Chapter 2: Problems

  1. The following field notes were taken at a nest of piezometers installed side by side at a single site:

    Piezometer a b c
    Elevation at surface (m.a.s.l.) 450 450 450
    Depth of piezometer (m) 150 100 50
    Depth of water (m) 27 47 36

    Let A, B, and C refer to the points of measurement of piezometers a, b, and c. Calculate:

    1. The hydraulic head at A, B, and C (m).
    2. The pressure head at A, B, and C (m).
    3. The elevation head at A, B, and C (m).
    4. The fluid pressure at B (N/m2).
    5. The hydraulic gradients between A and B and between B and C. Can you conceive of a hydrogeological situation that would lead to the directions of flow indicated by these data?

  1. Draw diagrams of two realistic field situations in which three piezometers installed side by side, but bottoming at different depths, would have the same water-level elevation.
  1. Three piezometers located 1000 m apart bottom in the same horizontal aquifer. Piezometer A is due south of piezometer B and piezometer C is to the east of the line AB. The surface elevations of A, B, and C are 95, 110, and 135 m, respectively. The depth to water in A is 5 m, in B is 30 m, and in C is 35 m. Determine the direction of groundwater flow through the triangle ABC and calculate the hydraulic gradient.
  1. Show that the fluid potential \Phi is an energy term, by carrying out a dimensional analysis on the equation \Phi = gz + p/\rho. Do so for both the SI system of units and the FPS system of units.
  2. Three formations, each 25 m thick, overlie one another. If a constant-velocity vertical flow field is set up across the set of formations with h = 120 m at the top and h = 100 m at the bottom, calculate h at the two internal boundaries. The hydraulic conductivity of the top formation is 0.0001 m/s, the middle formation 0.0005 m/s, and the bottom formation 0.0010 m/s.
  1. A geologic formation has a permeability of 0.1 darcy (as determined by a petroleum company for the flow of oil). What is the hydraulic conductivity of the formation for the flow of water? Give your answer in m/s and in gal/day/ft2. What kind of rock would this likely be?
    1. Four horizontal, homogeneous, isotropic geologic formations, each 5 m thick, overlie one another. If the hydraulic conductivities are 10–4, 10–6, 10–4, and 10–6 m/s, respectively, calculate the horizontal and vertical components of hydraulic conductivity for the equivalent homogeneous-but-anisotropic formation.
    2. Repeat for hydraulic conductivities of 10–4, 10–10, 10–4, and 10–8 m/s, and for hydraulic conductivities of 10–4, 10–10, 10–4, and 10–10 m/s. Put the results of the three sets of calculations in a table relating orders of magnitude of layered heterogeneity to resulting equivalent anisotropy.

    1. From the volumetric definitions of porosity and void ratio, develop the relationships given in Eq. (2.40).
    2. Is the porosity ever greater than the void ratio when both are measured on the same soil sample?

  1. The elevation of the ground surface at a soil-moisture measurement site is 300 cm. The soil is a sand and its unsaturated properties are represented by the drying curves of Figure 2.13. Draw a quantitatively accurate set of vertical profiles of moisture content, pressure head, and hydraulic head versus depth (as in Figure 2.12) for a 200-cm depth under the following conditions:
    1. The moisture content is 20% throughout the profile.
    2. The pressure head is –50 cm throughout the profile.
    3. The hydraulic head is 150 cm throughout the profile (static case).

    For cases (a) and (b), calculate the hydraulic gradients and the rates of flow through the profile. For case (c), determine the depth to the water table.

  1. Given a potentiometric surface with a regional slope of 7 m/km, calculate the natural rate of groundwater discharge through a confined aquifer with transmissivity, T = 0.002 m2/s.
  1. Show by dimensional analysis on the equation S = \rho g b (\alpha + n\beta) that the storativity is dimensionless.
    1. A horizontal aquifer is overlain by 50 ft of saturated clay. The specific weight (or unit dry weight) of the clay is 120 lb/ft3. The specific weight of water is 62.4 lb/ft3. Calculate the total stress acting on the top of the aquifer.
    2. If the pressure head in the aquifer is 100 ft, calculate the effective stress in the aquifer.
    3. If the aquifer is pumped and the hydraulic head at some point is reduced 10 ft, what will be the resulting changes in the pressure head, the fluid pressure, the effective stress, and the total stress?
    4. If the compressibility of the aquifer is 10–6 ft2/lb and its thickness is 25 ft, how much compaction will the aquifer undergo during the head reduction in part (c)?
    5. If the porosity and hydraulic conductivity of the aquifer are 0.30 and 10 gal/day/ft2, calculate the transmissivity and storativity for the aquifer. The compressibility of water is 2.1 × 10–8 ft2/lb.

  1. Review the problems that arise in the definition or use of the following classical groundwater terms: potentiometric surface, permeability, and groundwater flow velocity.