Some considerations regarding the radius or ...

Some considerations regarding the radius or influence or a pumping well

Walter DRAGONI

(1)

Considérations sur le rayon d'influence d'un puits de pompage

Hydrogéologie, n° 3, 1998, pp. 21-25, 3 fig., 2 tabl. Key-words: Methodology, Aquifer drawdown, Water wells, Radius of influence Mots-clés : Méthodologie, Rabattement nappe, Puits eau, Rayon d'influence

Abstract

Si on pose dans (1)

The case of a well pumping at a constant rate without recharge is examined. lt is shown that the drawdown s(cj) at the radius of injluence computed according to the Cooper - Jacob formula is always given by the formula s(cj) 0.03912 Q/T, where Q and T are the pumpedflow and rhe rransmissivity ofrhe aquifer. A radius of influence is defined according to the drawdown that can be detected with the usual devices used in the field. Some methods and some simple expressions to compute such a radius are given.

Tt

2.245842

r S

on obtient le rayon d'influence :

=

Résumé étendu Pour ce qui suir, on considèrera comme valables les symboles reportés dans le tableau l, tandis que l'on utilisera pour les tableaux, équations et figures la meme numération que do.ns le tate. Dans cet article, 011 examine la signification du rayon d'influence R(cj) dans le cas d'un puits pompé à débit constant, le cours des rabanements dans le temps et do.ns l'espace étant décrit par l'équarion de Theis. On suppose généralemenr que le rayon d'influence (c'esr-à-dire la distance du puits à laquelle le rabattemenr provoqué par le pompage est nul ou négligeable) peut erre calculé au moyen de l'équation de Cooper-Jacob : s

= ---.2...- in 2.24584Tt 4nT

2

r S

(I)

=l

2.24584Tt (2)

S

L'équation (1) n'est valable que lorsque u < 0.02, mais quand l'équation (la) est vérifiée, on démontre que l'on a roujours u = 0.561459 et par conséquenr l'usage de l'équation (2) peut mener à des erreurs non négligeables. Considérons l'équarion de Theis complète:

s=~w 41tI'

(5)

(u)

Quand u = 0.561459, on a Wru ) = 0.491535 ; si on englobe toutes les constantes présentes dans (5) en une constante unique, à la distance R(cj) le rabattement est fourni par l'équation suivanre:

5(.)

CJ

Q = 0.03912T

(5a)

Un puits en pompage peut avoir un rappor! Q/T supérieur à 25 - 30 mèrres .. la dépression s(Cj) peut alors ne pas etre négligeable et dépasser le mètre (fig. 1).

Dans de lwmbreux cas, il est plus comnwde de définir le rayon d'influence camme la disrance du puirs à laquelle le rabattemenr est égal à une valeur d - où d est une très petite dépression choisie au cas par cas (normalement, d pourrait erre de quelques centimèrres). Appelons R(d) ce nouveau rayon d'influence. R(d) (rayon d'influence selon le rabattement) peut erre calculé de différentes façons .. l'article en présenre trois. La premièrefaçon de calculer R(d) est par le biais de l'équation de Theis (voir équations (6) et (7) dans le texte) .. il est alors nécessaire d'avoir à disposition un tableau avec les valeurs de la fonction W(u) en fonction de u. Pour celte équation, on dispose également du code de calcul « Radius ». La deuxième méthode pour calculer R(d) utilise ce que ['on appelle la loi des temps (voir équation (IO) dans le texte), pour laquelle il est nécessaire d'avoir un piézomètre. La troisième méthode, qui donne une approximation d'environ 0,2 %, est d'utiliser les formules reportées dans le tableau 2, qui donnent R(dJ en fonction de R(cj) et du rapport QIT. Les formules du tableau (2) ont été obtenues par le meilleur ajusrement entre In(Q/T) et le rapport R(dIR(cj) (fig. 2). La figure 3 montre combien le rayon d'influence peut varier selon la définition utilisée et la valeur du rapport QIT.

(1) Dipartimento Scienze Terra, Università di Perugia - Piazza Università 1 - 06100 - Perugia, ltalia.

