Soil Mechanics Permeability of Soils and Seepage page 1 CHAPITRE 9. PERMEABILITY OF SOILS AND SEEPAGE...1

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1 Soil Mechanics Permeability of Soils and Seepage page 1 Contents of this chapter : CHAPITRE 9. PERMEABILITY OF SOILS AND SEEPAGE INTRODUCTION DARCY S LAW DEFINITION OF HEAD EXAMPLE - CALCULATION OF HEAD DARCY S EXPERIMENT TYPICAL PERMEABILITY RANGES EXERCISES...4 Chapitre 9. Permeability of Soils and Seepage 9.1 Introduction Seepage is the flo of a fluid through soil pores. The flo of ater in soils can be very significant, for example: it is important to kno the amount of ater that it ill be necessary to pump out of an excavation during construction, or the amount of stored ater that may be lost by percolation through or beneath a dam. The behaviour of soil is governed by the effective stress, hich is the difference beteen total stress and pore ater pressure. When ater flos the pore ater pressures in the ground change. A knoledge of ho the pore ater pressure changes can be important in considering the stability of earth dams, retaining alls, etc. 9.2 Darcy s la Because the pores in soils are so small the flo through most soils is laminar. This laminar flo is governed by Darcy's La hich ill be discussed belo Definition of Head 1 Let us consider a point P at a height z measured vertically UP from an arbitrary DATUM 2 Referring to Fig. 1 the head h at point P is defined by the Bernoulli la 2 u( P) v h( P) = + z( P) + (1) 2g In this equation (10 kn/m 3 ) is the unit eight of ater, and u (P) is the pore ater pressure. P z(p) Datum Figure 1 Definition of Head at a Point 1 charge 2 repère

2 Soil Mechanics Permeability of Soils and Seepage page 2 Note The quantity u(p)/ is usually called the pressure head. The quantity z(p) is called the elevation head (its value depends upon the choice of a datum). The velocity head v ²/2g is generally neglected. The only circumstances here it may be significant is in flo through rock-fill, but in this circumstance, the flo ill generally be turbulent and so Darcy's la is not valid Example - Calculation of Head Static ater table 2 m 1 m X P Impermeable stratum 1 m 5 m Figure 2 Calculation of head using different datums Datum choosen at the top of the impermeable stratum Calculation of Head at P u ( P ) = 4 z (P) = 1 m h( P ) = = Calculation of Head at X u ( X ) = 1 z (X) = 4 m h( X ) = + 4 = It appears that hen there is a static ater table the head is constant throughout the saturated zone. 5 m 5 m Datum choosen at the ater table Calculation of Head at P u ( P = 4 ) z (P) = - 4 m Calculation of Head at X u ( X = 1 ) z (X) = - 1 m h( P ) h( X ) = = = = 0 m 0 m

3 Soil Mechanics Permeability of Soils and Seepage page 3 When there is a static ater table the head is constant throughout the saturated zone, but its numerical value depends on the choice of datum. The use of imaginary observation pipes 3 can be helpful in visualising head. The head is then given by the height of the ater in the pipe above the datum Note also that it is differences in head (not pressure) that cause flo Darcy s Experiment 4 During his fundamental studies of the flo of ater in soil Darcy found that the flo Q as: h Soil Sample L Figure 3 Darcy's Permeameter proportional to the head difference h proportional to the cross sectional area A inversely proportional to the length L of the soil sample. Thus Darcy concluded that: h Q = k A (2a) l here k is the coefficient of permeability or hydraulic conductivity. Equation (2a) may be reritten: here i = h/ L is the hydraulic gradient v = Q/A is the velocity. Q = k Ai v = k i (2b) Typical permeability ranges Soils exhibit a very ide range of permeabilities and hile particle size may vary by about 3-4 orders of magnitude, permeability may vary by about 10 orders of magnitude. 3 Piézomètre = Dispositif de détermination du niveau de la surface d'une nappe souterraine. Il s'agit souvent d'un tube, crépiné sur une certaine longueur, introduit dans un forage 4 Perméamètre

Soil Mechanics Permeability of Soils and Seepage page 4 10 2 10 1 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 Gravels Sands Silts Homogeneous Clays Fissured & Weathered Clays Figure 4 Ranges of

4 Soil Mechanics Permeability of Soils and Seepage page Gravels Sands Silts Homogeneous Clays Fissured & Weathered Clays Figure 4 Ranges of Permeabilities of Soils (in cm/s) Under 10-6 cm/s soil can be considered as impermeable.. Drainage is allays possible up to 10-3 cm/s. 9.3 Exercises 1. Three soil samples are placed in an horizontal 100mmx100mm square tube. A constant head difference of 300mm is maintained beteen the inlet and the outlet of the tube. The coefficients of pearmeability of the soils are : K1=10-2 cm/sec K2= cm/sec K3= cm/sec Figure 5 Calculate : a) the flo of ater through the samples. b) the heights ha et hb of the ater level in observation pipes placed in A and B, Solution Darcy : v=k.i=k. h/ L The Flo through the three soils is constant, thus : Q = v.a = constant, and as the soils have the same length (150mm) K 1 h 1 = K 2 h 2 =K 3 h 3 And of course h 1 + h 2 + h 3 = 300mm h 1 = K 2 / K 1 h 2 = 0.3 h 2 h 3 = K 2 / K 3 h 2 = 6 h 2 (K 2 / K 1 + K 2 / K 3 + 1). h 2 = 300mm h 2 = 300/7.3 = 41.1 mm h 1 = 12.3 mm h 3 = mm

