BASIC EQUATION FOR SEEPAGE

S EEPAGE AND F LOW N ETS

6.4 BASIC EQUATION FOR SEEPAGE

DHARM

N-GEO\GE6-1.PM5 172

172 GEOTECHNICAL ENGINEERING

6.3.3 Flow through Earth Dam

The flow through an earth dam differs from the other cases in that the top flow line is not know in advance of sketching the flow net. Thus, it is a case of unconfined flow. The determination of the top flow line will be dealt with in a later section.

The top flow line as well as the flow net will be dependent upon the nature of internal drainage for the earth dam. Typical cases are shown in Fig. 6.8; the top flow line only is shown.

Assuming that the top flow line is determined, a typical flow net for an earth dam with a rock toe, resting on an impervious foundation is shown in Fig. 6.9:

B

A

Impervious D

Rock toe

C C

Fig. 6.9 Flow net for an earth dam with rock toe (for steady state seepage)

AB is known to be an equipotential and AD a flow line. BC is the top flow line; at all points of this line the pressure head is zero. Thus BC is also the ‘phreatic line’; or, on this line, the total head is equal to the elevation head. Line CD is neither an equipotential nor a flow line, but the total head equals the elevation head at all points of CD.

SEEPAGE AND FLOW NETS 173

Vertical component of flow

Z

Y X

dz

dy

dx (x, y, z)

Fig. 6.10 Flow through an element of soil By Darcy’s law,

q_z = k · i · A,

where A is the area of the bottom face and q_z is the flow into the bottom face.

= k_z

FHG IKJ

− ^{∂∂hz dx · dy,}

where k_z is the permeability of the soil in the Z-direction at the point (x, y, z) and h is the total head.

Flow out of the top of the element is given by:

q_z + ∆q_z = k k

z dz h z

h z dz

z + z

FHG IKJ F

^{− −}

HG I

∂

KJ

∂

∂

∂

∂

, ∂² .

2 . dx dy Net flow into the element from vertical flow:

∆q_z = inflow – outflow

= k_z

FHG IKJ

−^{∂∂hz dxdy –}

FHG

^{kz+ kzz dz}

IKJ F

^{−hz− zh dz}

HG I

∂

KJ

∂

∂

∂

∂

. ∂² .

2 dx dy

∴ ∆q_z = k h z

k h z

k

z dz h

z.∂ z z . z

∂

∂ ∂

∂

∂

∂

∂

∂

2

2 2

2

+ + 2

F HG I

KJ

^{dx dy dz}

Assuming the permeability to be constant at all points in a given direction, (that is, the soil is homogeneous),

∂

∂ k

z^z = 0

∴ ∆q_z = k h

z∂z

∂

2

F

2

HG I

KJ

^{dx dy dz}

Similarly, the net inflow in the X-direction is:

∆q_x = k h

x. ∂x

∂

2

F

2

HG I

KJ

^{dx dy dz}

For two-dimensional flow, q_y = 0

∴ ∆q = ∆q_x + ∆q_z = k h

x k h

x.∂ z. z

∂

∂

∂

2 2

2

+ 2

F HG I

KJ

^{dx dy dz}

DHARM

N-GEO\GE6-1.PM5 174

174 GEOTECHNICAL ENGINEERING

∆q may be obtained in a different manner as follows:

The volume of water in the element is:

V_w = S e e .

(1 + ) . dx dy dz

∆q = rate of change of water in the element with time:

= ^∂

∂ V

t^w = (∂/∂t) S e

e dx dy dz .

( ).

L

1 +

NM O

dx dy dz

QP

e

(1 + ) is the volume of solids, which is constant.

∴ ∆q = dx dy dz

e

(1 + ) (∂/∂t) (S . e) Equating the two expressions for ∆q, we have:

k h

x k h

x.∂ z. z

∂

∂

∂

2 2

2

+ 2

F HG I

KJ

dx dy dz = dx dy dz e

(1 + ) . (∂/∂t) (S . e)

or k_x ^∂

∂

2 2

h

x + k_z . ^∂

∂

2 2

h z = ¹

1

( ) .

+^e

FHG

^{e ∂∂St} +^S∂∂et

IKJ

^{…(Eq. 6.2)}

This is the basic equation for two-dimensional laminar flow through soil.

The following are the possible situations:

(i) Both e and S are constant.

(ii) e varies, S remaining constant.

(iii) S varies, e remaining constant.

(iv) Both e and S vary.

Situation (i) represents steady flow which has been treated in Chapter 5 and this chapter.