HYDROGÉOLOGIE, N° 3, 1998

21

RADIUS OF INFLUENCE OF A PUMPING WELL

Putting the Ieft member of equation (4a) in equation (3), one finds that when equation (3) is true, then u=0.561459. As the Cooper - Jacob equation is an approximation acceptable only if u is small (let us say < 0.02), it appears tbat, in generaI, equation (2) is not an acceptable approximation.

d = drawdown used lo define the radius of influence (m) In = naturallogarithm Q = pumped flow (m3/day) r = any dislance from the well (m) R(cj) = radius of influence according lo the Cooper - Jacob equalion (m) R(d) = radius of infiuence according lo the drawdown criteria (m) s =drawdown in generai (m) s(cj) = drawdown at the distance R(cj) from the well (m) S = storage coefficient (dimensionless) l = lime (days) T = transmissivity (m2/day)

Let us eonsider tbe Theis equation

Q s=--w 4n'f

(u)

(5)

2

u

=

~ (dummy variable in the Theis equation, dimensionless)

Il can be checked from any table giving tbe value oftbe Theis funclion lhat if u = 0.561459, then W(u) = 0.491535. In this condition equation (5) becomes

4Tt W(u) = well function of the Theis equation (dimensionless) Table \.- Symbols used in the lexl. Tabl. 1.- Symboles utilisés dans le texte.

Introduction Due lO the use of groundwaler modeIs, the concept of radius of int1uence of a well is of Iimited use today. However, as il is stili of some interest from a practical point of view to have a readiIy available idea about the size of the zone which is perturbed by a pumping well, the following considerations couId stili have some use. This note lakes into consideration a well pumping in a confined aquifer without recharge, in unsteady state, in sueh conditions that tbe Theis equation is valido However, it seems that similar considerations could be applied to uneonfined aquifers or to more eomplex situations. Table l gives the meaning of the symbols used in tbe text.

The radius of influence according to the CooperJacob equation The Theis equation implies tbat the radius of int1uence of a well pumped at a constanI radius extends to the infinite. Sinee tbis situation has little significance in tbe real world, where a very small drawdown (e.g. a few millimetres or a few centimetres) can be neglected, it is usually aeeepted that a realistic estimation of tbe radius of influence can be obtained from tbe Cooper - Jacob equation (Cooper and Jacob, 1946) 22

Q 2.24584Tt S=--in---2

4nT

(1)

r S

According to equation (1), at any lime t, tbe radius of influence R(cj) is given by 2.24584Tt

(2)

S Equation (2) implies that R(cj) is independent of the flow pumped out from tbe weI!. In the case of pumping tests carried out close to a boundary, formula (2) is sometimes used lO estimate the distanee between the imaginary well and the point where the drawdown is measured (de Marsily, 1986). Formula (2) relies on the fact tbat when and where s =O in equation (1) tbe foliowing must be true

Tt

2.24584-2- = l r S

(3)

In the Theis equation tbe dummy variable u is given by 2

r S

u=4Tl

(4)

which can also be written as TI 4u

=

r 2S

(4a)

Q

S(cj)

= O.03912~

(5a)

This last formula gives the true drawdown at tbe R(cj) distanee eomputed according to equation (2); as expected, it shows that the actual drawdown at R(cj) is proportional to tbe ratio Qff. It is also noteworthy to consider tbal, according to formula (5a), as long as the drawdown is described by the Theis equation, s(cj) does noI depend on S, l and R(cj)' To get an idea about the greatest order of magnitude of s(cj)' let us consider that in most cases, for a well of some importance, the value of Q/T is roughly between 1 m and 30 m, in rare cases being larger than 60 m or smaller than l m. This range is taken from my own personal experience and from a survey of published data about pumping wells (cf., for example, WaJton, 1970; Civita, 1975; Castany, 1982; Raghunath, 1982; Custodio and Llamas, 1983; Celico, 1986; Kruseman and de Ridder, 1990; Vukovic and Soro, 1992; Gichaba et al., 1996). Figure l is a plot of formula (1): it is clear that, in tbe noI extreme and not-so-uneommon case of a ratio Qff larger than 20 or 25 ID, s(cj) can be around l m or more: this means that equation (2) can easily give unsatisfaclory results, especially if one is dealing with problems regarding interference, superposition and boundaries. HYDROGÉOLOGIE, N· 3, 1998


drawdown d at tbe distance R(d) from tbe wel!.

························-····-·-···---1 5

At some time (t,) in tbe piezometer we shall have tbc drawdown (d) tbat defines tbe radius of influence R(d)' At t], the drawdown is described by Theis equation

I

i -~~~

- - _ ---

_.-

_ _---

--_.~-+-~~_

(8)

-_....

-~-c:;7""'------.

