Mythological Sciences—Journal—des Sciences Hydrologiques, 43(6) December 1998 859

Transition from transient Theis wells to steady Thiem wells

WILLEM J. ZAADNOORDIJK* /WACO, Consultants for Water and Environment, PO Box 8520, 3009 AM Rotterdam, The Netherlands e-mail: wjzaadnoordijk@iwaco.nl

Abstract Simple analytical calculations of groundwater flow around wells usually employ the Thiem well formula for steady calculations and the Theis formula for transient calculations. The superposition principle can be used with both formulas and even for combinations of both formulas. It is generally assumed that flow converges to a steady state when the boundary conditions remain constant for a long time. However, the heads in Theis's formula do not converge to steady heads for large times, although the (specific) discharges do converge to those of the steady Thiem well. This undesirable behaviour of the Theis solution can be compensated by means of a new function: the ring well. The ring well is a combination of Theis's well and Glover's well (solution for a transient well in a circular island). A second function, uniform recharge during a time interval, is needed to assure that the heads converge at a specific level. The functions also allow an analytical transition with Theis wells between any two steady states consisting of Thiem wells. The functions do not have a wide range of practical applications but they increase insight into the relationship between the two main functions in groundwater flow.

Passage du régime transitoire (Theis) au regime permanent (Thiem) dans les forages Résumé Les calculs analytiques simples des écoulements souterrains autour des forages s'effectuent généralement en employant la formule de Thiem pour des calculs en régime permanent et la formule de Theis pour des calculs en régime transitoire. Le principe de superposition peut être utilisé avec les deux formules et même lors de la combinaison de ces deux formules. Bien que cela soit généralement le cas, il est faux de penser que les écoulements convergent nécessairement vers un régime permanent lorsque les conditions aux limites restent constantes pendant un certain temps. C'est ainsi que les hauteurs (d'eau) dans la formule de Theis ne convergent pas vers des hauteurs permanentes après un certain temps, bien que les débits spécifiques convergent vers les valeurs obtenues en régime permanent avec la formule de Thiem. Ce comportement indésirable de la solution de Theis peut être compensé au moyen d'une nouvelle fonction: l'anneau de forage. Il s'agit d'une combinaison des méthodes de Theis et de Glover qui fournit des solutions pour un forage en régime transitoire. Une seconde fonction, recharge uniforme pendant un intervalle de temps, est nécessaire pour assurer la convergence des hauteurs vers un niveau spécifique. Les fonctions permettent aussi une transition analytique selon la méthode de Theis entre deux états permanents obtenus par la méthode de Thiem. Les fonctions n'ont pas un vaste domaine d'application pratique mais elles augmentent la connaissance de la relation entre les deux fonctions principales des écoulements souterrains.

'Also at: Section for Hydrology and Ecology, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands.

Open for discussion until I June 1999

860 Willem./. Zaadnoordijk

N O T A T I O N

C constant in discharge potential [L3 T"1] H thickness of aquifer [LJ j counter for steady wells [-] k hydraulic conductivity [L T 1 ] L distance to make logs of wells dimensionless [L] TV recharge [L T"1] m number of wells [-] n counter in summation of ring well [-] Q discharge of well [L3 T"1]

r radial coordinate for ring well ( r = ^J(x - x R ) 2 + (y - y R ) 2 ) [L] S storativity [-] t t ime [T] x Cartesian horizontal coordinate [L] y Cartesian horizontal coordinate [L] w counter for transient wells [-] a aquifer diffusivity ( a = kHIS) [L2 T"1] p factor for zero of Bessel function J() [1 L"1] y Euler ' s constant (y = 0.577 215 664 (see Abramowitz & Stegun, 1972) [-] 0 discharge potential [L3 T"1] cp piezometric head [LJ

Subscripts:

a initial well in basic problem 1 b final well in basic problem 1 d well in basic problem 2 / final steady state i initial steady state j index for steady wells TV recharge n index for summation in ring well R ring well s starting of recharge t ending of recharge w index for transient wells 1 basic problem 1 2 basic problem 2

