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A simple method of ground water direction measurement in a single bore hole
Journal of Hydrology 12 (1971) 387-410; © North-Holland Publishing Co., Amsterdam Not to be reproduced by pbotoprint or microfilm without written permission f r o m the publisher
A SIMPLE METHOD MEASUREMENT
OF GROUND WATER DIRECTION IN A SINGLE BORE HOLE
MOHAMMAD SALEEM
Physicist, Irrigation Research Institute, Lahore, Punjab (West Pakistan) Abstract: A simple technique to find both the magnitude and direction of ground water flow in a single bore hole, sited in a pervious water bearing medium, is described. The experiment may be performed with tracers which may not be radioactive (like inactive potassium bromide). At any given horizon where velocity measurements are required, the strainer of the bore-hole is divided into three regions, each about 10 cm long. The central region has slits all round as is usually the case for a strainer. The top region consists of two equal semicircular halves, each subtending an angle of 180 degrees at the axis of the strainer, such that one half has slits cut in it as usual while the second half is kept all blind during manufacture. The horizontal line separating the perforated and the blind segments is taken as the reference line. The lower region is similarly constructed as the top one except that its reference line leads or lags behind the reference line of the top region by an angle of 45 degrees. For proper separation of the above three regions, there are buffer zones between any two consecutive regions. Inactive potassium bromide solution is injected in all the three regions. Using three independent brass electrodes, one in each of the three regions, electrical conductivities at regular intervals of time are carried out by using low frequency and low voltage alternating current, the common electro de being the strainer itself. The magnitude of flow can be determined from the information collected from the central region; whereas the data from the top and lower regions give the direction of flow. Introduction W u r z e l and W a r d 1) have a n n o u n c e d a single well t ech n i q u e for the measu r e m e n t o f direction o f g r o u n d w a t e r flow by using special r a d i o a c t i v e tracers which readily a d s o r b on the m e tal surfaces. U si n g a special technique, a tracer is injected very slowly into the bore hole (an o b s e r v a t i o n well with p e r f o r a t e d screen). T h e injected tracer is slowly carried away, by the water currents, in the direction o f flow t o w a r d s the well screen. Th e a d s o r p t i o n o f the tracer is a l lo w e d n o t on the m a t e r i a l o f the screen b u t on a wire-gauze cylinder (length a b o u t 60 c m and d i a m e t e r slightly less t h a n the well screen). A f t e r some time, this cylinder is r e m o v e d f r o m the well an d rushed to the l a b o r a t o r y where it is scanned to find the relative a d s o r p t i o n o f the tracer on the wire gauze. T h e d ir e c t io n o f flow is the direction in which the ad so r p tion o f the tracer on the gauze is m a x i m u m . 387
388
MOHAMMAD SALEEM
A m o n g the non radioactive tracer techniques, the author 2) has described a technique in which the ground water flow is treated as a vector quantity. Using inactive potassium bromide solution as a tracer, the magnitude of flow can be found by measuring the rate of dilution of this tracer in the well by recording the electrical resistance of the saline column. The reciprocal of the resistance is a measure of the electrical conductivity. As regards the direction of flow, this can be evaluted by introducing another vector quantity of known magnitude and direction and then finally measuring the magnitude of the resultant flow in the same observation well where the original experiment was performed. This additional vector in the ground water movement can be introduced by pumping a small tube well at a known constant discharge in the vicinity of the observation well. This technique is subjected to the limitations that it is applicable only to the homogeneous and isotropic media. Secondly it is not strictly a single well technique because an additional tube well is also required for direction measurements. The present technique is strictly a single well non radioactive tracer technique and is a considerable advancement over the one already mentioned.
Experimental set up First of all a test bore is made at the site where velocity measurements are required. This would help in deciding the slit size of the strainer to be used in the bore hole and also for proper location of the observation points. A 10 cm diameter brass strainer is selected whose length should be equal to the length of the sand column to be encountered during installation. This strainer should be coupled with an equivalent diameter blind pipe whose lengths should correspond to the depths of any clay layers occurring in the strata. Thus the strainer should only be installed in the sandy regions and the blind pipe should take care of the clayey regions of the bore hole. Before installation, the brass strainer is laid on a level ground and all its lengths are coupled together to form one single length. The blind pipe is also coupled along with it. Starting from the upper end of the blind pipe we mark off lengths, which will correspond to depths from the top, where velocity measurements will be required. At each marked point we divide the screen into three regions, each about 10 cm long. The upper and lower regions are separated from the central one with the help of two buffer zones, each about 6 cm wide. No slits are cut in these zones during manufacture of this strainer piece. There are two more buffer zones one above the top region and the other below the lower region (Fig. 2). The central region has slits all-round. The upper region is divided into two semi-circular halves such that one half
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
389
has slits as usual but the second half is kept all blind during manufacture. Except for the selected points, where the strainer is specially prepared as explained above, the rest of the strainer length has slits as usual. The line AB (Fig. 1) separating the two halves is called the reference line. The lower region is similarly constructed except that its reference line A'B' makes an angle of 45 degrees with the line AB of the upper region. The line AB is taken as the standard reference line. To know the position of the line AB at the top we draw two parallel lines over the strainer surface along the length such that one line starts from the end A while the other from end B and run right upto the top end of the blind pipe. P E R F O R A T E ~
I~
- -- - ~ \
// /
",\
/
k
\
z
\
it A
B
....................
( a ) UPPER REGION
ALL PERFORATED -'2 / / /
f
\
"\ \ \ I
I
/
\
/ '-,\ /
~"~. ~ _ ~
-/"/"
y...
( b ) CENTRAL REGION
,\C PERFORA EO .
BL ND
/
....
"~.\
I B
II
.)"g
(C)LOWER REGION
Fig. 1. Diagram showing the three regions: upper, central and lower. The central being perforated all round while the upper and lower are identical except that the reference lines AB and A'B' make an angle of 45 degrees with each other. The reference line separates the two halves such that one half is all perforated while the other half all blind. The length and diameter of each region being 10 cm.
390
MOHAMMAD SALEEM
Similar regions are constructed at other selected horizons taking care that the lines AB and A'B' at all the horizons are kept individually parallel. THREE CORE SUSPENSION
CABLE BLIND PIPE
BORE HOLE STRAINER
j'
'ATER :ABLE
.
__~.__~.!
i
I
I
I
SUPPORTING HOOK
CENTRAL REGION
INSULA]ED WIRES
- ~ I~
BUFFER ZONES
LOWER REGION BRASS ELECTRODES ~
I h ~
\\\
I I
SOLID SUSPENSION ROD
~1-~ I I
I I I
SUSPENSION WEIGHT I
I I
BAIL PLUG
t i
I ~
PERSPEX DISCS DIA SLIGHTLY LESS THAN THE STRAINER
.....|
~\~'RESTING
ROD
lr--
I I
I
Fig. 2. Diagram showing the details of the bore hole along with the arrangement for measuring the conductivity in the three regions. There is a buffer zone between any two consecutive regions having plastic discs with diameter slightly less than the bore hole and thickness 6 cm. The bore hole strainer is also blind round each of the four plastic discs. The three regions are occupied automatically in the bore hole with the help of a suspension weight, connected with a solid suspension rod to the conductivity measuring assembly, when it rests on a horizontal resting rod installed in the bore hole strainer. The conductivity in the three regions is measured independently with brass electrodes, the common electrode being the brass bore hole strainer itself.
A SIMPLE METHOD OF G R O U N D WATER DIRECTION MEASUREMENT
391
While selecting the observation points at different horizons, it should be kept in mind that all the three regions at a given horizon remain in the same type of strata. Velocity measurements at more than one horizon are carried out to account for any heterogeneous conditions in the strata. To start the experiment, potassium bromide solution is injected as tracer in the bore hole at each selected horizon over a range of about 2 meters in the form of a cloud. The proper mixing of the tracer is done, at each of these depths either by a small propeller or by any other suitable mechanical means. The electrode assembly shown in Fig. 2 is then lowered into the bore hole. The assembly illustrated in this figure is only for one particular selected depth. Similar assemblies may be constructed for other selected horizons. There is a resting rod about 1 cm thick, fixed beforehand in the strainer along one of its diameters near the bottom end shown. For smooth lowering of the electrode assembly a heavy suspension weight attached to a solid rod (about 1 cm diameter and 25 cm long) is screwed into the lowest perspex disc as shown in Fig. 2. After lowering of the assembly when the suspension weight rests on the resting rod, the three electrodes lie automatically in their respective regions while the four perspex discs occupy their respective buffer zones. In case there are more observation points in the bore hole, then the various electrode assemblies are inter-connected by suspension rods of appropriate lengths so that when the suspension weight rests on the resting rod, the various electrodes and perspex discs automatically occupy their respective pre-designed positions. To take the observations, we simply record the electrical conductivities (reciprocal of electrical resistance of the saline column) versus time for each electrode, the c o m m o n electrode being the conducting bore hole itself. A Wheatstone bridge coupled with a two stage transistorised A.C. amplifier with a variable gain may be used for resistance measurements. The alternating current power supply at about 6 volts may be obtained from a vibrator system run b) a D.C. storage battery. The observations obtained with each electrode are plotted separately on semi-log papers with conductivity C on log scale and time t on linear scale. The curves that we get are all similar except that their slopes are different. There is a dip in the beginning z) (due to additional factors like diffusion in the horizontal direction and adsorption etc., responsible for extra dilution of the saline columns over and above the dilution due to flow) and then afterwards the curves become straight lines. By extrapolating the straight portions of the curves in the back-ward direction we get the conductivity C o at time t equal to zero in each case, had there been no extra dilution due to the additional factors mentioned above. The observations recorded by the central electrode will give us the magnitude of flow while the data obtained from the upper and lower regions will together
392
MOHAMMAD SALEEM
furnish the direction of flow. Now we proceed with the mathematical analysis of the technique involved.
