BASIC
CONCEPTS OF FLOW THROUGH POROUS MEDIA
3. 1 Types of Flow through
Porous Media:
The flow of fluids through porous media is usually very slow and hence can be considered as laminar flow. However, in many cases such as around small size covered drains or wells and around corners of embedded impervious objects, the velocity of flow may exceed the lower limits of turbulent flow.
The
fluid flows from a point to another in a porous media if the total energy of the
first point is higher than that of the second. The total energy E of a fluid at
a point lies higher than an arbitrary chosen datum by a distance z and having a
water pressure p while the water moves at it by an absolute velocity v as given
by equation 3.1
E = z + p/g_{f} + v^{2}/ (2g ) (3.1) 
where g_{f } is
the specific weight of the fluid and g is the gravity acceleration.
E= p / g_{f} +z (3.2) 
In 1856, Darcy showed experimentally that the velocity of fluid flow through porous media is linearly related to the hydraulic gradient i. By hydraulic gradient, it is meant the ratio between the difference in the total energy between any two points and the length of the fluid path. In his experiments, Darcy used an apparatus similar to that shown in Fig. 3.1
The equation suggested by Darcy, which is known as
Darcy’s Law has the following form:
v =  ki (3.3) 
where
k is a constant depends on soil and fluid properties, which was previously used
to be called the permeability and now is more frequently referred to as the
hydraulic conductivity.
The hydraulic gradient i = Dh/L, where
L is the length of the soil sample in the pipe. Thus, the fluid velocity
is given by equation (3.4)
v =  k{( Z_{B }+ H_{B})+ (Z_{A}+H_{A})}/L ( 3.4) 
The velocity given by equation (3.4) is the
average velocity assuming that the
area of flow equals the whole cross
sectional area of the tube “a”. Thus, the rate of flow Q can hence be
calculated from the following formula:
Q=
ka{(Z_{B}+H_{B}) (Z_{A}+H_{A})}/L (3.5) 
The actual area of flow equals (a.n) where n is
the two dimensional void ratio perpendicular to the direction of flow. Thus the actual velocity v_{s
}can
be obtained in terms of v and n from the relation :
v_{s} = v/n (3.6) 
Also,
the fluid through porous media flows through tortuous channels as shown in Fig.
3.2. Hence the actual hydraulic gradient i_{s } equals
Dh/L_{t }, where L_{t }equals the actual length of the tortuous
channel.
In
Equation (3.7), is is
written in terms of the average hydraulic gradient i.
is=i/T (3.7) 
where
T= Lt/L and is called the tortuousity coefficient. Using
the given definition of is, the actual velocity of flow through porous
media can be written as follows:
vs= v/(n.T) (3.8) 
Due to the difficulty in calculating T, it is
more practical in design purposes to work with both the average velocity
v and the average hydraulic gradient i.
It
is proven experimentally in numerous works that Darcy’s Law
is valid only for laminar flow through porous media. The laminar flow is
suggested to be that which corresponds to Reynolds number values £
1. In soil hydraulics, the Reynolds number Rn has the following definition
:
Rn = vd/n (3.9) 
where,
v
= average velocity of flow (l/t)
d=
average diameter of soil particles (l)
n= kinematic viscosity of the fluid (l2/t)
Several works are available concerning the flow
through porous media with Rn>1. These
cases of
turbulent flow through porous media can occur in
gravel or
coarse sand soil layers. The relation between the hydraulic gradient
and the
velocity
that governs
this
type of
flow
can be
put in a general power series of order
N.
i=S
Am vm
, m=0,1,2,.....,N
(3. 
where
Am are
constants to be determined experimentally
Actually,
numerous simpler relations are available for turbulent flow in porous media.
Among these, is equation
(3.11) by Leps ( 1973).
v=C R^{0.5} i^{0.54} (3.11) 
where
C=
grain shape and roughness factor
R=
mean hydraulic radius of soil particles which is defined as the ratio between
the void
cross sectional area
perpendicular to the flow and integrated
perimeter length of the void boundaries
It is clear that the application of the above
equation requires a tedious laboratory work in order to obtain a good estimate
for the values of C& R. Otherwise, imperical and less accurate values are
available.
