the filter surface changes with time as a result of filtration. Three filter experiments .... A nonzero value of the filtration coefficient indicates that filtration occurs and ...
Although the equations in Table 3 are shown to be significant, it would be interesting to see the standard error of estimate of the adjusted CN. If these errors are larger than the proposed adjustment, then the adjustment is questionable. The adjustment to CN for the normal range of CN (60-90), as shown by the Table 3 equations, is less than 5.
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FIG. 3
Fig. 3 shows values to the left of Q/P = O. The authors are correct in stating that only values to the right of Q/P are valid. Values to the left are for points where (P - 0.2S) < 0, which means that the rainfall is less than the initial abstraction and runoff must be zero. This should be clear if one follows the derivation of (2) (USSCS 1985). When the interrelationship between Q, P, and CN is displayed in the usual form shown in Fig. 5, it becomes more apparent that Q = 0 where P = 0.2S.
APPENDIX.
REFERENCES
Hawkins, R. H. (1993). "Asymptotic determinations of runoff curve numbers from data." J. lrrig. and Drain. Engrg., ASCE, 119(2), 334345. United States Soil Conservation Service. (1986). "Urban hydrology for small watersheds." Tech. Release 55. U.S. Dept. of Agric., Washington, D.C.
Closure by G. W. Titmarsh,6 I. Cordery,' and D. H. Pilgrim8 Those who contributed to discussion of this paper are thanked for their efforts. The writers reiterate that use of parameter values estimated using descriptive catchment characteristics-curve numbers for the USSCS Method and runoff coefficients for the rational method-should be phased out. This is because such an approach is inherently unreliable. If either method is to have continued use, maps similar to that shown in Fig. 1 of the paper should be developed for the region of interest. In the United States there are now adequate data from a number of sources to permit this. In the paper, the writers show that curve number (CN) varies with average recurrence interval (ARI). Examination of the results presented by Hjelmfelt (1980) also suggest that there may be some variation of CN with changing ARI. As pointed out by the discussors, the changes suggested by Table 3 of the paper are small, but they are based on a large data set (105 catchments). Similar results were obtained for a further 43 catchments located near Canberra, Australia (Hoesein et al. 1989).
6 Soil Conservationist, Queensland Dept. of Primary Industries, P.O. Box 102, Toowoomba, Australia, 4350. 7 Assoc. Prof., School of Civ. Engrg., Univ. of New South Wales, P.O. Box I, Kensington, Australia, 2033. sProf., School of Civ. Engrg., Univ. of New South Wales, P.O. Box 1, Kensington, Australia, 2033.
DESIGN RELATIONSHIP FOR FILTERS IN B BED PROTECTION
Discussion by Anders Worman 4 The authors provide a significant contribution to the hydraulic filter design of riprap protections for canal bottoms. This design principle implies that for a sufficiently small hydraulic load, the base material can be considered to be stable even if the geometrical filter criterion of Terzaghi type is not satisfied. The authors compare the results of Worman (1989) with experimental results conducted at Delft (The Netherlands) HydraulicslGeotechnics. However, these two investigations concern two different load situations: (1) filter protection of a local bottom area around a pier in streaming water; and (2) a general filter protection of canal bottoms. The difference may have to be considered in transforming (9) into (10) and comparing the result with (11). Eq. (11) states a relationship between the critical grain-size ratio and the hydraulic load represented in terms of the water depth initially introduced in (1). Contrarily, (1) is not a relevant expression for the bottom shear stress arising in the vicinity of a pier in water. A prime issue to hydraulic filter design in general is the definition of inception of motion of the base material. In the case of local scour around a pier, studies by Chiew (1984), Parola et al. (1993), and others indicate that the maximum local erosion depth is finite and decreases with decreasing hydraulic load. In the case of a general erosion along a canal bottom, the maximum erosion depth is not limited in the same way as for local scour. Experiments made by Worman and Olofsdottir (1992) of similar type as the one reported by the authors indicates that an absolutely stable bed does not exist (or cannot be verified) for bed shear stresses larger than zero. For uniform base soils the inception of motion state used by the authors seem to correspond to a small but finite transport rate of G/V d ss = l' 1O-s, where G is volume transport rate per unit width in units (m%) and V is mean pore velocity in the filter. A motivation for this transport parameter is given by Neill and Yalin (1969). For instance, if the pore velocity is 0.05 mls and the base material size is 0.001 m, the transport rate is G = 5 '10- 10• During a hundred-year period, over 4 t of the base material has then been transported per unit width through a fixed cross section of the canal. Depending on the problem variables (particularly the longitudinal boundary conditions for the canal), the long-term effect of a small but finite erosion rate might be essential to take into consideration in design. For a broadly graded base soil, successive clogging at the base/filter interface with time tends to decrease the base material surface available to erosion. This causes the transport rate to slowly decrease with time, thus, the long-term effect of erosion is counteracted. The relationship between grain-size ratio, hydraulic load, and transport rate (mobility state) along the filter surface changes with time as a result of filtration. Three filter experiments of similar type as described by Worman and Olafsdottir (1992) have recently been conducted in which the filter material was constant. Also the cumulative grain-size distribution of the base soil was constant in the grain size range (do = 0.25 mm; d ss = 0.04 mm) in all three experiments. The maximum grain size was altered between duX! = 0.5, 1.0, and 2.0 mm, hence resulting in varying gradation only ·September 1994, Vol. 120, No.9, by K. J. Bakker, H. J. Verheij, and M. B. de Groot (Technical note 5493). 4 Assoc. Prof., Inst. of Earth Sci., Uppsala Univ.• Norbyviigen 18B, 752 36 Uppsala, Sweden. JOURNAL OF HYDRAULIC ENGINEERING / MARCH 1996/177