Mathematical apparatus To start with let the bore hole consist of a simple strainer with outer and inner radii as al and o 2 respectively. Let K1 = Permeability of the aquifer (sandy region, called region (1)) in which the strainer has been installed, K2 --- Permeability of the strainer material (called region (2)) in which slits have been cut, I = Uniform natural gradient under which flowis taking place, v = KII = Unperturbed velocity of subsoil flow in region (1). Then at any field point P in region (1) with polar co-ordinates (r, 0), the pressure ~Pl is given* by
{ K2(a~+a2)-K~ (a~-a2)a2 =
K2
+
+ K,
r}IcosO
(1)
r
and if the point P is taken in region (2):
2a2K1 (a~ ) sP2=K~(aZ_a2)+Kz(a~-+a22).\r-r
I cos0.
(2)
If the strainer is replaced by a blind pipe with the same inner and outer radii then the pressure distribution in the two regions is given by bPt = --
(°: :,) +
bpZ -- aZ _ az
I al cos0
(3)
- r
(4)
(o))
I cosO.
Now let us consider the case when the bore hole is neither a pure screened well (perforated all round) nor a simple blind pipe but instead consists of two semi-circular equal segments such that one is perforated while the other is blind (Fig. 3). The reference line AB is taken along the direction of maximum ground water gradient. Let the axis of the well pass through the origin O. Take any field point P with polar co-ordinates (r, 0) in region (2). Now when the point P is in region (2a) then the macroscopic velocity * See A p p e n d i x 1
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
393
~V, in the r-direction is given by: sV, = - K2 a7 (sp2)
2aZKiK2
(a~ )
=Ki(a~--a~)+K2(a~+a~) \r z + 1
Icos0.
(5)
When the point P moves into region (2b) then the macroscopic velocity bV, in the r-direction is zero as the permeability of this region is zero too. Keeping r fixed let the point P move in the entire region (2). The velocity distribution curve obtained in this case may be considered to be the product of cos 0 with the velocity distribution curve shown in Fig. (4). PERFORATED
,/
i A
/
//(2o,// i
[
/
o/,.~ ;I ~/
X/f'r,e) ~,
\ \\ ~
. . . . . . . . . .
\ MAX MUM " OP-OUND WATER, GRADIENT
"~"BLIND
Fig. 3. D i a g r a m showing the details of the upper region. The u p p e r half is all perforated while the lower h a l f is all blind. The line A B separating the two halves is taken as the reference line.
A
ZERO LEVEL
. 0 Fig. 4.
1 IT 0 --
2~
Diagram showing the velocity distribution curve resembling a square wave, having period 2~r, usually generated in electrical circuits.
394
MOHAMMAD
SALEEM
Writing:
2a2vK2 A=-Kl(a~_a~)+K2(a
{a~ 2 + a 2 z) ~r 2 + 1
) (6)
Eq. (5) becomes sVr = A cos 0.
(7)
To find the interference effect of the regions (2a) and (2b), we apply Fourier Analysis to Fig. 4. The velocityf (0) at any point may be expressed in the form of a series as f(0)=½ao+
Z (a, c o s n 0 + b , sinn0)
(8)
n=l
where ao, a, and b. are constants to be determined. As the curve of Fig. 4 has a discontinuity when 0 equals re, we have a, = 0 ,
for a l l n , 2re
ao =
f (0)'d0 o
(9)
2n
b, =
f (0) sinnO.dO
7r o
Now
b, =
sin nO.dO + 0 7r o
=
A
(1 - cos nrc)
nT~
= 0, for all even values of n, 2A = - , for all odd values o f n . nzt
(10)
Again:
0
On substituting the various values, Eq. (8) becomes f(0)
A = 2 +
2A ~ TC
sin nO II
n odd
-.
(11)
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
395
Multiplying (11) by cos0, we get the macroscopic velocity Vr in r-direction at any point in region (2) as A ~ 2sinn0cos0 A cos 0 + 1/~=2 ~z. L. . . . n . n odd
A
cos0+
----2
A
{sin 20 + ½ (sin 40 + sin 20)
rC
+ + (sin 60 + sin40) + ~ (sin 80 + sin 60) + ---}. or
Vr=
A 2
cos0+
A 7r
( 3 4 s i n 2 0 + ~ - s s i n a 0 + 3 ~ s i n 6 0 + ---)
(12)
Now if the reference line AB makes an angle ~ with OX as shown in Fig. 5, then Eq. (14) may be written as Vr=
A 2
cos(0-~)+
A 7r
"(~sin2(0-~)+
+ ~3-11 sin 6 (0 - ~ ) + . . . . . . . .
~-~sin4(0-~)
} .
(13)
Now the horizontal discharge q. passing through the upper region per unit of its length is given by 3~t/2 pl
qu = /
[VJ . . . . "a 2 dO
n/2
PERFORATED
I
/
I /
/
/
/
,/
"-.
\\
\
\ \
I
DIRECTION OF
I t
MAX: GROUND WATER GRADIENT
BLIND
Fig. 5.
D i a g r a m s h o w i n g the position of the upper region w h e n the reference line A B m a k e s a n angle a with t h e direction o f m a x i m u m g r o u n d water gradient.
396
M O H A M M A D SALEEM
which on using Eq. (13) becomes 31U2
qu = [A],=,2"a2
½ cos (0 - c~) +
if{ rr/2
+ ~ =-
sin4(0-
c0 + ~
sin6(0-
, (-~ sin 2 (0 -
c~)
~)+ ...... )/d01"
[A],=,2"a2"cose + 0 + 0 + .........
On substituting the value of A from Eq. (6) this becomes :
4K2a~a 2 • v cos e
qu =
K1 (a~ - a~) + K2 (a~ + a~) or
qu = .
.
.
4a~a2v cos ~ .
K"(a,~ -
.
.
.
(14)
a b + a,~ + a~
Kz or
1(2 (a~ - a~) + a~ + a~
4~,~,,
= cos ~
(15)
Since we are not using any radioactive tracers so, on the basis of dilution of the tracer alone, it is not possible to differentiate between two directions of flow having the same magnitude but separated by an angle of 180 degrees. Let 0~
+~r ......
,~I i\I i\ i
71", /~ I /i / i
\
\ I -
Fig.
6.
b
, !J 1 P
',/
2/ .-
D i a g r a m s h o w i n g a cosine curve. There are f o u r values o f cx corresponding to the t w o values o f b.
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
397
draw the cosine curve as shown in Fig. 6, then it is quite evident that we get two values of ~ given by ~1 and ~2 corresponding to the value + b and two values of [ ] given by ~3 and ~4 corresponding to the value - b . These four values of ~ are such that they form two mutually intersecting straight lines like FOL and F ' O L ' as shown in Fig. 7. AOB, of course, is the position of the reference line in the upper region at a given horizon. The position of the reference line A ' O B ' in the lower region leading or lagging behind the line AOB by ¼~ is also shown in the figure. A'
\
•
AF ~
\
\
\
\
\
\
\
~L B
\
\
\
\
\
\B'
Fig. 7. D i a g r a m s h o w i n g the f o u r directions of flow given by the four values o f c~. These directions are given by the two m u t u a l l y intersecting straight lines F O L a n d F ' O L ' . A B is the reference line of the upper region a n d A ' B ' that o f the lower region.
Now q, is known experimentally with the help of the equation 2) .C = uCo exp
(q..t/uVo)
(16)
where uC = concentration of the tracer in the upper region at time t =t, uCo = tracer concentration in the upper region at time t =0, uV0 = constant volume of water in the upper region per unit length, = n (a 2 - a3z) where a 3 is radius of the brass electrode. Also the unperturbed sub soil velocity v can be calculated from the data obtained from the central region and using an equation similar to Eq. (16) cC = cCo exp
(q~t/~Vo)
(17)
where cC = tracer concentration in the central region at any time t=t, cCo = tracer concentration in the central region at time t =0, ¢Vo = constant volume of water in the central region per unit length =
398
MOHAMMAD SALEEM
and qc given* by
= horizontal discharge through the central region per unit length
8vaZa2 qo
=
(18)
-
~1
k2 (a 2 - a 2 2 ) + a l
2
2
+o 2
Thus v, the magnitude of velocity, is also known experimentally. Now substituting the values of the three unknowns qu, v, and the ratio** K1/K 2 into Eq. (1 5) we get four values of~, giving the directions of flow, which incidently form two mutually intersecting straight lines as explained above. But we have also performed a similar experiment in the lower region as well. This will furnish another pair of mutually intersecting straight lines. Now if we super-impose these two pairs of straight lines, taking care that for the upper region the reference line is AOB while for the lower region the line is A'OB' (Fig. 7), we will find that there will be only one common line among the two pairs. This common line is in fact the line of flow giving us, of course, two directions of flow separated by an angle of 180 degrees. We have thus come down from four possible directions to two. For further differentiation we may proceed as follows 2): The assembly of Fig. 2 is carefully taken out and then the saline column in the bore hole is pumped out by using the air compressor. After giving sufficient rest, the assembly of Fig. 8 is lowered in the bore hole with a solid
SUSPENSION ROD
PERFORATED BRASS TUBE FILLED WITH POT: BROMIDE
BRASS ELECTRODE Fig. 8.
Diagram showing the assembly for differentiation between the two directions of flow separated by an angle of 180 degrees.
* See Appendix 1. ** See Appendix 2.