To
understand the phenomenon of quick sand condition, consider the arrangement
shown in Fig. 3.3
A soil sample of cross sectional area A and a
length L is put in a vertical tube with the fluid total head at the sample
bottom higher than that of its top
by the value h. The water flows through the
sample from bottom to top. In case of granular soil particles, this may cause
the vertical movement of the soil grains if the upward force is high enough.
To avoid this, the downward force
which consists of the sample weight should exceed the upward force on the
sample bottom. This force occurs as a result of the difference in total
head between the bottom and top sides of the sample. In other words, if
the side friction with the tube is
neglected, the process of quick sand can be avoided if the following condition
is satisfied:
AL g_{sub} > hAg_{w}_{ }(3.12) 
_{ }where g_{sub}
and g_{w}
are respectively the submerged
specific weight of the soil and the specific weight of water.
The critical hydraulic gradient i_{c} is
defined as that which corresponds to the case in which both sides of relation
(3.12) are equal, or
i_{c} = g_{sub}/g_{w} (3.13) 
This means that if the hydraulic gradient in granular soil exceeds the ratio between the submerged
specific weight of soil and that of water, quick sand condition is
expected to occur.
The g_{sub} is defined in terms of the specific gravity of the solid
particles G and the void ratio e by
the relation:
g_{sub}
= g_{w}(G1)/(1+e) (3.14) 
The
substitution from (3.14) into (3.13) leads to equation (3.15)
i_{c }= (G1)/(1+e) (3.15) 
3.2 Computation of Hydraulic Conductivity in Terms
of Fluid and Soil Properties
The
hydraulic conductivity k is dependent on both the fluid and soil properties. The
density and viscosity of the fluid play the major role from the fluid side. The
list of soil factors which influence the value
of k include among less important factors the following:
· void
ratio
·
shape
and size of soil particles
·
orientation
of soil particles as well as macroscopic and
microscopic
The velocity v of fluid through a capillary tube
with constant cross sectional area a
can be determined according to HagenPoiseuille’s equation.
v = (g_{f}iR^{2})/(8m) (3.16) 
where,
i
= hydraulic gradient
g_{f}=
specific weight of fluid [m/(l^{2} t^{2})]
R=
hydraulic radius (l)
m= coefficient of dynamic
viscosity (m/lt)
This equation is extended to include the flow through soil by introducing the term of soil hydraulic radius
R_{h } which is defined by
the relation:
R_{h}=
(volume of soil voids)/( surface area of soil voids) (3.17)
v
= (g_{f}iR_{h}^{2})/(C_{s}m)
(3.18) 
where,
C_{s
}is a shape factor
The substitution of R_{h} in terms of void
ratio e, Tortuousity T and the surface area per unit volume S into
equation ( 3.18) leads to :
v= [{1/(C_{s }S^{2 } T^{2})}.{g_{f }/m}.{e^{3}/(1+e)}].i (3.19) 
From the analogy of equations (3.3) and (3.19),
the relation for the hydraulic conductivity can be obtained
k= {1/(C_{s}S^{2}T^{2})}.{g_{f }/ m}.{e^{3}/(1+e)} (3.20) 
For granular soil the shape factor C_{s } is
approximately 2.5 while the tortuosity factor T is close to Ö2
. On the other hand, care must be taken in the application of equation (3.20) as
it usually leads to an underestimation of
k values for clay and hence it may be used only to obtain a lower bound
for its value.
3.3
Determination of the Hydraulic Conductivity in the
Laboratory
There
are numerous direct and indirect methods to determine the hydraulic conductive
in the laboratory. What is mentioned here are the most widely used. They are not
necessarily the most accurate.
