## LESSON 17. Permeability of Soil

17.1 Introduction

Permeability of the soil quantitatively describes how easily water can flow through it. In loose soil, amount of pores within the soil grains is more. Water can flow easily through loose soils. However, in case of dense soil, amount of pores within the soil grains is less. Water can not flow easily through dense soils (as shown in Figure 17.1). Thus, permeability is high in case of loose soil whereas, it is low in case of dense soil. Fig.17.1. Flow of water a different type of soils

Darcy’s Law

Velocity (v) of flow is proportional to the hydraulic gradient (i). Thus,

V = Ki              (17.1)

where k is the coefficient of permeability (cm/sec) or hydraulic conductivity. If i is equal to one, the v = k. Thus, coefficient of permeability is velocity of water for unit hydraulic gradient. Hydraulic gradient can be expressed as: i = Δh / L , where Δh is the head loss and L is the length between two points (as shown in Figure 17.2). Fig.17.2. Flow of water

17.2. Permeability in layered soil

Figure 17.3 shows a layered soil system. The thickness of each layer is H1, H2, H3 ………. Hn. The coefficient of permeability is k1, k2, k3 …………… kn. In case of horizontal flow, it takes place through all the layers at the same time. Thus, hydraulic gradient is same for all the layers, i.e i1 = i2= i3= …… in = i, where n is the number of layers. However, the velocity of flow is different in different layer. Fig. 17.3. Permeability of layered soil

According to Darcy’s law, the average discharge velocity (vavg) can be written as:

${v_{avg}}={k_H}i={1 \over H}({v_1}{H_1} + {v_2}{H_2} + ............. + {v_n}{H_n})$                           (17.2)

where kH is the coefficient of horizontal permeability, v1, v2 …….vn is the velocity of flow in different layers. H is the total thickness of the layers, i.e H1+H2+………+Hn = H. The Eq. (17.2) can be written as:

${k_H}i={1 \over H}({k_1}i{H_1} + {k_2}i{H_2} + ............. + {k_n}i{H_n})$                                           (17.3)

${k_H}={1 \over H}({k_1}{H_1} + {k_2}{H_2} + ............. + {k_n}{H_n})$                                                (17.4)

In case of vertical flow, the hydraulic gradient is different in each layer. However, the velocity of flow is same in all the layers. The total head loss is h and the head losses in each layer is h1, h2, ………hn.

Head loss in each layer, ${h_1}={H_1}{i_1}$ ,  ${h_2}={H_2}{i_2}$.......................${h_n}={H_n}{i_n}$                  (17.5)

Thus, the total head loss $h={H_1}{i_1} + {H_2}{i_2} + .................. + {H_n}{i_n}$                                         (17.6)

The velocity of flow is same in all the layers. Thus,

$v={k_v}{h \over H}={k_1}{i_1}$                                      (17.7)

where kv is the coefficient of vertical permeability.

Thus,

${k_v}={{H{k_1}{i_1}} \over {{H_1}{i_1} + {H_2}{i_2} + ..........{H_n}{i_n}}}$                              (17.8)

${k_v} = \frac{{H{k_1}{i_1}}}{{\frac{{{H_1}}}{{{k_1}}} + \frac{{{H_2}}}{{{k_2}}} + .......... + \frac{{{H_n}}}{{{k_n}}}}}$             (17.9)

17.3. Factors affecting permeability

Some of the factors affecting permeability are:

• Grain Size

• Viscosity and temperature

• Void ratio

• Soil fabric of clay

Typical values of coefficient of permeability of various soils are presented as (Das, 1999): Appendix 17.1

Head loss in each layer,  ${h_1}={H_1}{i_1}$ ,  ${h_2}={H_2}{i_2}$...................${h_n}={H_n}{i_n}$              (17.10)

The total head loss $h={h_1} + {h_2} + .................. + {h_n}$                   (17.11)

The velocity of flow is same in all the layers. Thus,

$v={k_v}{h \over H}={k_1}{{{h_1}} \over {{H_1}}}={k_2}{{{h_2}} \over {{H_2}}}=\cdots={k_i}{{{h_i}}\over {{H_i}}}=\cdots={k_n}{{{h_n}} \over {{H_n}}}$                                                         (17.12)

where kv is the coefficient of vertical permeability. Rearranging,

$h=v{H \over {{k_v}}}\quad {h_1}=v{{{H_1}}\over{{k_1}}}\quad {h_2}=v{{{H_2}}\over{{k_2}}}\quad\cdots{h_i}=v{{{H_i}}\over{{k_i}}}\quad\cdots{h_n}=v{{{H_n}}\over{{k_n}}}$                      (17.13)

$h=v{H \over{{k_v}}}=v{{{H_1}}\over {{k_1}}}+v{{{H_2}}\over{{k_2}}}+\cdots v{{{H_i}}\over{{k_i}}}+\quad\cdots v{{{H_n}}\over {{k_n}}}$                         (17.14)

${H \over{{k_v}}}={{{H_1}}\over{{k_1}}}+{{{H_2}}\over {{k_2}}}+\cdots{{{H_i}} \over {{k_i}}}+\quad\cdots{{{H_n}}\over {{k_n}}}$                                       (17.15)

${k_v} = \frac{H}{{\frac{{{H_1}}}{{{k_1}}}+\frac{{{H_2}}}{{{k_2}}}+\cdots\frac{{{H_i}}}{{{k_i}}}+\quad\cdots\frac{{{H_n}}}{{{k_n}}}}}$                 (17.16)

References

Ranjan, G. and Rao, A.S.R. (2000). Basic and Applied Soil Mechanics. New Age International Publisher, New Delhi, India.

PPT of Professor N. Sivakugan, JCU, Australia.