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CHAPTER SIX
TWO-DIMENSIONAL FLOW OF WATER THROUGH SOILS
5. Introduction
Many catastrophic failures in geotechnical engineering result from instability of soil masses
due to groundwater flow. Lives are lost, infrastructures are damaged or destroyed, and major
economic losses occur. In this chapter, you will study the basic principles of two-dimensional flow
of water through soils. The topics covered here will help you to avoid pitfalls in the analyses and
design of geotechnical systems where groundwater flow can lead to instability. The emphasis in
this chapter is on gaining an understanding of the forces that provoke failures resulting from
groundwater flow. You will learn methods to calculate flow, pore water pressure distribution, uplift
forces, and seepage stresses.
When you complete this chapter, you should be able to:
 Calculate flow under and within earth structures.
 Calculate seepage stresses, pore water pressure distribution, uplift forces, hydraulic
gradients, and the critical hydraulic gradient.
 Determine the stability of simple geotechnical systems subjected to two dimensional flow
of water.
You will use the following principles learned from previous chapters and your courses in
mechanics.
 Statics.
 Hydraulic gradient, flow of water through soils (chapter 3).
 Effective stress and seepage (chapter 3).
Sample Practical Situation A deep excavation is required for the construction of a building. The
soil is silty sand with groundwater level just below ground level. The excavation cannot be made
unless the sides are supported. You, a geotechnical engineer, are required to design the retaining
structure for the excavation and to recommend a scheme to keep the inside of the excavation dry.
1. Definition of Key Terms
Equipotential Line is a line representing constant head.
Flow Line is the flow path of a particle of water.
Flow Net is a graphical representation of a flow field.
Seepage Stress is the stress (similar to frictional stress in pipes) imposed on a soil as water
flows through it.

Static Liquifaction is the behavior of a soil as a viscous fluid when seepage reduces the
effective stress to zero.
2. Two-dimensional Flow of Water Through Porous Media
The flow of water through soils is described by Laplace’s equation. The popular form of
Laplace’s equation for two-dimensional flow of water through soils is
(5.1)
where H is the total head and kx and kz are the coefficients of permeability in the X and Z
directions. Laplace’s equation expresses the condition that the changes of hydraulic gradient in
one direction are balanced by the changes in the other directions.
The assumptions in Laplace’s equation are:
 Darcy’s law is valid.
 The soil is homogeneous and saturated.
 The soil and water are incompressible.
 No volume change occurs.
If the soil were an isotropic material then kx = kz and Laplace’s equation becomes
(5.2)
In this chapter, we are going to emphasis in an approximate (graphical) solution technique for
Laplace’s equation called the flow net sketching. The flow net sketching technique is simple and
flexible and conveys a picture of the flow regime. It is the method of choice among geotechnical
engineers. But before we delve into this solution technique, we will establish some key conditions
that are needed to understand two-dimensional flow.
The solution if Eq. (5.1) depends only on the values of the total head within the flow field in the
XZ plane. Let us introduce a velocity potential (), which describes the variation of total head in a
soil mass as
(5.3)
where k is a generic coefficient of permeability. The velocities of flow in the X and Z directions are
(5.4)
(5.5)
The inference from Eqs. (5.4) and (5.5) is that the velocity of flow (v) is normal to lines of constant
total head (also called constant piezometric head or equipotential lines) as illustrated in Fig. 5.1.

