Drawdown due to ground-water abstraction with straight lines of wells

D R A W D O W N D U E T O G R O U N D - W A T E R

A B S T R A C T I O N W I T H S T R A I G H T L I N E S O F W E L L S

B Y I R . L . H U I S M A N

S E N I O R N E W W O R K S E N G I N E E R , M U N I C I P A L W A T E R W O R K S , A M S T E R D A M

P A S T M E T H O D S O F C A L C U L A T I O N

I n 1915 the M i n i s t e r o f t h e I n t e r i o r ordered the d r a w i n g up o f plans f o r a central d r i n k i n g water supply i n the provinces of South H o l l a n d , N o r t h H o l l a n d and U t r e c h t . T h e Government Institute f o r Water Supply car-ried out this w o r k and described its findings i n m i n u t e detail i n a report published i n 1919. Ground-water always has been - and still is - con-sidered the safest and most reliable source f o r h u m a n consumption and more or less as a matter o f course the above mentioned plans were, there-fore, based on ground-water abstraction, f o r w h i c h a line o f wells had been projected.

I n 1919, that is a little over 40 years ago, different catchments w i t h ground-water recovery f o r public and industrial water supplies were already i n operation, some even f o r many years. Due to lack of knowledge and understanding, the design o f a new system, however, still offered m a n y difficulties and ample opportunities f o r failure. These diflRculties were not so m u c h due to the circumstance that i n this western p a r t o f the Netherlands ground-water generally is salt or brackish. A l o n g the river Lek the d r i l l i n g of a large number o f testholes revealed the presence of fresh water pockets of sufficient magnitude w i t h regard to the proposed amount of abstraction and w i t h a chemical composition of the g r o u n d -water otherwise such that i t could easily be made fit as source f o r a potable supply. T h e real d i f f i c u l t y concerned the lowering o f the ground-water table inside the i n d i v i d u a l wells of the system d u r i n g operation. A t t h a t time submersible pumps had already been applied f o r many years, b u t

they were very expensive and rather unreliable, necessitating ground-water abstraction w i t h central pumps and suction lines. W i t h regard to the l i m i t e d amount of suction head available, this system is only feasible f r o m an economic p o i n t o f view w h e n d u r i n g operation the water level i n the i n d i v i d u a l wells does not drop more then 6 to 7 m below g r o u n d -surface. I t was the duty of the hydrologist concerned to prove that his design satisfied this requirement.

As catchment area for the proposed central d r i n k i n g water supply a site near Jaarsveld had been chosen. T h e geo-hydrologic profile here showed what nowadays w o u l d be called a two-storied leaky artesian aquifer between depths of 5 to 10 and 100 m below ground surface. A t the b o t t o m this aquifer is bounded b y a practically impervious layer o f clay, at the top overlain by strata of clay and peat and subdivided i n t o two parts by deposits of silty sand and clay roughly between 55 and 75 m below ground surface.

As to confined aquifers w i t h o u t recharge, bounded at top and b o t t o m by f u l l y impervious aquicludes, formulas of the d r a w d o w n curves ac-companying ground-water abstraction w i t h a single well are already k n o w n for many years. Starting f r o m D U P U I T ' S assumption (1863) that w i t h horizontal aquifers of constant thickness the vertical components of fiow can be neglected, these formulas are derived by a p p l y i n g D A R C Y ' S law (1856) and the equation of continuity. For a well i n the centre of a circular island (FIG. 1), this gives:

da9

D A R C Y Q = -2nr • H • k - - ^

dr C o n t i n u i t y Q = constant = Qo

E l i m i n a t i o n of Q f r o m both relations produces the general d i f f e r e n t i a l equation:

d = -^ 2nkH' r

By integration between the limits r = r, (p = (p and r = R, cp = 0 finally D U P U I T ' S f o r m u l a is obtained:

Qo , R

i n w h i c h tp is the steady state d r a w d o w n of the artesian water table at a distance r f r o m a w e l l pumped at a constant rate Qo. T h e radius o f the island is represented by R, while k, the coefficient of p e r m e a b i l i t y and H the saturated thickness of the aquifer are f o r m a t i o n constants. W h e n the above mentioned differential equation is integrated between the limits ?• = ; i , (p = (pi and r = r2, <p = <p2, the f o r m u l a of T H I E M (1870) is obtained:

Qo , »-2

InkH ri

g i v i n g the difference i n d r a w d o w n for 2 points at distances n and rg f r o m the well centre. W i t h good approximation this f o r m u l a proved true f o r all geo-hydrologic conditions, for steady as well as f o r unsteady flow and for every r a n d o m position of the w e l l . This success i n i t i a t e d the use o f D U P U I T ' S f o r m u l a also for other topographic conditions. There one could not speak of a circular island, and R was called the radius o f the sphere of influence, whatever the meaning of this t e r m m i g h t be. T h e fact that i n reality the value o f R depends on the effective distance to open water at a constant level and w i t h a given situation even varies f r o m one p o i n t of observation to another was not always f u l l y recognised. M i x i n g up steady w i t h near-steady flow also added to the confusion and as regards the value of R, formulas have been derived, w h i c h actually set back the progress o f ground-water hydrology.

