Transition from transient Theis wells to steady Transition from transient Theis wells to steady Thiem

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• 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

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 calcu- lated 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 func- tion 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 thick- ness 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

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

with a source term (the recharge N):

d2 d2 Id —-

(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 equa- tion (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.

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:

• 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

, - + C (6) 7 = 1

where where E, is the exponential integral which is defined as (Abramowitz & Stegun,

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