Soil Permeability and Seepage

• Module 6 covers permeability and seepage in soil mechanics.
• Learning outcomes include discussing methods for determining the coefficient of permeability, determining the rate of water flow through soils, and understanding seepage and flow nets to solve related problems.

Introduction

• Permeability dictates the ease with which water flows through soil.
• Seepage is the movement of water through soil due to hydraulic gradients.
• Applications of permeability and seepage knowledge include foundation design, slope stability analysis, and the construction of earth dams and tunnels.

Definition of Terms

• Groundwater fills soil pores under gravity.
• Head (H) is the mechanical energy per unit weight in soil.
• Hydraulic conductivity (k), or coefficient of permeability, is a constant that relates to the velocity of water flow through soils.
• Permeability reflects a soil's ability to allow the passage of fluids, an intrinsic property independent of fluid type.
• Hydraulic conductivity is a material-fluid-dependent property, influenced by both soil structure and water characteristics, and reflects the measure of ease for water flow.
• Pore water pressure (u) is the pressure exerted by water within soil pores; it equals zero at the water table and acts equally in all directions.
• Effective stress is the total stress minus the pore water pressure.
• Hydrostatic condition occurs when pore water pressure is constant over time and the water table is level, indicating no flow.
• Steady-state seepage condition assumes stationary soil particles, with the amount of flow constant over time, the water table is not level and the hydraulic gradient causes steady-state seepage flow.
• Transient seepage condition: Changes in effective stress result in soil deformation, and both pore water pressure and flow rate vary with time; the water table is not level.

Head and Pressure Variation in a Fluid at Rest

• Steady-state exists when water flow rate and direction remain constant at every mass point over time.
• Hydraulic head and gradient remain constant at any soil location in steady-state.
• This assumes steady and inviscid flow, incompressible, and irrotational fluid particles.

Darcy's Law

• Average flow velocity through soils is proportional to the total head gradient
• The flow in any direction, j:
  • Vj = kj * (dH/dxj)
  • where:
    • v is the average flow velocity
    • k is the hydraulic conductivity coefficient
    • dH change in total head
    • dx is distance
• The unit of measurement for k is length/time (cm/s).
• Darcy's law equation: Vx = kx * (ΔH/L) = kx * i
  • i = ΔH/L (hydraulic gradient).
• Darcy's law applies when flow is laminar.
• In most soils, flow is slow, it is considered laminar.
• Head loss increases linearly with velocity in laminar flow.
• Exceeding the transition zone initiates internal eddy currents and mixing, leading to a nonlinear relationship as energy dissipates at a greater rate.

Average Velocity

• Average velocity (v) is a macroscopic concept representing the apparent velocity of water flowing through a unit cross-sectional area of bulk soil.
• Imagine a pipe filled with soil: average velocity is the total water flow rate divided by the pipe's total cross-sectional area.
• Water flows through interconnected pore spaces, so the actual velocity of water molecules within these pores is greater than the average velocity.

Seepage Velocity

• Seepage velocity (vs) represents the actual average velocity of water flowing through interconnected pore spaces.
• It considers that water moves through a portion of the total cross-sectional area.

Soil Factors Affecting Hydraulic Conductivity

• Coarse-grained soils have higher hydraulic conductivities than fine-grained soils due to the water in the double layer reducing the seepage pore space.
• Hydraulic conductivity depends on D10 for coarse soils.
• Pore fluid properties, specifically viscosity: k₁/k₂ ≈ μ₂/μ₁, where is dynamic viscosity.
• Void ratio: k₁/k₂ ≈ e₁/e₂ for coarse grains.
• Pore size: Higher interconnected pore space results in higher hydraulic conductivity, related to the square of the pore size.
• Water tends to seep quickly through loose layers and fissures.
• Catastrophic failures can occur from seepage.
• Entrapped gases reduce conductivity, and are difficult to get rid of.
• Darcy's law is valid only for laminar flow (Reynolds number < 2100).
• Fancher et al. (1933) gave the following criterion for Darcy's Law: (vDρw)/µ ≤ 1

Other Factors Affecting Hydraulic Conductivity

• Fluid viscosity, pore and grain-size distribution, roughness and degree of saturation are factors.
• In clayey soils, structure plays an important role in hydraulic conductivity.
• The ionic concentration and the thickness of layers of water held to clay particles affect the permeability of clays.

Constant-Head Test

• The water supply is adjusted so that the head difference is constant during the test period
• Water is collected in a graduated flask for a duration once a constant flow rate is achieved.
• Q = Avt = A(ki)t
  • Q is the volume of water
  • A is the cross-sectional area
  • t is the collection time
  • i = h/L
  • h is the head difference
  • L, the specimen length
• Hydraulic conductivity is then k = (QL)/(Aht)
• where Q is voume of water collected, A is the cross sectional area of the specimen, t is duration and h is the head

Falling-Head Test

• The method consists of water from a standpipe, flowing through a soil specimen.
• The initial head difference, h1 at time t = 0, and the final head difference, h2 at time t = t2, are recorded
• The rate of flow: q = k(A/L)h = -a(dh/dt)
• To find conductivity, use, k = 2.303*(aL/At)*log10(h1/h2), with:
  • k = hydraulic conductivity
  • a = cross-sectional area of the standpipe
  • A is the cross-sectional area of the soil specimen
  • L is the length
  • t is time

Illustrative Problem 1

• Thickness of permeable soil layer = 1.1 m, angle = 15°, K = 3.75x10-2 cm/sec, e = 0.6
• Calculate:
  • Hydraulic gradient
  • Flow rate (cu.m/hr/m)
  • Seepage velocity

Calculating Flow Parameters

• Soil sample is 10 cm in diameter, 1 m long.
• 1 cm3 of water is collected every 10 seconds.
• Determine:
  • Hydraulic gradient
  • Flow rate
  • Average velocity
  • Seepage velocity
  • Hydraulic conductivity

Illustrative Problem 2

• Constant-head test, where:
  • L = 30 cm
  • A = 177 cm²
  • h = 50 cm
  • Water collected = 350 cm³ in 5 minutes.
  • Find hydraulic conductivity (cm/sec).

Illustrative Problem 3

• For a falling-head permeability test
  • Length = 8 in.
  • Soil area = 1.6 in²
  • Standpipe area = 0.06 in²
  • Head difference at t=0 = 20 in.
  • Head difference at t=180 sec = 12 in.
• Determine hydraulic conductivity (in/sec).

Equivalent Hydraulic Conductivity in Stratified Soils

• Consider a unit length cross-section passing through the n layer and perpendicular to flow direction.
• Total flow through the cross-section in unit time:
• q = v * 1 * H
• where: - v = average discharge velocity - v₁, v₂, v₃, ..., vₙ are discharge velocities of flow in layers
• Let's say that kH₁, kH₂, kH₃,..., kHₙ are the hydraulic conductivities of the individual layers in the horizontal direction and kH(eq) is the equivalent hydraulic conductivity in the horizontal direction
• From Darcy's law: v = kH(eq)*ieq; v₁ = kH₁ i₁; v₂ = kH₂ i₂; v₃ = kH₃ i₃; ... vₙ = kHₙ in