Jourr[ or Geopsysrcel Rlrrrrcs VoL. 68, No. 16 Aueusr 15. 1963

A Theoretical Analysis of Groundwater Flow in Small Drainage Basins' J. T6rn

Grourdwater Diuision, Research Caurrcil of Alberta E d,monLon,A Iber to, C anad,a

Abstract. Theoretically, three types of flow systems may occur in a srnall ba,sin: local, in- termediate, and regional. The local systems are separated by subvertical boundaries, and the systems of difrerent order are separated by subhorizontal boundaries. The higher the topo- graphic relief, the greater is the importance of the local sys0ems. The flow lines of large un- confined flow systems do not cross major topographic features. Stagnant bodies of groundwater occur at points where flow systems meet or branch. Recharge and discharge areas alternate; thus only part of the basin will contribute to the baseflow of its main stream. Motion of groundwater is sluggish or nil under extended flat areas, with little chance of the water being freshened. Water level fluctuations decreasewith depth, and only a small percentage of the total volume of the groundwater in the basin participates in the hydrologic cycle.

Introduction. Whereas it is important to but others are hidden and may not be revealed have a general understanding of the motion of even by expensivetest programs. The neglect groundwater in dealing n'ith groundwater prob- of these latter properties could lead to entirely lems, the careless and frequent use of the ex- wrong conclusionsregarding groundwater flow pression ma,y subvert its basic meaning. Until in small basins either in general or in any par- certain characteristics of the flow systems in- ticular case.Even if the theory is not used to volved are well defined, groundwater motion in obtain quantitative results, the qualitative ap- a given area cannot be conceived to be generally plication may still contribute to the general known. Among the numerous features, a knowl- understanding of groundwater flow in small edge of which is indispensableto the under- basins. standirg of groundwater motion in an area, the Before starting with the development of the following are thought to be the most important: theory a brief account will be given of the rea- the locations and extent of recharge.and dis. sons why a theoretical analysis is believed to be charge areas,the direction and velocity of flow best suited for an initial general study of ground- at any given point in the region, and the depths water flow in a given area. of penetration of the flow systems.It is easy to Gmeral. The methods of studying ground- appreciate the value of this information if one water motion can be either practical or theo- considersonly the difficulties which may arise retical. The group of practical methods includes in connection with problems such as outlining field investigations based on the principles of areas of potentially equal yield, tracing con- geology, geophysics,geochemistry, and hydrol- taminants, estimating baseflow of rivers, and ogy, and it is thus basedon obeervationsof phe- establishing groundwater budgets. nomena controlling or related to the flow of The purpose of this paper is to present a groundwater in nature. The theoretical meth- theory by means of which groundwater flow in ods, on the other hand, make use of electrical small drainage basins can be analyzed. Some of analogs, scale models, and mathematical models the properties of flow derived from this analy- to investigate phenomena resulting from ideal- sis are obvious and may be observedin the field, ized situations. In the final analysis the conclu- sions drawn from the data of both groups should be considered,and they must be in agreement. r Contribution 185 from the Research Council of Nevertheless, the writer believes the results ob- Alberta, Edmonton, Alberta. Presented at the Third Cana.dian Eydrolory Symposium, Calgary, tained by the application of the theoretical Alberta, November &9, 1962. methods to be the more weful in the initial 4795 4796 J. TOTE stages of an irnvestigation. This view is based still may remain the uncertainty, however, of on the presumption that an observedphenome- whether some of the decisivefeatures have been non is usually related to only one feature of a overlooked and whether there may be some given flow system, whereasit might be brought characteristicsthat cannot be measured at a]l. about by different causesin difierent situations; Whereas it is practically impossible to observe it may be the identica,l particular solution of separately all phenomenaconnected with a re- several problems. gime of groundwater