Optional Questions: 18-Geol-A6 Soil Mechanics, May 2015, Q6

Question 6: Optional Questions (5 marks each, 3 of 5 required — all 5 answered here)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Q6(1) — Darcy's Law

Darcy flow through a soil element of length $L$ and cross-sectional area $A$, driven by the head difference $h_1-h_2$.

Darcy's law states that the seepage discharge through a saturated soil is directly proportional to the hydraulic gradient driving it:

$$v=ki,\qquad q=vA=kiA$$

Here $v$ is the discharge velocity (a superficial, cross-section-averaged velocity, not the true pore velocity, since flow only occurs through the void space), $k$ is the coefficient of permeability (hydraulic conductivity) of the soil, and $i=\Delta h/L$ is the hydraulic gradient — the head loss $\Delta h=h_1-h_2$ between two points divided by the flow-path length $L$ between them, as shown in the figure. $A$ is the total (gross) cross-sectional area of soil perpendicular to flow, and $q$ is the total volumetric discharge. Darcy's law is empirical but is now understood as the low-Reynolds-number (laminar) limit of flow through a porous medium, valid for essentially all groundwater and seepage problems in soils, and it underlies every seepage, consolidation and flow-net calculation in this paper.

Q6(2) — Effective Stress

Conceptual model: total stress $\sigma$ carried partly by pore water pressure $u$ (acting equally in all directions) and partly by inter-granular contact forces.

Consider a horizontal plane cutting through a saturated soil mass at some points of grain-to-grain contact and elsewhere through the pore water. The total normal stress $\sigma$ on this plane is the total weight of everything above it (soil solids + water) divided by the gross area $A$. This stress is carried by two mechanisms in parallel: the pore water pressure $u$, acting equally in all directions on the (much larger) area of water at the plane, and the sum of the inter-granular contact forces $N'$, acting over the (much smaller) actual solid-to-solid contact area $A_c$. Summing vertical force equilibrium over the gross area $A$:

$$\sigma A = \sum N' + u(A-A_c)$$

Dividing through by $A$ and defining the effective stress as the average inter-granular force per unit gross area, $\sigma'=\sum N'/A$:

$$\sigma = \sigma' + u\left(1-\frac{A_c}{A}\right)$$

Since the actual contact area $A_c$ is experimentally found to be a negligibly small fraction of the gross area $A$ for real soils (contacts are near-point contacts), $A_c/A\approx0$, giving Terzaghi's effective stress principle:

$$\boxed{\sigma' = \sigma - u}$$

This is the single most important relation in soil mechanics: strength and volume-change behaviour of soil depend only on $\sigma'$, the stress actually carried grain-to-grain, not on the total stress $\sigma$ alone — which is why raising the pore pressure $u$ (holding $\sigma$ fixed) can trigger shear failure or excess settlement even with no change in the external load.

Q6(3) — Capillary Rise and Water-Retention Curves

Left: capillary rise $h_c$ in a fine tube of radius $r$ above a free-water surface. Right: a typical soil–water characteristic curve (SWCC) relating suction to saturation.

When a fine glass tube is inserted into a free water surface, surface tension $T_s$ acting around the tube's wetted perimeter pulls water up the tube against gravity, forming a curved meniscus. Vertical equilibrium of the suspended water column (weight balanced by the vertical component of the surface-tension force around the circumference) gives

$$h_c=\frac{2T_s\cos\alpha}{r\gamma_w}$$

where $\alpha$ is the contact angle (near $0^{\circ}$ for a clean glass–water interface) and $r$ is the tube radius: the narrower the tube, the higher the rise. In an unsaturated soil, the pore network behaves like a bundle of capillary tubes of varying effective radius (controlled by the pore/grain size), so the same physics produces matric suction — a negative pore pressure that holds water in the soil above the free water table. Because a real soil has a continuous distribution of pore sizes rather than one uniform radius, the relationship between suction and the resulting degree of saturation is not a single rise height but a continuous curve: the soil–water characteristic curve (SWCC). At low suction, nearly all pores (large and small) remain saturated; as suction increases, progressively smaller pores are able to resist drainage, so the largest pores desaturate first and saturation falls gradually, giving the SWCC's characteristic S-shape (steep desaturation once the air-entry suction of the coarsest pores is exceeded, flattening again as only the finest pores remain water-filled).

