Groundwater, Soils, and Sediments

A. Basic hydrogeology

Hydrogeology and Aquifer Properties

Confined, unconfined, and perched aquifers, aquitards, the porosity/specific-yield/specific-retention split, the water table vs the potentiometric surface, and recharge and discharge.

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B. Groundwater flow

Darcy’s Law and Seepage Velocity

Darcy’s law, hydraulic conductivity, transmissivity and storativity, and the crucial gap between the Darcy flux and the seepage velocity a contaminant actually travels at.

Almost every groundwater question on the FE Environmental exam stands on a single 1856 experiment: water moving through saturated soil discharges in direct proportion to the hydraulic gradient driving it. Get Darcy's law right and you can compute well yields, plume arrival times, capture zones, and seepage under dams; get the velocity definitions backwards and you will be off by a factor of three to five on every travel-time problem. The single most common — and most expensive — error in this whole topic is reporting the Darcy flux

q

when the question asks how fast the water (or a contaminant) actually moves. This concept builds the fluency to keep the discharge, the specific flux, and the seepage velocity cleanly separated. The governing relations are in the FE Reference Handbook — Civil Engineering under 'Darcy's Law'.

Darcy’s law and hydraulic conductivity

Darcy found that volumetric discharge

Q

through a porous medium is proportional to the cross-sectional area

A

and to the hydraulic gradient

-\,dh/dx

, with the proportionality constant being the hydraulic conductivity

K

. The minus sign encodes the physics: groundwater flows from high head to low head, so positive discharge accompanies a falling head.

K

(units of velocity, ft/s or m/s) bundles together both the medium — grain size, sorting, pore connectivity — and the fluid's density and viscosity. Clean gravel may carry

K \approx 10^{2}\,\text{m/d}

while a clay aquitard sits near

10^{-4}\,\text{m/d}

, a span of six orders of magnitude that explains why one layer is an aquifer and the one beneath it confines.

Q = -KA\,\frac{dh}{dx}

Specific discharge — the Darcy flux

Divide discharge by the full cross-sectional area and you get the specific discharge

q

, also called the Darcy velocity or superficial velocity. It has units of velocity but it is not a real velocity — it is the discharge that would occur if water moved through the entire face of the medium, solids included.

q

is exactly what you need for flux and mass-loading calculations (volume of water per unit total area per time), and it is what Darcy's law delivers directly. Treating

q

as the speed of a water particle is the canonical mistake, because water cannot flow through the grains, only around them.

q = \frac{Q}{A} = -K\,\frac{dh}{dx}

Seepage velocity — how fast water really moves

Water threads only through the connected pore space, so its actual average velocity is larger than

q

. Dividing the Darcy flux by the effective porosity

n_e

— the fraction of the bulk volume made of interconnected, flow-carrying pores — gives the average seepage velocity (also called the linear or pore velocity). Because

n_e

is typically

0.1

to

0.35

, the seepage velocity is roughly three to ten times the Darcy flux. This is the velocity that governs how fast a conservative tracer or the leading edge of a plume travels, and therefore the velocity behind every contaminant travel-time problem.

v = \frac{q}{n_e} = \frac{K}{n_e}\,\frac{dh}{dx}

Effective vs total porosity

Total porosity

n

counts all void space; effective porosity

n_e

counts only the interconnected voids that actually conduct flow. Dead-end pores and water bound to grain surfaces hold water but do not transmit it, so

n_e \le n

always — and the gap can be large in fine-grained or poorly connected media. Clay can have a total porosity near

0.5

yet an effective porosity an order of magnitude smaller. For seepage velocity you must use

n_e

; mixing in the total porosity inflates the denominator and underestimates how fast water moves.

Transmissivity and storativity

When you treat a whole aquifer rather than a point, two integrated properties take over. Transmissivity

T = Kb

is the conductivity times the saturated thickness

b

(units

L^2/T

, e.g. m

^2

/d); it measures how much water the full thickness of the aquifer can transmit and is the natural parameter in well equations. Storativity (storage coefficient)

S

is the volume of water released from or taken into storage per unit aquifer area per unit drop in head — dimensionless. In a confined aquifer the water comes from elastic expansion of water and compression of the skeleton, so

S

is tiny, roughly

10^{-5}

to

10^{-3}

. In an unconfined aquifer storage is mostly gravity drainage of pores, so

S

approaches the specific yield, typically

0.05

to

0.30

— hundreds of times larger.