flow, a correct theory dis- A few examples may prove helpful in clarify- closes every feature and draws attention to the ing this statement. For instance, a decreasein most important properties of the flow. hydraulic head with depth, commonly observed It is believed that small drainage basins are in water wells, may be produced either by head the most important units in the groundwater lossesdue to the vertical downward-component regime. A good understanding of groundwater of the water motion or by the water beiag movement in adjacent small basins makes pos- perched in the permeablelayers of a geological sible an accur&te representation of the motion formation consistingof a seriesof more and less of groundwater within the large basin that they pervious beds lMei,nzer, 1923, p. 411. Another form. In pelking from larger basins to smaller good example is a perennial body of impounded ba.sinsthe weight of the uncertainties increases, surface water. It may owe its existenceeither and a vague and possibly unreliable analysis is to poor underground drainage due to geologic obtained. Apart from this, a small basin is com- conditions or to continuous groundwater dis- monly much less complicatedthan a large basin chargecaused by the generalpattern of the flow with respect to geology and topography; there- systems.The causeof a relatively low baseflow fore, it lends itself much better to both practi- yield for a river may be even more uncertain cal and theoreticalstudies. becauseit could be explained by a basin-wide The definition of a small drainage basin as it low permeability, by good surface drainage, or will be understoodthroughout this paper is: on by the stream not beiag the only place of s,rea bulnded by topographit highs, its l,owest groundwater dischargein the basin. It is real- parts bei.ngoccupied bg an impounded body of ized, of course,that the larger the variety of in- surlace water or by the outlet ol a relatiaelE dependentinvestigations, the more preciselythe low order strearn and, hauing similnr phgsio- characteristicsof the flow can be outlined. There graphic conditions ouer the whoLeol its turface.

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Fig. 1. Idealized cross section of a valley flank in a small drainage basin. GROUNDWATER FLOW IN SMALL BASINS 4797 The upper limit of area for such basins is usu- slopes of the valley flanls greatly exceed the ally several hundred square miles. longitudinal slopesof the valley floors. This dif- Mathematical deuelopment. In the mathe- ference in slope causesthe longitudinal compo- matical analysis the region of groundwater flow nent of the flow to becomenegligible compared at one side of the valley is representedby a with the lateral component. rectangular area (Figure 1). This area is limited The distribution of the fluid potential in a by a horizontal impermeable boundary at its basin with boundaries as outlined above is de. base, by two vertical impermeable boundaries rived from the general expressionfor the fluid extending downward from the stream and the potential lHubbert, 1940,p. 8021: water divide, and by a horizontal line at the ele- vation of the stream along which line the fluid- 6 : * (1) potential distribution is supposedto be the samo sz [,".+ as that for the real water table. The assump- where { : fluid potential, g : accelerationdue tion of a horizontal impermeable boundary as to the earth's gravity field, z: elevation above justified the lower limit of the basin is because, the horizontal impermeable boundary as stand- in the interval above this in which no zuch ard datum, po = pressure of the atmosphere, boundary is known, all groundwater belongsto p - pressure in the flow region at any point, the flow region of the basin. If, however,a rela- p : density of water. tively impermeable boundary is present under- If the water table is defined as a specifi,c lying the whole basin, the water systemsunder pi,ezometricntrlace i,n the groundwater region it will not significantly interfere with the sys- at whi,ch the grauitg potential is a macimum tems within the basin. The assumption of the ond the pressure potmtial eqwls that ol the two vertical boundaries is, strictly speaking, atmosphere,(1) reducesto correct only if the surface drainage pattern is syrnmetrical-that is, if the basin is bounded 6t : gzt Q) by two parallel and equally removed surface- for the water table, where z, : the topographic water divides of equal topographic elevation. In elevation of the water table at any point ia the this casethe potential distribution at both sides basin. It