Chapter 8: Problems

    1. Show by dimensional analysis on Eq. (8.6) that u is dimensionless.
    2. Show by dimensional analysis on Eq. (8.7) that W(u) is dimensionless.
    3. Show that the values of the coefficients A and B given in connection with Eqs. (8.38) and (8.39) are correct for the engineering system of units commonly used in North America in which volumes are measured in U.S. gallons.
  1. A fully penetrating well pumps water from an infinite, horizontal, confined, homogeneous, isotropic aquifer at a constant rate of 25 \ell/s. If T is 1.2 × 10–2 m2/s and S is 2.0 × 10–4, make the following calculations.
    1. Calculate the drawdown that would occur in an observation well 60 m from the pumping well at times of 1, 5, 10, 50, and 210 min after the start of pumping. Plot these values on a log-log graph of h0h versus t.
    2. Calculate the drawdown that would occur in a set of observation wells at distances 1 m, 3 m, 15 m, 60 m, and 300 m from the pumping well at a time 210 min after the start of pumping. Plot these values on a semilog graph of h0h versus r.
  2. A confined aquifer with T = 7.0 × 10–3 m2/s and S = 5.0 × 10–4 is pumped by two wells 35 m apart. One well is pumped at 7.6 \ell/s and one at 15.2 \ell/s. Plot the drawdown h0h as a function of position along the line joining the two wells at a time 4 h after the start of pumping.
    1. Why is a 10-day pumping test better than a 10-h pumping text?
    2. Why are storativities for unconfined aquifers so much larger than those for confined aquifers?
    3. What kind of pumping-test arrangement would be required to determine the exact location of a straight, vertical impermeable boundary?
    1. List the assumptions underlying the Theis solution.
    2. Sketch two plots that show the approximate shape you would expect for the time drawdown curve from a confined aquifer if:
      1. The aquifer pinches out to the west.
      2. The overlying confining formations are impermeable, but the underlying formations are leaky.
      3. The pumping well is located near a fault that is in hydraulic connection to a surface stream.
      4. The well is on the shore of a tidal estuary.
      5. The pump broke down halfway through the test.
      6. The barometric pressure increased at the pump test site.
    1. Plot the values of u versus W(u) given in Table 8.1 on a log-log graph. It is only necessary to plot those values lying in the range 10–9 < u < 1.
    2. Plot these same values as 1/u versus W(u) on a log-log graph.
  1. The thickness of a horizontal, confined, homogeneous, isotropic aquifer of infinite areal extent is 30 m. A well fully penetrating the aquifer was continuously pumped at a constant rate of 0.1 m3/s for a period of 1 day. The drawdowns given in the attached table were observed in a fully penetrating observation well 90 m from the pumping well. Compute the transmissivity and the storativity by using:
    1. The Theis method of log-log matching [using the type curve prepared in Problem 6(b)].
    2. The Jacob method of semilog plotting.
t (min) h0h (m) t h0h t h0h t h0h
1 0.14 7 0.39 40 0.66 100 0.81
2 0.22 8 0.40 50 0.70 200 0.90
3 0.28 9 0.42 60 0.71 400 0.99
4 0.32 10 0.44 70 0.73 800 1.07
5 0.34 21 0.55 80 0.76 1000 1.10
6 0.37 30 0.62 90 0.79
  1. A homogeneous, isotropic, confined aquifer is 30.5 m thick and infinite in horizontal extent. A fully penetrating production well is pumped at a constant rate of 38 \ell/s. The drawdown in an observation well 30.5 m from the production well after 200 days is 2.56 m.
    1. Assume a reasonable value for the storativity and then calculate the transmissivity T for the aquifer.
    2. Calculate the hydraulic conductivity and the compressibility of the aquifer. (Assume reasonable values for any unknown parameters.)
    1. A well pumps at 15.7 \ell/s from a horizontal, confined, homogeneous, isotropic aquifer. The attached table lists the drawdown observed in an observation well 30 m from the pumping well. Plot these data on a semilogarithmic graph and use the Jacob method on the early data to calculate T and S.
