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Geotechnical Permeability and Testing Notes

Permeability, Seepage, and Testing — Lecture Notes

Key concepts and definitions

  • Permeability (k): the intrinsic ability of a soil to allow fluid to pass through its voids when a hydraulic gradient is applied.
    • Units: typically meters per second (m/s) in Darcy’s law form.
    • Related to soil properties: porosity (n), pore connectivity, grain size distribution, tortuosity, and saturation state.
  • Hydraulic head (h): the energy per unit weight of fluid at a point, combining elevation head and pressure head.
    • Total head at a point: h = he + hp, where he is elevation head and hp is pressure head.
    • Pressure head conversion: hp = \frac{p}{\gammaw}, where (\gamma_w) is the unit weight of water and (p) is the pressure.
  • Darcy’s law (one-dimensional seepage): the discharge (flow) per unit area is proportional to the hydraulic gradient.
    • Darcy velocity (discharge velocity, often denoted V): V = ki \; i, where (i = \dfrac{\Delta h}{L}) is the hydraulic gradient and (ki) is the permeability coefficient (often simply called k).
    • For a cross-section area (A), the volumetric flow rate is: Q = V \times A.
    • Substituting Darcy’s law: Q = k_i \left(\frac{\Delta h}{L}\right) A.
  • Flow velocities through real pore spaces vs. total cross-section:
    • Discharge velocity (through total cross-section): V. – Often used in simple calculations.
    • Seepage velocity (velocity through the void spaces): Vs. - Related to porosity via conservation of mass: Q = V \cdot A = Vs \cdot Av. where (Av = nA) (area of voids; (n) is porosity).
    • Therefore: Vs = \frac{V \cdot A}{Av} = \frac{V}{n}. Since (0 < n < 1), V_s > V.
  • Porosity (n): the ratio of the volume of voids to the total volume: n = \frac{V{void}}{V{total}}. In many 2D/3D discussions, a similar area-based relation gives (A_v = nA).
  • Practical implications:
    • Discharge velocity ((V)) is useful for estimating total flow rate and seepage through an excavation or structure.
    • Seepage velocity ((V_s)) is relevant when we care about travel time of contaminants or water through the pore network.
  • Applications of permeability in geotechnical design:
    • Settlement and consolidation; drainage and dissipation of pore water.
    • Design of dams, liners, landfill covers, slope stability, and tunnels.
    • Soil stabilization and seepage control via material selection (low permeability for barriers; high permeability for drains).

1D seepage and flow rate relationships

  • Core relationships:
    • Discharge rate: Q = V \cdot A. (volume per unit time)
    • Darcy’s law (permeability times hydraulic gradient): V = k_i \; i,\quad i = \frac{\Delta h}{L}.
    • Combine to express flow: Q = k_i \left(\frac{\Delta h}{L}\right) A.
  • Interpreting the area:
    • In 1D seepage problems we often treat the cross-sectional area as the total area through which water moves.
    • The actual flow moves through voids, but the total flow rate is the same whether you think in terms of total area or void area: Q = V \cdot A = Vs \cdot Av.
    • Therefore, Vs = \dfrac{V A}{Av} = \dfrac{V}{n}.
  • Relationship to porosity:
    • Since (n < 1), the seepage velocity is greater than the Darcy (discharge) velocity: V_s > V.
  • When to use which velocity:
    • For total discharge rate and volumetric calculations in a project (e.g., excavation drainage), use discharge velocity (V).
    • For travel time of fluids or contaminants (e.g., contaminant transport), use seepage velocity (V_s).
  • Concept of total head consistency:
    • If the total head is constant along a vertical/flow path, there is no flow (no head gradient) and thus no seepage.

