Geotechnical Engineering: Soil Permeability and Seepage Analysis

Introduction to Permeability

• Definition of Soil Mass Structure: A soil mass consists of solid particles of various sizes with interconnected void spaces.

• Fluid Flow Mechanism: The continuous void spaces in a soil permit water to flow from a point of high energy to a point of low energy.

• Permeability Definition: Permeability is the specific property of a soil that allows the seepage of fluids through its interconnected void spaces.

• Motto: "SEE the INVISIBLE and DO the IMPOSSIBLE."

Darcy’s Law and the History of Permeability

• Origin (1856): Henry Darcy reported experiment results used to enhance water flow through sand filter beds for the city of Dijon, France, for water treatment purposes.

• Engineering Objective: Darcy aimed to design sand beds that would efficiently filter the daily volume of water required by the city.

• Experimental Variables: To evaluate water volume filtered over time, Darcy experimented with:

  • The type of sand.

  • The area of the filter bed (tube diameter).

  • The thickness of the sand (length of the sample).

  • The force driving water through the filter bed.

• Mathematical Discovery: Darcy discovered that the steady-state flow rate ($Q$) through a circular sand filter was:

  • Directly proportional to the cross-sectional area of the filter ($A$).

  • Directly proportional to the difference in hydraulic head ($\Delta h$) on each side of the filter (measured from a datum).

  • Inversely proportional to the length of the filter material ($\Delta L$).

• Hydraulic Head: The elevation of the water level in piezometers is referred to as the hydraulic head.

• Fundamental Proportionality:     $Q \propto \frac{\Delta h}{\Delta L} A$

• The Darcy Equation:     $Q = k \frac{\Delta h}{\Delta L} A$

  • $Q$ = volumetric flow rate ($L^3/T$).

  • $k$ = hydraulic conductivity or permeability ($L/T$). It reflects the ease with which water flows through a material.

  • $\Delta h$ = difference in hydraulic head ($h_2 - h_1$) between two points ($L$).

  • $\Delta L$ = length along the flow path ($L$).

  • $\frac{\Delta h}{\Delta L}$ = gradient of hydraulic head (dimensionless).

  • $A$ = cross-sectional area perpendicular to the direction of flow ($L^2$).

• Infinitesimal Interval Formula:     $Q = k \frac{dh}{dl} A$

Flow Velocity and Flux Concepts

• Specific Discharge ($q$):

  • Also referred to as "groundwater flux," "Darcy flux," "Darcy velocity," or "apparent velocity."

  • Definition: The volume of water that flows through a unit cross-sectional area of porous media per unit time.

  • Formula:         $q = \frac{Q}{A} = k \frac{dh}{dl}$

  • Units: $L^3/(L^2 T)$, simplified to $L/T$.

• Average Linear Velocity ($v_s$):

  • Also known as "seepage velocity" or "average interstitial velocity."

  • In reality, water can only pass through the connected pore space ($n$), making the actual velocity higher than the specific discharge.

  • Formula:         $v_s = \frac{Q}{An} = \frac{q}{n} = \frac{k}{n} \frac{\Delta h}{\Delta L}$

• Hydraulic Gradient ($i$):

  • The difference in the hydraulic head over a distance along the flow.

  • Formula:         $i = \frac{\Delta h}{\Delta L}$

Hydraulic Conductivity Parameters and Typical Values

• Definition: The volume of water discharge through a unit cross-sectional area of porous medium under a unit hydraulic gradient.

• Standard Reference Temperature: Measurements are typically standardized to $20^\circ\text{C}$ to account for changes in water viscosity.

• Units: Unit length per unit time (e.g., $m/s$, $cm/s$, $mm/s$).

• Relevance: Crucial for groundwater flow analysis, contaminant transport modeling, and geotechnical design.

• Intrinsic/Absolute Permeability ($\bar{k}$):

  • Separates the properties of the fluid (viscosity and density) from the properties of the soil matrix.

  • Formula:         $k = \frac{\rho g}{\mu} \bar{k} = \frac{\gamma_w}{\mu} \bar{k}$

  • $k$ = hydraulic conductivity.

  • $\bar{k}$ = intrinsic permeability.

  • $\mu$ = dynamic viscosity of water.

  • $\gamma_w$ = unit weight of water.

  • $\rho$ = density of water.

  • $g = 9.81\,m/s^2$.

• Conversion Factor: $1\,\text{darcy} = 0.987 \times 10^{-12}\,m^2$. Approximately, $1\,\text{darcy} \approx 10^{-5}\,m/s$.

• Typical Coefficient of Permeability ($k$) Values (USBR):

  • Clean gravel: $10^{+1} \text{ to } 10^{+2}\,mm/s$ (Very Good Drainage).

  • Coarse and medium sands: $10^{-2} \text{ to } 10^{+1}\,mm/s$ (Good Drainage).

  • Fine sands and loose silts: $10^{-4} \text{ to } 10^{-2}\,mm/s$ (Fair Drainage).

  • Dense silt and clayey silt: $10^{-5} \text{ to } 10^{-4}\,mm/s$ (Poor Drainage).

