Stresses on soil and seepage

Soil carries load through grain-to-grain contacts, while pore water carries part of the stress without adding to shear strength. Terzaghi's effective stress principle — the most important idea in soil mechanics — explains settlement, strength, quicksand and seepage failure. This note also covers capillarity, seepage pressure and flow nets (used in design of dams, weirs and cofferdams common in Nepal's hydropower and irrigation projects), compressibility indices and compaction.

Key concepts

Effective stress

$\sigma' = \sigma - u$ $\sigma$ = total stress, $u$ = pore water pressure, $\sigma'$ = effective (intergranular) stress, all in kPa.

• Only effective stress controls strength and volume change.
• Factors affecting it: water table position, surcharge, capillary rise (negative pore pressure increases $\sigma'$), seepage direction (downward flow increases $\sigma'$, upward decreases it).
• Below water table at depth $z$ under water table at surface: $\sigma' = \gamma' z$.

Capillary rise

• Water rises above the water table in fine pores due to surface tension; pore pressure in capillary zone is negative ($u = -h\gamma_w$).
• Height: $h_c = \dfrac{4T\cos\alpha}{\gamma_w d}$; for water at 20 °C, $h_c \approx \dfrac{0.3}{d}$ (both $h_c$ and $d$ in cm; e.g. $d$ = 0.01 cm gives about 30 cm). Empirically $h_c = \dfrac{C}{eD_{10}}$.
• Larger in clays (metres) than in sands (cm).

Quicksand condition

• When upward seepage makes effective stress zero, the soil loses strength and behaves like a liquid. This is a condition, not a soil type — occurs in fine sands and silts, not in clays (cohesion).
• Critical hydraulic gradient: $i_c = \frac{\gamma'}{\gamma_w} = \frac{G-1}{1+e}\approx 1$
• Factor of safety against piping/boiling = $i_c/i_{exit}$; weirs on permeable soil are designed for a safe exit gradient with a large margin (Khosla's theory).

Seepage and seepage pressure

• Darcy's law: $v = ki$, $q = kiA$ ($k$ = permeability, m/s; $i$ = hydraulic gradient).
• Seepage pressure $p_s = h\gamma_w$ ; seepage force per unit volume $j = i\gamma_w$, acting in the direction of flow.
• Laplace's equation $\dfrac{\partial^2h}{\partial x^2} + \dfrac{\partial^2h}{\partial z^2} = 0$ governs 2-D steady flow in isotropic soil.

Flow nets

• Network of flow lines (paths of water) and equipotential lines (equal total head) intersecting at right angles forming curvilinear squares.
• Seepage discharge per unit width: $q = kH\frac{N_f}{N_d}$ $N_f$ = number of flow channels, $N_d$ = number of equipotential drops, $H$ = total head loss (m).
• Anisotropic soil: use $k = \sqrt{k_xk_z}$ and transform scale ($x\sqrt{k_z/k_x}$).
• Applications: seepage quantity, uplift pressure under dams/weirs, exit gradient and piping check, pore pressure in earth dams (phreatic line by Kozeny's parabola).

Soil compressibility

• Compression index $C_c$ = slope of $e$–$\log\sigma'$ curve on virgin compression; Skempton: $C_c = 0.009(w_L - 10)$ (undisturbed), $0.007(w_L - 10)$ (remoulded).
• Recompression/swelling index $C_r$ or $C_s$ ≈ (1/5 – 1/10)$C_c$.
• Coefficient of compressibility $a_v = -\Delta e/\Delta\sigma'$; coefficient of volume compressibility $m_v = \dfrac{a_v}{1+e_0}$.
• Coefficient of consolidation $c_v = \dfrac{k}{m_v\gamma_w}$.
• Overconsolidation ratio OCR = $\sigma'_p/\sigma'_0$ (NC: OCR = 1).

