THE IRRIGATION AGE. 



857 



an embankment of broken stone, or the rush of water through 

 the crevices and open spaces between large boulders, the safe 

 permissible velocity of flow must be confined to that which 

 the material will stand without the movement, displacement, 

 or abrasion of the particles which compose it. 



Turneaure and Russell* refer to numerous experiments 

 which have been carried out at various times to determine the 

 velocity at which water will flow through sands and gravels 

 of different grades of fineness. These experiments were made 

 for various purposes, such as to determine the dimensions of 

 sand filters, the probable supply of water from wells, etc. 



D'Arcy, Hagen, Hazen, and others, found that for grades 

 of sand varying from very fine sand to fine gravel, the velocity 

 of flow closely followed the law of flow through capillary 

 tubes. 



In considering the flow of seepage, or ground-waters, 

 we are not concerned in refinements of experimental data 

 which indicate the influence of temperature on the rate of 

 flow ; therefore, after eliminating these, the result of the ex- 

 periments is the following formula : 



V = 8,200 d-s. 



In which (" = the velocity of flow, in feet per day. 



d = the effective size of the sand grains (.varying from 

 0.1 mm., which is very fine sand, to 3 mm., which is fine 

 gravel) ; and 



j = the slope of the ground water surface (hydraulic- 

 gradient). 



According to this formula, the velocity of flow of ground- 

 water, through fine sand having an effective size of grains of 

 0.1 mm. and a general slope of ground-water surface of 

 5 ft. per mile (hydraulic gradient = 0.001), would be: 



y = 8,200 <fs = 8,200 X 0.01 X 0.001 = 0.082 ft. per day. 



The velocity of flow through gravel having an effective 

 diameter of grains of 3 mm. and a general slope of water 

 surface of approximately 50 ft. per mile (hydraulic gradient 

 = 0.01) by this formula would be: 



V 8,200 d~s = 8,200 X 9 X 0.01 738 ft. per day 

 =r approximately 0.0085 ft. per sec. 



The Report of the Massachusetts State Board of Health 

 for 1902 contains the results of a series of experiments to 

 determine the velocity of flow of water which it is possible 

 to obtain through screened gravel, of various grades of fine- 

 ness, assuming 40 per cent porosity. These results are shown 

 in Table I. 



From an examination of the figures in Table I, it be- 

 comes evident that all the foregoing experimental data are, 

 at best, only rough approximations. The formula : I' = 

 8,200 d~s, which was determined from experiments with va- 

 rious grades of sand, gives much larger results than the quan- 

 tities in Table I for screened gravel. To modify the formula 

 so as to produce the results given in that table would in- 

 volve a variation of the coefficient (8,200) between 6,222 (for 

 d = :!mm. and ^ = 0.0005) to 940 (for d = 35 mm. and 

 s = 0.01). 



For many years the water supply of the Citizens Water 

 Company of Denver, was secured from about a mile of tim- 

 ber crib, 30 in. square in section, and about a mile of per- 

 forated 30-in. pipe both submerged in the sands of the Platte 

 river at depths varying from 14 to 22 ft. The timber crib 

 was open at the bottom, so that, with the openings in the% 

 cribbing, perhaps one-half of its superficial area was open 

 for the inflow of water, while, for the pipe line, the net area 

 of the perforations would probably bear a much smaller ratio 

 to the circumferential area of the pipe. It may be assumed, 

 therefore, that on the two miles of combined crib and pipe 

 line the total area of inlet openings would approximate 25 

 per cent of the surface area exposed to contact with the 

 water-bearing sand, thus affording a total net area of inlet 

 openings of 26,000 sq. ft. The water was led away through 

 pipes by 'natural flow, and the supply thus secured was be- 

 tween 400,000 and 450.000 cu. ft. per day : which is equivalent 

 to 13 cu. ft. per day per square foot of inlet opening, or a 

 velocity of inflow of 0.00015 ft. per sec. 



