a point on the slope is the sum of the flows through the rock layer (q2) and over the rock layer (q3):
q = q2 + qa = V H2 + Va(( - Hz)n (2-17) 2 where V2 = flux in the rock layer V3 = flux over-top of rock H2 = thickness of riprap layer The flux over the top of the rock layer is calculated using the Darcy-Weisbach equation for flow resistance in open channels:
b n fy b
9 h y (2-18) y3,
where Rh = the hydraulic radius f = the Darcy-Weisbach friction factor The hydraulic radius is approximated as the water depth over the top of rock:
Rh = n(( - H )
(2-19) 2 An effective resistance factor K* for the total flow in and over the rock layer can be derived:
<d>g2 K* =
2 (2-20)
H (<d>/K')g + ( - H ' '8(( - H )' h 2
2 2
n
,t f
Equations 2-1, 2-2, and 2-3 are solved with the effective value K* substituted for K' when ( is greater than the rock layer thickness Hg.
Rating curves for flowrate versus water depth at steady state for the example are shown in Figure 2-3.
The much higher carrying ability of the over-top layer is evident from this figure.
2.2.5 Comparison of Model and Data Recognizing a lack of basic information, the Nuclear Regulatory Commission sponsored research at Colorado State University (CSU) to collect data on flow resistance and failure of rock armor. The experiments were conducted in flumes for a variety of rock sizes, layer thicknesses, and sicpes, both with and without a filter layer.
They are described in detail in Abt et al. (1987).
A large concrete outdoor flume, shown in Figure 2-4, was used to simulate a steep (20%) embankment; a smaller indoor flurte with a tiltable bed, shown in Figure 2-5, was used to simulate the flatter (1% to 10%) top slopes.
NUREG-1263 2-7 1
1 f i l
t 1
(pv(
b 9
Figure 2-3 Rating curve example t
i Nominal median stone sizes (dso) tested were 26, 56, 104, 130, and 157 mm (1, 2, 4, 5, and 6 inches) in diareter.
Riprap was obtained from a limestono quarry near Denver, Colorado, except for the smallest rock, which was crushed alluvial gravel.
The 26-mm riprap was also used for filter material for the 157-mm riprap; the 56-mm rock was used as the filter material for the 104-and 130-mm riprap.
Filter rock for the 26-mm and 157-mm riprap was gravel with d o of I
3 3.4 and 6.1 mm, respectively.
l Armor and riprap were dump placed in the fiumes, and leveled manually to form l
a level surface.
The thicknesses of the rock layers were measured by means of a flat plate to approximately the top of the largest rocks in the layer.
Riprap and filter layer properties are summarized in Table 2 1.
The porosity values of the filter rock for the 26-and 56-mm riprap have not been measured, but were l
estimated to be 30%, because of their wide gradations, i
The flow resistance formulas proposed for the present simulation models are compared in the following sections with the data collected from the CSU fiumes.
Data from the CSU studies are summarized in Tables 2-2 and 2-3.
[
2.2.5.1 Flow Below the Top of Rock
[
t Flows above the surface of the armor layer were shown by means of model experi-ments to occur only when the conveyance of the rock layer was exceeded.
The
{
l NUREG-1263 2-8
,,-,-,4-
I p ft Layer Thickness 1.2 1.5 f t Layer Thickness 1
/
//
1.0 e
A *'
s V
k 0.8 sf s
\\*
/
~
s 0.6
=
i 0.4
?
po9' 66 9
infinite Laye] cgs,,,
0.2 l
Top Slope i
e i
0 O
1.0 2.0 3.0 TRUE WATER LEVEL, ft i
l
(
L
<3 cw
\\
4 o
e c
u Oo O
O E
?w
.ha 4
eN boG LA.
4 NUREG-1263 2-9
A Stilling Basin B
\\
/
ll ll B
d) s ' le s 3.7 m 6.1 m C
3 13.9Fa.Y f.r?'?hs.. b -:ol %. ?. M u
/
N ff o
AM 30.5 m 9.1 m 15.2 m
=
=
=
=
=
=
PLAN C
/ amera Tower
/ eadwall / ee Detail C-C S
H Sediment Basin m
/
{
M-T ^'&
2.4 m m
u
. Embankm%ent "15 m p'
.,,.o.,
o l;
I
=l 1.5 m 4.6 m 7.6 m g
SECTION B-B Riprap
. j,g.
%]oP.,kh, '
T Baffle \\
q--
Filter Geo-Fabn. /
.- o i
c 5
M 0.15 m i
i Sand Piezometer SECTION A-A DETAIL C-C
- C L
r:
7
Y lu$
/
Figure 2-5 Diagram of indoor fiume velocities of water estimated from the model and the occurrence of flow concen-trations were strongly dependent on the conveyance below the surface of the rock.
The stage-discharge relationship represented by equations 2-14, 2-15, and 2-16 (Model 1) are compared in Figure 2-6 to the measured values for those cases in which the stage is below the top surface of the riprap, in terms of the dimen-sionless flowrate qi=
Q W(H + Hz)(gSd )
2 as is the flow rate where W is the width of the flume.
This comparison demon-strates that Model 1 generally overestimates flow for a given stage, especially for the smaller rock sizes.
The lack of agreement is not r.ecessarily surprising, since there was considerable scatter in the correlations performed by Stephenson (1979).
Discussion Direct measurements of interstitial velocity in the riprap layer by the tracer injection technique have been analyzed in order to understand the frictional relationships more thoroughly.
Friction depends approximately on the square of i
the velocity through the tortuous paths around the rocks.
Stephenson suggests l
that the interstitial velocity of water through the rock layer should be based on the flux divided by the porosity V/n.
Direct measurements of the movement of the salt water tracer down the fiume however, indicate that the water is moving faster than V/n.
Measurement of tracer velocity is analogous to the situation commonly encountered in the transport of dissolved tracers in groundwater.
The speed at which an inert tracer is transported through a porous medium is related to the flux divided by an effective porosity" n,, which takes into account the fact that not all of l
the voids that can be measured in the mediur are likely to carry flow.
Table 2-4 shows the values of porosity determined as the ratio of rock volume to total volume and effective porosity (back-calculated from measurements of tracer l
i NUREG-1263 2-10 l
t I
t
e s
t p
u%)v p
a a
m G
rT Ps(
o' p
a r
,l g
t p
e
_~
iR k
n n
t a
o s
l it e
B m
c T
e r
j i
e 2
S t
l 5
t F
1 se
\\
T A
1 n
1 m 'i o
e t
s e
1 n
r m
6 a
c t
r n
,y o
e e
s C
c m
n s
i o
e d
f r
i i
)
n r
t p
v a
O c
p S
e u
r
/
S S
k n
c t
I.
m n
e i
v
\\h e
n T
a 4
m W
N m
4 p
r 2
o j'-
n 6
m
\\
l t
e v
r 7
e e
D s
M u
w f
s f
o D
o fk i
lF c
/
4 o
EE!
R
\\
\\
\\
i il 1
J llI1
Table 2 2 Data summary for outdoor fiume II)
- dso, q, liters /
- StageIII, Av. velocity I3)
Run number am Slope see am am/sec Station 3W 104 0.2 122.9 0
219 22-24 t
3W 104 0.2 145.8 18 221 22-24 I
3W 104 0.2 226.5 41 235 22-24 F
3W 104 0.2 75.9
-117 129 22-24 4W 104 0.2 120.1 0
296 35-37 4W 104 0.2 150.4 41 280 35 37 4W 104 0.2 229.9 72 351 35-37 4W 104 0.2 73.1 99 238 35-37 6W 104 0.2 291.4 27 35 37 6W 104 0.2 368.1 37 35-37 6W 104 0.2 455.9 46 35-37 6W 104 0.2 496.1 55 35-37 I
N 104 0.2 399.3 34 35-37 N
104 0.2 531.5 56 35-37 N
104 0.2 584.4 55 35-37 I
N 104 0.2 616.7 51 35-37 8W 130 0.2 161.4 0
317 22-24 8W 130 0.2 171.9 18 320 22-24 BW 130 0.2 247.8 51 327 22 24 BW 130 0.2 323.1 76 376 22 24 8W 130 0.2 82.1
-152 381 22-24 BW 130 0.2 168.8 0
263 35-37 BW 130 0.2 204.4 25 268 35 37 8W 130 0.2 275.8 64 564 35-37 l
8W 130 0.2 86.6
-152 256 35-37 8W 130 0.2 247.8 24 10-12 e
8W 130 0.2 323.1 49 10-12 9W 130 0.2 290.5 35 35 37 9W 130 0.2 356.5 41 35 37 9W 130 0.2 411.4 51 35-37 9W 130 0.2 497.2 60 35-37 9W 130 0.2 547.1 64 35-37 9W 130 0.2 583.6 62 35-37 9W 130 0.2 677.6 72 35 37 9W 130 0.2 741.6 79 35 37 4
11W 157 0.2 222.8 13 22 24 11W 157 0.2 385.1 51 22-24
,i 11W 157 0.2 551.1 80 22 24 12V 157 0.2 353.9 46 22 24 12W 157 0.2 560.7 70 22-24 13W 157 0.2 750.4 100 22 24 13W 157 0.2 795.7 110 22 24 J
13W 157 0.2 880.6 107 22 24 l
1 14W 157 0.2 129.7
-18 343 35 37 l
14W 157 0.2 176.1 0
460 35 37 14W 157 0.2 246.6 36 464 35 37 l
14W 157 0,2 304.4 53 483 35 J7
'i 14V 157 0.2 376.0 66 564 35 37
[
IN 56 0.2 114 17 35 37 IN 56 0.2 133 22 35 37 IN 56 0.2 151 28 35-37 i
IN 56 0.2 158 30 35-37 18W 56 0.2 76.6 12 22-24 18W 56 0.2 108 19 22 24 18W 56 0.2 126 24 22-24 18W 56 0.2 171 34 22 24 j
t (1) Stage measured above apparent top of rock.
i (2) Average of available velocities measured 38 mm,114 sei, and 190 era below riprap surface.
'i (3) Station a feet downstream from end of diffuser or headwal).
j I
l NUREG-1263 2-11 3
Table 2-3 Data summary for indoor flume d o, q, liters /
Stage (1)
Av. velocity (2) 3 Run number mm Slope sec mm mm/see Station (3) 3I(4) 56 0.02 3.4
-7 A 58 120 31 56 0.02 9.3 72 120 31 56 0.02 17.8 24 7o 120 31 56 0.02 134 76
/8 120 3I 56 0.02 445 150 140 120 41 56 0.01 2.0
-71 27 120 41 56 0.01 6.5 0
46 120 41 56 0.01 17.8 25 49 120 41 56 0.01 106 76 75 120 41 56 0.01 351 152 126 120 61 26 0.01 1.4
-20 30 120 61 26 0.01 3.1 0
30 120 6I 26 0.01 18.7 25 27 120 6I 26 0.01 134 76 120 71 26 0.02 1.1
-36 34 120 7I 26 0.02 3.1 0
40 120
?!
26 0.02 177 76 120 71 26 0.02 385 122 120 81 26 0.02 0.6
-36 27 120 8I 26 0.02 2.3 0
34 120 81 26 0.02 18.7 25 27 120 81 26 0.02 134 71 120 81 26 0.02 268 102 120 81 26 0.02 283 96 120 91 26 0.1 2.8
-44 52 120 91 26 0.1 5.9 0
73 120 91 26 0.1 11.3 24 73 120 91 26 0.1 19.0 33 95 120 91 26 0.1 47.3 42 52 120 101 56 0.1 8.2
-74 101 140-142 101 56 0.1 15.9 0
110 140-142 101 56 0.1 19.3 4.4 113 140-142 l
101 56 0.1 60.0 32 113 140-142 101 56 0.1 142 54 102 140-142 111 56 0.1 283 7.9 148-150 i
111 56 0.1 8.2
-6.1 98 148-150 i
111 56 0.1 15.9 0
101 148-150 i
111 56 0.1 19 10.2 114 149-150 111 56 0.1 60 26.7 114 148-150 111 56 0.1 142 54.6 108 148-150 i
1 56 0.02 697 175 120 l
2 56 0.02 1376 255 120 3
56 0.02 1235 247 120 4
56 0.02 1328 260 120 5
56 0.02 1492 283 120 I
See footnotes at end of table.
1 NUREG-1263 2-12
i 4
1 Table 2-3 (Continued)
II)
Av. velocity (2)
- dso, q, liters /
Stage Run number m
Slope see mm am/sec Station (3) i 5
56 0.02 1602 305 120 6
56 0.02 1007 225 120 9
56 0.02 430 126 120 10 26 0.02 518 142 120 11 26 0.02 251 93 120 12 26 0.02 241 94 120 i
13 26 0.02 340 117 120 l
14 26 0.02 425 132 120 i
15 26 0.02 504 126 120
[
16 26 0.02 340 102 120 17 26 0.02 425 121 120 18 26 0.02 507 130 120 l
19 26 0.01 283 123 120 a
20 26 0.01 340 142 120 1
21 26 0.01 898 244 120 22 26 0.01 977 261 120 i
1 23 26 0.01 1133 285 120 t
24 26 0.01 1218 304 120 l
26 26 0.1 76 24 140-142 t
27 26 0.1 70 34 140-142
[
28 26 0.1 95 43 140-142 e
29 56 0.1 255
- 7. 0 148-150 i
30 56 0.1 283 7.4 148-150 t
31 56 0.1 283 7.9 148-150 t
1, 32 56 0.08 411 10.4 140-142 I
(1) Stage measured above apparent top of rock.
(2) Average of available velocitie, measured 38 mm, 114 mm, and 190 mm below 1
physical top-of-rock surface.
l l
(3) Station = feet downstream from end of diffuser or headwall, i
a (4) The suffix I indicates that the run was set up to measure interstitial velocity.
i j
These runs should not be confused with those without the suffix.
i velocity in the flumes), and the measured stage-discharge relationship for flows beneath the surface of the rock.
The effective porosity is defined:
i q
l "e
M
[
i r
l where q2 = flow through riprap layer V2 = average measured velocity of the tracer In the riprap layer
(
i l
I 1
L j
NUREG-1263 2-13 l
i
l j
u l
cat 6 l
Figure 2-6 Correlation of calculated and measured discharge for flow below surface of riprap Table 2-4 Effective porosities back-calculated from measured interstitial velocities h/sec II) n(2)
"e W, m F
Run 5
H,m 2
61 26 0.01 0.076 0.0031 2.4 0.079 0.03 0.44 0.52 71 26 0.02 0.076 0.0031 2.4 0.103
- 0. 0 /.
0.44 0.38 41 56 0.01 0.152 0.0065 2.4 0.045 0.046 0.45 0.37 31 56 0.02 0.152 0.0093 2.4 0.059 0.072 0.45 0.33 91 26 0.1 0.076 0.0059 f-
? 192 0.073 0.44 0.36 101 56 0.1 0.152 0.0159
.4
- 3. '.96 0.11
- 0.,5 0.36 1
111 56 0.1 0.152 0.159 2i S 96 0.101 0.45 0.30 f.~
4W 104 0.2 0.305 0.12 3.7 0.201 0.296 0.44 0.29 3W 104 0.2 0.305 0.123 3.7 a.201 0.219 0.44 0.4 F.
8W 130 0.2 0.305 0.161 3.7 0.171 0,317 0.46 0.37 8W 130 0.2 0.305 0.169 3.7 0.171 0.263 0.46 0.47 14W 157 0.2 0.305 0.176 3.7 0.11 0.460 0.46 0.30 Average 0.38 (1) F = estimated fraction of flow in filter layer.
(2) Measure directly from pore volumes.