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.
If lines are drawn that are tangent to the velocity of flow at every point in the flow field in the
XZ plane, we will get a series of lines that are normal to the equipotential lines. These tangential
lines are called streamlines or flow lines (Fig. 5.1). A flow line represents the flow path that a
particle of water is expected to take in steady state flow.
Figure 5.1: Illustration of flow terms.
Since flow lines are normal to equipotential lines, there can be no flow across flow lines. The
rate of flow between any two flow lines is constant. The area between two flow lines is called a
flow channel (fig. 5.1). Therefore, the rate of flow is constant in a flow channel.
3. Flow Net Sketching
1. Criteria for Sketching Flow Nets
A flow net is a graphical representation of a flow field that satisfies Laplace’s equation and
comprises a family of flow lines and equipotential lines.
A flow net must meet the following criteria:
1. The boundary conditions must be satisfied.
2. Flow lines must intersect equipotential lines at right angles.
3. The area between flow lines and equipotential lines must be curvilinear squares. A
curvilinear square has the property that an inscribed circle can be drawn to touch each
side of the square and continuous bisection results, in the limit, in a point.
4. The quantity of flow through each flow channel is constant.
5. The head loss between each consecutive equipotential line is constant.
6. A flow line cannot intersect another flow line.
7. An equipotential line cannot intersect another equipotential line.
An infinite number of flow lines and equipotential lines can be drawn to satisfy Laplace’s
equation. However, only a few are required to obtain an accurate solution. The procedure for
constructing a flow net is described next. A few examples of flow nets are shown in Figs. 5.2 to
5.3.
Figure 5.2: Flow net for a sheet pile.
2. Procedure for Sketching Flow Nets
6. Draw the structure and soil mass to a suitable scale.
7. Identify impermeable and permeable boundaries. The soil-impermeable boundary
interfaces are flow lines because water can flow along these interfaces. The soil-

permeable boundary interfaces are equipotential lines because the total head is constant
along these interfaces.
8. Sketch a series of flow lines (four or five) and then sketch an appropriate number of
equipotential lines such that the area between a pair of flow lines and a pair of
equipotential lines (cell) is approximately a curvilinear square. You would have to adjust
the flow lines and equipotential lines to make curvilinear squares. You should check that
the average width and the average length of a cell are approximately equal. You should
also sketch the entire flow net before making adjustments.
Figure 5.3: Flow net under a dam with a cutoff curtain (sheet pile) on the upstream end.
Figure 5.4: Flow net in the backfill of a retaining wall with a vertical drainage blanket.
4. Interpretation of Flow Nets
1. Flow Rate
The head loss (between each consecutive pair of equipotential lines is:
(5.6)
Whereis the total head loss across the flow domain and Nd is the number of equipotential drops
(number of equipotential lines minus one). From Darcy’s law, the flow through each flow channel
for an isotropic soil is
(5.7)
where B and L are sides of the curvilinear square. By construction B/L = 1, and therefore the total
flow is
(5.8)
where Nf is the number of flow channels (number of flow lines minus one). The ratio Nf/Nd is
called the shape factor. Both Nf and Nd can be fractional. In the case of anisotropic soils (different
permeabilities in X and Z directions), the quantity of flow is
(5.9)
2. Hydraulic Gradient
You can find the hydraulic gradient over each square by dividing the head loss by the length,
L, of the cell, that is
(5.10)

You should notice that L is not constant. Therefore the hydraulic gradient is not constant. The
maximum hydraulic gradient occurs where L is a minimum, that is,
(5.11)
Where Lmin is the minimum length of the cells within the flow domain. Usually, Lmin occurs at exit
points or around corners.
3. Static Liquefaction
Let us consider an element of soil at a depth z near the downstream end of a sheet pile wall
structure subjected to seepage forces with upward flow. As discussed in chapter 3, the vertical
effective stress is:
(5.12)
If the effective stress becomes zero, the soil loses its strength and behaves like a viscous fluid.
The soil state at which the effective stress is zero is called static liquefaction. Various other
names such as boiling, quicksand, piping, and heaving are used to describe specific events
connected to the static liquefaction state. A structure founded on a soil that statically liquefies will
collapse. Liquefaction can also be produced by dynamic events such as earthquake.
4. Critical Hydraulic Gradient
We can determine the hydraulic gradient that brings a soil mass (essentially, coarse grained
soils) to static liquefaction. Solving for i in Eq. (5.12) when , we get
(5.13)
where icr is the critical hydraulic gradient, Gs is specific gravity, and e is the void ratio. Since Gs is
constant the critical hydraulic gradient is solely a function of the void ratio of the soil. In designing
structures that are subjected to steady state seepage, it is absolutely essential to ensure that the
critical hydraulic gradient cannot develop.
5. Pore Water Pressure Distribution
The pore water pressure at any point j is calculated as follows:
1. Select a datum (for example, choose the downstream water level as the datum.)
2. Determine the total head at j: where is the number of equipotential drops at point j.
3. Subtract the elevation head at point j from the total head Hj to get the pressure head. For
point j below the datum (recall the datum is assumed to be the downstream water level),
let the elevation head hz is –z. The pressure head is then
(5.14)