Such was the state of hydrologie knowledge when D r . I r . J . VERSLUYS, m i n i n g engineer o f the Government Institute for W a t e r Supply, was called upon to design the ground-water catchment at Jaarsveld. O n one

hand VERSLUYS doubted the validity of D U P U I T ' S f o r m u l a i n the case under consideration, while on the other hand this f o r m u l a only yields numerical results when the values of the geo-hydrologic constants R and the product kH, the so-called coefficient of transmissibility, are k n o w n . For b o t h reasons VERSLUYS decided to carry out a test p u m p i n g . I n both parts of the aquifer f u l l y penetrating wells were set and p u m p e d one by one w i t h various capacities up to 2 4 0 m^/hour. The accompanying lower-ing o f t h e artesian water table was measured each time inside the pumped w e l l and i n the aquifer f r o m w h i c h abstraction occurred at distances f r o m

1 0 , 4 0 and 8 0 m f r o m the w e l l centre. F r o m the result o f this and other test pumpings VERSLUYS first drew two i m p o r t a n t conclusions:

1. i n any point of the aquifer, including the well itself, the d r a w d o w n of the piezometric surface is linearly p r o p o r t i o n a l to the capacity of abstraction;

2 . the actual amount o f d r a w d o w n depends on the shape and extent of the ground-water body and on the location o f t h e p o i n t of observation, but is independent o f the presence of existing ground-water flows. Reduced to an abstraction o f l m^/minute, that is 1 4 4 0 m^/day, VERSLUYS measured drawdowns of 0 . 3 5 m inside the w e l l and 0 . 2 7 m , 0 . 1 7 3 m and 0 . 1 0 m at distances of 10, 4 0 and 8 0 m f r o m the well respectively. F r o m these measurements he concluded:

3. the d r a w d o w n o f the piezometric surface decreases linearly w i t h the logarithm of the distance to the centre of the p u m p e d well.

T h e 3 conclusions together constitute D U P U I T ' S f o r m u l a . F r o m the data given, VERSLUYS derived the relation:

§ 0 , 2 5 0

w = • m

^ 2 7 r - 2 6 5 0 r

i n w h i c h (p and r are expressed i n m and Qo i n m^/day.

T o use this f o r m u l a for the calculation of the d r a w d o w n due to a number of wells, VERSLUYS otherwise had on the other hand to k n o w their m u t u a l influence. F r o m his first rule he concluded i n this respect:

4 . the d r a w d o w n d i s t r i b u t i o n due to p u m p i n g a w e l l system is the sum of the distributions due to p u m p i n g the i n d i v i d u a l wells.

Nowadays this rule is k n o w n as the principle of superposition and derived f r o m the circumstance that for the flow of artesian water the Laplace equation is linear.

W i t h this principle of superposition f o r m u l a t e d , the calculation o f the d r a w d o w n due to p u m p i n g a straight line o f wells is only a matter o f mathematics. W h e n the line contains n wells at constant intervals b and p u m p e d at capacities of Q i , Q2, Qn, the d r a w d o w n inside the various wells (neglecting w e l l losses) becomes:

Ql R Q2 . R ^ Qs . R ^ ^ Qn . R 27ikH b 2jikH ro 2nkH b 27ikH {?i-2)-b W i t h a central suction line of ample dimensions so that pipe losses are negiigeable, the d r a w d o w n inside the wells is the same for a l l units:

(pi = q>z = . . . = 95» = (po

W i t h the total capacity o f the system fixed at Q: Q = Ql + Q2 + ... + Qn

a set o f 1 linear equations i n (po and Qi to Qn must be solved. This is a time c o n s u m i n g j o b b u t i t d i d not offer any special difhculties to V E R S L U Y S .

D E V E L O P M E N T 1 9 2 0 - 1 9 6 0

L o o k i n g back 4 0 years, i t must be said that i n terms o f practical results obtained, the work of VERSLUYS is a complete failure. T o VERSLUYS i t is more or less a matter for congratulation that liis scheme for ground-water abstraction never materialized and consequently i t never appeared t h a t actual drawdowns were far higher than calculated by h i m .

