Statistical and analytical study of the propagation of flood-induced groundwater rise in an alluvial aquifer

Journal of Hydrology

E L S E V I E R Journal of Hydrology 205 (1998) I 12- 125

Statistical and analytical study of the propagation of flood-induced groundwater rise in an alluvial aquifer

Z. Vekerdy*, A.M.J. Meijerink International lnstitute jbr Aerospace Survey and Earth Sciences (1TC), P.O. Box 6, 7500 AA Enschede, The Netherlands

Received 22 January 1997; revised 22 October 1997; accepted 20 November 1997

Abstract

The propagation of stage rises of the river Danube in an adjoining alluvial aquifer (Kisalf'old, NW Hungary) has been studied by preparing a map which shows the lag times corresponding to maximum correlation values between the hydrographs of groundwater observation wells and the river stages. As expected, the lag times for the unconfined aquifer generally exceed those for the semi-confined part. The complex pattern suggests that other rivers in the area also play a role and that transmissivities and resistances of a cover layer are not sufficiently well known, despite the presence of a relatively dense network of bore holes. A systematic increase of the lag times with the distance from the river was noted at two sections. Nine flood events have been selected to study how well observed head rises could be predicted using equations for the calculation of the propagation of the head for phreatic and for confined conditions. The results suggest that the water levels in the complex flood plain and the river bed conductance of silted up branches during the rise of the flood play an important role. © 1998 Elsevier Science B.V.

Keywords: Alluvial aquifer; Groundwater; Bank storage

I. I n t r o d u c t i o n

A rise in the water level o f a river traversing an alluvial aquifer with permeable layers induces flow from the river into the aquifer. When the river is bordered by alluvial terraces, most o f the water is returned to the river during low stages. This is known as bank storage and was quantitatively mod- elled, e.g. by Moench et al. (1974) and Hunt (1990). When the river flows on an alluvial fan, groundwater flow is generally away from the river, with the result that only little water returns back to the river when the stage is falling. Hence there is a net flux from the river to the groundwater, as is the case in the studied area.

* Corresponding author.

0022-1694/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PH S0022- 1 6 9 4 ( 9 7 ) 0 0 1 4 8 - 0

Early works concentrated on the setup and analyti- cal solution o f formulae describing the river groundwater interaction, e.g. the draft on a river by pumping a well adjacent to it (e.g. Glover and Balmer, 1954). More recent developments include the use of a three-dimensional coupled surface water and ground- water flow model, based on the numerical solution o f open channel flow and groundwater flow equations and allowing the modell ing of the surface wa te r - groundwater interaction. An overview of the develop- ment o f such models is given e.g. by Vasil iev (1987). One o f the examples o f the recent developments in this field is shown by Swain (1994), who describes the coupling of US Geological Survey computer codes BRANCH and M O D F L O W to calculate the surface wate r -groundwater interactions.

Z. Vekerdy, A.M.J. Mei/erink/dournal o/ IJvdrolog~ 205 (1998) 112-125 t 13

An analysis of the effects of the river stage rises could precede the complex modelling to get informa- tion about the spatial inhomogeneity of hydrologic parameters for grid design or for the calibration of a groundwater model. The model calibration is usually difficult in alluvial areas in the zones along the river due to the complicated interactions between river stages and groundwater levels. Simplification and generalisation are usually required. It is useful to investigate whether simple analytical approaches can be of assistance.

A general method to understand the response of groundwater heads on stage changes in rivers may follow two steps: first a statistical description of the phenomenon is made, then a hydraulic analysis. This approach was used by Sophocleous (1991), who investigated the pressure propagation along buried paleochannels in Kansas, USA. He demonstrated by conceptual modelling that a pressure wave generated by a river may propagate over tens of kilometres in the confined paleochannels.