has been observedin Alberta lMene- of the stream is ryrnmetrical and the imper- leg, 7963,pp. 4-12f as well as elsewhere[Kang, meable boundaries may be drawn vertically at 1892, pp. 15-18; Meinzer, 7923, p. 341.Wisler the stream and at the divides. It will be shown and Brater,1959,Figure 85; Wieckowslta,1960, later that a small amount of asymmetry in the p. 641 that the water table is generally similar topography does not cause a significant devia- in form to the land surface.Thus z, is found to tion from the vertical of these boundaries. consistof three components:zo,2,, and z, (Fig- The potential distribution along the theo- ure 1). eois a constant,denoting the depth to retical surface, although identical with that of the horizontal impermeable boundary from the the water table, is along a horizontal surface, stream bottom. za : a tan a, where o is the and this restricts the validity of the numerical horizontal distance of any point in the flow re- results to small slopes of about 3o or less. For gion from the valley bottom and a is the aver- the topography within the basin, a sinusoidal age slope of the valley flank. As long as a is shape has been chosen,the highs and lows of small, z, may be approximated by which are thought to be representative of the hills and depressionsof the natural land surface. sin (bz/cos a) The analysis is also based zz: a on the assumption cosd that the geologic conditions in the basin are isotropicand homogeneous.'Whether where o is the amplitude of the sine curve, or not this - assumption is justified depends on the extent b zir/X is the frequency, and )" is the period to which a real casedeviates from the ideal con- of the sine wave. With the three components ditions. known, the equation of the water table is ob- The assumption that the problem can be tained: treated as a two-dimensional one is supported (bc/cos a) 2t: 20* x tana * asin by the recogrritionthat in most small basinsthe cos4 4798 J. TOTE Upon introducing the abbreviations tan q. : C, Darcy's law, (6) can be used to obtain the spe- - a/cos a d, ar.d b/cos a = b', the final form ci.ficmass dischargein the direction of r fHub- of zr is written as bert, 1940,p. Uzl.

2t:h*c't{a'sinb't (3) j,: -paog/&r (7a) From (2) (3) potential and the at the water or the total flow vector: table is found: -pn j: cradd Qb) 6' : g(zo* c'x * a' sin b'r) (4) where o - hp/n (& : coefrcient of permeabil- Owing to the natural equilibrium of the ground- ity, q - viscosity of fluid). water budget in a basin, the averagelevel of the Numeri,cal computations. To analyze the ef- water table is assumed to be constant. The fect of the geometry of the basin on the ground- problem is thus a steady-state potential prob- watrer flow, (6) has been solved for various lem which may be solved by applying the parameters. To facilitate visualization of the Laplace equation: flow, the numerical values of the potential are 'head o'6/oa'{026/0,2:g expressed in of water above standard datum.' The potential distributions and flow The four boundary conditionswill be as follows: patterns for the various casesare shown in Fig- 0g/0a:0 at s:0ands ures 2a to 2i. The horizontal distance between the water for 01z1zs (5a,5b) divide and the valley bottom is 20,000 feet in all computed cases.This distance seemsto be dg/02:0 at z:0for0(a(s (5c) fairly representative for the half-width of a 6, : g(zo* c'a * a' sin btn) at 2 : zo small basin. Three valueshave been aszumedfor the depth for 0(a(s (bd) to the impermeableboundary at the valley bot- where s is the horizontal distance between the tom: 1000,5000, and 10,fi10feet. The 1.000-foot valley bottom and the water divide. case is likely to be encountered in nature, The general solution of the Laplace equation whereasa relatively homogeneousbody of sedi- can be written in the following form: ment 10,000 feet thick is a rather hypothetical case.Flow patterns have, however, been evalu- : O e-b"(A cos kc * B sin fto) ated for this situation, for several reasons:first, * er"(M cos fto * N sin /co) it represents an extreme case, and therefore the generalvalidity of the conclusionsarrived at by The arbitrary constantsA, B, M, and N can be employing (6) can be checked; second,the gen- found from the boundary eonditions. eral features of the flow pattems are mote con- Upon performing the derivation we get tho spicuous in the deeper boundaries than in the following final equation for the fluid potential: shallow ones; and third, the measurethat is the most characteristic in determining the potential a: * - cos b's) distribution is tbe ratb n - (2"/s) of the depth 4"*t #u of the impermeable boundary to the horizontal distance between divide and stream. By em- ploying the three values of zo,& wide variety of potential distributions for values of zl up to 0.5 c's' .l +;*, (cos, rnr - 1)-l can be inferred at least qualitatively. Flow sgsterns i,n smalJ bosins based.on ,inter- pretati.on of the mathemnti,cal resits. Upo.t cos(mtra/s) eosh(rntrz/s)'t ,^, inspection of Figures 2a to 2i, we recognize a s.cosh(nurzo/s) '"' ) certain grouping of the flow lines. (In the fig- Equation 6 satisfiesboth the boundary condi- ures the solid lines are called lines of force. Un- tions and the Laplace equation. By means of der isotropic conditions the lines of force coin- GROUNDWATER FLOW IN SMALL BASINS 47gg

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-' il il il ,( ll o- l:"")'\, \ 4806 J. TOTE cide with the flow lines.) Such a group of flow Figure 2 that the greatest flow-line densitiesare Iines is said to form a flow system if it satisfies found at shallow depths of the local systems. the following definition: a flow systemis a set Except at placeswhere local stagnant bodies of of flow li,nesin which any two flow lines adja- water occur, the density of the flow lines de- cent at one point ol the flow regi,onrenxai,n ad- creasesrapidly with depth and with the transi- jacent through the whole region; they con be tion from the local to the intermediate region i,ntersecteilonEwhere bE an unf,nterrupted,sur- and reachesits minimum in the regional system, face acrosswhich flow takes plonein one direc- provided the latter exists. tion onIE. This interpretation of the theoretical results Further investigation of the figures shows is very much in agreementwith views expressed that three distinctly different tlpes of flow sys- by Noruatou and, Popou [1961, p. 21]. They 'three tems can occupy a basin, na.m.ely,local, inter- recognize well pronounced vertical zones mediate,and regionalsystems (Figure 3) . L local of groundwaterflow': systernof groundwater flow has its rechargearea 1. 'upper zone of active flow, whose geo- high and its dischargearea at a at a topographic graphical zonality coincideswith climatic belts topographic low that are located adjacent to The lower boundary of this zone coincideswith each other. Local systems ca.n be readily ob-. the local baselevels of rivers; servedon eachdiagram of Figure 2. The major 2. 'medium zone of delayed flow, subject to an intermediate system of. characteristic ol lesser climatic effect but also geographically groundwater is that, although its recharge flow zonal. The lower boundary of this zone is the and dischargeareas do not occupy the highest baselevel of large rivers; and lowest elevated places, respectively, in the 3. 'lower zone (of relativelystapant water), and lows basin, one or more topographic highs geographically azor'al and lying below the base may be located between them. Very-well-de- level of large strea.msystems.' fined intermediate systems can be seen in Fig- ares 2e, l, g, h, and rl. The apparent lack of Taking into account the extent of the re- intermediate systems in those casesfor which charge areas of the regional systems (which are zo is 1000 feet does not mean that no zuch sys- small relative to those of the local systems),we tems may exist in basins of relatively shallow see that flow in the regional system is influ- depth. As soon as the real land surface departs enced by climatic effects to a much lesser de- from the regularity of the sine curve, the sym- gree than flow in the upper zone. Climatic or metrical flow pattern of Figure 2c, for instance, geographicalzonalities are, therefore, a straight- will be somewhat modified, and flow will occur forward consequenceof the present theory. between intermediate highs and lows also. A In the next few paragraphs an analysis will sysbemof groundwater flow is consideredto be be given of the effects of geomorphologicalfac- regional if its recharge area occupiesthe water tors on the flow of groundwater. These factors, divide and its dischargearea lies at the bottom or parametersin (6), are (o) the ratio rzof the of the basin. Regional systemscan be observed depth zo to the impermeable boundary to the in all the deep casesand in Figure 2c, where zo half-width s of the basin (for convenience,in is 1000feet. discussingthe effect of n, only the depth to the Whereastheoretically the boundariesbetween impermeableboundary will be referredto, s be- different flow systems are very well defined, ing the same in all cases); (b) the average they do not signify an abrupt changeof