CHAPTER EIGHT

SEEPAGE

8.1 Introduction The flow of water through soil is not in one direction only, nor is it uniform over the entire area perpendicular to the flow. In such cases, the groundwater flow is generally calculated by the use of graphs referred to as flow nets. The concept of the flow net is based on Laplace’s equation of continuity.

8.2 Laplace’s Equation of Continuity

With Darcy’s law, the discharge velocities can be expressed as:

where kx and kz are the hydraulic conductivities in the horizontal and vertical directions, respectively.

8.3 Flow Nets

The continuity equation in an isotropic medium represents two orthogonal families of curves: The flow lines: lines along which water particles will travel from upstream to the

downstream side in the permeable soil medium. The equipotential lines: lines along which the potential head at all points is

equal. Thus, if piezometers are placed at different points along an equipotential line, the water level will rise to the same elevation in all of them.

The flow net: is a combination of a number of flow lines and equipotential lines. flow nets are constructed for: • the calculation of ground-water flow • the evaluation of heads in the media. To complete the graphic construction of a flow net, one must draw the flow and equipotential lines in such a way that: 1. The equipotential lines intersect the flow lines at right angles 2. The flow elements formed are approximate squares.

ab and de are equipotential lines so all the flow lines intersect them at right angles.

fg and acd are flow lines,

Thus The equipotential lines intersect them at right angles.

8.4 Seepage Calculation from a Flow Net

The flow channel: is the strip between any two adjacent flow lines

• Let h1, h2, h3, h4, . . ., hn be the piezometric levels corresponding to the equipotential lines.

• The rate of seepage through the flow channel per unit can be calculated as follows:

because there is no flow across the flow lines.

Δq1 = Δq2 = Δq3 = …….. Δq

From Darcy’s law, the flow rate q = k i A. Thus:

• the potential drop is the drop in the piezometric level between any two adjacent equipotential lines

• If the flow elements are drawn as approximate squares, the potential drop is the same.

where H : head difference between the upstream and downstream sides Nd: number of potential drops

the total rate of flow through all the channels per unit length can be given by:

In case of rectangular mesh for a flow channel:

For example

• flow channels 1 and 2 have square elements • flow channel 3 has rectangular elements. These elements have a width-to-length ratio of about 0.38;

a.

b.

c.

8.4 Flow Nets in Anisotropic Soil

• seepage calculation have been based on the assumption that the soil is isotropic. • in nature, most soils exhibit some degree of anisotropy.

• The rate of seepage per unit length can be calculated by

8.5 Uplift Pressure Under Hydraulic Structures

• Flow nets can be used to determine the uplift pressure at the base of a hydraulic structure.

Example • Nd = 7 • H =7 m • The head loss for each potential drop is : Δh = H/7 = 7/7= 1 m. • The uplift pressure at “a” = Pressure head at “a” x γw

= [(7+2) -1 (1)] γw = 8 γw

• The uplift pressure at “b”

= [(7+2) -2(1)] γw = 7 γw

• The uplift pressure at “f”

= [(7+2) -6(1)] γw = 3 γw

8.6 SAFETY AGAINST PIPING & HEAVE In addition to calculating seepage losses and uplift pressures below hydraulic structures, Flow nets can also be used to determine gradients especially at certain critical points: • At the toe of a dam in order to assess the potential for erosion and piping • Or for upward seepage adjacent to a sheet pile in order to assess the potential

for heaving in the soil.

1. PIPING: • Is a phenomenon where seeping water progressively erodes or washes away soil particles, leaving large voids in the soil. • If uncontrolled collapse of structure • Piping usually occurs at water stream exit

Exit

Exit gradient : iExit = Δh/ l Factor of safety against piping

F.S = i cr/iexit ≥ 5-6 Where Δh: H/Nd l: is the length of the square in the flow direction at exit

2- Heaving

underground seepage may cause heaving of soil on the downstream side. Terzaghi concluded that heaving generally occurs within a distance of D/2 from the sheet piles (when D equals depth of embedment of sheet piles into the permeable layer).

Factor of safety against heave: F.S = i cr / i avg ≥ 4-5

i avg = average hydraulic gradient at the bottom of the block of soil (deduced from the flow net.) i avg = n Δh/l , where n is the number of drops from down stream to middle of block of soil

i avg n = 1.8

• One way to increase the factor of safety against heave is to use a filter in the downstream side of the sheet-pile structure. • A filter is a granular material with openings small enough to prevent the movement

of the soil particles upon which it is placed and, at the same time, is pervious enough to offer little resistance to seepage through it

• the thickness of the filter material is D1. • The factor of safety against heave is thus:

γ'F = effective unit weight of the filter material. γ‘ = effective unit weight of soil

8.7 Use of Filters to Increase the Factor of safety against heave

8.7.1 Filter Design • When seepage water flows from a soil with relatively fine grains into a coarser

material, there is danger that the fine soil particles may wash away into the coarse material.

• Over a period of time, this process may clog the void spaces in the coarser material.

• Hence, the grain-size distribution of the coarse material should be properly manipulated to avoid this situation.

• A properly designed coarser material is called a filter.

• For proper selection of the filter material, two conditions should be kept in

mind: Condition 1: The size of the voids in the filter material should be small enough to hold the larger particles of the protected material in place. Condition 2: The filter material should have a high hydraulic conductivity to prevent buildup of large seepage forces and hydrostatic pressures in the filters.