Groundwater Monitoring and Management (Proceedings of the Dresden Symposium, March 1987). IAHS Publ. no. 173,1990.

Unsteady flow to a multi-aquifer artesian well

G. C. MEHRA & S. CHANDRA National Institute of Hydrology, Roorkee, Uttar Pradesh, India

Abstract Analysis of unsteady flow to a multi-aquifer well, which is open to a number of aquifers, has been made using a discrete kernel approach. The aquifers are separated by aquicludes and have different potentiometric surfaces prior to pumping. The following flow characteristics have been presented for a case in which the well taps three confined aquifers: (a) the exchange of flows that takes place through the well screens among the aquifers prior to pumping owing to the differences in piezometric surfaces, (b) the contributions of each of the aquifers to the discharge of the well during pumping, (c) the exchange of flows that takes place among the aquifers after stoppage of pumping, and (d) variations of drawdown in the piezometric surfaces with time at the well point. It is found that prior to pumping, aquifers having equal initial hydraulic head and diffusivity, receive water in proportion to their respective transmissivity values from the aquifer having highest potentiometric surface.

INTRODUCTION

In a sedimentary groundwater basin, the occurrence of multiple aquifers separated by confining layers of low and negligible permeability is quite common. A water well in such a basin may have to be constructed tapping more than one aquifer in order to provide the requisite yield. If the aquifers are separated by confining layers of negligible permeability (aquicludes) interaction among the aquifers tapped by the well, takes place only through the well screens. Wells, which are open to two or more water bearing strata, which have different hydraulic properties and which are not closely connected except by the well itself, are referred to as multi-aquifer wells (Papadopulos, 1966). A solution for unsteady flow to a well, tapping two confined aquifers having different potentiometric surfaces prior to well construction was obtained by Papadopulos. The Laplace transform technique has been used to obtain an exact expression for head distribution but the solution is intractable for numerical calculation. Subsequently, asymptotic solutions for both head and discharge distribution, amenable to computation which yield results accurate enough for practical application, have been derived by Papadopulos (1966); however, no numerical results were presented. Using the integral transform technique, unsteady flow to a multi-aquifer well, which is open to two aquifers, has also been analysed by Khader &

293

G. C. Mishra & S. Chandra 294

Veerankutty (1975), who have presented numerical results for the contributions of individual aquifers to the total discharge of a well. Using Laplace transform technique, solutions for unsteady flow to a multi-aquifer well which is open to several aquifers (more than two) have been obtained by Wikramaratna (1984).

In this, Wikramaratna assumed all the aquifers to have equal potentiometric surfaces prior to pumping and has presented results for a two aquifer system. A solution of unsteady flow to a multi-aquifer well which taps two aquifers, by a discrete kernel approach, has been given by Mishra et al. (1985). In their analysis, it has been assumed that the initial potentiometric surfaces prior to pumping are the same in all the aquifers. In the present paper, unsteady flow to a well which is open to several aquifers where different potentiometric surfaces prevail prior to well construction, has been analysed.

STATEMENT OF THE PROBLEM

A schematic cross section of a well tapping a number of confined aquifers which are separated by aquicludes is shown in Fig. 1. Each of the aquifers is homogeneous, isotropic, and infinite in areal extent. The potentiometric surfaces in the aquifers are different from each other, and prior to the well construction all the aquifers were at rest. After construction the well remains unpumped for a period t during which exchange of flow occurs

AQUiCLUDE

AQUIFER I

AQUICLUDE

AQUIFER 2

T 2 , * 2

AQUICLUDE

AQUIFER 3

AQUICLUDE

Fig. 1 Schematic section of a multiaquifer well.