( 3 ) n, =
NUREG-1263 2-14
I lij 03 0
m 8
2 2
2 2
2 2
m 5
5 5
5 5
1 1
1 1
1 0
H r
e 6
lt iF m
2 0
m 4
1 6
6 4
)
3 6
2 2
3 2
h m
d 4
S 2
I m
9 0
m 0
9 I
I 2
5 5
5 31 2
m 6
5 0
0 0
H 7
1 3
3 3
2 f
</
9
+
p H
2 1
a 0
i r
& /+
ip
=
H m
(
R m
0 2
5 0
3 9
6 6
4 0
2 4/
x 1
1 1
0
+
Q 0 /
1
/
8 t
1 o
I 4
/
b 0
m O a +
X f-
/
m S
y
'q 6
1 D
/
V m
I E
0 C
R
/
5 U
ls/
/
S X
4 1
A
/
f m
0 E
/
m M
g 6
/
2 2
1
/ /
yO f
/
0 34
/
- /
/
+
0 1
f 0
/
8 f
0 f
0 o/
y-6 0
0 1
0 70#
0 f
/O/0 l[
40 1
0
/0 4
/
2
/0 0
f 0
f F
O i =
2 0
8 6
4 2
4 2
2 2
1 1
0 0
0 0
0 0
0 0
0 0
0 a
_ ;53S$
e i
l l
l1 Il, i
l
The flowrate q2 in the riprap layer is estimated from the fraction of flow in each layer calculated by equations 2-14, 2-15, and 2-16, because flow in the filter layer was not measured directly.
The fraction of flow through the filter layer is estimated to be up to a third of the total, and generally cannot be neglected.
This estimated correction inserts a possible source of error.
Results of the calculation indicate that the effective porosity is significantly smaller than the measured porosity.
The average value of n, is 0.38, as compared to measured values of n' ranging from 0.44 to 0.46.
Measurements within tha rock layer indicate that velocity may be correlated to stage.
Figure 2-7 demonstrates this apparent relationship between stage and the velocity measured at two levels and stages, for the 56-mm riprap in the indoor flume.
Neither the Stephenson (1979) nor Leps (1973) formulations for interstitial friction indicate that flows confined below the surface of the rock would be dependent on stage.
This phenomenon is a possible explanation for the deviation demonstrated in Figure 2-6 of measured and predicted runoff as stage approaches the surface of the rock layer.
It appears that there may be a vertical velocity gradient established within the rock layer, with the lowest velocities near the bottom and the highest velocities near the surface.
l In overtopping flows, velocities above the top of the rock are appreciably 3
greater than the interstitial velocities, and are also highly sensitive to stage.
a
]
This high velocity boundary condition seems to influence the velocity closer to the surface (38 mm below the surface) than the velocity closer to the bottom (114 mm below surface).
The influence of stage on interstitial velocity is not as evident for flows that do not overtop the riprap.
The conveyance of the rock layer relative to the total flow decreases for increasing stage once the rock is overtopped however, reducing the significance of this potential error on the total conveyance.
2.2.5.2 Combined Flow Relationships
(
The stage-discharge relationships calculated from Model 1 for the complete l
range of flows are shown in Figures 2-8 through 2-12 for the range of rock sizes studied.
Agreement between the model and data is generally excellent.
The results of Model 2 are not shown, but demonstrate higher stage for a given flowrate, especially at low flow and for c.ases in which flow through the filter layer is appreciable, notably the experiments with the 4-and 5-inch-diameter rock.
- 2. 3 Numerical Solution The numerical solution of the two-dimensional model as presently implemented i
employs the "leapfrog" explicit algorithm (Roach, 1972).
Tt.is method was chosen because it was easily programmed, and appeared to give acceptable results.
The staggered finite difference grid employed for the two-dimensional model is illustrated in Figure 2-13.
The finite difference grid blocks are square, and of equal size throughcut.
The variables in the finite difference equations are defined on the corners of the grid blocks as shown in Figure 2-13.
NUREG-1263 2-15
cet 1 Figure 2-7 Velocity in 56-mm riprap layer vs. stage Ihe continuity equation, equation ?-1, is represented in finite difference form:
/n nT Ot l +j " l,j + Rat n1 n
(gi-1,j+l,jjU"i,j2nax 4
i i,
i n
1,j 2nax fn(i,j-1 g1,j 1,j-1 2nax
-[e,i+1,j.gi,j at n
n at n
n yn g
y x
-(6,j-1 + l,j)V t
1 i
i,j 2 x (2-21)
NUREG-1263 2-16
}
h 2.4 2.2
---O Lower Station,114 mm Below Surface t
2.0
+ Upper Station,38 mm Below Surface 4
I i
1.8 l
1.6 1
l S
.4 1
Yl
[8 m
o d
1.2 8
=
e
- l Wi e
1.0 oI i
4l 0
E l
Z
+
9 0.8 l
+
0 e2
/
low 1) o 0.6 O
"o
/,,,, +b i
o Q
0.4
" -O o O
P 0.2 l
l 1
O O
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 L
z-H i DIMENSIONLESS STAGE, H2
.s T b.
u
-i i
I
-l I
i i
I i
t Figure 2-8 Stage vs. disc?,arge for 26-mm riprap l
i The subscripts i and j refer to the locations on the finite-difference grid, i
Figure 2-13.
The superscripts n and n+1 refer to the time level, either nat or l
(n+1)At.
i 1
The relationships for velocity, equations 2-2 and 2-3, are coupled through the ab,otute flux term (U2 + V2)h.
Since the gradient down the embankment is i
I greatt:S than that across the embankment, the flux in the y direction (V) will be 13rg & than the flux in the x direction (U) in these runoff calculations, j
i Ne& G-1263 2-17 i
\\
i>.
,aK-.-...---..,-.-.__....,,,
,_,_.,,-----,,.,--.-c~.e------,-.---,.
v 2.25 Symbol Run Slope Station O
61 0.01 120 I
a 71 0.02 120 t
o 2.W
+
si 0.02 120 a
X 91 0.1 1?0 O
t 18 0.02 120
,s v
to 24 0.01 120 j
1.75 O
26-28 0.1 140 142 J
8 1.50
% [
Eq.7 o
N g
= = === = E q. 6, a = 11.08 0
4 1.25 X
0 M
x M
X b
1 z
1.00 - - - - - +
0 Physical top of rock M
o ZW 4
I 5
0.75 0.50 du. filter layer - 3.4 mm Hi = 152 m m dg. riprap layer = 26 mm H - 76 m m l
'I 0
10 2 10'1 10 10 0
1 4
DIMENSIONLESS DISCHARGE, (H + H )(gSdh2)%
i 2
i i
I 1
Figure 2-9 Stage vs. discharge for 56-mm riprap Therefore, equation 2-2 is solved for V, using the U and V fluxes from the previous timestep in a correction factor:
Vg=
SGN(S)
(2-22)
NUREG-1263 2-18
Symbol Run Slope Station o
41 0.01 120 2.25 a
31 0.02 120
/
+
17 0.02 120
/
o X
32 0.08 140 142
, ' +
2.00 0
101 0.1 140 142 sI+
9 111 0.1 148 150 D
29 31 0.1 148-150
,# +
1.75 17w 0.2 36 3/ (outdoor fiume)
,/
e 18W 0.2 22 24 (outdoor fiume)
/
l+
l'N
~
Eq.7 I $
= = ---- E q. 6. a = 11.08 W
C 1.25 U
Ph j
1.00 _ _ ysical top of rock,
9 mZ W
v E
0.75 0#
O dg.
filter layer = 0.1 mm 0.50 Hi = 152 mm dg. riprap layer = 54 mm H2 = 152 mm 0.25 0
0 1
10 2 10'l 10 10 4
DIMENSIONLESS DISCHARGE, (H + H )(gSd i
2 h2Ih
.4 J
(/ ',
10 Figure g.3n age ys, dischar# IOP 'han pf 7 rap where ci,
y (yxy),' g (2-23)
SGN = t'he sign or 3 U #'
'A l'1,1
- U"s,3 3 + u,_
(g.g4y n
NUREG-1263 2-19
1.4 Symbol Run Slope Station O
6W 0.2 36-37 6
7W 0.2 36 37
+
3W 0.2 22 24 1.3 X
4W 0.2 36 37 Eq.7
-- ~ ~ - E q. 6. a = 11.08 X
a O ok
-1 2 N
1.1 X
+
uJ o
O aC
+
Physical top of rock 1.0 mm Nz O
E 0.9 Zw E
5 0.8 X
dg.
filter layer = 66 mm
+
Hi = 305 m m du-
'I 'ap layer = 144 mm 0.7 P
H2 = 306 mm 0.6 0.7 0
5x10 2 10'1 10 4
DIMENSIONLESS DISCHAPGE, (H + H )(gSdh2)"
i 2
a 1,
Cut
d I
i Figure 2-11 Stage vs. discharge for 130-mm riprap
\\2 (2-25) y*Y =
u*Y\\2 n
+1 y
( I'd )
((
/
/
I +1i,j+1
- b +1 n
n i,j (2-26) 3 _- 3 y.j ax NUREG-1263 2-20 A
l4
~
Symbol Run Stor e Station O
8W 0.2 22 24 a
SW 0.2 36 37
+
WW 0.2 10 12 1.3 X
9W 0.2 36-37 I"' I 1.2
- --- E q. 6. o = 11.08 0
h [
0 X XX 1.1 g
x+
4 O
Physical top of rock dz 9
mZ 0.9 m
5 0.8 d o. filter layer = 66 mm 5
H, = 305 mm dso. riprep layer = 130 mm 0.7 H2 = 305 mm M
0.6 0.55x10'2 10'1 100 4
DIMENSIONLESS DISCHARGE, (H + H )(gSdh2)"
3 2
'i e.
\\
figure 2-12 Stage vs. discharge for ?.57-m riprap The U fluxes are then solved once all of the y values have been generated:
I.]
on y
~
f z
i+1 f Uj
=
K' < d> d (2-27) 04 NUREG-1263 2-21 t
i I
1.4 Symbol Run Slope Station O
11W 0.2 22 24 a
12W 0.2 22 24
+
13W 0.2 22 24 X
14W 0.2 36 37
$+
1.2 o
= = = = = E q. 6, o = 11.08 3
E A
o X 0 d
1.1 O
X M
M Physical top of rock m
1.0 aZ X
9 M
0.9 I
5 0.8 da. filte-layer - 26 mm Hi = 306 r:.m du. riprep layre = 157 mm H2
- 306 mm 0.6
I
'I i
0.5 5x10 2 10'1 100 4
DIMENSIONLESS DISCHARGE, (H + H )(gSdh2)"
i 2
e
13 l
Figure 2-13 Finite-difference grid l
?Uj'J\\2+ [y Xy 2 h where V =
(2-28)
.(
/
\\
/
y +j, y ++1i 1,j, y +1i,j-1 y -1,J-1 (2-29) n1 n
n n+1 i,
1 yxy 4
Fluxes normal to all borders except the downstream boundary are defined as zero. The gradient a(/8x is zero at the ends of each row vector.
The gradient a(/ay is zero at the end of each column vector.
Normal flow (i.e., gravita-tional forces exactly balance frictional forces) in the +y direction is assumed at the downstream end of each column vector, and is implemented in the finite difference solution as:
At
'I I
~ [E +1,N
- I Nju n
n hn
$n+1
- Rat
- 2nax n
n n
I-1,N*S,NfUi-1,N 1,N n
i i
(i t
i,N g
.N-1
- I.N Y
I (2-30) lN K
.N-1 The maximum flow will occur along the centerline of the embankment.
The verti-cal velocity and stage are'not calculated exolicitly along the centerline, since it is a boundary. The nearest points at which vertical velocity and stage are calculated are ax/2 away.
The centerline flowrate is estimated by assuming the symmetry boundary conditions, DV/ax = 0 and a(/ax = 0 apply, and fitting a NUREG-1263 2-22
______________a
i f
i i
l i
i s
6 (i...,
l
+
+
=
=
I l
t
~
ll jf If I
5 - 1.1 E l-
'a+i I
i
+1 u' "
+
=
=
I i
AX f
if
%l i p p
l 5.j +,
6
=
=
l
=
3x
=
l
\\3 l
i L
I i
(
a p
. l v
parabola to the first two points on its right.
For an equally spaced grid, the relationship for the centerline variables (e.g., q) is:
qi,j " (1 + S)q2.j ' 09,j (2'31) 3 where p = a factor equal to 0.125 2.4 Precipitation Model The rate and duration of precipitation onto the embankment is one of the most important factors determining the runoff, which in turn, determines the design requirements of the armor.
The Probable Maximum Precipitation (PMP) is the most severe precipitation event that can reasonably be expected to occur at the site, and it is this precipitation that is suggested for the design criterion, It is axiomatic that precipitation events that cover a small area of land can i
be very intense, but short lived.
Conversely, precipitation events covering larger areas may be less intense, but ultimately produce greater amounts of rainfall over longer periods of time.
The PHP chosen for a particular site depends on the characteristic time at which the drainage basin responds to a precipitation event.
That is, a large drainage basin would respond much more slowly than would the drainage basin for a small tributary stream.
The PMP for the former would be a storm of long duration, with only moderate rates of rain-fall.
The PHP for the small tributary, however, would be a storm of short du-ration but of intense rates of precipitation.
This time constant is generally called the "time of concentration." The time of concentration for large rivers could be weeks or even months; that of small tributary streams would be hours or fractions of hours.
Typical tabulttions for the PMP give rainfall intensi-ties for periods no shorter than 15 minutes.
The drainage area for typical embankments is generally not more than a few tens of acres.
The time of concentration would be measured in minutes.
There are no widespread estimates for the PMP which are tabulated for cases in which the times of concentration are so small.
Therefore, the rainfall-duration relation-ships for the model have been developed from estimates made by the staff of the U.S. National Weather Service (NWS) for durations less than 15 minutes (Hansen, 1985).
The NWS estimated that the 5-minute duration PMP for the area covered by Hydrometerological Report 49 (HR-49) (NOAA, 1977) was 45 i ST,of the 1-hour PMP.
For durations shorter than 5 minutes, NWS advised that the maximum rainfall rates could be estimated from record rainfall amounts otasured at mid-latitudes on the globe.
NRC therefore used the U.S. record for 1 minute of 1.23 inches, measured at Unionville, Md, on July 4, 1956.
The rainfall-duration curve for durations of 15 minutes to 2 hours2.314815e-5 days <br />5.555556e-4 hours <br />3.306878e-6 weeks <br />7.61e-7 months <br /> was estimated from HR-49.
This report is most suited to the Colorado and Great Basi.) drainages of the western United States, but rainfall-duration relationships for other regions of the United States could be developed along similar lines.
The above estimates have been interpolated by means of a cubic spline.
The spline equation and coefficients are given in Table 2-5.
Standard practice in performing flood estimates dictates that rainfall within the period of the PHP is not temporally correlated; i.e., the rainfall can be arranged any way within the time period of the PHP, as long as the cumulative amounts over the period are the same.
Recognizing that conditions that saturate NUREG-1263 2-23
r i
Table 2-5 Spline curve for rainfall intensity vs. duration j
t, Rangej g
C C
C F
i min min 4,1 i,2 g,3 9
1 0.0 0.0 - 1.0 0.1616205E+00 0.0
-0.7620493E-0 2 0.0 2
1.0 1.0 - 5.0 0.1381590E+00 0.2346148E-01
-0.7820493E-02 0.1538 i
3 5.0 5.0 - 15.0 0.3967789E+01 -0.1158800E-02
-0.7820493E-02 0.45 4
15.0 15.0 - 30.0 0.1923221E-01 -0.8857681E-03
-0.7820493E-02 0.74 5
30.0 30.0 - 45.0 0.4822097E-02 -0.7490628E-04
-0.7820493E-02 0.89 6
45.0 45.0 - 60.0 0.3749401E-02 -0.1460676E-04
-0.7820493E-02 0.95 Equation:
R' =
C x0+C x0+C x0+F I
g,3 i,2 9,3 g
where R' = fraction of 1 hour1.157407e-5 days <br />2.777778e-4 hours <br />1.653439e-6 weeks <br />3.805e-7 months <br /> PMP accumulation D = duration - t, min l
j Standard practice in performing flood estimates dictates that rainfall within the period of the PHP is not temporally correlated; i.e., the rainfall can be arranged any way within the time period of the PHP, as long as the cumulative amounts over the period are the same.
Recognizing that conditions that saturate the rock layers are likely to produce the greatest runoffs,'the design-basis rate of precipitation for embankments was formulated so that there would be an increasing intensity of precipitation, and that the last 2.5 minutes of the first hour would be the most intense.
