Seepage Effects on Stresses: 1D Seepage, Downward/Upward Seepage, and Quick Condition (Worked Example)

Overview: seepage effects on stresses in one-dimensional flow

Goal: understand how seepage alters stresses in soils by changing pore water pressures while total stresses may remain the same (static load does not change with seepage unless the boundary conditions change).
Key quantities:
- Total vertical stress,
  \sigma_v: stress carried by both water and soil above the point.
- Pore water pressure,
  u: pressure within the pore water at that point.
- Effective vertical stress,
  \sigma'v = \sigmav - u: stress carried by the soil skeleton.
Static (hydrostatic) condition recap:
- Consider a vertical column: water head $hw$ on top and a saturated soil column of length $L$ with saturated unit weight $\gammas$.
- Point A at depth $x$ from the soil surface. Under static conditions:
- Total stress: \sigmav = \gammaw hw + \gammas x
- Pore pressure: u = \gammaw(hw + x)
- Effective stress: \sigma'v = \sigmav - u = (\gammas - \gammaw)x
Hydrostatic pore pressure and stress profiles:
- Pore pressure increases linearly with depth, slope controlled by $\gamma_w$.
- Total stress increases with a combination of the water column and the soil column, with a slope that changes at the soil top due to $\gamma_s$.
Seepage basics (1D):
- Seepage occurs when there is a difference in total head across the soil layer: top total head $H1$, bottom total head $H2$, with $\Delta h = H1 - H2$.
- Hydraulic gradient for the soil length $L$ is
  I = \frac{\Delta h}{L}
- Downward seepage: water moves downward (top head higher than bottom head).
- Upward seepage: water moves upward (bottom head higher).
- Under seepage, pore water pressure within the soil becomes a linear function of depth, and the total head in water and soil parts remains consistent with the boundary heads.
Piezometer observation under seepage:
- In static case, total head is the same across piezometers (no flow).
- With seepage, pore pressure distribution changes while total stress can remain unchanged (for a fixed boundary load).
Downward seepage: derivation and results
- Boundary heads produce a head loss across the soil layer: delta $h$.
- For a point at depth $x$ from the top of the soil, the pore pressure under downward seepage is:
  u{down}(x) = \gammaw hw - \gammaw\Delta h\left(\frac{x}{L}\right)
  (linear variation from the top value to a reduced bottom value).
- The total stress remains the same as in the hydrostatic case:
  \sigmav(x) = \gammaw hw + \gammas x
- Therefore the effective stress under downward seepage is:
  \sigma'{v,down}(x) = \sigmav(x) - u{down}(x) = \gammas x + \gamma_w\Delta h\left(\frac{x}{L}\right)
  which shows an increase in effective stress due to seepage-induced reduction in pore pressure.
Upward seepage: derivation and results
- Pore pressure increases with depth due to upward flow:
  u{up}(x) = \gammaw hw + \gammaw\Delta h\left(\frac{x}{L}\right)
- Effective stress under upward seepage:
  \sigma'{v,up}(x) = \sigmav(x) - u{up}(x) = \gammas x - \gamma_w\Delta h\left(\frac{x}{L}\right)
- Result: pore pressures rise relative to hydrostatic, so the effective stress decreases; this can lead to instability if $\sigma'_{v,up}$ approaches zero.
Quick condition / piping concept
- If the effective stress becomes zero or negative, soil particles lose contact and piping (quick condition) can occur.
- Critical hydraulic gradient for quick condition can be expressed as
  I{crit} = \frac{\gammas - \gammaw}{\gammaw}
  (a form given in phase-diagram discussions; relates soil properties and water unit weight).
- If the actual hydraulic gradient $I$ reaches or exceeds $I_{crit}$, piping is possible; design must prevent this.
Worked example (1D seepage, 3-layer setup)
- Given data (example values):
- Water unit weight: \gamma_w = 9.81\ \text{kN/m}^3
- Saturated soil unit weight: \gamma_s = 18\ \text{kN/m}^3
- Top water head: h_w = 0.10\ \text{m}
- Soil layer length: L = 0.40\ \text{m}
- Point of interest from top of soil: x = 0.20\ \text{m}
- Boundary heads for seepage example: $H1 = 0.50\ \text{m}$, $H2 = 0.05\ \text{m}$, so head loss across soil: \Delta h = H1 - H2 = 0.45\ \text{m}
- Static (hydrostatic) case at center (x = 0.20 m):
- Pore water pressure: u{static,c} = \gammaw(h_w + x) = 9.81(0.10 + 0.20) = 2.943\ \text{kPa}
- Total stress: \sigma{v,static,c} = \gammaw hw + \gammas x = 9.81(0.10) + 18(0.20) = 0.981 + 3.60 = 4.581\ \text{kPa}
- Effective stress: \sigma'{v,static,c} = \sigma{v,static,c} - u_{static,c} = 4.581 - 2.943 = 1.638\ \text{kPa}
- Downward seepage case (top head higher than bottom; downward flow):