Measurements and tests of permeability

  • Permeability is measured by laboratory tests on soil samples extracted from the field; two common tests are:
    • Constant head test: best for coarse-grained soils (rapid flow, large pores).
    • Falling head test: suitable for fine to coarse soils (slower flow; can be used across a wider range).
  • Test setup (constant head):
    • A sample is placed in a vertical cylinder with a porous stone at the top and bottom.
    • A water reservoir maintains a constant head across the sample.
    • Piezometers measure the total head at different locations along the sample.
    • The incoming water flow through the sample is collected at the bottom in a graduated container; the volume collected over a time interval gives the discharge.
  • Test setup (falling head):
    • Similar cylinder and sample, but the head is not maintained; water is allowed to fall from the standpipe as the test proceeds.
    • The head difference across the sample changes with time; measure H1 at time t1 and H2 at time t2, with water outflow through the sample.
  • Measurement instruments:
    • Piezometers to measure total head at various locations.
    • A standpipe to measure the head of water and to determine the head drop (for falling head) or constant head (for constant head).
  • Key relationships used in testing:
    • Two-piezometer method (constant head): measure head difference ((\Delta h)) and the distance across the sample (L); area of the specimen (A); volume collected in time (Q) to compute permeability.
    • Head difference and geometry yield hydraulic gradient: i = \frac{\Delta h}{L}.
    • Total head across the sample is used to determine the driving force for flow.
  • Important practical notes:
    • For large-grained materials (coarse sand, gravel), maintain a constant head to ensure measurable flow rates.
    • For finer materials (clays), falling head tests are often more practical due to slower flow rates.
    • Sample size should be compatible with particle size; larger aggregates require larger specimens.
    • The cross-sectional area of the standpipe (a) and the cross-sectional area of the soil specimen (A) are crucial in calculations for falling head tests.

Constant head permeability test — setup and formula

  • Setup (conceptual):
    • Sample enclosed in a cylinder with porous stones, connected to a water reservoir whose head is kept constant.
    • Water flows through the sample from top to bottom; standpipes with piezometers measure the head at different points.
    • Collect the effluent in a graduated cylinder for a known time interval.
  • Geometry and measurements:
    • Standpipe cross-sectional area: a (often derived from diameter of the standpipe).
    • Soil specimen cross-sectional area: A.
    • Length of the soil specimen: L.
    • Head difference across the specimen: Δh (constant in this test).
    • Discharged volume collected in time t: Q (volume units; flow rate is Q/t).
  • Formula for constant head:
    • The discharge rate is related to hydrostatic head driving the flow by Darcy’s law:
      Q = ki \left(\frac{\Delta h}{L}\right) A \quad\Rightarrow\quad ki = \frac{Q \; L}{A \; \Delta h \; t}.
    • If you express Q as the volume collected in a known time t, you can calculate k from the data.
  • Practical notes:
    • If you have a very permeable material, the head must be maintained carefully to allow measurable flow.
    • The relation shows that the permeability increases with larger L and Q, and decreases with larger A or (\Delta h) (for a fixed t).