T = K\,b \qquad S = S_s\,b\ \text{(confined)},\quad S \approx S_y\ \text{(unconfined)}

Gradient and flow direction

The hydraulic gradient is the change in head per unit distance,

i = -\,dh/dx

, and flow follows the steepest descent of head, perpendicular to the equipotential (equal-head) contours. With three monitoring wells you can triangulate both the magnitude and the compass direction of the gradient: contour the heads, then drop a perpendicular from high to low. Field problems often give you head differences across a known distance — keep the gradient dimensionless by carrying head and distance in the same length units, and remember the gradient points downhill in head, which is the direction the plume will go.

Exam strategy

Read the verb: 'flux', 'specific discharge', or 'Darcy velocity' means

q = Ki

; 'how fast', 'travel time', 'pore' or 'seepage velocity' means

v = q/n_e

. Write down which one is asked before you compute. Build the gradient as a clean dimensionless ratio (equal length units top and bottom), and watch

K

's time unit — handbook

K

is often ft/s while the problem wants ft/day, a factor of

86{,}400

. For travel time across a distance

L

, use

t = L/v = L\,n_e/(Ki)

, never

L/q

. For aquifer-scale discharge through a strip of width

w

, combine with transmissivity as

Q = T\,i\,w

, which already folds in the thickness.

Key equations

Darcy’s law (discharge)

Q = -KA\,\dfrac{dh}{dx}

$Q$ = volumetric discharge (m³/s, ft³/s); $K$ = hydraulic conductivity (m/s, ft/s); $A$ = total cross-sectional area; $dh/dx$ = hydraulic gradient. Minus sign: flow goes down-gradient.

Specific discharge (Darcy flux)

q = \dfrac{Q}{A} = -K\,\dfrac{dh}{dx} = K\,i

Discharge per unit total area; units of velocity but NOT a particle speed. Use for flux/mass-loading. $i = -dh/dx$ is the gradient (dimensionless).

Seepage (pore) velocity

v = \dfrac{q}{n_e} = \dfrac{K\,i}{n_e}

Actual average water velocity through the interconnected pores; $n_e$ = effective porosity. This is the velocity for travel-time and plume problems.

Hydraulic gradient

i = -\dfrac{dh}{dx} = \dfrac{h_1 - h_2}{L}

Head drop per unit distance along the flow path, dimensionless. Use consistent length units for head and distance.

Transmissivity

T = K\,b

Conductivity times saturated thickness $b$ ; units $L^2/T$ (m²/d, ft²/s). Aquifer-scale measure of water-transmitting capacity.

Aquifer discharge through a width

Q = T\,i\,w = K\,b\,i\,w

Discharge through a vertical strip of aquifer of width $w$ and thickness $b$ ; folds thickness into $T$ .

Storativity (storage coefficient)

S = S_s\,b \quad (\text{confined}), \qquad S \approx S_y \quad (\text{unconfined})

Volume released per unit area per unit head drop, dimensionless. Confined $S \sim 10^{-5}\text{–}10^{-3}$ ; unconfined $S \approx S_y \sim 0.05\text{–}0.30$ .

Travel time

t = \dfrac{L}{v} = \dfrac{L\,n_e}{K\,i}

Time for water (or a non-sorbing tracer) to travel distance $L$ at the seepage velocity. Uses $n_e$ , not $q$ .

Worked examples

Darcy flux, seepage velocity, and travel time

Problem. A confined sand aquifer has hydraulic conductivity $K = 15\,\text{m/d}$ and effective porosity $n_e = 0.25$ . Two wells $500\,\text{m}$ apart along the flow path show a head difference of $2.0\,\text{m}$ . Find (a) the Darcy flux, (b) the seepage velocity, and (c) the time for a conservative tracer to travel between the wells.