Q6(4) — Sand-Cone Field Compaction Test

Given. Excavated hole volume $V=1165\text{ cm}^3$; wet soil mass $=2230$ g; dry soil mass $=1852$ g.

Find. (a) Field dry density. (b) Field water content.

(a) Field dry density. $$\rho_{d,field}=\frac{M_{dry}}{V}=\frac{1852\text{ g}}{1165\text{ cm}^3}=\boxed{1.590\text{ g/cm}^3}\ \left(=1590\text{ kg/m}^3,\ \gamma_{d,field}=15.6\text{ kN/m}^3\right)$$
(b) Field water content. $$w_{field}=\frac{M_{wet}-M_{dry}}{M_{dry}}=\frac{2230-1852}{1852}=\boxed{20.4\%}$$

Quantity	Value
Field dry density	1.590 g/cm³ (15.6 kN/m³)
Field water content	20.4%

Q6(5) — Groundwater Table and Total Head Profile

Elevation, pressure and total head vs.\ depth through a 10 m sand layer with the water table 2 m below the surface (datum $z=0$ at the base of the layer).

The groundwater table (water table, phreatic surface) is the level in a soil deposit at which pore water pressure equals atmospheric pressure ($u=0$); below it the soil is saturated and pore pressure increases hydrostatically with depth, while above it (in the absence of significant capillary rise) the soil is unsaturated and pore pressure is taken as zero. For a 10 m thick sand layer with the water table 2 m below the ground surface, taking the elevation datum $z=0$ at the base of the layer: the ground surface is at $z=10$ m and the water table at $z=8$ m.

Below the water table, in a static (no-flow) groundwater condition, total head is constant with depth and equal to the elevation of the water table itself, $h_{total}=z_{WT}=8$ m; elevation head $z$ decreases linearly with depth while pressure head $\psi=h_{total}-z$ grows correspondingly, reaching $\psi=8$ m at the base of the layer ($z=0$). Above the water table, pressure head is taken as zero (ignoring capillary rise for this plot), so total head simply equals elevation head and varies from $8$ m (at the water table) up to $10$ m (at the ground surface) — this region is not part of the same hydrostatically-connected system, which is exactly why total head is not continuous across the water table in this simplified picture.

Concept & Theory

Topic. Darcy's law, Terzaghi's effective stress principle, capillarity/SWCC, field compaction control, and the head components of static groundwater — the five foundational relations of seepage and phase behaviour in soil mechanics.

Key relations. $q=kiA$; $\sigma'=\sigma-u$; $h_c=2T_s\cos\alpha/(r\gamma_w)$; $\rho_d=M_{dry}/V$, $w=(M_{wet}-M_{dry})/M_{dry}$; $h_{total}=z+\psi$.

Why this works. All five parts describe how water moves through, and interacts mechanically with, the soil skeleton: Darcy's law governs the rate of movement; effective stress governs how pore pressure changes the mechanical behaviour of the skeleton; capillarity explains how water is retained above a free water table without external pressure; the sand-cone test is the standard field check that a compacted fill has actually achieved its target dry density; and the total-head decomposition is what makes flow-net seepage analysis (Q5) possible in the first place.

Common pitfall. In Q6(4), dividing by the wet mass instead of the dry mass when computing water content (the definition is always mass of water over mass of solids, i.e.\ dry mass, not over total wet mass).

Source. Das, Principles of Geotechnical Engineering, Ch. 3 (phase relations, compaction) and Ch. 8 (seepage); Craig, Craig's Soil Mechanics, Ch. 2 (effective stress, capillarity).