1.4 Groundwater Characterization & Hydrogeology

Fundamentals of Hydrogeology & Bernoulli Head

Groundwater dynamics heavily govern soil strength, effective stress, slope stability, and foundation performance. Water movement through porous soil media is driven by gradients in total hydraulic head ($h$), defined by Bernoulli's equation (neglecting fluid velocity head):

$$h = z + h_p = z + \frac{u}{\gamma_w}$$

Where:

• $z$ = Elevation head relative to an arbitrarily chosen horizontal datum ($m$).
• $h_p = u / \gamma_w$ = Pressure head ($m$), where $u$ is pore water pressure and $\gamma_w$ is unit weight of water ($9.81\text{ kN/m}^3$ or $62.4\text{ pcf}$).

Flow occurs in the direction of decreasing total head across a hydraulic gradient ($i$):

$$\ni = \frac{\Delta h}{L}$$

Where $\Delta h$ is total head loss over flow path length $L$.

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Piezometer Types & Pore Pressure Monitoring

Accurate measurement of in situ pore pressure $u$ and piezometric surfaces requires selecting appropriate field instrumentation:

1. Open Standpipe (Casagrande) Piezometers: Consists of a perforated screen surrounded by a sand intake zone in an isolated borehole zone sealed above with bentonite. Measures static water levels in medium-to-high permeability soils. Limitation: High hydrodynamic time lag in tight clays.
2. Vibrating Wire Piezometers (VWP): Contains a metallic diaphragm attached to a tensioned wire. Fluid pressure distorts the diaphragm, altering wire resonant frequency. Features near-zero volume displacement, providing rapid response times in low-permeability clays and silts.
3. Pneumatic Piezometers: Uses gas pressure to flex a flexible rubber diaphragm until it opens a valve port, reading pressure directly. Ideal for construction monitoring.

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Laboratory and Field Permeability Determination

Hydraulic conductivity ($k$) quantifies fluid flow through soil under unit hydraulic gradient (Darcy's Law: $v = k \cdot i$, $Q = k \cdot i \cdot A$).

Laboratory Tests

• Constant-Head Test (ASTM D2434): For coarse-grained soils ($k > 10^{-4}\text{ cm/s}$): $$k = \frac{Q \cdot L}{A \cdot h \cdot t}$$
• Falling-Head Test: For fine-grained soils ($k < 10^{-4}\text{ cm/s}$): $$k = \frac{a \cdot L}{A \cdot t} \ln \left( \frac{h_1}{h_2} \right)$$ Where $a$ is cross-sectional area of standpipe, $A$ is soil specimen area, and $h_1, h_2$ are initial and final head levels over time interval $t$.

In Situ Field Pumping Tests (Thiem Equations)

Field pumping tests eliminate boundary artifacts by pumping a central well at steady discharge rate $q$ while recording drawdowns in observation wells at radial distances $r_1$ and $r_2$.

1. Confined Aquifer (Thickness $b$): $$k = \frac{q \ln(r_2 / r_1)}{2 \pi b (h_2 - h_1)}$$
2. Unconfined Aquifer (Resting on Impervious Strata): $$k = \frac{q \ln(r_2 / r_1)}{\pi (h_2^2 - h_1^2)}$$

Where $h_1, h_2$ are piezometric water column heights in observation wells located at radii $r_1, r_2$.

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Two-Dimensional Seepage & Flow Nets

Steady isotropic two-dimensional groundwater flow is governed by Laplace's differential equation:

$$\frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial z^2} = 0$$

Flow nets provide a graphical solution consisting of two orthogonal families of curves:

• Flow Lines: Path lines along which water particles travel.
• Equipotential Lines: Lines connecting points of equal total hydraulic head $h$.

Governing Flow Net Equations

For a flow net constructed with square elements (where element length equals width):

$$q = k \cdot \Delta H \cdot \frac{N_f}{N_d}$$

Where:

• $q$ = Total seepage rate per unit width of structure ($m^3/s/m$).
• $k$ = Soil hydraulic conductivity ($m/s$).
• $\Delta H$ = Total hydraulic head difference across the flow net ($m$).
• $N_f$ = Number of flow channels.
• $N_d$ = Number of equipotential drops.

Computing Pore Pressure at Flow Net Grid Points

The total head drop per equipotential stage is $\Delta h = \Delta H / N_d$. At any arbitrary point $X$ located $n_d$ drops downstream from the upstream boundary:

$$h_X = h_{upstream} - n_d \cdot \left( \frac{\Delta H}{N_d} \right)$$ $$\nu_X = (h_X - z_X) \cdot \gamma_w$$

Where $z_X$ is the elevation head of point $X$ relative to the designated datum.

