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RadialFlow to Pumping Wells 3 - 5

Prob l em 1THEIS EQUATION FOR DISTANCEVERSUS DRAWDOWNFire Protection Well, River Bend Station NuclearPower Plant, St. Francisville, Louisiana

Overv i ewRadial flow of groundwater to a pumping well can be mathematically expressedby combining various forms of Darcy's Law with equations of continuity. Flowto wells in confined or unconfined aquifers and under steady-state or transientconditions can be addressed mathematically. One of the most basic methods andsimplest sets of conditions describes a confined aquifer with transient flow to awell that penetrates the entire saturated thickness of the aquifer (i.e., the well isfully penetrating). Theis (1935) was the first to represent these conditions inmathematical terms. Figure 3.4 shows flow to a fully penetrating well in aconfined aquifer where the flow is horizontal and radial.

- Pumping Well

InitialPotentiometricSurface \ _

. ]%" Si

S*

' Confining Layer

H|

, ri !0 h hn \ Confined Aquifer

Scr eenConfining Layer

Figure 3.4. Conceptua liz at ion of radial flow to a fully pene tr at ingpumping well showing initial and pumping potentiometric surfaces. Twoobservation we lts at radii r% and r2 i ll us tr ate the spa ti al va ri at ion indrawdowns s, and s2. The saturated thickness of the aqui fe r is b anddrawdown inthe pumping well is sw.

In 1935 C.V. Theis published his original paper, which described theanalogous transient behavior of heat flow to a line sink in an infinite conductive

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3 - 6 Chapter Threesolid and groundwater flow to awell in an infinite aquifer. The nrathematicalovulation he presented described the amount of dn^towoCi)jted byDumping well at any time (/) after pumping commences, at any radial distanceWThfs mathematical model became known as the Theis equation and requirescertain assumptions about the nature of the aquifer, the pumping rate, and thedesign of the well. The Theis equation is as follows

where

hoh

QT

s-h h=-4tvTG du (3-1)

is drawdown at any time and distance from the pumping wellis initial hydraulic head at any distance [t =0] (L),is hydraulic head at the same distance after elapsed time [/ - tj(L), 3is discharge from pumping well (L /T),is the transmissivity ofthe aquifer (L /T), andis the Theis equationparameter.

The Theis equation parameter () is defined asr2S (3-2)ATt

where r is radial distance from the pumping well to any distance (L),S is storage coefficient (L /LJ),T is transmissivity (L IT), and/ is elapsed time since the beginning of pumping stress (I).

The assumptions inherent in the Theis equation are as follows:Aquifer - isotropic, homogeneous, uniform thickness flat lying, infiniteinareal extent, overlain and underlain by impermeable layers, releaseswater instantaneouslyfrom storage.Well -fully penetrates the aquifer, discharges at a constant rate, noborehole storage.

The Theis equation (equation 3-1) cannot be integrated directly but can beapproximated using the following expansion.

S=h'h=-^YQ -0 .5772 - In u +- 2-2! 3-3! 4-4!(3-3)

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Radial FlowtoPumping Wells 3 - 7

To simplify the expression, the entire series expansion is usually denoted by theterm W(u), which is called the Theis well function. Thus, the Theis equation canbewritten in a simplified form.

Q_4ttTs = K-h =- f - W{u) (3_4)i where