Functions:

E, Exponential integral J0 Bessel function of first kind of order zero J, Bessel function of first kind of order one log natural logarithm

Transition from transient Theis wells to steady Thiem wells g61

INTRODUCTION

Most textbooks on groundwater flow include simple calculations with analytical solutions (see e.g. Verruijt, 1970). This is an illustration of the fact that simple analytical calculations are useful to get a basic understanding of groundwater flow. The use of simple analytical calculations is not limited to the classroom, but is common in engineering practice.

Analytical calculations are a good starting point to improve understanding before data collection or complicated modelling studies are carried out. Moreover, analytical calculations involving wells are typically used in the design of pumping for small aquifer remediation and building pit dewatering projects. The time scales of these problems are such that the transition between the transient flow and the steady state is an important issue. A steady state calculation is carried out first. When the calculated drawdowns are unacceptable, a transient calculation is performed to find out whether these drawdowns will be reached during the period of pumping. The transient calculation should give the transition from the initial situation to the calcu­lated steady state when the pumping period is long.

Many analytical calculations involve steady flow around wells, for which the Thiem well formula is used. This function represents steady flow to a well, the final steady state that will be reached ultimately after pumping has started. The question remains, how soon after the well has been switched on is the flow approximately steady? The answer to this question involves transient groundwater flow.

The transient well function used most is the Theis formula. However, this func­tion cannot be used directly to answer the question of convergence to a steady state, since the function does not converge to a steady function for large values of time. The fact that Theis's well does not converge to a steady well for large times is well known (see e.g. Haitjema, 1995). For large times, the piezometric head decreases logarithmically. The decrease is less at greater distances from the well and remains zero in the limit to infinity, which does cause a distinct different behaviour from real situations in which the influence of a well is limited to a finite area.

Rather than trying to derive a function that could be used instead of the Theis formula, this paper presents a solution whereby this most common transient well formula can still be used and convergence to a specified steady state can be assured.

GOVERNING EQUATIONS

In the description of the groundwater flow, the following simplifying assumptions are used (Bear & Verruijt, 1987). The properties of the groundwater and the matrix of the aquifer are constant. Darcy's law is valid and the hydraulic conductivity k is isotropic and constant. The base of the aquifer is horizontal and the saturated thick­ness H is constant. The flow is shallow so that the Dupuit assumption is adopted. The storage and release of water in the aquifer is linear, occurs instantaneously and is fully reversible, so that the amount stored per unit area per unit time is equal to Sdip/dt, where S is the storativity and cp the piezometric head. The storativity S is

862 Willem ,1. Zaadnoordijk

constant. With these assumptions, the governing differential equation is the heat equation

with a source term (the recharge N):

d2 d2 Id —-<P + — - ® = -N + 0) (1) dx~ ay a at

where the aquifer diffusivity a is equal to the quotient of the transmissivity and the storativity a = (kH)IS and x and y are horizontal Cartesian coordinates.

A discharge potential <t> is used as dependent variable (Strack, 1989) which has the following relation to the head cp (measured from the base of the aquifer):

O = kHq> (2)

In the absence of recharge the partial differential equation (1) becomes the heat equation:

d2 d2 Id

ox" oy a at

and the corresponding steady state is governed by the Laplace equation:

d2 d2

-T® + zrT® = 0 (4) ox~ ay~

The following discussion will focus on transition from one steady state to another. The potentials of the steady states are solutions to the differential equa­tion (4), while the potential during the transition is a solution of equation (1). The discussion in this paper will be limited to Thiem and Theis wells. Many other solutions to the differential equations (see Strack, 1989; Zaadnoordijk & Strack, 1993) show a similar behaviour so that the conclusions can easily be extended.