Total precipitation for the first, hour was 203 mm (8 inches).
Precipitation for the second hour was 1 4 of that foi' the first
[
hour.
A tabulation of rainfall intensities versus time ir given in Table 3-6.
t i
Table 2-6 Rainfall rate for Probabir
[
Maximum Precipitation i
f Multiplier for Multiplier for Time, sec 3-hr rate (1) 1-hr rate (2) 0 - 1800 0.22 0.22 I
1800 - 2700 0.6 0.6 i
2700 - 3000 1.43 1.43 l
3000 - 3300 2.05 2.05 l
3300 - 3450 3.25 5.4
~
3450 - 3600 7.55 5.4 t
3600 3750 1,06 0.753 i
3750 - 3900 0.445 0.753 3900 - 4200 0.286 0.286 4200 - 4500 0.2 0.2 4500 - 5400 0.084 0.084 l
5400 - 7200 0.031 0.031 (1) 2.5-minute minimum duration.
(2) 5-minute minimum duration, j
NUREG-1263 2-24
l l
The sensitivity of the maximum rate of runoff to the choice of the duration of the shortest, most intense segment will be demonstrated in Sectien 2.5.
j 2.5 Model Results and Sensitivity Experiments An example is presented to demonstrate the use of the model for estimating peak I
runoffs.
The modeled embankment is typical of those found at uranium mill tailings sites.
The embankment is assumed to be of triangular shape and sym-i metrical around the vertical centerline, similar to that shown in Figure 2-14a.
It is 213 m (700 feet) long from top to bottom, and 266 m (1200 feet) wide at the base.
The topaportion of the embankment is 134 m (440 feet) long, the r
slope is 2%, and the rock thickness of the layer is 1 foot.
The lower portion of the embankment is 79 m (260 feet) long, the slope is 20%, and the rock layer is 0.46 m (1.5 feet) thick.
The harmonic mean diameters of the rock are 0.305 i
to 0.1 m (0.1 foot), and 0.3 foot for the top and side slopes, respectively.
The de, diameters are 0.1 to 0.23 m (0.32 and 0.75 foot), respectively.
The rock is crushed quarry material, and is assumed to have a friction factor for flow of K = 4.0.
Other properules of the riptap and embankments are given in
(
Table 2-7.
Rainfall intensity for the design is given in Table 2-6.
i I
Table 2-7 Inputs for sample design of stable rock j
Parameter Top slope Side slope
[
Friction index, k 4
4 I
Diameter, d 49 mm (0.16 ft) 143 mm (0.47 ft) h da4 73 mm (0.24 ft) 201 mm (0.66 ft)
H 305 mm (1.0 ft) 457 mm (1.5 ft) j n
0.35 0.35 S
0.02 0.2 q
22.91 liters /sec 22.91 liters /sec (0.81 ft /sec)
(0.81 ft /sec) 3 3
C factor (0.22 for smooth 0.27 0.27 i
rock. 0.27 for crusheo, eq. 3-4)
Angle of repose. 0 40' 41.5' Specific gravity of rock 2.65 gm/cc 2.65 gm/cc Safety factor 1.5 1.5 d o, safety factor method 43.6 mm (0.143 ft) 488 mm (1.6 ft)
[
3 dso, Stephenson method 12.5 mm (0.041 ft) 72.8 mm (0.239 ft)
(after multiplying by factor of 1.5) f 2.5.1 Benchmark Case Runoff per unit width from the toes of the top and side slopes of the satople embankment is shown in Figure 2-15.
These and subsequent results are also summarized in Table 2-8.
In the present case, the top and side slopes are assumed to be unfalled.
Peak flow from the top slope is nearly coincident with the peak precipitation rate.
Runoff from the side slope shows a small disturbance af ter its peak, which is caused by the routing of the peak flow from the top slope.
NUREG-1263 2-25
The sensitivity of the maximum rate of runoff to the choice of the duration of the shortest, most intense segment will be demonstrated in Sec. tion 2.5.
2.5 Model Results and Sensitivity Experiments An example is presented to demonstrate the use of the model for estimating peak runoffs.
The modeled embankment is typical of those found at uranium mill tailings sites.
The embankment is assumed to be of triangular shape and sym-metrical around the vertical centerline, similar to that shown in Figure 2-14a.
It is 213 m (700 feet) long from top to bottom, and 2C6 m (1200 feet) wide at the base.
The top portion of the embankment is 134 m (440 feet) long, the slope is 2%, and the rock thickness of the layer is I foot.
The lower portion of the embankment is 79 m (260 feet) long, the slope is 20%, and the rock layer is 0.46 m (1.5 feet) thick.
The harmonic mean diameters of the rock are 0.305 to 0.1 m (0.1 foot), and 0.3 foot for the top and side slopes, respectively.
The d.
diameters are 0.1 to 0.23 m (0.32 and 0.75 foot), respectively.
The rock is crushed quarry material, and is assumed to have a friction factor for e
flow of K = 4.0.
Other properties of the riprap and embankments are given in Table 2-7.
Rainfall intensity for the design is given in Table 2-6.
Table 2-7 Inputs for sample design of stable rock Parameter Top slope Side slope Friction index, k 4
4 Diameter, d 49 mm (0.16 ft) 143 mm (0.47 ft) h ds4 73 mm (0.24 ft) 201 mm (0.66 ft)
H 305 mm (1.0 ft) 457 mm (1.5 ft) i 0.35 0.35 nn 5
0.02 0.2 q
22.91 liters /sec 22.91 liters /sec 3
8 (0.81 ft /sec)
(0.81 ft /sec)
C factor (0.22 for smooth 0.27 0.27 rock, 0.27 for crushed, eq. 3-4)
Angle of repose. 0 40*
41.5" Specific gravity of rock 2.65 gm/cc 2.65 gm/cc Safety factor 1.5 1.5 dso, safety factor method 43.6 mm (0.143 ft) 488 mm (1.6 ft) d o, Stephenson method 12.5 mm (0.041 ft) 72.8 mm (0.239 ft) 3 (after multiplying by factor of 1.5) 2.5.1 Benchmark Case Runoff per unit width from the toes of the top and side slopes of the sample embankment is shown in Figure 2-15.
These and subsequent results are also su:tnarized in Table 2-8.
In the present case, the top and side slopes are assteed to be unfailed.
Peak flow from the top slope is nearly coincident with the peak precipitation rate.
Runoff from the ride slope shows a small disturbance after its peak, which is caused by the routing of the peak flow from the top slope.
NUREG-1263 2-25 i
g I'l Figure 2-14 Embankment failure scenarios Table 2-8 Summary of model experiments 3
Peak runoff, ft /sec/ft Scenario Top slope Side slope Benchmark, 0% siump 0.27 0.31 Halved d 0.57 0.28 h
Doubled d 0.17 0.39 l
h 1/2 layer thickness 0.56 0.94 1/2% slump 0.81 0.44 l
1% slump 1.65 0.99 Infinite layer, 1% slump 0.19 0.42 l
Filled rock, 1/2 <d> 1ayer thickness 0.52 0.85 5-min minimum duration, benchmark 0.25 0.31 l
l 5 min minimum duration, 1/2% slump 0.69 0.43 l
l l
I NUREG-1263 2-26
-- ~
--e
(,
W s
(
i (a) Benchmark slope (b) Single failure
%, '.,,, ri)
Q'fti
. 'i l
%..s,' h'It s "., k'i,
\\
- N
's p ',,
s s
-x. ',
2 's 's
, 'r N,
i
~^y c
'~
l i,
,i1 I,/
i
~.
. 'y
', 1 r,
' i
/ '4
/,
.1i
/
- 3. 's
', 2~
=
-i a
/ -
i w'
\\
>'\\
4 gi,,,<
)
fc) Multiple failure l
id) Trench failure i'
.s i
e i
I i
[
i 4
~ -, _, _.. _ _.,.. _,. -. _. _. _ _ - _ _ -, - -. _
A
.U M
b I
s.
4 E.
t 1
E S
J 'I e
C*
E 2-N A
EoO E
NUREG-1263 2 27
_ _ _ _ _ _ _ _ _ _ _ _ _ - - - - - - - ' - - - ~ - - - ~ ~ '
/
/
/
/
/
,/
,.=*"'"#
. =
_ _ _ _ _ _ _ _. _ s g
x
\\
\\
\\\\- R
\\
\\
\\
\\- l4
\\
I 3
\\
\\
a
\\- kE
\\
\\
l 1 -
\\
E.
E ae 4
i o.#E l
1 g
I O
i
='
e O
m N
d d
\\$
2;foss/ct)'ddONOH I
(jv
2.5.2 Flow Concentration Flow concentration is a term that describes the preferential flowpaths on the embankments caused by nonuniformities of the embankment profile. The analysis l
of runoff presented above is for flat surfaces with uniform slopes.
Construc-
[
tion practices on the embankment earthworks will prosumably strive to maintain i
flat or crowned surfaces and uniform placement of the rock layer.
Nonuniformity of the embankments, however, could lead to concentration of runoff, causing higher flowrates than would otherwise be predicted.
Conditions that could lead to I
flow concentration include:
i (1) non-uniform slope grading l
(2) uneven placement of rock (3) gu11ying caused by erosion l
(4) slumping of earthwork i
There is evidence that natural slopes often erode when runoff through under-ground channels leads to collapse.
Erosion of soil at the surface will be inhibited by the protection of the rock armor and filter layers.
It is not J
clear at this time whether observations of erosion on natural unprotected i
j slopes are relevant to erosion on armored, well-engineered embankments.
l A likely cause of flow concentration, given that good grading and rock place-t ment practices are followed, is a failure or differential settlement of the earthwork with subsequent subsidence or slumping.
Such a failure could create l
a depression toward which water running off the embankment would collect.
The
[
]
nature of such a failure is highly speculative.
l There are at least two compensating factors tending to resist flow concentration:
(1) If the rock layers are thick enough, water will flow beneath the rock layer surface, and the uniformity of the layer should be less important.
4 i
(2) Tailings enhankments are of ten narrow at +he top and wide at the bottom.
This condition leads to a natural hydraulic gradient out from the center-j line of the slope, tending to disperse rather than concentrate flow.
I The smaller grade and lower water-carrying ability on the top slope would ac-centuate the effects of settlement on flow concentration, Settlement of from f
one to several feet might be possible (Wardwell, 1984). Good engineering and l
j construction practice probably can reduce the effect of settlement.
{
Nevertheless, several scenarios of embankment failure have been postulated and i
studied with the numerical runoff model, as illustrated in Figure 2-14.
Figure t
l 2-14b shows the embankment as built.
Figure 2-14b shows a uniform inward slump-l ing of the embankment toward the centerline.
Multiple failures as illustrated l
in Figure 2-14c probably would cause less severe flow concentration, because i
i the drainage area for each sub-basin is smaller than the single-failure case.
l Other failures are possible, such as the opening of a trough by slumping and erosion of an otherwise-unfailed embankment (Figure 2-14d).
l 3
t 1
I i
i 1
l l
NUREG-1263 2-28 l
l i
l
r 2.5.2.1 Embankment Slumping Two cases of embankment slumping of the type illustrated in Figure 2-14b are presented in order to demonstrate flow concentration:
(1) uniform inward slope of 1/2% toward centerline and (2) uniform inward slope of 1%.
Flow concentrations resulting frcm steady rainfall on the slumped embankments are presented in Figure 2-16 as thi ratio of runoff at the embankment center-line to that runoff to the same embsnkment with no slumping.
E d
4 j
i t
df gju I
Figure 2-16 Flow concentration for steady rate of precipitation t
Flow concentration for the 1/2% and 1% slump scenarios are all greater than unity and depend on the rainfall intensity.
The high degree of flow concentration from the top slope is explained largely by the saturation and overtopping of the rock layer.
Resistance to flow is greatly reduced once overtopping occurs.
In addition, the inward slope in each case is a significant fraction of the 2%
downward Gradient of the original slope.
There is significantly less flow concentration on the steep side slopes.
Overtopping would occur only at point.
above the slope break.
Peak flow rates are attenuated within the rock layer of the side slope.
Transient runoffs from the top and side slopes resulting from the local PMP are presented in Figure 2-17 for the 1/2% slump scenario.
There is a considerable degree of flow concentration for the slumped case, particularly on the top i
slope.
An interesting observation is that peak runoff may occur at the toe of the top slope rather than at the toe of the side slope.
The design of tne rock layer on the side slope may therefore be controlled by runoff from the top slope.
l NUREG-1263 2-29
8 1
p p
m
%m
%u u
%l 1
lS S
6 I
1 p
4 f
1 2
I 1
p ruo h
/s 0
e f
1 h
c p
n i
ETA R
8 L
I LA p
F N
/
/
t R
n m
/
e 6
I
/
kna
/
b m
/
E k
/
e e r
p a 4
I p
o m o
l lS S hc e
p n
d o
e i
T S B 2
I O
5 4
3 2
1 O
[o$4,zOl4xlzoU2oo t
94 I
i Figure 2-17 Transient runoff for 1/2% inward slump 2.5.2.2 Reduced Conveyance Peak runoff is sensitive to the ability of the flow to remain confined to the rock layer rather than overtop it.
The ability of the rock layer to store and transport most of the runoff is a critical factor in the attenuation of peak flow from the slope.
This effect will be diminished, however, if the rock layer is too thin, its friction too great, or its porosity too small, d
The capacity of the rock layer to conduct flow is related to its thickness and i
flux.
A convenient grouping of terms is the conveyance l
[
IgSdh q' = Hnl l
( K, j Reducing porosity n or increasing K would reduce the water-carrying ability of the rock iayer.
The flood peak will be attenuated as long as the flow remains confined below the surface of the rock.
If overtopping should occur, however, the friction is decreased dramatically, and the conveyance of the slope increases.
[
The net effect of friction on peak flood flow is highly nonlinear and cannot be l
expressed as a simple csusal relationship, NUREG-1263 2-30 i
l 1
a a
42,_-.
a I
/
/
/
/
/
E
/
4
/
1
...=dE_,,_,,a.**#-
f l
_,s
\\
\\
\\
\\- R
\\
\\
\\
\\- @z
\\
n e
\\
8
\\
8
\\
a I~k h i
\\
\\
8
=e s.
IR E
a 1-e.
4 i
4m I
I i
E 1
g I
I o
o n
g o
o 6
6 0
);/oes/c)) 'ddONnB
~
The effect of doubling and halving the estimate of d is given in Table 2-8.
h Doubling d lowers the internal friction, increasing the peak runoff.
h l
Interestingly, halving d increases friction, but causes the flow on the top h
slope to exceed the carrying capacity of the rock layer, resulting in an increase in the peak runoff at the toe of the top slope.
The transient case was rerun for the 1/2% slump scenario, but with an essentially infinite layer thickness which eliminates the possibility of overtopping. The results of this run are shown in Figure 2-18, along with the runoff for the normal thickness of the rock layer.
Peak runoffs for this case are lowered dramatically. Furthermore, the peak runoff occurs at the toe of the side slope and is no longer controlled by runoff from the top slope.
The maximum thick-ness of the rock layer necessary for the sample embankment to completely con-tain the peak flows are about 3.3 feet on the top slope and 1.4 feet on the side slope.
\\A Figure 2-18 Runoff reduction on thick armored embankment 2.5.2.3 Peak Intensity of Precipitation The "time of concentration" (TOC) is a measure of the response time or frequency response of a flow system (Overton, 1976).
The TOC for flow on an embankment in response to a precipitation event depends on length of the embankment and the speed at which disturbances propagate.
The usefulness of the time of NUREG-1263 2-31
1.1 i
i I
I i
1.0 l
,}
0.9 Top Slope -
ll 1 Ft Thick ie 0.8 l 1 5
l1 l
{
g 0.7 I
g iI 1
l t
z i
i Side Slope -
3 0.6 i
L 1.5 Ft Thick s
I i
F 1
g 0.5 l
p g
3 i
Side Slope.
f i
g infinite g
u.
0.4 I
\\
Thickness O
\\
z
\\
D" 0.3 g
s t
8 s
8
"~~~~~~.;%
0'2 s,
, i \\
s/
~
side SIO9' gop f.9, Top Slope.
