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Many researches indicated that
failure of embankment dams due to seepage alone stands for about 25% of the total failure
cases, apart from overtopping, piping, internal erosion, etc (Singh, 1995). The flownet sketching technique is straightforward and adaptable, and it represents the flow regime. It is the preferred method of analysing flow through soils for geotechnical engineers. Before delving into these solution techniques, however, we will establish a few key conditions necessary to comprehend two-dimensional flow. Any differential equation requires knowledge of the boundary conditions in order to be solved.

The average pore pressure ratio, ru, for the whole embankment of the earth dam model have
been calculated in accordance to statistical procedure and algorithms / equations that were carried
out. The detailed presentation of the statistical analysis steps are not included, as this is out of the
scope of this experimental research paper. However, the equations used for this calculation are
presented below. Figure 2 shows a flownet for a sheet pile wall, and Figure 3 shows a flownet beneath a dam. In the case of the retaining wall, the vertical drainage blanket of coarse-grained soil is used to transport excess porewater pressure from the backfill to prevent the imposition of a hydrostatic force on the wall. The interface boundary, is neither an equipotential line or a flow line.

It is only applied to problems with simple and ideal

Flow nets were originally used for determining how much water could flow under a dam. In
case that there is a substantial amount of water flow under the body of a dam, it can create a lot of
pressure on the alluvium / sediments. Over time the groundwater can erode the sediments, and the
dam can collapse, causing a disastrous flood. To solve this problem analytically it is difficult, but
flow nets can be used to give a graphical answer (Sachpazis et al, 2005). Calculation of the pore pressure ratio for an embankment is highly important, as this value is
extremely useful in embankment stability analysis problems (Smith, 2006).

With reference to Figures 15 & 17, the following terms
may be defined in order to estimate the quantity of seepage through the earth dam model. Mathematically, the process of making out a flownet consists of contouring the two
harmonic or analytic functions of potential and flow line function. These functions both satisfy
the Laplace equation and the contour lines represent lines of constant head, i. Together, the potential function and the stream
function form the complex potential, where the potential is the real part, and the stream function
is the imaginary part.

Electrical Analogy Method:

The inference from Equations (4a) and (4b) is that the velocity of flow (v) is normal to lines of constant total head, as illustrated in Figure 1 The direction of v is in the direction of decreasing total head. The head difference between two equipotential lines is called a potential drop or head loss. The method consists of filling the flow area with stream and equipotential lines, which are everywhere perpendicular to each other, making a curvilinear grid. As mentioned earlier the main application of flow net is that it is employed in estimating
quantity of seepage. If H is the net hydraulic head of flow (i. the difference in total head between
the first and last equipotential), the quantity of seepage due to flow may be estimated by drawing
the flow net, which is shown in Figure 15.

• However, the equations used for this calculation are
  presented below.
• As mentioned earlier the main application of flow net is that it is employed in estimating
  quantity of seepage.
• Continuity equation for
  steady state and Darcy’s equations and for the case of isotropic soil, the permeability coefficient is
  independent of direction (Craig, 2004).
• Finally, using the above mentioned procedure the average pore pressure ratio, ru, for the whole
  embankment of the earth dam model was calculated equal to 0.
• In this diagram, the pore water pressure
  contours along with the flow lines are also presented.
• The condition that the changes in hydraulic gradient in one direction are balanced by changes in the other directions is expressed by Laplace’s equation.

Structville is a media channel dedicated to civil engineering designs, tutorials, research, and general development. At Structville, we stop at nothing in giving you new dimensions to the profession of civil engineering. For instance, the portion of the flownet beneath the base of the sheet pile in Figure 2 is not composed of curvilinear squares. Check these sections to ensure that repeated bisection results in a point for a precise flownet. The second flow net pictured here (modified from Ferris, et al., 1962) shows a flow net being used to analyze map-view flow (invariant in the vertical direction), rather than a cross-section.

Flow net results

We can also attempt to replicate the flow through the actual structure using physical models. The first is an approximation known as flownet sketching, and the second is the finite difference method. The uplift pressure at any point within the soil mass can be found using the undermentioned formula. Different methods have been identified to study the extent of seepage in earth dams.

Due to
its relative simplicity, flow net is the most commonly used amongst these methods. These points are mathematical artifacts of the equation used to solve the real-world problem, and do not actually mean that there is infinite or no flux at points in the subsurface. These types of points often do make other types of solutions (especially numeric) https://personal-accounting.org/github/ to these problems difficult, while the simple graphical technique handles them nicely. Unconfined seepage problems are commonly encountered in geotechnical
engineering. In these problems, the determination of the free surface and the calculation of
seepage usually requires sophisticated numerical techniques, unfamiliar to most engineers.

As such, this experimental research also assesses the validity of the ratio value obtained. Pore pressure ratio, ru, is the ratio of pore water pressure at any given point of the earth dam
model to the weight of the soil material acting on unit area at that point ( Tsuyoshi, 2006). The
concept of ru is relevant to both granular and cohesive soils.

• Note that this problem has symmetry, and only the left or right portions of it needed to have been done.
• In the case of the retaining wall, the vertical drainage blanket of coarse-grained soil is used to transport excess porewater pressure from the backfill to prevent the imposition of a hydrostatic force on the wall.
• An infinite number of flow lines and equipotential lines can be drawn to satisfy Laplace’s equation.
• Historically, Stelzer et al, 1987, presented an introductory scheme for plotting contours that
  are traced along paths of constant function values.

Where H is the total head and kx and kz are the hydraulic conductivities in the X and Z directions. The condition that the changes in hydraulic gradient in one direction are balanced by changes in the other directions is expressed by Laplace’s equation. Since the head drops are uniform by construction, the gradient is inversely proportional to the size of the blocks. Big blocks mean there is a low gradient, and therefore low discharge (hydraulic conductivity is assumed constant here).

Extensions to standard flow nets

A flow net represents the graphical solution of the equations of the steady / continuous flow of
groundwater. Two sets of lines constitute a flow net, which should be always orthogonal to each
other. The flow lines indicate the direction of groundwater flow and the equipotential lines or head
lines (lines which represent the constant head), indicate the distribution of potential energy.

The area between two flow lines is called a flow channel (Figure 1). The following diagram in figure 18 shows a static ground water level within the earth dam
model as derived from the Flow Net shown in Figure 15. In this diagram, the pore water pressure
contours along with the flow lines are also presented. When Laplace equation is solved
graphically the equation gives flow net consisting two sets of curves intersecting at right angles
known as flow lines (or stream lines) and equipotential lines, as presented in previous section. Flow lines represent the path of flow along which the water will seep through the soil. Equipotential lines are formed by connecting the points of equal total head.

Foundation for Bridges Over Water

Note that this problem has symmetry, and only the left or right portions of it needed to have been done. To create a flow net to a point sink (a singularity), there must be a recharge boundary nearby to provide water and allow a steady-state flowfield to develop. Dams are constructed to impound water for irrigation, water supply, energy generation,
flood control, recreation as well as pollution control.

The
flow nets are usually built through a trial and error procedure with sketches. The solution of Laplace equation requires knowledge of complex boundary conditions. Geotechnical problems usually have complex boundary conditions for which it is difficult to
obtain a closed form solution. For this reason, approximate methods such as graphical methods
and numerical methods are often employed. Flow net technique is a graphical method, which
satisfies Laplace equation. A flow net is a graphical representation of a flow field (Solution of
Laplace equation) and comprises a family of flow lines and equipotential lines, as presented in
previous section.