##### Document information

CE 240 Soil Mechanics & Foundations

Lecture 4.3

**Permeability I
(Das, Ch. 6)**

**Outline of this Lecture
1. Permeability in Soils
2. Bernoulli’s Equation
3. Darcy’s Law
4. Hydraulic Conductivity
5. Hydraulic Conductivity Tests**

**Due to the existence of the inter-connected
voids, soils are permeable. The permeable
soils will allow water flow from points of high
energy to points of low energy.
Permeability is the parameter to characterize
the ability of soil to transport water. **

Permeability in Soils • Permeability is the measure of the soil’s

ability to permit water to flow through its pores or voids

• It is one of the most important soil properties of interest to geotechnical engineers

Soil Permeability

Physical (Soil Characteristics) Mechanical

Moisture Content Unit Weight

Compressibility**Permeability**SpecificGravity Gradation
Atterberg

Limits Strength (Shear)

Soil Properties

Compaction

**1 – Constant-Head Test
2 – Falling-Head Test**

Importance of permeability • The following applications illustrate the

importance of permeability in geotechnical design: – Permeability influences the rate of settlement of a

saturated soil under load. – The design of earth dams is very much based upon

the permeability of the soils used. – The stability of slopes and retaining structures can be

greatly affected by the permeability of the soils involved.

– Filters made of soils are designed based upon their permeability.

Use of Permeability • Knowledge of the permeability properties

of soil is necessary to: – Estimating the quantity of underground

seepage (Chapter 7); – Solving problems involving pumping seepage

water from construction excavation; – Stability analyses of earth structures and

earth retaining walls subjected to seepage forces.

**Bernoulli’s equation
**

**The total pressure in terms of water head is
formed from 3 parts: 1), pressure head; 2),
dynamic head; and 3), elevation head. This is
known as the Bernoulli’s equation:
**

2

2*w
*

*P vh Z
g
*

**h: total head in m, or ft;
P: water pressure in Pa, or psi;
**

**w: unit weight of water, in kg/(s2m2), or lb/(ft3);
v: velocity of water, in m/s, or ft/s;
g: gravity acceleration m/s2 or ft/s2;
Z: elevation head in m, or ft.**

**The surface of the water column (the head)
is the water table. Water Table in an
Unconfined Aquifer is the surface along
which the hydrostatic pressure is equal to
the atmospheric pressure.
**

Atmospheric pressure

**Confined Aquifer:
**• **Water in confined aquifer is separated from air by
**

**impermeable layers known as aquiclude. This type
of aquifer forms an artesian system;
**

• **The well drilling into confined aquifer then could be
an artesian well (the water level in the well is above
the height of the ceiling aquiclude).
**

Piezometric surface

**Ceiling aquiclude
**

**Floor aquiclude**

**The dynamic head is usually negligible since
the water flow velocity is usually small. The
elevation head is accounted from the datum
to the elevation of the bottom of the well, and
the pressure head is the portion above the
well bottom to the water table.
**

Piezometric surface

Elevation head Z

Pressure head P/ w

datum

Again, since the seepage flow velocity in soil is small, the dynamic head (velocity head) can be neglected, so that the total head at any points is

*hi
L
*

*w
*

*Ph Z
*

Hydraulic gradient:

**in most soil we found the
following relation, i.e., the water
**

**flow velocity in the soil is proportional
to the hydraulic gradient
**

*v i
*

may exist in fractured rock, stones, gravels, and very coarse sands

Darcy’s Law • The coefficient of permeability, or hydraulic

conductivity, *k*, is a product of Darcy’s Law.
• In 1856, Darcy established an empirical

relationship for the flow of water through porous
media known as Darcy’s Law, which states: *v =
v = -ki or q = -kiA
*

*q *= flow rate (cm3/s)

*k *= coefficient of permeability (cm/s)

*A *= cross-sectional Area (cm2)

*i *= hydraulic gradient

**The parameter q in Darcy’s law is called the
flow rate or simply the flow (flux). It describes
in a unit time, over a unit cross-section area,
how much water in terms of volume has been
flowed through.
**

, ( )*volume lengthq vA area
time time
*

v

A

**The flow rate q is in the unit of velocity (L/t).
Examination of the Darcy’s law make us be
aware that the permeability k is also in the unit
of velocity.**

