For old exams (without answers) checkout Previous exams.

For old exams (with answers) checkout Previous exams with answers.

Welcome to TransientGroundwaterFlowIHEDelft’s documentation!

Table of Contents:

• 1. Nomenclature
  • 1.1. General
  • 1.2. Hydraulic and mechanical properties
  • 1.3. Heat properties and flow
  • 1.4. Aquifer system
  • 1.5. Groundwater waves
  • 1.6. Physics, math and mechanics
• 2. Introduction
  • 2.1. Objectives of the course
  • 2.2. Exam
  • 2.3. Assignment
  • 2.4. Gradiing
  • 2.5. What to learn / understand
  • 2.6. Note with respect to the exercises
• 3. Introduction to transient phenomena in groundwater
• 4. Irreversible transient phenomena
  • 4.1. Consolidation
  • 4.2. Liquefaction
  • 4.3. Intrusion of salt water
  • 4.4. Questions
• 5. Reversible groundwater storage
  • 5.1. Phreatic storage (water table storage, specific yield, Sy)
    • 5.1.1. Phreatic responses
    • 5.1.2. Questions
  • 5.2. Elastic Storage
    • 5.2.1. Introduction
    • 5.2.2. Loading efficiency
    • 5.2.3. Barometer efficiency
    • 5.2.4. How much are the loading efficiency and the barometer efficiency when expressed in the properties of the water and the porous medium ?
    • 5.2.5. Specific (elastic) storage coefficient
    • 5.2.6. Application (not for exam)
    • 5.2.7. Questions
  • 5.3. Earth tides (not for exam)
• 6. One-dimensional transient groundwater flow
  • 6.1. Scope
  • 6.2. Governing equations
  • 6.3. Sinusoidal fluctuations of the groundwater head and flows
    • 6.3.1. Groundwater fluctuations due to sinusoidal tides
      • 6.3.1.1. What is the velocity of the wave, or what is the delay of the wave at any distance \(x\) from the sea or river?
      • 6.3.1.2. How much is the wavelength in the ground?
    • 6.3.2. Using characteristic time and length
    • 6.3.3. Fluctuations of temperature in the subsurface
    • 6.3.4. Concluding remarks
    • 6.3.5. Questions
      • 6.3.5.1. Heat flow
  • 6.4. Non-fluctuating interaction with surface water
    • 6.4.1. Basins of half-infinite lateral extent
      • 6.4.1.1. Exercise:
    • 6.4.2. Questions
    • 6.4.3. Higher-order solutions (not for exam)
      • 6.4.3.1. Example:
    • 6.4.4. Questions
    • 6.4.5. Superposition in time, half-infinite aquifer
      • 6.4.5.1. Example
  • 6.5. Groundwater basins as land strips of limited width between straight head boundaries
    • 6.5.1. Introduction
    • 6.5.2. Water level at the left-hand side suddenly rises over a height of \(A\) m, while the water level at the right-hand side remains at zero
      • 6.5.2.1. Shifting the zero point of the \(x\)-axis
    • 6.5.3. Arbitrary non-symmetrical case
    • 6.5.4. Symmetrical case, \(A=B\)
      • 6.5.4.1. Another perspective, drainage of a basin in which the head is initially at distance A above the fixed heads at either side of the strip
    • 6.5.5. Questions
  • 6.6. Symmetrical drainage from a land strip bounded by straight head boundaries (characteristic time of flow basins)
    • 6.6.1. Analytical solution
      • 6.6.1.1. Exercise:
    • 6.6.2. Long-term drainage behavior, characteristic drainage time
    • 6.6.3. Questions
• 7. Transient flow to wells
  • 7.1. Introduction