0.1 Infinite Thickness 0**~~~~I,,,,
I I
I I
I 1500 2000 2500 3000 3500 4000 4500 5000 TIM E, seconds i
r l
,l
\\
, _ _ _ ~. _ _ _ _, _ _ _ _,.... _
1 concentration is that it sets a limit to the duration of the PHP that must be j
considered in the analysis; i.e., the peak flowrate should be relatively insensitive to the duration of disturbances shorter than the time of concentra-tion.
The shorter the time of concentration, the shorter, and therefore more intense, the periods of precipitation that must be considered.
For example, a large embankment will be less sensitive than a small embankment to intense but short-lived precipitation.
i The speed at which the disturbances propagate along the embankment is assumed to be the kinematic velocities of flow through and over the rock, determined by balancing the forces of gravity and friction.
The kinematic velocity increases l
once the rock layer is overtopped.
The time of concentration, therefore, should be smaller for higher rates of precipitation, or for conditions of the embank-ment that are likely to cause overtopping of the rock layer.
Although there are a number of relationships for times of concentration (Overton, 1976), they were derived largely for impermeable plane surfaces, and require difficult-to-j define parameters such as Manning's coefficient.
Their adequacy for the present situation of flow on compound armored embankments has not been demonstrated, t
The PMP used in the present analysis considars periods in the rainfall duration i
curve as short as 2.5 minutes.
To test the sensitivity of the peak runoff to 3
duration and intensity of the rainfall, the benchmark and 1/2% slump scenarios 1
were rerun, but under the influence of the most intense 5-ninute periods instead of the 2.5-minute periods, The revised rainfall-duration curve is presented in i
i Table 2-6.
The benchmark case shows modest sensitivity to the change.
The case 3
for the 1/2% slump shows a greater difference between the two rainfall-duration
[
curves:
the 5-minute duration case gives peak runoff values up to 15% lower l
than for the 2.5-minute case.
This indicates that the time of concentration is shorter than 5 minutes for the slumped scenarios, and the 2.5-minute rainfall-j duration curve would be more acceptable.
2.5.2.4 Infilling of Rock l
i Some embankment designs call for the interstices of the riprap to oe filled with soil.
Even where this is not being done deliberately, it is conceivable that natural processes such as rock weathering and windblown transport of soil may cause the interstices to clog.
Much of the attenuation of the PHP is due l
to the capacity for flow beneath the surf ace of the rock layer, and this would be lost should the interstices become filled.
l Two additionel runs were made to demonstrate the effects on peak flow of a diminished thickness of rock layer on an unslumped slope.
The first run di-minishes the rock layer thickness by half.
In the second run, the rock layer thickness is reduced to one-half of the <d> rock diameters.
Peak runoffs in-
]
crease for both cases, as presented in Table 2-8.
Interestingly, the more i
significant runoff occurred for the former case where the rock layer thickness i
was diminished by half, rather than the later case where the thicknesses of the
{
j layers were much smaller.
This phenomenon probably is caused by the timing of
[
the rainfall onto the slope and the speed at which it runs off.
In the later l
j case, the higher speed of runoff allowed the water to drain off the top slope; i
in the former case, water accumulated and ran off coincidentally with the peak flow. This somewhat counterintuitive finding points to the complexity of run-off from armored slopes, and the need to study the designs carefully.
l t
NUREG-1263 2-32
l l
2.6 Conclusions Runoff from armored compound slopes on tailings embankinents resulting from intense precipitation has been studied by means of a mathematical model for kinematic flow.
Several interesting conclusions can be drawn frota the mathematical experiments with the model:
(1) The calculation of runoff must consider flow both through and over the top of the armor layer.
(2) Irregularities in the surface of the slopes may lead to large concen-trations of flow along preferential paths.
(3) The peak runoff from the gentler top slope could be greater than the peak runoff from the steeper side slope, thereby controlling the design of the armor on both slopes.
This condition may occur when the ability of the rock layer to carry the fiov is inadequate, as illustrated in Section 2.5.2.2, forcing the flow to overtop the rock layer.
The most severa hydrologic stresses on the armor are likely to occur at the break between j
the top and side slopes for this situation.
This observation indicates that the larger armor used on the side slope should extend a distance l
above the break in the slope, onto the less steep slope, i
(4) The use of larger diameter rock and thicker rock layers tends to diminish peak runoff from the top slope.
l (5) The effects of flow concentration caused by geotechnical failure or slump-ing can be greatly diminished by having an adequate rock layer thickness.
(6) For typical embankments, the rainfall duration should consider durations as short as 2.5 minutes, especially when evaluating cases for slumped embankments.
Experiments with a 5-minute minimum duration showed up to IST, lower results for peak runoff than produe?d for the same embankment with a 2.5-minute minimum duration.
Results were less critical for un-slumped embankments.
[
(7) Attenuation in the armor layer is lost if a soil-filled rock is used, leading to significantly higher flood peaks because of a shorter time of concentration.
t h
i l
NUREG-1263 2-33
3 DETERMINING THE SIZE OF THE RIPRAP 3.1 7ntroduction 1
This chapter deals with one possible method that could be used by the designer to determine the size of rock n9cessary to resist the forces generated by run-off from severe precipitation.
It is presumed that standard design principles and good design practices will be followed for the overall design of the riprap and filter layers.
Such specifications are outside the scope of this report.
The present report provides only the necessary tools for the hydrologic stabil-ity analysis.
The demonstration of suitability presented in this chapter will be based on the 4
following procedures:
1 (1) Select initial riprap layer specifications for tha entire embankment.
Using the methodology for runoff calculations discussed in Chapter 2 calculate the peak flowrate and stage at key points on the embankment for a given design.
(2) Utilize the safety factor method (Stevens, 1971) and Stephenson method (Stephenson, 1979) to determine the size of rock necessary to withstand the forces generated by the peak flows.
(3) Check to see that the given rock sizes meet or exceed the rock sizes calculated in step 2.
(4)
If the given rock sizws tre smaller than sizes needed, modify the design; e.g., decrease the slope, increase the rock size, increase the layer thick-nesses.
It may be necessary to recalculate peak flows from step 1, if it is suspected that they might increase under the new design.
The safety factor and Stephenson methods will be described in the sections that follow, and will be coopared with the results of fiume studies conducted at Colorado State University.
Other methods for determining the rock size are re-t viewed in Abt et al. (1987).
Finally, a brief computer reogram will be described, which will aid the investigator in applying the formulas presented in this chapter for determining the rock size.
i 3.2 Safety Factor Method The safety factor method was developed to determine '.he stability of rock riprap in flowing water in the absence of wave and seepage forces (Stevens,1971).
The method relies primarily on the observation that rocks on side slopes tend to i
roll rather than slide.
The stability of the rock in determined by summing the moments produced by gravity, buoyancy, drag force, and lift force around the j
axis of rotation.
l I
NUREG-1263 3-1
i i
For overtopping flows principally down the gradient of an embankment, the safety factor method can be simplified.
For flowing water at steady state, the tractive force on the rock surface (t,), must just balance the force of gravity:
I
- TIS (3'1) i s
y where y = density of water (62.4 lb/ft3)
( = depth of water over the top of rock (ft)
S = slope of the embankment face y
The depth of water ( is determined either from a stage-discharge rating curve or from a simple formula for conveyance (such as Manning's equation).
i The representative diameter of the stable rock d can be determined if the trac-j tive force on the rock is balanced against the natural tendency of the rock to remain in place:
)
21t d = (5, - 1)yq (3-2)
=cosoh-y where q j
g3,3)
S, = specific gravity of rock a = angle of grade = tan 1Sy
$ = angle of reposs for dumped rock
[
SF = safety factor A safety factor of unity theoretically means that the rock is just on the verge l
of stability.
A typical safety factor for the design of riprap is SF = 1.5.
[
The angle of repose is an empirical relationship shown in Figure 3-1, and is the f
1 measure of the maximum stable angle for a slope without external forces acting on it.
It is a function of the median rock diameter d o and rock angularity 3
(i.e., crushed, angular, or very round).
i 3.3 Stephenson's Method
(
f Stephenson's method was developed for calculating the stability of rock-fill l
l dams in rivers (Stephenson, 1979).
One of the main differences between the Stephenson and safety factors methods is that the former considers the stability of the rock layer as a whole but the latter considers the stability of individual i
L rocks.
It has been observed empirically that the stability of rock layers is greater than the stability of individual rocks in the layer treated in isolation.
Consequently, the Stephenson method generally is less conservative than the l
safety factor method.
[
l I
Using the Stephenson method, the rock diameter that would just begin to move l
under the influence of flowing water has been empirically determined to be:
l l
NUREG-1263 3-2 I
i 1
N Figure 3-1 Angle of repose for typical rock armor 7/6" /6 2/3 1
d=
(3'4)
[Cg\\[(1 - n)(5, - 1)cos a(tan $ - S ))S/8 y
where q = flowrate on the embankment n = rock space fraction (i.e., porosity)
[
C = a factor that accounts for the angularity of the rock (determined i
to range from 0.22 for gravel to 0.27 for crushed granite) i g = acceleration of gravity l
S, = specific gravity of rock l
4 = angle of repose of the rock i
The diameter determined from equation 3-4 is for the "threshold" flowrate, which is the flowrate at which the rock will just start to move.
At flowrates just above the threshold, the rock will rearrange to a more stable configuration. At much higher velocities, the structure will collapse.
Olivier (1967) reports that the flowrate for collapse is from 120% of threshold flow for gravel to 180% of threshold flow for crushed ledge rock (the computer program ROCKSIZE, presented in Appendix B calculates the diameter for threshold flowrate, not failure).
i 3.4 Discussion
+
Both the safety factor and Stephenson methods are presented in terms of a "representative rock diameter
<d>", rather than a typical measure such as the median dso.
Richardson et al. (1975) report that the representative diameter j
for riprap is larger than the median.
Experimental data on scour of submerged
+
rock armor using rock materials of widely different gradations showed that the i
larger rocks had a dominant effect on the determination of stability, and there-fore should be more heavily weighted in determining tne representative rock i
diameter.
The ratio of representative diameter <d> to dso ranged from 1.06 to j
2.25 in several experiments performed by St6vens (1971).
Richardson et al.
L (1975) recommend that the riprap be thick enough t.o permit the loss of fines
{
without uncovering the protected material or filter.
The above discussion sug-
[
gests that the use of dso for the representative diameter in the safety factor and Stephenson methods is probably conservative.
The possibility of loss of
[
fines reducing the thickness of the riprap should be borne in mind, however, j
especially for those cases in which the riprap layers would be constructed from i
material with a large coefficient of gradation; i.e., with a significant frac-
[
tion of fines.
I NUREG-1263 3-3 l
t f
r
3 dols 30lS
>t t
t d
l-I i
I i
Q
{
3
)
s R
o i.
k
'I
??-
~
Iii e*o E
E S
55E E
~
o
"*4 E g 5
of
@h
'C, O 4*
o 7
o w
Og 4
E 4
m
< 3<0 u.
b A'e'i<
j d
o 5
2 -
5
- e A
E
\\
n 4
7 w
.Y v.
+
O e
g C:g g
W 8
14 R
R (ss)ysul 'H313WV10 NV3W A
6 b
s i
i 1
The reliability of the safety factor and Stephenson methods to determine the i
stability of riprap layers is demonstrated using the experimental data collected in the Colorado State University flumes, and presented in Table 3-1.
The repre-sentative diameters for failure for the safety factor method have been calculated i
using a safety factor of unity, and the computed stage-discharge relationship 3
(Model 1) for the flowrate at which actual failure was observed.
Similarly, i
the representative diameters are calculated from the Stephenson formula, assuming that the observed flowrate q at failure in equation 3-4 represents 120% of the i
i flowrate for incipient movement.
Also presented in Table 3-1 are the rock diameters d o, which would have been chosen using typical design factors.
A 3
safety factor of 1.5 was chosen for the safety factor method; a 50% increase in diameter was used in the case of the Stephenson method.
Table 3-1 Modeled and measured flowrstes for riprap failure dso(I)
Q Stage (2), Stage (3) d(4) d(5) d(6) d(7)
[
]
Slope i n, fh/sec/ft ft ft in.
in, in, in.
0.01 1.02 1.5 0.467 0.503 0.75 1.13 0.34 0.51 l
0.02 1.02 1.1 0.353 0.304 1.06 1.61 0.48 0.72 l
)
0.10 1.02 0.36 0.129 2.16 3.47 0.90 1.35 i
I 0,02 2.2 4.5 0.89 0.74 2.67 4.05 1.21 1.82 i
0.08 2.2 1.81 0.386 0.338 5.0
- 7. 9 2.12 3.18
)
0.1 2.2 1.25 0.303 0.259 5.04 8.1 2.02 3.03 l
0.2 2.2
- 0. 5 0.161 6.25 11.0 2.24 3.36 1
0.2 4.1 1.81 0.355 0.167 14.2 24.9 5.1 7.7 l
{
0.2 5.1 3.55 0.543 21.7 38.1 8.43 12.6 i
0.2 6.2 4.43 0.632 25.3 44.3 9.8 14.7
{
(1) Riprap actually used.
t
]
(2) Calculated from Model 1.
(3) Measured where available.
I (4) Safety factor method, SF = 1.
)
(5) Safety factor method, SF = 1.5.
]
(6) Stephenson method for incipient motion (not slope failure).
)
(7) 1.5 times the Stephenson method for incipient motion.
[
The watir level necessary for the calculations by the safety factor method originated from the stage-discharge rating curves derived from equations 2-14 l
through 2-16 (Model 1).
Heasured values of stage are presented where they are l
)
available.
There is often a significant difference between the predicted and j
measured stage, since the observed water level over the top of the rock in sev-j eral of the runs was small.
This error may be compounded becsuse the datum for measuring stage is unclear.
The computational model defines the datum for "top s
of rock" as the depth at which the frictional forces for flow through the rock 2
just equal the frictional forces over the top of the rock (see Section 2.2.3).
l The difference between the measured top of rock and apparent datum is about 30 i
to 40% of d o for the present data.
[
3 l
)
The safety factor method proved to be best suited at the lower slope angles j
(less than 10%), but overestimated the rock size on the steeper slopes.
Some NUREG-1263 3-4 4
of the rock sizes were underestimated with a safety factor of unity, but a safety factor of 1.5 always produced acceptably conservative results.
Some of the rock sites predicted for the steep slopes were greatly overestimated, however.
The Stephenson method was more suited to steep slopes, and did not overestimate the necessary rock size by as large a margin as did the safety factor method.
The Stephenson method tariously underestimated the rock size needed on the gentler slopes, even when the predicted rock size was increased by 50%.
- 3. 5 Example Calculations for Rock Armor The Stephenson and safety factor methods are formalized into a BASIC language computer program ROCKSIZE, described in Appendix 8.
The interactive run with this program will be illustrated below.
Assume fcr the example that an independent geotechnical analysis has determined that the 1/2% inward slope scenario with the local Probable Maximum Precipita-tion would be the design-basis event.
Other properties of the embankment are thoss given in Table 3-1.
Determine the adequacy of the rock to resist the calculated runoff.
The interactive sessions with program ROCKSIZE are shown in Figures 3-2 and 3-3 for the side and top slopes, respectively.
The calculations indicate that for the safety factor method on the top slope and the Stephenson method on the side slope (in accordance with the discussion of Section 3.4), the chosen rock sizes would be adequate to protect the embankment.
NUREG-1263 3-5
e PROGRAM ROCKSIZE DETERMINE THE STABLE DIAMETER FOR RIPRAP ON ARMORED SLOPES BY STEPHENSON AND SAFETY FACTOR METHOD U.S. NUCLEAR REGULATORY COMMISSION, WASHINGTON D.C.