A SIMPLE METHOD OF G R O U N D WATER DIRECTION MEASUREMENT
399
rod so that the perforated tube along with the two side electrodes now lie in the central region having slits all around. The suspension rod is now so rotated that the line formed by the perforated brass tube and the side electrodes becomes pmallel to the line of flow which we have already calculated above. Potassium bromide crystals will go into solution and will be carried away by the flowing water currents in the direction of flow. We measure electrical conductivities between the central brass tube and the two electrodes on either side after regular intervals of time in the same way as we did previously. There will be a relatively increased electrical conductivity (fall of electrical resistance) between that pair of electrodes along which flow is actually taking place. As the solubility of KBr is high, density currents are likely to be generated and so the conductivity of the other pair will also be affected. However the density currents in the back-ward direction will be opposed but in the forward direction re-inforced by the forward ground water flow. Moreover as the aim of this experiment (Fig. 8) is only a qualitative information to differentiate between the two already known directions of flow, on the basis of previous experiments, the density currents will not create any serious difficulty in finding the true direction of flow. In case the flow rate is extremely low then in place of KBr some other less soluble salt (say gypsum) is recommended. Conclusions
Both magnitude and direction of sub soil flow, under some natural gradient, can be determined by noting the rate of dilution of a tracer (may be non radioactive like inactive potassium bromide) in a single bore hole. The dilution rate, however, is computed by measuring the electrical conductivity (reciprocal of electrical resistance) of the saline column. The magnitude of flow is known by performing the dilution experiment in that section of the bore which has slits all around whereas the direction measurements are carried on by performing similar experiments in two adjacent sections which are perforated over one half and blind over the other half, the only difference in the last two sections being that they are relatively rotated by an angle of n/4. From these experiments we get two directions of flow separated by an angle of n. Further differentiation is carried out by conducting the fourth experiment, in the section with all round perforations, to note the direction along which the rate of change of electrical resistance is maximum. This, in fact, is the direction of flow. It is purely a single well technique and requires no costly equipment, no radioactive tracers and there are no health hazards too. However in regions where the ground water is highly saline radioactive tracers have to be used.
400
MOHAMMAD SALEEM
References 1) P. Wurzel and P. R. B. Ward, A simplified method of ground water direction measurements in a single bore hole. J. Hydrol. 3 (1965) 97-105 2) M o h a m m a d Saleem, An in-expensive method for determining the direction of natural flow of ground water. J. Hydrol. 9 (1969) 73-89
Appendix 1 Let a well-screen (strainer), with inner and outer radii as az and a I respectively, with appropriate length be installed in the aquifer. Let the flow be due to the uniform natural gradient L Further let K 1 be the permeability of region (1) viz. aquifer, K2 the permeability of region (2) which in our case is the material of the well-screen in which slits have been cut and K3 the permeability of region (3) which is a region of sand free water (Fig. 9). K 3 is such that in the limit both K~/K 3 and K2/K 3 approach zero. With the axis of the
I
KL
k \1
II
\\K,~ ' \\ K~ \ \',, X "
/
/
/
/
#/ GP,ADIENT.
/
/ ./
Fig. 9. Diagram showing the details of a well-screen (strainer) resembling the central region of Fig. 2./(1 is the permeability of medium 1 (aquifer), Ks that of the material of the strainer in which slits have been cut and/£3 that of sand free water within the bore hole. 1(3>:>1(2 and/(1.
well passing through the origin O, take any field point P in region (1) with polar co-ordinates (r, 0). Then in the absence of the screened well, the pressure p at the point P is - r I cos 0. When the well-screen is installed, the pressure at the point P is perturbed. Assuming that there is no flow in the vertical direction (Z-direction) the perturbation pressure can be found 2) by solving the Laplace equation in cylindrical polar co-ordinates (r, 0). Thus using subscripts (1), (2) and (3) we write the pressure distribution in the three
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
401
regions as:
Pl = - r l c o s O +
~ (AncosnO+B. sinnO).r-",
r>>.al
(i)
n=l
P2= ~
(C. c o s n 0 + D ,
sinn0)'r"
.~= 1 + ~ (E. cosnO+F, sinnO)'r -~,
(ii) a2<<.r<<.al
n=l
P3= ~
r<~a2.
(G. c o s n 0 + H , s i n n 0 ) ' r " ,
(iii)
n=l
Since the pressure distribution is symmetrical over the line OX, we re-write the above equations by omitting the sine terms:
Pl = - rl cosO + ~ A..r-" cosnO,
r >~a 1.
(iv)
n=l
p2 = ~
C.r"cosnO + ~ E.'r-"cosnO,
n=l
P3 = ~
a2<~r<<.al.
(v)
•=1
G . ' r " cos nO,
r <~a 2.
(vi)
n= 1
Applying the first boundary condition at r = a t , Pl =P2, so
- a,I cosO + ~ A.'aC" cosnO = ~ C.a~" cosnO + ~ E.a-~" cosnO. n=l
n=l
n=l
Equating coefficients of cosine terms on both sides we get n = 1: - a~I + (A~/a,) = C~al + (El~a1)
(vii)
n > 1: (A./a])
(viii)
= C.a] + (E./a~)
Again at r = az, P2 =P3, and so we get
{C.a"2 + (E./a~)} cos nO = ~ G.a"2cos . 0 . n=l
n=l
Equating coefficients of cosine terms we get n ) 1:C.a"2 + E.a~" = G.a~. Now applying the second boundary condition 63p I
at we get
t3p2
r = al, K1 8 r = K 2 ~ r '
- KlI cosO-- K~ ~ A..nal"-~'cosnO
= r2 ~ n=l
n(C.a"~ -~ - E . ' a ? " - ~ ) c o s n O .
(ix)
402
MOHAMMAD SALEEM
Equating coefficients of cosine terms on both sides n = 1: - K I I - K1Ala-; 2 = K 2 (C1 - E t a ; z) n>l:
- K 1 A n . n . a ~- n - 1
(x)
= K 2 n ( C n a tn- 1 - E n a ~ n - 1 ) .
(xi)
Also at t~P2
r = a2,
•P3
K 2 t~r
K3 -0r
=
and hence:
K2
n(Cna
-' - E n a ; ° - ' ) c o s n 0 = K3
n=l
a -"cosn0. n:l
Equating coefficients of cosine terms on both sides we get n >1 1 : KEn (Cna~- ~ -- Ena~"- l) = K a n G . a ~ 2.
(xii)
N o w f r o m (vii): - a ~ I + A~ = C~ a~ + E~ and f r o m (x) : - Kla~ I - K IA t = K2 Cl a~ -- K 2 E 1 on adding these we get: - a ~ I ( 1 + K 1 ) + A 1 (1 - K 1 ) = C , a l (1 + K2) + E~ (l - K2) or
A~ = C l a t ' l
1+ g _ g
2
+
a2i. 1 + K1 1 ~ V ~ K.I + E ~ ' ~ _
-
-
K 2
K .
(xiii)
Since Lim K2 = 0, K3~aO K3
so f r o m (xii) we get G~=0,
n/> 1.
(xiv)
,
(xv)
F r o m (ix) En=-C,,a2
2n
n/> 1.
Substituting (xv) in (viii) we get A, = C. (a~"
2,, -a2),
n>
1.
(xvi)
F r o m (vii) we have A~ = a2tl + Cla~ + Et .
(xvii)
Using (xv) for n = 1, Eqs. (xvii) and (xiii) become A~ = a~I + C t (a 2 - a~) and A 1 = Cla~" 1 + K 2 i =K, +a2I
I + K~ 1 --g 1
1 -- K 2 2 CI'I-K1 "a2
(xviii)
A SIMPLE METHOD OF G R O U N D W A T E R DIRECTION MEASUREMENT
403
or
A1 ~ C1 ( a211 +- K2K, Ii - K, K2 a2) + a2I'll
-+ KI"K 1
(xix)
Equating Eqs. (xviii) and (xix) we get { 2 K1 + K2
2 K 1 - K2X~ a2" i ~ k I ) ~- - 202K1[/(1 - g l )
C, ka," ~ Z K 1 OF
- 2a2KxI CI - gx (a 2 - a22) + (a 2 + a22) Kz
.
(xx)
.
(xxi)
F r o m (xv) for n = 1 2 2 2ala2KlI
E1
=
K l (a 2 - a~) + (a 2 + a 2) K 2
Since K 1, K2, a 1 and a 2 are all non-zero, so on combining (xi) and (xvi) we get on using (xv) C n = 0, n > 1 (xxii) E.=0,
n>l
(xxiii)
An=0,
n > 1.
(xxiv)
and f r o m (xvi)
Substituting (xx) in (xviii) we get
A l = a2I -
2a2KtI (a2 - a22) K, (a 2 -- a 2) + K 2 (a 2 + a~)"
(xxv)
Substituting Eqs. (xx) to (xxv) and Eq. (xiv) in Eqs. (iv) to (vi) we get the values of p l , P2 and P3 in the three regions as
pt = - rI cosO +
2alKlI (al - a2) a ~ I - K l ( a ~ - - a ~ ) + - K 2 ( a ~ +a~)
cosO
r
which on simplification gives:
[K2 (a~ + a~) - Kl (a~ - a~) a~
t
Pl = [ K 2 (a 2 + a22) ~ KI (a 2 - a2)" r -- r . I cos 0.
(xxvi)
Like-wise
P2 -- K1 (a 2 _ a~) + K2 (a 2 + a2)" P3 = 0.
( )cos0 r
xxvii (xxviii)
T o find the horizontal discharge q per unit length of the bore hole passing in and out of it, we proceed as follows.
404
MOHAMMAD SALEEM
The radial velocity ~ in region (2) is Vr = _ k 20P2
~r 2a~K1K2I
.(a!
)
=K,(a 2 - a 2)+K2(a~+a 2) \r z + 1 cos0, and ....
