WATER RESOURCES RESEARCH, VOL. 35, NO. 12, PAGES 3967-3968, NOVEMBER 1999

Tidal inflow to aquifers

Benjamin Ross Disposal Safety Incorporated, Washington, D.C.

Abstract. Where an aquifer discharges into a tidal estuary, flow can reverse at high tide. When the aquifer and estuary waters differ chemically, it is useful to know whether flow reverses and, if so, how much. For the case of a confined aquifer discharging into a tidal estuary, a formula for the ratio of inflow to outflow during one tidal cycle is derived.

Because manufacturing plants are frequently located near harbors, it is not rare for contaminated groundwater to dis- charge into tidal estuaries. At high tide, groundwater flow near the discharge point can temporarily reverse, introducing un- contaminated water into the aquifer. During the next low tide the estuary water will be flushed out of the aquifer. Similar exchanges ofwater with differing chemistry can occur when the aquifer is fresh and the estuary is brackish. To interpret chem- ical analyses in such situations, one must determine whether there is inflow from the estuary to the aquifer and, if so, how much.

Ferris [1951] solved for the head in a confined aquifer ad- joining a stream with cyclically varying water levels. Ferris's formula is reprinted by Walton [1984, equation (5.210)]. This note uses Ferris's solution to obtain an explicit formula for the ratio of inflow to outflow in one tidal cycle. The result is expressed directly in terms of high and low water levels mea- sured in a well near the bank and in the stream.

The solution is for a perfectly confined aquifer discharging into a stream with sinusoidally varying water levels. For con- venience, the mean stream stage is set to zero. A cross section is depicted in Figure 1.

Ferris's solution can be written as

s * ( t ) = s he -X/X sin ( tot -- • --) (1)

oo = 2 =/ts (2)

12= tsT/•rS, (3)

where the symbols have the following meanings: s* difference between instantaneous and mean head in

observation well [L]; t time[T];

S h amplitude (or half range) of stream-stage change [L]; x distance from edge of stream to observation well [L]; t s period of tidal fluctuations (time between high tides) [T]; T transmissivity of aquifer [L2T-1]; S storage coefficient of aquifer [dimensionless].

If there is a uniform hydraulic gradient i toward the stream, the mean head in the well is ix and the instantaneous head in the

well s is

s(t) = she-X/X sin ( tot -- • --) +ix. (4) Copyright 1999 by the American Geophysical Union.

Paper number 1999WR900246. 0043-1397/99/1999 WR900246509.00

Now the flux of water entering the aquifer from one bank of the stream, per unit length of stream, which we denote Q [L2T-1], is

= - = T •-(sintot+costot)-i (5) Q T •xx or

cos tot- -i . (6)

This has a maximum value of 21/2Sh/)k- i, which occurs at dimensionless times tot = vr/4, 9,r/4, '-'. If this maximum value is negative, there will be no inflow from the stream to the aquifer. If the maximum value is positive, the zeros of (6) occur at times tot = ,r/4 +_ arccos b, 9,r/4 _+ arccos b, ..., where b is the dimensionless quantity

b = 2 -1/2-- . (7) Sh

The total inflow of stream water into the aquifer during one tidal cycle Q st .... [L2] is

stream "- f(1/to)(vr/4+arccos b) 1 dt d(1/to)(vr/4- arccos b)

(8)

Qstream --- Ti •-• sin (arccos b) - -- arccos b (9)

Qstream - to •-5 1 - arccos b . (10)

The total outflow of water from the aquifer into the stream during one tidal cycle Q aquifer [L2] is

Qaquifer •- Ti f(1/tø)(9vr/4-arccøs b) d (1/to)( vr/4 + arccos b)

1

(11)

Q aquifer '- to •-•- 1 + ,r- arccos b , (12)

and the ratio of inflow to outflow is

where

3967

Q stream

Q aquifer

I

•+I (13)

1 )1/2 I: •-1 - arccos b. (14)

3968 ROSS: TECHNICAL NOTE

Stream Observation Well

• Aquiclude Aquiclude

Uniformly porous aquifer

Aquiclude

Figure 1. Cross section of confined aquifer discharging into tidal stream.

This formula can be applied most easily if b is expressed in terms of the quantities measured in the field, which are the high and low water levels in the stream and in wells. This can be done by using Walton's [1984] equation (5.211) for the range of groundwater fluctuation in a well st:

S r = 2Sh e-x/x (15)

or

X S r -ln (16) • 2S h

Since the mean hydraulic gradient away from the stream is uniform, the average head difference between the well and the stream Sav e is simply

Sav e --ix. (17)

Combining (7), (16), and (17) gives

2-1/2Save = Sh In 2Sh/Sr'

or, if we replace the half range of tidal variation in the stream Sh by the full range ss,

2 •/2s ave b = (19) ss In ss/Sr'

If b in (19) is more than 1, groundwater discharges to the estuary throughout the tidal cycle. If b is less than 1, estuary

water flows into the aquifer at some times during the cycle, and (13) gives the ratio of the inflow during an entire tidal cycle to the outflow during the cycle. By using (14) and (19), the result (13) is expressed in terms of the minimum, maximum, and average water levels in the estuary and a well, with interpreted or estimated quantities such as transmissivity and storage can- celed out. Thus the ratio of inflow to outflow can be calculated

directly from measurements in a single well on a single day. Because water levels in estuaries fluctuate from day to day and the subsurface is variable, larger data sets should be used to obtain more reliable estimates.

Acknowledgment. This work was supported by the San Francisco BayKeeper.

References

Ferris, J. G., Cyclic fluctuations of water levels as a basis for deter- mining aquifer transmissibility, Int. Assoc. Sci. Hydrol. Gen. Assem. Brussels, Publ. 33, 2, 1951.

Walton, W. C., PracticalAspects of Groundwater Modeling, Natl. Water Well Assoc., Columbus, Ohio, 1984.

B. Ross, Disposal Safety Incorporated, 1701 K Street NW, Suite 510, Washington, DC 20006. (easi@worldnet.att.net)

(Received March 1, 1999; revised July 29, 1999; accepted July 30, 1999.)