A simplification of a complex regional problem is possible by reducing the number of dimensions. This is feasible in the case of rivers since the flood- induced head wave propagates in a well-defined direc- tion, namely perpendicularly away from the river, superposed on the general groundwater flow pattern. In case of a single aquifer with uniform vertical distribution of the heads, a one-dimensional analysis is feasible. A further simplification in the presence of much spatial variation of hydrogeologic para- meters, often not known from available data sets, is to use equations with generic parameters (in the following referred to as simple equations). Complex

ones would require much guesswork in the absence of a proper observation network. The results obtained by the simple equations could help in finding out what aspects should be included for further improve- ment. Analytical solutions of the flow equations are known for such cases (e.g. Edelman, 1947; Heij, 1989; Leenen, 1994; Zhang, 1992).

The first objective of this study is to analyse the pattern of the propagation of high water levels in the aquifer away from the river after a linear rise of the stage by using lag correlation. After that, the results of equations for the propagation of the heads for phreatic (Edelman, 1947) and for confined (Leenen, 1994) conditions are compared with observed heads at selected regions.

2. River-aquifer boundary on alluvial regions and description of the study area

In case the aquifer is predominantly composed of coarse grained materials, essentially four types of boundary conditions can be recognised along a river, as sketched in Fig. 1. Most alluvial areas are seldom uniform in plan and profile because of changing climatic conditions in the provenance areas during the time of deposition. Lithologic logs of shallow boreholes often show important variations at rela- tively short distances. Therefore, the type of boundary condition along the river can be difficult to define and the same is true for the variations in transmissivity.

The presence of a cover layer [Fig. l(c)] with fine grained deposits resting on thick, coarser grained, braided river deposits is characteristic for many

river river river river

I \ . o o ' . o l ~ " • D"

12. a b c d

a.) Highly resistant riverbed b.) River in highly resistant cover layer c.) (Semi-)confined groundwater, permeable riverbed d.) Phreatic groundwater, permeable riverbed

Fig. 1. Possible contacts between a river and the groundwater table in alluvial regions.

114 Z. Vekerdy, A.M.J. Meijerink/Journal ~?]' Hydrolog, y 205 (1998) 112-125

alluvial areas. The cover layer varies usually in thick- ness and has variations in textures because it is com- posed of overbank suspended load deposits and channel fill deposits. This situation is found on the alluvial fan of the Danube river in the KisaW61d region (Small Plain, in the north-western part of Hungary). In the studied area the cover layer is coarse in the upstream part of the fan and the groundwater is phreatic. In the downstream part of the fan the cover layer is fine grained, resulting in semi-confined groundwater. The available data indicate that mainly river-aquifer boundary types (c) and (d) of Fig. 1 occur, i.e. the main branch of the Danube is not sealed from the groundwater by an impervious river bed. It should be mentioned that, despite the availability of a fairly dense network of shallow (approximately l0 m deep) boreholes with lithologic descriptions, there remains much uncertainty concerning the local spatial variation of the hydrogeologic variables (Vekerdy, 1996).

heavier textures of the cover layer. As expected, the propagation under confined or semi-confined condi- tions was faster (i.e. at the downstream part of the alluvial fan) than in the phreatic (upstream) part. There was a tendency for lag times to increase with distance away from the river. However, some erratic values can be noted on the map. These can be attrib- uted mainly to the influence of the Moson-Danube River (see Fig. 2). Other factors can also have effects, such as the spatially variable transmissivity of the lower gravels and sands; the spatially variable resistivity of the cover layer; non-steady flow at the beginning of the head rises; and the transformation of phreatic to (semi-)confined conditions at places, where rising groundwater from the lower gravels enters into the cover layer during the rise.

Along two sections, line #6, located in the (semi-)confined part, and line #2, located in the phrea- tic part of the aquifer, the lag time increases essen- tially regularly with distance. These two lines have been selected for further scrutiny.

3. Statistical analysis of the groundwater head propagation

Records of daily water levels in observation wells, of the entire period from 1979 to 1989, have been used for lag correlation. The correlation coefficients (Pearson's r) have been calculated for the stage heights of the river and the heads in the wells with different time lags (1,2,...,31,33,35 ..... 85 days) for each individual well.