I n the w o r k of VERSLUYS for the ground-water catchment at Jaarsveld three parts can be distinguished:

1. the carrying out of a test p u m p i n g to determine experimentally the relation between the amount of d r a w d o w n and the distance to the point of abstraction;

2 . the analysis o f the test p u m p i n g for purposes o f i n t e r p o l a t i o n and extrapolation of the drawdowns observed;

3. the application o f t h e method of superposition to calculate the draw-d o w n draw-d i s t r i b u t i o n draw-due to the projectedraw-d line of wells.

Present hydrologie knowledge can f i n d no f a u l t i n the t h i r d step o f his work, b u t as regards the first two the f o l l o w i n g m a j o r deficiences can be distinguished:

1. the test p u m p i n g carried out by VERSLUYS was completely insufficient to determine the shape o f the d r a w d o w n curve w i t h any accuracy or certainty. The test w e l l was situated at a short distance f r o m the river Lek (explaining the otherwise too small d r a w d o w n inside the p u m p e d w e l l ) , resulting i n aberrations w h i c h even VERSLUYS must have f o u n d d i f f i c u l t to accept;

2. when analysing the result of the test p u m p i n g VERSLUYS d i d not take the presence of the river Lek into account. This is the more strange as j u d g i n g f r o m his second conclusion, VERSLUYS was well aware of the influence o f size and shape o f t h e ground-water body on the actual amount of d r a w d o w n and zero lowering at a distance of 250 m f r o m a w e l l i n an infinite aquifer certainly must have looked strange to h i m . This neglect had the more serious effects as the final catchment was projected at a far greater distance f r o m the river;

3. w i t h a leaky artesian aquifer the relation between d r a w d o w n and distance to the pumped well is not governed by a semi-logarithmic expression b u t by a Bessel f u n c t i o n or a combination o f these func-tions. W i t h the d r a w d o w n curves according to b o t h expressions co-i n c co-i d co-i n g co-i n the near v co-i c co-i n co-i t y o f the well, at greater dco-istances to the p o i n t of abstraction the Bessel functions give a m u c h larger lowering of the artesian water table.

VERSLUYS may be excused f o r the last deficiency, but as regards the two first mentioned ones, he certainly can be held liable. Even i n his time w i t h a mathematical treatment of ground-water flow problems still to come, f a r better results could have been obtained.

T h e development of ground-water hydrology i n the years after the w o r k of VERSLUYS f o r the proposed catchment at Jaarsveld shows many aspects. As first m a j o r gain can be mentioned the way to carry out a test p u m p i n g . Even w i t h such an i m p o r t a n t project as supplying a quarter o f a m i l l i o n people w i t h ground-water taken f r o m a single catchment near Jaarsveld, VER.SLUYS measured the d r a w d o w n only i n 3 observation holes, to a m a x i m u m distance of no more than 80 m f r o m the p u m p e d well and d i d not pay any attention to the accompanying lowering o f the water table

i n the bounding aquifers. Nowadays w i t h such a case, the d r a w d o w n w o u l d have been measured w i t h about 30 observation holes, set i n 4 direc-tions at 90° interval, at distances f r o m 10 to 1000 m f r o m the p u m p e d w e l l and equipped w i t h different screens placed i n the various aquifers. T h e mistake of locating the pumped w e l l close to the river Lek (not only complicating the analysis of test p u m p i n g results enormously b u t even m a k i n g this well-nigh impossible w i t h complex profiles) w o u l d not occur and the well w o u l d have been pumped w i t h a far greater capacity as corresponds w i t h a d r a w d o w n inside the w e l l of 1.4 m only, m a k i n g u n -avoidable n a t u r a l variations i n piezometric level (due to showers, fluc-tuations i n barometric pressure, changes i n the level of b o u n d i n g streams, etc.) negiigeable.

A test p u m p i n g as described above gives a f a i r l y accurate relation between d r a w d o w n and distance to the pumped well. W h e n the d r a w d o w n curves i n a l l 4 directions are about the same, so that any influence o f boundary conditions may be assumed absent, this relation can be used w i t h o u t f u r t h e r analysis to calculate the d r a w d o w n caused by a given w e l l system. N o t w i t h s t a n d i n g all precautions, however, this relation is troubled w i t h different errors due to inaccuracies i n observation, n a t u r a l fluctuations i n piezometric level, local variations i n geo-hydrologic conditions etc. and f u r t h e r m o r e u n k n o w n at larger distances f r o m the well. A t a large dis-tance f r o m a number of wells however, the d r a w d o w n may still be consid-erable. A n analysis of a test p u m p i n g is, therefore, still necessary to smooth out the d r a w d o w n curve and to calculate the drawdowns lbr greater distances, up to i n f i n i t y . Such an analysis, however, is only possible w h e n the formulas for the d r a w d o w n curves of wells under various geo-hydro-logic conditions are k n o w n .

