flow net anisotropic and nonhomogeneous

1. Flow net in
anisotropic soils

2. Flow net:
definition:A set of flow lines and equipotential lines is called a
flow net.
 a flow line is a line along which
a water particle will travel.

3.  An equipotential line is a line joining the points that show the
same piezometric elevation.
 The flow lines and the equipotential lines are drawn by trial and
error.
 It must be remembered that the flow lines intersect the
equipotential lines at right angles.
 The flow and equipotential lines are usually drawn in such a way
that the flow elements are approximately squares. that need not
always be the case.

4.  Flow net is a time-consuming process because trail and error.
 Once a satisfactory flow net has been drawn, it can be traced
out.
Laplace equation:
 In many practical cases, the nature of the flow of water through
soil is such that the velocity and gradient vary throughout the
medium.

5.  The concept of the flow net is based on Laplace’s equation of
continuity.
 which describes the steady flow condition.
 To derive the equation of continuity of flow, consider an
elementary soil prism at point “A”.
 The flows entering the soil prism in the x, y, and z directions can
be given from Darcy’s law as
qx = kxixAx = kx
𝜕ℎ
𝜕𝑥
dy dz

6. qy = kyiyAy = ky
𝜕ℎ
𝜕𝑦
dx dz
qz = kzizAz = kz
𝜕ℎ
𝜕𝑧
dx dy
qx qy qz = flow entering in directions x, y, and z, respectively
kx ky kz = coefficients of permeability in directions x, y, and z,
respectively h = hydraulic head at point A
 The respective flows leaving the prism in
the x, y, and z directions are

7. qx +dqx = kx( ix +dix)Ax
=kx
𝜕ℎ
𝜕𝑥
+
𝜕2
ℎ
𝜕𝑥2 𝑑𝑥 dy dz
qy +dqy=ky
𝜕ℎ
𝜕𝑦
+
𝜕2
ℎ
𝜕𝑦2 𝑑𝑦 dx dz
dz +dqz=kz
𝜕ℎ
𝜕𝑧
+
𝜕2
ℎ
𝜕𝑧2 𝑑𝑧 dx dy

8.  For steady flow through an incompressible medium, the flow
entering the elementary prism is equal to the flow leaving the
elementary prism.
qx +qy +qz = (qx +dqx)+(qy +dqy)+(qz +dqz)
We can solve above equation we get
kx
𝜕2
ℎ
𝜕𝑥2 + ky
𝜕2
ℎ
𝜕𝑦2+ kz
𝜕2
ℎ
𝜕𝑧2=0

9. For two dimensional
kx
𝜕2
ℎ
𝜕𝑥2 + ky
𝜕2
ℎ
𝜕𝑦2=0
If the soil is isotropic with respect to permeability, kx = kz = k, and
the continuity equation simplifies
𝜕2
ℎ
𝜕𝑥2 +
𝜕2
ℎ
𝜕𝑦2=0
kx
𝜕2
ℎ
𝜕𝑥2 + ky
𝜕2
ℎ
𝜕𝑦2=0
This is generally referred as Laplace equation

10. Flow nets in anisotropic material:
 Let us now consider the case of constructing flow nets for
seepage through soils that show anisotropy with respect to
permeability.
 For two-dimensional flow problems,
kx
𝜕2
ℎ
𝜕𝑥2 + ky
𝜕2
ℎ
𝜕𝑧2=0
kx
𝜕2
ℎ
𝜕𝑥2 + ky
𝜕2
ℎ
𝜕𝑧2=0
𝜕2
ℎ
(kz/kx)𝜕𝑥2
+
𝜕2
ℎ
𝜕𝑧2=0

11. L𝑒𝑡 𝑥′
=
𝐾𝑧
𝑘𝑥
𝑥 𝑡ℎ𝑒𝑛,
𝜕2
ℎ
(kz/kx)𝜕𝑥2=
𝜕2
ℎ
𝜕𝑥′2
𝜕2
ℎ
𝜕𝑥′2+
𝜕2
ℎ
𝜕𝑧2=0
The above equation like Laplace equation which governs the flow
in isotropic soils and should represent two sets of orthogonal lines
in the x z plane.

