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**Construction of Flow Net for Seepage Analysis!**

**Laplace’s Equation: **

The equation is used in the construction of flow net.

**Assumptions: **

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1. Flow is two dimensional.

2. Water and soil are incompressible.

3. Soil is isotropic and homogeneous.

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4. The soil is fully saturated.

5. The flow is steady, i.e., flow conditions do not change with time.

6. Darcy’s law is valid.

Let us consider an element of soil of size dx, dz through which flow is taking place. The third dimension along y-axis is large.

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For convenience, it is taken as unity.

As the flow is steady and the soil is compressible, the discharge entering the element is equal to that leaving the element.

Equation (1) is the continuity equation for two-dimensional flow.

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Let h be the total head at any point. The horizontal and vertical component of the hydraulic gradient are, respectively.

Equation (2) is the Laplace’s equation in terms of head h. Sometimes, the Laplace’s equation can be represented in terms of velocity potential ɸ, given by –

is the Laplace’s Eqn. 3 in terms of velocity potential.

Laplace’s Eqn. 3 can be solved if the boundary conditions at the inlet and exit are known.

**Determining Seepage Discharge: **

Let b and L be the dimensions of the field and Δh be the head drop through this field.

Also, let Δq represent the discharge passing through the flow channel, per unit length of structure (perpendicular to paper).

Then from Darcy’s law, we can write –

Where k = permeability of the soil.

Δh = head loss in the field, i.e., potential drop between two equipotential forming the field.

b x 1 = cross-sectional area of the field, considering 1 m length perpendicular to paper.

N_{d} represents the total number of potential drop in the complete flow net, then –

Δh – N_{d }= H_{L} …(5)

Where H_{L} is the total hydraulic head causing flow, and is equal to the difference of the upstream and the downstream heads.

Substituting the value of Δh form Eqn. (5), we get –

From the drawn flow net, N* _{f}* and N

_{d}can be easily counted, and hence, the seepage discharge can be easily computed by using Eqn. (9).

**Construction of Flow Net (Graphical Method): **

The accuracy of the computation of hydraulic quantities, such as discharge and pore water pressure, does not depend much on the exactness of the flow net.

**The following points should be kept in mind while sketching the flow net: **

1. Too many flow channels to distract the attraction from the essential features. Normally, three to five flow channels are sufficient.

2. The appearance of the entire flow net should be watched and not that of a part of it. Small details can be adjusted after the entire flow net has been roughly drawn.

3. The curves should be roughly elliptical or parabolic in shape.

4. All transitions should be smooth.

5. The flow line and equipotential lines should be orthogonal and form approximate squares.

6. The size of the square in a flow channel should change gradually from the upstream to the downstream.

**The procedure for drawing the flow net can be divided into the following steps:**

1. First identify the hydraulic boundary conditions. In Fig. 8.3, the upstream bed level GDA represents 100% potential line and the downstream bed level CFJ, 0% potential line. The first flow line KLM is formed by the flow of water on the upstream of the sheet pile, the downstream of the sheet pile and at the interface of the base of the dam and the soil surface. The long flow line is indicated by the impervious stratum NP.

2. Draw a trial flow line ABC adjacent to boundary line. The line must be at right angles to the upstream and downstream beds.

3. Starting from the upstream end, divide the first flow channel into approximate squares. The size of the squares should change gradually. Some of the squares may, however, be quite irregular. Such squares are called singular squares.

4. Extend the equipotential lines downward forming the sides of the squares. These extensions point out appropriate width of the squares, such as squares marked (1) and (2).

Other sides of the squares are set equal to the widths as determined above. Irregularities are smoothed out, and the next flow line DF is drawn joining these bases. While sketching the flow line, care should be taken to make flow fields as approximate squares throughout.

5. The equipotential lines are further extended downward, and one more flow line GHJ is drawn, representing the step (4).

6. If the flow fields in the last flow channel are inconsistent with the actual boundary conditions, the whole procedure is repeated after taking a new trial flow line.

**To Draw Phreatic Line in an Earth Dam (Casagrande Method): **

Case 1 – When horizontal drainage filter is provided-

In order to draw the flow net, it is first essential to find the location and shape of the phreatic line or the top flow line separating the saturated and unsaturated zones.

Phreatic line is a seepage line as the line within a dam section below which there are positive hydrostatic pressures in the dam. The hydrostatic pressure on the phreatic line itself is atmospheric.

AB is the u/s face. Its projection be L. On the water surface, measure a distance BC = 0.3L. Point C is the starting point of the base parabola. Assume F point of filter EF as focus of parabola. Take C as centre take an arc of radius CF. This should cut the line CB at D.

Draw a vertical DH which cuts filter at H. This line is the directrix of parabola.

The last point G on the parabola will lie midway between F and H.

i. Draw a line FN parallel to DH.

ii. Cut this line FN equal to FH at S.

iii. Now draw a parabola joining points GSC.

iv. Correct the parabola at B.

v. In case more point to be located say P, from vertical line QP at any distance x from F. Measure QH. With F as the centre and QH as the radius, draw an arc to cut vertical line through Q in point P. Now join all the points G, S, P, B to get parabola. The phreatic line must start from B and not from C. Also, the phreatic line is a flow line, and must start perpendicularly to the u/s face AB which is a 100% equipotential line.

Hence, the portion of the phreatic line at B is sketched free hand in such a way that it starts perpendicularly to AB, and meets the rest of the parabola tangentially without any kink. The base parabola should also meet the ∂/s filter perpendicular at G.

Seepage per unit length of dam, q = ks

Where S = FH = focal distance between parabola and directrix

k = coefficient of permeability

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