Assuming VERSLUYS had not made the obvious mistake o f l o c a t i n g the test well close to the river, then indeed D U P U I T ' S f o r m u l a could have been used as a first a p p r o x i m a t i o n to analyse the results obtained. Strictly spoken however, this f o r m u l a only holds true f o r artesian aquifers w i t h -out recharge, confined at top and b o t t o m by f u l l y impervious layers. I n Jaarsveld the bounding layers i n the meanwhile are not impervious, only less pervious, resulting i n a recharge f r o m above and f r o m below. For the most simple case of such a leaky artesian aquifer, recharged only f r o m the unconfined aquifer w i t h constant water table above the semi-pervious

layer (FIG. 2 ) , DE G L E E ( 1 9 3 0 ) gave as derivation o f t h e d r a w d o w n f o r m u l a : d(p D A R C Y C o n t i n u i t y Q = -2m-H-k dr dQ dr -2m-• «O i ^ ^ ^ ^ ^ ^ 1 • H • . ; . 1 • • 1 ' ' ^ < / / / / / / / / / / / / / / / / ,

phreatic water table artesian water table

F I G . 2 W e l l i n a l e a k y artesian a q u i f e r

/y ///

i n w h i c h c is the resistance of the semi-pervious layer against vertical water movement. D i f f e r e n t i a t i n g D A R C Y ' S law to r and substituting the value of dQ/d?- i n the continuity equation gives after rearranging terms:

kHc ^

W i t h the boundary conditions r ^ oo, 97 = 0 and r ro, Q differential equation has as solution:

d > 1

d r a ' ^ T ' d ^

Qo this

Ko w i t h 1 = Vkllc ' 2nkH \r,

and Ko a modified Besselfunction o f t h e second k i n d , order zero.

T h e proper situation i n Jaarsveld, where the aquifer i n w h i c h the test well was set, is recharged f r o m the phreatic water ( w i t h constant table) above as w e l l as f r o m the artesian water below (FIG. 3 ) , has been investigated by

H U I S M A N and K E M P E R M A N ( 1 9 5 1 ) . A c c o r d i n g to their calculations:

upper aquifer: d(pi D A R C Y C o n t i n u i t y Ql d Q i dr -2m--Hi-ki Cl dr (pi—i C2. 1 0 4

_L1_

C2

E l i m i n a t i o n o f Qi and Q2 f r o m both sets of equations gives:

d^cpi 1 d f l (pl _ y i — _ Q

d?-2 dr kiHiCi kiHiCt d^ffi-a 1 d(p2 (p2 — (pi Q

d/'3 r dr k2H2C2

W i t h the boundary conditions ?• ^ 0 0 , 991 = 0, ^2 = 0 and r -> ro,

Ql = Qo, Qz = 0, these differential equations have as solution:

f2 = (pi = ^ { ( A i - a a ) • KoiHi • r) + ( « 2 - A 2 ) • Ml^h • r)} 2jikiHi Al—A2 QO «2

2nkiHi X1-X2

k2H2C2' ^ kiHiC2

I t is w i t h these formulas, that the analysis o f test p u m p i n g results i n Jaarsveld properly should have taken place. H U I S M A N and K E M P E R M A N have shown how such an analysis can be carried out w i t h success, pro-vided sufficient observations about the lowering o f the artesian water level i n b o t h confined aquifers are available.

For the near v i c i n i t y of the well, the formulas of the d r a w d o w n curves w i t h the different types of artesian aquifer can be w r i t t e n w i t h good approximation i n simplified f o r m : R cp D U P U I T 1 " D E G L E E 2nkH Qo In 1.123A • I n -.123-Z H U I S M A N / K E M P E R M A N cpi : 2nkiHi r w i t h the geo-hydrologic constant L determined f r o m :

In Z = In Vh H In

V X2

of the distance to the p o i n t o f a b s t r a c t i o n

T h a t is to say i n all cases o f confined aquifers there exists here a linear relationship between Q and the l o g a r i t h m of the distance to the point of abstraction. VERSLUYS Hmited his observations to the near v i c i n i t y of the well and even i f the concept o f a leaky artesian aquifer had been k n o w n to h i m , he w o u l d have been unable to choose between the dif-ferent possibilities. W i t h the 3 d r a w d o w n curves i n the near v i c i n i t y of the w e l l coinciding, at greater distances however the remaining draw-106

d o w n is larger to m u c h larger as the recharge f r o m the bounding aquifers is more i m p o r t a n t (FIG. 4 ) . This again w o u l d have given VERSLUYS an unpleasant surprise.