    2. What kind of boundary is indicated by the break in slope? Measure the slope of the two limbs and note that the second limb has twice the slope of the first limb. In this case, how many image wells must be needed to provide an equivalent aquifer of infinite extent? Draw a sketch showing a possible configuration of pumping well, image well(s), and boundary, and note whether the image wells are pumping wells or recharge wells.
t (min) h0h (m) t h0h t h0h t h0h
11 2.13 21 2.50 52 3.11 88 3.70
14 2.27 28 2.68 60 3.29 100 3.86
18 2.44 35 2.80 74 3.41 112 4.01
130 4.14

  1. The straight-line portion of a semilog plot of drawdown versus time taken from an observation well 200 ft from a pumping well (Q = 500 U.S. gal/min) in a confined aquifer goes through the points (t = 4 × 10–3 day, h0h = 1.6 ft) and (t = 2 × 10–2 day, h0h = 9.4 ft)
    1. Calculate T and S for the aquifer.
    2. Calculate the drawdown that would occur 400 ft from the pumping well 10 h after the start of pumping.
    1. The hydraulic conductivity of a 30-m-thick confined aquifer is known from laboratory testing to have a value of 4.7 × 10–4 m/s. If the straight-line portion of a Jacob semilogarithmic plot goes through the points (t = 10–3 day, h0h = 0.3 m) and (t = 10–2 day, h0h = 0.6 m) for an observation well 30 m from a pumping well, calculate the transmissivity and storativity of the aquifer.
    2. Over what range of time values is the Jacob method of analysis valid for this observation well in this aquifer?
  2. You are asked to design a pump test for a confined aquifer in which the transmissivity is expected to be about 1.4 × 10–2 m2/s and the storativity about 1.0 × 10–4. What pumping rate would you recommend for the test if it is desired that there be an easily measured drawdown of at least 1 m during the first 6 h of the test in an observation well 150 m from the pumping well?
    1. Venice, Italy, has subsided 20 cm in 35 years; San Jose, California, has subsided 20 ft in 35 years. List the hydrogeological conditions that these two cities must have in common (in that they have both undergone subsidence), and comment on how these conditions may differ (to account for the large difference in total subsidence).
    2. The following data were obtained from a laboratory consolidation test on a core sample with a cross-sectional area of 100.0 cm3 taken from a confining clay bed at Venice. Calculate the compressibility of the sample in m2/N that would apply at an effective stress of 2.0 × 106 N/m2.
    3. Load (N) 0 2000 5000 10,000 15,000 20,000 30,000
      Void ratio 0.98 0.83 0.75 0.68 0.63 0.59 0.56
    4. Calculate the coefficient of compressibility, av, and the compression index, Cc, for these data. Choose a K value representative of a clay and calculate the coefficient of consolidation, cv.
  1. It is proposed to construct an unlined, artificial pond near the brink of a cliff. The geological deposits are unconsolidated, interbedded sands and clays. The water table is known to be rather deep.
    1. What are the possible negative impacts of the proposed pond?
    2. List in order, and briefiy describe, the methods of exploration you would recommend to clarify the geology and hydrogeology of the site.
    3. List four possible methods that could be used to determine hydraulic conductivities. Which methods would be the most reasonable to use? The least reasonable? Why?
  2. An undisturbed cylindrical core sample of soil 10 cm high and 5 cm in diameter weights 350 gm. Calculate the porosity.
  3. If the water level in a 5-cm-diameter piezometer standpipe recovers 90% of its bailed drawdown in 20 h, calculate K. The intake is 0.5 m long and the same diameter as the standpipe. Assume that the assumptions underlying the Hvorslev point test are met.
  4. Assume that the grain-size curve of Figure 8.25(a) is shifted one \phi unit to the left. Calculate the hydraulic conductivity for the soil according to both the Hazen relation and the Masch and Denny curves.
    1. Develop the transient finite-difference equation for an internal node in a three-dimensional, homogeneous, isotropic nodal grid where \Delta x = \Delta y = \Delta z.
    2. Develop the transient finite-difference equation for a node adjacent to an impermeable boundary in a two-dimensional homogeneous, isotropic system with \Delta x = \Delta y. Do so:
      1. Using the simple approach of Section 8.8.
      2. The more sophisticated approach of Appendix IX.
  1. Assume that resistors in the range 104–105 \Omega and capacitors in the range 10–12–10–11 F are commercially available. Choose a set of scale factors for the analog simulation of an aquifer with T \simeq 105 U.S. gal/day/ft and S \simeq 3 × 10–3. The aquifer is approximately 10 miles square and drawdowns of 10’s of feet are expected over 10’s of years in response to total pumping rates up to 106 U.S. gal/day.