Situation (ii) represents ‘Consolidation’ or ‘Expansion’, depending upon whether e decreases or increases, and is treated in Chapter 7. Situation (iii) represents ‘drainage’ at constant volume or ‘imbibition’, depending upon whether S decreases or increases. Situation (iv) includes problems of compression and expansion. Situations (iii) and (iv) are complex flow conditions for which satisfactory solutions have yet to be found. (Strictly speaking, Eq. 6.2 is applicable only for small strains).

For situation (i), Eq. 6.2 reduces to:

k_x . ^∂

∂

2 2

h

x + k_z . ^∂

∂

2 2

h

z = 0 …(Eq. 6.3)

If the permeability is the same in all directions, (that is, the soil is isotropic),

∂

∂

∂

∂

2 2

2 2

h x

h

+ z = 0 …(Eq. 6.4)

This is nothing but the Laplace’s equation in two-dimensions. In words, this equation means that the change of gradient in the X-direction plus that in the Z-direction is zero.

From Eq. 6.3,

(∂/∂x) k h

x.∂x

FHG IKJ

∂ ^{+ (∂/∂z)}

FHG IKJ

^{kz.∂∂hz = 0}

SEEPAGE AND FLOW NETS 175 But k_x . ∂h/∂x = v_x and k_z . ∂h/∂z = v_z, by Darcy’s law.

∴ ^∂

∂

∂

∂ v

x v

z

x + z = 0 …(Eq. 6.5)

This is called the ‘Equation of Continuity’ in two-dimensions and can be got by setting

∆q = 0 (or net inflow is zero) during the derivation of Eq. 6.2.

The flow net which consists of two sets of curves – a series of flow lines and of equipotential lines–is obtained merely as a solution to the Laplace’s equation – Eq. 6.4. The fact that the basic equation of steady flow in isotropic soil satisfies Laplace’s equation, suggests that, the flow lines and equipotential lines intersect at right-angles to form an orthogonal net – the ‘flow net’. In other words, the flow net as drawn in the preceding sections is a theoretically sound solution to the flow problems.

The ‘velocity potential’ is defined as a scalar function of space and time such that its derivative with respect to any direction gives the velocity in that direction.

Thus, if the velocity potential, φ is defined as kh, φ being a function of x and z,

Similarly,

∂φ

∂

∂

∂φ ∂

∂

∂

∂ x k h

x v z k h

z v

x

y

= =

= =

U V|

W||

.

. …(Eq. 6.6)

In view of Eq. 6.4 for an isotropic soil and in view of the definition of the velocity poten-tial, we have:

∂ φ

∂

∂ φ

∂

2 2

2

x + z2 = 0 …(Eq. 6.7)

This is to say the head as well as the velocity potential satisfy the Laplace’s equation in two-dimensions.

The equipotential lines are contours of equal or potential. The direction of seepage is always at right angles to the equipotential lines.

The ‘stream function’ is defined as a scalar function of space and time such that the partial derivative of this function with respect to any direction gives the component of velocity in a direction inclined at + 90° (clockwise) to the original direction.

If the stream function is designated as ψ(x, z),

and

∂ψ

∂ψ∂

∂ z v

x v

x

z

=

= −

U V|

W||

^{…(Eq. 6.8)}

by definition.

By Eqs. 6.6 and 6.8, we have:

and

∂φ ∂ ∂ψ ∂

∂φ ∂ ∂ψ ∂

/ /

/ /

x z

z= x

= −

UVW

^{…(Eq. 6.9)}

These equations are known as Cauchy-Riemann equations. Substituting the relevant values in terms of ψ in the continuity equation (Eq. 6.5) and Laplace’s equation (Eq. 6.7), we can show easily that the stream function ψ(x, z) satisfies both these equations just as φ(x, z) does.

DHARM

N-GEO\GE6-1.PM5 176

176 GEOTECHNICAL ENGINEERING

Functions φ and ψ are termed ‘Conjugate harmonic functions’. In such a case, the curves

‘‘φ(x, z) = a constant’’ will be orthogonal trajectories of the curves ‘‘ψ(x, z) = a constant’’.

Flow nets will be useful for the determination of rate of seepage, hydrostatic pressure, seepage pressure and exit gradient. These aspects have already been discussed in Sec. 6.2.

The following important properties of the flow nets are useful to remember:

(i) The flow lines and equipotential lines intersect at right angles to each other.

(ii) The spaces between consecutive flow and equipotential lines form elementary squares (a circle can be inscribed touching all four lines).

(iii) The head drop will be the same between successive equipotentials; also, the flow in each flow channel will be the same.

(iv) The transitions are smooth, being elliptical or parabolic in shape.