INPUT FRICTION INDEX, K
?4 ENTER DBAR, D84. FT
? 0.47,0.66 ENTER LAYER THICKNESS, FT
? 1.5 ENTER SLOPE 7 0.2 ENTEft EFFECTIVE POROSITY
? 0.35 l
CORRECTION TO LAYER THICKNESS :
.1915754 FEET ENTER PEAK RUNOFF, CFS/FT
? 1.13 STAGE ABOVE ROCK SURFACE :
.3224664 FT ENTER ANGLE OF REPOSE, DEGREES? 41.5 ENTER SPECIFIC GRAVITY OF ROCK, GM/CC 7 2.65 t
ENTER SAFETY FACTOR ? 1.0 STABLE ROCK DIAMETER BY SAFETY FACTOR METHOD =
1.031614 FEET I
ENTER SMOOTHNESS FACTOR, C IN STEPHENSON FORMULA (0.22 FOR SMOOTH ROCK AND 0.27 FOR ANGULAR CRUSHED ROCK)
? 0.27 STABLE ROCK DIAMETER BY STEPHENSON METHOD =
.1993453 FEET Figure 3-2 Sample problem - interactive session for rocksize on side slope with program ROCKSIZE NUREG-1263 3-6
l l
I PROGRAM ROCKSIZE DETERMINE THE STABLE DIAMETER FOR RIPRAP ON ARMORED SLOPES l
BY STEPHENSON AND LAFETY FACTOR METHOD U.S. NUCLEAR REGUIATORY COMMISSION, WASHINGTON D.C.
INPUT FRICTION INDEX, F
?4 l
ENTER DBAR, D84, FT
' 0.16,0.24 ENTER LAYER THICKNELS, FT
? 1.0 ENTER SLOPE
? 0.02 l
ENTER EFFECTIVE POROSITY
? 0.35 CORRECTION TO LAYER THICKNESS =
6.942815E-02 FEET ENTER PEAK RUNOFF. CFS/FT
? 1.13 STAGE ABOVE ROCK SURFACE =
.4278018 FT ENTER ANGLE OF REPOSE, DEGREES? 40 ENTER SPECIFIC GRAVITY OF ROCK, GM/CC
? 2.65 i
i ENTER SAFETY FACTOR ? 1.0 l
l STABLE ROCK DIAMETER BY SAFETY FACTOR METHOD =
.1115765 FEET ENTER SMOOTHNESS FACTOR. C IN STEPHENSON FORMULA (0.22 FOR SMOOTH ROCK AND 0.27 FOR ANGULAR CRUSHED ROCK)
L
? 0.27 STABLE ROCK DIAMETER BY STEPHENSON METHOD =
3.451491E-02 FEET f
l i
l I
e l
Figure 3-3 Sample problem - interactive session for rock size on I
top slope with program ROCKS!ZE l
NUREG-1263 3-7
4 CONCLUSIONS The design of rock armor for embankments to resist the local Probable Maximum Precipitation (PMP) involves the calculation of runoff and the determination of the properties that help the rock resist movement by the calculated runoff.
The staff has developed a set of mathematical models and associated computer programs to calculate runoff from armored embankments.
The models take into account the resistance to flow both through and over top of the armor layer.
The techniques developed here can be used to study the effects of various designs of the embankments on the runoff caused by intense precipitation.
Runoffs calculated from the models are employed with empirical techniques to determine if the embankment slopes will be stable under the design-basis precipitation events.
Some of the conclus?ons that can bt drawn from the experimentation with the runoff models are listed below:
(1) The calculation of runoff must consider the flow both through and over the i
top of the armor layer, unless the rock is filled with soil or otherwise impervious.
(2)
Irregularities in the surface of the embankments may lead to large concen-trations of flow along preferential paths; such concentrations of flow would place more severe loads on the rock armor.
(3) The peak runoff from the gentler top slope can be more severe than the peak runaff from the steeper side slope, thereby controlling the design of the armor on both slopes.
This condition may occur when the capacity of the rock layer to carry the flow is inadequate, forcing the flow over the top of the rock layer.
The most severe hydraulic stresses on the armor are likely to occur at the break between the top and side slopes for this situation.
(4) Design factors that tend to diminish the peak runoff from the top slope include larger rock diameter and thicker layers.
Degradation of the rock over the design lifetime of the embankments should be taken into considera-tion, and the size of the rock should be adjusted accordingly.
The effects of flow concentration caused by geotechnical failure can be eliminated almost entirely by having a large thick rock layer.
( 5 '. The characteristic time for runoff on a typical embankment appears to be i
on the order of minutes.
Therefore, short periods of very intense rainfall must be included in the design-basis PHP, For the embankments studied in the present report, periods as short as 2.5 minutea were required.
(6) Friction of flow on armored embankments is expressed adequately by a compound resistance curve, using a square law for flow beneath the surface of the rock layer and a Darcy-Weisbach law for flow that overtops the rock layer, i
NUREG 1263 4-1 I
i (7) Flume tests with crushed rock indicate that the safety factor method ade-quately describes the stability of the rock for slopes of less than 10%.
i The Stephenson method is suited for slopes greater than 10%.
l Several shortcomings of the procedures presented in this report must be men-tiened.
The peak runoff to which the embankment is likely to be subjected is i
strongly dependent on the shape of the surface.
One of the largest uncertainties in the application of the design principles presented in this report is the t
prediction of possible future states of the embankment.
The scenarios studied l
in this report for various failure states were offered for illustrative purposes only and are highly speculativa.
Evidence of failure modes for embankments, other than those caused by hydrologic forces, should be compiled and analyzed.
Measures that tend to offset the concentration of flow should be used to reduce t
the sensitivity of the peak runoff to future, unknown states of embankment shape.
Peak values of runoff were difficult to predict for cases of slumping and com-binations of other factors that tend to diminish the capacity for flow within the rock layer.
Severe oscillations of flowrate tended to occur for high rates of flow.
These oscillations are probably computational artifacts, but real oscillation might occur also.
Experimentation with other forms of solutions f
to the differential equations (such as implicit methods) and sensitivity to l
parameter values (such as time step and grid spacing) should be pursued.
(
The future development of model tests to demonstrate the phenomena of flow con-l centration should be considered.
Such experiments could consider the irrigation l
of a scale model of a typical embankment for various shapes and parameter values.
Results of this scale model experiment could serve to validate the mathematical models presented in this report.
l t
i l
t L
I i
i NUREG-1263 4-2 s
5 REFERENCES Abt. S. R., J. F.Ruf f, M. S. Khattak, R. J. Wittler, A. Shaikh, J. D. Nelson, D. W. Lee, and N. E. Hinkle, "Development of Riprap Design Criteria by Riprap Testing in Flumes:
Phase 1," MUREG/CR-4651, U.S. Nuclear Regulatory Commission, Washington, D.C., May 1987.
Envir' nmental Protection Agency, Title 40, code of Federal Regulations, Part 192, o
Washington, D.C.
1985.
Hey, R. D., "Flow Resistance in Gravel Eed Rivers," J. Hydraulics Division, ASCE, no. HY4 pp. 365-379, 1979.
3 Leps, T. M., "Flow Through Rockffil," in Embankment Dam Engineer. Casagrande Volume, R. C. Hirshfeld and S. J. Poulos, Ed' tors, John Wiley and Sons, New York, pp.87-108, 1973.
Morris, E. M., and D. A. Woolhiser, "Unsteady One Dimensional Flow Over a Plane,"
Water Resources Research, Vol. 16, no. 2; pp. 355-360, 1980.
National Oceanic and Atmospheric Administration, Hydrometeorological Report No. 49, "Probable Maximum Precipitation Estimates Colorado River and Great Basin Drainages," Silver Spring, Maryland, U.S. Dept. of Commerce,1977.
Olivier, H., "Through and Overflow Rockfill Dams--New Design Techniques,"
Proceedinas, Institute of Civil Engineers, pp. 433-471, March 1967.
Overton, D. E., and M. E. Meadows, Stormwater Modelina, Academic Press, New York, 1976.
Richardson, E. V., D. B. Simons, S. Krek, K. Mahmood, and M. A. Stevens, "Highways in the River Environment--Hydraulics and Environmental Design Consid-erations," U.S. Dept. of Transportation Available from Publications Office, Engineering Research Center, Colorado State University Fort Collins, Colorado, 1975.
Roache, P. J., Computational Fluid Mechanics, Hermosa Publishers, Albuquerque N.M., 1972.
Sherard, J. L, J. W. Richard, W. Stanley, and A. C. Williams, Earth and Earth Rock Dams, John Wiley and Sons, New York, 1963.
Simons, D. B., and F. Senturk, Sediment Transport Technology, Water Resources Publications, Fort Collins, Colorado, 1977.
Stephenson, D., Rockfill in Hydraulic Engineerina, Elsevier, Amsterdam, 1979.
Stevens, M. A., ant D. B. Simons, "Stability Analysis for Coarse Granular Mate-rial on Slopes " Chapter 17 in River Mechanics, Edited and published by H. W. Shen, P.O. Box 606, Fort Collins, Colorado, 1971.
NUREG-1263 5-1
Wardwell, R. C., J. D. Nelson, S. R. Abt, and W. P. Staub, "Design Considerations for Long-Term Stabilization of Uranium Mill Tailings," in Management of Uranium Mill Tailinas, Colorado State University, Fort Collins, Colorado, 1984.
I L
NUREG-1263 5-2 l
APPENDIX A USER'S GUIDE FOR SLOPE 2D Program SLOPF20 is a finite-difference computer code that computGs the time-dependent runoff along the centerline of an armor-covered embankment according r
to the mathematical relationships presented in Chapter 2.
The embankment consists of four subareas or quadrants, and is assumed to be symmetrical around the centerline, as illustrated in Figure A-1.
Rainfall rates are specified in a tabular fashion.
The output of the program is the runoff rate per unit width along the centerline of the embankment at the i
base of the embankment and also at the break point between the upper and lower slopes.
Peak runoff rates may then be used to design suitable rock armor covers.
l The program is written in FORTRAN 77.
It is set up to run on an IBM compatible f
personal computer.
The program disk contains the source code (file SLOPE 20.FOR) l and a compiled version for computers with the 8087 mathematics coprocesser (file SLOPE 20.EXE).
A sample data file is also included on the disk (file SAMP.DAT),
I as well as program ROCKSIZE. BAS, described in Appendix B.
Program SLOPE 2D can be used on a mainframe computer with minor revisions.
Limitations on speed make the use of program SLOPE 20 on a typical personal cos-i puter somewhat tedious, making the use of a high-speed microcomputer or mainframe l
computer desirable.
i When run on a personal computer, the default input and output files are the keyboard and screen, respectively.
The input file, however is generally a disk l
file created with a text editor such as EDLIN.
The name of the input file is
.i specified at run time using the standard 005 redirection method.
For example,
[
if the data were specified in file SAMP.DAT, the execution step would be.
SLOPE 20<SAMP.DAT Output from the program is directed to the screen, but can be listed on the f
printer using the "Control" and "Printscreen" keys on the keyboard, i
Data inputs to and outputs from the program are presented below.
All data are input in a free format fashion.
Individual data points are separated by commas for each line.
Decimal points are optional.
Data Inputs l
t The following data are read by program SLOPE 20:
}
Line 1 Title line - up to 80 characters for run title
}
Line 2 Number of entries in rainfall table, NRAIN Next NRAIN Rainfall table.
FR(!) = multiplier for the average lines 1-hout reinfall rate R8 j
t NUREG-1263 A-1 Appendix A l
l l
I
e l
t t
f I
i l
i Figure A-1 Four quadrants of armored embankanent TR(I) = time, seconds, at which FR(!) becomes effective
{
Next line DX = grid spacing, ft N1 = effective porosity of riprap
(
051 = median rock diameter for top slope, ft l
052 = median rack diameter for side slope, ft THICK 1 = thickness of riprap layer on top slope, f t l
THICK 2 = thickness of riprep layer on side slope, ft SX1 = slope in the +x direction for quadrants 1 and 3 l
SX2 = slope in the +x direction for quadrants 2 and 4 i
SY1 = slope in the +y direction for quadrants 1 and 2 l
SY2 = slope in the +y direction for quadrants 3 and 4 A1 = index of the bottommost grid block in quadrants 2 and 4 61 = index of the rightmost grid block in quadrants 1 and 3 Next line DT = initial timestep for model, seconds NT = n'eber of iterations of model l
KP = iterations between printouts i
TSTART = time in rainfall table corresponding to commencement of simulation, seconds NUREG-1263 A-2 Appendix A f
TOP IOfp 9
d 3
t O
Substope 1 Subslope 2 6,
S,1 S,2 S,1 S,1 y
y x
x 2
y m
Subslope 3 Substopo 4 S,2 S,2 S,2 2
S,j y
y x
x O
m BOTTOM ko<
'~ NORMAL Flow 99 UND4Ry 1
1 l
l l
l l-!
. i, 9 :
i i
l Next line TCH = time at which timestep is changed, seconds l
OTCH = new timestep, seconds l
Next line K = roughness index k for riprap, i.e., 1 = smooth marbles l
2 = smooth rock, 4 = angular rock R8 = rainfall in 1-hour PHP, inches (typ. 8 in./hr) 081 = ds, diameter for rock on top slope, ft D82 = da4 diameter for rock on side slope, ft Next line NCOL = number of columns in grid NROW = number of rows in grid th Next (NCOL-1) JSTART(!) = topmost grid block for the i column (greater lines than or equal to 2) th JEND(I)
= bottomost grid block for the i column (less than or equal to 40) th i
l Next (NROW-1)
ISTART(J) leftmost grid block for the j rew lines (greater than or equal to 2) th IEND(J)
= rightmost grid block for the j row (less than or equal to 35)
Proaram Outputs All of the data input to the program are specified at the start of the output
- listing, The flowrates along the centerline (i.e., x = 0) at the slope break (QBRK) and the bcttom of the side slope (QCCENT) are then output as a function of time.
Finally, the highest. values of QBRK and QCCENT are given.
SETTING UP THE GRID The embankment is assurned to ve symmetrical around the vertical axis, and to be represented oy four quadcants.
The slope in the x and y directiels can be specified in each quadrant, as illustrated in Figure A-1.
The finite difference grid for an embankment in the example is represented by 6 rews and 5 columns.
The finite difference grid has cells of dimension DX (ft) on a side.
Only the righthar.d side of the embankment is represented, because of the assumption of symetry.
Column and row indexes start with number P.,
as illustrated in Figure A-2; also illustrated is the specification of the down-slope and horizontal slopa breaks A-1 and A-1, respectively.
The maximum row dimension is 35; the maximum coluten dimension is 40.
Determining Starting and Ending Times The object of program SLOPE 20 is generally to determins the peak runoff rates from the embankment.
The PHP used in the calculations optimizes the rainfall rate so that the most intense rainfall occurs at the end of the first hour.
Generally, the runoff from the top slope will peak right after one hour.
The peak runoff from the foot of the side slope will peak at a somewhat later time.
In order to reduce the computational burden, it is desirable to determine the shortest period that the simulation needs to be run in order to ensure that the peak runoffs will result.
For a typical embanUnent, a starting time of 900 l
seconds into the PHP lasting until 4000 seconds appetrs to be adequate to allow maximum buildup of the flow and passage of the flood peaks Some experimente-
'i tion may be necessaty to ensure that the croper time bounds are chosen.
f NUREG-1263 A-3 Appenoix A
4 l
i' i
V f
i f
I i
l i
I i
I
[
l
(
l l
i J
i i
j Figure A-2 Finite-difference grid for example probler
(
j Determining Timestep Size i
The finite-difference equations are solved by an explicit algorithm.
The i
stability Itait for a linear system states that the velocity times the j
timestep should be less than the spatial distance between grid points.
Veloc-e ities over or through the riprap are not expected to exceed 3 f t/sec under any likely conditions.
For a 20 foot grid spacing, a timestep of OT = 7 seconds would probably satisfy the linear stability criterion, In practice, however, j
the equations are not linear, especially when the rock is overtopped.
For the example problem presented later, a 5 second DT was used for the first 3100 s
i seconds of the PMP.
After 3100 seconds, a 2-second DT was found to be suitable i
i for the unfallad slope; a 0,5-second DT was used for the failed scenarios.