3.1. The regional pattern of the propagation as expressed by lag correlation

The characteristic time lag was determined as the highest value of the correlation coefficient for each observation well. A choice was made to consider only those wells that have a correlation coefficient of more than 0.2, because with lower values of the coefficient it was difficult to define the characteristic time lag. The characteristic time lags have been plotted on a map (Fig. 2) together with the identifica- tion numbers of the wells and well-lines.

When the result of the time lag analysis is com- pared with the lithological description of bore logs, it is apparent that lower lag times are associated with

3.2. Time lags along selected lines

The lag correlation coefficients have been plotted against the lag times for wells at various distances from the Danube, along line #6 (Fig. 3) and line #2 (Fig. 4). In both sections, the curves of the correlation coefficients show a clear maximum or culmination only at short distances from the river, particularly for the confined condition. Considerable dampening of the effect of the pressure propagation takes place with increasing distances, which is expressed by a decrease of the lag correlation coefficient and by a broadening of the culmination of the curves.

For the unconfined part of the aquifer, culminations of the curves can be easily noted for the three wells closest to the river. For wells further away the culmi- nations are broad and the maximum value was determined by eye. The time lag corresponding to the culmination (maximum value) was considered as the characteristic lag time and is plotted on the map of Fig. 2. The generally flat culmination of these curves is due to the dampening effect, and possibly to the effect of surface recharge.

When the characteristic lag times, as deduced from the graphs of Fig. 3 and Fig. 4, are plotted against the distance of the wells from the river along line #6

Z. Vekerdy, A.M.J. Me(jerink/Journal qf'Hydrology 205 (1998) 112-125 115

#1

#2

Well-line #3

~ , , Well-line #4 /

N

0 lO km

~ Groundwater well identifier Characteristic time lag

/ , ® / I

d I

Well-line #5

Well-line #6 /

Well-line #7

Fig. 2. Characteristic time lags between the river stages in the Danube and the groundwater levels measured in the observation wells.

and line #2, a consistent pattern emerges (Fig. 5). The lag time for the phreatic conditions increases with distance. The overall relationship is not fully linear. For the semi-confined conditions sufficient data are not available to express the relationship for larger distances. However, the graph shows that a rise in the stage of the river is propagated faster than under the phreatic condition.

4. Analysis of flood-induced groundwater head rise based on hydraulic equations

The selection of wells along the two lines, where the heads showed some consistency in the response to rises in the river stage, has been done to allow comparison of

observed heads with heads predicted by equations based on theory. They will now be discussed.

4.1. Semi-confined condition

The analytical solution of the equation of the one- dimensional groundwater flow given by Leenen (1994) was used for the analysis of the groundwater response to the stage rise in the Danube. The deriva- tion of the equation is described in Appendix A. The solution gives the groundwater head rise caused by the rise of the flood stage. Analysis showed that the heads are sensitive to the distance from the river, the transmissivity of the aquifer and the resistance of the cover layer (Fig. 6), but less sensitive to the elastic storage of the aquifer.

[ 16 Z Vekerdy, A.M.J. Meijerink/Journal q/Hvdrology 205 (1998) 112 125

0.8

0.6

0.4

0.2

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 lag [days]

Well 63 (1630 m from the Danube) ~ Well 64 (1930 m from the Danube)

Well 65 (2170 m from the Danube) ~ Well 66 (2050 m from the Danube)

Well 62 (2450 m from the Danube) ~ Well 68 (3640 m from the Danube)

- - I - Well 69 (4680 m from the Danube) ~ Well 71 (5700 m from the Danube)

Well 72 (5800 m from the Danube)

Fig. 3. Lag correlation coefficients (r) calculated with different time lags along well-line #6 (semi-confined part of the aquifer).