=
4aEKa I cos 0 (K1/K2)(a~ -- a 2) + a~ +
a 2"
Now 3~t/2
q= f
[V~]. . . . "a2 dO
n/2 or
q=
8rata2 (K,/K2) (a~ - a~) + a~ + a~
(xxix)
where
v = KII = unperturbed velocity of sub-soil flow in the aquifer. The ratio K1/K 2 can be found experimentally by a technique described in appendix 2. FOR A BLIND PIPE
Now suppose that the strainer of Fig. 9 is replaced by a blind pipe having the same inner and outer radii. We have, in this case, the following limiting conditions K2
and
-*0
K1 K2 -
(xxx) --~O.
K3
Using (xxx), the Eqs. (xxvi) to (xxviii) become b P l = --
bP2
(o: :) nt-
Ial
cos 0
(xxxi)
a~ -- a~
bP3 = 0.
(xxxiii)
The perusal of Eqs. (xxviii) and (xxxiii) shows that the pressure P3 at any
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
405
point within the pipe (sand-free region) is zero irrespective of the fact whether the pipe in question is a pure strainer or a pure blind pipe. Likewise when the pipe is neither a pure strainer nor a pure blind pipe but instead consists of two segments like upper or lower region of Fig. 1 then again we have the limiting condition: K z / K 3 -+ 0
and hence P3 is zero in this case too. Consequently the pressure at any point within the entire bore hole of Fig. 2 is zero irrespective of the mode of perforations made along the length. This excludes the possibility of any vertical inter-regional flow as a result of non uniform perforations in the three regions of Fig. 2. Appendix 2 In these pages a simple technique for the determination of both K 1 and K 2, as defined in appendix 1, is described. EXPERIMENTAL SET UP
An impermeable concrete tank measuring 3 meters × 3 meters × one meter is selected. A 20 cm wide and one meter deep gravel filled region is build round the inner periphery of the tank as shown in fig. 10. The remaining portion of the tank is filled with some medium grain sand. A 1.2 meters long and 10 cm diameter brass strainer with proper slit size is installed at the centre of this tank to work as tube well. Two observation wells which are also filter pipes (diameter about 2.5 cm and length 1.2 meters each) are installed FILLED
O
OBSERVATION WELL 2 O OBSERVATION WELL 1
_J bJ
TUBE WELL
m~ !~ I
co
SATURATED SAND
I I It ]
I
J FOP, WATER SUPPLY
Fig. 10.
Diagram showing the details of the experimental tank for determining the ratio
K1/K~. There is a gravel filled region near the outer periphery of the tank for constant water supply to the tubewell installed in the saturated sand. The observation wells 1 and 2 are meant for measuring the draw-down at known distances from the tubewell.
406
MOHAMMAD SALEEM
in the sandy region at different distances fron the tube well as shown in the figure. Water is supplied into the gravel filled region with the help of syphon arrangements so that a constant supply level is maintained in this region. This region in turn feeds water into the sandy zone. When steady conditions are set up in the whole tank pumping is started. The pumping, however, is an open well pumping done with the help of a syphon. When the tube well starts functioning, the water level both in the tube well and observation wells falls down. To record accurately the water levels in all the wells, we again employ the syphon principle. This is illustrated in Fig. 11. Clearly the water level
GLASS TUBING~ //
~]
MEASURING-~ SCALE
j~j, "
i
"~TUBEWELL "~ WATER LEa,C:L
RTUBING 7PLUG BAIL Fig. 11.
Diagram showing the arrangement for knowing accurately the water level in the tubewell during its working. This is based on simple syphon method.
in the glass tubing is the same as in the tubewell. While setting the experiment it should be made clear that there are no air bubbles in the rubber tubing. To remove the air bubbles from the rubber tubing, we take off its end from the glass tubing and allow water to flow out of the rubber tubing for some time. When all bubbles are expelled then its end is slipped on to the glass tubing. Similar arrangements are made to record water levels in the observation wells. Care is taken that the recording glass tubings for the three wells are placed parallel to each other and have a common zero. Another tubing to record the water supply level in the gravel filled region ia also placed parallel to the above recording tubes over the same scale. When the tubewell operates for some time, the water levels in the various pipes will attain a constant value. These constant levels are recorded along with the constant discharge of the tubewell. Then (lPe-Pw) denotes the
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
407
difference in levels between the observation well (l) and the tubewell. Likewise ( 2 P e - - P w ) denotes the similar difference with regard to observation well (2). After this the discharge Q of the tubewell is varied to some other constant value and similar observations are made after steady conditions have been reached in the system, and so on. We now plot two curves: p Q versus ( i p e - P w ) and (2P~-Pw) on the same graph paper (# being coefficient of viscosity of water at the temperature of the experiment). Selecting a common value of pQ corresponding nearly to the middle of these curves, we find the corresponding values of (lPe--Pw) and (2Pe--Pw). The calculations for the determination of the permeability of the given brass strainer are carried out as detailed below. MATHEMATICAL APPARATUS
Let al and a 2 be the outer and inner radii of the brass strainer respectively. Let K 1 be the permeability of the water bearing medium, called region (1), in which the strainer has been installed. Let the average permeability of the material of the brass strainer be g 2. This is denoted as region (2). As the well is tapping the full depth of the aquifer, there is no flow in the vertical direction (z-direction) after pumping is started. The flow is assumed to be purely radial so that it is independent of the variation of angle 0. We start with the Laplace equation in cylindrical polar coordinates and solve it when the flow is purely radial. We thus have V 2 p = r ~r
r
= 0
or
?p r
ar
= const.
Integrating p = C l o g r + C o where p is pressure at the point at distance r from the axis of the well and C, Co are constants. We therefore write the pressure distributions in regions (1) and (2) b y p l andp2 as P i = C1 l o g r + C2
(1)
P2 = C3 l o g r + C4.
(2)
Applying the first boundary condition at (::3pi
r=al:K
0p2
l ~r = K2 ~r '
408
MOHAMMAD
SALEEM
we get
1
K1CI.--
1 = K 2 C 3.
a1
al
or
K1 C3 = K2 C1 = KC1
(3)
where K =
K1
.
(4)
K2 Substituting this value of C3 in (2) we get P2 = KC1 l o g r + C 4.
(5)
Now applying the second boundary condition at r = a l :Pa =P2, we get Ca l o g a l + C2 = KC~ l o g a t + C4 or
(1 - K) C, logal + C2 --- C4.
(6)
Substituting this value of (74 in (5) we get P2 = C1 { K l o g r + (1 - K) logal} + C2.
(7)
The next two boundary conditions are at r = a~: px = p~ and at r = a2:P2 = Pw
(8)
where the distance a¢ is the radius of influence - a distance where a reasonable estimate o f p , can be made. Substituting these conditions in Eqs. (1) and (7), we get pC -- C a -loga~ + C 2 (9) Pw = C1 ( K l o g a 2 + (1 - K ) loga~} + C2 = Ca log {a[" a ] ' - r ) } + C2 or
P , = Ca logaw + C2
(10)
a. =
(ll)
where Subtracting (9) and (1 O) C 1 -
P~ - Pw log(a Jaw)
A SIMPLE M E T H O D OF G R O U N D W A T E R D I R E C T I O N MEASUREMENT
409
and Pw" log a¢ - pc log aw C 2
~
-
log(ae/aw)
Substituting these values of C1 and C2 in (1) and (7) we get pw'log ae - Pe'log aw Pe - Pw log(ae/a~) + lo~(aJa~..) " l ° g r
P'or
Pe -- Pw r Pl = P,~ + l o g ( a J a w ) log aw .
.
.
.
.
(12)
and P2 = Pw +
Pe - Pw log (a J a w )
R
.log
(13)
aw
where R = r K" a~l - r )
(14)
The radial velocity Vr is given by
Vr
~
--
KI ~Pl /* Or "
-
"~-
--
K1 P~--Pw 1 p log(ae/aw) r "
- -
. . . .
"
~---
.
.
.
Q 27thr .
.
.
where Q is the discharge pumped by the tubewell, h the depth of aquifer and /~ the viscosity of pumped water, or log (a J a w ) = + KI" 27zh (Pe - pw)/pQ or
aw = a¢.exp { - 2rthK 1 (p¢ - pw)/pQ} Or using (4) and (1 l) a2 tall
= a~.exp { - 2~hK~ .(p~ - Pw)/PQ} al
or
r,,/K2 log(a2/a )
= l o g ( a ~ / a l ) - 2=hK~
(po -
pw)/pQ.
(t5)
Corresponding to the two observation wells, we re-write Eq. (15) as K~ log a2 = log la° - 2K~=h (~p¢ - pw)/pQ K2 al a~
(16)
and KI K2
log a2 = log [ae _ 2K,rch. (2P~ - pw)/pO a~ a 1
(17)
MOHAMMADSALEEM
410
where Xae and 2ae are respectively the distances of the observation wells (1) and (2) from the tubewell. Writing
2nh ~Q
~av
t i p ° - pw) = b x ,
(2Pe - Pw) =
bz,
we have from (16) and (17)
(
1
a2)'ae
K, ab l +K21oga, and
(
,a 0
K1 ab2+KzlOgam
=l°g
al
= l ° g al .
(18)
(19)
Eliminating K t from (18) and (19) we get
K2 =
log (2ae/lae).log
(ai/al)
a (b2 log ('adam) - b, log (2a.la,)
or
K2 =
log (Zao/'ao).log
(a2/a,)
27th
(20)
~/Q" {(2Pe -- Pw) log ('ae/al) - (lPe - Pw)'l°g (Za,/al)} In Eq. (20) all the quantities are known and so Kz, the permeability of the brass strainer material, can easily be calculated. To find the permeability KI of the sand, we subtract Eqs. (18) and (19) and get /~Q log (2ae/lae) K, . . . . . . . (21) 2roll (2Pe - 1P~) It may be noted over here that the sand used in Fig. 10 may not be the same as obtained during sampling at the time of installation of the bore hole. However such samples may be tested for permeability determination by other simple laboratory experiments. These values of K m may then be inserted in the formulae for velocity measurements. The above technique for the measurement of K2 can well be applied to even more complicated strainers like those obtained by tightly winding a coir string on an iron cage.