Being submerged under the bed of the river at depths 

 varying from 14 to 22 ft., these cribs are entirely surrounded 

 by water-bearing sand ; therefore, as the water may approach 

 radially from every direction, it must be assumed that the 

 maximum velocity of flow through the sand is less than one- 

 fourth of the velocity of inflow through the perforations 

 and openings. 



Assuming an average head of 16 ft. as acting on these 



cribs, the spouting velocity of water entering the pipe through 

 one of the perforations from a free body of water under the 

 pressure due to a 16- ft. head (\ 7 = VZgh) would be 32 ft. 

 per sec., while the actual average velocity of inflow is only 

 1/200000 of such rate. 



All formulas for the velocity of flow of water, under 

 special conditions, are modifications of V = V 2 g h, with 

 coefficients introduced to provide for 'the special conditions of 

 frictional resistances encountered. In working backward from 

 the results indicated above, so as to determine the special 

 coefficient which must be applied to the formula to adapt it 

 to this special case, we obtain the following: 



V = 0.00004 VJT 



The lack of harmony in the results indicated in all the 

 foregoing data makes it impracticable to devise even an ap- 

 proximate formula for the determination of the magnitude 

 of the force produced by the flow of seepage water under a 

 line of sheet-piling. Even if it were possible to determine 

 the relative effects of frictional resistance and capillary at- 

 traction on the flow of water through interstices of such an 

 infinite variety as is afforded by the range of materials from 

 fine clay to large boulders, such "hair-splitting"' refinements 

 of calculation would have very little practical value for this 

 purpose, on account of the uncertainties which attend the de- 

 termination of the variations of actual conditions to be met. 



Causes of Failure of Sheet-Pile Dams. Failures in sheet- 

 pile dams have generally occurred in those of the spillway or 

 overflow type, which are commonly called diversion weirs. 

 The seepage water which finds its way under a line of sheet- 

 piling tends to buoy up the particles of sand on the down- 

 stream side, while the surface of the river bed is frequently 

 subjected to a -variable degree of scouring action from the 

 current of water which spills over the crest. These two forces 

 combined move the finer particles of sand first, then, as the 

 interstices in the sand gradually increase is size, the upward 

 flow of seepage water increases, both in force and volume, 

 and the destruction of the dam is in progress. 



*In "Public Water Supplies." 



Fig. 1. 



It is customary with many engineers to make the length 

 of sheet-piling such that the depth of penetration below the 

 bed of the stream will be approximately equal to the head of 

 water to be impounded by the dam or weir, and it is likely 

 that many of the failures of such structures, which have been 

 attributed to "unknown" causes, were due to the inadequacy 

 of the cut-off wall under the structure more often, however, 

 on account of carelessness in driving and faulty alignment 

 than from insufficient depth of penetration. 



Length of Sheet-Piling Required. Capillary attraction 

 and surface friction, in the interstices between the grains, 

 are forces of such magnitude in retarding the velocity of 

 seepage flow through sand, or other pervious materials, that, 

 with reasonable depth of penetration of a curtain-wall of 

 sheet-piling, the direct pressure which may result from any 

 reasonable head of water impounded by the dam cannot pro- 

 duce - a velocity of seepage flow sufficiently rapid to move 

 the fine particles of sand without the added force of the 

 overflow current flowing on the surface of the river bed : 

 therefore, in designing the dam or diversion weir, adequate 

 provision must be made to counteract the effects of both 

 of these forces. 



Assuming that the force of capillary attraction effec- 

 tively resists the pressure head of the impounded water, in 

 its natural function of producing an acceleration of seepage 

 flow, then the body of sand on the down-stream side may- 

 be considered as a mass which is held together by the force 

 of capillary attraction, and that the pressure due to the head 

 of water on the up-stream side passes under the line of sheet- 

 piling and acts on a wedge-shaped mass of sand on the 

 down-stream side. 



The hydrostatic pressure at the bottom of the sheet- 