  • Silt Clay and Clay: $10^{-8} \text{ to } 10^{-5}\,mm/s$ (Very Poor Drainage).

Temperature Correction Factors for Permeability

To standardize measurement to $k_{20}$, apply the correction factor based on the actual water temperature ($T$):

Laboratory Methods for Determining Coefficient of Permeability

1. Constant Head Permeability Test

• Applicability: Relatively more permeable soils (coarse-grained soils).

• Sample: Disturbed ($2.5\,kg$) or undisturbed ($85\,mm$ diameter, cylindrical).

• Formula Derivation:     $Q = kiA$     $Q = k \frac{\Delta h}{\Delta L} A$     $k = \frac{QL}{Ah}$     Since $Q = \frac{\forall}{t}$, where $\forall$ is volume:     $k = \frac{\forall L}{Aht}$

2. Falling (Variable) Head Permeability Test

• Applicability: Relatively less permeable soils (fine sands, silty, and clayey soils).

• Sample: Disturbed ($2.5\,kg$) or undisturbed ($85\,mm$ diameter).

• Formula Derivation:     Equating flow through standpipe to flow through soil:     $q = a \left(-\frac{dh}{dt}\right)$     $q = k i A = k \frac{h}{L} A$     $k \frac{h}{L} A dt = a (-dh)$     Integrating from $t_1$ to $t_2$ and $h_1$ to $h_2$:     $k = \frac{aL}{A(t_2 - t_1)} \ln\left(\frac{h_1}{h_2}\right)$     Standard manual form:     $k = \frac{2.303 aL}{A(t_2 - t_1)} \log_{10}\left(\frac{h_1}{h_2}\right)$

Empirical Formulas for Hydraulic Conductivity

• Slichter Method (1899):     $k = \frac{10.22 d^2}{\mu C_s}$

  • $d$ = mean grain diameter ($cm$).

  • $C_s$ = constant for a given porosity ($n$). Vukuvic and Soro (1992) suggest $C_s = \frac{1}{n^{3.287}}$.

• Hazen Method (1911):     $k = C (d_{10})^2$

  • $d_{10}$ = effective size (in $cm$).

  • $C$ ranges from $40$ to $150$ depending on sand sorting and cleanliness.

• Terzaghi’s Method (1925): Accounts for porosity and effective grain diameter while adjusting for fluid viscosity.

• USBR Method (1978): Developed for medium sands with uniformity coefficient ($C_u$) less than $5$.

• Kozeny-Carman Method: Relates $k$ to void ratio ($e$) and specific surface area.

• Carrier (2003) Method:     Uses Shape Factor ($SF$). General form involves terms like:     $\frac{e^3}{1+e}$

• Chapuis (2004) Method:     $k = 2.4622 (D_{10})^2 \left[ \frac{e^3}{1+e} \right]^{0.7825}$

• Amer and Award (1974) Method:     $k = 35 \left[ \frac{e^3}{1+e} \right] C_u^{0.60} D_{10}^{2.32}$

• Samarasinghe et al. (1982) Method: For normally consolidated clays.     $k = C \frac{e^n}{1+e}$

• Mesri and Olson (1971) Method:     $\log(k) = A' \log(e) + B'$

Permeability in Stratified Soil

In natural deposits, hydraulic conductivity often varies in different directions due to layering.

1. Equivalent Horizontal Hydraulic Conductivity ($k_{H(eq)}$)

• Mechanism: Flow is parallel to the layers. Change in total flow is the sum of flow in individual layers.

• Formula:     $k_{H(eq)} = \frac{1}{H} (k_1 H_1 + k_2 H_2 + ... + k_n H_n)$     Where $H$ is the total thickness ($H = \sum H_i$).

2. Equivalent Vertical Hydraulic Conductivity ($k_{V(eq)}$)

• Mechanism: Flow is perpendicular to the layers. Flow rate is the same through each layer, but head loss varies.

• Formula:     $k_{V(eq)} = \frac{H}{\frac{H_1}{k_1} + \frac{H_2}{k_2} + ... + \frac{H_n}{k_n}}$

Sample Problem Scenarios

• Scenario 1 (Seepage Rate): A soil layer ($k = 5.3 \times 10^{-5}\,m/s$) above an impervious layer. Calculate seepage per unit width for a thickness $H=3\,m$ and angle $\alpha=8^\circ$.

• Scenario 2 (Darcy Calculation): Find flow rate in $m^3/s/m$ given head difference $h=4\,m$, length $S=50\,m$, and $k=0.08\,cm/s$.

• Scenario 3 (Levee Seepage): A $650\,m$ long levee with $2.5\,m$ thick sand layer. Flow collected is $13.5\,m^3/hr$. Solve for $k$.

• Scenario 4 (Concrete Dam): Seepage through a sandy layer ($H_3 = 0.75\,m$, $k = 0.009\,cm/s$, $e = 0.8$) under a dam. Determine rate per unit length.

• Scenario 5 (Lab Test): Constant head test with $15\,cm$ distance between tappings, $40\,cm$ head difference, and $500\,ml$ collected in $900\,s$. Includes calculation of seepage velocity ($v_s$) using sample dry mass ($486\,g$) and $G_s = 2.654$.