Darcy's law and hydraulic conductivity are crucial concepts in understanding soil water movement. They explain how water flows through porous materials like soil and rock, helping us predict groundwater behavior and design drainage systems.

Hydraulic conductivity measures how easily water moves through soil. It's affected by soil properties like particle size and void ratio. Understanding these factors is key to managing water flow in construction, agriculture, and environmental projects.

Darcy's Law and its Assumptions

Fundamental Equation and Principles

• Darcy's law describes fluid flow through porous media (soil or rock)
• Flow rate proportional to hydraulic gradient and hydraulic conductivity
• Mathematical expression: $Q = -KA(dh/dL)$
  • Q: flow rate
  • K: hydraulic conductivity
  • A: cross-sectional area
  • dh/dL: hydraulic gradient
• Negative sign indicates flow from high to low hydraulic head
• Valid for most natural groundwater flow conditions
• May break down in cases of very high flow velocities or extremely fine-grained or fractured media

Assumptions and Limitations

• Flow remains laminar and steady-state
• Porous medium fully saturated with fluid
• Fluid considered incompressible
• No chemical or thermal gradients affecting flow
• Limitations in applicability:
  • Very high flow velocities (turbulent flow)
  • Extremely fine-grained media (non-Darcian flow)
  • Fractured media (preferential flow paths)

Hydraulic Conductivity of Soil

Definition and Units

• Measures porous medium's ability to transmit water under hydraulic gradient
• Expressed in units of length per time (m/s or cm/s)
• Depends on properties of both porous medium and flowing fluid
• Relationship with intrinsic permeability: $K = (k*ρ*g)/μ$
  • k: intrinsic permeability
  • ρ: fluid density
  • g: gravitational acceleration
  • μ: fluid viscosity

Influencing Factors

• Soil properties affecting hydraulic conductivity:
  • Particle size distribution (coarser-grained soils generally have higher K)
  • Void ratio (higher void ratios typically result in higher K)
  • Degree of saturation (fully saturated soils have higher K)
• Fluid properties influencing hydraulic conductivity:
  • Density (affected by temperature)
  • Viscosity (affected by temperature)
• Examples of hydraulic conductivity values:
  • Clean gravels: $10^{-2}$ m/s
  • Fine sands: $10^{-5}$ m/s
  • Silts: $10^{-8}$ m/s
  • Clays: $10^{-12}$ m/s

Discharge Velocity and Flow Rate

Velocity Calculations

• Discharge velocity (v): apparent velocity of water flow through porous medium
  • Calculated as: $v = Q/A$
  • Using Darcy's law: $v = -K(dh/dL)$
• Actual seepage velocity (vs): higher than discharge velocity
  • Calculated as: $vs = v/ne$
  • ne: effective porosity
• Example:
  • For a soil with K = $10^{-5}$ m/s, hydraulic gradient of 0.01, and effective porosity of 0.3:
    • Discharge velocity: v = $-(10^{-5})(0.01) = 10^{-7}$ m/s
    • Seepage velocity: vs = $(10^{-7})/(0.3) = 3.33 10^{-7}$ m/s

Flow Rate Calculations

• Flow rate (Q) calculated using Darcy's law: $Q = -KA(dh/dL)$
• Considerations for anisotropic soils:
  • Hydraulic conductivity varies with direction
  • Principal directions of hydraulic conductivity must be considered
• Multi-dimensional flow problems:
  • Darcy's law extended using vector notation and partial derivatives
• Example:
  • For a soil layer with K = $10^{-4}$ m/s, cross-sectional area of 10 m², and hydraulic gradient of 0.05:
    • Flow rate: Q = $-(10^{-4})(10)(0.05) = 5 10^{-5}$ m³/s

Determining Hydraulic Conductivity

Laboratory Methods

• Constant head permeameter test:
  • Suitable for coarse-grained soils
  • Measures flow rate under constant hydraulic head
  • Example: sand sample with steady-state flow of 2.5 cm³/s under 30 cm head difference
• Falling head permeameter test:
  • Appropriate for fine-grained soils
  • Measures rate of head drop over time
  • Example: clay sample with initial head of 100 cm dropping to 50 cm in 24 hours

Field Methods

• Pumping tests:
  • Involve pumping water from well and measuring drawdown in observation wells
  • Example: continuous pumping at 100 L/min for 72 hours, monitoring water levels in surrounding wells
• Slug tests:
  • Rapidly change water level in well and measure recovery rate
  • Example: instantaneous removal of 1 L of water from well, recording water level rise over time
• Packer tests:
  • Used in boreholes to isolate and test specific intervals of soil or rock
  • Example: testing 1 m sections of a borehole at different depths

Additional Considerations

• Empirical correlations (Hazen formula for sands) estimate K based on grain size distribution
• Scale effect in hydraulic conductivity measurements:
  • Laboratory tests may not accurately represent field-scale behavior
  • Soil heterogeneity and anisotropy influence results
• Proper sample preparation and test procedures crucial for accurate K values
• Multiple methods often employed to obtain representative K values for a site