# Chapter 5: Problems

1. Consider a saturated, homogeneous, isotropic, rectangular, vertical cross section ABCDA with upper boundary AB, basal boundary DC, left-hand boundary AD, and right-hand boundary BC. Make the distance DC twice that of AD. Draw a quantitatively accurate flow net for each of the following cases.
1. BC and DC are impermeable. AB is a constant-head boundary with h = 100 m. AD is divided into two equal lengths with the upper portion impermeable and the lower portion a constant-head boundary with h = 40 m.
2. AD and BC are impermeable, AB is a constant-head boundary with h = 100 m. DC is divided into three equal lengths with the left and right portions impermeable and the central portion a constant-head boundary with h = 40 m.
2. Let the vertical cross section ABCDA from Problem 1 become trapezoidal by raising B vertically in such a way that the elevations of points D and C are 0 m, A is 100 m, and B is 130 m. Let AD, DC, and BC be impermeable and let AB represent a water table of constant slope (on which the hydraulic head equals the elevation).
1. Draw a quantitatively accurate flow net for this case. Label the equipotential lines with their correct h.
2. If the hydraulic conductivity in the region is 10-4 m/s, calculate the total flow through the system in m-5 m/s (per meter of thickness perpendicular to the section).
3. Use Darcy’s law to calculate the entrance or exit velocity of flow at each point where a flowline intersects the upper boundary.
1. Repeat Problems 2(a) and 2(b) for the homogeneous, anisotropic case where the horizontal hydraulic conductivity is 10-4 m/s and the vertical hydraulic conductivity is 10-5 m/s.
2. Draw the hydraulic conductivity ellipse for the homogeneous, anisotropic formation in part (a). Show by suitable constructions on the ellipse that the relation between the direction of flow and the direction of the hydraulic gradient indicated by your flow net is correct at two points on the flow net.
3. Repeat Problem 2(a) for the case where a free-flowing drain (i.e., at atmospheric pressure) is located at the midpoint of BC. The drain is oriented at right angles to the plane of the flow net.
1. Repeat Problems 1(a), 1(b), and 2(a) for the two-layer case where the lower half of the field has a hydraulic conductivity value 5 times greater than that of the upper half.
2. Repeat Problem 1(b) for the two-layer case where the upper half of the field has a hydraulic conductivity value 5 times greater than that of the lower half.
4. Sketch a piezometer that bottoms near the center of the field of flow in each of the flow nets constructed in Problems 2, 3, 4, and 5, and show the water levels that would exist in these piezometers according to the flow nets as you have drawn them.
1. Redraw the flow net of Figure 5.3 for a dam that is 150 m wide at its base, overlying a surface layer 120 m thick. Set h1, = 150 m and h2 = 125 m.
2. Repeat Problem 7(a) for a two-layer case in which the upper 60-m layer is 10 times less permeable than the lower 60-m layer.
5. Two piezometers, 500 m apart, bottom at depths of 100 m and 120 m in an unconfined aquifer. The elevation of the water level is 170 m above the horizontal impermeable, basal boundary in the shallow piezometer, and 150 m in the deeper piezometer. Utilize the Dupuit-Forchheimer assumptions to calculate the height of the water table midway between the piezometers, and to calculate the quantity of seepage through a 10-m section in which K = 10-3 m/s.
6. Sketch flow nets on a horizontal plane through a horizontal confined aquifer:
1. For flow toward a single steady-state pumping well (i.e., a well in which the water level remains constant).
2. For two steady-state pumping wells pumping at equal rates (i.e., producing equal heads at the well).
3. For a well near a linear, constant-head boundary.