A SIMPLE METHOD MEASUREMENT
OF GROUND WATER DIRECTION IN A SINGLE BORE HOLE
MOHAMMAD SALEEM
Physicist, Irrigation Research Institute, Lahore, Punjab (West Pakistan) Abstract: A simple technique to find both the magnitude and direction of ground water flow in a single bore hole, sited in a pervious water bearing medium, is described. The experiment may be performed with tracers which may not be radioactive (like inactive potassium bromide). At any given horizon where velocity measurements are required, the strainer of the bore-hole is divided into three regions, each about 10 cm long. The central region has slits all round as is usually the case for a strainer. The top region consists of two equal semicircular halves, each subtending an angle of 180 degrees at the axis of the strainer, such that one half has slits cut in it as usual while the second half is kept all blind during manufacture. The horizontal line separating the perforated and the blind segments is taken as the reference line. The lower region is similarly constructed as the top one except that its reference line leads or lags behind the reference line of the top region by an angle of 45 degrees. For proper separation of the above three regions, there are buffer zones between any two consecutive regions. Inactive potassium bromide solution is injected in all the three regions. Using three independent brass electrodes, one in each of the three regions, electrical conductivities at regular intervals of time are carried out by using low frequency and low voltage alternating current, the common electro de being the strainer itself. The magnitude of flow can be determined from the information collected from the central region; whereas the data from the top and lower regions give the direction of flow. Introduction W u r z e l and W a r d 1) have a n n o u n c e d a single well t ech n i q u e for the measu r e m e n t o f direction o f g r o u n d w a t e r flow by using special r a d i o a c t i v e tracers which readily a d s o r b on the m e tal surfaces. U si n g a special technique, a tracer is injected very slowly into the bore hole (an o b s e r v a t i o n well with p e r f o r a t e d screen). T h e injected tracer is slowly carried away, by the water currents, in the direction o f flow t o w a r d s the well screen. Th e a d s o r p t i o n o f the tracer is a l lo w e d n o t on the m a t e r i a l o f the screen b u t on a wire-gauze cylinder (length a b o u t 60 c m and d i a m e t e r slightly less t h a n the well screen). A f t e r some time, this cylinder is r e m o v e d f r o m the well an d rushed to the l a b o r a t o r y where it is scanned to find the relative a d s o r p t i o n o f the tracer on the wire gauze. T h e d ir e c t io n o f flow is the direction in which the ad so r p tion o f the tracer on the gauze is m a x i m u m . 387
388
MOHAMMAD SALEEM
A m o n g the non radioactive tracer techniques, the author 2) has described a technique in which the ground water flow is treated as a vector quantity. Using inactive potassium bromide solution as a tracer, the magnitude of flow can be found by measuring the rate of dilution of this tracer in the well by recording the electrical resistance of the saline column. The reciprocal of the resistance is a measure of the electrical conductivity. As regards the direction of flow, this can be evaluted by introducing another vector quantity of known magnitude and direction and then finally measuring the magnitude of the resultant flow in the same observation well where the original experiment was performed. This additional vector in the ground water movement can be introduced by pumping a small tube well at a known constant discharge in the vicinity of the observation well. This technique is subjected to the limitations that it is applicable only to the homogeneous and isotropic media. Secondly it is not strictly a single well technique because an additional tube well is also required for direction measurements. The present technique is strictly a single well non radioactive tracer technique and is a considerable advancement over the one already mentioned.
Experimental set up First of all a test bore is made at the site where velocity measurements are required. This would help in deciding the slit size of the strainer to be used in the bore hole and also for proper location of the observation points. A 10 cm diameter brass strainer is selected whose length should be equal to the length of the sand column to be encountered during installation. This strainer should be coupled with an equivalent diameter blind pipe whose lengths should correspond to the depths of any clay layers occurring in the strata. Thus the strainer should only be installed in the sandy regions and the blind pipe should take care of the clayey regions of the bore hole. Before installation, the brass strainer is laid on a level ground and all its lengths are coupled together to form one single length. The blind pipe is also coupled along with it. Starting from the upper end of the blind pipe we mark off lengths, which will correspond to depths from the top, where velocity measurements will be required. At each marked point we divide the screen into three regions, each about 10 cm long. The upper and lower regions are separated from the central one with the help of two buffer zones, each about 6 cm wide. No slits are cut in these zones during manufacture of this strainer piece. There are two more buffer zones one above the top region and the other below the lower region (Fig. 2). The central region has slits all-round. The upper region is divided into two semi-circular halves such that one half
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
389
has slits as usual but the second half is kept all blind during manufacture. Except for the selected points, where the strainer is specially prepared as explained above, the rest of the strainer length has slits as usual. The line AB (Fig. 1) separating the two halves is called the reference line. The lower region is similarly constructed except that its reference line A'B' makes an angle of 45 degrees with the line AB of the upper region. The line AB is taken as the standard reference line. To know the position of the line AB at the top we draw two parallel lines over the strainer surface along the length such that one line starts from the end A while the other from end B and run right upto the top end of the blind pipe. P E R F O R A T E ~
I~
- -- - ~ \
// /
",\
/
k
\
z
\
it A
B
....................
( a ) UPPER REGION
ALL PERFORATED -'2 / / /
f
\
"\ \ \ I
I
/
\
/ '-,\ /
~"~. ~ _ ~
-/"/"
y...
( b ) CENTRAL REGION
,\C PERFORA EO .
BL ND
/
....
"~.\
I B
II
.)"g
(C)LOWER REGION
Fig. 1. Diagram showing the three regions: upper, central and lower. The central being perforated all round while the upper and lower are identical except that the reference lines AB and A'B' make an angle of 45 degrees with each other. The reference line separates the two halves such that one half is all perforated while the other half all blind. The length and diameter of each region being 10 cm.
390
MOHAMMAD SALEEM
Similar regions are constructed at other selected horizons taking care that the lines AB and A'B' at all the horizons are kept individually parallel. THREE CORE SUSPENSION
CABLE BLIND PIPE
BORE HOLE STRAINER
j'
'ATER :ABLE
.
__~.__~.!
i
I
I
I
SUPPORTING HOOK
CENTRAL REGION
INSULA]ED WIRES
- ~ I~
BUFFER ZONES
LOWER REGION BRASS ELECTRODES ~
I h ~
\\\
I I
SOLID SUSPENSION ROD
~1-~ I I
I I I
SUSPENSION WEIGHT I
I I
BAIL PLUG
t i
I ~
PERSPEX DISCS DIA SLIGHTLY LESS THAN THE STRAINER
.....|
~\~'RESTING
ROD
lr--
I I
I
Fig. 2. Diagram showing the details of the bore hole along with the arrangement for measuring the conductivity in the three regions. There is a buffer zone between any two consecutive regions having plastic discs with diameter slightly less than the bore hole and thickness 6 cm. The bore hole strainer is also blind round each of the four plastic discs. The three regions are occupied automatically in the bore hole with the help of a suspension weight, connected with a solid suspension rod to the conductivity measuring assembly, when it rests on a horizontal resting rod installed in the bore hole strainer. The conductivity in the three regions is measured independently with brass electrodes, the common electrode being the brass bore hole strainer itself.
A SIMPLE METHOD OF G R O U N D WATER DIRECTION MEASUREMENT
391
While selecting the observation points at different horizons, it should be kept in mind that all the three regions at a given horizon remain in the same type of strata. Velocity measurements at more than one horizon are carried out to account for any heterogeneous conditions in the strata. To start the experiment, potassium bromide solution is injected as tracer in the bore hole at each selected horizon over a range of about 2 meters in the form of a cloud. The proper mixing of the tracer is done, at each of these depths either by a small propeller or by any other suitable mechanical means. The electrode assembly shown in Fig. 2 is then lowered into the bore hole. The assembly illustrated in this figure is only for one particular selected depth. Similar assemblies may be constructed for other selected horizons. There is a resting rod about 1 cm thick, fixed beforehand in the strainer along one of its diameters near the bottom end shown. For smooth lowering of the electrode assembly a heavy suspension weight attached to a solid rod (about 1 cm diameter and 25 cm long) is screwed into the lowest perspex disc as shown in Fig. 2. After lowering of the assembly when the suspension weight rests on the resting rod, the three electrodes lie automatically in their respective regions while the four perspex discs occupy their respective buffer zones. In case there are more observation points in the bore hole, then the various electrode assemblies are inter-connected by suspension rods of appropriate lengths so that when the suspension weight rests on the resting rod, the various electrodes and perspex discs automatically occupy their respective pre-designed positions. To take the observations, we simply record the electrical conductivities (reciprocal of electrical resistance of the saline column) versus time for each electrode, the c o m m o n electrode being the conducting bore hole itself. A Wheatstone bridge coupled with a two stage transistorised A.C. amplifier with a variable gain may be used for resistance measurements. The alternating current power supply at about 6 volts may be obtained from a vibrator system run b) a D.C. storage battery. The observations obtained with each electrode are plotted separately on semi-log papers with conductivity C on log scale and time t on linear scale. The curves that we get are all similar except that their slopes are different. There is a dip in the beginning z) (due to additional factors like diffusion in the horizontal direction and adsorption etc., responsible for extra dilution of the saline columns over and above the dilution due to flow) and then afterwards the curves become straight lines. By extrapolating the straight portions of the curves in the back-ward direction we get the conductivity C o at time t equal to zero in each case, had there been no extra dilution due to the additional factors mentioned above. The observations recorded by the central electrode will give us the magnitude of flow while the data obtained from the upper and lower regions will together
392
MOHAMMAD SALEEM
furnish the direction of flow. Now we proceed with the mathematical analysis of the technique involved.
