CE 473/573 Groundwater
Fall 2007
Homework 4
Due Wednesday October 10
*19. Derive equation (4.44) in Fetter’s book. State all assumptions.
20. Suppose for the steady 1D flow in a confined aquifer the head and flow per unit width
at x = 0 are known (i.e., h = h0 and Q = Q0 at x = 0). Compute the hydraulic head
as a function of x.
*21. For the same flow in a confined aquifer, which has a larger gradient—coarse sand or
silt? For the same gradient in a confined aquifer, which has a larger flow—coarse sand
or silt?
22. Do analysis problem a on p. 147 of Fetter’s book.
23. Do problem 9 on p. 148 of Fetter’s book.
*24. The hydraulic conductivity in a heterogeneous confined aquifer varies as K = K0 (1 −
αx/L). The hydraulic heads at x = 0 and x = L are h0 and hL , respectively.
R
a. Compute the hydraulic head as a function of x. (Hint: du/u = ln u.)
b. Suppose h0 = 40 m, hL = 36 m, L = 4 km, K0 = 5 × 10−6 m/s, and α = 0.9. Plot
the hydraulic head and specific discharge as a function of x.
c. Explain (i) why the absence of K0 in your answer from part a makes sense, (ii)
why the gradient plotted part b is greater in some parts of the aquifer than others,
(iii) why the variation of specific discharge with x in part b makes sense, and (iv)
why your answer from part a is correct by considering the limit as α → 0.
d. (Optional extra credit challenge) Compute the effective hydraulic conductivity of
the aquifer and show that it results as a limiting case of problem 12.
*25. An aquifer of coarse sand is confined by a layer of clay. Why is the flow in the aquitard
nearly vertical if the flow in the aquifer is nearly horizontal?
26. Reconsider the example from lecture of a confined aquifer with variable thickness
b = b0 exp(−ax).
a. Fill in the details of applying the boundary conditions to find the constants.
b. Plot the head as a function of x for b0 = 10 m, L = 5 km, a = 1.5 × 10−4 m−1 ,
h0 = 5 m, K = 8 × 10−7 m/s, and hL = 3.8 m.
c. Compute the flow per unit width at x = 3.141592 km.