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Groundwater and Wells

5% of exam

Aquifer properties, Darcy groundwater flow, and steady/transient well drawdown analysis (Thiem, Theis, Dupuit) for confined and unconfined aquifers.

3 concepts

A. Aquifers

Darcy's Law and Aquifer Properties

Confined vs unconfined aquifers, hydraulic conductivity and transmissivity, storativity and specific yield, and the difference between Darcy and seepage velocity.

Groundwater is the quiet half of water resources: it moves slowly, invisibly, and almost always laminarly, which is exactly why one nineteenth-century experiment by Henry Darcy still governs the whole subject. Master a handful of ideas here — what an aquifer is, the single linear law

Q = KiA

that drives flow through it, and the property bookkeeping (

K

,

T

,

S

,

S_y

, porosity) that turns that law into numbers — and the rest of the Groundwater and Wells topic (flow nets, well drawdown, time of travel) becomes substitution. Where examinees lose points is not the algebra but the conventions: mixing up discharge with velocity, conductivity with transmissivity, or the bulk Darcy velocity with the much faster velocity a tracer actually travels. Everything in this concept lives in the NCEES PE Civil Reference Handbook under

\S 6.6

Groundwater and Wells (with the parallel seepage treatment in

\S 3.16

).

Confined vs unconfined aquifers

An aquifer is a saturated geologic formation transmissive enough to yield usable water. In an

\textbf{unconfined}

(water-table) aquifer the upper surface is the water table itself, at atmospheric pressure; pumping physically dewaters the pores, so the saturated thickness

h

shrinks and the flow problem is nonlinear in head. In a

\textbf{confined}

(artesian) aquifer a low-permeability aquitard caps the water, holding it under pressure above the top of the formation; the saturated thickness

b

is fixed, water is released elastically (not by draining pores), and the head is represented by an imaginary

\textbf{piezometric}

(potentiometric) surface that can sit well above the aquifer. This distinction sets which well equation you use — Dupuit for unconfined, Thiem for confined — and which storage term applies.

Darcy's law and hydraulic gradient

Darcy found that flow through a saturated porous medium is proportional to the cross-sectional area and to the head loss per unit length — the hydraulic gradient

i = \Delta h/\Delta L

— with the proportionality constant being the hydraulic conductivity

K

. The discharge

Q

is a volumetric flow (

\text{ft}^3/\text{s}

or

\text{m}^3/\text{s}

); dividing by the gross area gives the specific discharge

q

(also called the Darcy or superficial velocity). Written here in the handbook's magnitude form with

i

taken positive, the law returns a positive discharge directed from high head toward low head (a leading minus sign is sometimes added to make the vector point down-gradient explicitly). The law holds only in the laminar regime (low Reynolds number), which covers nearly all natural groundwater flow.

Q = K\,i\,A = K\,\frac{\Delta h}{\Delta L}\,A, \qquad q = \frac{Q}{A} = K\,i

Hydraulic conductivity and intrinsic permeability

Hydraulic conductivity

K

bundles two things: a property of the porous medium (the intrinsic or specific permeability

k

, units of area,

\text{ft}^2

or darcys) and properties of the fluid (its specific weight

\gamma

and dynamic viscosity

\mu

). Separating them matters whenever temperature or fluid changes: warm water is less viscous, so the same soil conducts it faster. Typical

K

spans about nine orders of magnitude — from

10^{-9}\,\text{cm/s}

in clay to

1\,\text{cm/s}

in clean gravel — which is why a single careless order-of-magnitude error swamps everything else.

K = \frac{k\,\gamma}{\mu} = \frac{k\,\rho\,g}{\mu}

Transmissivity

For a regional flow or well problem we rarely care about the point conductivity; we care about how much water the full thickness of the aquifer can transmit.

\textbf{Transmissivity}

T

is conductivity integrated over the saturated thickness —

T = Kb

for a confined aquifer of thickness

b

, or

T = K h

using the saturated thickness

h

in an unconfined one. It carries units of

\text{length}^2/\text{time}

(

\text{ft}^2/\text{s}

,

\text{m}^2/\text{s}

, or the field unit

\text{gpd/ft}

). Transmissivity is the single property that the Thiem, Theis, and Cooper-Jacob well equations are written in.

T = K\,b

Storativity, specific yield, and porosity

Two media of identical

T

can store wildly different amounts of water.

\textbf{Storativity}

(storage coefficient)

S

is the volume of water released from storage per unit aquifer area per unit decline in head. In a confined aquifer water comes only from elastic expansion of water and compression of the skeleton, so

S = S_s\,b

is tiny (

5\times10^{-5}

to

5\times10^{-3}

). In an unconfined aquifer storage is dominated by actual draining of pores, so

S \approx S_y

, the

\textbf{specific yield}

(commonly

0.1

to

0.3

). The total porosity splits into the water that drains and the water held back by capillarity and adsorption:

n = S_y + S_r

, with

S_r

the specific retention. The drainable (effective) porosity

S_y

— not the total porosity

n

— is what governs how fast a contaminant front moves.