W i t h the results o f the test p u m p i n g analysed and the l o w e r i n g o f the artesian water table k n o w n at any distance f r o m the point of abstraction, VERSLUYS was able to calculate the d r a w d o w n distribution due to the proposed straigth line of wells by a simple application of the method of superposition. Nowadays the same method is still used w h e n the wells are few i n number and must still be applied w i t h a random position o f the wells, when the location of the different wells cannot be described w i t h simple mathematical formulas. For a large number of wells o f con-stant capacity equally spaced i n a straight line, however, new formulas have been developed which i n m u c h shorter time give the most i m p o r t a n t results. They are treated i n the t h i r d part of this paper.

Reviewing the development of ground-water hydrology i n the past 4 0 years, i t w i l l be clear f r o m the preceding that the most i m p o r t a n t aspect is the successful application of mathematics to problems of ground-water f l o w . This application not only provided formulas to describe mathemat-ically the dilferent types o f flow, b u t also assisted greatly i n o b t a i n i n g a better understanding of flow phenomena. I n the last decade this applica-tion reached a. new high i n the co-operaapplica-tion between the Netherlands Hydrologie C o l l o q u i u m and the M a t h e m a t i c a l Centre i n A m s t e r d a m . T h e p u b l i c a t i o n of this C o l l o q u i u m about the steady flow of ground-water to wells ( 1 9 6 2 ) is there to show how m u c h this co-operation has achieved. Outside the Netherlands a similar development can be noticed for w h i c h only the names of T H E I S , J A G O B and especially H A N T U S H need to be mentioned as proof.

Nowadays the situation is such that as regards steady flow o f g r o u n d -water i n schematized geo-hydrologic proflles, any problem can be solved. For most cases formulas are available or can be derived, i n exceptional cases only recourse has to be taken to numerical methods or model tests. M o d e l tests, especially tests w i t h the glass-plates model developed b y D I E T Z and S A N T I N G for the Government Institute o f D r i n k i n g W a t e r Supply still play a m a j o r role i n the calculation of non-steady flow problems and i n problems of steady flow where a larger number o f b o u n d

-ary conditions must be investigated or the values of the geo-hydrologic constants vary strongly over short distances. I t may be expected, however, that i n the near f u t u r e they w i l l be superseded by the apphcation of electronic computers, w h i c h i n shorter time are able to give more accurate results.

S u m m i n g up, the most i m p o r t a n t gain i n the development o f the science of ground-water hydrology is the co-operation between hydrologie practi-tioners and professional mathematicians. I n this respect, however, ground-water hydrology does not differ f r o m any other field of science.

P R E S E N T M E T H O D S OF C A L C U L A T I O N

W i t h ground-water abstraction by lines of wells, central pumps and suction lines nowadays are superseded by a system where each w e l l is equipped w i t h its o w n p u m p (usually o f the deep w e ü c e n t r i f u g a l type). W i t h these well pumps each well delivers the same amount o f water and

" T

u

t

constant phreatic water table :;;;:.v'. .v/.wv,',"| l ' ; ; r ' r : ; . ' . v . i : ^ / . : resistance c 11 . ' • • . _ transmissivity k H V///////////^:^///////////, F I G . 5 I n f i n i t e s t r a i g h t l i n e o f wells i n a l e a k y artesian a q u i f e r o f u n l i m i t e d e x t e n t 1 0 8

w i t h an infinite number of such wells i n a straight line, the dr a w d o w n cp i n a random point {x,y) o f a leaky artesian aquifer w i t h unUmited extent

(FIG. 5) is given b y :

2nkH z_j \ A ?; = — CO

i n w h i c h A is the characteristic length of the f o r m a t i o n : A = Vk-H-c

and the wells are supposed to penetrate the f u l l saturated thickness o f the aquifer.

W i t h the i n t e r v a l b equal to or larger t h a n A, the serie converges r a p i d l y and already a few terms suffice to give accurate results. I n most cases, however, b is small compared to X and computing 99 w i t h the f o r m u l a above w o u l d involve a l o t o f figure-work. W i t h b smaller t h a n about 1/2 to 3 / 4 , the afore mentioned publication o f t h e Netherlands H y d r o l o g i e CoUoquium gives as good approximation (fully penetrating wells):

. _ i i ^ f i _ 2 e - ' - ' ' c o s ^ + e 2jTkH\ b

-•inxlb

This f o r m u l a looks complicated, but can be simplified i n d i f f e r e n t ways: 1. at greater distances f r o m the line of wells, w i t h larger values oï x, the

factors g"^"'" and e^'^'"''' r a p i d l y approach zero, reducing the draw-d o w n f o r m u l a t o :

_ Qo _^ix ^ Qo ^ _.v/A

~ 2nkH • T • ^ ~2J>'kH''

W h e n the fine of wells is replaced by the equivalent gallery, that is a gallery w i t h the same capacity per lineal meter:

Qo ? o = y

the d r a w d o w n 93I due to ground-water abstraction w i t h this gallery can easily be calculated at:

T h a t is to say, at greater distances f r o m the line o f wells, the draw-downs equal those o f the equivalent gallery. A t small distances there exists a difference between both drawdowns, but w i t h x > bj2, this difference is less than 1 % and negiigeable.