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Seepage Forces, Quick Conditions, and Piping Stability

Water flowing through soil exerts a dynamic drag force termed the seepage force per unit volume ($j$):

$$j = i \cdot \gamma_w$$

Critical Hydraulic Gradient ($i_{cr}$)

When upward seepage flow occurs (such as downstream of sheet piles or dam toes), the upward seepage force opposes effective overburden stress. When upward seepage force equals buoyant soil weight, effective stress vanishes ($\sigma' = 0$), initiating a quicksand or boiling state:

$$\ni_{cr} = \frac{\gamma'}{\gamma_w} = \frac{G_s - 1}{1 + e}$$

Where $\gamma' = \gamma_{sat} - \gamma_w$ is submerged unit weight, $G_s$ is specific gravity of solids, and $e$ is void ratio. For typical soils ($G_s \approx 2.65, e \approx 0.65$), $i_{cr} \approx 1.0$.

Factor of Safety Against Piping Heave ($FS_{piping}$)

To prevent catastrophic piping erosion at the exit zone downstream of a structure:

$$FS_{piping} = \frac{i_{cr}}{i_{exit}} \ge 3.0$$

Where $i_{exit} = \frac{\Delta h_{last}}{L_{last}}$ is the exit hydraulic gradient across the last flow net square adjacent to the ground surface.

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Detailed Worked Numerical Examples

Problem Statement

A sheet pile wall penetrates $6.0\text{ m}$ into a homogeneous sand layer ($k = 2.5 \times 10^{-4}\text{ m/s}$, $G_s = 2.68$, $e = 0.70$, $\gamma_{sat} = 19.5\text{ kN/m}^3$). The water level is $5.0\text{ m}$ above the ground surface on the upstream side and flush with the ground surface on the downstream side.

A sketched flow net yields $N_f = 4$ flow channels and $N_d = 10$ equipotential drops. The dimension of the downstream exit flow element is $L_{last} = 1.2\text{ m}$.

Calculate:

1. Total hydraulic head difference $\Delta H$.
2. Seepage discharge per meter length of sheet pile wall ($q$).
3. Critical hydraulic gradient $i_{cr}$.
4. Exit hydraulic gradient $i_{exit}$ and Factor of Safety against piping ($FS_{piping}$).

Solution

Step 1: Calculate Total Head Difference $\Delta H$ Setting datum at the downstream ground surface:

• Upstream total head $h_{up} = 5.0\text{ m}$.
• Downstream total head $h_{down} = 0.0\text{ m}$.
• Head difference $\Delta H = 5.0 - 0.0 = 5.0\text{ m}$.

Step 2: Calculate Seepage Rate $q$

$$q = k \cdot \Delta H \cdot \frac{N_f}{N_d} = (2.5 \times 10^{-4}\text{ m/s}) \times 5.0\text{ m} \times \frac{4}{10}$$ $$q = 1.25 \times 10^{-3} \times 0.4 = 5.0 \times 10^{-4}\text{ m}^3/\text{s/m} = 0.50\text{ L/s/m}$$

Step 3: Calculate Critical Hydraulic Gradient $i_{cr}$ Using index parameters $G_s = 2.68$ and $e = 0.70$:

$$\ni_{cr} = \frac{G_s - 1}{1 + e} = \frac{2.68 - 1}{1 + 0.70} = \frac{1.68}{1.70} = 0.988$$

Alternatively using unit weight: $\gamma' = 19.5 - 9.81 = 9.69\text{ kN/m}^3$. $i_{cr} = 9.69 / 9.81 = 0.988$.

Step 4: Calculate $i_{exit}$ and $FS_{piping}$ Head drop per square $\Delta h = \frac{\Delta H}{N_d} = \frac{5.0\text{ m}}{10} = 0.50\text{ m}$.

$$\ni_{exit} = \frac{\Delta h}{L_{last}} = \frac{0.50\text{ m}}{1.20\text{ m}} = 0.417$$ $$FS_{piping} = \frac{i_{cr}}{i_{exit}} = \frac{0.988}{0.417} = 2.37$$

Since $FS_{piping} = 2.37 < 3.0$, an engineered downstream filter berm or deeper sheet pile penetration is required to satisfy safety standards.