STEADY STATE

The potential for the steady flow to a well at location (xy, yj) with a discharge Qt is adapted from the Thiem formula and written as:

<D. =—Moi ' 4TT

r(x-xj?+{y-yj)^

L1 (5)

where L is a length that has been introduced in order to keep the (natural) logarithm dimensionless. The value of L is arbitrary. A convenient choice will be discussed later.

The potential (equation (5)) is a solution to the differential equation (4) in the entire plane except the point (x;, yj) and infinity. Therefore another condition is required besides the discharge Q:i at (x-, y). If the distance L is not used to meet

Transition from transient Theis wells to steady Thiem wells 863

this condition, a constant should be added to provide the degree of freedom to meet it. When several wells are superimposed, only one constant C is needed (and not a constant per well, since infinity is a singular point that is common to all wells):

m

<D = 2>, - + C (6) 7 = 1

where <&,• are m Thiem wells (equation (5)). The constant C can be determined from a given head at some location (not at a well), the so-called reference point and reference head (Strack, 1989).

TRANSIENT STATE

The potential for a transient well at location (xw, y J with a discharge Qw starting at time tw is adapted from the Theis formula and written as:

*-~*E{ 4 „ , - , , ) J (7> where E, is the exponential integral which is defined as (Abramowitz & Stegun, 1972):

E,(s)=]—du (8) v U

The potential 0„, equation (7) is a solution to the partial differential equation (3) except at (x„„ )'„,). These wells will be superimposed on the initial steady state, for which the potential has the form of equation (6) and is a solution to equation (4). Thus the resulting potential also is a solution to equation (3) except at the location of the wells.

The exponential integral (equation (8)) has the following behaviour for small values of the argument:

E,CY)«-logi--Y ,v« 1 (9)

where y is Euler's constant. The error is less than 0.25% for s < 0.01. The approximation (equation (9)) gives the value of the potential (equation (7))

for large values of the time t:

cb « d> + — 1 • ' • p / 1 « / - » » • +

4 7 t A-y^) + y (x-xn)

2 +(v-v...)2

I » " • (10) 4a

where 0 ; is the potential of a steady well (equation (5)). The potential still depends on t and does not converge to a steady state, although the (specific) discharges approach those of the Thiem well (equation (5)). This is illustrated in Fig. 1.

864 Willem J. Zaadnoordijk

«* w Thiem - — Theis (time increasing logarithmically) Fig. 1 Heads due to Thiem and Theis well.

BASIC PROBLEMS OF STEADY STATE TRANSITIONS

There are two basic problems in the transition of one steady state to another, with respect to wells. The first basic problem is a shift of the location. A well at location (x„, ya) is stopped and another well of the same discharge Qx is started at another location (xb, yh). The potential for the initial steady state is given by:

<* S . ((x~xa)2+(y^ya)

2)

and for the final steady state by:

°"=^ l 0< û The indices / and / indicate the initial and final steady states respectively. The constants Cn and Cfl are not necessarily equal. They can be determined from a reference point or points with reference heads for the initial and final steady states. One has to be careful in deciding which point(s) to use, since the reference point is not a physical boundary condition (Haitjema, 1995); it is merely a convenient way of expressing the boundary condition at infinity, since the potential at infinity is infinite.

The second basic problem is the starting of a new well. For the sake of simplicity the potential for the initiai steady state is a constant:

+ C (12)

4>/2 = CL (13)

Transition from transient Theis wells to steady Thiem wells 865

and in the final steady state, one well is active at location (xd, yd):

Qi, ((x^x.f+iy-y,,)2

0/2 = ^ l o g ^ — ^ _ _ ^ J + C / 1 (14)

Again, the constants Ca and Cp_ can be determined from reference point(s). The transition of the potential On to <$>n and of O a to <bf2 will be discussed next. These transitions will be described by solutions of the differential equation (1) or (3).

TRANSITION FOR SHIFTED ABSTRACTION LOCATION

In this section the transition from a steady state with one well active to another steady state with a well active at a different location is discussed.

