Chapter 18: Groundwater and Seepage - Free

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18-18 The Civil Engineering Handbook, Second Edition8 ftAFree surface(a)10 ftB1 2DrainC25 ftQk8(Q/k) 1(b)642(Q/k) 2B =14 ft0 0.5 1.0 1.5BTFIGURE 18.18 Example 18.5.10 ftyb =20 fta 1a 23020a 2 = 18.3 ftEq. (17.36)h 1 = 70 ft31ah1 2(a)L13b3a 2100x 50h = 52.7 ftEq. (17.38)55 60h(b)FIGURE 18.19 Example 18.6.With tailwater absent, h 2 = 0, the flow in region 3, a type IX fragment, with cot b = 3 producesQ---k=a---- 23(18.33)Finally, from the geometry of the section, we haveL = 20 + cot b [ h d – a 2 ] = 20 + 3 [ 80 – a 2 ](18.34)The four independent equations contain only the four unknowns, h, a 2 , Q/k, and L, and hence providea complete, if not explicit, solution.© 2003 by CRC Press LLC
Groundwater and Seepage 18-19Combining Eqs. (18.32) and (18.33) and substituting for L in Eq. (18.34), we obtain, in general (b =crest width),a 2=----------- b+cot bh – Ê ----------- b +d Ëcotbh d ¯ˆ2– h 2(18.35)and, in particular,20a 2 = ---- + 803Ê20---- + 80 – h 2Ë 3 ¯ˆ2(18.36)Likewise, from Eqs. (18.31) and (18.33), in general,h da ----------------- 2 cot a= (cot bh – h ) ln -------------1 – hh d(18.37)and, in particular,(18.38)Now, Eqs. (18.36) and (18.38), and (18.35) and (18.37) in general, contain only two unknowns (a 2 and h),and hence can be solved without difficulty. For selected values of h, resulting values of a 2 are plotted forEqs. (18.36) and (18.38) in Fig. 18.19(b). Thus, a 2 = 18.3 ft, h = 52.7 ft, and L = 205.1 ft. From Eq. (18.33),the quantity of flow per 100 ft is18.5 Flow in Layered Systems80a 2 = ( 70 – h ) ln-------------- 80 – hQ = 100 ¥ 2 ¥ 10 – 3¥18.3 --------- = 1.22 ft 3 § min = 9.1 gal § min3Closed-form solutions for the flow characteristics of even simple structures founded in layered mediaoffer considerable mathematical difficulty. Polubarinova-Kochina [1962] obtained closed-form solutionsfor the two layered sections shown in Fig. 18.20 (with d 1 = d 2 ). In her solution she found a cluster ofparameters that suggested to Harr [1962] an approximate procedure whereby the flow characteristics ofstructures founded in layered systems can be obtained simply and with a great degree of reliability.hhd 1sBk 1 k 1k 2d 2d 1k 2d 2(a)(b)FIGURE 18.20 Examples of two-layered systems.© 2003 by CRC Press LLC
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