Geotechnical Engineering

DHARM

SOIL MOISTURE–PERMEABILITY AND CAPILLARITY 135

v

v

v

v

v kn

k 3

k 2

k 1

in

i 3

i 2

i 1

hn

h 3

h 2

h 1

hh

Fig. 5.10 Flow perpendicular to bedding planes
Let h 1 , h 2 , h 3 ...hn be the thicknesses of each of the n layers which constitute the deposit,
of total thickness h. Let k 1 , k 2 , k 3 ... kn be the Darcy coefficients of permeability of these layers
respectively.
In this case, the velocity of flow v, and hence the discharge q, is the same through all the
layers, for the continuity of flow.

Let the total head lost be ∆h and the head lost in each of the layers be ∆h 1 , ∆h 2 , ∆h 3 , ...

∆hn.

∆h = ∆h 1 + ∆h 2 + ∆h 3 + ... ∆hn.

The hydraulic gradients are :

i 1 = ∆h 1 /h 1 If i is the gradient for the deposit, i = ∆h/h

i 2 = ∆h 2 /h 2

in = ∆hn/hn

Since q is the same in all the layers, and area of cross-section of flow is the same, the

velocity is the same in all layers.

Let kz be the average permeability perpendicular to the bedding planes.

Now kz. i = k 1 i 1 = k 2 i 2 = k 3 i 3 = ... knin = v

∴ kz∆h/h = k 1 ∆h 1 /h 1 = k 2 ∆h 2 /h 2 = ...

kh

h

nn

n

∆

= v

Substituting the expression for ∆h 1 , ∆h 2 , ... in terms of v in the equation for ∆h, we get :

vh/kz = vh 1 /k 1 + vh 2 /k 2 + ... + vhn/kn

or kz =

h
( /hk hk 11 +++ 22 / ... h knn/ ) ...(Eq. 5.34)
This is the equation for average permeability for flow perpendicular to the bedding
planes.

Flow Parallel to the Bedding Planes

Let the flow be parallel to the bedding planes as shown in Fig. 5.11.