Solution. Gradient:

i = \dfrac{2.0\,\text{m}}{500\,\text{m}} = 0.004

(dimensionless). (a) Darcy flux:

q = K\,i = 15 \times 0.004 = 0.060\,\text{m/d}

. (b) Seepage velocity:

v = \dfrac{q}{n_e} = \dfrac{0.060}{0.25} = 0.240\,\text{m/d}

. (c) Travel time:

t = \dfrac{L}{v} = \dfrac{500}{0.240} = 2.08\times10^{3}\,\text{d} \approx 5.70\,\text{yr}

. Sanity check:

v

is

4\times

the flux because

1/n_e = 1/0.25 = 4

; using

q

instead of

v

would have wrongly given

8{,}330

days. Units:

(\text{m/d})/(\text{dimensionless}) = \text{m/d}

, and

\text{m}\div(\text{m/d}) = \text{d}

.

t = \frac{L\,n_e}{K\,i} = \frac{500 \times 0.25}{15 \times 0.004} = 2.08\times10^{3}\ \text{d}

Transmissivity and aquifer underflow

Problem. An unconfined aquifer is $12\,\text{m}$ thick with $K = 25\,\text{m/d}$ . Estimate its transmissivity, then find the groundwater underflow passing through a $200\,\text{m}$ -wide section where the gradient is $0.003$ .

Solution. Transmissivity:

T = K\,b = 25 \times 12 = 300\,\text{m}^2/\text{d}

. Underflow through the strip:

Q = T\,i\,w = 300 \times 0.003 \times 200 = 180\,\text{m}^3/\text{d}

. Final:

Q = 1.80\times10^{2}\,\text{m}^3/\text{d}

. Sanity check via Darcy's law directly:

A = b\,w = 12 \times 200 = 2400\,\text{m}^2

,

q = Ki = 25\times0.003 = 0.075\,\text{m/d}

, so

Q = qA = 0.075 \times 2400 = 180\,\text{m}^3/\text{d}

. Agreement confirms

T = Kb

simply repackages thickness. Units:

(\text{m}^2/\text{d})(\,)(\text{m}) = \text{m}^3/\text{d}

.

Q = T\,i\,w = 300 \times 0.003 \times 200 = 180\ \text{m}^3/\text{d}

Common pitfalls

•Reporting the Darcy flux $q$ when the question asks for velocity or travel time — divide by $n_e$ to get the seepage velocity $v$ . This factor of 3–10 is the single most common error in the topic.
•Using total porosity $n$ instead of effective porosity $n_e$ in $v = q/n_e$ ; bound water and dead-end pores do not transmit flow, so $n_e \le n$ .
•Forgetting the time-unit mismatch in $K$ : handbook conductivities are often ft/s or m/s, but problems want ft/day or m/day — convert before computing ( $\times 86{,}400$ s/day).
•Confusing confined and unconfined storativity: confined $S$ is tiny ( $\sim10^{-4}$ , elastic storage), while unconfined $S \approx S_y \sim 0.2$ (gravity drainage) — hundreds of times larger.
•Treating transmissivity as a velocity or a conductivity; $T = Kb$ has units $L^2/T$ and already includes the saturated thickness, so do not multiply by $b$ again.
•Dropping the sign/direction of the gradient: flow goes toward LOW head. Building $i$ with head and distance in different length units (ft head over m distance) silently corrupts every result.

References

NCEES FE Reference Handbook — Civil Engineering: Darcy's Law, Transmissivity, Storativity
Fetter, Applied Hydrogeology — Darcy flux vs seepage velocity, aquifer properties
Davis & Cornwell, Introduction to Environmental Engineering — groundwater flow fundamentals

C. Drawdown

Well Drawdown: Theis, Jacob, Thiem, and Dupuit

Steady drawdown in confined (Thiem) and unconfined (Dupuit) aquifers, the cone of depression and radius of influence, transient drawdown (Theis/Cooper-Jacob), and specific capacity.

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D. Remediation of soil, sediment, and/or groundwater

Contaminant Transport and Remediation

Advection-dispersion, the retardation factor that slows a sorbing plume, partitioning from foc and Koc, and the matching of remediation technologies to contaminant behavior.

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