2 . at the face of each well, x = 0 and j = rQ-\-n.-b, the f o r m u l a f o r an infinite straight line of f u l l y penetrating wells gives as d r a w d o w n :

Qo X Qo

In 2 sin

2b kH 2nkH \ b

W i t h the d r a w d o w n at the face of the equivalent f u l l y penetrating gallery equal t o :

^ _ ^ ^ Qo X

~ Y ' k H ~ 2 b ' k H and w i t h rojb small, thus:

. 71)0 TD'O

the d r a w d o w n at the face of the different wells is given b y :

1 , Qo 1 ^

90 = + 7 r —27ikH 2mo I n

-halfway between two wells, ,v = 0, j = i / 2 + nb the d r a w d o w n is given b y :

\nX 2nkH\b

w h i c h also can be w r i t t e n as

n!2 = TTTTA-r ~ i l n 4

4. according to 2 and 3, i n the line o f wells the d r a w d o w n at the well face is larger and h a l f w a y between the wells smaller t h a n that o f the 110

equivalent gallery. Somewhere i n this line of wells (.v = 0), at a dis-tance w f r o m each well, the d r a w d o w n must consequently equal the d r a w d o w n of the equivalent gallery:

f r o m w h i c h follows W = bjS or: <Pi>/6 =

A c c o r d i n g to the formulas given above, the d r a w d o w n due to p u m p i n g an infinite straight line o f f u l l y penetrating wells i n a one-storied leaky artesian aquifer can be split up i n two parts, the d r a w d o w n y i of a f u l l y penetrating gallery w i t h the same capacity per lineal meter and for the near v i c i n i t y of the w e l l corrections A95 to compensate for p o i n t abstrac-t i o n . T h e magniabstrac-tudes o f abstrac-these correcabstrac-tion facabstrac-tors only depend on abstrac-the conditions i n the near v i c i n i t y of the well. As has been shown i n the second part o f this paper, f o r m a t i o n losses here are only a f u n c t i o n o f the coefficient of transmissibility kH of the aquifer concerned ( f o r m u l a o f T H I E M ) , and are the same for confined or unconfined flow, w i t h or w i t h -out recharge, for steady as w e l l as for unsteady conditions, etc. T h e for-mulas derived above for the additional drawdowns due to p o i n t abstrac-t i o n f r o m a one-sabstrac-toried leaky arabstrac-tesian aquifer, abstrac-therefore, are v a l i d under all circumstances, independent of geo-hydrologic profile, boundary condi-tions, etc. W i t h this proposition, however, the split up mentioned above gives a very simple method to calculate the dra.wdown d i s t r i b u t i o n caused by p u m p i n g an i n f i n i t e straight line of f u l l y penetrating wells at intervals b w i t h equal capacities Qo under a l l circumstances:

1. determine the d r a w d o w n (p^ due to p u m p i n g a f u l l y penetrating gallery w i t h the same capacity per lineal meter. This d r a w d o w n takes into account the general conditions such as geo-hydrologic profile, boundary conditions, etc. The formulas for the d r a w d o w n curves o f f u l l y penetrating galleries are simple and easy to handle;

2 . at b o t h sides o f the line of wells, at distances larger than bl2, the drawdowns equal those o f t h e equivalent gallery;

3. i n the line o f wells (FIG. 6) the d r a w d o w n at the well face is larger b y an a m o u n t :

cross-section longitudinal sect F I G . 6 D r a w d o w n curves o f l i n e o f wells a n d e q u i v a l e n t g a l l e r y F I G . 7 F u l l y a n d p a r t i a l l y p e n e t r a t i n g w e l l

r

r

h a l f w a y between two wells smaller by an a m o u n t : Qo

2nkH 0.693

while at a distance i / 6 f r o m each well the d r a w d o w n is the same as for the equivalent gallery.