5 Soil Mechanics Permeability of Soils and Seepage page 5 a) And thus h A = 300- h 1 = 287.7mm and h B = h A - h 2 = 246.6mm b) And the flo = Q = v.a = K 1 h 1 / L 1. A = / = 82 mm³/sec 2. A horizontal stratum of sandy soil 9 m thick overlies a bed of impermeable material. In order to determine the in-situ permeability of the soil, a test ell as driven to the bottom of the stratum. The initial level of the ater in the test ell as found to be 2 m belo ground level. Water as pumped from the test ell at the rate of 3 x 10-3 m³/s until the ater level became steady. The ater levels in to observation boreholes situated 12.5 m and 25 m from the ell ere found to be 4.75 m and 2.5 m belo ground level respectively. Calculate the coefficient of permeability of the sandy soil. Determine the radius of influence 5 at this pumping rate. Solution : When ater is pumped from the ground it has the effect of loering the ater table over a large area. This is shon diagrammatically in fig. 6. Consider the flo inards through the boundary of a cylindrical section a radius r here the ater table is at a height h above the impermeable surface and express that that flo is equal to the flo pumped out of the ell (steady state 6 ). The quantity floing in unit time Q/t = q = area x velocity and assuming Darcy's la v=k.i. 5 The radius of influence is the distance from the ell here the pumping has no effect on the existing ater level. 6 régime permanent

6 Soil Mechanics Permeability of Soils and Seepage page 6 Figure 6

7 Soil Mechanics Permeability of Soils and Seepage page 7 Commentaire [b1] : a) Darcy : v=k.i=k. h/ L The Flo through the three soils is constant, thus : Q = v.a = constant, and as the soils have the same thicknesses (1 m) K 1 h 1= K 2 h 2=K 3 h 3 The ater level in an observation pipe placed in D is 2.1m above the ground level (u D/ =5.1m). 3. Three layers of sandy soils are above a highly permeable gravel substratum (Fig. 7). In an observation pipe placed at the third layer/gravel interface the ater goes 2.10 m above the ground surface. a) Calculate the head difference h D - h A. Is there any flo of ater in the soil (seepage)? If yes, describe it. b) Calculate the pore ater pressure at each layer interface. Take the top of the gravel layer as datum. Dra the diagram of the pore ater pressure in the sand layers. Compare it to diagram you should have if there ere no seepage? c) If there is seepage, calculate the coefficient of permeability of an equivalent isotropic soil of the same total thickness. An observation pipe placed in A ould have a ater level at 1m above the ground level (application of the la of interconnected vessels 1 )(u A/ =1m). Thus h A=3+1=4m and h D=0+5.1=5.1m. Thus h D- h A = h 1+ h 2+ h 3 = 1.1m. As h D- h A >0, there is a flo from D to A (going upards). b) h 1= K 2 / K 1 h 2 = 0.05 h 2 h 3= K 2 / K 3 h 2 = 0.05 h 2 (K 2 / K 1 + K 2 / K 3 + 1). h 2 = 1.1m h 2 = 1.1/1.1 =1 m h 1 = 0.05 m h 3 = 0.05 m And thus h B = h A+ h 1 = 4.05m and h C = h D- h 3 = = 5.05m Figure 7 as, at any point P, e have: u( P) h( P) = + z( P) e can deduce the pore ater pressures : u A = 10 kn/m² u B = (h B-z B) = (4.05-2).10=20.5 kn/m² and u C = (h C-z C) = (5.05-1).10=40.5 kn/m² and u D = 51 kn/m² c) The velocity of the ater through any of the sand layers is v = K 1 h 1/ L 1 = /1 = m/sec An equivalent isotropic soil of the same total thickness ould have the same velocity thus : v = K eq (h D- h A )/3 = m/sec thus K eq = /1.1 = [1]

8 Page 7: [1] Commentaire [b1] boeraeve 22/11/ :45:00 a) Darcy : v=k.i=k. h/ L The Flo through the three soils is constant, thus : Q = v.a = constant, and as the soils have the same thicknesses (1 m) K 1 h 1 = K 2 h 2 =K 3 h 3 The ater level in an observation pipe placed in D is 2.1m above the ground level (u D / =5.1m). An observation pipe placed in A ould have a ater level at 1m above the ground level (application of the la of interconnected vessels 1 )(u A / =1m). Thus h A =3+1=4m and h D =0+5.1=5.1m. Thus h D - h A = h 1 + h 2 + h 3 = 1.1m. As h D - h A >0, there is a flo from D to A (going upards). b) h 1 = K 2 / K 1 h 2 = 0.05 h 2 h 3 = K 2 / K 3 h 2 = 0.05 h 2 (K 2 / K 1 + K 2 / K 3 + 1). h 2 = 1.1m h 2 = 1.1/1.1 =1 m h 1 = 0.05 m h 3 = 0.05 m And thus h B = h A + h 1 = 4.05m and h C = h D - h 3 = = 5.05m u( P) as, at any point P, e have: h( P) = + z( P) e can deduce the pore ater pressures : u A = 10 kn/m² u B = (h B -z B ) = (4.05-2).10=20.5 kn/m² and u C = (h C -z C ) = (5.05-1).10=40.5 kn/m² and u D = 51 kn/m² c) The velocity of the ater through any of the sand layers is v = K 1 h 1 / L 1 = /1 = m/sec An equivalent isotropic soil of the same total thickness ould have the same velocity thus : v = K eq (h D - h A )/3 = m/sec thus K eq = /1.1 = m/sec

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

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