**Velocity and seepage velocity**

**in the field, the gradient of the head is the
head difference over the distance
separating the 2 wells.
**

2 1*H HdHv k k
dx x
*

x

H1 H2

Water flow

**Darcy’s law states that how fast the
groundwater flow in the aquifer depends on two
parameters:
1, how large is the hydraulic gradient of the
water head ( i=dH/dx); and
2, the parameter describing how permeable the
aquifer porous medium – the coefficient of
permeability (hydraulic conductivity) k.
**

**The minus sign in the equation denotes that the
direction of flow is opposite to the positive
direction of the gradient of the head.**

**The physical description of groundwater
flow in soil is the Darcy’s law. The
fundamental premise for Darcy’s law to work
are:
**

**1, the flow is laminar, no turbulent flows;
2, fully saturated;
3, the flow is in steady state, no temporal
variation.**

**Hydraulic conductivity k and
absolute permeability
**

**The absolute permeability is in the unite of LL
(length square); and the expression for the
relation is
**

*wk K
*

*K*

**Units of the coefficient of Permeability k
**

**The permeability k is in the dimension of velocity.
However, in deferent field people prefer use different
units for permeability simply because different fields
deal different scales of subsurface fluid flow. In
hydrogeology a used to be popular unit is meinzer; in
geotechnical world is cm/sec; and in petroleum
engineering people just use the unit of darcy. Here are
the conversions:
**

**1 cm/sec = 864 m/day
1 darcy = 1 cm3 of fluid with viscosity of 1 centipose
in 1 sec, under a pressure change of 1 atm. over a
length of 1 cm through a porous medium of 1 cm2 in
cross-sectional area.
1 Meinzer = 1gal/day/ft2**

**(West, 1995)**

Hydraulic Conductivity • The coefficient or permeability is also

known as hydraulic conductivity;
• Hydraulic Conductivity, *k, *is a measure

of soil permeability; • k is determined in the lab using two

methods: – Constant-Head Test – Falling-Head Test

Hydraulic Conductivity (Cont.)

• Hydraulic conductivity of soils depends on several factors: – Fluid viscosity – Pore size distribution – Grain size distribution – Void ratio – Degree of soil saturation

Constant Head Test • The constant head test is used primarily

for coarse-grained soils; • This test is based on the assumption of

laminar flow where *k *is independent of *i
(low values of i);
*

• *ASTM D 2434;
*• *This test applies a constant head of water
*

*to each end of a soil in a “permeameter”.*

Permeameter

Constant-head hydraulic conductivity test with permeameter

( )*Q Avt A ki t*

Procedure (Constant head)

1. Setup screens on the permeameter
2. Measurements for permeameter, (D), (L), H1
3. Take 1000 g passing No.4 soil (M1)
4. Take a sample for M.C.
5. Assemble the permeameter – ** make sure seals are air-tight
**6. Fill the mold in several layers and compact it as prescribed.
7. Put top porous stone and measure H2
8. Weigh remainder of soil (M2)
9. Complete assembling the permeameter. (keep outlet valve closed)
10.Connect Manometer tubes, but keep the valves closed.
11.Apply vacuum to remove air for 15 minutes (through inlet tube at

top) 12.Run the Test (follow instructions in the lab manual) ….. 13.Take readings

– Manometer heads h1 & h2 – Collect water at the outlet, Q ml at time t 60 sec.

**Calculation (Constant head)
**• Determine the unit weight;
• Calculate the void ratio of the compacted

specimen;

• Calculate *k *as:

• Calculate

( )*hfrom Q Akit A k t
L
*

*QLget k
Aht
*

*C
*

*CT
*

*CT
kk C
*

020

0

0020

Falling Head Test • The falling head test is used both for

coarse-grained soils as well as fine- grained soils;

• Same procedure in constant head test except: – Record initial head difference, h1 at t = 0 – Allow water to flow through the soil specimen – Record the final head difference, h2 at time

t = t2 – Collect water at the outlet, Q (in ml) at time t

60 sec

**Calculation (Falling head)
**

• Calculate *k *as

• Where:
*A *= inside cross sectional area of the water tank
*a *= inside cross sectional area of the standing pipe
*h*1 = distance to bottom of the beaker before the test
*h*2 = distance to bottom of the beaker after the test

• Calculate

2

1ln
*h
h
*

*At
aLk
*

*C
*

*CT
*

*CT
kk C
*

020

0

0020

**Falling Head Test**

**Example 6.4
Figure 6.7**

**Example 6.5
**

**Figure 6.8**

**Reading Assignment:
**

**Das, Ch. 6
**

**Homework:
**

**6.3, 6.4, 6.7, 6.8, 6.12**