Two transient wells (equation (7)) are needed, one to end the old abstraction and one to start the new abstraction. If the ending and starting times are equal to ta and tb

respectively, the potential of these two wells superimposed on the initial steady state (equation (11)) is equal to:

Ô, , l(x-xa)2Hy-ya)^

O . ^ l o g ! 4TT °V Ll + C„

a „ ((x~-Xg)2+(y-y„)2 | , ^ , , + ^ H 4 o ^ ^ 1 ' > ' - ' * <15>

^a_ E Ux-xb)2+(y-yby

47t ' I 4a(t-th)

This potential does not necessarily converge to the potential of the final steady state (equation (12)) for large values of time:

r ^ n Q\ i f(x^y />)2 +(y-yk) hm 0), = Cn + — log -,

/ , , N ( 1 6 )

Q\, \(x-Xb) +(y-y„y * c ^ + ^ l 0 \ Û

where equation (10) has been used. From this result it is clear that the two transient wells do not suffice to describe the transition from equation (11) to equation (12) since in general the constant of the initial steady state C,-, is not equal to the constant of the final steady state Cn.

The difference between these constants corresponds to a uniform difference of the potential. This difference can be eliminated by adding a function to the potential (equation (15)) that changes the potential uniformly. Such a function is the transient uniform recharge for a limited period of time (see Zaadnoordijk, 1988):

866 Willem J. Zaadnoordijk

<D,

0 / < ts

NNa(t-tJ ts <t <t,

NNa(t,-t,) t>t,

(17)

where ts and t, are the starting and ending time of the recharge respectively, and the amount is equal to NN = (Cfl - C,,)/((î, - ts)a). The potential <t>A, is a solution to the partial differential equation (1) with N = NN, while up to this correction function a zero recharge N = 0 has been used (solutions to equation (3)). The sum of <Î>N and potentials fulfilling equation (3) is also a solution to equation (1) with N = NN.

Thus the transition from the steady state with one well to the steady state with a similar well in a different location has to be changed from equation (15) to the following expression:

0 > , = - l o g l j 2 +Cn

fi>r ((x-xa)2+(y-yg)2

4TI \ 4a(t-ta)

a.E| (izaiii<£ziL|t<t>A, 4TI ' I 4a(.t-t„) '

(18)

»,: (',"',)«

CONVERGENCE TO A STEADY STATE WITH A NEW ABSTRACTION

In this section the transition from a steady state without any wells to a steady state with one active well is discussed. One transient well is needed. It starts the well of the final steady state at time td. The potential of this well (equation (7)) is superimposed on the potential of the initial steady state (equation (13)):

<&•, Cl2 .92, {x-xdy +(y-y,,Y

4 a (t-tj) (>(, (19)

However this expression does not converge to a steady potential as can be seen by applying the approximation (equation (10)) to the potential (equation (19)):

cD, ,., Q-,, (x-xyy +(y~y,,y C, 2 + —logl J} 1+7

4% log[

L1

t » {x-x^f +(y-yiiy

4a (20)

An explanation for this behaviour, in which the potential of the transient well (equation (7)) does not stop to decline, is the fact that this solution satisfies the boundary condition that the head does not change at infinity. This behaviour could be

Transition from transient Theis wells to steady Thiem wells 867

changed if the boundary condition at infinity is replaced by a boundary condition at a finite distance. This can be done by adding the potential of a so-called ring well <f>R

(Zaadnoordijk, 1988):

* , OIL An

-X

log! Û Aa(t~tR)

•To(fV)e •K<*('-'«)

(P„Z)2J^(P„I)

r = ^(x-xlty +(y~~yKy < L (21)

where J0 and J, are the Bessel functions of the first kind of order zero and one respectively, the factors /?„ correspond to the nth zero of J0 (see Abramowitz & Stegun, 1972), and r is the radius measured from the centre of the ring well (xR, yR). The ring well is derived from the solution of Glover (1974) for a transient well in the centre of a circular constant equipotential and a Theis well (see Fig. 2). The ring well provides a recharge at a ring with radius L around the centre (xR, yR) in such a way that it lets a Theis well converge to a Thiem well both with a discharge QR. The recharge starts at time tR.