3.3.1
Constant Head Permeameter
The
undisturbed soil sample is put in a tube and is bounded from the two sides
by two thin porous plates . Both water
levels at the tube ends are kept
constant. The water flowing through
the sample during a certain period of time is collected in a container
and hence the flow rate can be calculated. There are several setups of constant
head permeameter. In Fig. 3.4 one of these setups is presented. According to
Darcy’s Law
v=Q/a=k(H/L) (3.21) 
where
a is the internal cross sectional area of the tube
The hydraulic conductivity k is hence obtained
from the relation:
K=(Q.L)/(H.a) (3.22) 
The use of constant head permeameter suits the
case of soils with high permeability values such as sand. In the case of soil
with low permeability values, the experiment needs a very long time to be
completed or otherwise needs the
choice of very high H values and very small a and L values.
3.3.2
Falling Head Permeameter:
The
falling head permeameter is an arrangement specially designed to suit soils with
low hydraulic conductivity values. It consists of a thick rigid vertical tube
with internal cross sectional area A. The tube contains the undisturbed soil
sample of length L which is bounded from bottom by a thin porous plate and from
the top by an impervious plate.
The
lower part of the tube is immersed in a container containing water, which is
kept at constant level. A vertical thin tube with internal cross sectional area
“ a “ contains water whose level lies initially above the level of the water
in the lower tube by a distance h_{1}. As time elapses the level of
water in the thin tube starts to drop
At time t, the water level has the value h and it falls down by a
distance Dh during a time
interval Dt.
Thus at time t the rate of flow q is given by the relation :
q=a.dh/dt (3.23) 
Also, from Darcy’s Law
q=A.v=Ak.(h/L) (3.24) 
Equating equations (3.23) and (3.24) leads to:
dh/dt =  (hAk)/(La) (3.25) 
Integrating the equation between t=0 and t, the
integral relation (3.26) results.
_{0}ò^{t
} dt
= 
_{h1}ò^{h} 
_{ }
or,
t
= {(aL)/(Ak)}.
ln (h_{1}/h)
(3.27) 
If the measured head at time t_{2 }is h_{2},
equation (3.27) takes after rearrangement the following form:
k= (aL)/(At_{2}) . ln(h_{1}/h_{2}) (3.28) 
Relation (3.28) is used to calculate the hydraulic conductivity._{ }
In
this type of tests, the phenomenon of capillary action is used to obtain the
value of k experimentally. The used apparatus can be as that illustrated in Fig.
3.6.
The apparatus consists of a horizontal relatively
thick rigid tube connected to elevated containers located higher than the tube
centerline by distances h_{1}& h_{2}. The connection
contains two vertical thin pipes, one for each container and a combined thin
horizontal pipe. Each of the vertical pipes includes a valve.
The soil sample is inserted in the horizontal
thick tube and is bounded from both sides by thin porous plates. The side of the
pipe that is not connected to the elevated containers is provided by a short
horizontal thin pipe in order to allow for the drainage of flowing water to the
open air.
Before
the experiment is started both valves are kept closed. At t= 0, the valve A is opened
allowing the water to flow through
the soil creating a front surface
which reaches at time “t” a distance “x” from the inlet side.
The water pressure at a point just before the
water front equals the capillary pressure h_{c}. The actual velocity v_{a
}equals dx/dt while the average velocity can be obtained from Darcy’s
Law.
v= k(h_{1}+h_{c})/x (3.29) 
Since v=n.v_{a}
, where n is the porosity, then
n.dx/dt= k(h_{1}+h_{c})/x (3.30) 
Integrating
equation 3.30 leads to:
n/{k(h_{1}+h_{c})}. _{0}ò^{x} ^{ }x
dx = _{0}ò^{t} 
or,
t = [n/{2k(h_{1}+h_{c})}]. x^{2 }(3.32) 
This means
that t is linearly proportional to x^{2}.
In other words, if the values
of x are measured for different time values, the relation between t and x^{2}
should be a straight line passing through the origin similar to the first part
of the curve shown in Fig. 3.7.
When x reaches about half of the sample length L,
valve A is closed while
simultaneously valve B is opened. Assuming that the time then is t_{1 } and
x= x_{1}, the experiment is continued until the value of x approaches L.
