Permeability and Darcy's Law

• Permeability refers to the property of a porous material that allows fluids to pass through it

Hydraulic Conductivity (k)

• Hydraulic conductivity is a measure of how easily water can flow through soil or rock
• Factors such as pore size and distribution, particle size and shape, soil structure, void ratio, degree of saturation, and fluid properties (viscosity and density) influence hydraulic conductivity
• Expressed using Darcy's Law: v = ki, where v is the discharge velocity and i is the hydraulic gradient

Darcy's Law

• Darcy's Law describes fluid flow through a porous medium using the equation Q = k i A
  • Q is the flow rate, k is the hydraulic conductivity, i is the hydraulic gradient, and A is the cross-sectional area
• Assumptions for Darcy's Law include laminar flow, saturated soil, steady flow, and incompressible fluid
• Limitations include that it is not valid for turbulent flow or partially saturated soils without modification

Factors Affecting Permeability

• Grain Size: Larger particles generally provide higher permeability due to larger pores facilitating easier water flow
• Void Ratio: Higher void ratio increases permeability, it is proportional
• Soil Structure: Flocculated structures have higher permeability compared to dispersed structures
• Soil Composition: Clay minerals reduce permeability with their small size and swelling when wet
• Degree of Saturation: Full saturation is required for Darcy's Law to be directly applicable; partial saturation with air voids reduces permeability
• Temperature: Affects fluid viscosity; higher temperatures decrease viscosity, increasing permeability

Determination of Hydraulic Conductivity

• Constant Head Test: Primarily for coarse-grained soils, maintaining a constant water level and measuring the flow rate to calculate hydraulic conductivity using Darcy's Law
• Falling Head Test: Suited for fine-grained soils, connecting a standpipe to the soil sample and measuring the water level drop rate over time to calculate hydraulic conductivity
• Pumping Tests (Field Methods): Pumping water from a well and observing drawdown in nearby wells to analyze and determine aquifer hydraulic conductivity
• Empirical Formulas:
  • Hazen's Formula (for sands): k = C(d10)^2
    • k is the hydraulic conductivity (cm/s)
    • C typically ranges between 100 and 150
    • d10 is the effective particle size (mm)
  • Kozeny-Carman Equation: k = (γw/μ) * (e^3 / (1 + e)) * (1 / (Cs * St^2))
    • γw is the unit weight of water
    • μ is the dynamic viscosity of water
    • e is the void ratio
    • Cs is the shape factor
    • St is the specific surface area

Permeability in Stratified Soils

• Horizontal Flow: The equivalent hydraulic conductivity (keq) is the weighted average, using the formula keq = (Σ(ki * hi)) / (Σhi)
  • ki is the hydraulic conductivity of layer i
  • hi is the thickness of layer i
• Vertical Flow: The equivalent hydraulic conductivity (keq) is the harmonic mean, using the formula keq = (Σhi) / (Σ(hi / ki))
  • ki is the hydraulic conductivity of layer i
  • hi is the thickness of layer i

Seepage

• Seepage refers to water flow through soil under a hydraulic gradient
• Occurs beneath dams, around sheet pile walls, and through natural soil slopes
• Seepage Velocity (vs):
  • Indicates the average velocity, using the formula vs = v / n
    • v is the discharge velocity (Darcy velocity)
    • n is the porosity of the soil

Seepage Analysis

• Flow Nets: Are a graphical representation of flow paths and equipotential lines to estimate seepage quantities and pore water pressures
  • Composed of flow lines and equipotential lines intersecting at right angles
• Uplift Pressure: Seepage can generate uplift pressure beneath hydraulic structures, reducing effective stress and potentially causing instability
• Exit Gradient: Refers to the hydraulic gradient at the downstream end, with high exit gradients potentially leading to piping or soil erosion

Piping

• Piping is a form of soil erosion caused by water flowing through soil
• Occurs when the exit gradient is high enough
• Can lead to the formation of underground channels or pipes, which can undermine the stability of structures
• Filter Design:
  • Filters prevent piping with a specific particle size distribution
  • Filter criteria (Terzaghi's filter criteria):
    • D15(filter) ≤ 4 to 5 * D85(soil)
    • D15(filter) ≥ 4 to 5 * D15(soil)
      • D15 and D85 are the particle sizes for which 15% and 85% of the material is finer by weight, respectively

Applications of Permeability

• Groundwater Hydrology: Used for assessing groundwater flow rates and directions, designing well systems, and modeling contaminant transport
• Geotechnical Engineering: Used for designing drainage systems, analyzing slope stability, and predicting settlement rates
• Environmental Engineering: Designing landfill liners, remediating contaminated soils, and assessing soil suitability for septic systems