Chapter seven

PERMEABILITY

Soils are permeable due to the existence of interconnected voids through which water can flow from points of high energy to points of low energy.

The study of the flow of water through permeable soil media is important for estimating the quantity of underground seepage :

To solve problems involving the pumping of water for

underground construction for making stability analyses of earth dams and earth-retaining

structures that are subject to seepage forces.

Bernoulli’s Equation

According to Bernoulli’s equation in fluid mechanics, the total head at a point in water under motion (flow line) is given by the sum of pressure, velocity, & elevation head:

Example Determine the total heads at points P and X

Datum

Assuming the datum at top of rock level, the total head at P is estimated as follows:

• At point P:

• At point X:

Thus the total head at point “X” equals that at “P” No flow or hydrostatic state

SEEPAGE – WATER FLOW

• For static conditions (no flow), all points have the same

energy • During seepage, water will flow from points of higher head to points of lower head • Energy is dissipated in overcoming the soil resistance and hence is the head loss.

Darcy’s Law

Based on observations of flow of water through clean sands, Darcy found that the flow (volume per unit time) was: proportional to the head difference Δh proportional to the cross-sectional area A inversely proportional to the length of sample L

q = K i A

Darcy published a simple equation for the discharge velocity of water through saturated soils, which may be expressed as:

Where: = discharge velocity, which is the quantity of water flowing in unit time = seepage velocity k = hydraulic conductivity or coefficient of permeability Av = area of void in the cross section of the specimen

As = area of soil solids in the cross section of the specimen.

Vv = volume of voids in the specimen

Vs = volume of soil solids in the specimen n = porosity

e = void ratio

COEFFICIENT OF PERMEABILITY

The hydraulic conductivity of soils depends on several factors: fluid viscosity Pore size distribution grain-size distribution void ratio roughness of mineral particles degree of soil saturation. Coefficient of permeability is generally expressed in cm/sec or m/sec in SI units and

in ft/min or ft/day in English units (like the velocity units)

Laboratory Determination of Hydraulic Conductivity Laboratory Determination of Hydraulic Conductivity

Two standard laboratory tests are used to determine the hydraulic conductivity of soil: the constant-head test the falling-head test

1- constant head

the difference of head between the inlet and the outlet remains constant during the test period. After a constant flow rate is established,

water is collected in a graduated flask for a known duration.

The total volume of water collected may be expressed as:

2- Falling head Water from a standpipe flows through the soil. The initial head difference h1 at time t = 0 is recorded, and water is allowed to flow through the soil specimen such that the final head difference at time t = t2 is h2.

The flow (volume/time) in the standpipe = flow in soil

For a time interval δt

• The flow in the pipe:

• The flow in the soil:

Equivalent Hydraulic Conductivity in Stratified Soil

In a stratified soil deposit where the hydraulic conductivity for flow in a given direction changes from layer to layer, an equivalent hydraulic conductivity can be computed to simplify calculations.

• Flow in the horizontal direction: The total flow through the cross section in unit time can be written as: • Q 1 = kH1 i1A1 = kH1 (Δh1/Lo) A1

• Q 2 = kH2 i2A2 = kH2 (Δh2/Lo) A2

• Q n = kHn inAn = kHn (Δhn/Lo) An

• Q = kequivalent i A = keq (ΔH/Lo)A

Hence, Q = Q1 +Q2+…Qn noting that Δh1 = Δh2 = Δhn = ΔH

L0

K eq (ΔH/Lo)A = kH1 (Δh1/Lo) A1 + kH2 (Δh2/Lo) A2+…kHn (Δhn/Lo) An

where A1 = L0.H1; A2 = L0.H2; A = L0.H (same area was considered)

K eq = ( kH1 H1 + kH2H2+…… kHnHn) / H

• Flow in the vertical direction:

In this case, the velocity of flow through all the layers is the same. However, the total head loss, h, is equal to the sum of the head losses in all layers. Thus,

• Q 1 = kv1 i1A1 = kv1 (Δh1/Lo) A1

• Q 2 = kv2 i2A2 = kv2 (Δh2/Lo) A2

• Q n = kvn in An = kvn (Δhn/Lo) An

• Q = k equivalent i A = keq (ΔH/Lo)A

Hence, ΔH = Δh1 + Δh2 +… Δhn

Q L /Keq A = Q1L1/ Kv1A1 + Q2 L2/ Kv2A2

Knowing that A= A1 = A2 Q = Q1 = Q2

L /Keq= L1/K1 + L2 / K2