  • 7.2. Wells and well functions overview
  • 7.3. General relation between the well-induced head change in a water-table and confined aquifer and the relation between the different analytical solutions for drawdown by a fully penetrating pumping well
  • 7.4. Theis and Hantush wells in an infinite aquifer with constant transmissivity and storativity; the governing partial differential equation
    • 7.4.1. Introduction
    • 7.4.2. The governing partial differential equation
    • 7.4.3. The Theis and Hantush well functions
    • 7.4.4. Implementation of the Theis and Hantush well function in Python
      • 7.4.4.1. Theis’ well function (exponential integral)
      • 7.4.4.2. Hantush well function
    • 7.4.5. Type-curves for the Theis and Hantush well functions
      • 7.4.5.1. Example application
      • 7.4.5.2. Unconfined flow approximation by Theis:
      • 7.4.5.3. Inflection points in the graphs of the Hantush well functions on half-logarithmic scales (well functions versus log of \(1/u\))
    • 7.4.6. Power approximation of Theis well function
    • 7.4.7. Radius of influence
    • 7.4.8. Graphical illustration of the radius of influence
    • 7.4.9. Relation between the transient Theis drawdown and the well-known Thiem solution for the drawdown in the steady-state situation. Time to reaching steady state.
    • 7.4.10. Flow at distance \(r\) from the well in the Theis and Hantush situations
      • 7.4.10.1. Theis situation
      • 7.4.10.2. Hantush situation
  • 7.5. Pumping-test analyses
    • 7.5.1. Introduction
    • 7.5.2. Analysis of the pumping test in the Theis situation using the logarithmic approximation of the Theis well function
      • 7.5.2.1. Example
    • 7.5.3. Theis and Hantush classic pumping-test analysis
      • 7.5.3.1. Can we improve the accuracy?
    • 7.5.4. Exercises
    • 7.5.5. Superposition in time
    • 7.5.6. Superposition in space
      • 7.5.6.1. Exercise:
    • 7.5.7. Questions
  • 7.6. Partial penetration of well screens
    • 7.6.1. Huisman’s method
    • 7.6.2. Hantush’s solution for partial penetrating screens
    • 7.6.3. Example
      • 7.6.3.1. Python implementation of partial penetration
    • 7.6.4. Questions
  • 7.7. Delayed yield (delayed water-table response)
    • 7.7.1. Introduction
    • 7.7.2. Water-table aquifer
    • 7.7.3. Generalization to semi-confined aquifers
    • 7.7.4. The two Theis bounding curves
    • 7.7.5. Influence of partial penetration of the well screen
    • 7.7.6. Questions
  • 7.8. Large-diameter wells (not for the exam)
• 8. Convolution
  • 8.1. What convolution is and how it works
  • 8.2. Examples
    • 8.2.1. Arbitrarily fluctuating river stage
    • 8.2.2. Impulse response, block response, and step response comparison
    • 8.2.3. Arbitrarily fluctuating extraction of multiple Theis wells
    • 8.2.4. Convolution using recharge as input time series to simulate varying groundwater levels
  • 8.3. Questions
• 9. Laplace solutions (illustration, not for exam)
  • 9.1. Sudden head rise at the boundary of a one-dimensional semi-infinite aquifer
  • 9.2. Laplace solution for the Theis well function
  • 9.3. Stehfest for numerical back-transformation of Laplace solutions
• 10. Bibliography

