CVE 464/564 Groundwater Hydrology - HW1 (100)
Due: 6 pm, Wed, Sep 8, 2021
1
CSU ID#: Last Name: First Name:
A. Attach this problem sheet as a cover on top of your answer sheets.
B. Clearly write your calculation procedures and answers in your answer sheets.
1. A soil particle size distribution is given below. The total porosity of the soil sample was
estimated as 0.4. Estimate both the hydraulic conductivity (K) for water and intrinsic
permeability (k) of the soil using Hazen (1911), Harleman et al. (1963), and Kozeny-Carmen
Bear XXXXXXXXXXequations given below. Use the relationship between K and k when you have
only either K or k equation. Fill out the table to compare the values you calculated. Assume
the water temperature is 15 °C and clearly specify your related variables (e.g., density,
dynamic viscosity, etc.), and be careful of unit.
- Hazen XXXXXXXXXXhydraulic conductivity: K (cm/s) = ???10
2
, where CH = 120 s
-1·cm-1, d10 in cm
- Harleman et al XXXXXXXXXXk (cm2) = 6.5E-4×(d10)
2, where d10 in cm.
- Kozeny-Carmen Bear (1972): k (cm2) =
?3
180∙(1−?)2
??
2
, where n = porosity, dm = d60 in cm.
Equation k (cm2) K (m/d)
Hazen
Harleman
Kozeny-Carmen Bear
0
10
20
30
40
50
60
70
80
90
100
XXXXXXXXXX
Grain size, mm
P
e
c
e
n
t
p
a
s
s
in
g
y
w
e
ig
h
t,
%
CVE 564 Groundwater Hydrology - HW1 (100)
Due: 6 pm, Wed, Sep 8, 2021
2
2. Consider the two formations of five different soil layers with their own hydraulic
conductivities. The values of K1 to K5 are 5, 10, 8, 25, and 2 m/day, respectively. The
thicknesses (b) of them are 3, 2, 6, 4, and 1 m, respectively. Assume that a concrete layer is
impermeable. Perform the following tasks.
(1) Prove that the equivalent hydraulic conductivity KH =
( )
=


i i
h
i
K
K
for the following formation, and compute
the KH using the values given (Hint: The hydraulic
gradient i is constant for each soil layer).
(2) Prove that the equivalent hydraulic conductivity KV =
( )∑
∑
ii
i
K
for the following formation, and compute the
KV using the values given (Hint: The flux q is constant
for each soil layer).
3. Consider the following formation of five different soil layers.
K1 to K5 are 5, 10, 8, 25, and 2 m/day (same as Question 2).
Perform the following tasks.
(1) Prove that
1+1+ tan
tan
=
i
i
i
i
α
α
K
K
(Hint: The flux q is constant in
vertical flow and the hydraulic gradient is constant in
horizontal flow).
(2) Compute all αi and draw the flow path from layer 1 to 5
assuming the flow direction in the upper most layer (α1) is
25° under steady flow conditions. Display an inclined flow
angle for each layer.
4. The volume of a wet soil sample is 2000 cm3. The volume of water drained from the sample
y gravity was 500 cm3. After drying the drained soil in the oven, the volume was 1350 cm3.
Compute the specific retention, specific yield, and porosity of this soil.
K1
K2
K3
K4
K5
Concrete
Concrete
h1
h2
L
K
1
K
2
K
3
K
4
K
5
Concrete
Concrete
h1
h2L
K1
K2
K3
K4
K5
q1 α
CVE 564 Groundwater Hydrology - HW1 (100)
Due: 6 pm, Wed, Sep 8, 2021
3
5. A ground water basin with an area of 10 mi2 has two aquifer systems; (1) an upper unconfined
aquifer with 200 feet thickness and the specific yield of 0.2, and (2) a lower confined aquifer
with an available head drop of 100 feet and the storage coefficient of XXXXXXXXXXEstimate the
amount of recoverable water from this ground water basin.
6. The following hydraulic conductivity K (m/s) were tested from a geological layer.
4x10-8 8x10-9 2.5x10-8 2x10-10
1x10-7 1x10-7 2x10-7 1x10-7
3x10-8 1x10-7 2x10-7 8x10-8
4x10-7 1x10-8 5x10-8 4x10-8
1x10-6 8x10-7 2x10-7 3x10-7
1.5x10-7 5x10-8 2.5x10-8 6x10-7
6x10-8 4x10-7 1x10-7 2.5x10-7
1.5x10-6 5x10-7 1x10-6 4x10-8
Perform the following tasks.
(1) Compute the arithmetic mean, geometric mean, and harmonic mean of K values.
(2) Plot the frequency histogram for both K and ln(K) values using a bin size 1.5E-7 m/s and 2,
espectively. Which histogram is more close to a normal (Gaussian) distribution? Note that ln
is the natural logarithm.
(3) If the distribution of ln(K) is close to a normal distribution, calculated the mean K value.
Note that Kavg = exp(E[ln(K)]), where E[ · ] represents the expectation operation.
(4) Which averaging method would give a more reasonable mean?