(v) The smaller the size of the elementary square, the greater will be the velocity and the hydraulic gradient.

These are correct for homogeneous and isotropic soils.

*6.5 SEEPAGE THROUGH NON-HOMOGENEOUS AND ANISOTROPIC SOIL

Although Eq. 6.2 was derived fro general conditions, the preceding examples considered only soil that does not vary in properties from point horizontally or vertically–homogeneous soil–

and one that has similar properties at a given location on planes at all inclination–isotropic soil. Unfortunately, soils are invariably non-homogeneous and anisotropic.

The process of formation of sedimentary soils is such that the vertical compression is larger than the horizontal compression. Because of the higher vertical effective stress in a sedimentary soil, the clay platelets tend to have a horizontal alignment resulting in lower permeability for vertical flow than for horizontal flow.

In man-made as well as natural soil, the horizontal permeability tends to be larger than the vertical. The method of placement and compaction of earth fills is such that stratification tend to be built into the embankments leading to anisotropy.

Non-homogeneous Soil

In case of flow perpendicular to soil strata, the loss of head and rate of flow are influenced primarily by the less pervious soil whereas in the case of flow parallel to the strata, the rate of flow is essential controlled by comparatively more pervious soil.

Figure 6.11 shows a flow channel and part of a flow net, from soil A to soil B. The permeability of soil A is greater than that of soil B. By the principle of continuity, the same rate of flow exists in the flow channel in soil A as in soil B. By means of this, relationship between the angles of incidence of the flow paths with the boundary of the two flow channels can be determined. Not only does the direction of flow change at the boundary between soils with different permeabilities, but also the geometry of the figures in the flow net changes. As can be seen from Fig. 6.11, the figures in soil B are not squares as in soil A, but are rectangles.

SEEPAGE AND FLOW NETS 177

lB

b_B

_A

_B _B

k_B lA

b_A k_A

Fig. 6.11 Flow at the boundary between two soils q_A = q_B

But q_A = k_A . ∆h

l_A . b_A q_B = k_B . ∆h

l_B . b_B

∴ k_A . ∆h

l_A . b_A = k_B . ∆h l_B . b_B l

b

l b

A

A A

B

B B

=tanα_ =tanα_

k_A k

A

B

tanα_ =tanαB_

tan tan





α α

A B

A B

k

= k Anisotropic Soil

Laplace’s equation for flow through soil, Eq. 6.4, was derived under the assumption that per-meability is the same in all directions. Before stipulating this condition in the derivation, the equation was:

k_x . ^∂

∂

∂

∂

2 2

2

2 0

h

x k h

z z

+ . = …(Eq. 6.3)

This may be reduced to the form:

∂

∂

∂

∂

2 2

2

2

h z

h k k^z x

x

+

F

HG I

KJ

= 0 …(Eq. 6.10)

By changing the co-ordinate x to x_T such that x_T = k

k^z_x . x, we get

∂

∂

∂

∂

2 2

2 2

h z

h x_T

+ = 0 ...(Eq. 6.11)

which is once again the Laplace’s equation in x_T and z.

In other words, the profile is to be transformed according to the relationship between x and x_T and the flow net sketched on the transformed section.

DHARM

N-GEO\GE6-1.PM5 178

178 GEOTECHNICAL ENGINEERING

From the transformed section, the rate of seepage can be determined using Eq. 6.1 with exception that k_e is to be substituted for k (see Fig. 6.12):

l b k_x

k_z

Flow Flow

Natural scale Transformed scale

l. k /k_{x z}

Fig. 6.12 Flow in anisotropic soil Transformed Section:

q_T = k_e .iA = k_e . ∆h

l . b = k_e . ∆h Natural Section:

q_N = k_x . i_N . A = k_x . ∆h

l k k_x/ _z . b = k_x . ∆h k k_x/ _z Since q_T = q_N,

k_e ∆h = k_x . ∆h

k k_x/ _z …(Eq. 6.12)

k_e is said to be the effective permeability.

The transformed section can also be used to determine the head at any point. However, when determining a gradient, it is important to remember that the dimensions on the trans-formed section must be corrected while taking the distance over which the head is lost. To compute the gradient, the head loss between equipotentials is divided by the distance l_N, the perpendicular distance between equipotentials on the natural scale, and not by l_T,the distance between equipotentials on the transformed scale (Fig. 6.13).

Square lT

Flow

Transformed section Natural section

Flow

lN

Parallelograms

k_z k_x

Fig. 6.13 Portion of flow net in anisotropic soil

Also, note that flow is perpendicular to equipotentials in only isotropic soils.