12. steps for construction of a flow
net in an anisotropic medium:
1. To plot the section of the hydraulic structure, adopt a vertical
scale.
2. Determine
𝐾𝑧
𝑘𝑥
3. Adopt a horizontal scale such that (scalehorizontal )=
𝐾𝑧
𝑘𝑥
(scalevertical).
4. With the scales adopted in steps 1 and 3, plot the cross-section
of the structure.

13. 5. Draw the flow net for the transformed section plotted in step 4 in
the same manner as is done for seepage through isotropic soils.
6. Calculate the rate of seepage as q = 𝑘𝑥𝑘𝑧 ℎ
𝑁𝑓
𝑁𝑑
Construction of flow nets for
nonhomogeneous subsoils:
Homogeneous condition rarely occurs in a nature. In most cases,
we encounter stratified soil deposits.
When a flow net is constructed across the boundary of two soils
with different permeabilities, the flow net deflects at the boundary.
This is called a transfer condition.

14.  Figure 5.33 shows a general condition where a flow channel
crosses the boundary of two soils.
 Soil layers 1 and 2 have permeabilities of k1 and k2,
respectively.
 The dashed lines drawn across the flow channel are the
equipotential lines.
 Let Δh be the loss of hydraulic
head between two consecutive
equipotential lines..
 Considering a unit length
perpendicular to the section
shown.

15. the rate of seepage through the flow channel is
∆𝑞 = 𝑘1
∆ℎ
𝑙1
𝑏1 × 1 = 𝑘2
∆ℎ
𝑙2
𝑏2 × 1
𝑘1
∆ℎ
𝑙1
𝑏1 × 1 = 𝑘2
∆ℎ
𝑙2
𝑏2 × 1
𝑘1
𝑘2
=
∆ℎ
𝑙1
𝑏1
∆ℎ
𝑙1
𝑏1
𝑘1
𝑘2
=
𝑏1
/𝑙1
𝑏2
/𝑙2

16. where 𝑙1 and 𝑏1 are the length and width of the flow elements in soil
layer 1 and 𝑙2 and 𝑏2 are the length and width of the flow elements
in soil layer 2.
Referring again to Figure 5.33,
𝑙1 = AB sin 𝜃1 = AB cos𝛼1
𝑙2 = AB sin 𝜃2 = AB cos𝛼2
b1 = AC cos 𝜃1 = AC sin𝛼1
b2 = AC cos 𝜃2 = AC sin𝛼2
b1
𝑙1
=
cos 𝜃1
sin𝜃1
=
sin𝛼1
cos𝛼1

17. b1
𝑙1
=
1
𝑡𝑎𝑛𝜃1
= tan 𝛼1 (3)
Also,
b2
𝑙2
=
cos 𝜃2
sin𝜃2
=
sin𝛼2
cos𝛼2
b2
𝑙2
=
1
𝑡𝑎𝑛𝜃2
= tan 𝛼2
(4)
Equation(2) and (3) in equation (1)
𝑘1
𝑘2
=
𝑡𝑎𝑛𝜃1
𝑡𝑎𝑛𝜃2
=
tan 𝛼2
tan 𝛼1
(5)

18. It is useful to keep the following points in mind while constructing
the flow nets:
1. If k1 > k2, we may plot square flow elements in layer 1. This
means that 𝑙1 = 𝑏1 in equation (1) thenSo 𝑘1/𝑘2 = 𝑏2/𝑙2
• Thus, the flow elements in layer 2 will be rectangles and their
width-to-length ratios will be equal to k1/k2.
2. If k1 < k2, we may plot square flow elements in layer 1. This
means that 𝑙1 = 𝑏1 in equation (1) thenSo 𝑘1/𝑘2 = 𝑏2/𝑙2Thus,
the flow elements in layer 2 will be rectangles.

19. Thank you……