0.8

0,6

"0.4

0.2

i

. . . . . . . =I= :I I : : : I : : : I ,==:I : ; : : , , , :=I : : : : I : : :==;=;=I====I=:; ;1===~ . . . . :: : : : : I ; = : ; : : :==:==;=:

0 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 Lag [day]

Well #129 ( 250 m from the Danube)

Well #126 (1150 m from the Danube)

- = - Well #135 (3700 m from the Danube)

Well #133 (7600 m from the Danube)

--v-- Well #112 (16 500 m from the Danube)

Well #128 ( 480 m from the Danube)

Well #137 (1600 m from the Danube)

- + - Well #134 (6100 m from the Danube)

- , * - Well #132 (8800 m from the Danube)

Fig. 4. Lag correlation coefficients (r) calculated with different time lags along well-line #2 (phreatic part of the aquifer).

Z. Vekerdy. A M.J. Mei/erink/Journal ()['Hvdrologv 205 (1998) 112 125 117

70

60

50

40

A

. • . ' . . .

--I ~ 3 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i , •

o ~ • : n . ~ z , i , i , i 0 2000 4000 6000 8000

Distance from the Danube [m]

O Well-line # 6 (confined aquifer) I • Well-line # 2 (phreatic aquifer) I

Fig. 5. Dependence o f the characteris t ic t ime lag on the dis tance

f rom the Danube.

Some observation wells on the semi-confined, downstream part of the Danube fan [along and close to well-line #6, Fig. 7(b)] have been selected for comparing heads calculated by Eq. (6), with the observed ones. The confining cover layer in this part of the aquifer has a high resistance (80-400 day), as determined by Eq. (2), based on lithologic logs of the cover layer• Data from literature describing infiltration measurements and other field data of the Kisalf61d ( Erd61yi, 1971; Maginecz, 1993; Papp, 1988; Sz~kely, 1977; (lj faludy, 1986) were used for the

determination of Kz•cV The hydraulic properties along the well-line were determined using a map of hydraulic terrain units (HU), which was compiled by crossing of the map with classes of the cover layer resistance (Fig. 8) and the map with classes of aquifer transmis- sivity (Fig. 9) in a geographical information system (GIS). A schematic cross-section (Fig. 10) illustrates the derivation of HUs. Since the pressure head propa- gation in a semi-confined aquifer is fast (exceeding several kilometres per day), it has been assumed that in the neighbourhood of the river the characteristic times are equal in each consecutive HU:

t , l ~ t , 2 --~ . . . ~ l , j 7

where n is the number of HUs. Elastic storage was assumed to be 0.001, and the river-aquifer boundary as sketched in Fig. l(c).

Nine flood waves were selected for the analysis of the head propagation (Table 1), using the following criteria:

1. continuous (quasi-)linear rise of the river stage; 2. relatively stable low water period preceding the

flood; 3. the selected floods represent a range of different

maximum stages.

The results (Fig. 11) show a reasonable fit for smaller floods, but there is a systematic deviation of

® g

-~ 1 ~ I

0.8 ~ 0.8 .~0 .6 .~0.6

0.4 ~ 0.4 o P

0.2 "o 0.2

= 0 = 0 ®

0 z z t)

J Fig. 6. Sensit ivity o f the normal ised g roundwa te r head to the different parameters in the calcula t ion method of Leenen (1994) (S~ = elastic s torage o f the aquifer, T = t ransmissivi ty o f the aquifer, t* = normal ised durat ion o f the flood, col = resistance o f the cover layer).

118 Z. Vekerdy, A.M.J. Meijerink/Journal o[" Hydrology 205 (1998) 112-125

groundwater ~ river ~25 observation well

dike ~" small stream

b

0 10 km

Fig. 7. Location of the wells used in the hydraulic equation-based analysis of flood-induced groundwater head rise: (a) phreatic part of the aquifer; (b) semi-confined part of the aquifer.

calculated heads compared to the observed ones for higher floods. The main explanation of the deviation lies in the flood plain geometry (Fig. 12). When the flood peak is not higher than the river bank, the distance between the infiltration front of river water and the observation wells is rather constant. At higher

stages the flood waters start spreading over the flood plain, first in the flood plain channels and later over the full width of the flood plain, up to the dikes. Because of the complex geometry this process is gradual, but on average the distance from the infiltra- tion front to the wells is shortened. It is difficult to

N

0 10km

Groundwater • observation well

Dike

Confining layer resistance:

?iii~i~ 1 - 20 day a 20- 40 day

40- 80 day BIB 80- 160 day

160- 320 day

320-640 day

Fig. 8. Confined parts of the aquifer with the resistance of the confining cover layer. The map was compiled from data provided by the Geological Survey of Hungary and the Water Resources Research Centre (VITUKI), Budapest.