1 - —

295 Unsteady flow to a multi-aquifer artesian well

among the aquifers through the well screens owing to the difference in initial heads. The multi-aquifer well is pumped subsequently at a constant rate for a period t . It is required to find the following: (a) the exchange of flows that takes place through the well screen among

the aquifers prior to pumping due to the difference in piezometric surfaces,

(b) the contributions of each of the aquifers to the discharge of the well during pumping,

(c) the exchange of flows that takes place among the aquifers after stoppage of pumping, and

(d) drawdowns in the piezometric surfaces at various times after well construction, during pumping and after stoppage of pumping.

ANALYSIS

The following assumptions have been made in the analysis: (a) at any time, the drawdowns in all the aquifers at the well face are

equal, (b) the time parameter is discrete; within each time step, the abstraction

rates of water derived from each of the aquifers and from well storage are separate constants. The set of differential equations which describes the axially symmetric,

radial, unsteady flow in the aquifers is given by

d2ht ldhj 4>t dht

^^-TtlT' i = ia>-M r > r » (1)

in which rw = radius of the well screen, M = total number of aquifers tapped by the well, h. = head at a distance r from the well at time t in the /th aquifer, Ti = transmissivity of the /th aquifer, and 4>{ = storage coefficient of the /th aquifer. Solution to equation (1) is to be found for the initial conditions

h.(r,Q) = H-f i=l,2,-M (2)

in which Hi is the initial head in the /th aquifer prior to well construction. The boundary conditions to be satisfied are:

h.("/) = Hi (3)

hx(rwfy = h2(rw>t) = ... = hM{rw,t) = hjf) (4)

G. C. Mishra & S. Chandra 296

l"i**Ji 8r - nP

dhw(t)

8/ = 2/0 (5)

In equation (5), hjt) represents the head in the well and Q (t) the pumping rate.

Let the duration tQ and / be discretized to m and n units of equal time step respectively. Let the pumping rate Q if) be equal to Q. Let Q((7) and QWQ) be the contributions of the ith aquifer and well storage respectively during the 7th unit time period. Using the second assumption, the boundary condition prescribed by equation (5) can be rewritten as:

QPiD

QX{D + Q2(D * Q3(T) * ••• * QJ.D + QJJ) = Qp(l) (6)

0 for I S m

= Q for m < I « (m + n)

= 0 for I > (m + n)

Had the aquifers been tapped separately, for the initial condition h^rfi) = H., and the boundary condition h.(<°fy = Hp solution to differential equation (1) if unit quantity of water is withdrawn from the ith aquifer at time f = 0, is (Carslaw & Jaeger, 1959)

Ht - hJirfi exp 4712)/

Defining a unit impulse kernel

1

r2

46,./ J

fc,(0 4nr./

exp 46,/J

(7)

(8)

and designating Ht - hfrjt) = sfr,t), which is the drawdown in the piezometric surface in the z'th aquifer, drawdown for variable withdrawal from the ith aquifer can be written in the form (Morel-Seytoux, 1975)

s^t) = f Q.(c) kft - c) dc (9)

Q[c) being the withdrawal rate from the ith aquifer at time c. Dividing the time span into discrete time steps and assuming that the aquifer discharge is constant within each time step but varies from step to step, the drawdown at the end of time step I in the ith aquifer at a distance r from the well can be written as (Morel-Seytoux, 1975)

si(rJ) = ]J/__1 6 ^ ( 7 - 7 + 1)0,(7) (10)

297 Unsteady flow to a multi-aquifer artesian well

in which the discrete kernel coefficient dri(T) is defined as

dr/F)= !l W-Qàc

4JlT: £,

4B.I -E,

r2

L40J</- 1)JJ (11)

The exponential integral Ex(x) appearing in equation (11) is defined as (Abramowitz & Stegun, 1970)

/CO

exp(-w) uldu X

If M aquifers are tapped by a single well, at any time after the well construction and before the commencement of pumping, the zth aquifer will either loose or gain water depending on whether the composite hydraulic head at the well at that time is less than the initial hydraulic head in the zth aquifer or more. During pumping there will be contribution from each of the aquifers through the respective well screen.

