L06-flow_nets - 1 6 FLOW NETS 6.1 Introduction Let us...

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6. FLOW NETS 6.1 Introduction Let us consider a state of plane seepage as for example in the dam shown in Figure 1. Drainage blanket Phreatic line Unsaturated Soil Flow of water z x Fig. 1 Flow through a dam For an isotropic material the head satisfies Laplace's equations, thus analysis involves the solution of: 0 2 2 2 2 = + z h x h subject to certain boundary conditions. 6.2 Representation of Solution At every, point (x,z) where there is flow there will be a value of head h(x,z). In order to represent these values we draw contours of equal head as shown on Figure 2. Flow line (FL) Equipotential (EP) Fig.2 Flow lines and equipotentials 1
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These lines are called equipotentials. On an equipotential (EP). by definition: constant ) , ( = z x h (1a) it thus follows h x dx h z dz + = 0 (1b) and hence the slope of an equipotential is given by dz dx h x h z EP = - / / (1c) It is also useful in visualising the flow in a soil to plot the flow lines (FL), these are lines that are tangential to the flow at a given point and are illustrated in Figure 2. It can be seen from Fig. (2) that the flow lines and equipotentials are orthogonal. To show this notice that on a flow line the tangent at any point is parallel to the flow at that point so that: [ ] [ ] dx dz v v x z : : (2a) it follows immediately that: dx dz v v now from Darcy s law v k h x v k h z thus dx dz h x h z FL x z x z FL = = - = - = ' / / (2b) and so dx dz dx dz FL EP = - 1 (3) and thus the flow lines and equipotentials are orthogonal in an isotropic material. 2
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6.3 Some Geometric Properties of Flow Nets Consider a pair of flow lines, clearly the flow through this flow tube must be constant and so as the tube narrows the velocity must increase. Suppose now we have a pair of flow lines as shown in Figure 3. Q X y z t T Y Z X FL FL Q h h+ h h+2 h EP Fig. 3 Equipotentials intersecting a pair of Flow Lines Suppose that the flow per unit width (in the y direction) is, Q, then the velocity v in the tube is given by v Q yx = (4a) Also let us assume that the potential drop between any adjacent pair of equipotentials is h then it follows from Darcy’s law: v k h zt = (4b) It thus follows that: Q k h yx zt = (4c) using an identical argument to that used in developing equation(4c) it can be shown that: Q k h YX ZT = (4d) and hence that: 3 x
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yx zt YX ZT = (5) Thus each of the elemental rectangles bounded by the given pair of flow lines and a pair of equipotentials (having an equal head drop) have the same length to breadth ratio. Next consider a pair of equipotentials cut by flow tubes each carrying the same flow
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This note was uploaded on 08/30/2011 for the course CIVL 2410 taught by Professor Dairey during the Three '11 term at University of Sydney.

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L06-flow_nets - 1 6 FLOW NETS 6.1 Introduction Let us...

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