10 Exercises
1) A 100 cubic centimeter (cm^{3}) sample of soil has an initial weight of 227.1 grams. It is oven dried at 105°C to a constant weight of 222.0 grams. The sample is then saturated with water and has a weight of 236.6 grams. Next, the sample is then allowed to drain by gravity in an environment of 100% humidity and is reweighted at 224.4 grams. Assuming that 1 cm^{3} of water = 1 gram at 15.5°C:
a) Calculate the porosity;
b) Calculate the specific yield;
c) Calculate the specific retention;
d) Evaluate whether the resulting particle density is reasonable;
e) Calculate the void ratio;
f) Calculate the initial moisture content;
g) Calculate the initial degree of saturation.
Click for solution to exercise 1.
2) Show that , where n is porosity and e is the void ratio.
Click for solution to exercise 2.
3) Rank the materials from the sediment with the largest value (1) to the material with the smallest value (3) for hydraulic conductivity, K, specific yield, S_{y}, and porosity n.
K | S_{y} | n | |
Silty Sand | _________ | _________ | _________ |
Clay | _________ | _________ | _________ |
Equally mixed sand and gravel | _________ | _________ | _________ |
Click for solution to exercise 3.
4) A regional unconfined sand aquifer was developed during 1990. As a result of the extraction of water the water table dropped about 40 m over a 1 square km area. If the porosity is 34% and the specific retention is 12%, how much water (m^{3}) was withdrawn from the impacted area? How might this volume differ if the aquifer were a 100 m thick confined sand with water at a temperature of 8°C?
Click for solution to exercise 4.
5) A factory is disposing of hot waste water by injecting it into a 1000 m deep well that penetrates a confined aquifer. The sandstone aquifer contains water at 35°C, a similar temperature as the waste water. A regulator wanted the company to model how far the contaminated water would travel in the aquifer over a 10-year period. Hydraulic conductivity values were sparse. They required the company to core part of the 50 m thick confined aquifer and determine a representative value of hydraulic conductivity. A portion of the core was placed in a constant head permeameter as illustrated in the accompanying figure.
a) If an average of 34.0 ml/min was collected at the outlet, what is K in cm/s?
b) If the laboratory experiment was completed using 15°C water, what is the intrinsic permeability of the sandstone in cm^{2}?
c) What is the hydraulic conductivity of the sandstone if the water temperature is 35°C (in cm/s)?
Click for solution to exercise 5.
6) K can be estimated for unconsolidated samples using empirical equations. Grain size distribution data for Sample A, a coarse sand with a porosity of 0.26 (red line), and Sample B, a fine sand with a porosity of 0.3 (yellow line), are shown on the accompanying graph.
a) By inspection of the graphs, which sample is likely to have the highest hydraulic conductivity? Why?
b) Compute the uniformity coefficient for each sample. Compare and contrast the results. Do the coefficients appear to support your observations in part a?
c) Using the Hazen approximation and Slichter method, estimate K in cm/s.
d) Are the K values computed in part c characteristic of sands? Do both methods produce similar values? Why or why not?
Click for solution to exercise 6.
7) Storage in confined aquifers relies on squeezing water in and out of the saturated skeleton. Explain the effects on the effective stress and pore water pressure when water is pumped into a confined aquifer and the water level in tightly cased wells penetrating the aquifer rises 1 m.
Click for solution to exercise 7.
8) Show that the units of S_{s} are 1/L.
Click for solution to exercise 8.
9) A confined aquifer underlies a 10 km^{2} area. The average water level in a number of wells penetrating the confined system rose 2.5 m from April through June. An overlying unconfined aquifer showed an average water table rise of 2.5 m over the same period of time.
Assume the storativity for the confined system is 3.6 × 10^{-5}, and specific yield is 0.12 for the unconfined system. How much water (in m^{3}) recharged each aquifer based on the responses of each potentiometric surface?
Click for solution to exercise 9.
10) Hydraulic head controls the flow of groundwater. A sand filled column is set up as shown in the accompanying figure using a datum of 1000 m. If O=9 cm, K=5 cm, P=5 cm, S=12 cm, A=12 cm, G=5 cm, R=4 cm, M=4 cm, B=7cm, J=9cm, Q=15 cm, C=6 cm, answer the lettered items below. To answer b through d write the answer first as the letters for the relevant line segments, and then as the total value, e.g., for G+M write, 5 cm + 4 cm = 9 cm.
a) Which way is the water flowing?
b) What is the total head at the white mark labeled X relative to the 1000 m datum?
c) What is the elevation head at the white mark labeled Y relative to the 1000 m datum?
d) What is the pressure head at the white mark labeled X?
Click for solution to exercise 10.
11) Write the governing equation that describes two-dimensional (x and y), steady-state flow in a confined anisotropic and homogenous aquifer.
Click for solution to exercise 11.
12) What conditions does the following governing equation represent? Explain the features of the terms of the equation that indicate each condition.
Click for solution to exercise 12.
13) Draw and clearly label a cross section of a water table aquifer overlying a confined aquifer. Label the confining units as aquitards. Place two observation wells in each aquifer and show the water level in each of the wells so that groundwater flow is from the left to the right for the water table aquifer and from the right to the left for the confined aquifer. The illustration should suggest that leakage from the confined aquifer is upward into the unconfined aquifer. Show the corresponding water table and potentiometric surface positions.
Click for solution to exercise 13.
14) Ten wells are located in a valley setting. Glacial material underlies the land surface and a confined sandstone aquifer underlies the region. All wells shown in the accompanying diagram tap the confined aquifer. DATA: All water levels (WL) are measured depths below the top of casing (TOC) elevation. TD is the total well depth below land surface. The head measurement is at the bottom of each well.
Well A TD 150 m, TOC 1105 m, WL 57 m
Well B TD 160 m, TOC 1100 m, WL 39 m
Well C TD 170 m, TOC 1108 m, WL 76 m
Well D TD 150 m, TOC 1100 m, WL 59 m
Well E TD 180 m, TOC 1098 m, WL 67 m
Well F TD 160 m, TOC 1090 m, WL 57 m
Well G TD 180 m, TOC 1080 m, WL 53 m
Well H TD 170 m, TOC 1079 m, WL 41 m
Well I TD 180 m, TOC 1070 m, WL 50 m
Well J TD 170 m, TOC 1100 m, WL 41 m
a) Create a potentiometric surface map view.
b) If the confined aquifer water at the orange rectangle became contaminated with dissolved nitrate from downward leakage of water in the overlying unconfined aquifer, would any other monitoring wells become contaminated under steady state flow conditions? Construct equipotential lines and flow lines to support your answer.
Click for solution to exercise 14.
15) A map view of a heterogeneous and isotropic field site containing a complex confined aquifer system is presented in the accompanying figure. If a contaminant is released at A and each zone of hydraulic conductivity is isotropic and homogeneous:
a) Map the flowline from A to the row of houses. Would wells at any of the houses be impacted?
b) Now assume that only the northern portion of the aquifer is anisotropic K_{x} = 10 m/d and K_{y} = 1 m/d. Starting at A construct the anisotropic flow path and show which, if any, of the houses (blue rectangles) will be affected by this new flow path. Present your construction and show how you determined the flow path in the northern portion of the aquifer.
Click for solution to exercise 15.
16) In the accompanying figure flow lines (black lines) form two flow tubes, A and B. If conditions are isotropic and homogeneous and flow is at steady state, create equipotential lines under the following conditions.
a) For A, assume the aquifer thickness and hydraulic conductivities are constant.
b) For B, assume the thickness of the aquifer increases from left to right while all other conditions remain constant.
Click for solution to exercise 16.