Mathematical apparatus To start with let the bore hole consist of a simple strainer with outer and inner radii as al and o 2 respectively. Let K1 = Permeability of the aquifer (sandy region, called region (1)) in which the strainer has been installed, K2 --- Permeability of the strainer material (called region (2)) in which slits have been cut, I = Uniform natural gradient under which flowis taking place, v = KII = Unperturbed velocity of subsoil flow in region (1). Then at any field point P in region (1) with polar co-ordinates (r, 0), the pressure ~Pl is given* by
{ K2(a~+a2)-K~ (a~-a2)a2 =
K2
+
+ K,
r}IcosO
(1)
r
and if the point P is taken in region (2):
2a2K1 (a~ ) sP2=K~(aZ_a2)+Kz(a~-+a22).\r-r
I cos0.
(2)
If the strainer is replaced by a blind pipe with the same inner and outer radii then the pressure distribution in the two regions is given by bPt = --
(°: :,) +
bpZ -- aZ _ az
I al cos0
(3)
- r
(4)
(o))
I cosO.
Now let us consider the case when the bore hole is neither a pure screened well (perforated all round) nor a simple blind pipe but instead consists of two semi-circular equal segments such that one is perforated while the other is blind (Fig. 3). The reference line AB is taken along the direction of maximum ground water gradient. Let the axis of the well pass through the origin O. Take any field point P with polar co-ordinates (r, 0) in region (2). Now when the point P is in region (2a) then the macroscopic velocity * See A p p e n d i x 1
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
393
~V, in the r-direction is given by: sV, = - K2 a7 (sp2)
2aZKiK2
(a~ )
=Ki(a~--a~)+K2(a~+a~) \r z + 1
Icos0.
(5)
When the point P moves into region (2b) then the macroscopic velocity bV, in the r-direction is zero as the permeability of this region is zero too. Keeping r fixed let the point P move in the entire region (2). The velocity distribution curve obtained in this case may be considered to be the product of cos 0 with the velocity distribution curve shown in Fig. (4). PERFORATED
,/
i A
/
//(2o,// i
[
/
o/,.~ ;I ~/
X/f'r,e) ~,
\ \\ ~
. . . . . . . . . .
\ MAX MUM " OP-OUND WATER, GRADIENT
"~"BLIND
Fig. 3. D i a g r a m showing the details of the upper region. The u p p e r half is all perforated while the lower h a l f is all blind. The line A B separating the two halves is taken as the reference line.
A
ZERO LEVEL
. 0 Fig. 4.
1 IT 0 --
2~
Diagram showing the velocity distribution curve resembling a square wave, having period 2~r, usually generated in electrical circuits.
394
MOHAMMAD
SALEEM
Writing:
2a2vK2 A=-Kl(a~_a~)+K2(a
{a~ 2 + a 2 z) ~r 2 + 1
) (6)
Eq. (5) becomes sVr = A cos 0.
(7)
To find the interference effect of the regions (2a) and (2b), we apply Fourier Analysis to Fig. 4. The velocityf (0) at any point may be expressed in the form of a series as f(0)=½ao+
Z (a, c o s n 0 + b , sinn0)
(8)
n=l
where ao, a, and b. are constants to be determined. As the curve of Fig. 4 has a discontinuity when 0 equals re, we have a, = 0 ,
for a l l n , 2re
ao =
f (0)'d0 o
(9)
2n
b, =
f (0) sinnO.dO
7r o
Now
b, =
sin nO.dO + 0 7r o
=
A
(1 - cos nrc)
nT~
= 0, for all even values of n, 2A = - , for all odd values o f n . nzt
(10)
Again:
0
On substituting the various values, Eq. (8) becomes f(0)
A = 2 +
2A ~ TC
sin nO II
n odd
-.
(11)
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
395
Multiplying (11) by cos0, we get the macroscopic velocity Vr in r-direction at any point in region (2) as A ~ 2sinn0cos0 A cos 0 + 1/~=2 ~z. L. . . . n . n odd
A
cos0+
----2
A
{sin 20 + ½ (sin 40 + sin 20)
rC
+ + (sin 60 + sin40) + ~ (sin 80 + sin 60) + ---}. or
Vr=
A 2
cos0+
A 7r
( 3 4 s i n 2 0 + ~ - s s i n a 0 + 3 ~ s i n 6 0 + ---)
(12)
Now if the reference line AB makes an angle ~ with OX as shown in Fig. 5, then Eq. (14) may be written as Vr=
A 2
cos(0-~)+
A 7r
"(~sin2(0-~)+
+ ~3-11 sin 6 (0 - ~ ) + . . . . . . . .
~-~sin4(0-~)
} .
(13)
Now the horizontal discharge q. passing through the upper region per unit of its length is given by 3~t/2 pl
qu = /
[VJ . . . . "a 2 dO
n/2
PERFORATED
I
/
I /
/
/
/
,/
"-.
\\
\
\ \
I
DIRECTION OF
I t
MAX: GROUND WATER GRADIENT
BLIND
Fig. 5.
D i a g r a m s h o w i n g the position of the upper region w h e n the reference line A B m a k e s a n angle a with t h e direction o f m a x i m u m g r o u n d water gradient.
396
M O H A M M A D SALEEM
which on using Eq. (13) becomes 31U2
qu = [A],=,2"a2
½ cos (0 - c~) +
if{ rr/2
+ ~ =-
sin4(0-
c0 + ~
sin6(0-
, (-~ sin 2 (0 -
c~)
~)+ ...... )/d01"
[A],=,2"a2"cose + 0 + 0 + .........
On substituting the value of A from Eq. (6) this becomes :
4K2a~a 2 • v cos e
qu =
K1 (a~ - a~) + K2 (a~ + a~) or
qu = .
.
.
4a~a2v cos ~ .
K"(a,~ -
.
.
.
(14)
a b + a,~ + a~
Kz or
1(2 (a~ - a~) + a~ + a~
4~,~,,
= cos ~
(15)
Since we are not using any radioactive tracers so, on the basis of dilution of the tracer alone, it is not possible to differentiate between two directions of flow having the same magnitude but separated by an angle of 180 degrees. Let 0~
+~r ......
,~I i\I i\ i
71", /~ I /i / i
\
\ I -
Fig.
6.
b
, !J 1 P
',/
2/ .-
D i a g r a m s h o w i n g a cosine curve. There are f o u r values o f cx corresponding to the t w o values o f b.
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
397
draw the cosine curve as shown in Fig. 6, then it is quite evident that we get two values of ~ given by ~1 and ~2 corresponding to the value + b and two values of [ ] given by ~3 and ~4 corresponding to the value - b . These four values of ~ are such that they form two mutually intersecting straight lines like FOL and F ' O L ' as shown in Fig. 7. AOB, of course, is the position of the reference line in the upper region at a given horizon. The position of the reference line A ' O B ' in the lower region leading or lagging behind the line AOB by ¼~ is also shown in the figure. A'
\
•
AF ~
\
\
\
\
\
\
\
~L B
\
\
\
\
\
\B'
Fig. 7. D i a g r a m s h o w i n g the f o u r directions of flow given by the four values o f c~. These directions are given by the two m u t u a l l y intersecting straight lines F O L a n d F ' O L ' . A B is the reference line of the upper region a n d A ' B ' that o f the lower region.
Now q, is known experimentally with the help of the equation 2) .C = uCo exp
(q..t/uVo)
(16)
where uC = concentration of the tracer in the upper region at time t =t, uCo = tracer concentration in the upper region at time t =0, uV0 = constant volume of water in the upper region per unit length, = n (a 2 - a3z) where a 3 is radius of the brass electrode. Also the unperturbed sub soil velocity v can be calculated from the data obtained from the central region and using an equation similar to Eq. (16) cC = cCo exp
(q~t/~Vo)
(17)
where cC = tracer concentration in the central region at any time t=t, cCo = tracer concentration in the central region at time t =0, ¢Vo = constant volume of water in the central region per unit length =
398
MOHAMMAD SALEEM
and qc given* by
= horizontal discharge through the central region per unit length
8vaZa2 qo
=
(18)
-
~1
k2 (a 2 - a 2 2 ) + a l
2
2
+o 2
Thus v, the magnitude of velocity, is also known experimentally. Now substituting the values of the three unknowns qu, v, and the ratio** K1/K 2 into Eq. (1 5) we get four values of~, giving the directions of flow, which incidently form two mutually intersecting straight lines as explained above. But we have also performed a similar experiment in the lower region as well. This will furnish another pair of mutually intersecting straight lines. Now if we super-impose these two pairs of straight lines, taking care that for the upper region the reference line is AOB while for the lower region the line is A'OB' (Fig. 7), we will find that there will be only one common line among the two pairs. This common line is in fact the line of flow giving us, of course, two directions of flow separated by an angle of 180 degrees. We have thus come down from four possible directions to two. For further differentiation we may proceed as follows 2): The assembly of Fig. 2 is carefully taken out and then the saline column in the bore hole is pumped out by using the air compressor. After giving sufficient rest, the assembly of Fig. 8 is lowered in the bore hole with a solid
SUSPENSION ROD
PERFORATED BRASS TUBE FILLED WITH POT: BROMIDE
BRASS ELECTRODE Fig. 8.
Diagram showing the assembly for differentiation between the two directions of flow separated by an angle of 180 degrees.
* See Appendix 1. ** See Appendix 2.