W i t h the formulas given above, the w e l l is supposed to penetrate the f u l l y saturated thickness o f t h e aquifer (FIG. 7). I n many cases the depth of the aquifer is so large, that f u l l penetration is not j u s t i f i e d f r o m an economic p o i n t of view. W i t h p a r t i a l penetration, however, ground-water velocities i n the immediate v i c i n i t y of the w e l l are higher t h a n accounted for, resulting i n an additional loss of head. T h e influence of p a r t i a l pen-etration on d r a w d o w n d i s t r i b u t i o n again is l i m i t e d to the near vicinity o f t h e well and consequently can also be taken i n t o account w i t h a correc-t i o n faccorrec-tor. T h e moscorrec-t common sicorrec-tuacorrec-tion o f a p a r correc-t i a l l y penecorrec-tracorrec-ting w e l l is shown i n F I G . 8 to the left. F r o m the w o r k o f D E G L E E i t can be derived that at the face of such a well the additional d r a w d o w n due to p a r t i a l penetration is given w i t h good approximation b y :

L I J Hi U-2rn ^ ' T i r F I G . 8 P a r t i a l l y p e n e t r a t i n g wells Ayo = Qo 1 - ^ ^^{\.2-p)h ïnkH p ro i n w h i c h is /) is the amount o f penetration:

p = hjH

W i t h a r a n d o m position of the well screen, as shown i n FIG, 8 on the right hand side, an exact f o r m u l a for the additional d r a w d o w n due to p a r t i a l penetration is given by:

A990 = Qo 2nkH \~p P I n -ah ?o

w i t h a a f u n c t i o n o f t h e amount of penetration p = hlH and o f t h e amount of eccentricity e = öjH. T h e relation between a and these two parameters has first been given i n the publication o f the Netherlands H y d r o l o g i e C o l l o q u i u m and is reproduced below:

e = 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.1 d = 0.54 0.54 0.55 0.55 0.56 0.57 0.59 0.61 0.67 1.09 0.2 0.44 0.44 0.45 0.46 0.47 0.49 0.52 0.59 0.89 0.3 0.37 0.37 0.38 0.39 0.41 0.43 0.50 0.74 0.4 _{0.31 0.31 0.32 0.34 0.36 0.42 0.62} 0.5 0.25 0.26 0.27 0.29 0.34 0.51 0.6 _{0.21 0.21 0.23 0.27 0.41} 0.7 0.16 0.17 0.20 0.32 0.8 0.11 0.13 0.22 0.9 0.06 0.12

T h e formulas given above describe the d r a w d o w n distribution due to p u m p i n g an infinite line o f wells. N o doubt they are very interesting, but their practical significance is small as i n reality a line of wells always extends over a l i m i t e d distance only, resulting i n smaller drawdowns. I n the foregoing i t has been shown that w i t h an i n f i n i t e line o f wells the d r a w d o w n (p can be split up i n two parts:

(p = y^ + A y

i n w h i c h y i is the d r a w d o w n due to the equivalent gallery and A y a correction to take into account the influence o f point abstraction. The magnitude of A93 only depends on the condidons i n the near v i c i n i t y of the well, that is on the value o f t h e coefficient of transmissibility kH, the distance b between wells and the diameter 2ro o f t h e well screen (including gravel pack i f present). These factors, however, r e m a i n the same whether the line o f wells is i n f i n i t e or finite and w i t h a finite number of wells the formulas f o r the additional drawdowns A y , therefore, r e m a i n unchanged. W i t h a finite number o f wells meanwhile the equivalent gallery also has a l i m i t e d length only, and consequently the d r a w d o w n y i due to p u m p i n g a gaUery of i n f i n i t e length may only p a r t l y be taken into account. W i t h a l i m i t e d number of wells, the d r a w d o w n becomes

y = / 5 ' y ^ + A y w i t h /3 always smaller as unity.

C o n t i n u i n g a method of calculation given by E D E L M A N ( 1 9 4 6 ) , i t has proved possible to determine the value of/3 for points on the line o f wells. T h e value of ji, however, now also depends on the geo-hydrologic profile as w e l l as on boundary conditions. W h e n i n a one-storied leaky artesian aquifer (FIG. 5 ) , the line of wells contains n units at interval b and the length L of this line is defined as:

L = n-b

of w h i c h lengths Li and Z 2 extend on either side o f the point o f observa-t i o n (FIG. 9 ) :

L = L1 + L2

then the value of /3 is given b y :

J . F I G . 9 F i n i t e l i n e o f wells

w i t h A = VkHc

and Fi(«) = — / Ko{u)-A, _u

71 I

0

W i t h a confined or an unconfined aquifer w i t h o u t gain or loss o f water t h r o u g h leaky aquicludes, the value of /3 w i t h a similar line o f wells at a distance / parallel to open water at a constant level equals:

T h e values of the f u n c t i o n F i and F2 are tabulated overleaf.