any of slopeof the valley flanks; and (c) the local re- the physical properties of the flow. Relatively lief. rapid changes of the chemical composition of In analyzing the effect of zo on the flow of the water acrossthe boundaries,however, could groundwater, a comparison of the diagrams of be expectedbecause of the different locations of Figure 2 is helpful. Let those diagrams be con- the recharge and the different lengths of the sidered for which all parameters but zo are flow paths of the different systems.In a small equal,for instance,Figures 2d, e, tnd f. It ap- basin of moderate relief the amount of recharge pears that the spacing of the equipotential lines water is directly proportional to the area of re- is closer in the shallow casethan in the deeper charge.With this in mind it is obyious from ones.The flow lines are more arcuateas depth 3 o

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laaJ ur un+op pJopuo+s a^oqD Ja+oa Jo poaq puD uor+D^al3:rqdo:6odo1 4808 J. T6TH increasesand there is room for intermediate and retarded; neither regional nor local systemscan regional systems to form in those cases for develop. Groundwater may be discharged only which eo is 5000 or 10,000feet. These features by evapotranspiration; discharge of this type indicate a more evenly distributed flow, there- will possibly rezult in water-logged areas. If a fore a lessintense motion if zoincreases. A com- relation between mineralization of the water parison of Figures 2c and 2h shows that a re- and velocity of flow can be a,ssumed,water in gional system is aJso possible in a reiatively those areaswill have high concentrationsof solu- shallow case but that, with other parameters ble salts. remaining the same, a much larger amount of 2. If the local relief is negligible and if there water is transmitted through the regional sys- is a general slope only, a regional system will tem if zois large. To zummarizethe effect of ao develop.Theoretically, if the line located on the on the flow system it can be stated that, as the surface midway between and parallel to the depth of the flow region increases,the water valley bottom and the water divide is called the movement will slow down. A slow motion will 'midline,' the recharge and discharge areas of probably result in higher mineralization of the this regional system are located between the grouadwater. midline and divide, and betweenthe midline and A general increasein the slope of the valley valley bottorn, respectively lT6th, 19621. Be- flank will result in an increased lateral flow to- cause of the decreasing velocities, a gradual ward the bottom of the valley. The surficial increaso of dissolved mineral constituents with areas of those systems whose flow lines are di- depth is to be expected.A good example of po- rected toward the center of the basin extend, tential distribution and flow pattern in this situ- and those of the other systems decrease.Local ation has been produced by meazurementsin flow systems may degenerate to stagnant areas Long Island, New York, in connection with and may even vanish, thus allowing intemediate attempts to solve conta,minationproblems. Fig- or regional systemsto form. Figures 2a, 2c,21, ure 4 shows the results of the meazurements and 2h illustrate this changewell. Owing to the lGeraghty,1960, p. 381. generally increasedvelocity of motion, a larger 3. If the topography has a well-defined range of fluctuation of the piezometric surface relief, local flow systems originate. The higher is to be expected for the steeper slope of the the relief, the deeper are the local systems. At valley flank. the boundary betweentwo adjacent systemsthe Increasing topographic relief will tend to in- flow is subvertical and downward or upward in creasethe depths and the intensities of the local direction. Such a boundary is located under 'Whereas flow qvstems. in Figure 2c a well- the highest and lowest elevated parts of local developed regional flow system is observed,in hills and depressions,respectively. Thus, where- Figure 2d bhe entire basin is occupied by local as in the basin the underground drainage is not systems. The depth ranges of the local system strictly symmetrical, imaginary impermeable in Figure 2f arc approximately 1000 feet shaJ- boundaries may be thought to be located at lower than those in Figure 29. fn the extreme local lows and highs, at least to the depths of case where the relief is negligible no local sys- the local systems.Figure 5 is a good sxample of tems will form. Ilowever, sincewell-defined local an inferred local system lBack, 7960, p. 