S OIL M OISTURE –P ERMEABILITY AND C APILLARITY
5.5 THE DETERMINATION OF PERMEABILITY
5.5.5 Determination of Permeability—Field Approach
The average permeability of a soil deposit or stratum in the field may be somewhat different from the values obtained from tests on laboratory samples; the former may be determined by pumping tests in the field. But these are time-consuming and costlier.
A few terms must be understood in this connection. ‘Aquifer’ is a permeable formation which allows a significant quantity of water to move through it under field conditions. Aqui-fers may be ‘Unconfined aquiAqui-fers’ or ‘Confined aquiAqui-fers’. Unconfined aquifer is one in which the ground water table is the upper surface of the zone of saturation and it lies within the test stratum. It is also called ‘free’, ‘phreatic’ or ‘non-artesian’ aquifer. Confined aquifer is one in which ground water remains entrapped under pressure greater than atmospheric, by overly-ing relatively impermeable strata. It is also called ‘artesian aquifer’. ‘Coefficient of Transmis-sibility’ is defined as the rate of flow of water through a vertical strip of aquifer of unit width and extending the full height of saturation under unit hydraulic gradient. This coefficient is obtained by multiplying the field coefficient of permeability by the thickness of the aquifer.
When a well is penetrated into a homogeneous aquifer, the water table in the well ini-tially remains horizontal. When water is pumped out from the well, the aquifer gets depleted of water, and the water table is lowered resulting in a circular depression in the phreatic surface. This is referred to as the ‘Drawdown curve’ or ‘Cone of depression’. The analysis of flow towards such a well was given by Dupuit (1863) and modified by Thiem (1870).
In pumping-out tests, drawdowns corresponding to a steady discharge are observed at a number of observation wells. Pumping must continue at a uniform rate for an adequate time to establish a steady state condition, in which the drawdown changes negligibly with time.
The following assumptions are relevant to the discussion that would follow :
(i) The aquifer is homogeneous with uniform permeability and is of infinite areal extent.
(ii) The flow is laminar and Darcy’s law is valid.
(iii) The flow is horizontal and uniform at all points in the vertical section.
(iv) The well penetrates the entire thickness of the aquifer.
(v) Natural groundwater regime affecting the aquifer remains constant with time.
(vi) The velocity of flow is proportional to the tangent of the hydraulic gradient (Dupuit’s assumption).
Unconfined Aquifer
A well penetrating an unconfined aquifer to its full depth is shown in Fig. 5.6.
Let r0 be radius of central well,
r1 and r2 be the radial distances from the central well to two of the observation wells,
z1 and z2 be the corresponding heights of a drawdown curve above the impervious boundary,
z0 be the height of water level after pumping in the central well above the impervi-ous boundary,
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d0, d1 and d2 be the depths of water level after pumping from the initial level of water table, or the drawdowns at the central well and the two observation wells respectively,
h be the initial height of the water table above the impervious layer (h = z0 + d0, obviously) and,
R be the radius of influence or the radial distance from the central well of the point where the drawdown curve meets the original water table.
Impervious boundary
r2 r2
rr r1 r1
r0 z2
z2 zz zz11 h
h
z0 z0 d2
d1 dzdr
Observation wells
d0 d0
R Original water tableR
Drawdown curve Ground level Central well
q
Fig. 5.6 Flow toward a well in an unconfined aquifer
Let r and z be the radial distance and height above the impervious boundary at any point on the drawdown curve.
By Darcy’s law, the discharge q is given by : q = k.A.dz/dr,
since the hydraulic gradient, i, is given by dz/dr by Dupuit’s assumption.
Here,
k is the coefficient of permeability.
But A = 2πrz.
∴ q = k.2πrz.dz/dr
or k.zdz = q dr
r
FHG IKJ
2π .Integrating between the limits r1 and r2 for r and z1 and z2 for z, k z
z 2 z
2 1
RS
2T UV W
= (q/2π) loge rr
RS|
rT| UV| W|
12
SOIL MOISTURE–PERMEABILITY AND CAPILLARITY 127
∴ k z z q r
e r .( 22 12) ( / ) log 2
2 2 1
− = π
RS T UV W
∴ k = q
z z
r r
q z z
r
e r
π( )log
. ( )log
22
12 2
1 22
12 10 2
1 36 1
− =
− ...(Eq. 5.18)
k can be evaluated if z1, z2, r1 and r2 are obtained from observations in the field. It can be noted that z1 = (h – d1) and z2 = (h – d2).
If the extreme limits z0 and h at r0 and R are applied, Equation 5.18 reduces to
k = q
h z
R 1 36. ( 2 02). log10 r0
− ...(Eq. 5.19)
This may also be put in the form
k = q
d d z
R r 136. 0( 0 2 0). log10 0
+ ...(Eq. 5.20)
For one to be in a position to use (Eq. 5.19) or (Eq. 5.20), one must have an idea of the radius of influence R. The selection of a value for R is approximate and arbitrary in practice.
