Contents preface (VII) introduction 1—37

15.3. ESTIMATION AND CONTROL OF SEEPAGE
15.3.1. Seepage Analysis
The theory of flow through porous media is utilised for the estimation of seepage through an embankment dam and its foundation. As discussed in Sec. 9.3, the governing equation for two-dimensional seepage, as would occur in an embankment dam and its foundation, is the Laplace equation

∂^2h	+	∂^2h	= 0	(15.6)
∂x^2	∂y^2

Here, h is the seepage head. Equation (15.6) is valid for homogeneous, isotropic, and incompressible soil which is fully saturated with incompressible water. Equation (15.6) can be solved by graphical, analytical, numerical, or some other suitable methods, such as analogue methods. The graphical solution of Eq. (15.6) involves drawing of flownet which has been discussed in Sec. 9.3.
Most of the seepage problems related to embankment dams can be analysed by drawing flownets for sections with single permeability. For example, if the outer shells of a dam are many times pervious than the core, the analysis of the seepage conditions in the core alone may be adequate for such cases. However, in many seepage problems one has to analyse seepage through sections of different permeabilities. For such conditions, a basic deflection rule must be followed in passing from a soil of one permeability to a soil of different permeability. Seepage water needs less energy to flow through a region of relatively higher permeability. Therefore, when water flows from a region of high permeability to one of lower permeability, the flow takes place in such a manner that it remains in the more permeable region for the greatest possible distance. In other words, water seeks the easiest path to travel in order to conserve its energy. Another way of appreciating the seepage behaviour in sections of different permeabilities is that, other factors being the same, smaller area of flow cross-section is needed in the higher permeability region.

b

a = b

a = b

b

k1

k1

a

b

a

b

b

a

b

a

c

k

^k2

c

d

2

c

d

d

c

c/d = k2/k1

(a) k 2 < k1

(b) k2 > k1

Fig. 15.7 Change in shape of flownet squares on account of regions of different permeability

In seepage through porous medium, the hydraulic gradient is the measure of the rate of energy loss. One would, therefore, expect steep hydraulic gradients in zones of low permeability.

EMBANKMENT DAMS

499

Figure 15.7 shows the deflection of flow lines when they cross boundaries between soils of different permeabilities. The flow lines bend in accordance with the following relationship (8):

tan β	=	K1	(15.7)
tan α	K2

Also, the areas formed by the intersecting flow and equipotential lines either elongate or shorten according to the following relationship (8):

c	=	K2	(15.8)
d	K1

While drawing flownets for sections with different permeabilities, one must regularly measure the lengths and widths of the figures to ensure that Eq. (15.8) is satisfied.
At times, the compacted embankments and natural soil deposits are stratified rendering them more permeable in the horizontal direction than in the vertical direction. For this anisotropic condition, the velocity components, Eq. (9.42), for two-dimensional flow will be

u = – K		∂h	and v = – K	∂h	(15.9)
	x	∂x	y	∂y

Thus, for two-dimensional flow, the continuity equation, Eq. (9.43), with ∂w/∂z = 0 and combined with Eq. (15.9), yields

K	∂^2 h	+ K y	∂^2h	= 0	(15.10)
	∂y^2
x ∂x2

Equation (15.10) can, alternatively, be written as

∂ ^2h	+	∂ ^2h	= 0	(15.11)
2	∂y	2
∂xt

where, x_t = x K_y/K_x
Equation (15.11) is the familiar Laplace equation with transformed coordinate system involving x_t and y, and governs the flow in anisotropic seepage condition. To draw a flownet for the anisotropic condition, one needs only to shrink the dimensions of the given cross-section in the direction of greater permeability (8). Having drawn the flownet for the transformed section, it is then reconstructed on the cross-section drawn to the original scale. Obviously, the flownet on the original cross-section would not be composed of squares but of rectangles elongated in the direction of greater permeability.
The effective permeability K can be determined by comparing the discharge ∆q through any one figure of the transformed section and the corresponding figure of the original section (Fig. 15.8). If ∆h is the drop in head between adjacent equipotential lines, then

b

b kx /ky

b

b

(I)

(II)