941. flow systems are found even with the relatively 4. The very pronounced efrect of the relief low relief of approximately 100 feet per 2500 on the formation of local systems suggeststhat feet (a gradient of 0.04), neglecting their ex- no extended,unconfined regional systemsof flow istence in theoretical or practical problems can can exbend across valleys of large rivers or hardly be justified. highly elevated watersheds. Consequmcesof the theorg. The major fea- 5. As a result of the local systems,alternat- tures of groundwater flow and flow systemsde- ing recharge and discharge areas are found rived from the theory presented here will be acrossa valley. This means that the origins of discussedin the following paragraphs. waters obtained from closelylocated place may 1. Recalling the efrects of the general slope not even be related. Rapid change in chemical and local relief on the flow, we see that under quality may thus be expected. extended flat areas groundwater movement is 6. At points where three flow systems meet GROUNDWATER FLOW IN SMALL BASINE 4809

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Fig.5. Diaglanunatic closs section through southern Maryland shorving the lines of gound- water flow [after Baclc, 1960]. (Figure 2c), an area of stagnant water is if the two flanks of a basin are of low relief, so formed. A high accumulation of mineral con- that there are no local systems,the bulk of the stituents is probable at these places.Below such basin discharge will take place between the mid- a stagnant body of groundwater, flow occurs line and the valley bottom and only a small por- again and may result in a better quality of tion will appear as baseflow. water than that from shallower depths. L A further consequence of the theory is 7. Decreasing potential with depth in re- that the water levels at shallow depths are the charge areas and increasing potential in dis- most affected by seasonal recharge and dis- charge areas are direct consequencesof the the- charge.The small intake and outlet areasof the ory and can be observed in all diagrams of intermediate and regional zones.prevent the Figure 2. It should be noted, however,that such water levels from fluctuating widely. Plotnilcou a configuration of the equipotential lines is pro- and, Bogomolou 1L958,p. 901 make a distinc- nounced only in the immediate vicinities of the tion between two zoneson the basis of fluctua- highs and lows. The midlines and their vicinities tion of the water levels.They call the first 'zone are locations of relatively straight, vertical equi- of oscillationsof underground water levels.' Ac- potential lines. cording to them the volume of water that occu- 8. From Figure 2 it can safely be stated that pies the zone of oscillation undergoesseasonal the major stream of the basin receivesground- variations. This volume would control ground- water contributions only from the adjacent water dischargea.nd therefore they call it'con- topographic highs and from possible regional trol reservesof underground waters.'Their sec- flow. The latter is probably unimportant in ond zoneincludes all the water that is below the most easesbecause of the low rate of flow. It zone of oscillation, both that in the deeper,still is conceivable,then, that the methods in which homogeneousparts of the basin and that in baseflowdata are usedfor computation of basin- artesian aquifers; these are the 'secular re- wide characteristics (average recharge,permea- sources.'It seemsthat the 'zone of oscillation' 'secular bility, etc.) are misleading or erroneous.Even and the portion of the resourceg'that GROUNDWATER FLOW IN SMALL BASINS 4811 is above the first impermeable boundary coin- on the general situation for which local topogra- 'local cide very well with the systems'and with phy plays a part in controliing groundwater 'intermediate the and regional systems' of tho motion. The distribution of the flow systems presentpaper. will, in turn, have its effect on the chemica.r 10. Another result of the analysisis that only quality of local occurrences of groundwater. The a small portion of the total amount of water areally unrelated origin of local systems, asso- occupying the basin participates in the hydro- ciated with local topographic highs and lows, logic cycle. The deeperthe basin, the smaller is may result in abrupt changes in the chemical this portion. This is easily conceivedwhen one composition of relatively shallow groundwater. considersthat the greatest pari of the surface Vertical changes in quality may be the result of of the basin is occupied by the recharge