S = S_y + S_s\,b \;\;(\text{uncon.}\;S\approx S_y), \qquad n = S_y + S_r

Darcy velocity vs seepage velocity

This is the single most-missed idea in the topic. The Darcy velocity

q = Ki

is a fictitious bulk velocity — it assumes water moves through the entire cross-section, solids included. But water can only travel through the pore space, so the actual average velocity of a water particle (or a dissolved tracer) is faster, by the reciprocal of the effective porosity. The

\textbf{seepage}

(average linear, pore) velocity

v_s = q/n_e

is what you use for travel time and contaminant transport; the Darcy velocity is what you use in flux and discharge. Confusing the two routinely understates plume arrival times by a factor of three to five.

v_s = \frac{q}{n_e} = \frac{K\,i}{n_e} = \frac{K}{n_e}\frac{\Delta h}{\Delta L}

Layered aquifers: equivalent conductivity

Real aquifers are stratified. For flow parallel to the layering, the high-

K

beds short-circuit the flow and the equivalent conductivity is the thickness-weighted arithmetic mean (dominated by the most permeable bed). For flow perpendicular to the layering, the low-

K

beds throttle everything and the equivalent is a harmonic mean (dominated by the least permeable bed). The horizontal value is always the larger, which is why aquifers are far more transmissive sideways than vertically.

K_{x}=\dfrac{\sum K_i b_i}{\sum b_i}, \qquad K_{z}=\dfrac{\sum b_i}{\sum (b_i/K_i)}

Exam strategy

Read the units of the answer first: a velocity (

\text{ft/s}

) means

q=Ki

or

v_s=Ki/n_e

, a discharge (

\text{ft}^3/\text{s}

or

\text{gpm}

) means

Q=KiA

, and a

\text{ft}^2/\text{s}

or

\text{gpd/ft}

answer is transmissivity. Decide early whether the problem wants the bulk Darcy velocity (flux) or the seepage velocity (travel time) — the word 'tracer', 'contaminant', or 'how long' signals seepage velocity, so divide by effective porosity. Keep

K

in consistent units; field

K

often comes in

\text{gpd/ft}^2

or

\text{cm/s}

, so convert before substituting. For confined problems use

T=Kb

with fixed

b

; for unconfined,

b

is the variable saturated thickness. And remember storativity is dimensionless and minuscule for confined aquifers but near

0.2

for unconfined.

Key equations

Darcy's law (discharge)

Q = K\,i\,A = K\,\dfrac{\Delta h}{\Delta L}\,A

Volumetric flow $Q$ ( $\text{ft}^3/\text{s}$ , $\text{m}^3/\text{s}$ ) from conductivity $K$ , gradient $i=\Delta h/\Delta L$

Specific (Darcy) velocity

q = \dfrac{Q}{A} = K\,i

Bulk/superficial flux across the full cross-section (length/time). Not the velocity of a water particle.

Seepage (pore) velocity

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

Transmissivity

T = K\,b

Conductivity times saturated thickness $b$ ( $\text{ft}^2/\text{s}$ , $\text{m}^2/\text{s}$

Conductivity vs intrinsic permeability

K = \dfrac{k\,\gamma}{\mu} = \dfrac{k\,\rho\,g}{\mu}

Storativity (confined)

S = S_s\,b

Volume released per unit area per unit head drop; $S_s$ = specific storage ( $\text{ft}^{-1}$

Porosity split (unconfined)

n = S_y + S_r, \qquad S \approx S_y

Layered equivalent conductivity

K_x=\dfrac{\sum K_i b_i}{\sum b_i}, \quad K_z=\dfrac{\sum b_i}{\sum (b_i/K_i)}

Worked examples

Darcy discharge, Darcy velocity, and seepage velocity

Problem. A confined sand aquifer has $K = 25\,\text{ft/day}$ and an effective porosity $n_e = 0.25$ . The piezometric (potentiometric) gradient is $0.004\,\text{ft/ft}$ and the cross-sectional area of flow is $500\,\text{ft}^2$ . Find (a) the Darcy velocity, (b) the discharge, and (c) the seepage velocity.