A SIMPLE METHOD OF G R O U N D WATER DIRECTION MEASUREMENT
399
rod so that the perforated tube along with the two side electrodes now lie in the central region having slits all around. The suspension rod is now so rotated that the line formed by the perforated brass tube and the side electrodes becomes pmallel to the line of flow which we have already calculated above. Potassium bromide crystals will go into solution and will be carried away by the flowing water currents in the direction of flow. We measure electrical conductivities between the central brass tube and the two electrodes on either side after regular intervals of time in the same way as we did previously. There will be a relatively increased electrical conductivity (fall of electrical resistance) between that pair of electrodes along which flow is actually taking place. As the solubility of KBr is high, density currents are likely to be generated and so the conductivity of the other pair will also be affected. However the density currents in the back-ward direction will be opposed but in the forward direction re-inforced by the forward ground water flow. Moreover as the aim of this experiment (Fig. 8) is only a qualitative information to differentiate between the two already known directions of flow, on the basis of previous experiments, the density currents will not create any serious difficulty in finding the true direction of flow. In case the flow rate is extremely low then in place of KBr some other less soluble salt (say gypsum) is recommended. Conclusions
Both magnitude and direction of sub soil flow, under some natural gradient, can be determined by noting the rate of dilution of a tracer (may be non radioactive like inactive potassium bromide) in a single bore hole. The dilution rate, however, is computed by measuring the electrical conductivity (reciprocal of electrical resistance) of the saline column. The magnitude of flow is known by performing the dilution experiment in that section of the bore which has slits all around whereas the direction measurements are carried on by performing similar experiments in two adjacent sections which are perforated over one half and blind over the other half, the only difference in the last two sections being that they are relatively rotated by an angle of n/4. From these experiments we get two directions of flow separated by an angle of n. Further differentiation is carried out by conducting the fourth experiment, in the section with all round perforations, to note the direction along which the rate of change of electrical resistance is maximum. This, in fact, is the direction of flow. It is purely a single well technique and requires no costly equipment, no radioactive tracers and there are no health hazards too. However in regions where the ground water is highly saline radioactive tracers have to be used.
400
MOHAMMAD SALEEM
References 1) P. Wurzel and P. R. B. Ward, A simplified method of ground water direction measurements in a single bore hole. J. Hydrol. 3 (1965) 97-105 2) M o h a m m a d Saleem, An in-expensive method for determining the direction of natural flow of ground water. J. Hydrol. 9 (1969) 73-89
Appendix 1 Let a well-screen (strainer), with inner and outer radii as az and a I respectively, with appropriate length be installed in the aquifer. Let the flow be due to the uniform natural gradient L Further let K 1 be the permeability of region (1) viz. aquifer, K2 the permeability of region (2) which in our case is the material of the well-screen in which slits have been cut and K3 the permeability of region (3) which is a region of sand free water (Fig. 9). K 3 is such that in the limit both K~/K 3 and K2/K 3 approach zero. With the axis of the
I
KL
k \1
II
\\K,~ ' \\ K~ \ \',, X "
/
/
/
/
#/ GP,ADIENT.
/
/ ./
Fig. 9. Diagram showing the details of a well-screen (strainer) resembling the central region of Fig. 2./(1 is the permeability of medium 1 (aquifer), Ks that of the material of the strainer in which slits have been cut and/£3 that of sand free water within the bore hole. 1(3>:>1(2 and/(1.
well passing through the origin O, take any field point P in region (1) with polar co-ordinates (r, 0). Then in the absence of the screened well, the pressure p at the point P is - r I cos 0. When the well-screen is installed, the pressure at the point P is perturbed. Assuming that there is no flow in the vertical direction (Z-direction) the perturbation pressure can be found 2) by solving the Laplace equation in cylindrical polar co-ordinates (r, 0). Thus using subscripts (1), (2) and (3) we write the pressure distribution in the three
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
401
regions as:
Pl = - r l c o s O +
~ (AncosnO+B. sinnO).r-",
r>>.al
(i)
n=l
P2= ~
(C. c o s n 0 + D ,
sinn0)'r"
.~= 1 + ~ (E. cosnO+F, sinnO)'r -~,
(ii) a2<<.r<<.al
n=l
P3= ~
r<~a2.
(G. c o s n 0 + H , s i n n 0 ) ' r " ,
(iii)
n=l
Since the pressure distribution is symmetrical over the line OX, we re-write the above equations by omitting the sine terms:
Pl = - rl cosO + ~ A..r-" cosnO,
r >~a 1.
(iv)
n=l
p2 = ~
C.r"cosnO + ~ E.'r-"cosnO,
n=l
P3 = ~
a2<~r<<.al.
(v)
•=1
G . ' r " cos nO,
r <~a 2.
(vi)
n= 1
Applying the first boundary condition at r = a t , Pl =P2, so
- a,I cosO + ~ A.'aC" cosnO = ~ C.a~" cosnO + ~ E.a-~" cosnO. n=l
n=l
n=l
Equating coefficients of cosine terms on both sides we get n = 1: - a~I + (A~/a,) = C~al + (El~a1)
(vii)
n > 1: (A./a])
(viii)
= C.a] + (E./a~)
Again at r = az, P2 =P3, and so we get
{C.a"2 + (E./a~)} cos nO = ~ G.a"2cos . 0 . n=l
n=l
Equating coefficients of cosine terms we get n ) 1:C.a"2 + E.a~" = G.a~. Now applying the second boundary condition 63p I
at we get
t3p2
r = al, K1 8 r = K 2 ~ r '
- KlI cosO-- K~ ~ A..nal"-~'cosnO
= r2 ~ n=l
n(C.a"~ -~ - E . ' a ? " - ~ ) c o s n O .
(ix)
402
MOHAMMAD SALEEM
Equating coefficients of cosine terms on both sides n = 1: - K I I - K1Ala-; 2 = K 2 (C1 - E t a ; z) n>l:
- K 1 A n . n . a ~- n - 1
(x)
= K 2 n ( C n a tn- 1 - E n a ~ n - 1 ) .
(xi)
Also at t~P2
r = a2,
•P3
K 2 t~r
K3 -0r
=
and hence:
K2
n(Cna
-' - E n a ; ° - ' ) c o s n 0 = K3
n=l
a -"cosn0. n:l
Equating coefficients of cosine terms on both sides we get n >1 1 : KEn (Cna~- ~ -- Ena~"- l) = K a n G . a ~ 2.
(xii)
N o w f r o m (vii): - a ~ I + A~ = C~ a~ + E~ and f r o m (x) : - Kla~ I - K IA t = K2 Cl a~ -- K 2 E 1 on adding these we get: - a ~ I ( 1 + K 1 ) + A 1 (1 - K 1 ) = C , a l (1 + K2) + E~ (l - K2) or
A~ = C l a t ' l
1+ g _ g
2
+
a2i. 1 + K1 1 ~ V ~ K.I + E ~ ' ~ _
-
-
K 2
K .
(xiii)
Since Lim K2 = 0, K3~aO K3
so f r o m (xii) we get G~=0,
n/> 1.
(xiv)
,
(xv)
F r o m (ix) En=-C,,a2
2n
n/> 1.
Substituting (xv) in (viii) we get A, = C. (a~"
2,, -a2),
n>
1.
(xvi)
F r o m (vii) we have A~ = a2tl + Cla~ + Et .
(xvii)
Using (xv) for n = 1, Eqs. (xvii) and (xiii) become A~ = a~I + C t (a 2 - a~) and A 1 = Cla~" 1 + K 2 i =K, +a2I
I + K~ 1 --g 1
1 -- K 2 2 CI'I-K1 "a2
(xviii)
A SIMPLE METHOD OF G R O U N D W A T E R DIRECTION MEASUREMENT
403
or
A1 ~ C1 ( a211 +- K2K, Ii - K, K2 a2) + a2I'll
-+ KI"K 1
(xix)
Equating Eqs. (xviii) and (xix) we get { 2 K1 + K2
2 K 1 - K2X~ a2" i ~ k I ) ~- - 202K1[/(1 - g l )
C, ka," ~ Z K 1 OF
- 2a2KxI CI - gx (a 2 - a22) + (a 2 + a22) Kz
.
(xx)
.
(xxi)
F r o m (xv) for n = 1 2 2 2ala2KlI
E1
=
K l (a 2 - a~) + (a 2 + a 2) K 2
Since K 1, K2, a 1 and a 2 are all non-zero, so on combining (xi) and (xvi) we get on using (xv) C n = 0, n > 1 (xxii) E.=0,
n>l
(xxiii)
An=0,
n > 1.
(xxiv)
and f r o m (xvi)
Substituting (xx) in (xviii) we get
A l = a2I -
2a2KtI (a2 - a22) K, (a 2 -- a 2) + K 2 (a 2 + a~)"
(xxv)
Substituting Eqs. (xx) to (xxv) and Eq. (xiv) in Eqs. (iv) to (vi) we get the values of p l , P2 and P3 in the three regions as
pt = - rI cosO +
2alKlI (al - a2) a ~ I - K l ( a ~ - - a ~ ) + - K 2 ( a ~ +a~)
cosO
r
which on simplification gives:
[K2 (a~ + a~) - Kl (a~ - a~) a~
t
Pl = [ K 2 (a 2 + a22) ~ KI (a 2 - a2)" r -- r . I cos 0.
(xxvi)
Like-wise
P2 -- K1 (a 2 _ a~) + K2 (a 2 + a2)" P3 = 0.
( )cos0 r
xxvii (xxviii)
T o find the horizontal discharge q per unit length of the bore hole passing in and out of it, we proceed as follows.
404
MOHAMMAD SALEEM
The radial velocity ~ in region (2) is Vr = _ k 20P2
~r 2a~K1K2I
.(a!
)
=K,(a 2 - a 2)+K2(a~+a 2) \r z + 1 cos0, and ....
=
4aEKa I cos 0 (K1/K2)(a~ -- a 2) + a~ +
a 2"
Now 3~t/2
q= f
[V~]. . . . "a2 dO
n/2 or
q=
8rata2 (K,/K2) (a~ - a~) + a~ + a~
(xxix)
where
v = KII = unperturbed velocity of sub-soil flow in the aquifer. The ratio K1/K 2 can be found experimentally by a technique described in appendix 2. FOR A BLIND PIPE
Now suppose that the strainer of Fig. 9 is replaced by a blind pipe having the same inner and outer radii. We have, in this case, the following limiting conditions K2
and
-*0
K1 K2 -
(xxx) --~O.