and local stagnant bodies and of the vertical arrange- dischargeareas of local s5rstemswhich are usu- ment of different flow systems. ally shallow. But, even when the local systems It is thought that (6) may be used for ob- reach the horizontal impermeable boundary taining quantitative information about ground- (Figures 2a, b, d), approximately90 per cent water flow in an area, the surface of which can of the total recharge water never penetrates be approximated by a harmonic function. It is deeper than 250 to 300 feet. A similar view is hoped also that the results of the above analysis expressedby Ubell [1962, p. 96] who believes wiil be useful in test programs planning well 'static that about 80 to 90 per cent of the sup- fields, solving pollution and tracer problems, ply does not participate in the natural hydro- making baseflow studies, and setting up water logical cycle.' IIis experiments, on the basis of budgets. which this conclusionwas drawn, indicated that 'below a certain depth in loose sedimentary Aclmowledgments. I am greatly indebted to rocks . . . water doesnot move in the voids until Messrs. R. Newton and P. Redberger, of the pe- troleum Division, their state of stressis disturbedby boring.' Research Council of Alberba, who adapted the flow equation described in the Summary. It is the writer's belief that in paper for solution by digital computer. Only the drainagebasins, down to depths at which basin- availability of the computer solutions made pos- wide extendedlayers of contrasting low permea- sible the derivation of the many flow diagrarns. bility are found, groundwater motion may be treated as an unconfined flow through a homo- Rnrpnswcps geneousmedium. On the basis of this principle Back, W., Origin of hydrochemicalfacies of ground a mathematical model of a small drainagebasin water in the Atlantic coastal plain, Intern. Geol. Congr., 21st, (as definedin the paper) has been constructed. Copenhagen, 1960,Rept. Session, Norden, part 1, pp.87-95, Copenhagen,1g60. Potential distributions have been computed for Geraghty, J. J., Mouernent of Contaminants basinsof different geometricalparameters. These through G eologi,c F ormati,ons,90 pp., Technical computations have led to a number of conclu- Division Activities, National Water Well As- sions regarding features of the groundwater sociation,Urbana, Ill., 1960. Hubbert, M. K., groundwater flow. The theory of mo- tion, "I. Geol.,48(8), part l, 78b-g44,1940. In the most general case,groundwater flow in King, F. I[., Obscvations and experiments on the a basin can be thought to be apportioned among fluctuations in the level and rate of movement three types of flow systems,the regional,inter- of ground-water on the Wisconsin Acricultural Experiment mediate, and local systems.The three systerns, Station Farm, U. S. Dept. Agr. Weather Bureau, Bu,Il.no. 5,75 pp., l8g?. being the results of combinations of three par- Meinzer, O. E., Outline of ground-water hydrol- ticular solutions of the Laplace equation, can be ogy with definitions, U. S. Geol. 9uru. Water superimposedon one another. If the local varia- Supply Paper /19!, 71 pp., 1923. tions in topographyare negligiblethe flow con- Meneley, W. A., The occurrenceand movement of groundwater in Alberta, in Early Contributions sists of the combination of only two particular to the Grutndwater Hgdrology ol Alberta, Res. solutions, and no local systemsoccur. This case Cwncil Alberta, Can., BuII. 19, l2Z pp., Edmon- has been found in nature by Geraghtg 11960l ton, 1963. and has been theoretically treated in detail else- Norvatov, A. M., and O. V. Popov, Laws of the formation of minimum stream flow, BuIl. wherelT6th,1962l. Intern. ,4ssoc. Sci. Hydrol., vol. 6, no. l, pp. 2G-27, The emphasisin the present paper has been Louvain,1961. 48,12 J. TOTE 'Wieckowska, Plotnikov, N. A., and G. B. Bogomolov, Classifica. 8., Zorea geographiqueedes eaux tion of underground waters r€sourcesand their phreatiques,Intern..r{ssoc. Scr. Hydrol., Gencral reflection on m&ps,Intent. Assoc.Sci. Hydrol., Asse?nblyof Eelsi,nki, l;o. 52, 6(D pp., Gent- Geturol Assembly of Tmonto, vol. 2, no. 44, brugge, 1960. 525 pp., Gentbrugge,1958. Wisler, Ct. O., and E. F. Brater, Hudrolow' Znd T6th, J., A theory of groundwatermotion in small ed., 408 pp., John Wiley & Sons, New York, drainage basins in Central Alberts, Canada, J. 1959. Geophg s. Res.,€l (lL), 4925-4387,L%2. Ilbell, K., A felszin alatti vizk6sulet,Ei,ilrol. Kozl., 49(2), 94-lM; Budapesb (Eungary), April 1962 (Manuscript received Februa,ry2, 19ft3; (T'lnglinh ilmmary). revised May 29, 1963.)