4. The pore water pressure is
(5.15)
6. Uplift Forces
Lateral and uplift forces due to groundwater flow can adversely affect the stability of structures
such as dams and weirs. The uplift force per unit length (length is normal to the XZ plane) is
found by calculating the pore water pressure at discrete points along the base (in the X direction)
and then finding the area under the pore water pressure distribution diagram, that is,
(5.16)
where Pw is the uplift force per unit length, uj is the average pore water pressure over an interval ,
and n is the number of intervals. It is convenient to use Simpson’s rule to calculate Pw:
(5.17)
EXAMPLE 5.1
A dam, shown in Fig.E5.1a, retains 10 m of water. A sheet pile wall (cutoff curtain) on the
upstream side, which is used to reduce seepage under the dam, penetrates 7 m into a 20.3 m
thick silty sand stratum. Below the silty sand is a thick deposit of clay. The average coefficient of
permeability of the silty sand is 2.0×10-4 cm/s. Assume that the silty sand is homogeneous and
isotropic.
a. Draw the flow net under the dam.
b. Calculate the flow, q.
c. Calculate and draw the pore water pressure distribution at the base of the dam.
d. Determine the uplift force.
e. Determine and draw the pore water pressure distribution on the upstream and
downstream faces of the sheet pile wall.
f. Determine the resultant lateral force on the sheet pile wall due to the pore water.
g. Determine the maximum hydraulic gradient.
h. Will piping occur if the void ratio of the silty sand is 0.8.
i. What is the effect of reducing the depth of penetration of the sheet pile wall?
Strategy: Follow the procedures described in section 5.2 to draw the flow net and calculate the
required parameters.
FigureE5.1a
5. Flow Through Earth Dams

Flow through earth dams is an important design consideration. We need to ensure that the
pore water pressure at the downstream end of the dam will not lead to instability and the exit
hydraulic gradient does not lead to piping. The major exercise is to find the top flow line called the
phreatic surface (Fig.5.2). The pressure head on the phreatic surface is zero. Casagrande (1937)
showed that the phreatic surface can be approximated by a parabola with corrections at the
points of entry and exit. The focus of the parabola is at the toe of the dam, point F (Fig.5.2).
Figure 5.5: Phreatic surface within an earth dam.
The procedure to draw a phreatic surface within an earth dam, with reference to Fig. 5.5, is as
follows.
1. Draw the structure to scale.
2. Locate a point A at the intersection of a vertical line from the bottom of the upstream face
and the water surface, and a point B where the waterline intersects the upstream face.
3. Locate point C, such that BC=0.3AB.
4. Project a vertical line from C to intersect the base of the dam at D.
5. Locate the focus of the basic parabola. The focus is located at the toe of the dam.
6. Calculate the focal distance, , where b is the distance FD and H is the height of water at
the upstream face.
7. Construct the basic parabola from .
8. Sketch in a transition section BE.
9. Calculate the length of the discharge face, a, using
For , use Fig.5.7 and (a) measure the distance TF, where T is the intersection of the basic
parabola with the downstream face; (b) for the known angle , read the corresponding factor from
the chart; and (c) find the distance .
10. Measure the distance a from the toe of the dam along the downstream face to point G.
11. Sketch in a transition section, GK.
12. Calculate the flow using where k is the coefficient of permeability. If the downstream
slope has a horizontal drainage blanket as shown in Fig. 5.3, the flow is calculated using .
Figure 5.6: A horizontal drainage blanket at the toe of an earth dam.
Figure 5.7: Correction factor for downstream curve

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