Z. 1,2,kera~v, A.M.J. Me(/erink/Journal o/" Hydrolog3' 205 (1998) 112 125 119

i 0 10 km

Groundwater e observation well

Dike

Transmissivity of the aquifer:

1 - 200 m2/day

i i i! i 200 - 400 m2/day

'!~,~i, 400 - 800 m2/day

800 - 1600 m2/day

1600 - 3200 m2/day

' I 3 2 0 0 - 6400m2/day

I 6400 - 12 800 m2/day

I 12 800 - 25 600 m2/day

I 25 600 - 51 200 m2/day

I > 51 200 m2/day

Fig. 9. Transmissivity of the aquifer. The map was compiled from data provided by the Geological Survey of Hungary and the Water Resources Research Centre (VITUKI), Budapest.

consider this complex geometry in the calculations, but it could explain why the calculated heads are lower than the measured ones.

4.2. P h r e a t i c cond i t ion

Edelman (1947) gives an analytical solution o f the equation o f one-dimensional groundwater flow in an

aquifer bordered by a water body. The most important steps of the solution are given in Appendix A. The solution gives the rise of the groundwater table during the period of the river stage rise (when c~ is positive, see Appendix A), or river stage decline [when c~ is negative in Eq. (12)].

The phreatic model is sensitive to transmissivity and distance from the river, as shown in Fig. 13. It

HU1 HU2 HU3

X l " x2 : x3 " ~ "~'~ ~ ~ ~_"

° T 1 ° i . "T 2 "i. = "T3 ° o ~

a.) b.)

Fig. 10. Scheme of the derivation of the hydraulic terrain units (HU): (a) plan; (b) section.

120 Z. Vekerdy, A.M.J. Meijerink/Journal ()/'Hydrologo, 205 (1998) 112 125

Table 1

Main characteristics of the floods selected for the analysis

Flood duration

From To

Duration of rise, tp (day) Full rise, Hp (m)

01-Feb-85 05-Feb-85 4

19-Jul-81 25-Jul-81 6 18-Dec-87 2 l-Dec-87 3 10-Mar-81 16-Mar-81 6 12-Jan-86 16-Jan-86 4 02-Feb-80 08-Feb-80 6 04-Feb-81 1 O-Feb-81 6 28-Feb-83 03-Mar-83 3 28-Nov-83 0 l-Dec-83 3

4.5 4.45 4.09 3.88 2.98 2.74 2.63 1.58 1.35

is not particularly sensitive to the variations in the specific yield, for which a general value can be adopted.

In the upstream part of the Danube alluvial fan the groundwater is mostly phreatic. The main river reach, where direct river stage measurements are available, is surrounded by a wide flood plain up to the dikes. Large side branches of the river are located on the flood plain just at the foot of the dikes. The selected wells #125, #128, #129 and #140 are in the vicinity of the side branches of the river on the protected side of the dikes, in the phreatic part of the aquifer [Fig. 7(a)]. The river stage data of a nearby gauging

station (at Rajka) were projected to the side branches. Distances from the corresponding river branch to the wells were taken perpendicularly to the river. Transmissivity values encountered along the flow lines have been calculated based on data provided by the Water Resources Research Centre (VITUKI), Budapest. The specific yield was assumed to be 0.25.

The calculated head rises in the wells have been plotted against the measured ones, for the nine events mentioned above, see Fig. 14.