K3
Using (xxx), the Eqs. (xxvi) to (xxviii) become b P l = --
bP2
(o: :) nt-
Ial
cos 0
(xxxi)
a~ -- a~
bP3 = 0.
(xxxiii)
The perusal of Eqs. (xxviii) and (xxxiii) shows that the pressure P3 at any
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
405
point within the pipe (sand-free region) is zero irrespective of the fact whether the pipe in question is a pure strainer or a pure blind pipe. Likewise when the pipe is neither a pure strainer nor a pure blind pipe but instead consists of two segments like upper or lower region of Fig. 1 then again we have the limiting condition: K z / K 3 -+ 0
and hence P3 is zero in this case too. Consequently the pressure at any point within the entire bore hole of Fig. 2 is zero irrespective of the mode of perforations made along the length. This excludes the possibility of any vertical inter-regional flow as a result of non uniform perforations in the three regions of Fig. 2. Appendix 2 In these pages a simple technique for the determination of both K 1 and K 2, as defined in appendix 1, is described. EXPERIMENTAL SET UP
An impermeable concrete tank measuring 3 meters × 3 meters × one meter is selected. A 20 cm wide and one meter deep gravel filled region is build round the inner periphery of the tank as shown in fig. 10. The remaining portion of the tank is filled with some medium grain sand. A 1.2 meters long and 10 cm diameter brass strainer with proper slit size is installed at the centre of this tank to work as tube well. Two observation wells which are also filter pipes (diameter about 2.5 cm and length 1.2 meters each) are installed FILLED
O
OBSERVATION WELL 2 O OBSERVATION WELL 1
_J bJ
TUBE WELL
m~ !~ I
co
SATURATED SAND
I I It ]
I
J FOP, WATER SUPPLY
Fig. 10.
Diagram showing the details of the experimental tank for determining the ratio
K1/K~. There is a gravel filled region near the outer periphery of the tank for constant water supply to the tubewell installed in the saturated sand. The observation wells 1 and 2 are meant for measuring the draw-down at known distances from the tubewell.
406
MOHAMMAD SALEEM
in the sandy region at different distances fron the tube well as shown in the figure. Water is supplied into the gravel filled region with the help of syphon arrangements so that a constant supply level is maintained in this region. This region in turn feeds water into the sandy zone. When steady conditions are set up in the whole tank pumping is started. The pumping, however, is an open well pumping done with the help of a syphon. When the tube well starts functioning, the water level both in the tube well and observation wells falls down. To record accurately the water levels in all the wells, we again employ the syphon principle. This is illustrated in Fig. 11. Clearly the water level
GLASS TUBING~ //
~]
MEASURING-~ SCALE
j~j, "
i
"~TUBEWELL "~ WATER LEa,C:L
RTUBING 7PLUG BAIL Fig. 11.
Diagram showing the arrangement for knowing accurately the water level in the tubewell during its working. This is based on simple syphon method.
in the glass tubing is the same as in the tubewell. While setting the experiment it should be made clear that there are no air bubbles in the rubber tubing. To remove the air bubbles from the rubber tubing, we take off its end from the glass tubing and allow water to flow out of the rubber tubing for some time. When all bubbles are expelled then its end is slipped on to the glass tubing. Similar arrangements are made to record water levels in the observation wells. Care is taken that the recording glass tubings for the three wells are placed parallel to each other and have a common zero. Another tubing to record the water supply level in the gravel filled region ia also placed parallel to the above recording tubes over the same scale. When the tubewell operates for some time, the water levels in the various pipes will attain a constant value. These constant levels are recorded along with the constant discharge of the tubewell. Then (lPe-Pw) denotes the
A SIMPLE METHOD OF GROUND WATER DIRECTION MEASUREMENT
407
difference in levels between the observation well (l) and the tubewell. Likewise ( 2 P e - - P w ) denotes the similar difference with regard to observation well (2). After this the discharge Q of the tubewell is varied to some other constant value and similar observations are made after steady conditions have been reached in the system, and so on. We now plot two curves: p Q versus ( i p e - P w ) and (2P~-Pw) on the same graph paper (# being coefficient of viscosity of water at the temperature of the experiment). Selecting a common value of pQ corresponding nearly to the middle of these curves, we find the corresponding values of (lPe--Pw) and (2Pe--Pw). The calculations for the determination of the permeability of the given brass strainer are carried out as detailed below. MATHEMATICAL APPARATUS
Let al and a 2 be the outer and inner radii of the brass strainer respectively. Let K 1 be the permeability of the water bearing medium, called region (1), in which the strainer has been installed. Let the average permeability of the material of the brass strainer be g 2. This is denoted as region (2). As the well is tapping the full depth of the aquifer, there is no flow in the vertical direction (z-direction) after pumping is started. The flow is assumed to be purely radial so that it is independent of the variation of angle 0. We start with the Laplace equation in cylindrical polar coordinates and solve it when the flow is purely radial. We thus have V 2 p = r ~r
r
= 0
or
?p r
ar
= const.
Integrating p = C l o g r + C o where p is pressure at the point at distance r from the axis of the well and C, Co are constants. We therefore write the pressure distributions in regions (1) and (2) b y p l andp2 as P i = C1 l o g r + C2
(1)
P2 = C3 l o g r + C4.
(2)
Applying the first boundary condition at (::3pi
r=al:K
0p2
l ~r = K2 ~r '
408
MOHAMMAD
SALEEM
we get
1
K1CI.--
1 = K 2 C 3.
a1
al
or
K1 C3 = K2 C1 = KC1
(3)
where K =
K1
.
(4)
K2 Substituting this value of C3 in (2) we get P2 = KC1 l o g r + C 4.
(5)
Now applying the second boundary condition at r = a l :Pa =P2, we get Ca l o g a l + C2 = KC~ l o g a t + C4 or
(1 - K) C, logal + C2 --- C4.
(6)
Substituting this value of (74 in (5) we get P2 = C1 { K l o g r + (1 - K) logal} + C2.
(7)
The next two boundary conditions are at r = a~: px = p~ and at r = a2:P2 = Pw
(8)
where the distance a¢ is the radius of influence - a distance where a reasonable estimate o f p , can be made. Substituting these conditions in Eqs. (1) and (7), we get pC -- C a -loga~ + C 2 (9) Pw = C1 ( K l o g a 2 + (1 - K ) loga~} + C2 = Ca log {a[" a ] ' - r ) } + C2 or
P , = Ca logaw + C2
(10)
a. =
(ll)
where Subtracting (9) and (1 O) C 1 -
P~ - Pw log(a Jaw)
A SIMPLE M E T H O D OF G R O U N D W A T E R D I R E C T I O N MEASUREMENT
409
and Pw" log a¢ - pc log aw C 2
~
-
log(ae/aw)
Substituting these values of C1 and C2 in (1) and (7) we get pw'log ae - Pe'log aw Pe - Pw log(ae/a~) + lo~(aJa~..) " l ° g r
P'or
Pe -- Pw r Pl = P,~ + l o g ( a J a w ) log aw .
.
.
.
.
(12)
and P2 = Pw +
Pe - Pw log (a J a w )
R
.log
(13)
aw
where R = r K" a~l - r )
(14)
The radial velocity Vr is given by
Vr
~
--
KI ~Pl /* Or "
-
"~-
--
K1 P~--Pw 1 p log(ae/aw) r "
- -
. . . .
"
~---
.
.
.
Q 27thr .
.
.
where Q is the discharge pumped by the tubewell, h the depth of aquifer and /~ the viscosity of pumped water, or log (a J a w ) = + KI" 27zh (Pe - pw)/pQ or
aw = a¢.exp { - 2rthK 1 (p¢ - pw)/pQ} Or using (4) and (1 l) a2 tall
= a~.exp { - 2~hK~ .(p~ - Pw)/PQ} al
or
r,,/K2 log(a2/a )
= l o g ( a ~ / a l ) - 2=hK~
(po -
pw)/pQ.
(t5)
Corresponding to the two observation wells, we re-write Eq. (15) as K~ log a2 = log la° - 2K~=h (~p¢ - pw)/pQ K2 al a~
(16)
and KI K2
log a2 = log [ae _ 2K,rch. (2P~ - pw)/pO a~ a 1
(17)
MOHAMMADSALEEM
410
where Xae and 2ae are respectively the distances of the observation wells (1) and (2) from the tubewell. Writing
2nh ~Q
~av
t i p ° - pw) = b x ,
(2Pe - Pw) =
bz,
we have from (16) and (17)
(
1
a2)'ae
K, ab l +K21oga, and
(
,a 0
K1 ab2+KzlOgam
=l°g
al
= l ° g al .
(18)
(19)
Eliminating K t from (18) and (19) we get
K2 =
log (2ae/lae).log
(ai/al)
a (b2 log ('adam) - b, log (2a.la,)
or
K2 =
log (Zao/'ao).log
(a2/a,)
27th
(20)
~/Q" {(2Pe -- Pw) log ('ae/al) - (lPe - Pw)'l°g (Za,/al)} In Eq. (20) all the quantities are known and so Kz, the permeability of the brass strainer material, can easily be calculated. To find the permeability KI of the sand, we subtract Eqs. (18) and (19) and get /~Q log (2ae/lae) K, . . . . . . . (21) 2roll (2Pe - 1P~) It may be noted over here that the sand used in Fig. 10 may not be the same as obtained during sampling at the time of installation of the bore hole. However such samples may be tested for permeability determination by other simple laboratory experiments. These values of K m may then be inserted in the formulae for velocity measurements. The above technique for the measurement of K2 can well be applied to even more complicated strainers like those obtained by tightly winding a coir string on an iron cage.