A similar procedure can be adopted to obtain a
relation similar to (3.32). In this case, the initial condition is t= t_{1}
for x= x_{1}, the new relation has the following form :
(t t_{1}) = [n/{2k(h_{2}+hc)}].(x^{2}x_{1}^{2}) (3.33) 
The measurement of x
for different t values and hence, plotting t against x^{2} for x ³ x_{1} produces a straight line
similar to the second part of the curve in Fig. 3.7.
For known values of
x and t, equation (3.32 ) contain 2 unknowns, namely , h_{c} and
k. Assuming that the slope of the first straight line is a and that of the
second is b, then it is easy to prove that:
a=(2k/n).(h_{1}+h_{c}) (3.34) 
&b=(2k/n).(h_{2}+h_{c}) (3.35) 
The simultaneous solution of equations (3.34) and
(3.35) requires only the knowledge of the
porosity n and the determination of a&b from the constructed curve between t
and x^{2}.
3.3.4
Hydraulic Conductivity and Consolidation
Tests:
The
consolidation tests are conducted mainly to study the stress  strain relations
of compressive soils. However, the results can be used to determine the value of
the hydraulic conductivity of the tested soil. This is achieved using the
definitions and equations that are used in the theory of consolidation.
The volume coefficient m_{v} can be
obtained in terms of the initial void ratio e and De which is the change of void ratio due to the application of the
incremental pressure Ds.
m_{v}
= De /{Ds(
1+e)}
(3.36) 
The time factor T_{v} for a certain degree of consolidation can be obtained from table 3.1. The actual time t corresponding to T_{v }is obtained from the test results. Equation (3.37) can be used to calculate the coefficient of consolidation C_{v. }
C_{v }= (T_{v}H^{2})/t (3.37) 
where H is the length of the drainage path.
Table
3.1 Values of
T_{v }for different Percentages of Consolidation
U% 
T_{v} 
U% 
T_{v} 
U% 
T_{v} 
0 
0 
45 
0.159 
75 
0.478 
10 
0.008 
50 
0.197 
80 
0.567 
20 
0.031 
55 
0.238 
85 
0.684 
30 
0.071 
60 
0.287 
90 
0.848 
35 
0.096 
65 
0.342 
95 
1.127 
40 
0.126 
70 
0.403 
100 
®
¥ 
The hydraulic conductivity can finally be obtained
from the following formula:
k = g_{w }m_{v }C_{v }_{ }(3.38) 
where g_{w} is the specific weight of
water.
The previous equations (3.36) to (3.38) are
concerned with one dimensional linear consolidation tests for a homogeneous
isotropic soil. In other cases, the corresponding relations can be applied.
3.4
Field Determination of the Hydraulic Conductivity:
The
value of the hydraulic conductivity that is
obtained in the laboratory for a certain sample
represents only that sample . What is actually needed is a value that
represents a region where a project is to be constructed or some engineering
works have to be executed.
Therefore, despite the ease and relative accuracy
of laboratory results, field tests are often needed in order to have values that
represent the region of interest or
at least part of it Several methods
of field testing are being used to estimate the values of the regional hydraulic
conductivity.
The
interest in this section is restricted to the determination of the regional
hydraulic conductivity of the soil. Conducting steady state
pumping tests can satisfy this purpose. The procedure of executing the
test and the analyses of the test results depend on the condition of the water bearing
formation and the depth of the test well.
3.4.1.1 Pumping Test of a confined Aquifer:
Consider
a horizontal aquifer of uniform thickness D that extends indefinitely in all
horizontal directions and is bounded from top and bottom by impermeable layers.
The head of water H as measured from the bottom surface of the aquifer may
exceed the value of D In this case, the water in the aquifer is defined to be in
artesian condition.
To execute the pumping test, an arrangement
similar to that shown in Fig. 3.8 is used. The used arrangement should include:
i
A vertical pumped well with a
screen that
extends through the whole thickness of the aquifer D
ii
Two vertical piezometers located at distances r_{1 }and r_{2 }from
the well center.