Old exams:

• Previous exams
  • Open-book exam (1h), Feb 7, 2022
    • Question 1:
    • Question 2:
  • Closed-book exam (1h), Feb 23, 2021
    • Question 1:
    • Question 2:
    • Question 3:
    • Question 4:
  • Closed book exam (1h), Feb 4, 2020
    • Question 1:
    • Question 2:
    • Question 3:
    • Question 4:
    • Question 5:
  • Closed book reexam (1h), March 2018
    • Question 1
    • Question 2
    • Question 3
  • Closed book exam (1h), Feb 7, 2017
    • Question 1
    • Question 2
    • Question 3
  • Closed book reexam (1h), 2016
    • Question 1
    • Question 2
    • Question 3
  • Closed-book exam (1h), Feb 1, 2016
    • Question 1: (16 points)
    • Question 2: (14 points)
  • Closed-book exam (1h), Feb 2015
    • Question 1
    • Question 2
    • Question 3
    • Question 4
  • Closed-book reexam (1h), March 2015
    • Question 1
    • Question 2
    • Question 3
  • Closed book exam, Feb 2014
    • Question 1
    • Question 2
    • Question 3
    • Question 4
  • Closed-book exam (1h), Feb 3, 2011
    • Question 1: Pressure in confined aquifer
    • Question 2
    • Question 3
    • Question 4: wells
  • Closed-book exam (1h), Feb 2010
    • Question 1
    • Question 2
    • Question 3
    • Question 4
  • Closed-book exam (3h), Feb 2009
    • Question 1
    • Question 2
    • Question 3
    • Question 4
    • Question 4
    • Question 5
    • Question 6
  • Closed-book exam (3h), Feb 2007
    • Question 1: General
    • Question 2: Diffusion equation
    • Question 3: Fluctuation groundwater
    • Question 4: Flow to an extraction canal
    • Question 4: Well in semi-confined aquifer
    • Question 6: Drawdown due to a pumping station in an unconfined aquifer
  • Closed-book exam (3h), Feb 2006
    • Question 1: Conceptual
    • Question 2: Characteristic time of groundwater basin
    • Question 3: Tides in groundwater
    • Question 4: Aquifer with river
    • Question 5: Well in a confined aquifer
    • Question 6: Well in a leaky aquifer
    • Question 7: Well in an unconfined aquifer
• Previous exams with answers
  • Open-book exam (1h), Feb 7, 2022
    • Question 1
    • Question 2
    • Question 3:
  • Closed-book exam (1h), Feb 23, 2021
    • Question 1
    • Question 2
    • Question 3:
    • Question 4:
  • Closed book exam (1h), Feb 4, 2020
    • Question 1
    • Question 2
    • Question 3
  • Closed book reexam (1h), March 2018
    • Question 1:
    • Question 2:
    • Question 3
  • Closed book exam (1h), Feb 7, 2017
    • Question 1:
    • Question 3:
  • Closed book reexam (1h), Feb 2016
    • Question 1:
    • Question 2:
    • Question 3:
  • Closed-book exam (1h), Feb 1, 2016
    • Question 1: (16 points)
    • Question 2: (14 points)
  • Close-book exam (1h), Feb 2015
    • Question 1:
    • Question 2:
    • Question 3:
    • Question 4:
  • Closed-book reexam (1h), March 2015
    • Question 1:
    • Qustion 2:
    • Question 3:
  • Closed-book exam (1h), Feb 2014
    • Question 1:
    • Question 2:
    • Qestion 3:
    • Question 4
  • Closed-book exam (1h), Feb 3, 2011
    • Question 1: Pressure in confined aquifer
    • Question 2: Tidal waves
    • Question 2: Characteristic time of groundwater systems
    • Question 4: Wells
  • Closed-book exam (1h), Feb 5, 2010
    • Question 1:
    • Question 2:
    • Question 3:
    • Quesiton 4:
  • Closed-book exam (3h), Feb 2009
    • Question 1:
    • Question 2:
    • Question 3:
    • Question 4:
    • Question 5:
    • Question 6:
    • Quesiton 7:
  • Closed-book exam (3h), Feb 2007
    • Question 1: Conceptual
    • Question 2: Diffusion equation
    • Question 3: Fluctuation groundwater
    • Question 4: Flow to an extraction canal
    • Question 5: Well in semi-confined aquifer
    • Question 6: Drawdown due to a pumping station in an unconfined aquifer
  • Closed-book exam (3h) Feb 2006
    • Question 1: Conceptual
    • Question 2: Characteristic time of groundwater basin
    • Quesiton 3: Tides in groundwater
    • Question 4: Aquifer with river
    • Question 5: Well in a confined aquifer
    • Question 6: Well in a leaky aquifer
    • Question 7: Well in an unconfined aquifer

Old Assignments:

• Previous Assignments
  • Assignment Jan 2022. Wells along a river.
    • Problem statement
    • How to work out the assignment
  • Assignment Jan 2021. Wells along a river in a closed valley.
    • Assignment questions
    • The pumping-test data
  • Assignment Jan 2020. Same as Jan 2018
  • Assignment Jan 2019. A building pit next to a river.
    • Problem statement
    • Questions
    • Hints
  • Assignment Jan 2018. Water company uses ASR system to prevent river inflow during summer.
  • Assignment Jan 2017. Water company uses ASR system to prevent extraction from tiver during summer
    • Problem statement
    • Hints and further information:
    • Steps to take:
    • Suggestions
    • The data as a csv (comma separated values)
  • Assignment Jan 2016. Several exercises based on the syllabus.
    • Setup
    • Capillary rise
    • Tidal fluctuations
    • Temperature variation in the subsurface:
    • Effect of a sudden change of the water level in a river
    • Decay of head in a strip of land of given width (characteristic time of the groundwater system)
    • Implement the Theis and Hantush well functions \(\mathrm{W}(u)\) and \(\mathrm{W}(u,r/\lambda)\) as a function of \(1/u\)
    • Pumping test Dalem
• Assignment 2016 2022 with answers
• Assignment Jan 2022. Wells along a river
  • Problem statement
  • Convenient plot functions
  • The data
    • 1. Make a picture of the water table at a line through the pumping well perpendicular to the river at t the following times 0.001, 1day, 1 week, 1 month, 1 year, 10 years after the start of the extraction.
    • As can be seen, the difference between the situation at t=356 d and t=3650 d is very small. Hence, we can assume that the situation at t=3650 d is practically steady-state.
    • 2. Make a picture of the head contours in top view after 1 month, 1 year and 10 years.
    • 4 Inflow from the river.
    • Q8: Find the point where the inflow is just zero. (This may be done analytically)
    • Q9: River stage varies by 2 m with T = 1 year = 365 days
    • Q10: plot the envelopes around the actual situation. Just use steady-state drawdown for convenience
    • 11 the delay of the wave at the point where the amplitude has declined by a factor of 20
    • Q10: By how much will the water table rise due to this sudden recharge is it is assumed that all this precipitation will add to the groundwater?
    • Q11: Show the development of the water table over time (for a few times after the shower took place) by adding its effect to the steady-state situation (pumping station has been pumping continuously for at least 10 years?
• Assignment Jan 2021. Wells along a river in a closed valley
  • Problem statement
  • Assignment questions
  • Import the modules that we will need
  • Get and show our data (pumping test data)
  • Plot of drawdown versus \(t/r^2\) instead of versus \(t\)
  • Our data on versus \(\log(t)\)
  • Define the three wells and compute the dd at one point as a function of time
  • Drawdown along a lines parallel to the axis of the valley (y-axis) for different times
  • To compute the drawdown we need to mirror the well in the center of the valley with respect to be the river on the left and the closed side at the right
    • Continue with just one well, at the center of the three of which the extraction is the sum. This is ok, because the datails of the three invidual wells don’t matter at larger distances. (The valley is 5 km wide, and the distance between the wells was only 500 m).
  • These results show the correct behavior:
  • What is the influx from the river into the aquifer?
  • Implement the flow across the midline between a well and its opposite mirror to simulate a river or canal.
  • The inflow as a function of time (to prove that it will approach the extraction from the well)
  • Verfication of the mirroring scheme using cross section of drawdown and specific discharge
    • Below is a streamplot and quiver based on the specific discharge vectors, but they are inaccurate when there are singularities like wells in the flowfied.
  • Required depth of the well screens
  • Conclusion
• Assigment Jan 2020
• Assignment Jan 2019. A building pit next to a river
  • Problem statement
  • Questions
  • Hints
  • Modules you wil need are imported here
  • Convenient plotting functions
  • Setup the situation, define the wells and observation point
    • Fulfilling the required drawdown
    • River-stage change
    • Impact of the month-long wave in the river stage
  • Impact of river tide
  • How much water is extracted from the river?
• Assignment Jan 2017 preventing extraction from river
  • Problem statement: water company uses ASR system to prevent extraction from river during summer
    • Hints and further information:
    • Steps to take:
    • Suggestions:
  • First show the heads and the specific discharge to gain overview
  • Compute the infiltration from the river if the extration were continuous after \(t=0\)
  • Simulating the actual flow regime, with winter injection and summer extraction.
    • Set up the regime
  • Conclusion
• Assignment Jan 2016. Several exercises based on the syllabus
  • Intro
  • Capillary rise
  • Tidal fluctuations
  • Temperature variations in the subsurface
  • Effect of a sudden change of the water level in a river
  • Decay of head in a strip of land of given width (characteristic time of the groundwater system)
  • Pumping test Dalem
    • Implement the Theis and Hantush well functions and show their type curves
    • Loading (importing) the pumping test data for the assignment from excel.
    • Showing the pumping test data on different scales
    • Showing the pumping test data as a function of \(t/r^2\) instead of \(t\)
    • Working out the pumping test
    • First method, use the straight part of the drawdown
    • Results
  • Using the traditional method, matching the data and the Hantush curves on double log paper (shifting until the best fit is obtained.)
  • Generating test data for the pumping test analyses, so that each students will have his/her unique test data.
• Some python techniques useful for your assignment
  • Introduction
  • Strings (unchangable series of characters)
  • Tuples (unchangeble lists)
  • Lists are just like tuples, but can be changed in place
  • Numpy arrays, so-called ndarrays (n-dimensional arrays)
  • Some ways to generate arrays
  • For-loops with zipping and enumerate
  • Superposition, using logical indexing to handle switch times
  • Some elegant logical indexing, which is extremely useful
  • List comprehensions to generate and filter lists
  • Conclusions

Indices and tables

• Index

• Module Index

• Search Page