The potential <t>R (equation (21)) fulfils the partial differential equation (3) for 0 < r <L. The initial condition is a value of zero for all r. The boundary conditions are no extraction at the centre (r 4- 0) and a potential at r = L equal to [gR/(47i)]E,{L2/[4a(f - tR)]} (the negative potential of a Theis well at a distance L). The centre (xR, yR) and the radius L have to be such that the ring encloses the entire area of interest. The limit for infinite times of a ring well does not involve a change of constant, since the radius is chosen equal to the distance L in the expression of the Thiem well (equation (5)).

Thus the potential of a ring well (equation (21)) can be added to the potential (equation (19)) to ensure convergence to the desired final steady state (equation (14)). The resulting transient potential is:

2 '- An '

(x-x,,)- +(y-yllY

4<x(/-<,,) + * , t >t,„ I >t, (22)

r Q R

+ r

QR

Çrlover's wel l Theis 's wel l Fig. 2 Construction of ring well.

r ing wel l

868 Willem J. Zaadnoordijk

GENERALIZATION

The solutions for the basic problems of steady state transitions that have been presented in the preceding sections make it possible to describe the transition between two arbitrary steady flow fields around Thiem wells. A combination of Theis wells and the functions for recharge (equation (17)) and the ring well (equation (21)) can model the transition of any steady state consisting of Thiem wells to any other such steady state (equation (6)). The recharge function can still be used since the Thiem wells do not influence the constant due to the choice of the distance L. The ring well can provide for convergence in case of a net change of discharge due to a number of wells in the same way as it does for the addition of a single well.

Let the potential for the initial steady state be given by:

*,=ï 7 = 1

9L 471

log (x-Xjf+jy-yj)2

L2 + C, (23)

where m,- is the number of wells and let the potential of the final steady state with mf

wells be given by:

% = X Qj, (x-xj)2+(y-yj)2

4TT log L1 + C, (24)

and let the recharge NN in the recharge function (equation (17)) be determined in the same way as presented before:

®N =

0 c,

t <t.. •C;

cf-c, -{t - / , ) ts <t <t, (25)

t>t.

and let the discharge QR of the ring well function (equation (21)) be equal to the difference between the total well discharge of the initial and the final steady state:

1>«

0

ÏQJ-ZQJ

4% log + E, "I

J0(P,/)e ~ M ( ' - ' K

4a«-tR)J t ? ( P „ £ r J , ( P „ £ )

(<t,

t>tR

(26)

then the transient potential

cD, = 0 ) . + X C ^ U ^ + ^JV +<&h (27)

describes the transition from the initial state equation (23) to the final steady state (equation (24)). The potentials Ow are transient wells (equation (7)) that terminate the

Transition from transient Theis wells to steady Thiem wells 869

wells of the initial steady state (Qw = -Qtfor j ~ w = 1,2,..., m-) and start the wells of the final steady state (Qw = Qjfor y = w = m,• + 1, in, + 2, ..., ml + m^. The choice of the parameters ts and t„ xR, yR, tR, and L represents the control on the groundwater during the transition. The values can not be derived directly from physical properties, just as the reference points of the steady states.

EXAMPLE

A simple model is presented to illustrate the recharge and ring well functions in the transition between two steady states. The aquifer has a hydraulic conductivity k = 10 m day', thickness H = 10 m and storativity S = 0.1. The length L is chosen equal to 500 m. A layout of the model is given in Fig. 3.

The initial steady state consists of three wells (the data are given in Table 1). The piezometric head at (45.25, 150) is equal to 20.0054 m above the base of the aquifer. The piezometric contours of the initial steady state are given in Fig. 4.

