Chapter 18: Groundwater and Seepage - Free

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18-8 The Civil Engineering Handbook, Second Editiony 1y 2y 3f 1f 2f 3f 4f 5FIGURE 18.4 Streamlines and equipotential lines.yn yynqy + dydxdsqdy0dQn xxFIGURE 18.5 Flow between streamlines.anddQ=dy(18.21)Hence the quantity of flow (also called the discharge quantity) between any pair of streamlines is aconstant whose value is numerically equal to the difference in their respective y values. Thus, once asequence of streamlines of flow has been obtained, with neighboring y values differing by a constantamount, their plot will not only show the expected direction of flow but the relative magnitudes of thevelocity along the flow channels; that is, the velocity at any point in the flow channel varies inverselywith the streamline spacing in the vicinity of that point.An equipotential line was defined previously as the locus of points where there is an expected level ofavailable energy sufficient to move a particle of water from a point on that line to the tailwater surface.Thus, it is convenient to reduce all energy levels relative to a tailwater datum. For example, a piezometerlocated anywhere along an equipotential line, say at 0.75h in Fig. 18.6, would display a column of waterextending to a height of 0.75h above the tailwater surface. Of course, the pressure in the water along theequipotential line would vary with its elevation.0.75 h hTailwaterdatumbaEquipotentialline, 0.75 hFIGURE 18.6 Pressure head along equipotential line.© 2003 by CRC Press LLC
As the quantity of flow between any two streamlines is a constant, DQ, and equal to Dy (Eq. 18.21)we haveEquations (18.22) and (18.23) are approximate. However, as the distances Dw and Ds become verysmall, Eq. (18.22) approaches the velocity at the point and Eq. (18.23) yields the quantity of dischargethrough the flow channel.In Fig. 18.8 is shown the completed flow net for a common type of structure. We first note that thereare four boundaries: the bottom impervious contour of the structure BGHC, the surface of the imperviouslayer EF, the headwater boundary AB, and the tailwater boundary CD. The latter two boundariesdesignate the equipotential lines h = h and h = 0, respectively. For steady state conditions, the quantityGroundwater and Seepage 18-918.3 The Flow NetThe graphical representation of special members of thefamilies of streamlines and corresponding equipotentiallines within a flow region form a flow net. The orthogonal∆QB h + ∆hnetwork shown in Fig. 18.4 represents such a system.Although the construction of a flow net often requires∆shtedious trial-and-error adjustments, it is one of the moreCAvaluable methods employed to obtain solutions for twodimensionaly + ∆ yflow problems. Of additional importance,even a hastily drawn flow net will often provide a check on∆wthe reasonableness of solutions obtained by other means.Noting that, for steady state conditions, Laplace’s equationDalso models the action (see Table 18.1) of thermal, electrical,yacoustical, odoriferous, torsional, and other systems,FIGURE 18.7 Flow at point.the flow net is seen to be a significant tool for analysis.point, according to Darcy’s law, will bek DhDyv ª --------- ª-------Ds D w(18.22)DQ ª k------ DwDhD s(18.23)60 ftA h = 16 ft h = h B50 fth = h 1h 2HGy BGHCC3h = 01Dh 3E2y EFFFIGURE 18.8 Example of flow net.© 2003 by CRC Press LLC
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