If Q,(7)> 7 = 1,2,.../, are the contributions by the zth aquifer, drawdown at the well face in the zth aquifer at the end of time step I is given by

*,(V/)=I/=1 0,(7) 3Wi-(/-7 + D (12)

in which the discrete kernel coefficient 3 ,-(i) is defined as

*nJ!) = 4 7lT.

rr2

r w -E, r2..

UB,.(/-1)J (13)

Thus the head at the well face in the zth aquifer at the end of time step I can be expressed by the relation

h,(rwJ) = Hi- y=lQi{y)Qm.{I-y+l) (14)

Since the heads at the well face at the end of any time step I in all the aquifers are equal, therefore,

Hx- y=1 gjOO dml (I-y + l)=H2- Z/Bl <22(7) 6 ^ ( 7 - 7 + 1)

= 7f3 - Zy=1 Q3(7) 3 ^ (/ " 7 • 1) = HM-^sl QMiJ) dmMd-7 - 1)

(15)

G. C. Mishra & S. Chandra 298

The above set of equations can be written as

-GiCOa^d)+ Qi^rJV = H2-HX + y:\ j21(7)an)4(/ - 7 * i)

- rJ13l02(7)aniea-7 + l) (16)

- iJiiGsOoa^tf-y + i) (i7)

-Q1(/)8nvl(D + Q ^ a ^ D = " w - " i + 5>i fiiOOV - 7 + 1)

- ^ 2 ^ ( 7 ) 3 ^ / - 7 H-1) (18)

Let # m a x be the maximum value of H-. Head in the well in consequence to abstraction from well storage is given by

^ = # m a x - ^ l — T ( 1 9)

K hJJ) is equal to h ft J) for all values of i and I at r = rw. Using this relation one more equation can be written as

H, - I^=1 Qt{y)dml (I - 7 + 1) = J/m a x- 1 ^ — (20)

Rearranging,

- 0 1 ^ 9 ^ ( 1 ) + ~ — = HmaK-H1+ 2^-l1 Qx(7)awl (/ - 7 + 1)

; Qw(7) (21)

In matrix notation, equations (6), (16), (17), (18) and (21) can be written as

[A] [B] = [C] (22)

in which

299 Unsteady flow to a multi-aquifer artesian well

[A] =

[C]

1

-9w l(D

-9w l(D

-9w l(D

-9w l(D

1

9^(1)

0

0

0

1

0

9™l(l) ••

0

0

1 0

0

*WD 0

1 0

0

0

V(fy)

[B] = [Q^D, Q2{I), Q3(I), ..., Qjjil), QJfi] T

H2-H1+ Xf=1 Qp)dml(I - 7 + 1) " Zjli 02(7)9^ / - 7 + 1)

H3-H1+ E£ a QjC^g^C/ - 7 + 1) - If l i G3(7)9re3(/ - 7 + 1)

f/M- ^ + Z & d o o a , ^ / - 7 +1) - z ^ Qjy)dmAii - 7 • i)

^max- # i + lyli 2i(7)9wl(/ - 7 + 1) - l/(/>) Zjli QJ7)

In particular, for time step 1

[C] = [0, H2 - Hv H, - HV...HM- HvHms^ - H^

QxQ), Q$)> Q?,(!)> Qf/JO a n d QJf> c a n b e s o l v e d i n s u c c e s s i o n starting from time step 1 using the relation

[B] = [AY1 [C] (23)

Knowing Qt(I) values, the drawdown in the rth aquifer can be calculated using equation (10).