Solution. (a) Darcy/specific velocity:

q = K\,i = (25)(0.004) = 0.100\,\text{ft/day}

. (b) Discharge:

Q = q\,A = (0.100)(500) = 50.0\,\text{ft}^3/\text{day}

v_s = \dfrac{K\,i}{n_e} = \dfrac{(25)(0.004)}{0.25} = 0.400\ \text{ft/day}

Transmissivity and regional discharge through a confined aquifer

Problem. A confined aquifer is $30\,\text{m}$ thick with $K = 2\times10^{-4}\,\text{m/s}$ . Over a $1{,}000\,\text{m}$

Common pitfalls

•Reporting the Darcy velocity when the problem wants travel time. Words like 'tracer', 'contaminant', or 'how long' demand the seepage velocity $v_s = q/n_e$ , which is larger than $q$ .
•Dividing by total porosity instead of effective (drainable) porosity $n_e$ for seepage velocity — total porosity overstates the flow area and underestimates the true velocity.
•Confusing conductivity $K$ (length/time) with transmissivity $T = Kb$ (length $^2$ /time). Well equations need $T$ ; point flux needs $K$
•Mixing unit systems for $K$ : field values arrive in $\text{gpd/ft}^2$ , $\text{cm/s}$ , or $\text{ft/day}$ . Convert everything to one system before substituting.
•Using total porosity $n$ as the storativity. For a confined aquifer $S = S_s b$ is around $10^{-3}$ or smaller, nothing like $n$
•Treating an unconfined aquifer's thickness as fixed. Saturated thickness $h$ shrinks as the water table drops, making the flow nonlinear ( $h^2$ terms) — that is why Dupuit, not a constant- $b$ formula, applies.
•Mishandling flow direction: water always moves from high head to low head, so confirm which head is upstream before trusting a computed direction (the optional leading minus sign in Darcy's law only encodes that down-gradient direction — keep $i$ positive in magnitude calculations).

References

NCEES PE Civil Reference Handbook — §6.6 Groundwater and Wells
NCEES PE Civil Reference Handbook — §3.16 Groundwater and Seepage
Fetter, Applied Hydrogeology — aquifer properties, Darcy's law
Freeze and Cherry, Groundwater — conductivity, storativity, seepage velocity

B. Groundwater flow

Flow Nets, Seepage and Uplift

Two-dimensional flow nets, seepage quantity from flow channels and equipotential drops, exit gradient and the factor of safety against piping, and uplift on hydraulic structures.

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C. Well and drawdown analysis

Well Drawdown: Thiem, Theis and Dupuit

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

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(taken positive), and gross area

A

. Flow is directed toward lower head.

Average linear velocity through the pores; $n_e$ = effective (drainable) porosity. Use for travel time and contaminant transport.

, or

\text{gpd/ft}

). Used directly in well equations.

$k$ = intrinsic permeability (area, darcys); $\gamma,\mu$ = fluid specific weight and dynamic viscosity. Separates soil and fluid effects.

). Dimensionless,

5\times10^{-5}

to

5\times10^{-3}

.

Total porosity = specific yield (drainable) + specific retention (held). Unconfined storativity $\approx S_y$ ( $0.1$ – $0.3$ ).

Arithmetic (thickness-weighted) mean parallel to layers; harmonic mean perpendicular. $K_x \ge K_z$ always.

. (c) Seepage velocity:

v_s = q/n_e = 0.100/0.25 = 0.400\,\text{ft/day}

. Sanity check:

v_s > q

because water threads only through

25\%

of the area, so it must move four times faster — consistent with

1/n_e = 4

. A tracer would cross

100\,\text{ft}

of this aquifer in

100/0.400 = 250

days, not

1000

days. Final:

q = 0.100\,\text{ft/day}

,

Q = 50.0\,\text{ft}^3/\text{day}

,

v_s = 0.400\,\text{ft/day}

.

wide flow front the piezometric gradient is

0.002

. Find the transmissivity and the total groundwater discharge.

Solution. Transmissivity:

T = K\,b = (2\times10^{-4})(30) = 6.0\times10^{-3}\,\text{m}^2/\text{s}

. The flow area is the thickness times the front width,

A = b\,W = 30 \times 1000 = 30{,}000\,\text{m}^2

, so

Q = K\,i\,A = T\,i\,W = (6.0\times10^{-3})(0.002)(1000)

.

Q = 0.0120\,\text{m}^3/\text{s}

. Sanity check: dimensions

[\text{m}^2/\text{s}][\,-\,][\text{m}] = \text{m}^3/\text{s}

. As a daily volume,

0.0120 \times 86{,}400 = 1.04\times10^{3}\,\text{m}^3/\text{day}

— a believable underflow for a sandy aquifer. Final:

T = 6.00\times10^{-3}\,\text{m}^2/\text{s}

,

Q = 0.0120\,\text{m}^3/\text{s}

.

Q = T\,i\,W = (6.0\times10^{-3})(0.002)(1000) = 0.0120\ \text{m}^3/\text{s}

.

; only an unconfined

S \approx S_y

approaches

0.1

–

0.3

.