The piezometer bottom should penetrate the aquifer
to a point where the water is allowed to flow freely between the
aquifer and the piezometer. The
distances of the piezometers from the well should be different, i.e. r_{1}¹ r_{2}. Also it is preferable not to
have the well and the two piezometers located on a straight line. This is done
to take care of the influence of regional soil anisotropy.
The well is pumped with a constant discharge Q
until a steady state condition is reached which is noticed when the water heads
h_{1} and h_{2} in the piezometers stop changing with time.
Consider a cylinder of radius r in the aquifer.
Assume that the head at the surface of the cylinder is h, hence the hydraulic
gradient equals dh/dr. The discharge Q can be calculated by Darcy’s Law.
Q=2prD.(kdh/dr) (3.39) 
Integrating equation (3.39) between r_{1}
and r_{2},
{Q/( 2pDk)} _{r1}ò ^{r2}^{ }dr/r =^{ }k^{ } _{h1}ò^{h2} dh (3.40) 
k=
{Q/(2pD)}.
ln(r_{2}/r_{1})/(h_{2}h_{1}) (3.41) 
In the cases in which only one piezometer is
available the well radius r_{w} and the head in the well h_{w}
can respectively replace r_{1} and h_{1 }of equation (3.41).
However, this is not recommended as the well losses influence
the calculation accuracy in
this case.
Moreover, the radius of influence R and the
original head H can be used in place of r_{2} and h_{2}
respectively. In this case, equation (3.41) reduces
to the form:
k
= {Q/(2pD)}. ln(R/r_{w})/(Hh_{w})
(3.42) 
The application of equation (3.42) requires the
knowledge of the value of the radius of influence at the time in which the
steady state condition is practically reached. Although the application of this
formula has the advantage of canceling the need for observation wells, it is not
recommended to use it in projects that need reasonable degree of computation
accuracy.
This is due to the reason mentioned before about
well losses as well as the fact that
most available relations for R determination lead often to very
approximate or even misleading
results. Actually, most of these relations are developed empirically and their
application is recommended to be done with precaution.
3.4.1.2 Pumping Test in an Unconfined Aquifer:
Consider
the water bearing formation shown in Fig. 3.9 which is bounded from the bottom
by an infinite horizontal impermeable surface. The original water table in the
aquifer lies a distance H above the impermeable layer.
Similar to the case of the confined aquifer, a well with radius r_{w}
and 2 piezometers are needed to
conduct the test. The piezometers are located at distances r_{1 }and r_{2
} from the well. The conditions
mentioned in section 3.4.1.1 regarding the location and depth of the well and
piezometers apply here .
Here also, the well is pumped until a steady state
condition is reached. The final water levels in the piezometers are measured and
hence h_{1 }and h_{2 }can be measured.
Assuming that the head at a cylinder of radius r
is h, the hydraulic gradient of flow is dh/dr. Applying Darcy’s Law as in the
case of section 3.4.1.1
Q = (2prh).(kdh/dr) (3.43) 
Integrating between r_{1} and r_{2},
Q/(2pk)_{
r1}ò^{r2} dr/r
= _{h1}ò^{h2} 
Thus,
k = {Q ln(r_{2}/r_{1})}/ {p( h_{2}^{2}  h_{1}^{2})} (3.45) 
Using the heads at the
radius of influence R and at well radius r_{w} instead of those
at r_{1} and r_{2}, the equation (3.46) results.
k = {Q ln(R/r_{w})}/{ p( H^{2} h_{w}^{2})} (3.46) 
In some cases, the well does not penetrate the
whole unconfined aquifer as illustrated
in Fig. 3.10 In this case, the value of k can be obtained for partially
penetrated wells from equation (3.47). Other relations are also available.
k={Q ln(r_{2}/r_{1})}/[ p{(h_{2} s)^{2} (h_{1}s)^{2 }}.{1+ (0.3+10r_{1}/h_{2}).sin (1.8s/h_{2})}] (3.47) 
where s is
the distance between the bottom of the well and the impermeable layer.