RESULTS

Though the analysis has been done for a multi-aquifer well which is open to any number of aquifers, results have been presented for a well which is open to three confined aquifers only. The duration starting from completion of the well to commencement of pumping and the pumping period are discretized to m and n units with equal time steps. For known values of Tv <pv T2, <t>2> Ty

<P3, and r , the discrete kernel coefficients, dmj(I), are generated making use of equation (13) for different integer values of /. For known values of m, n, Hi and Q (7), the values of Q{(1) and QJJ) have been found in succession,

G.C. Mishra & S. Chandra 300

0-251-

Fig. 2 Variation of -Q/t)/[T(Hmax - H)] and Q{t)/[T(Hmax -H J] with 4Tt/($r2J for T^T^T^ 3:2:1, and <t>j:<P2:<P3= 3:2:1 and

starting from step 1. _ _ _ The variations of QJt)/ p W m a t - HJ], and Q2(t)/ [T(Hmax - HJ] with

ATtl^P;) for Q(t) = 0 are shown in Fig. 2_for (H - HJ/ffl^ - H2) = 1, T^.T2\T?i = 3:2:1 and 4>l-4>2-'P3 = 3:2:1. T and $ are the arithmetic mean values of transmissivities and storage coefficients respectively. These results pertain to the case where (i) all the aquifers have equal hydraulic diffusivity, and (ii) the initial hydraulic heads in the first and in the second aquifer are the same and less than that of the third aquifer. It is found from Fig. 2 that when the aquifers have equal hydraulic diffusivity values and have same potentiometric surfaces, the flow quantities received by them from the third aquifer are in proportion to their respective transmissivity values. In other words if aquifer 1 and aquifer 2 have equal hydraulic diffusivity and have same initial potentiometric surface and if they are receiving water from the third aquifer in which the potentiometric surface is at a higher level, then at all times Q1(0/Q2(

r) v a l u e s a r e e £ lu a l t o TJTr The response of a multi-aquifer system to pumping can conveniently be

decomposed to the following two parts: Part 1: Response due to difference in potentiometric surfaces as existing in

the field but Qp{t) = 0. Part 2: Response due to pumping when all the initial hydraulic heads are

equal to the lowest initial hydraulic head i.e.:

Table 1 Contributions of each aquifer and head at the well point evaluated for TJ = 500 m2 day'1, T2 = 400 m2 day'1, T, = 300 m2

day1, 4>2 = 0.003, <p2= 0.002, <p3= 0.0001, r = 0.1, H = 200.0 m, H 2= 201.0 m, H 3= 202.00 m, and Q (t)=0

t Q, ( t ) Q ( t ) Q (t) h. (r , t ) -L -, £ ~5 ~ 1 W

(day) (m /day) (m^/day) (m /day) (m)

1

2

3

5

10

11

12

14

16

18

20

21

25

35

40

-283

-272.'

-267

-260

-251

-250.

-249.

-247.

-246.

-244.

-243.

-243.

-241.

-237.

-236.

.359

993

. 301

.480

.782

.634

.593

.771

.216

.859

,658

,107

,156

,479

,051

59

56

55,

53,

51.

51.

51.

50.

50.

49.

49.

49.

48.

48.

47.

.668

.987

.533

.805

.622

,338

079

627

244

909

612

476

996

097

749

223

216

211

206

200

199,

198.

197.

195.

194.

194.

193.

192.

189.

188.

.691

.005

.768

.675

.159

.295

.514

.144

.971

.950

,046

.631

,160

,382

302

200

200

200

200,

200,

200.

200.

200.

200.

200.

200.

200.

200.

200.

200.

.7897

.7912

.7920

.7930

.7943

,7945

.7946

.7949

7951

,7953

,7955

,7956

7959

7964

7966

Table 2 Contributions of each aquifer and head at the well point evaluated for T1 = 500 m2 day'1, T2 = 400 m2 day'1, T3 = 300 m2

day1, <j>2 = 0.003, 4>2 = 0.002, <p3 = 0.0001, rw=0.1 m, H.=H2 = H3

= 200.0 m, t0 = 10 days, t = 10 days, Q = 1000 m3day-1

t Q ^ t ) Q 2 ( t ) Q 3 ( t ) h i ( r (day) (m3/day) (m3/day) (m3/day) (m)

10

11

12

14

16

18

20

21

25

35

40

i

433

432

432

432

431

431

-1.1

0

.511

.923

.377

. 078

.870

.712

36490

-.77452

-.33449

-.25906

1

343

342

342

342

342

342

-l.i

D

.237

.907

.601

.429

.315

.226

34855

-.43699

-.18934

-.14970

0

223.

