Flow of Water in Soil Permeability and Seepage. Water flows through the voids in a soil which are. seepage, since the velocities are very small.

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1 Flow of Water in Soil Permeability and Seepage Water flows through the voids in a soil which are interconnected. t This flow may be called seepage, since the velocities are very small. Water flows from a higher energy to a lower energyand behaves according to the principles of fluid mechanics

2 An Energy Equation from Fluid Mechanics 3 An Energy Equation for Water Flow in Soils The velocities of water flowing through the voids in a soil are very small, and the velocity head in the previous equation may be neglected. Therefore, for flow of water in soil the equation is: 4 2

3 Basic Principles Calculation of Heads Total Head = Pressure Head + Elevation Head The pressure head is zero at a water surface. The head loss in the water is assumed to be zero. All head loss occurs in the soil. 5 Elevation, pressure and total head Pore pressure at a given point (e.g. point A in the diagram) can be measured by the height of water in a standpipe located at that point. The height of the water column is the pressure head (hw) hw = u γw or hw = P γw 6 3

4 The elevation head (hz) of a point is its height above the datum line. The height above the datum of the water level in the standpipe is the total head (h). h = hz + hw 7 Energy Loss Equation Water is flowing in the direction indicated in the figure. 8 4

5 Energy Loss Equation 9 Darcy's law The rate of flow of water q (volume/time) through cross sectional area A is found to be proportional to hydraulic gade gradient i according to Darcy's acyslaw: 10 5

6 Darcy's law 11 Darcy's law v = q = k.i A i = Dh ds The value of the coefficient of permeability k depends on the average size of the pores and is related to the distribution of particle sizes, particle shape and soil structure. 12 6

7 Coefficient of Permeability The notation for coefficient of permeability is k. It is sometimes called hydraulic conductivity. Typical Values k k Soil Type cm/sec ft/min Clean Gravel 1.0 to to 200 Coarse Sand 0.01 to to 2.0 Fine Sand to to 0.02 Silty Clay to to Clay Less Than Less Than Laboratory Tests to Determine Coefficient of Permeabilty Constant Head Permeability Test (Granular soils) Falling Head Permeability Test (Fine grained soils) Both of these tests will be conducted as laboratory exercises. 14 7

8 Constant head test Recommended for coarse grained soils. Steady total head drop h is measured across gauge length L, as water flows through a sample of crosssection area A. 15 Constant head test 16 8

9 Falling head test Recommended for fine grained soils. Total head h in standpipe of area a is allowed to fall; heads h1 and h2 are measured at times t1 and t2. Hydraulic gradient Dh/L varies with time. 17 Falling head test 18 9

10 Two Dimensional Flow Seepage and Flow Nets In the previous section the seepage problems discussed were all lab models consisting of onedimensional flow. In field construction, structures used for water barriers generally involve two or three dimensional seepage flow, such as: (1) Cofferdam cells (sheet pile wall) and Concrete dams (2) Earth dams and levees. 19 The purposes of studying the seepage conditions under or within these structures are: to estimate the rate of flow (reservoirs for keeping water cut off ability) to estimate seepage force (uplift force) (erosion) to estimate pore pressure distribution for effective stress analysis 20 10

11 Two Dimensional Flow Laplace's Equation of Continuity A. Anisotropic Soil B. Isotropic Soil

12 From Darcy s law: For isotropic permeability, Kz = Kz = k, giving 23 Solutions to Laplace's Equation Computer solutions using finite element or finite difference techniques. Graphical solutions known as flow nets

13 Solutions to Laplace's equation for twodimensional seepage can be presented as flow nets. Two orthogonal sets of curves form a flow net: flow lines equipotentials 25 Flow Lines and Equipotential Lines A flow line is the path a water particle would follow in moving from upstream to downstream. An equipotential line is a line along which the total head, h, is a constant value. It is similar to a contour line, except that total head is constant, rather than elevation. A flow net is a combination of flow lines and equipotential lines that satisfy Laplace's equation and the boundary conditions

14 27 Flow Lines and Equipotential Lines 28 14

15 29 If standpipe piezometers were inserted into the ground with their tips on a single equipotential then the water would rise to the same level in each standpipe. (The pore pressures would be different because of their different elevations.) There can be no flow along an equipotential, because there is no hydraulic gradient, so there can be no component of flow across a flow line. The flow lines define channels along which the volume flow rate is constant

16 Flow Net Construction Constructing a flow net In the example, water is flowing under a sheet pile wall that t is driven di partially through h a permeable isotropic i soil. Draw the situation to scale. Establish the equipotential boundary conditions. For this example, ba and de are boundary equipotential lines. The total head is 56' everywhere along line ba. The total head is 39' everywhere along line de

17 Rules for Flow Net Construction Establish the flow boundary conditions. For this example, acd and fg are boundary flow lines. Flow lines and equipotential lines must always intersect at right angles. The fields should be "square". By this we mean that the length and width of a field should be equal. Fields are the spaces in a flow net formed by the intersecting flow lines and equipotential lines. The sides of the fields are commonly curved, so the term "square" is a stretch of the normal definition. 33 Construction of the Flow Net The flow net for this example follows. Equipotential lines areredred and dashed, while flow lines are green and solid. The length and width of one of the fields is shown. The length in the direction of flow is L, while the width is b. To be "square" ( L= b )

18 Calculation of total flow For a complete problem, the flownet has been drawn with the overall head drop h divided into Nd equal intervals: Δh = h / Nd h: total head Δh: loss of head for each drop Nd: number of equipotential drops with Nf flow channels. Then the total flow rate per unit width is 35 Flow Rate 36 18

19 Flow Net Calculations A. Example The flow net from the previous section will be used for the calculations. The coefficient of permeability for the soil is 0.04 ft/s

20 39 For the example: 40 20

21 C. Heads and Water Pressures We will calculate the water pressure at point A in the flow net. 41 or Pressure point A = (21+20) - 4.5(17/6)= ft PA= 28.25(62.4)= 1763 lb/ft square 42 21

22 Gradient calculation Gradients are different for each field in a flow net. Head losses across each field will be the same, but the flow length depends on the size of the field. Small fields will have a small flow length and a large gradient. Flow length may be measured directly from the flow net, since it is drawn to scale. Following is a gradient calculation for the field with " L = 11' " printed inside. The length, L, was determined by direct measurement and multiplication by the scale factor

23 critical hydraulic gradient (ic) The hydraulic gradient at which effective stresses becomes zero; with upward seepage, sand may become quicksand. 45 Quick condition and piping If the flow is upward then the water pressure tends to lift the soil element. If the upward water pressure is high enough the effective stresses in the soil disappear, no frictional strength can be mobilised and the soil behaves as a fluid. This is the quick or quicksand condition and is associated with piping instabilities around excavations and with liquefaction events in or following earthquakes

24 Critical hydraulic gradient The quick condition occurs at a critical upward hydraulic gradient ic, when the seepage force just balances bl the buoyant weight ihof an element of soil