These timesteps were determined by experimentation, If there is a question
(
about the size of the timestep, a smaller timestep should be chosen and the 1
results should be compared, i
1 4
l MUREG-1263 A-4 Appendix A
{
L i
r 4
i
? I l=2 3
4 5
6 Top of Embank.T *,at il J
(Slope Break B1 =3)
J=2
+
3
+
+
(Quadrant 2)
[
(Quadrant 1)
>/
4 g 4
+
+
4 S
(Slope Break)
Y E
ll I
Al=4 o
s
+
+
+
+
~
-.4 _ - - _ -._. _ _ l._
(Quadrant 3)
(Quadrant 4)
I i
e
+
l
+
+
i
+
i
+
t i
I 1
?
__._.__.___.._p._.___. j L._ ___ _.
l 2
7
+
Y
+
l
+
1
+ '
?*
ll l
l I
l I
l l f V
Too of Embankment i
q q(2)
(centerline) 1 6
ca.t ? I
SAMPLE PROBLEMS I
Consider the triangular embankment as illustrated in Figure 2-14a.
The upper section is 440 feet long, has a 2% grade, and is covered with a 1-foot-thick layer of riprap, consisting of e. rushed rock.
The side slope is 260 feet long, has a 20% grade, and is covered by a 1.5-foot-thick layer of riprap, also of crushed rock.
The ef fective porosity of the rock is estimated to be 0.35.
The gradations for the rocks are given in Figure A-3.
Calculate the peak runoff at the bottom of the top slope and the bottom of the side slope for the case of
]
an unfailed embankment and the design-basis PMP.
Repeat the calculations for the case of a uniform slump of 1/2% toward the centerline, and the case of a 200-foot wide trench in an otherwise unfailed embankment, with a slump of R.
)
l p, & [ T l
l Figure A-3 Gradation of rock armor for example problem Benchmark Erbankeent The grid si2e DX was chosen to be 20 feet.
The downhill slope break occurs at A1 = 22.
The horizontal slope break is imaterial in this case, but is set at B1 = 6.
The rainfall table has 14 lines (NRAIN = 14) and is based on the design-casis PMP presented in Table 2-6.
The 1-hour FMP of 8 inches (R8 = 8) is assumed.
A friction index of k = 4 is chosen becawse the riprap is crushed rock, and therefore angular.
The slopes SX1 and SX2 are zero for the benchrrark slope.
The slopes in the downhill direction are 5Y1 = 0.02 and SY2 = 0.2.
The do huREG-1263 A-5 Appendis A
100 i
ii i
i i
i ;
i i
90 80 Top Slope Side Slope 1 70 f
x wzE 50 H2 40 5'
30 20 10 0
III I
I I
I I I I
I 987 6 5
4 3
2
.9.8.7.6
.5
.4
.3 10.0 1.0 ROCK SIZE, inches
.,. - -, - ~ _,,. -,. - - - _ _ _ - _ -
... _. - - - -, - _,. _ - - -. - - - -,.... ~... _.,, - - -,,. _,
rock diameters are read directly from the gradation curves presented in Figure A-3 to be 0.24 and 0.66 foot for the top and side slopes, respectively.
The characteristic rock diameter <d> is represented by the harmonic mean d '
h as calculated by equation 2-8.
The calculations for the harmonic mean are given in Table A-1.
The harmonic means are 0.16 and 0.47 foot for the top and side slopes, respectively.
Table A-1 Calculation of harmonic mean rock diameters Percentile d, in.
d, in.
range (large rock) 1/d (small rock) 1/d 90 - 100 9
0.111 3.3 0.303 80 - 90 8
0.125 2.9 0.345 70 - 80 7.2 0.139 2.7 0.37 60 - 70
- 6. 6 0.152
- 2. 5 0.4 50 - 60 6.2 0.161
- 2. 3 0.435 40 - 50 6.0 0.167 2.1 0.476 30 - 40 5.6 0.179 1.95 0.513 20 - 30 5.0 0.2
- 1. 7 0.588 10 - 20 4.2 0.238
- 1. 5 0.667 0 - 10 3.4 0.294 0.99 1.01 Harmonic mean 5.66 1.96 The calculation commences at TSTART = 900 seconds into the PMP.
An initial timestep of DT = 2.0 seconds is chosen, but is switched to OTCH = 0.5 second after TCH = 3100 seconds into the PMP.
A total of NT = 4000 steps is used to cover the occurrence of the peak flows from the two slopes.
A print interval of KP = 20 is chosen.
The input file is illustrated in Figure A-4.
The output for the run is illus-trated in Figure A-5.
The peak runoff rates for the unfailed embankment are 3
0.27 and 0.31 ft /sec/ft for the top and side slope, respectively.
Results for this case are also plotted in Figure 2-18a.
l 1/2% Slump The input file for this run differs from the benchmark embankment example because the horizontal slopes must now be specified as SX1 and SX2 =
.005.
The hori-zontal breakpoint for the slope is immaterial for this case (as well as for the benchmark case), but is set to B1 = 6.
The timestep DTCH = 0.5 second becomes effective at TCH = 3100 seconds.
The revised input file is shown in Figure A-5(b) l with the changes highlighted.
The peak flows for this case were computed to be l
0.81 fta/sec/ft from the top slope and 0.44 ft /sec/ft from the side slope.
3 Results for this case are also plotted in Figure 2-17.
l i
i NUREG-1263 A-6 Appendix A l
I
n (a) Benchmark embankment BENCHMARK CASE 2/27/87 14
-100,0.0 0,.22 1800,.6 2700,1.43 3000,2.05 3300,3.25 3450,7.55 3600,1.06 3750,.445 3900,.286 4200,.2 4500,.084 5400,.0308 7200,0 20,.35,.16,.43,1,1.5,0,0,0.02,0.2,22,22 2,4000,20,900 3100,O.5 4,8,.24,.66 31,36 2,36
~
2,36 3,36 4,36 6,36 7,36 8,36 9,36 11,36 12,36 13,36 14,36 15,36 17,36 18,36 19,36 20,36 21,36 23,36 24,36 25,36 26,36 27,36 29,36 30,36 31,36 32,36 33,36 Figure A-4 Inputs to computer program SLOPE 20 NUREG-1263 A-7 Appendix A
p 34,36 35,36 2,3 j
2,4 -
i 2,5
.l 2,5 2,6 2,7 2,8 1
2,9 2,9
'2,10 2,11 2,12 2,13 2,14 2,14 2,15 2,16 2,17 2,18 2,19 2,20 2,20 2
2,21 2,22 2,23 2,24 2,25 2,25 2,26 2,27 2,28 2,29 r
2,30 t
2,31 l
2,31 r
(b) Changes for 1/2% slump (in box) 0.005 inward slump case 2/27/87 1
14
-100,0.0 0,.22 1800,.6 2700,1.43 3000,2.05 3300,3.25 3450,7.55 3600,1.06 i
l' 3750 445 j
Figure A-4 (Continued)
NUREG-1263 A-8 Appendix A l
3900,.286 4200,.2 4500,.084 5400. 0308 7200,0 I
I I 20,.35,.16,.43,1,1.5,.00E,.005 02,.2,22,22 I I
I 2,4000,20,900 3100,0.5 4,8,.24,.66 31,36 2,36 (c) Changes for trench case (in box) 200 FT WIDE 1% TRENCH SECENARIO 14
-100,0.0 0,.22 1800,.6 2700,1.43 3^00,2.05 3300,3.25 3450,7.55 3600,1,06 3750,.445 3900,.286 4200,.2 4500,.084 5400,.0308 7200,0 l
l l 20,.35,.16,.43,1,1.5,.01,0,.02,.2,22,6 I
I I
2,4000,20,900 3100,0.5 4,8. 24,.66 31,36 2,36 Figure A-4 (Continued)
NUREG-1263 A-9 Appendix A
(a) Benchmark case PROGRAM SLOPE 20 - RUN0FF FROM SLOPES BENCHMARK CASE 2/27/87 GRID SIZE, DX =
20.0 FEET EFFECTIVE POROSITY, N1 =
.350 D50 ROCK DIAMETER ON TOP, 051 =
.1600 FEET 050 ROCK DIAMETER ON SIDE SLOPE, 052 =
.4300 FEET THICKNESS OF TOP LAYER, THICK 1 =
1.00 FEET THICKNESS OF SIDE LAYER, THICK 2 =
1.50 FEET Y SLOPE ON TOP, SY1 =
.020 Y SLOPE ON SIDE, SY2 =
.200 X SLOPE ON TOP, SX1 =
.000 X SLOPE ON SIDE, SX2 =
.000 POSITION OF SLOPE BREAK D0WN SLOPE, A1 =
22 POSITION OF SLOFE BREAK IN Y DIRECTION, B1 =
22 DT =
2.00 SECONDS NUMBER OF STEPS, NT =
4000 PRINT INTERVAL KP =
20 TIME AT WHICH COMPUTATIONS COMMENCE, TSTART =
900.00 SECONDS 7IME AT WHICH TIMESTEF CHANGES, TCH =
3100.0 SECONDS NEW TIMESTEP, DTCH =
.500 SECONDS FRICTION FACTOR INDEX, K =
4.0 1 HOUR RAINFALL AMOUNT, R8 =
8.00 INCHES 084 DIAMETER FOR TOP SLOPE, 081 =
.240 FEET 084 DIAMETER FOR SIDE SLOPE, D82 =
.660 FEET TIME-SECONDS QCCENT - CFS/FT QBREAK - CFS/FT 940.00
.0012
.0006 980.00
.0026
.0009 1020.00
.0039
.0011 1060.00
.0053
.0014 1100.00
.0066
.0016 1140.00
.0080
.0018 1180.00
.0093
.0021 1220.00
.0106
.0023 1260.00
.0117
.0026 1300.00
.0124
.0028 (Output from 1340 to 3300 deleted from this listing) 3300.00
.1385
.0568 3310.00
.1416
.0585 3320.00
.1446
.0600 3330.00
.1477
.0614 3340.00
.1506
.0627 3350.00
.1536
.0639 3360.00
.1564
.0650 3370.00
.1592
.0661 3380.00
.1619
.0671 Figure A-5 Outputs f<+m computer program SLOPE 2D NUREG-1263 A-10 Appendix A
3390.00
.1646
.0681 3400.00
.1672
.0691 3410.00
.1698
.0702 3420.00
.1723
.0712 3430.00
.1748
.0722 3440.00
.1772
.0732 3450.00
.7197
.0742 3460.00
.1684
.0792 3470.00
.1974
.6836 4
3480.00
.2064
.0876 3490.00
.2154
.0912 3500.00
.2244
.0948 3510.00
.2334
.0986 3520.00
.2424
.1033 3530.00
.2514
.1090 3540.00
.2604
.1157 3550.00
.2694
.1296 3560.00
.2784
.1515 3570.00
.2874
.1778 3580.00
.2963
.2052 a
3590.00
.3052
.2328 3600.00
.3141
.2576 L
3610.00
.3133
.2686 3620.00
.3121
.2715 3630.00
.3107
.2686 3640.00
.3093
.2618 3650.00
.3078
.2527 3660.00
.3061
.2418 3670.00
.3044
.2303 3680.00
.3025
.2188 3690.00
.3005
.2076 3700.00
.2983
.1972 3710.00
.2960
.1877 3720.00
.2936
.1797 3730.00
.2910
.1728 3740.00
.2883
.1666 3750.00
.2854
.1608 3760.00
.2816
.1548
[
1-3770.00
.2775
.1489 3780.00
.2734
.1433 3790.00
.2693
.1378 l
3800.00
.2652
.1325 l
i (Output from t = 3800 to 4500 deleted from this listing)
I 4500.00
.0848
.0667 4510.00
.0841
.0664 i
4520.00
.0836
.0662
.0831
.0659 4530.00 4540.00
.0826
.0657 4550.00
.0822
.0654 4
MAX FLOW AT BASE =.314494 CFS/FT MAX FLOW, TOP SLOPE =.271536 CFS/FT Figure A-5 (Continued)
NUREG-1263 A-11 Appendix A F
(b) 1/2% inward slump case PROGRAM SLOPE 2D - RUN0FF FROM SLOPES 0.005 inward slump case 2/27/87 GRID SIZE, DX =
20.0 FEET EFFECTIVE POROSITY, N1 =
.350 D50 ROCK DIAMETER ON TOP, 051 =
.1600 FEET D50 ROCK DIAMETER ON SIDE SLOPE, 052 =
.4300 FEET THICKNESS OF TOP LAYER, THICK 1 =
1.00 FEET THICKNESS OF SIDE LAYER, THICK 2 =
1.50 FEET Y SLOPE ON TOP, SY1 =
.020 Y SLOPE ON SIDE, SY2 =
.200 X SLOPE ON TOP, SX1 =
.005 X SLOPE ON SIDE, SX2 =
.005 POSITION OF SLOPE BREAK DOWN SLOPE, A1 =
22 POSITION OF SLOPE BREAK IN Y DIRECTION, B1 =
22 DT =
2.00 SECONDS NUMBER OF STEPS, NT =
4000 PRINT INTERVAL KP =
20 TIME AT WHICH COMPUTATIONS COMMENCE, TSTART =
900.00 SECONDS TIME AT WHICH TIMESTEP CHANGES, TCH =
3100.0 SECONDS NEW TIMESTEP, DTCH =
.500 SECONDS FRICTION FACTOR INDEX, K =
4.0 1 HOUR RAINFALL AMOUNT, R8 =
8.00 INCHES 084 DIAMETER FOR TOP SLOPE, D81 =
.240 FEET 084 DIAMETER FOR SIDE SLOPE, 082 =
.660 FEET TIME-SECONDS QCCENT - CFS/FT QBREAK - CFS/FT 940.00
.0012
.0006 980.00
.0027
.0009 1020.00
.0042
.0012 1060.00
.0058
.0015 1100.00
.0074
.0019 1140.00
.0091
.0022 1180.00
.0109
.0026 1220.00
.0126
.0030 1260.00
.0141
.0035 1300.00
.0151
.0039 (Output from t = 1310 to 3400 deleted from this listing) 3400.00
.2081
.1047 3410.00
.2110
.1090 3420.00
.2138
.1125 3430.00
.2166
.1182 3440.00
.2194
.1263 3450.00
.2222
.1372 3460.00
.2312
.1564 3470.00
.2406
.1804 l
3480.00
.2499
.2090 Figure A-5 (Continued) i NUREG-1263 A-12 Appendix A
3490.00
.2593
.2452 3500.00
.2688
.2856 3510.00
.2782
.3243 3520.00
.2877
.3406 3530.00
.2973-
.5173 3540.00
.3068
.6731 3550.00
.3164
.7919 3560.00
.3260
.7997 3570.00
.3356
.7325 3580.00
.3451
.6774 3590.00
.3547
.6355 3600.00
.3643
.6556 3610.00
.3643
.6580 3620.00
.3638
.6829 3630.00
.3632
.6924 3640.00
.3625
.6438 3650.00
.3618
.5758 3660.00
.3611
.5317 3670.00
.3603
.5063 3680.00
.3596
.4856 3690.00
.3589
.4614 3700.00
.3583
.4347 3710.00
.3579
.4112 3720.00
.3578
'.3916 3730.00
.3580
.3773 3740.00
.3587
.3650 3750.00
.3600
.3537 3760.00
.3610
.3389 3770.00
.3629
.3248 3780.00
.3656
.3498 3790.00
.3693
.3455 3800.00
.3739
.3416 (Output from t = 3810 to 4500 deleted from this listing) 4500.00
.1602
.1111 4510.00
.1587
.1105 4520.00
.1573
.1098 4530.00
.1559
.1091 4540.00
.1545
.1084 4550.00
.1531
.1077 MAX FLOW AT BASE =.443591 CFS/FT MAX FLOW, TOP SLOPE =.811763 CFS/FT (c) 200-ft-wide, 1% inward slump trench PROGRAM SLOPE 2D - RUN0FF FROM SLOPES 200 FT WIDE 1% TRENCH SECENARIO GRID SIZE, DX =
20.0 FEET EFFECTIVE POROSITY, N1 =
.350 Figure A-5 (Continued)
NVREG-1263 A-13 Appendix A
050 ROCK DIAMETER ON TOP, 051 =
.1600 FEET D50 ROCK DIAMETER ON SIDE SLOPE, D52 =
.4300 FEET THICKNESS OF TOP LAYER, THICK 1 =
1.00 FEET THICKNESS OF SIDE LAYER, THICK 2 =
1.50 FEET Y SLOPE ON TOP, SY1 =
.020 Y SLOPE ON SIDE, SY2 =
.200 X SLOPE ON TOP, SX1 =
.010 X SLOPE ON SIDE, SX2 =
.000 POSITION OF SLOPE BREAK DOWN SLOPE, A1 =
22 POSITION OF SLOPE BREAK IN Y DIRECTION, B1 =
6 DT =
2.00 SECONOS NUMBER OF STEPS, NT =
4000 PRINT INTERVAL KP =
20 TIME AT WHICH COMPUTAYIONS CO MENCE, TSTART =
900.00 SECONDS TIME AT WHICH TIMESTEP CHANGES, TCH =
3100.0 SECONDS NEW TIMESTEP, DTCH =
.500 SECONDS FRICTION FACTOR INDEX, K =
4.0 1 HOUR RAINFALL AMOUNT, R8 =
8.00 INCHES 084 DIAMETER FOR TOP SLOPE, 081 =
.240 FEET 084 DIAMETER FOR SIDE SLOPE, D82 =
.660 FEET TIME-SECONDS QCCENT - CFS/FT QBREAK - CFS/FT 940.00
.0013
.0006 980.00
.0028
.0010 1020.00
.0045
.0013 1060.00
.0063
.0017 1100.00
.0082
.0021 1140.00
.0103
.0026 1180.00
.0125
.0031 1220.00
.0148
.0038 1260.00
.0168
.0044 1300.00
.0182
.0052 (Output from t = 1340 to 3500 deleted from this listing) 3500.00
.3067
.5972 3510.00
.3167
.8466 3520.00
.3267
.9511 3530.00
.3368
.8940 3540.00
.3471
.7883 3550.00
.3575
.7445 3560.00
.3680
.9106 3570.00
.3787 1.0941 3580.00
.3895 1.2325 3590.00
.4004 1.0154 i
3600.00
.4116
.9212 3610.00
.4133 1.3787 3620.00
.4147 1.0800 3630.00
.4163
.7665 Figure A-5 (Continued) i NUREG-1263 A-14 Appendix A
3640.00
.4179 1.0779 3650.00
.4197 1.0749 3660.00
.4216
.8026 3670.00
.4238
.7534 3680.00
.4265
.8000 3690.00
.4299
.7712 3700.00
.4344
.7093 3710.00
.4402
.6794 3720.00
.4475
.6482 3730.00
.4560
.6159 3740.00
.4656
.5904 3750.00
.4759
.5707 3760.00
.4855
.5472 3770.00
.4947
.5239 3780.00
.5033
.5010 3790.00
.5107
.4789 3800.00
.5169
.4578 (Output from t = 3810 to 4500 deleted from this listing) 4500.00
.2155
.1387 4510.00
.2135
.1379 4520.00
.2115
.1369 4530.00
.2094
.1359 4540.00
.2074
.1348 4550.00
.2054
.1335 MAX FLOW AT BASE =.526569 CFS/FT a
MAX FLOW, TOP SLOPE = 1.41945 CFS/FT 4
Figure A-5 (Continued)
Trench Failure The input file for this case differs from the benchmark embankment example in that the horizontal slope for the first and third quadrant must now be specified as SX1 = 0.01, and the horizontal breakpoint must be set to B1 = 6 to represent the width of the failed trench.