224.

225.

225.

225.

226.

,250

,170

,021

,493

.815

,062

2.91547

1.21180

.52395

.40880

2 CO

198.

198.

198.

198.

198.

198.

199.

199,

199,

199.

.7966

.7505

,7044

.6774

.6582

.6434

.8404

.9269

.9660

.9730

G. C. Mishra & S. Chandra 3 02

Table 3 Contributions of each aquifer and head at the well point evaluated for T1 = 500 m2 day'1, T2 = 400 m2 day1, T3 = 300 m2

day1, 4>2 = 0.003, <p2 = 0.002, 4>3 = 0.0001, rw = 0.1 m , Hi = 200.0 m, H2 = 201.0 m, H3 = 202.00 m, and Qp(t) = 1000 m3

day'1, to = 10 days, t = 10 days

(day ) (m / d a y )

Q 2 ( t ) Q 3 ( t ) h i ( r w , t )

( m J / d a y ) ( m J / d a y ) (m)

1

2

3

5

10

11

12

14

16

18

20

21

25

35

40

-283.359

-272.993

-267.301

-260.480

-251.782

182.879

183.332

184.608

185.862

187.012

188.056

-244.971

-241.929

-237.812

-236.308

59.668

56.987

55.533

53.805

51.622

394.576

393.988

393.229

392.674

392.221

391.839

48.429

48.560

47.909

47.600

223

216

211

206

200

422

422

422

421

420

420

196

193

189

188

691

005

768

675

159

542

678

163

464

767

105

544

369

903

708

200

200

200

200

200

199

199

199

199

199

199

200

200

200

200

7897

7912

7920

7930

7943

5911

5451

4993

4725

4536

4389

6360

7228

7624

7697

e/o =

0 for 0 < t « r0

Q for tQ < r < (t0 + g

o for t > (r0 + y

The response of an aquifer corresponding to part 1 and 2 when added would give its response to the pumping for the case when the potentiometric surfaces are at different levels. This can be seen from the results presented in Tables 1, 2 and 3.

CONCLUSIONS

A procedure that uses a unit response function has been described to analyse unsteady flow to a well opened to several aquifers, which are separated by aquicludes and in which the potentiometric surfaces are at different levels. The solution is tractable for numerical calculation. The contributions of each aquifer and composite hydraulic head at the well point have been evaluated

303 Unsteady flow to a multi-aquifer artesian well

when a well is open to three confined aquifers. Aquifers having the same initial hydraulic head and equal hydraulic diffusivity, receive water from the aquifer of higher potentiometric surface in proportion to their respective transmissivity values.

REFERENCES

Abramowitz, M & Stegun, I. A. (1970) Handbook of Mathematical Functions. Dover, New York. Carslaw, H. S. & Jaeger, J. C. (1959) Conduction of Heat in Solids. Oxford University Press,

New York. Khader, A. & Veerankutty, M. K. (1975) Transient well flow in an unconfined-confined

aquifer system. /. Hydrol. 26, 123-140. Mishra, G. C, Nautiyal, M. D. & Chandra, S. (1985) Unsteady flow to a well tapping two

aquifers separated by an aquiclude. /. Hydrol. 19, 357-369. Morel-Seytoux, H. J. (1975) Optimal legal conjunctive operation of surface and ground

water. Proc. Second World Congress, Int. Water Resour. Assoc. (New Delhi) 4, 119-129. Papadopulos, I. S. (1966) Non-steady flow to multiaquifer well. /. Geophys. Res. 71,

4791-4797. Wikramaratna, R. S. (1984) An analytical solution for the effects of abstraction from a

multiple-layered confined aquifer with no cross flow. Wat. Resour. Res. 20 (8), 1067-1074.