Compaction

• Definition: densification of soil by expelling air using mechanical energy (unlike consolidation, which expels water slowly under static load).
• Proctor test: standard (2.6 kg rammer, 310 mm drop, 3 layers, 25 blows, ~600 kN·m/m³) and modified (4.89 kg, 450 mm, 5 layers, 25 blows, ~2700 kN·m/m³) → OMC and MDD.
• Factors affecting: water content, compactive effort (more effort → higher MDD, lower OMC), soil type (well-graded coarse soils reach high MDD, clays lower), method of compaction, and admixtures.
• Dry side: flocculated structure, higher strength, more swelling; wet side: dispersed structure, lower permeability, more shrinkage.
• Field equipment: smooth-wheel rollers (granular base), sheep-foot rollers (clays), pneumatic-tyred, vibratory rollers (sands); field control by relative compaction (e.g. ≥ 95% of MDD).
• Zero-air-voids line: $\gamma_d = \dfrac{G\gamma_w}{1+wG}$.

High-yield points

• Only effective stress controls strength and volume change.
• $i_c = (G-1)/(1+e) \approx 1$; quicksand is a condition (fine sands/silts), not a soil type.
• Upward seepage reduces $\sigma'$ by $i\gamma_wz$; downward seepage increases it.
• Skempton: $C_c = 0.009(w_L - 10)$ for undisturbed clay.
• Standard Proctor: 2.6 kg rammer, 310 mm drop, 3 layers × 25 blows; modified: 4.89 kg, 450 mm, 5 layers.
• More compactive effort → higher MDD, lower OMC.

Formulas

• $\sigma' = \sigma - u$; capillary rise $h_c = 4T\cos\alpha/(\gamma_wd)$.
• $i_c = (G-1)/(1+e)$; Darcy $q = kiA$; flow net $q = kHN_f/N_d$.
• $m_v = a_v/(1+e_0)$; $c_v = k/(m_v\gamma_w)$; zero-air-voids $\gamma_d = G\gamma_w/(1+wG)$.

Worked example

A 4 m sand layer ($\gamma_{sat} = 20$ kN/m³) lies with water table at the surface. Find effective stress at 4 m. Then, if upward seepage occurs with head loss of 3 m across the layer, find the new $\sigma'$ and the factor of safety against quicksand. ($\gamma_w = 10$ kN/m³)

• Static: $\sigma' = (20 - 10)\times4 = 40$ kPa
• $i = 3/4 = 0.75$; seepage force reduces $\sigma'$ by $i\gamma_w z = 0.75\times10\times4 = 30$ kPa → $\sigma' = 10$ kPa
• $i_c = \gamma'/\gamma_w = 10/10 = 1.0$; FS = 1.0/0.75 = 1.33

Exam tips

• $\sigma' = \sigma - u$; upward flow reduces $\sigma'$, downward flow increases it.
• Critical gradient $i_c \approx 1$ for most sands; quicksand is a hydraulic condition, not a soil type.
• Flow lines and equipotentials meet at 90°; $q = kHN_f/N_d$.
• Compaction removes air (quick); consolidation removes water (slow).
• Higher compactive effort → higher MDD and lower OMC.

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. A flow net beneath a dam has 4 flow channels and 12 equipotential drops. The head difference between upstream and downstream is 6 m and the coefficient of permeability is $1 \times 10^{-5}$ m/s. The seepage discharge per metre length of the dam is

• A. $5 \times 10^{-6}$ m³/s
• B. $6 \times 10^{-5}$ m³/s
• C. $2 \times 10^{-5}$ m³/s
• D. $1.8 \times 10^{-4}$ m³/s

Answer: C. $q = kH\dfrac{N_f}{N_d} = 1\times10^{-5} \times 6 \times \dfrac{4}{12} = 2 \times 10^{-5}$ m³/s per metre.

2. A 3 m thick sand layer of submerged unit weight 10 kN/m³ carries upward seepage under a hydraulic gradient of 0.4. Taking $\gamma_w = 10$ kN/m³, the effective stress at the bottom of the layer is

• A. 42 kPa
• B. 30 kPa
• C. 18 kPa
• D. 12 kPa

Answer: C. With upward flow, $\sigma' = z\gamma' - iz\gamma_w = 3 \times 10 - 0.4 \times 3 \times 10 = 30 - 12 = 18$ kPa.

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