These
kinds of tests are much simpler to execute and less expensive than pumping
tests. However, the obtained results represent only a small region around the
borehole location while the pumping tests results represent the whole area of
influence of the pumped well.
A
borehole of diameter d is excavated to about the center of the layer whose
hydraulic conductivity is to be estimated. A pipe casing of the same diameter d
is used to line the borehole sides leaving the bottom unlined to allow the flow
of water as shown in Fig. 3.11.
Water is added through the pipe with a constant
rate Q which keeps a constant level of water in the pipe
. If the difference between the water
level in the pipe and the groundwater table is h, then k can be obtained from
relation (3.48)
k = Q/(2.75d.h) (3.48) 
This
test gives good results if the tested aquifer has a thickness > 10 d and the
bottom of the hole is kept clean through out the experiment. If the hydraulic
conductivity is very low or the groundwater table is close to the ground
surface, the water can be pumped through the tube under pressure p and hence h of equation (3.48) should be
taken equal to the pumping pressure head.
3.4.2.2
Variable Head Tests
For
very thick layers, the variable head tests
are more suitable than the constant head tests. As in the case of
constant head tests, the borehole is required to be deep enough in order to approach the stratum center. Two
possible arrangements of variable head tests are illustrated in Fig. 3.12.
In Fig. 3.12 a, a borehole is dug to a few meters below the groundwater
level. The sides of the hole are
lined with a pipe with the same diameter
d leaving only the bottom of the hole unlined. In Fig. 3.12b, the other
arrangement is introduced. In that arrangement, the pipe lines only the top part
of the hole leaving a length L > 4d of the hole unlined.
In both arrangements, a quantity of water is taken
from the hole. The water in the hole starts
to rise. The time t required for the water in the hole
to rise from h_{1} to h_{2} is measured.
In case
of arrangement (a), the hydraulic conductivity can be obtained from
equation (3.49).
k = {(pd)/(11t)}.ln(h_{1}/h_{2})
(3.49) 
On the other hand, equation (3.50) applies for
arrangement (b)
k
= {d^{2}/(8Lt)}. ln(2L/d).ln(h_{1}/h_{2})
(3.50) 
This
type of tests can be used to determine the hydraulic conductivity for a certain
strip of soil of thickness L at any level above or below the groundwater level.
A borehole is excavated to the bottom surface level of the tested strip. No pipe
lining of the hole is required.
A packer is inserted in the hole at the level of
the top surface of the strip. A packer is a sealing device, which forms a
watertight contact against the sides of the borehole, which prevents the water
from moving from one side of the packer to the other.
In Fig. 3.13,
the described arrangement is illustrated. In case a, which is shown in
the L.H.S.(left hand side), the groundwater
level is assumed to be above the level of the tested strip while in the
R.H.S., the ground water level is below the level of the strip.
If
a strip lies higher than the borehole bottom is required to be tested,
two packers are needed, one at the bottom surface and one at the top
surface of the strip. The cases of groundwater level above and below the strip are
shown in Fig. 3.14 a,b.
The water is pumped with a rate Q in the confined
region of the borehole and due to the existence of packers, water flows out only
to the tested strip of the soil. The hydraulic conductivity is given by
relations (3.51) and (3.52):
k={Q/(2 pLh)}.log(2L/d) for L ³5d (3.51) 
and,
k = {Q/(2pLh)}.sinh^{1 }(L/d) for 5d >L ³d/2 (3.52) 
Where
,
k=
hydraulic conductivity of the soil strip (l/t)
L=
length of tested soil strip (l)
Q= discharge (l^{3}/t)
h=
difference in head (l)
d=
diameter of borehole(l)
The above equations are applicable whether the
groundwater level is higher or lower than the strip level. However, care
must be taken that in case of lower groundwater level pressure is assumed to be
atmospheric at the center of the tested strip.