Sichart gives the following approximate relationship between R, d0 and k;
R = 3000 d0 k ...(Eq. 5.21)
where,
d0 is in metres, k is in metres/sec, and R is in metres.
One must apply an approximate value for the coefficient of permeability here, which itself is the quantity sought to be determined.
Two observation wells may not be adequate for obtaining reliable results. It is recom-mended that a few symmetrical pairs of observation wells be used and the average values of the drawdown which, strictly speaking, should be equal for observation wells, located sym-metrically with respect to the central well, be employed in the computations. Several values may be obtained for the coefficient of permeability by varying the combination of the wells chosen for the purpose. Hence, the average of all these is treated to be a more precise value than when just two wells are observed.
Alternatively, when a series of wells is used, a semi-logarithmic graph may be drawn between r to the logarithmic scale and z2 to the natural scale, which will be a straight line.
From this graph, the difference of ordinates y, corresponding to the limiting abscissae of one cycle is substituted in the following equation to obtain the best fit value of k for all the obser-vations :
k = q/y ...(Eq. 5.22)
This is a direct consequence of Eq. 5.18, observing that log10 (r2/r1) = 1 and denoting (z22 – z12) by y.
Confined Aquifer
A well penetrating a confined aquifer to its full depth is shown in Fig. 5.7.
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r2 r2
r1 r1 rr
r0 z2
z2 zz zz11 zz00 HH Aquifer
Impervious boundary Drawdown curve
d2 dzdr Observation
wells
d1
d0 d0
Central well
Original water table Ground level
Impervious stratum Impervious stratum h
h
Fig. 5.7 Flow toward a well in a confined aquifer
The notation in this case is precisely the same as that in the case of the unconfined aquifer; in addition, H denotes the thickness of the confined aquifer, bounded by impervious strata.
By Darcy’s law, the discharge q is given by : q = k.A.dz/dr, as before.
But the cylindrical surface area of flow is given by A = 2πrH, in view of the confined nature of the aquifer.
∴ q = k . 2πrH.dz/dr
or k.dz = q
H dr
r 2π . .
Integrating both sides within the limits z1 and z2 for z, and r1 and r2 for r, k z q
H r
z z
e r
L
rNM O
QP
=L
NMM O
QPP
1 2
1 2
2π log
or k(z2 – z1) = q
H
r
e r 2π . log 21
or k = q
H z z
r
e r 2π ( 2 1). log 21
−
Since z1 = (h – d1) and z2 = (h – d2), (z2 – z) = (d1 – d2) Substituting, we have :
k = q
H d d
r r
q H d d
r
e r
2 1 2 2 72
2
1 1 2 10 2
π ( ). log 1
. ( ). log
− =
− ...(Eq. 5.23)
Since the coefficient of transmissibility, T, by definition, is given by kH,
T = q
d d r r
2 72. ( 1 2). log ( / )10 2 1
− ...(Eq. 5.24)
SOIL MOISTURE–PERMEABILITY AND CAPILLARITY 129 If the extreme limits z0 and h at r0 and R are applied, we get :
k = q
H h z e R r 2π ( 0). log ( / )0
− But (h – z0) = d0
∴ k = 2π .H dq 0 . log ( / )e R r0 2 72qHd0 10 R r0 . . log ( / )
= ...(Eq. 5.25)
Since T = kH,
T = q d
R r
2 72. 0 . log10 0 ...(Eq. 5.26)
The field practice is to determine the average value of the coefficient of transmissibility from the observation of drawdown values from a number of wells. A convenient procedure for this is as follows:
A semi-logarithmic graph is plotted with r to the logarithmic scale as abscissa and d to the natural scale as ordinate, as shown in Fig. 5.8 :
1 10 100 1000
0 1 2 3 4 5 6
Dd Dd
Radial distance, r (log scale)
Drawdown,d
Fig. 5.8 Determination of T From Equation 5.24,
T = q d
2 72. .∆ ...(Eq. 5.27)
if r2 and r1 are chosen such that r2/r1 = 10 and ∆d is the corresponding value of the difference in drawdowns, (d2 – d1).
Thus, from the graph, d may be got for one logarithmic cycle of abscissa and substituted in Eq. 5.27 to obtain the coefficient of transmissibility, T.
The coefficient of permeability may then be computed by using the relation k = T/H, where H is the thickness of the confined aquifer.
Pumping-in tests have been devised by the U.S. Bureau of Reclamation (U.S.B.R.) for a similar purpose.
Field testing, though affording the advantage of obtaining the in-situ behaviour of soil deposits, is laborious and costly.
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