CELE Geotechnical Engineering — Permeability and SeepageRevision Notes

Formulas

Exam Tips

• Board exams almost always give head loss h and length L separately — compute i = h/L first as a distinct step.
• If given void ratio e instead of porosity n, convert: n = e/(1+e) before computing seepage velocity.
• Watch for problems asking for Q in litres/second: 1 m³/s = 1000 L/s.
• Memorise the order-of-magnitude range of k for gravel, sand, silt, and clay — some questions ask which test is appropriate for a given soil type.

Key Points

• Darcy's Law applies only to laminar (viscous) flow — the dominant regime in fine-grained soils and sands.
• The hydraulic gradient i = h/L is dimensionless; h is the head loss (m) and L is the flow path length (m).
• Discharge velocity v = ki acts over the gross cross-sectional area A, NOT just the pore area.
• Seepage velocity vs = v/n is the actual velocity of water through the interconnected pore space; it is always greater than v because n < 1.
• Reynolds number Re < 1 (some references use Re < 10) confirms laminar flow and validity of Darcy's Law.
• Units of k: m/s (SI), also expressed as cm/s or mm/s in lab reports — always verify unit consistency.

Definitions

Section Title

Darcy's Law — Fundamental Flow Equation

Common Mistakes

• Using seepage velocity vs instead of discharge velocity v in the Q = kiA formula — always use v = ki for discharge.
• Forgetting that i = h/L uses total head loss h (not pressure head) and the actual seepage path length L.
• Mixing units: if L is in mm and A is in mm², then Q comes out in mm³/s — convert consistently to SI (m).
• Applying Darcy's Law to turbulent flow in coarse gravels — the law is only valid for laminar conditions (Re < 1 to 10).

Formulas

Exam Tips

• Board exam problems typically state soil type or flow rate implicitly — if soil is sand/gravel, use constant-head; if silt/clay or very small flow, use falling-head.
• In the falling-head formula, remember: a is standpipe area, A is sample area — both are given; do not confuse them.
• For the constant-head formula, Q = V/t is the flow rate, so k = QL/(Ah) is an equivalent form — use whichever is more convenient with the given data.
• Check your answer against typical k ranges: sand 10⁻⁵ to 10⁻³ m/s, clay < 10⁻⁸ m/s — unreasonable answers signal a unit or formula error.

Key Points

• Constant-head test is used for coarse-grained soils (gravel, sand) because flow rates are high enough to measure easily.
• Falling-head test is used for fine-grained soils (silt, clay) where flow rates are very low.
• In the constant-head test, head h remains constant throughout; volume V is collected over time t.
• In the falling-head test, the head in the standpipe drops from h₁ to h₂ over time t; the log ratio h₁/h₂ accounts for the changing gradient.
• Both tests must be performed on saturated samples; air bubbles invalidate results.
• Temperature affects viscosity and thus k; test results should be corrected to a standard temperature (usually 20°C) using a viscosity-ratio correction.

Definitions

Section Title

Laboratory Permeability Tests

Common Mistakes

• Swapping the constant-head and falling-head formulas — constant-head has NO log term; falling-head DOES.
• Forgetting the 2.303 factor: the derivation of the falling-head formula uses natural log; multiplying by 2.303 converts to log base 10 for easier hand calculation.
• Using the sample area A instead of the standpipe area a (or vice versa) in the falling-head formula.
• Not maintaining consistent units throughout — mix-up of mm and m is the most common arithmetic error in board problems.
• Confusing 'h' in the constant-head formula (constant head across sample) with h₁ or h₂ in the falling-head formula (standpipe head levels).

Formulas

Exam Tips

• Memory aid: PARALLEL = Arithmetic average (high k dominates); PERPENDICULAR = Harmonic-type (low k dominates). Think: a chain is only as strong as its weakest link for perpendicular flow.
• Board exams frequently give two or three layers and ask for both keq values in one problem — compute both before checking the answer choices.
• In Philippine deltaic and alluvial plains, horizontal seepage for dam-underseepage uses parallel keq; vertical drainage for consolidation uses perpendicular keq.
• A quick sanity check: keq(perpendicular) ≤ smallest kᵢ ≤ largest kᵢ ≤ keq(parallel).

Key Points

• Real soil profiles consist of multiple horizontal (or inclined) layers, each with a different k; an equivalent single-layer k must be computed for design.
• For flow PARALLEL to layers (horizontal flow in a horizontally-layered profile): the layers act like electrical resistors in PARALLEL — high-k layers dominate, and keq is the thickness-weighted arithmetic average.
• For flow PERPENDICULAR to layers (vertical flow downward through horizontal layers): the layers act like resistors in SERIES — the low-k layer dominates, and keq is the harmonic mean.
• In most natural deposits, the horizontal permeability kH (parallel) is 2 to 10 times greater than the vertical permeability kV (perpendicular).
• Understanding which direction flow travels relative to layering is the key to choosing the correct formula.

