Determination of unconfined aquifer parameters using partially penetrating wells

Journal of Hydrology, 36 (1978) 225--231

225

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

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DETERMINATION OF UNCONFINED A Q U I F E R PARAMETERS USING PARTIALLY P E N E T R A T I N G WELLS

SATISH K. SONDHI and SITA RAM SINGH

Department of Soil and Water Engineering, Punjab Agricultural University, Ludhiana, Punjab (India) (Received April 22, 1977; accepted for publication June 15, 1977)

ABSTRACT Sondhi, S.K. and Singh, S.R., 1978. Determination of unconfined aquifer parameters using partially penetrating wells. J. Hydrol., 36: 225--231. An analytical solution is developed to determine unconfined aquifer parameters by conducting pump tests on partially penetrating wells. The aquifer is assumed to be homogeneous, isotropic and infinite in areal extent. A new concept of "effective penetration" is introduced. A dimensionless curve is developed to find the effective penetration of the well. A circular sand tank model is used to check the applicability of the relationships developed. INTRODUCTION

Frequently production wells do not completely penetrate the aquifer from which they are pumping. Therefore, the hydraulics of such wells is different from that of wells that penetrate fully the aquifer. The problem of partial penetration has long been recognised, and approximate steady-state solutions for various field conditions have been advanced by Forchheimer (1898), DeGlee (1930), Kozeny (1930) and Boreli (1955). The use of these solutions is limited by the fact that they describe the flow under the equilibrium condition only, a situation attained rarely during actual periods of well operation. The nonlinearity of the basic differential equation which governs the transient groundwater flow towards a well in an unconfined aquifer has made it difficult to obtain an analytical solution. In the past, solutions obtained for confined and leaky conditions have been applied to unconfined aquifer transient-flow problems. Various approximations have been proposed which make the basic differential equation linear (Bear, 1972). Boulton (1954) gave the solution for flow to a line sink of constant strength per unit length, partially penetrating an infinitely thick aquifer. The surface of seepage was neglected. Dagan (1967) derived an analytical solution for the problem of unsteady flow to a partially penetrating well in an unconfined aquifer of finite thickness. He represented the well by a line sink and solved

226

the linearized perturbation expansion problem for small drawdown and constant discharge. The results were generalized for anisotropic aquifers and example pump test analyses were presented. He presented a result similar to the result of Boulton for a well, screened over part of its length, so that no surface of seepage existed. Kipps (1973) solved the problem of unsteady flow to a single partially penetrating well of finite radius in a rigid semi-infinite homogeneous isotropic aquifer by neglecting the seepage face. He used the perturbation expansion technique to linearize the free-surface boundary condition. The mathematical model, thus developed, is suitable for studying the well behaviour for a relatively short time only. At present, no simple method is available to determine the aquifer parameters from partially penetrating wells in homogeneous isotropic unconfined aquifers. Kipps' method involves a tedious trial-and-error search to determine the parameters. Therefore, the present study was undertaken to develop a method of determining the parameters of an unconfined aquifer by conducting pump tests on partially penetrating wells. The Boussinesq (1904) equation was chosen as the mathematical model of well--aquifer interaction. The new concept of "effective penetration" has been introduced. The accuracy of the solution has been ascertained by conducting pump tests on a circular sand tank model. THEORETICAL ANALYSIS

The definition sketch of the flow system is shown in Fig.1. The fundamental differential equation in radial coordinates governing transient groundwater flow in an unconfined aquifer is (Boussinesq, 1904):

l~(r h ~h) S~h

r ~r

~r

- g ~t

(1)

In an one-dimensional flow field over a horizontal impervious layer, eq. 1 is applicable (Murray and Monkmeyer, 1973) if: S ~h K ~t

(~h]2 << 1 \~r ] rO

(2)

G.S.

Fig.1. Definition sketch o f a partially penetrating gravity well.

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in which h = the hydraulic head at time, t, at any distance r from the well axis; S = the specific yield; and K = the hydraulic conductivity. The assumptions made in the derivation of eq. 1 are as following: (1) The Dupuit-Forchheimer assumptions are valid; (2) the water-bearing material is homogeneous and isotropic; and (3) the aquifer material drains instantaneously with decline in hydraulic head to give a constant value for the specific yield. Additional assumptions made to determine the solution are: (a) the aquifer is of infinite extent; (b) the initial water table has a low slope; (c) the diameter of the pumped well is small compared to other dimensions of the system; (d) the seepage face in the pumped well and the head losses through the gravel pack and well screen are disregarded; and (e) the discharge of the pumped well is constant. The initial and boundary conditions to the flow system, depicted in Fig.l, are:

h (r,O) = h0

(3a)

lim h ( r , t ) = ho

(3b)

p - - > co

lim r ( b h / O r ) = Q / [ 2 n K (h - d)]