Timesteps are as in the 1/2% slump example above. The revised input file is shown in Figure A-Sc with the changes high-lighted.
The peak flows for this case were computed to be 1.42 ft /sec/ft from 3
3 the top slope and 0.53 ft /sec/ft from the side slope.
An oscillation of the flowrate from the top slope is evident.
It is not known whether this is a real phenomenon or a computational artifact.
If the latter is the case, the peak flow from the top slope would be considerably smaller, representing a time-L averaged value.
A listing of program SLOPE 2D is given in Figure A-6.
r f
NUREG-1263 A-15 Appendix A
..,n_,-
---,.-r--
PROGRAM SLOPE 20 C
USNRC 12/12/86 SLOPE 2DI FORTRAN VERSION C
R CODELL C
20 RUN0FF FROM SLOPES C
C INPUT VARIABLES CXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX C
DX = GRID SPACING, FT C
N1 = EFFECTIVE POROSITY OF ROCK LAYERS C
051 = D50 FOR TOP SLOPE, FT C
D52 = 050 FOR SIDE SLOPE, FT C
THICK 1 = THICKNESS OF TOP SLOPE ROCK C
THICK 2 = THICKNESS OF SIDE SLOPE ROCK, FT C
SX1 = X SLOPE ON TOP C
SX2 = X SLOPE ON SIDE C
SY1 = Y SLOPE ON TOP C
SY2 = Y SLOPE ON SIDE C
A1 = POSIT 0N OF BREAK IN SLOPE DOWN THE HILL C
B1 = POSITION OF BREAK ACROSS HILL C
DT = INITIAL TIME STEP, SECONDS C
NT = TOTAL NUMBER OF TIME STEPS C
KP = NO. OF STEPS BETWEEN PRINTS OR PLOT POINTS C
TR = TIME ORDINATE FOR RAINFALL TABLE, SECONDS C
FR = FRACTION OF 1 HOUR PHP FOR RAINFALL TABLE C
TSTART = STARTING TIME FOR SIMULATION, SECONDS C
TCH = TIME AT WHICH SMALLER TIMESTEP BECOMES EFFECTIVE, SECONDS C
DTCH = SMALLER TIMESTEP, SECONDS C
K = FRICTIONAL INDEX FOR ROCK, E.G., 1 FOR SMOOTH ROCK, C
2 FOR ROUNDED, 4 FOR ANGULAR C
R8 = RAINFALL AMOUNT IN 1 HOUR PHP, INCHES C
081 = 084 ROCK DIAMETER FOR TOP SLOPE, FEET C
082 = 084 ROCK DIAMETER FOR SIDE SLOPE, FEET C
NCOL = NUMBER OF COLUMNS C
NROW = NUMBER OF R0WS C
ISTART = LEFT GRID BLOCK IN A R0W C
IEND = RIGHT GRID BLOCK IN A R0W C
JSTART = TOP GRID BLOCK IN A COLUMN C
JEND = BOTTOM GRID BLOCK IN A COLUMN C
C OUTPUT VARIABLES CXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX C QBRK = FLOWRATE AT SLOPE BREAK ALOMG CENTERLINE, CFS/FT C QCCENT = FLOWRATE AT BOTTOM OF SIDE SLOPE ALONG CENTERLIHE INTEGER A1.B1 COMMON HY(40),D50(40),5X(35),5Y(40),FR(20),TR(20),H(40),
1 08(40),N1,K T9,ITIME.R8,R,DT,DX,052,5Y2, CON 1, 2 CON 2, CON 3, CON 4, CONS A1,051. THICK 1,081,5Y1,082, 3 THICK 2 NROW,NCOL,B1,5X1,5X2,YBAR,KFAC DIMENSION SE(35,40),SEP(35,40),U(35,40),UP(35,40),
1 V(35,40),VP(35,40) ISTART(40),IEND(40),JSTART(40),
2 QC(40),JEND(40),QST(500),QSTB(500),TST(500)
REAL N1,K,KFAC,KF2 CHARACTER *80 TITLE Figure A-6 Listing of program SLOPE 2D NUREG-1263 A-16 Appendix A
CHARACTER *15 PTIT READ (5,'(A)') TITLE WRITE (6,111) TITLE 111 FORMAT (10X, PROGRAM SLOPE 20 - RUN0FF FROM SLOPES',/
1 5X,A)
C C
READ IN THE RAINFALL TABLE C
READ (5,*) NRAIN do 1 i=1,nrain 1 READ (5,*) TR(I),FR(I)
READ (5,*) DX,N1,051,052, THICK 1, THICK 2,SX1,SX2,SY1,SY2,A1,81 WRITE (6,03) DX,N1,051,052 99 FORMAT (10X,' GRID SIZE, DX = ',F10.1,'
FEET'/
1 10X,' EFFECTIVE POROSITY, N1 = ',F10.3,/
2 10X,'D50 ROCK DIAMETER ON TOP, D51 = ',F10.4, '
FEET',/
3 10X,'D50 ROCK DIAMETER ON SIDE SLOPE, 052 = ',F10.4, 4 ' FEET')
WRITE (6,100) THICK 1, THICK 2 100 FORMAT (10X,' THICKNESS OF TOP LAYER, THICK 1 = ',
1 F10.2,'
FEET',/10X,' THICKNESS OF SIDE LAYER, THICK 2 = ',
1 F10.2,'
FEET')
WRITE (6,101) SY1,5Y2,5X1,5X2 WRITE (6,106) A1,81 106 FORMAT (10X,' POSITION OF SLOPE BREAK DOWN SLOPE, A1 ='
1,Il0,/10X, 2 ' POSITION OF SLOPE BREAK IN Y DIRECTION, B1 = ',110) 101 FORMAT (10X,'Y SLOPE ON TOP, SY1 = ',F10.3,/
1 10X,'Y SLOPE ON SIDE, SY2 = ',F10.3,/
2 10X, 'X SLOPE ON TOP, SX1 = ',F10.3,/
3 10X,'X SLOPE ON SIDE. SX2 = ',F10.3)
READ (5,*) DT,NT,KP,TSTART WRITE (6.102) DT,NT,KP,TSTART 102 FORMAT (10X,'DT = ',F10.2,'
SECONDS',/10X, 1 ' NUMBER OF STEPS, NT = ',Il0,/
1 10X,' PRINT INTERVAL KP = ', 110,/
2 10X,' TIME AT WHICH COMPUTATIONS COMMENCE, TSTART = ',
3 F10.2,'
SECONDS')
READ (5,*) TCH,DTCH WRITE (6,103) TCH,DTCH 103 FORMAT (10X,' TIME AT WHICH TIMESTEP CHANGES, TCH = ',
1 F10.1,'
SECONDS',/10X,'NEW TIMESTEP, DTCH = ',
2 F10.3,' SECONDS')
KTCH=0 READ (5,*)
K,R8,081,082 WRITE (6,104) K,R8,081,082 104 FORMAT (10X,' FRICTION FACTOR INDEX, K = ',F10.1,/
1 10X,'1 HOUR RAINFALL AMOUNT, R8 = ',F10.2 '
INCHES',/
2 10X,'084 DIAMETER FOR TOP SLOPE, 081 = ',F10.3,'
FEET',/
3 10X,'084 DIAMETER FOR SIDE SLOPE, D82 = ',F10.3, '
FEET',/
4)
Figure A-6 (Continued)
NUREG-1263 A-17 Appendix A
C INPur GRID DEFINITION C
READ (5,*) NCOL,NROW DO 3 I=2,NCOL 3 READ (5,*) JSTAR7(I),JEND(I)
DO 4 J=2,NROW 4 READ (5,*) ISTART(J),IEND(J)
R8=R8/(3600*12)
CALL SETUP CALL SETCON data kk,np1,kkk/3*0/
data qmax,qbreax/2*0.0/
t itime=0 r=0 C
INITIALIZE GRID 00 5 I=1,NCOL 00 5 J=1,NR0W SE(I,J)=0 SEP(I,J)=0 t
U(I,J)=0 V(I,J)=0 UP(I,J)=0 t
VP(I,J)=0 5 CONTINUE
- L C
C BEGIN NUMERICAL SOLUTION C
i T9=TSTART j
R=0 WRITE (6,6) 6 FORMAT (10X,' TIME-SECONDS',8X, 'QCCENT - CFS/FT',8X, 1 'QBREAK - CFS/FT')
DO 77 LO=1,NT IF (T9.LT.TCH) GOTO 800 IF(KTCH.EQ.1) GOTO 800 KTCH=1 DT=DTCH CALL SETCON 800 CONTINUE C
GET RAINFALL RATE ONTO SLOPE CALL RAIN C
GET START AND FINISH OF EACH COLUMN VECTOR l
00 7 I=2,NCOL IM1=I-1 IPl=I+1 JS=JSTART(I)
JE=JEND(I) l C
CALCULATE SEP AT EACH WET POINT 00 8 J=JS JE-1 JM1=J-1 Figure A-6 (Continued) 1 NUREG-1263 A-18 Appendix A
i DIFX=(SE(IM1,J)+SE(I,J))*U(IM1,J)-(SE(IP1,J)+SE(I,J))*U(I,J)
DIFY=(SE(I,JM1)+SE(I,J))*V(I,JM1)-(SE(1,J+1)+SE(1,J))*V(I,J)
SEP(I,J)=SE(I,J)+R* CON 2+ CON 1*(DIFX+DIFY)
Ib C
NORMAL FLOW BOUNDARY CONDITION AT COLUMN BOTTOMS DO 9 I=2,NCOL IM1=I-1 IPl=I+1 JE=JEND(I)
QC(I)=0 KFAC=1 YBAR=SE(I,JE)
J=JE C
CHANGE FLOW RESISTANCE WHEN ROCK TOP OVERFLOWS IF (YBAR.GT.H(JE)) CALL SKFAC(J)
IF(SE(I,JE).GT.0) QC(I)=SE(I,JE)*SQRT(CON 5/KFAC)
DIFX=(SE(IM1,JE)+SE(I,JE))*U(IM1,JE)-(SE(IP1,JE)+SE(I,JE))*U(I JE)
DI FY=(SE( I.J E-1)+ S E(I,J E ))*V(I, J E-1)
SEP(I,JE)=SE(I,JE)+R* CON 2+ CON 1*(DIFX+DIFY)-QC(I)* CON 3 9 CONTINUE C
CALCULATE VELOCITY DO 10 I=2,NCOL im1=i-1 JS=JSTART(I)
JE=JEND(I) 00 11 J=JS,JE-1 JM1=J-1 JPl=J+1 UBRBR=(U(I,JP1)+U(I,J)+U(IM1,J)+U(IM1,JM1))/4 VABS=V(I,J)**2+UBRBR**2 KF2=1 KFAC=1 YBAR=(SEP(I,J)+SEP(I,JP1))/2 IF(YBAR.GT.H(J)) CALL SKFAC(J)
Yl=SEP(I,J)
C CALCULATE VIRTUAL DEPTH IF(Y1.GT.HY(J).AND.KFAC.LT.1) Yl=HY(J)+(Y1-HY(J))*N1 Y2=SEP(I,JP1)
IF(Y2.GT.HY(JP1).AND.KFAC.LT.1) Y2=HY(JP1)+(Y2-HY(JP1))*N1 IF (VABS.GT.O.0) KF2=SQRT(1+UBRBR**2/VABS)*KFAC ARG=$Y(J)-(Y2-Y1)/DX VP(I,J)=5QRT(CON 4/KF2*D50(J)* ABS (ARG))* SIGN (1.0,ARG) 11 CONTINUE VP(I,JE)=VP(I,JE-1) 10 CONTINUE 00 12 J=2,NROW jm1=j-1 jp1=j+1 VP(1 J)=VP(2,J)
C GET START AND FINISH OF EACH R0W VECTOR IS=ISTART(J)
IE=IEND(J)
Figure A-6 (Continued)
NUREG-1263 A-19 Appendix A
I D0 13 I=IS,IE-1 IPl=I+1 IM1=I-1 KFAC=1 YBAR=(SEP(I,J)+SEP(IP1,J))/2 C
REDUCE FRICTION IF OVERTOPPING OCCURS IF (YBAR.GT.H(J)) CALL SKFAC(J)
C CALCULATE VIRTUAL DEPTH Y3=SEP(I,J)
IF(Y3.GT.HY(J).AND.KFAC.LT.1.0) Y3=HY(J)+(Y3-HY(J))*N1 Y4=SEP(IP1,J)
IF(Y4.GT.HY(J).AND.KFAC.LT.1.0) Y4=HY(J)+(Y4-HY(J))*N1 ARG=SX(I)-(Y4-Y3)/0X vbrbr=(vp(i,j)+vp(ipl,j)+vp(1,jml)+vp(im1,jml))/4 vab=sqrt(vbrbr**2+u(1,j)**2) up(i,j)=arg* con 4/kfac*d50(j)/vab 13 continue 12 centinue do 14 i=1,ncol do 15 j=1.nrow u(i,j)=up(i,j)
SE(I,J)=SEP(I,J)
V(I,J)=VP(I,J) 15 CONTINUE 14 CONTINUE T9=T9+0T C
FLOWRATE AT BREAK BETkEEN TOP AND SIDE SLOPE QBRK=(.5625*(SE(2,A1)+SE(2,A1+1))
.0625*(SE(3,A1)+
1 SE(3,A1+1)))*(1.125*V(2,A1),125* V(3,A1))
QCCENT=1.125*QC(2).125*QC(3)
IF(QMAX.LT.QCCENT) QMAX=QCCENT IF(QBRMAX.LT.QBRK) QBRMAX=QBRK KKK=KKK+1 IF(KKK.LT.KP) GOTO 16 WRITE (6.90) T9,QCCENT,QBRK 90 FORMAT (12X,F8.2.10X,f8.4,14X,F8.4,2F8.4)
C STORE VALUES OF RUNOFF TO BE PLOTTED LATER NPL=NPL+1 l
TST(NPL)=T9 QST(NPL)=QCCENT QSTB(NPL)=QBRK i
i KKK=0 16 CONTINUE I
77 CONTINUE WRITE (6,*) '
MAX FLOW AT BASE = ',QMAX,' CFS/FT' l
WRITE (6,*)'
MAX FLOW, TOP SLOPE = ',QBRMAX,' CFS/FT' STOP END C
SUBROUTINE SKFAC(J) l l
INTEGER A1,81 COMMON HY(40),050(40),5X(35),5Y(40),FR(20),TR(20),H(40),
i Figure A-6 (Continued) i NUREG-1263 A-20 Appendix A
1 08(40),N1,K,T9,ITIME,R8,R,DT.DX,052 SY2, CON 1, 2 CON 2, CON 3, CON 4, CONS,A1,D51, THICK 1,081,SY1,082, 3 THICK 2,NROW,NCOL,81,SX1,5X2,YBAR,KFAC REAL N1,K,KFAC C
CALCULATE REDUCTION IN K FOR OVERTOPPING C
DARCY-WEIS8ACH FLOW ASSUMED C
HIGH REYNOLDS NUMBER ASSUMED DH=(YBAR-H(J))*N1 DWF=.00001 DH3=3.85*DH IF(DH3.GT.D3(J)) DWF=0.881*ALOG(DH3/08(J))
ALPHA =H(J)*SQRT(D50(J)/K)+DH/N1*SQRT(8*DH)*DWF KFAC=D50(J)*(YBAR/ ALPHA)**2/K IF(KFAC.GT.1.0) KFAC=1 RETURN END C
SUBROUTINE RAIN INTEGER A1,81 CO M N HY(40),050(40),5X(35),5Y(40),FR(20),TR(20),H(40),
1 08(40),N1,K,T9,ITIME,R8 R.DT,0X,052,SY2, CON 1, 2 CON 2, CON 3, CON 4, CONS,A1,051 THICK 1,081,SY1,082, 3 THICK 2,NROW,NCOL,B1,5X1,5X2,YBAR,KFAC i
C C
GENERATE RAINFALL RATE C
l IF(T9.LT.TR(ITIME+1)) RETURN
]
ITIME=ITIME+1 R=R8*FR(ITIME) 4 RETURN END C
SUBROUTINE SETCON 1
C C
SET UP CONSTANTS WHICH ARE TIMESTEP DEPENDENT s
C INTEGER A1.B1 l
COEN HY(40),050(40),5X(35),5Y(40),FR(20).TR(20),H(40),