I

'··T··-j 20

40

60

80

100

where W(u\) is tbe well function for

120

r,

2

Qff (m) u)

Fig. 1.- Plot of the ratio Qff vs. slei)' For values of Qff larger than 20 m the drawdown at tbe radius of influence according to tbe Cooper - Jacob equation can be around one metre or more. Fig. 1.- Relation entre Qff el s(cjJ- Pour les valeurs de Qff supérieures à 20 m, le raballemeni à une disrance égale au rayon d'influence calculé par l'équation de Cooper· Jacob peul elre d'un mètre ou plus.

d = 0.01

R(d) = R(cj) (-O.OI40X 2 + 0.3673X + 1.5522)

d = 0.02

R(d)

= R(cj) (-O.0161X 2

+ 0.3957X + 1.2826)

(lOb)

d = 0.03

R(d)

= R(cj) (-0.0174X 2

+ 0.4138X + 1.1152)

(lOc)

d = 0.04

R(d)

= R(cj) (-0.0182X 2

+ 0.4270X + 0.9923)

(lOd)

= 0.05

R(d)

= R(cj) (-O.0188X2

+ 0.4373X + 0.8945)

(10e)

d

S =-4Tt]

(8a)

If we want to know R(d) at tbe time tz, which is larger tban t" we can consider tbat at tbe unknown distance R(d) the drawdown is stili given by tbe Theis equation, i.e.

(lOa)

(9)

where

W(uz)

is tbe well function for

(9b)

Table 2.· Approximate formulas for computing ~d) for 1 ":;;Qrr ,.:; 35 m. In tbe table, X=ln(Qrr). TabL 2.- Formules donnant R(dJ pour l SQIl' 535 m avec X=ln(Q:T).

Comparing formulas (8) and (9) shows that W(u\) W(uz); tbis implies that u j = ~, so that by combining (8a) and (9b) the following can easily be obtained

=

The radius of influence according to the drawdown criteria From a practical point of view, we can define tbe "radius of influence" as tbe distance beyond which tbe drawdown is so small tbat it cannot be detected; let us define such a small drawdown as d and the corresponding radius as R(d)' When T, S, and Q are known, R(d) can be computed in various ways. For instance, it is possible to compute W(u) directly from tbe Theis equation 4nTd W(u) = - -

(6)

Q When W(u) is known, it is possible to obtain the corresponding value of u by means of tables or witb some computer program, easy to implement even on a hand-held calculator (cf., for instance, Dragoni, 1985, 1986). When u is known we have HYDROGÉOLOGIE, N° 3,1998

(7)

From equation (6) it is obvious that R(d) depends also on Q. The procedure bere outlined is performed by tbe code "Radius", based on tbe Bolzano bisection metbod (Dragoni, 1985; tbis code can be freely downloaded at the web site indicated at tbe end of the present paper). In the field, with a piezometer, another metbod for computing R(d) is by using tbe so calied "law of times". Let us define tbe following symbols: r [ = distance between tbe well and tbe piezometer; t] = time, from tbe beginning of pumping, corresponding to a drawdown d in tbe piezometer.

tz = time when we want to know R(d)' i.e. time when tbe pumping generates a

(lO)

Since rl' t] and d can be measured, and tz depends on the choice of the operator (as well as d), equation (8) allows immediate computation of R(d)' Anotber easy metbod (only a handheld calculator is necessary) for estimating R(d) is to apply the appropriate formula from among those listed in Table 2. The formulas in Table 2 have bcen obtained by polynomial regression between In(Qrr) and R(d)lR(cj)' and witbin tbe ranges indicated they give an error no greater then 0.2% of tbe "true value", where tbe "true value" is tbe one tbat can be obtained computing u from W(u) by some numerical metbod and

23


2.80

4S00

A

2.ffJ

C

2.40

3S00 3000

221 -

g

2.00

~

A

4000

B

o:

].g)

2500

B

2000 D

IS00

I.ffJ

1000

1.40

!/ ~

C

SOO~

121 1.00

6

lO

12

Time (day,)

o.m O

lO

15

20

25

30

35

40

QIT (ml

Fig. 2.- Relationship between the ratio Q/T and the ratio R(d/R(ej)' I) "exact" values of R(d/R(ej): 2) value obtained by formulas in Table 2. Fig. 2.- Relation entre QfT et R(dIR(e})' 1) Valeurs "exactes" de RldIR(e}); 2) Valeurs données par lesformules du tableau 2.

Fig. 3.- Evolution in time of the radius of inf1uence in different conditions and according to different definitions. The simulations have been carried out supposing an ideai aquifer having T = 100 m2/day, S = 0.001 and d = 0.01 m. A) R(O.OI) for Q = 3500 m3/day, Q/T = 35; B) R(o.OI) for Q = 100 m3/day, Q/T = l; Cl R IO .Ol ) for Q = lO ml/day, Q/T = 0.1; D) R(ej)' independent from the pumping rate. lt is interesting lO note that the drawdown s(ej) at ~ej) depends on the ralio Q/T: thus s(ej) = 1.37 m for Q/T = 35, s(ej) = 0.391 m for Q/T = lO m, and s(ej) = 0.0039 m for Q/T= 0.1 m (cf. equalion (5a)).

