CHAPTER 3

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.                                       

3.1.1 Laminar 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/gf + v2/ (2g )   (3.1)

where gf  is the specific weight of the fluid and g is the gravity acceleration. In the flow through porous media, v is very small and hence   v2/2g<<(p/gf+z). Thus, for all practical purposes, this term can be neglected and equation (3.1) reduces to the following form, E @ p / gf + z, or simply,

   E= p / gf +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. In Fig. 3.1, the water pressure at point B is less than that at point A. In spite of that, the fluid flows from point B to point A, since point B has the higher total energy as       (ZA +HA) < (ZB+HB ).

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{-( ZB + HB)+ (ZA+HA)}/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{(ZB+HB) -(ZA+HA)}/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 vs can be obtained in terms of v and n from the relation :  

       vs = 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 is  equals Dh/Lt , where Lt 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.                            

3.1.2 Turbulent Flow

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              (3.10) 

  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 R0.5 i0.54                     (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.           

3.1.3 Quick sand Condition                                           

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 gsub >  hAgw                       (3.12)

  where gsub and gw are  respectively the submerged specific weight of the soil and the specific weight of water.

The critical hydraulic gradient ic is defined as that which corresponds to the case in which both sides of relation (3.12) are equal, or

   ic = gsub/gw                   (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  gsub  is defined in terms of the specific gravity of the solid particles G and the void ratio  e by the relation:

  gsub = gw(G-1)/(1+e)            (3.14)  

  The substitution from (3.14) into (3.13) leads to equation (3.15)

   ic =  (G-1)/(1+e)                      (3.15)

    The later is the more common relation used for ic.

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  composition of soil structure. 

The velocity v of fluid through a capillary tube with constant cross sectional area  a    can be determined according to Hagen-Poiseuille’s equation.  

 

  v = (gfiR2)/(8m)                           (3.16) 

                                                                                  

where,

i = hydraulic gradient

gf= specific weight of fluid [m/(l2 t2)]

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 Rh  which is defined by the relation:

Rh= (volume of soil voids)/( surface area of soil voids)   (3.17) The velocity of flow is hence given by the relation  

     v = -(gfiRh2)/(Csm)                (3.18)  

 

where,

Cs is a shape factor

The substitution of Rh in terms of void ratio e, Tortuousity T and the surface area per unit volume S into  equation ( 3.18) leads to :

  v= -[{1/(Cs S2  T2)}.{gf /m}.{e3/(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/(CsS2T2)}.{gf  m}.{e3/(1+e)}     (3.20)

For granular soil the shape factor Cs  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 h1. 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.     

  0òt dt =  -   h1òh  (aL)/(A.k)/(h)dh         (3.26)

    

or,  

 

   t  = {(aL)/(Ak)}. ln        (h1/h)        (3.27)  

If the measured head at time t2 is h2, equation (3.27) takes after rearrangement the following form:  

 

    k= (aL)/(At2) . ln(h1/h2)        (3.28)

   Relation (3.28) is used to calculate the hydraulic conductivity.

3.3.3 Capillary Test:-                                                  

 

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 h1& h2. 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 -hc. The actual velocity va equals dx/dt while the average velocity can be obtained from Darcy’s Law.

 

  v=  k(h1+hc)/x                (3.29)

Since  v=n.va , where n is the porosity, then

  n.dx/dt= k(h1+hc)/x         (3.30)  

                                                                 

Integrating equation 3.30 leads to:

  n/{k(h1+hc)}. 0òx   x dx =   0òt dt    (3.31)  

   or,  

 

  t = [n/{2k(h1+hc)}]. x2           (3.32)

                    

This  means that t is linearly proportional to x2.  In other words, if the    values of x are measured for different time values, the relation between t and x2 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 t1  and x= x1, 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= t1 for x= x1, the new relation has the following form :  

  (t- t1) = [n/{2k(h2+hc)}].(x2-x12)            (3.33)

                                           

The measurement of x  for different t values and hence, plotting t against x2 for x ³ x1 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 , hc 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).(h1+hc)               (3.34)

 

  &b=(2k/n).(h2+hc)              (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 x2. 

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 mv 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.  

  mv = De /{Ds( 1+e)}           (3.36)

  The time factor Tv for a certain degree of consolidation can be obtained from table 3.1. The actual time t corresponding to Tv is obtained from the test results. Equation (3.37) can be used to calculate the coefficient of consolidation Cv.  

   Cv = (TvH2)/t                  (3.37)

 

where H is the length of the drainage path.

 

Table  3.1   Values of  Tv for different Percentages of Consolidation  

 

U%

Tv  

U%  

Tv  

U%  

Tv  

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 = gw mv Cv                      (3.38)

where gw 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.

3.4.1 Well Pumping Tests:-                                           

 

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 r1 and r2 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. r1¹ r2. 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 h1 and h2 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 r1 and r2,                                                 

  {Q/( 2pDk)} r1ò r2   dr/r = k    h1òh2 dh        (3.40)

  or,       

  k= {Q/(2pD)}. ln(r2/r1)/(h2-h1)              (3.41)  

                                                                     

In the cases in which only one piezometer is available the well radius rw and the head in the well hw can respectively replace r1 and h1 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 r2 and h2 respectively. In this case, equation (3.41) reduces  to the form:  

  k = {Q/(2pD)}.  ln(R/rw)/(H-hw)         (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 rw and 2 piezometers are  needed to conduct the test. The piezometers are located at distances r1 and r2  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 h1 and h2 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 r1 and r2,                      

   Q/(2pk) r1òr2  dr/r = h1òh2  hdh   (3.44)

 Thus,  

 

   k = {Q ln(r2/r1)}/ {p( h22 - h12)}              (3.45) 

                               

Using the heads at the  radius of influence R and at well radius rw instead of those at r1 and r2, the equation (3.46) results.  

   k = {Q ln(R/rw)}/{ p( H2- hw2)}             (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(r2/r1)}/[ p{(h2 -s)2- (h1-s)2 }.{1+ (0.3+10r1/h2).sin (1.8s/h2)}]                                                              (3.47)

 

where s  is the distance between the bottom of the well and the impermeable layer.

3.4.2 Open Borehole Tests:-                                  

 

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.

3.4.2.1 Constant Head Test :-

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 h1 to h2 is measured.

In  case  of arrangement (a), the hydraulic conductivity can be obtained from equation (3.49).

  k = {(pd)/(11t)}.ln(h1/h2)               (3.49)  

   On the other hand, equation (3.50) applies for arrangement (b)  

  k = {d2/(8Lt)}. ln(2L/d).ln(h1/h2)            (3.50)  

3.4.3. Packer Tests:-

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 (l3/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.              

                    Problems on Chapter 3:-

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/cm2, 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  cv = 0.04 m2/d, t1 =2.42 hr, t2 = 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 20m3/s. The average soil particle  diameter is 3 mm, the shape and roughness factor is 1.7 and the fluid viscosity  is 1.1x10-5gm.s/cm2.

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 m3/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, rw = 0.46 m, Re = 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: Re =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 with the soil  profile shown in figure clay                          

 

                                                       

calculate the maximum possible depth of the trenchwithout the danger of sand quick sand condition   

for clay: e=0.4, gs =2.82,

water content = 16% , thickness=11.00 m

for sand: water pressure below clay layer = 7.00 m