(3c)

where h0 = hydraulic head in the aquifer at the start of the pumping; Q = the constant discharge of the p u m p e d well; and d = height of well b o t t o m above impervious layer. Eq. 1 is a non-linear partial differential equation of the second order which cannot readily be solved. It is linearized by substituting a new dependent variable H = h2/2, and then by considering that the variation in h(r, t) over space and time is so small that it could be replaced by its average value D. Then linearized eq. 1 becomes: ~2H 1 5H S bH -- + . . . . . . . ~r 2

r

~r

KD

(4)

~r

The independent variables of eq. 4 can be combined into a single variable by introducing the transformation u = r 2 S / 4 K D t . It is advantageous to use this transformation when groundwater flow takes place in an infinite medium, having an initially constant hydraulic head in the entire region; in this situation, the independent variables of the initial and boundary conditions are expressible in terms of u alone. Boltzmann (1894) was the first to use this t y p e of transformation, which is named after him. Substitution of the Boltzmann transformation in eq. 4 and simplification thereof, results in a second-order ordinary differential equation: u ( d 2 H / d u 2) + (1 + u) (dH/du) = 0

(5)

Solving eq. 5 and using the initial and boundary conditions (3), we obtain: QD H°--H = 4~K(D--d)

ff ~

exp (--a) c~

d~

(6)

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or 2

h°--h where

QD 2n K(D--d)

2 _

W(u)

(7)

(~

W(u) = f

exp (--a) da

(8)

U

The function W(u) is known as the "well function" in the field of groundwater hydraulics. Re-arranging the transformation, we obtain: S = (4KDut)/r 2 (9) Eqs. 7 and 9 do not give the direct computation of aquifer parameters. The standard matching technique, proposed by Theis (1935), can be used to find the aquifer parameters. For d = 0, eqs. 7 and 9 are reduced to the Theistype curve method, adapted for fully penetrating wells in an unconfined aquifer (Kriz, 1967). DETERMINATION OF THE EFFECTIVE PENETRATION

A circular sand tank model was used to test the applicability of the mathe, matical model. Values of aquifer parameters were determined by installing a fully penetrating well. A plot of ( h ~ - h 2) vs. r2/t was superimposed on the Theis-type curve, holding the coordinate axes of the two curves parallel in such a way that the plotted data best fit the type curve. An arbitrary match point was selected on the overlapping portion which gives (h0:-h2), W(u), r2/t and u. Substitution of these values in eqs. 7 and 9 gave K and S values of 12.01 m/day and 0.0625, respectively. For different percents of well penetration, [ ( H 0 - d ) / H o ] • 100, values of K and S have been calculated and are given in Table i. Table I shows that for different percents of the penetration ratio, the variation in K and S is 12--146% and 30--223%, respectively. This contradicts the fact that K and S are constant for a homogeneous and isotropic aquifer. The reason for this variation is that the eqs. 7 and 9 were derived using the horizontal fl0w assumption. Obviously, the flow pattern to such wells differs from that of fully penetrating wells. The average length of a flow line into a partially penetrating well exceeds that into a fully penetrating well, so that a greater resistance to the flow is encountered. In the derivation of the Boussinesq equation, the entire thickness of the aquifer participates in the flow, but in the case of the partially penetrating well, especially for small penetrations, the whole column does not contribute towards the flow. However, the thickness of the column contributing towards flow is more than the actual penetration, because of the curvilinear nature of flow lines near the well bottom. To circumvent this restriction, the concept

229

of "effective penetration, (D--de)" has been introduced. The effective penetration is the value of (D--d) which, when substituted in solutions based on Dupuit-Forchheimer assumptions, will result in the correct relationships. A similar concept named "equivalent depth to impervious layer" has been widely used in the area of seepage towards partially penetrating drains (Hooghoudt, 1940). This concept was originally used in analysing steady-state flow problems b u t it has been used for non-steady flow towards drains by Van Schilfgaarde (1974). Use of this concept has also been suggested by Bouwer (1969) in analysing seepage from partially penetrating canals. The flow towards a fully penetrating gravity well meets almost all the assumptions inherent in the derivation of the Boussinesq equation, therefore, the value of K and S, obtained by pump test on such a well, could be considered to be the true values of the parameters. Substituting these values of K and S and other observed values like Q, D and (h 2 - h 2) in eq. 7, a value of effective penetration ( D - - d e) can be obtained. Intensive studies were conducted on a sand tank model for different penetrations to develop a dimensionless graph between effective percent well penetration ratio and actual percent well penetration (Fig.2), under different operating conditions. Knowing the actual penetration, the effective penetration can n o w be obtained with the help of Fig.2. The shape of the curve shows that for large values of d, i.e. for small penetrations, the effective penetration almost approaches a constant value. The nature of the curve is similar to curves obtained for tile drains. In the preceding analysis d has been used in the derivation; in the numerical computations however, it should be replaced by d e . APPLICATION TO THE ANALYSIS OF WELL TESTS