The final steady state consists of two wells (see Table 1) and the same reference point is used as for the initial steady state. The piezometric contours of the final steady state are given in Fig. 5.

reference

initial well 1

initial well 2 initial well 3

final well 1 final well 2

,R=500m (=L)

Fig, 3 Layout of example model.

870 Willem J. Zaadnoordijk

reference X p 0 0 ; 1 5 Ô )

initial well 1

{ |' ( initial well 2 ^ 19.6m initial well 3

\ \ \ ® /'" 19Jm

final well 2 T9.8m final well 1

19.9m o n n 20.0m /

20.1m

20.2m j(-2QQ--150) Fig. 4 Piezometric contours for initial steady state.

20.1m reference A ' '

/ ' ,20.0m 20.0m

initial welll 1 9-9 m

19.8m 19.7m

initial well 2 initial well 3 19.6m

19.5 m

final well 2 final well 1

|̂ r200;-150) Fig. 5 Piezometric contours for final steady state.

Transition from transient Theis wells to steady Thiem wells 871

sjia^axAj, -u.% -p-oay sjia^axxA -u.% froay oxu-^aiMtozaxcC

872 Willem J. Zaadnoordijk

Table 1 Data of wells.

Well

Initial 1 Initial 2 Initial 3 Final 1 Final 2

X

(m)

-50 -100

50 50

-50

(m)

50 0 0

-50 -50

(m3 day"1)

100 75 75

125 100

^transient

(m3 day"1) -100 -75 -75 125 100

'start

(days)

100 200 300 150 350

The transition is modelled by means of five transient wells, recharge, and a ring well. The transient wells terminate the wells of the initial steady state and start the wells of the final steady state (see Table 1).

The recharge is active from time 5-40 days with a strength of -0.00013 m day"1. The ring well has a strength QR of 25 m3 day4 and the radius is equal to the length L (500 m). Time lines of the piezometric head in a row of points between (0, 0) and (0, -400) as a function of time are given in Fig. 6.

In order to show the influence of the recharge and the ring well, the time lines of the transition are also given without these functions (see Fig. 7).

CONCLUDING REMARKS

Using the functions for a ring well and transient uniform recharge, the transition from any steady state with Thiem wells to another can be simulated by means of Theis wells.

Instead of the uniform recharge function, one can use a function that gives recharge outside of the area of interest only. This function is constructed as the combination of the transient uniform recharge and a transient area sink, which provides a uniform evaporation inside a rectangle (Zaadnoordijk & Strack, 1993).

The practical use of these functions is limited due to the fact that the parameters are not easily linked to physical quantities. They represent the control on the piezo­metric head due to all features that are not modelled explicitly, such as surface waters. However, the functions clarify the relation between the well functions used most in geohydrological practice.

REFERENCES

Abramowitz, M. & Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover, New York. Bear, J. & Verruijt, A. (1987) Modelling Groundwater Flow and Pollution. Reidel, Dordrecht, The Netherlands. Glover, R. E. (1974) Transient Groundwater Hydraulics. Dept of Civil Engng, Colorado State Univ., Fort Collins,

Colorado, USA. Haitjema, H. M. (1995) Analytical Element Modeling of Groundwater Flow. Academic Press. San Diego, California,

USA. Strack, O. D. L. (1989) Groundwater Mechanics. Prentice Hall, Englewood Cliffs, New Jersey, USA. Verruijt, A. (1970) Theory of Groundwater Flow. Macmillan, London. Zaadnoordijk, W. J. (1988) Analytical elements for transient groundwater flow. PhD Thesis, Univ. of Minnesota,

Minneapolis, USA.

Transition from transient Theis wells to steady Thiem wells 873

Zaadnoordijk, W. J. & Strack, O. D. L. (1993) Area sinks in the analytical element method for transient groundwater flow. Wat. Resour. Res. 29, 4121-4129.

Received 16 June 1997; accepted 22 March 1998