1) A falling head permeameter test is conducted on a soil sample of length 200 mm and diameter 100 mm. The water level in the thin tube, which was originally 500 mm above that in the downstream basin, dropped by 200 mm in 20 min. Calculate the hydraulic conductivity if the diameter of the thin tube is 15 mm. How much time is needed for the water to drop more 150 mm.
Answer : k=1.92x10^{6} m/s,
t= 14 min.
2)
A sample of the soil, whose hydraulic conductivity equals 1.8x 10^{7}
mm/s, is tested in a consolidation
apparatus. At an incremental pressure of 1.5Kg/cm^{2}, the void ratio
reduced from 0.90 to 0.89, calculate the coefficient of consolidation of the
soil. If the sample in the test is 100 mm thick and is allowed to drain from the
top only, Calculate the time required for the sample to reach 70% and 95%
consolidation.
Answer
c_{v} = 0.04 m^{2}/d, t_{1} =2.42 hr, t_{2}
= 7.6 hr
3)Excavation is conducted to a depth of 5.00 m below the ground level in a sandy soil with thickness is 7.00 m, saturated specific gravity = 1.70 and a groundwater level 5.00m below the ground level. The sand is overlaying a thin clay layer that confines an artesian aquifer with pressure head that exceeds the ground level by 3.00 m. Test the excavation against quick sand condition. Calculate the thickness of plain concrete layer to be placed above the excavation bottom to overcome the quick sand condition.
4)
A soil sample is tested under 10.00 m head in a vertical tube with diameter 20
cm. The sample length is 15.00 cm. The rate of flow is found to be 20m^{3}/s.
The average soil particle diameter
is 3 mm, the shape and roughness factor is 1.7 and the fluid viscosity
is 1.1x10^{5}gm.s/cm^{2}.
a)
Does the
soil obey Darcy's Law
b)
Calculate
the hydraulic radius of the soil specimen
5)The
following table gives the results of a capillary test:
T(min.) 
X(mn) 
Remarks 
0 
0.00 
*
CONTAINER A (200 mm ABOVE SAMPLE) OPENED, CONTAINER B
KEPT CLOSED 
10 
48 

20 
72 

30 
85 

40 
103 
*
CONTAINER A CLOSED, CONTAINER B (300 MM ABOVE SAMPLE) OPENED 
50 
122 

60 
142 

70 
158 

80 
174 
Calculate
the hydraulic conductivity of the soil and the capillary head assuming that the
porosity is 0.35.
Answer
: k= 0.57 m/d
6)
A pumping test is conducted on an unconfined aquifer using a discharge rate of
1.5 m^{3}/s from a full penetrating well. The original thickness of the
groundwater body is 10 m. After reaching a steady state condition, the water
level dropped by 1.50 m in the well. At 2 observation wells located 5.00 m and
20.00 m, the water level dropped by 0.55
m, and 0.04 m respectively.
Calculate the hydraulic conductivity, the radius of the well and the radius of
influence.
Answer:
k=0.0669 m/s, r_{w }= 0.46 m, R_{e} = 22.38 m
7)
A partially penetrating well having the same diameter of the well in problem(6)
and extends 5.00 m below the
groundwater level of the same aquifer. When the well is
discharged with the same rate, a drawdown in the well of 3.00 m took
place. Calculate the radius of
influence.
Answer:
R_{e} =51 m
8)
A borehole with diameter 150 mm is excavated to a depth of 10 m in a thick
aquifer. The groundwater level is 1.00 m below ground level. A pipe is lowered
to the end of the hole. A quantity of water is removed from the hole . It is
recorded that the water in the whole needed 20 min. to rise from 200 mm to 100
mm below groundwater level. After reaching
the steady state condition
the pipe in the hole is raised 3 m and another amount of water is removed
from the pipe . Calculate the time
needed for the water surface in the pipe to
travel the same distance.
Answer:
k=1.48x10^{3} m/min, t=1.62 min
9) A trench is to be dug in a location
calculate
the maximum possible depth
for
clay: e=0.4,
g_{s}
=2.82,
water
content = 16% , thickness=11.00 m
for sand: water pressure below clay layer = 7.00 m