The results of this study can be used to analyse well pump tests in order to determine the aquifer parameters K and S from partially penetrating gravity wells. The pump tests were conducted on a sand tank model for different

TABLE I Variation in K and S at different penetrations because of the curvilinear flow near the well b o t t o m Actual penetration (%)

K (m/day)

S

100.0 75.0 62.5 50.0 34.4

12.01 13.46 15.51 19.62 29.69

0.065 0.085 0.102 0.135 0.210

230

50 oo ×

,.,

60

~

7o

&,-

~u

W

loo

,oo

9'0

8'0

7'o

6'o

H 0 -d Actual penetration ~

M0

5'0 xl00

4'0

3'0

Fig.2. Dimensionless curve to determine the effective penetration.

well penetration to find the variation in K and S after incorporating the effective penetration concept. The variation in K was found to be less than 10% and that in S was less than 30% (Table II). SUMMARY AND CONCLUSIONS

The present study's objective was to find an analytical solution to the linearized Boussinesq equation for partially penetrating gravity wells. The equation was linearized by making use of the second method of linearization. The well was treated as a line sink. The concept of effective penetration was introduced to take care of the curvilinear nature of flow lines near the well bottom. The applicability of the solution was studied by using a circular sand tank model. The values thus obtained were compared with those of a fully penetrating well obtained by the adaptive Theis-type curve method.

T A B L E II

Values of K and S obtained after incorporating the effective penetration concept Actual penetration

Effective penetration

K

(%)

100.00 68.75 56.25 43.75 34.40

100.00 76.30 74.80 74.40 74.10

12.01 12.16 12.86 12.51 12.82

(%)

S

(m/day)

0.065 0.077 0.078 0.086 0.086

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An approximate method which has been developed can be used to find the values of hydraulic conductivity and specific yield of an unconfined aquifer, using partially penetrating wells. Within the limitations of the initial assumptions, good results can be achieved by using the technique. REFERENCES Bear, J., 1972. Dynamics of Fluids in Porous Media. American Elsevier, New York, N.Y., 764 pp. Boltzmann, L., 1894. Zur Integration der Diffusions-gleichung bei variabeln Diffusionskoeffizienten. Ann. Phys., Leipzig, 53: 959--964. Boreli, M., 1955. Free-surface flow toward partially penetrating wells. Am. Geophys. Union, Trans., 36(4): 664--672. Boulton, N.S., 1954. The drawdown of the water-table under non-steady conditions near a pumped well in an unconfined formation. Proc. Br. Inst. Cir. Eng., 3(3): 564--579. Boussinesq, J., 1904. Recherches th~oriques sur l'~coulement des nappes d'eau infiltr~es dans le sol. J. Math. Pures Appl., Set. 5, 10 (Fasc. 1): 363--394. Bouwer, H., 1969. Theory of seepage from open channels. In: V.T. Chow (Editor), Advances in Hydroscience, Vol.5, Academic Press, New York, N.Y., pp. 121--172. Dagan, G., 1967. A method of determining permeability and effective porosity of unconfined, anisotropic aquifers. Water Resour. Res., 3(4): 1059--1071. DeGlee, G.J., 1930. Over Grondwater-stroomingen bij Wateronttrekking, door middel van Putten. Waltman, Delft, 175 pp. (Thesis.) Forchheimer, P., 1898. Grundwasserspiegel bei Brunnenanlagen. Z. ()sterr. Ing. Archit. Ver., 50 (45), 645 pp. Hooghoudt, S.B., 1940. Bijdragen tot de kennis van eenige natuurkundige grootheden van den grond, 7. Algemeene beschouwing van het probleem van de detail ontwatering en de infiltratie door middel van parallel loopende drains, greppels, slooten en kanalen. Versl. Landbouwk. Onderz., 46: 515--707. Kipps, Jr., K.L., 1973. Unsteady flow to a partially penetrating, finite radius well in an unconfined aquifer. Water Resour. Res., 9(2): 448--462. Kozeny, J., 1930. Theorie und Berechnung der Brunnen. Wasserkr. Wasserwirtsch., 28: 88--92; 101--105; and 113--116. Kriz, G.J., 1967. Determination of unconfined aquifer characteristics. J. Irrig. Drain. Div., A.S.C.E., 93(IR2): 37--47. Murray, W.A. and Monkmeyer, P.L., 1973. Validity of the Dupuit-Forchheimer equation. J. Hydraul. Div., A.S.C.E., 99(HY9): 1573--1583. Sondhi, S.K., 1975. Hydraulics of partially penetrating gravity wells. M.Tech. Thesis, Punjab Agricultural University, Ludhiana, 67 pp. Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground water storage. Am. Geophys. Union, Trans., 16: 519--524. Van Schilfgaarde, J., 1974. Non-steady flow to drains. In: J. Van Schilfgaarde (Editor), Drainage for Agriculture. Am. Soc. Agron., Madison, Wisc., pp.245--270.