Falling head permeability test — setup and formula

  • Setup (conceptual):
    • Similar sample and standpipe setup, but the head in the standpipe is allowed to fall (no replenishment).
    • The head in the standpipe decreases from H1 to H2 over the time interval as water drains through the sample.
    • The head change is monitored at two times t1 and t2, with corresponding heads H1 and H2.
  • Key measurements:
    • Cross-sectional area of standpipe: a.
    • Cross-sectional area of the soil specimen: A.
    • Length of the specimen: L.
    • Initial and final heads: H1 and H2.
    • Time interval: t2 − t1.
  • Derivation outline (constant mass through the sample):
    • Inflow to the sample (through standpipe) equals the outflow through the soil: Qin = Qout.
    • Qin relates to the rate of drop in head in the standpipe: Qin = A_standpipe × (−dH/dt).
    • Qout through the sample (Darcy’s law): Qout = k_i × (H/L) × A.
    • Integrating over the time interval from t1 to t2 yields:
      t2 - t1 = \frac{A L}{A ki} \ln\left(\frac{H1}{H_2}\right).
    • Rearranging gives the falling head permeability equation:
      ki = \frac{A L}{A \,(t2 - t1) \; \ln\left(\dfrac{H1}{H_2}\right)}.
  • Important note on symbols (standard form):
    • Let (a) be the cross-sectional area of the standpipe and (A) the cross-sectional area of the soil specimen; then the standard formula is:
      k = \frac{a L}{A (t2 - t1) \; \ln\left(\dfrac{H1}{H2}\right)}.
  • Example problem (falling head permeability):
    • Given: initial head (H1 = 1.0\,\text{m}), final head (H2 = 0.35\,\text{m}), time interval (t2 - t1 = 3\,\text{h}) = 10800 s.
    • Standpipe diameter: (ds = 5\,\text{mm}) ⇒ standpipe area (a = \pi ds^2/4 = \pi (0.005)^2/4 \approx 1.9635\times 10^{-5}\,\text{m}^2).
    • Specimen diameter: (D = 100\,\text{mm}) ⇒ cross-sectional area (A = \pi D^2/4 = \pi (0.10)^2/4 \approx 7.854\times 10^{-3}\,\text{m}^2).
    • Specimen length: (L = 200\,\text{mm} = 0.20\,\text{m}).
    • Compute:
    • (\ln(H1/H2) = \ln(1.0/0.35) \approx 1.0498).
    • Numerator: (a L = (1.9635\times 10^{-5}) (0.20) = 3.927\times 10^{-6}\,\text{m}^3).
    • Denominator: (A (t2 - t1) = (7.85398\times 10^{-3})(10800) \approx 84.823\,\text{m}^2\cdot\text{s}).
    • Denominator with log factor: (84.823 \times 1.0498 \approx 89.05).
    • Permeability: k = \frac{3.927\times 10^{-6}}{89.05} \approx 4.4\times 10^{-8}\;\text{m/s}. (Order of magnitude consistent with low-permeability clays; sands would yield higher values, e.g., ~10^{-4} m/s).
  • Interpretation of the example:
    • The calculated permeability is very low, typical of clays or slightly compacted soils.
    • Falling head tests are more challenging to perform for highly permeable soils because the head falls quickly, making measurement more difficult; constant head tests are preferable for such materials.

Connections to the course and real-world relevance

  • Why permeability matters:
    • Governs seepage through soils in foundations, embankments, dams, slopes, and excavations.
    • Affects rate of settlement and consolidation because drainage and pore-water expulsion depend on permeability.
    • Determines infiltration/recharge and drainage design in landfills, liners, and covers.
    • Critical for groundwater contamination studies: seepage velocity informs contaminant travel times and risk assessment.
  • How permeability links to broader geotechnical principles:
    • Combines with porosity to describe flow behavior; porosity alone does not determine flow rate because pore connectivity and tortuosity affect flow paths.
    • Interacts with soil structure, density, and effective stress; anisotropy in natural soils can lead to directional permeability differences.
  • Practical implications for design:
    • Materials with low permeability are used as barriers (e.g., dam cores, landfill liners) to impede seepage.
    • Materials with higher permeability enable drainage layers and rapid seepage control.
    • In consolidation and settlement analyses, seepage rates influence how fast water can leave the soil under load, impacting settlement timelines.

Quick reference: common formulas (LaTeX notation)

  • Hydraulic gradient: i = \frac{\Delta h}{L}.
  • Darcy’s law (Darcy velocity): V = ki \; i = ki \left(\frac{\Delta h}{L}\right).
  • Discharge rate: Q = V \; A.
  • Constant head permeability: k = \frac{Q \; L}{A \; \Delta h \; t}.
  • Falling head permeability: k = \frac{a \; L}{A \; (t2 - t1) \; \ln\left(\dfrac{H1}{H2}\right)}.
  • Seepage velocity relation: V_s = \frac{V}{n}.
  • Porosity: n = \frac{V{void}}{V{total}}.
  • Total head relation: $$h = he + hp,\