Definitions

Section Title

Layered Soils — Equivalent Permeability

Common Mistakes

• Swapping the parallel and perpendicular formulas — parallel uses Σ(kH) in the numerator; perpendicular uses Σ(H/k) in the denominator.
• Not recognizing that the low-k layer almost entirely controls perpendicular keq — a 1 mm clay layer can reduce vertical k by orders of magnitude.
• Computing the total thickness incorrectly — always verify ΣHᵢ before dividing.
• Using different units for k in the same calculation — convert all kᵢ to the same unit (m/s) before applying the formula.

Formulas

Exam Tips

• Board exam flow-net problems always give you Nf and Nd — you do not need to draw the flow net. Just plug into Q = kH(Nf/Nd).
• Uplift pressure problems require computing head at a specific point: count the number of potential drops from the upstream side to that point, multiply by Δh = H/Nd.
• Remember: Nf and Nd are SPACES, not LINES. This distinction eliminates a common off-by-one error.
• Total seepage for a dam of length B: Qtotal = Q × B (m³/s).

Key Points

• A flow net is a graphical solution to Laplace's equation for two-dimensional steady-state seepage; it consists of flow lines (streamlines) and equipotential lines that are mutually perpendicular.
• Flow lines show the direction of seepage; no water crosses a flow line.
• Equipotential lines connect points of equal total head; water flows perpendicular to them (from high to low head).
• Nf = number of flow channels (the spaces between adjacent flow lines); Nd = number of equipotential drops (the spaces between adjacent equipotential lines).
• Each flow channel carries the same discharge; each equipotential interval dissipates the same head = H/Nd.
• The seepage quantity formula Q = kH(Nf/Nd) per unit width is derived from Darcy's Law applied to one 'square' of the flow net.
• Typical flow-net shapes: under concrete dams (with and without cut-off walls), around sheet piles, through earth embankments.
• For anisotropic soils (kH ≠ kV), use a transformed section and an equivalent k = √(kH · kV).

Definitions

Section Title

Seepage and Flow Nets

Common Mistakes

• Counting flow LINES instead of flow CHANNELS (Nf) — if there are 5 flow lines including the two boundary flow lines, there are only 4 flow channels.
• Similarly, counting equipotential LINES instead of equipotential DROPS (Nd) — subtract 1 from the count of lines.
• Using Nf/Nd > 1 for seepage under a long dam with many drops — always verify the ratio makes physical sense (Nf is typically 3–6, Nd is typically 10–16 for dam problems).
• Forgetting to apply the per-unit-width qualifier — Q from the flow-net formula is per metre of dam; multiply by dam length for total seepage.
• Using kH alone for anisotropic soils instead of keq = √(kH·kV).

Formulas

Exam Tips

• Memorise the derivation pathway: σ'= 0 when upward seepage force equals submerged weight → icr = γ'/γw = (Gs−1)/(1+e). Understanding the derivation prevents formula-recall errors.
• If a problem says 'determine if quicksand occurs,' compute icr and compare with the given i; if i > icr, quicksand occurs.
• For FS problems, the exit gradient iexit is often determined from the flow net: iexit ≈ Δh/Δl where Δl is the size of the last flow net square at the exit.
• Typical board answer choices for icr cluster around 0.90–1.10; if your answer is wildly different, recheck Gs and e substitution.
• FS ≥ 3 is the commonly cited minimum for piping safety in Philippine dam engineering practice.

Key Points

• Upward seepage exerts a drag force on soil grains (seepage force = γw·i per unit volume).
• When the upward seepage force equals the buoyant weight of the soil skeleton, effective stress becomes zero and the soil loses all shear strength — this is the QUICK (BOILING) condition.
• The critical gradient icr is derived by setting effective stress σ' = 0: icr = γ'/γw = (Gs − 1)/(1 + e).
• For most soils, Gs ≈ 2.67 and e ≈ 0.65–0.80, giving icr ≈ 0.95 to 1.05 (approximately 1.0).
• The exit gradient iexit is the hydraulic gradient at the downstream toe of a dam or sheet pile — the most dangerous location for piping initiation.
• Factor of safety against piping: FS = icr / iexit. Philippine practice (and most codes) requires FS ≥ 3 to 4 for dams.
• Piping failures have caused numerous dam disasters worldwide — understanding this topic has direct life-safety implications in Philippine infrastructure.

Definitions

Section Title

Quick Condition (Boiling/Quicksand) and Piping Safety

Common Mistakes

• Using Gs = 2.65 (standard value) when the problem specifically gives a different Gs — always use the given value.
• Forgetting to compute γ' = (Gs − 1)γw/(1+e) separately and instead using total unit weight in the icr formula.
• Using FS = iexit/icr (inverted) — the factor of safety is always capacity over demand, i.e., icr/iexit.
• Assuming icr is always exactly 1.0 — this is only an approximation; board problems that give Gs and e expect you to compute the exact value.
• Confusing the SEEPAGE FORCE per unit volume (j = γw·i) with the hydraulic gradient i itself.