Vízügyi Közlemények, 1959 (41. évfolyam)

4. füzet - V. Kisebb közlemények-Ismertetések

(55) meable layers and vertical flow in the covering layers are only of significant influence on quantitative properties of seepage

, It follows from this simplification, that along the path of percolation the resistance against seepage within the permeable layer results in a certain pressure loss on the downstream side of the structure. A further pressure loss is due to the circumstance, that a certain amount of water escapes in every section and percolates from the permeable layer to the surface in proportion with the seepage resistance of the covering layers. The process is reversed on the upstream side of the structure. Advancing here towards the structure, a part of the available entire differential pressure — increasing with the ratio of the seepage resistance of the cover and of the magnitude of the pressure prevailing already in the permeable layer — is used to deliver water through the cover into the permeable layer, and in turn, under the structure. Mathematical conditions of the simplified flow pattern may, according to Fig. 6, be formulated separately for the upstream and the downstream sides. Considering for the time being no more than the downstream side, the differential equation in Eq. (21) can be developed from Eqs. (14)—(17). In this equation the characteristics of various layers are represented by a single value, the quantity Bi computed according to Eq. (19). The solution of the differential equation for the downstream side of the structure is given by Eqs. (22) and (25) which yield the relationship between the pressure loss and the water quantity percolating in the permeable layer at any distance x. The constants Cj and C

2 in the equation can be determined from the relations x = 0, q = q

0 and h = h

0, as well as x — q — 0 and h = 0, see Eqs. (26)—(32). Referring to Fig. 7, the relationship between the water quantity emerging from under the structure and the magnitude of the pressure is given by the final formula expressed by Eq. (32). Equations derived above apply also to changes in pressure ahead of the structure. Now the head h„ at the upstream toe of the structure represents the total pressure loss required to convey a water quantity q

0 through the upstream cover into the permeable layer, and as far as the toe of tbe structure within the latter. However, in addition to the upstream and downstream sides, a pressure loss occurs under the structure as well. This can be expressed by the simple Darcg formula, Eq. (35). Water quantities entering and emerging from under the structure must be equal. Therefore, the water quantity q

0 can be computed from the three expressions, Eqs. (32, 33, 35). On the other hand, the total pressure loss can not exceed the entire differential head. Eq. (37) for computing the volume of seepage under the structure, as well as Eqs. (40, 41 and 42) for determining the total pressure loss at the upstream and subsequently at the downstream side of the structure, can be obtained by these conditions. As to be seen from the expressions, the influence of the covering layers has been coverled during the derivation by the factors B

, and IS

2 into distances to be covered by the water in the permeable layer to suffer a pressure loss that is proportionate to the resistance of the cover. This equivalent percolation distance is expressed by Eq. (38). The compulation has thus been reduced to the diagrammatical hydraulic arrangement shown in Fig. 3, however with the difference, that now the resistances of the two extreme layers have been reduced to the permeability coefficient of the central layers and expressed by the quantities B

x and B

2. Adopting this arrangement and using B

0 for the entire percolation distance, all characteristics of seepage flow can be computed by the simple Darcy formula, Eqs. (39)—(44). Changes in pressure and in the rate of flow at any distance x ahead of, or behind the structure are given by Eq. (45) respectively Eq. (49). In cases, where the cover is built up of several layers, the hydraulic gradient for each of the layers can be determined in any section by Equations (51) and (52). The application of the computation method is illustrated by Example 1 (Fig. 8). The case of simple seepage is treated in 1 a, the effect of colmatation of the upstream cover is illustrated in 1 b, while no upstream cover is assumed in 1 c.