5. TWO-DIMENSIONAL FLOW OF WATER THROUGH SOILS 5.1 INTRODUCTION

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1 5. TWO-DIMENSIONAL FLOW OF WATER TROUG SOILS 5.1 INTRODUCTION In many instances the flo of ater through soils is neither one-dimensional nor uniform over the area perpendicular to flo. It is often necessary to kno events associated ith to dimensional flo of ater through soil media especially in hydraulic and earth retaining structures. Seepage is closely associated ith effective stress. Many catastrophic failures happened due to seepage stresses. 5. SEEPAGE STRESSES If ater is seeping through soil, the effective stress in a soil mass ill differ from that in the static case. To cases shall be considered: a. Upard seepage, and b. Donard seepage The figure belo shos a granular soil mass in a permeability setup here an upard seepage is maintained. In reference to the figure above, the total stress at a point located z distance from the top of the soil specimen is σ γ + zγ = 1 sat At this same location the pore ater pressure is

2 h u = z z γ Thus the effective pressure ill be given by h σ ' = σ u = zγ ' zγ The value h/ is the hydraulic gradient of the flo and thus σ ' = zγ ' izγ This equation shos that in the case of an upard seepage the value of the effective stress decreases by an amount izγ. Increasing the rate of seepage, i.e. the hydraulic gradient, ill at some point result in a critical state here zγ ' icr zγ = 0 γ ' Gs 1 icr = = γ 1+ e This value of the hydraulic gradient is knon as the critical hydraulic gradient. At this stage the soil ill have no resistance, i.e. σ =0 and acts like a fluid. This phenomenon is knon as boiling, or a quick condition and it usually results in failures of structures like dams and retaining alls. It should be taken into account hen designing ater and earth retaining structures. It is also important to consider boiling hen planning excavations in soil strata underlain by artesian aquifers. The critical hydraulic gradient is approximately 1.0 for most soils. Let us no consider a case of donard seepage as shon belo

3 The hydraulic gradient is h/. At any depth z in the soil specimen the total, pore ater and effective stresses are σ γ + zγ = 1 u = 1 + z sat h σ ' = σ u = zγ ' + z γ h zγ ence the effective stress in the case of a donard seepage increases along ith an increase in depth unlike the case of an upard seepage. 5.3 SEEPAGE AND LAPLACE S EQUATION OF CONTINUITY Ground ater flo is generally calculated by the use of graphs referred to as flo nets. The concept of the flo net is based on Laplace s equation of continuity, hich governs the steady flo condition for a given point in the soil mass. Let us consider a single ro of sheet piles that have been driven into a permeable soil layer shon belo. The steady state flo of ater from the upstream to the donstream side through the permeable soil layer is a to-dimensional flo. An elemental soil ith dimensions dx, dy, and dz is taken from the flo path is selected. dx v x dz v z Let vx and vz be the components of the discharge velocity in the horizontal and vertical directions. The rate of inflo to the elemental block is given by v x dydz and v z dxdy in the to directions. The rate of outflo is also given by vx vz vx + dx dzdy and vz + dz dxdy z v here x v and z are rates of change of discharge velocity in the x and z directions respectively z The total rate of inflo should be equal to the total rate of inflo, i.e

4 vx vz vx + dx dzdy + vz + dz dxdy z x zdxdy or vx vz + = 0 z With Darcy s la, the discharge velocities are expressed as [ v dzdy + v ] = 0 and Substituting these values the previous continuity equation becomes h h k x + k = 0 z z If the soil is isotropic, k x = k z. ence the equation ill assume the form h h + = 0 z i.e. the function h(x,z) satisfies the Laplace equation FLOW NETS The previous continuity equation for an isotropic medium represents to orthogonal families of curves. These curves are knon as flo lines and equipotential lines. A flo line is a line along hich a ater particle travels from the upstream to the donstream side. An equipotential line is a line along hich the potential head at all points is the equal. If piezometers are installed at different points along an equipotential line, the same ater level ill be observed in all. A combination of a number of flo lines and equipotential lines is called a flo net. Flo nets are important to estimate flo of groundater. In constructing flo nets the folloing requirement must be met. 1. Equipotential lines intersect flo lines at right angles.. Flo elements formed are approximate squares

5 SEEPAGE CALCULATION FROM FLOW NETS In any flo net, the strip beteen to adjacent flo lines is called a flo channel. The figure belo shos a flo channel ith the equipotential lines forming square elements. Let h1, h, h3,, hn be the piezometric levels in each equipotential line. Since there is no cross flo beteen flo channels, the flo can be given as Δq1 = Δq = Δq3 = K = Δq According to Darcy s la, q = kia. Thus h h Δq = k l h h k = l = l1 l => h 1 -h = h -h 3 = = N d Where = head difference beteen the upstream and donstream sides Nd = number of potential drops and Δ q = k N d... If the number of flo channels in a flo net is Nf, the total rate of flo through all the channels per unit N f length can be given by Δ q = k N d An example of trials of flo nets for a single ro of sheet piles is shon belo. The flo net shon in (b) is a trial ith obvious mistakes hile a correct one is shon in (c)

6 (c) In the flo net above, the number of flo channels is 4.3 and the number of equipotential drops is 1; thus the ration N f /N d is The equipotentials are numbered from zero at the d/s boundary; this number denoted by nd. The loss in total head beteen any to adjacent equipotentials is h 4 Δ h = = = 0.33m N d 1 The total head at every point on an equipotentail numbered n d is n d Δh. The total volume of ater floing under the piling per unit length of piling is given by N f Δ q = k = k x 4.00 x 0.36 = 1.44k m 3 /s N d A piezometer tube is shon at a point P on the equipotential denoted by n d = 10. The total head at P is nd 10 hp = h = 4.00 = 3. 33m Nd 1 i.e. the ater level in the tube is 3.33m above the datum

7 The point P is at a distance zp belo the datum, i.e. the elevation head is zp. The pore ater pressure at P can then be calculated from Bernoulli s theorem: u p = γ [ hp ( z p )] = γ h + z ( ) p p The hydraulic gradient across any square in the flo net involves measuring the average dimensions of the square. The highest hydraulic gradient (and hence the highest seepage velocity) occurs across the smallest square and vice versa. The condition adjacent to a sheet piling subjected to seepage is best studied using flo nets. A ro of sheet piles ith upard seepage on the donstream face is depicted belo. Model test have shon that a volume of soil d x d/ just adjacent to the sheet pile may become unstable to support the all. Failure first shos in the form of a rise or heave at the surface associated ith an expansion of the soil hich results in an increase in permeability. This in turn leads to increased flo, surface boiling in the case of sands and complete failure. The average hydraulic gradient for this condition is given by i m = Since failure due to heaving (quick condition) occurs hen the hydraulic gradient becomes ic, the factor of safety against heaving may be expressed as h d i FS = i c m m

8 Example

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*** ***! ) * % )!( & ' % # $. 0 1 %./ +, - 7 : %8% 9 ) 7 / ( * 7 : %8% 9 < ;14. > /' ;-,=. / ١ ١ ******!" #$ % & '!( ) % * ") +,-./ % 01. 3 ( 4 56 7/4 ) 8%9 % : 7 ;14 < 8%9 % : *7./ = ;-, >/'." Soil Permeability & Seepage ٢ Soil Permeability- Definition ٣ What is Permeability? Permeability is the