W i t h groundwater abstraction by means of a straight fine of wells, w i t h -out any doubt the most i m p o r t a n t d r a w d o w n is the m a x i m u m d r a w d o w n occuring at the face of the centre well(s). Easily and i n short time this d r a w d o w n can be calculated w i t h the formulas given above, w h i c h also provide a clear insight into the composition of the d r a w d o w n , the i n -fluence of u n i t capacities and amount of penetration, very i m p o r t a n t for design purposes. W i t h a l i m i t e d number of wells, these formulas i n the meanwhile cannot be used for a random p o i n t outside the line of wells and here the d r a w d o w n has still to be f o u n d by application o f t h e method of superposition. A t greater distances f r o m the line o f wells, the amount of figurework can be reduced by arranging the wells i n a few sections and concentrating the abstractions i n the respective centres of g r a v i t y . w i t h Fzi'ii) = — I n — -| arc tan u

T A B L E for Fi{ii) = — / Ko{ii) • du and F 2 («) = — In 1 + — = — arc tan a nJ n \ u^ I n 0 u tl FM F.{u) 0.01 0.036 0.036 0.70 0.690 0.637 0.02 0.064 0.063 0.75 0.710 0.654 0.03 0.088 0.086 0.80 0.729 0.669 0.04 0.111 0.107 0.85 0.747 0.684 0.05 0.131 0.127 0.90 0.763 0.697 0.06 0.150 0.146 0.95 0.778 0.709 0.07 0.169 0.163 1,0 0.791 0.721 0.08 0.186 0.180 1,1 0.816 0.741 0.09 0.202 0.195 1.2 0.838 0.759 0.10 0.218 0.210 1.3 0.857 0.775 0.12 0.248 0.239 1.4 0.874 0.789 0.14 0.276 0.265 1.5 0.888 0.801 0.16 0.302 0.289 1.6 0.901 0.812 0.18 0.326 0.312 1.7 0.912 0.822 0.20 0.349 0.333 1.8 0.922 0.831 0.22 0.371 0.353 1.9 0.931 0.839 0.24 0.391 0.372 2.0 0.938 0.847 0.26 0.411 0.390 2.2 0.951 0.860 0.28 0.430 0.407 2.4 0.961 0.871 0.30 0.447 0.424 2.6 0.969 0.880 0.32 0.465 0.439 2.8 0.976 0.889 0.34 0.481 0.454 3.0 0.980 0.896 0.36 0.497 0.468 3.5 0.989 0.910 0.38 0.512 0.482 4.0 0.994 0.921 0.40 0.526 0.494 4.5 0.996 0.930 0.42 0.540 0.507 5 0.998 0.937 0.44 0.554 0.519 6 0.999 0.947 0.46 0.566 0.530 7 1.000 0.955 0.48 0.579 0.541 8 1.000 0.960 0.50 0.591 0.551 9 1.000 0.965 0.55 0.619 0.576 10 1.000 0.968 0.60 0.645 0.598 20 1.000 0.984 0.65 0.668 0.618 50 1.000 0.994

A t a very great distance the d r a w d o w n due to p u m p i n g a line o f wells even equals the d r a w d o w n due to p u m p i n g one well i n the centre w i t h the combined capacity. A t small distances f r o m the line of wells (but larger than bj2), the d r a w d o w n equals that of the equivalent gallery of finite length, that is

As final words, the author of this paper w o u l d like to add that the des-c r i p t i o n given above about the development o f the various methods f o r calculation of the d r a w d o w n due to ground-water abstraction w i t h a straight line of wells is only meant as an example to demonstrate the general advancement of ground-water hydrology i n the past 40 years. T h i s advancement d i d not come o f its o w n accord, b u t arose f r o m sheer necessity when wellsystems f o r d r a i n i n g large and deep b u i l d i n g sites (as w i t h the Y m u i d e n lock) had to be designed such that success was assured, w h e n the diflFerent aspects o f reclaiming the Zuiderzee polders had to be predicted w i t h fair accuracy, when a reliable forecast had to made about the influence of large ground-water catchments f o r public water supplies, etc. etc. Professor THIJSSE to w h o m this paper is dedicated, contributed to this advancement i n no small way and the author retains pleasant m e m -ories o f the discussions about the f u t u r e ground-water table i n A m s t e r d a m after the southern Zuiderzee polders have been drained. Ground-water hydrology is an exact science owing to its mathematical origin, b u t i t is also an exacting science as soon after completion o f the work under con-sideration the outcome verifies the prediction. Both aspects must have appealed to professor THIJSSE.