J. Hydraul. Eng. 1996.122:177-178.
100
-I' 10
[ID 3. Exp2 +
+
o
0
• ..2L clss 1-11
dd
.1
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.01
,001 ,01
,1
1
10
100
We agree on the two different load conditions: (1) filter protection of a local area around a pier; and (2) a filter protection of a canal bottom. We also agree that the first equation, (1), is not a relevant expression for the shear stress near piers. Nevertheless, the resulting equations, (10) and (11), have the same structure and, unfortunately, also the coefficients are nearly equal (1.06 versus 1.09). The resemblance concerning the coefficient is mainly due to the applied value for the parameter Co. Based on the test results and taking into account a safety factor, we selected a value of 15. The average value of Co as a result of the tests was 6, If we substitute this value (more or less an expected value from a statistical point of view) into (11), the result is
..l.. cis.
FIG. 4. Typical Filtration Parameter (Ordinate) As Function of Relative Erosion Depth (Abscissa)
for the grain-size fraction larger than d s,. Fig. 4 shows how a filtration parameter defined by W6rman and Skoglund (1992) decreases with increasing erosion depth in the base soil, ~. The grain size, d s" concern the parent base material, the mean grain size of the clogged grains = del, the volume fraction of these grains in the parent material = Tl, and e = empirical coefficient varying within the range 0 < e < 2. A nonzero value of the filtration coefficient indicates that filtration occurs and causes a decreasing transport rate with time. Thus, in all experiments the transport rate slowly decreases with time and no significant differences in the filtration parameter could be verified. The fact that the filtration parameter decreases as erosion proceeds implies that filtration causes the transport rate to decrease with time/erosion depth. This behavior can be explained if the filtration occurs only in a limited number of pore channels that are successively filled up. Consequently, a completely stable base material could not be experimentally verified in these filtration experiments. The long-term effect of a small but finite erosion rate may be essential to take into account in hydraulic filter design, particular, in canal design.
APPENDIX.
DIU
ab
db"
a, D,"
R
(27)
-=2.72--
REFERENCES
Chiew, Y. M. (1984). "Local scour at bridge piers." Rep. No. 355, Dept. Civ. Engrg., Auckland Univ., Auckland, New Zealand. Neill, C. R., and Yalin, M. S. (1969). "Quantitative definitions of beginning of bed movement." J. Hydr. Div., ASCE, 95(1), 585-588. Parola, A. C. (1993). "Stability of riprap at bridge piers." J. Hydr. Engrg., ASCE, 119(10).1080-1093. Worman, A., and Skoglund, M. (1992). "Overtopping of the core in rockfill dams-internal erosion." HydroPower92, A. A. Balkema, Rotterdam, The Netherlands. Worman, A., and Olafsdottir, H. (1992). "Erosion in a granular medium interface." J. Hydr. Res., 30(5).
Closure by K. J. Bakker,S H. J. Verheij,6 and M. B. de Groot' The writers thank the discusser for his interest in the paper. Firstly, because it gives us the opportunity to clarify some aspects; secondly, the discusser's ideas on transport of base material are interesting. SHead of Struct. Res., Public Works and Water Management, P.O. Box 20.000, 3502 LA Utrecht, The Netherlands. •Sr. Advisor, Delft Hydraulics, P.O. Box 152, 8300 AD EMMELOORD, The Netherlands. 7 Consultant, Delft Geotechnics, P.O. Box 69, 2600 AB DELFT, The Netherlands.
Eqs. (10) and (27) now differ by a factor of 2.5 which is attributable to the different load conditions, for instance the different turbulence [see also (25) for Co that includes the turbulence factor r). The resemblance with respect to the formula structure is less surprising, as in Stephenson (1979), where a method was published to develop a stability criterion on the basis of the analogy between the erosion of base material and armor layer material. The method results in the following equation:
-DIU = 8.0 -ab - R db"
(28)
a, D,"
which can be transposed, assuming D,..
a
DIU b R -=6.4-db" a, D,"
= 1.25 D,'s, into (29)
Our (8) may be written in the same way. Therefore, we substitute the values e = 0.24, Co = 6 and assume t!Jb = t!J,. This results in (30)
Thus, the structure of the formula seems correct. We tried in our research to incorporate into the factor Co the influence of turbulence r and relative layer thickness dJR. We are still doing some research on this subject. Furthermore, the discusser mentions that the maximum erosion depth of a scour hole is limited and that the maximum erosion depth along a canal bottom is not limited in the same way as for local scour. We agree that for bed shear stresses larger than zero, an absolutely stable bed does not exist. However, the transport rate of base material is small and decreasing with time/erosion depth, as shown implicitly in Fig. 4 presented by the discusser. We did not have the intention to develop a formula to predict the transport of base material. We had the objective to present a stability criterion for the design of granular filters of bed protections near structures, for instance downstream sluices or near piers. Therefore, we accepted a critical shear stress accounting for phenomena such as armoring and clogging at the base/filter interface and assumed that the long-term effect of a small but finite erosion may be neglected, particularly because the protected area near structures is relatively small compared with the protection of a canal bed. On the contrary, the discusser has the intention to predict the transport of base material. The information provided by the discusser on transport rates of uniform and broadly graded base material is very interesting and we endorse his approach.
APPENDIX.
REFERENCE
Stephenson, D. (1979). Rockfill in hydraulic engineering. Elsevier Publishing Company, Amsterdam, The Netherlands.
178/ JOURNAL OF HYDRAULIC ENGINEERING / MARCH 1996
J. Hydraul. Eng. 1996.122:177-178.