- Depth ratio for the center: $x/L = 0.20/0.40 = 0.5$
- Pore pressure at center under downward seepage:
  u{down,c} = \gammaw hw - \gammaw \Delta h \left(\frac{x}{L}\right) = 0.981 - 9.81(0.45)(0.5) = 0.981 - 2.207 = -1.226\ \text{kPa}
- Effective stress at center under downward seepage:
  \sigma'{v,down,c} = \sigma{v,static,c} - u_{down,c} = 4.581 - (-1.226) = 5.807\ \text{kPa}
- Interpretation: downward seepage reduces pore pressure in the soil, which increases the soil skeleton load, hence the effective stress increases.
- Upward seepage case (upward flow; bottom boundary higher):
- Pore pressure at center under upward seepage:
  u{up,c} = \gammaw hw + \gammaw \Delta h \left(\frac{x}{L}\right) = 0.981 + 2.207 = 3.188\ \text{kPa}
- Effective stress at center under upward seepage:
  \sigma'{v,up,c} = \sigma{v,static,c} - u_{up,c} = 4.581 - 3.188 = 1.393\ \text{kPa}
- Interpretation: upward seepage raises pore pressure and reduces the effective stress; risk of instability increases as $\sigma'_v$ approaches zero.
- Summary of the center (midpoint) results for the example:
- Static: u{static,c}=2.943\ \text{kPa},\ \sigma{v,static,c}=4.581\ \text{kPa},\ \sigma'_{v,static,c}=1.638\ \text{kPa}
- Downward seepage: u{down,c}=-1.226\ \text{kPa},\ \sigma'{v,down,c}=5.807\ \text{kPa}
- Upward seepage: u{up,c}=3.188\ \text{kPa},\ \sigma'{v,up,c}=1.393\ \text{kPa}
- Observations from the example:
- Total stress at a given point remains the same as in hydrostatic loading when boundary loads are fixed; seepage changes only the pore pressure distribution.
- Downward seepage generally increases the effective stress (potentially causing more settlement tendency but fewer flow-related instabilities), whereas upward seepage reduces the effective stress and can lead to instability or piping if the gradient is high enough.
Practical implications and connections
- 1D seepage provides a foundation for understanding real soils where seepage affects vertical stress distribution and soil behavior.
- In practice, soils are often anisotropic and 2D/3D seepage effects become important; the 1D analysis here introduces key concepts that carry into multi-dimensional problems.
- The coefficient of permeability and laboratory methods to measure seepage properties are essential to estimate $\Delta h$, $I$, and to build equivalent permeabilities for stratified soils.
- Design considerations:
- Ensure the hydraulic gradient remains below the critical gradient to prevent piping.
- Account for seepage-induced changes in effective stress to avoid stability failures (e.g., settlement under downward seepage, or piping under upward seepage).
Final recap of key formulas to memorize
- Static case:
  \sigmav = \gammaw hw + \gammas x
  u = \gammaw(hw + x)
  \sigma'v = \sigmav - u = (\gammas - \gammaw) x
- Seepage (1D, with layer length $L$ and point at depth $x$):
- Downward seepage:
  u{down}(x) = \gammaw hw - \gammaw \Delta h \left(\frac{x}{L}\right)
  \sigma'{v,down}(x) = \sigmav(x) - u{down}(x) = \gammas x + \gamma_w \Delta h \left(\frac{x}{L}\right)
- Upward seepage:
  u{up}(x) = \gammaw hw + \gammaw \Delta h \left(\frac{x}{L}\right)
  \sigma'{v,up}(x) = \sigmav(x) - u{up}(x) = \gammas x - \gamma_w \Delta h \left(\frac{x}{L}\right) \right.
- Critical hydraulic gradient for quick condition (piping):
  i{crit} = \frac{\gammas - \gammaw}{\gammaw}
Connections to the broader module
- These one-dimensional insights form the basis for 2D/3D seepage analyses to be covered in the next topics.
- The discussion includes practical lab methods for permeability and strategies for handling stratified soils with equivalent permeability concepts.
- Ethical and practical note: ensure designs account for seepage to prevent catastrophic failures (e.g., piping) and to maintain soil stability under varying hydraulic conditions.
Example prompts you should be able to solve after this section
- Given $h_w$, $L$, $x$, and soil/gwater unit weights, compute:
- Static pore pressure $u$ at a depth $x$.
- Static total stress $\sigma_v$ at depth $x$.
- Static effective stress $\sigma'_v$ at depth $x$.
- Given seepage with boundary heads, compute for a point at depth $x$ within the soil:
- Downward seepage: $u{down}(x)$ and $\sigma'{v,down}(x)$.
- Upward seepage: $u{up}(x)$ and $\sigma'{v,up}(x)$.
- Determine whether piping is a risk by evaluating $I$ and $I_{crit}$ for the given soil and water conditions.

Seepage Effects on Stresses: 1D Seepage, Downward/Upward Seepage, and Quick Condition (Worked Example)

Overview: seepage effects on stresses in one-dimensional flow

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