FE Review Sessions

FE Exam

Prepare for the Fundamentals of Engineering (FE) exam, taken in your fourth year. The FE exam is generally the first step in the process to becoming a professional licensed engineer (PE).

The FE exam is administered by NCEES, the National Council of Examiners for Engineering and Surveying.

It is designed for recent graduates and students who are close to finishing an undergraduate engineering degree from an EAC/ABET-accredited program. The FE exam is a computer-based exam administered year-round at NCEES-approved Pearson VUE test centers.

Review Session Videos

Michigan Tech offers lecture review sessions on civil and environmental engineering topics.

Topic Instructor Duration
Structures Wiliam M. Bulleit 01:59:59
Circuits Glen E. Archer 01:59:56
Environmental Engineering Neil J. Hutzler 01:51:50
Soil Mechanics Stanly J. Vitton 01:45:51
Thermodynamics Gretchen Hein 01:47:08
Fluid Mechanics/Water Resources Brian D. Barkdoll 01:53:03
Hydrogeology John S. Gierke 01:16:07

Hydrogeology/Water Resources Review Notes

Download Review Notes

NRCS (SCS) Rainfall-Runoff


  • runoff
  • maximum basin retention
  • curve number


  • precipitation
  • maximum basin retention
  • runoff
  • curve number

Rational Formula


  • peak discharge


  • watershed area
  • runoff coefficient
  • rainfall intensity
  • peak discharge

Darcy’s Law


  • discharge rate
  • specific discharge
  • average seepage velocity


  • discharge rate
  • hydraulic conductivity
  • hydraulic head
  • cross-sectional area of flow
  • specific discharge (also called Darcy velocity or superficial velocity)
  • average seepage velocity
  • effective porosity


  • Unit hydrograph
  • Transmissivity
  • Storativity or storage coefficient of an aquifer

Well Drawdown


  • Unconfined aquifer on an impermeable layer

Dupuit’s Formula


  • flow rate of water drawn from well


  • flow rate of water drawn from well
  • coefficient of permeability of soil
  • height of water surface above bottom of aquifer at perimeter of well
  • height h of water surface above bottom of aquifer as a distance r from well centerline
  • radius to well at perimeter of well, ie, radius of well
  • r which is the radius to water surface whose height is h above the bottom of the aquifer
  • specific capacity
  • well drawdown


  • Confined aquifer on an impermeable layer


  • flow rate of water drawn from well
  • transmissivity


  • flow rate of water drawn from well
  • transmissivity
  • thickness of confined aquifer
  • heights of piezometric surface above bottom of aquifer
  • radii from pumping well

Groundwater Flow

  • Flows in pores and fractures of aquifers
  • Flow is induced by hydraulic gradients
  • Flow is in direction of decreasing head
  • Flow velocity is proportional to the hydraulic gradient


  • head
  • hydraulic gradient
  • groundwater flow
  • hydraulic conductivity


  • groundwater flow
  • area perpendicular to flow
  • specific discharge
  • Darcy velocity
  • hydraulic conductivity
  • intrinsic permeability (properties of the geologic formation)
  • fluid properties
  • mass density of the fluid
  • gravitational acceleration
  • dynamic viscosity of the fluid
  • length
  • time
  • mass


  • Limitation: applies for laminar flow
  • The negative sign is because flow is in the direction of decreasing head (ie, a negative hydraulic gradient)

Intrinsic Permeability

Most important properties affecting the magnitude of intrinsic permeability include:

  1. Sizes and numbers of pores
  2. Pore shape and “connectedness” (packing)
  3. Surface texture

Intrinsic permeability is proportional to the square of the grain diameter. A table is given for the median size of fine, medium, and course sand to show the difference by orders of magnitude in the intrinsic permeability.

Seepage Velocity

Darcy velocity or specific discharge (ie, flow rate per total area) is used for water supply (volume).

Average pore or seepage velocity, which reflects the average velocity of groundwater in pore spaces, is of interest for contaminant transport and geotechnical problems.


Aquifers are geological formations that are saturated with water.


  • degree of saturation
  • porosity


  • degree of saturation
  • volume of water
  • volume of voids
  • porosity
  • total (bulk) volume


  • Volumetric water content will equal porosity when the degree of saturation is 100% (saturated).
  • Groundwater flow occurs in pores and fractures
    • Unconsolidated systems are composed of broken rock pieces, and pores are the spaces between the grains/pieces.
    • Consolidated systems are whole rock formations, and pores can exist among the cemented grains that form rocks or as fractures/cracks that subsequently occur as a result of tectonics and/or weathering.
    • Unconsolidated formates are usually more porous and more permeable. Darcy’s Law often applies.
    • Flows in many consolidated formations occur primarily in fractures/conduits. Darcy’s Law often does NOT apply.

Well Hydraulics (Ideal)


  • governing equation for groundwater flow for a confined aquifer
  • groundwater flow under steady state conditions


  • governing equation for groundwater flow for a confined aquifer
  • distance from the center of the pumping well
  • aquifer thickness
  • aquifer compressibility
  • fluid (water) compressibility

Aquifer Behavior

Unconfined aquifers have a “free” water surface or groundwater table, which is at atmospheric pressure and below which the pressures are normally hydrostatic.

Confined aquifers are overpressured, and the potentiometric surface is above the top of the aquifer. The top of the aquifer is a confining unit (aquitard or aquiclude).


  • Unconfined (phreatic)
  • Confined (artesian)
  • Special case: flowing well


Solutions to the groundwater flow equation for a single pumping well in a fully confined aquifer:


drawdown, a potentiometric surface from static (unpumped, initial) level


  • homogeneous, isotropic properties
  • no boundaries
  • uniform thickness
  • constant pumping rate
  • fully confined
  • fully penetrating
  • flat SWL
  • Solutions for Theis and Cooper-Jacob approximations


  • Theis solution is valid for all t
  • Cooper-Jacob solution error diminishes as t goes up


  • Fully confined aquifer pumped @ 1 cubic meter per minute
  • Given:
    • porosity
    • aquifer thickness
    • aquifer compressibility
    • water compressibility
    • intrinsic permeability
    • water density
    • water viscosity
    • flow rate
  • Calculate T, S, s(r,t) @ r = 100 m, t = 1000 min
  • Use the Theis approach and then the Cooper-Jacob approximation

Special Condition: Steady State (Equilibrium)

In addition to the assumptions for the Theis solution (ie, ideal aquifer and pumping conditions), assume that the potentiometric surface has stabilized. In this case, the Theim solution applies, which comes from integrating Darcy’s Law for axisymmetric flow to the pumping well.


Application of the Thiem solution to an unconfined aquifer:


  • Darcy’s Law
  • Dupuit’s Formula

Special case for a pumping well.

Example of Steady State Well Hydraulics

Calculate the specific capacity for confined and unconfined.

Soil and Groundwater Remediation

Contaminant Phases

  1. Pure or “neat” or “free” or “nonaqueous” (eg, gasoline or fuel oil)
  2. Dissolved or aqueous
  3. Vapor or gaseous (for volatile contaminants)
  4. Sorbed

Sites that pose a threat to people, either because concentrations exceed drinking water standards (MCLs or max contaminant levels) or other exposures (eg, contact, inhalation, etc.), or to the environment, such as discharge to surface water, must undergo corrective action (remediation or “plume control”).

Plumes can be captured (“pump and treat”).

Source zones can be cleaned up (source control) using flushing or enhanced flushing techniques and/or chemical treatment (eg, advanced oxidation) and bioremediation.