t 1 08(40),N1,K,T9,ITIME.R8,R,DT,DX,052,5Y2, CON 1, 2 CON 2, CON 3. CON 4, CONS,A1,051. THICK 1,081,5Y1,082, l
1 3 THICK 2 NROW,NCOL,B1,5X1 SX2,YBAR,KFAC REAL N1,K L
l CON 1=DT/(2*N1*DX)
CON 2=DT/N1 l
CON 3=D's A DX*N1)
CON 4=32.2*N1*N1/K CON 5=32.2*D52*N1*N1*SY2/K L
RETURN END SUBROUTINE SETUP Figure A-6 (Continued) l l
NUREG-1263 A-21 Appendix A
[
l l
C C
SETUP GRID, SET SLOPES, ROCK DIA AND THICKNESS C
INTEGER A1,B1 Com0N HY(40),050(40),SX(35),SY(40),FR(20),TR(20),H(40),
1 D8(40),N1,K,T9,ITIME.R8,R.DT,0X,D52 SY2, CON 1, 2 CON 2 CON 3, CON 4, CON 5,A1,051, THICK 1,081,5Y1,082, 3 THICK 2,NROW,NCOL,81,SX1,5X2,YBAR,KFAC REAL N1,K c
correct layer thickness to match through and under flow RH1=D51 DO 7 I=1,10 RH1=(3.5*D81/13.486)*EXP(SQRT(051/(8*RH1*K))*N1/0.881) 7 CONTINUE RH2=D52 DO 8 I=1,10 RH2=(3.5*D82/13.486)*EXP(SQRT(052/(8*RH2*K))*N1/0.881) 8 CONTINUE THICK 1= THICK 1-RH1 THICK 2= THICK 2-RH2 00 1 J=1,Al-1 050(J)=D51 H(J)= THICK 1 08(J)=D81 SY(J)mSY1 1 CONTINUE 050(A1)=(051+052)/2 08(A1)=(081+D82)/2 H(A1)=(THICKl+ THICK 2)/2 SY(A1)=(SYl+SY2)/2 00 2 J=A1+1,NROW 050(J)=D52 H(J)= THICK 2 08(J)=D82 SY(J)=SY2 2 CONTINUE 00 3 I=1,B1-1 SX(I)=SX1 3 CONTINUE SX(B1)=($X1+SX2)/2 00 4 !=Bl+1,NCOL SX(I)=SX2 4 CONTINUE 00 5 !=1,NROW-1 5 HY(I)=(H(I)+H(IP1))/2 RETURN ENO Figure A-6 (Continued)
NUREG-1263 A-22 Appendix A
APPENDIX B PROGRAM ROCKSIZE Program ROCKSIZE is a BASIC language computer code to aid in the evaluation of the stable rock sizes for the hydrologic protection of tailings embankments from the effects of runoff.
Maximum runoff rates predicted from program SLOPE 2D or elsewhere are input to program ROCKSIZE, along with the physical attributes of the riprap and the slope of the embankment.
The stable rock diameter is determined from the safety factor method and the Stephenson method, as discussed in Chapter 3.
The water level above the top of the rock layer surface, necessary for calculat-ing of the safety factor method, is determined iteratively by evaluating the formule.:
2 2/3 I+1 =
a + (1 - a)y i (B-1) y g
3.85y)V8gs i
0.881 log ds4
/
i where a is the convergence factor less than unity (typically 0.2) and q3 is the flow over the top of the rock.
If the water level is not higher than the surface of the rock layer, then the safety f : tor method is not used.
i i
The stable rock diameter determined by the safety factor method includes the design safety factor specified in the input.
The diameter determined by the Stephenson method must be scaled up manually for the desired factor of safety.
Program ROCKSIZE is listed in Figure B-1.
The program is interactive and requests the following information:
K friction index k for flow through rock layer; i.e., K a 1 for i
smooth marbles, K =2 for smooth gravel, K a 4 for crushed, l
angular rock OBAR average rock diameter, usually the harmonic mean, ft
~
084 84% finer rock diameter, ft H1 thickness of rock layer, ft l
5 slope of embankment N1 effective porosity of rock 3
Q peak downhill runoff to which rock is subjected, ft /sec/ft SF safety factor for SF method only 2
PH angle of repose, degrees, from Figure 3-1 l
C smoothness factor for Stephenson method only 1
i NUREG-1263 B-1 Appendix B i
l
10 REM PROGRAM ROCKSIZE 20 REM TO DETERMINE THE STABLE ROCK DIAMETER FOR A GIVEN FLOWRATE 30 REM DOWN AN ARMOR-COVERED SLOPE 40 REM USING STEPENS% AND SAF ACTOR FORMULAS 50 REM REFERENCE W REG-1263, 1988 SEPTEMBER 7 60 REM R CODELL, US NUCLEAR R RY CO MISSION 70 REM WASHINGTON DC 20555 80 REM 90 PRINT" PROGRAM ROCKSIZE" 100 PRINT" DETERMINE THE STABLE DIAMETER FOR RIPRAP ON ARMORED SLOPES" 110 PRINT" BY STEPHENSON AND SAFETY FACTOR METH00" 120 PRINT" U.S. NUCLEAR REGULATORY COMISSION, WASHINGTON 0.C."
130 REM INPUT DATA 140 REM K = FRICTION INDEX FOR ROCK t
150 REM DBfR = MEDIAN ROCK DIAMETER, FT 160 REM 084
- 84 PERCENTILE FINER ROCK DIAMETER, FT 170 REM N1 = EFFECTIVE POROSITY 180 REM H1 = RIPRAP LAYER THICKNESS, FT 190 REM S = SLOPE 200 REM PH = ANGLE OF REPOSE 210 REM SS = SPECIFIC GRAVITY OF THE ROCK l
220 REM C = STEPHENSON CONSTANT = 0.22 FOR SMOOTH ROCK 230 REM 0.27 FOR CRUSHED, ROCK 240 REM i
250 PRINT"INPUT FRICTION INDEX, K ";
260 INPUT K 270 PRINT"ENTER OBAR, 084, FT ";
280 INPUT OBAR,084 290 PRINT"ENTER LAYER THICKNESS, FT ";
I 300 INPUT H1 310 PRINT"ENTER SLOPE ";
320 INPUT S 330 PRINT"ENTER EFFECTIVE POROSITY ";
340 INPUT N1 t
4 350 REM l
360 REM CORRECT ROCK LAYER THICKNESS FOR AGREEMENT OF THROUGH AND OVERFLOW" 370 REM 380 RH=0BAR t
390 FOR I=1 TO 10 400 RH=(3.5*084/13.46)*EXP(SQR(DBAR/(8*RH*K))*N1/.881) 410 NEXT I 420 PRINT "CORRECTION TO LAYER THICKNESS = ";RH;" FEET" 430 H1=H1-RH 440 PRINT"ENTER PEAK RUNOFF, CFS/FT ";
4 450 INPUT Q i
460 V=SQR(S*32.2*N1'2*DBAR/K) 470 Q0=H1*V 480 IF Q<=QO THFN 490 ELSE 520 490 PRINT"FLOW IS BELOW ROCK SURFACE - SAFETY FACTOR METHOD DOES NOT APPLY" 500 GOTO 710 i
$10 REM l
520 REM CALCULATE STAGE ABOVE ROCK SURFACE t
Figure B-1 Listing of program ROCKSIZE f
NUREG-1263 B-2 Appendix B r
530 REM 540 DQ=Q-Q0 550 Y=D84 560 C1=8*32.2*S 570 C2=SQR(C1) 580 REM A1 = CONVERGENCE FACTOR FOR CALCULATION OF STAGE ABOVE ROCK 590 Al=.2 600 FOR J=1 TO 100 610 DWF=.881* LOG (3.85*Y/084) 620 YP=((DQ/(DWF*C2))'.66667 *A1)+(1-A1)*Y 630 IF YP>D84/3.85 THEN 640 Y=YP 650 GOTO 670 660 ELSE Y=1.01*D84/3.85 670 REM CONTINUE 680 NEXT J 690 PRINT 700 PRINT "STAGE ABOVE ROCK SURFACE = ";Y;" FT" 710 REM CONTINUE 720 REM 730 REM SAFETY FACTOR METHOD I
740 REM 750 ELSE 760 PRINT"ENTER ANGLE OF REPOSE, DEGREES";
770 INPUT PH 4
780 PRINT"ENTER SPECIFIC GRAVITY OF ROCK, GM/CC ";
790 INPUT SS 800 IF Q<=Q0 THEN 910 810 TS=62.4*Y*5 820 AL=ATN(S) i 325 PRINT"ENTER SAFETY FACTOR ";
827 INPUT SF 830 ET=COS(AL)*(1/SF-5/ TAN (PH/57.3))
840 01=21*TS/((SS-1)'62.4*ET) 850 PRINT 860 PRINT 870 PRINT"STABLE ROCK OIAMETER BY SAFETY FACTOR METHOD = ";D1;" FEET" i
88u PRINT 890 REM CONTINUE 900 REM i
1 910 REM STEPHENSON METHOD 920 REM ROCK OIAMETER FOR INCIPIENT MOTION 930 C1=Q*S'(7/6)*N1'.166667 940 PRINT"ENTER SMOOTHNESS FACTOR, C IN STEPHENSON FORMULA" l
950 PRINT"(0.22 FOR SMOOTH ROCK AND 0.27 FOR ANGULAR CRUSHED ROCK)"
r 960 YNPUT C j
970 C2= C*SQR(32.2)*((1-N1)*(SS-1)*COS(AL)*(TAN (PH/57.3)-SY))*1.66667 980 02=(C1/(1.2*C2))'.66667 1
990 PRINT 1000 PRINT 1010 PRINT "STABLE ROCK OIAMETER BY STEPHENSON METHOD $ ";D2;" FEET "
1020 END l
Figure B-1 (Continued)
J
)
NUREG-1263 B-3 Appendix B l
test Po#88 35 W & 4WCLlast etIGUL.ToA Y Coe.wisetoe, i aecomi =uweaa seu gw e, r,0c, ese re, he, w e,s (2 ede Ifo',"E:
Bl800 GRAPHIC DATA SHEET NUREG-1263 lei 1481mWCTso4s om feet DE vt858 2 title.=o sheisits alt.vg36...
Hydrologic Design for Riprap on Embankment Slopes
. o.T g stront Cow' stye D wo*s t es vgam i w. oaise May 1988 8 o.f t 26Po47 88Sulo R.B. Codell wo=1 September 1988
'77e_o.wioo.a.mi.,,o==.wi.~o.is*or,o en,..
e,c,
s raoac, vai. woa. e., wwu a Division of High-Level Waste Management Of ff ee of Nuclear Material Safety and Safeguards U.S. Nuclear Regulatory Commission Washington, DC 20555 iuro soa =o o
.+4.v.o= =.wi.=o
.i6 o.ooae n,~w. e. c,
ii.1,,e o, a i *o.,
Technical Same as box 7 e ein oo co.iano n.
II Sv PLlwl=Tamy=of33 e
13 assim.CT IA10 eeres er oss, Waste impoundsents for uranium tallings and other hazardous substances are often protected by compacted earth and clay, covered with a layer of loose rock (rip-rap). The report outlines procedures that could be followed to design riprap te withstand forces caused by runoff resulting from extreme rainfall directly on the eebanksent. The Probable Maximum 8recipitation fer very small areas is developed froa considerations of severe stores of short duration at mid latitudes.
A two-dimensional finite difference model is then used to calculate the runoff from severe rainfall events. The procedure takes into account flow both beneath and above the rock layer and approximates the concentration in flow which could be caused by a non level or s'*eped embanksent. The sensitivity h various assumptions, such as the shape and site of the rock, the thickness of the layer, and the shape of the embanksent, suggests that peak runoff from an armored slope could be attenuated with proper design. Frictional relationshipt for complex flow re01ees are developed on the basis of flow through rock filled daes and in mountain streams. These relationships are tested against experimental data collected in laboratory f1wees; the tests preside en:ellent results. The re*
sultieg runoff is then used in either the Stephenson or safety factor method to find the sMie rock disteter. The rock slies determined by this procedure for a given flow have taen compared with data on the failure of rock layers in ex-perleental flumes, again with oncellent results. Computer programs are included for implementing the method.
.. ooc s...... 6.........c a c nc. - o..
,. 7,. g,.
mine tailings, riprap, embankments, hydrologic models Probable Unlimited Maximum Precipitation (PMP), Mathematical Models
' S $4 Cve #T, C t. ll<. iC.f sca, a..,.,,
Unclassified
. es=,... oei=e=eso,s ws
,r...,,
Unclassified
, wwua ce.. sis i s, a.4.
_