Fig. 3.- Variation au cours du temps du rayon d'injluence sous dif.férentes conditions et selon dif.féremes définitions. !..es simulations ont été réalùées en supposam un aquifère idéal où T = 100 m 2/j, S = 0,001 et d = 0,01 m. A) R(O.Ol) pour Q = 3500 m 3/j, Q/T = 35; B) R(O.Ol) pour Q = 100 m 3/j, QIT = 1; C) R(O.Ol) pour Q = lO m 3/j, Q/T = O, l; D) R(e})o indépendant du débit pompé. 11 faut noter que le rabattement s(e}) à R(e}) dépend de Q/T. Ainsi s(e)) = 1,37 m pour QfT = 35, s(e}) = 0,391 m pour QIT = lO m et s(e}) = 0,0039 m pour Q/T = 0,1 m (cf équation (5a)).

applying equation (7). Here the abovementioned code "Radius" was used; comparison between the values given by the code and the results that are obtained using the tabu1ated values of W(u) and u, has shown that the difference is less than 0.01 % (cf., for example, the tab1e m Custodio and Llamas, 1983, p. 945). It is interesting to note that the formulas in Tab1e 2 app1y for any time and any value of T and S, as long as R(cj) has been computed in the proper way, i.e. by means of equation (2). This is possib1e

because the ratio of equations (7) and (2) lS

according to the equations in Table 2; Figure 3 gives an example of the evolution in time of R(cj) and R(d)' Figures 2 and 3 show that:

In formula (11), the value of u(d) depends on the value of W(u) obtained from equation (6), which does not depend on t, T and S, but only on d and the ratio Qrr. Figure 2 shows a plot of the ratio Qrr against the ratio R(d)lR(cj)' and the best fit

a - for small values of d, and large va1ues of the ratio Qrr, R(d) can be a1most three times larger than R(cj); b - it may happen that R(cj) > R(d): according to equation (5a) this occurs when the drawdown at R(cj) is greater than d, i.e. when d < O.039IQrr.

References Castany G. (1982) - Principes et méthodes de l'hydrogéologie. Dunod Université, pp. 238. Celico P. (1986) - Prospezioni idrogeologiche. VoI. I, Liguori, pp. 735. Civita M. (1975) - Idrogeologia. In "Geologia tecnica per ingegneri e geologi", Ippolito E, Nicotera P., Lucini, Civita M., de Riso R., ISEDI - Milano, pp. 443. Cooper H.H., Jacob c.E. (1946) - A generalized graphical method for evaluating formation constants and summarising well-field history. Trans. Am. Geoph. Union, 27, 526-534. Custodio E., Llamas M.R. (1983) - Hidrologia subterranea. v.l. I, Ediciones Omega, pp. 1157. de Marsily G. (1986) - Quantitalive Hydrogeology - Groundwater Hydrology for Engineers. Academic Press, pp. 440.

Publication n. 1879 of CNR-GNDCI, U.O. 4.8. The research was partially supported by MURST 60% funding. Most of the computations in this paper were carried out by using the code "Radius", which can be freely downloaded at the web site ..http://www.gndci.pg.cnr.itf•.

24

HYDROGÉOLOGIE, N" 3, 1998


Dragoni W. (1985) - Contributo al calcolo dei parametri idrogeologici tramite prove di pompaggio. - Geologia Applicata e Idrogeologia, XX (I), 125-136. Dragoni W. (1986) - Sul calcolo della trasmissività e del coefficiente d'immagazzinamento dalla risalita del cono di depressione. Mem. Soc. Geol. /t., 35, 987-990. Gichaba C.M., Anyumba J., Peloso G.F. (1996) - Groundwater potential in Kidiani area, Kwale district, Kenya. Acque sotterranee, 2, 13-26. Kruseman G.P., de Ridder N.A. (1980) - Analysis and Evaluation of Pumping Test Data. ILRl, Wageningen, pub. no. 47, pp. 377. Raghunath H.M. (1982) - Ground Water. Wiley Eastem Limited. pp. 456. Vukovic M., Soro A. (1992) - Hydraulics of Water Wells, Theory and Application. Water Resources Publications, pp. 353. Walton

w.c.

(1970) - Groundwater Resources Evaluation. McGraw-Hill Kogalrusha, pp. 664.

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