The positions of the curves reflect the lower heads at increasing distance from the river. The increase of calculated head tallies almost with that of the

J

i ,,"/

"E 2 ':/"

"o Q

/ , . ~ . ," X X

. / : o ........ o

0 " ~ , L ~ i , , J , ~ , , t

0 I 2 3 Measured rise [m]

Wel164 (1930 m from the Danube) ~ Wel165 (2170 m from the

Well 66 (2050 m from the Danube)

Fig. l l. Comparison of" measured and calculated groundwater rise in wells as a result of nine selected floods of the Danube (semi-confined

aquifer). Every symbol represents one flood event.

Z. k2~kerdv, A.M.J. Me!/erink/Journal qf 14vdrolo~, 205 (I 998) 112-125 121

X2 :< ~.

; X 1

~2

H p l < Hp2; X l > x2

groundwater head

Fig. 12. Scheme of the different distances (x) between the infiltration front of the river water and the observation well at different flood peaks (Hp).

measured head, because the slopes of the curves show an approximate one-to-one relation, except for the farthest well at 820 m distance, where the calculated rise was nil. However, the graph shows that small floods of the river, represented by data points on the left side of Fig. 14, do not affect the observed water levels in the wells• This can be explained by the pre- sence of low dams across the upstream end of the side branches, so small floods cannot enter these branches from their upstream end. The river stage in the branches rises as a result of backwater from their

downstream end. As Fig. 7(a) shows, the upstream ends of the branches have no contact with the main channel.

For larger floods, the calculated values are higher than the observed ones, particularly for the wells at distances of 150 m (well #140) and 480 m (well #128) from the Danube. This can be partly explained by the delay of the actual water rise in the branches caused by the low dams at the upstream end of the branches, and partly by the fact that the river bed conductance was neglected in the calculations. The branches are

O)

r- Ca

t-

t-

O

t9

5 ~ 5

4 ~" ~ 4 (.1

3 "~3 2

0 "

+,,t the otl~ river [~]

Fig. 13. Sensitivity of the groundwater head to the changes of different parameters in the calculation method of Edelman (1947).

122 Z Vekerdv, A.M.J. Me!ierink/Journal of Hydrology 205 (1998) 112 125

3

0 × 1 : ,

<×

0.~== ', ', i : i , 0 1 2 3

Measured [m]

i Wel1140 (150 m far from the Danube) Well 129 (250 m far from the Danube) Well 128 (480 m far from the Danube) Well 125 (820 m far from the Danube)

Fig. 14. Comparison of measured and calculated groundwater rise in wells as a result of nine selected floods of the Danube (phreatic aquifer). Every symbol represents one flood event.

partly silted up because they are protected from the main current. In the main river bed current velocities are high and the gravels are dredged for the mainte- nance of the navigational route. Lower actual river bed conductance of the branches may result in over- estimation of the groundwater rise in the wells using Eq. (12). This is in agreement with the conclusion of Hantush (1965) who developed a procedure for the calculation of drawdown distribution and the rate and total volume of river depletion resulting from pumping near a stream with semi-pervious bed. He points out that if the effect of semi-perviousness of the river bed is ignored, the calculated head will be too high.

5. S u m m a r y and c o n c l u s i o n s

A data set of river level rises after periods with low stages of the Danube River in north-western Hungary was used to study the propagation of high river stages in the aquifer. Relatively great numbers of geologic bore hole logs and observation well records were available.

The spatial pattern of the groundwater head changes caused by the Danube was analysed statisti- cally using lag correlation. A complex pattern was obtained when the time lag for the highest lag correla- tion between river stages and the selected well rises

was plotted for the groundwater wells on a map of the study area. It was possible to separate the response for the phreatic part of the aquifer and that for the part where confined or semi-confined conditions occur. The latter could be judged from the textures and the groundwater table with respect to the thickness of the cover layer. Deviations of observed characteristic time lags from expected spatial pattern could be attributed to the presence of smaller rivers, drainage channels and - - unknown - - local variables of hydrogeological conditions.

Study of the lag times, as observed along lines of wells in the phreatic and in the semi-confined part, showed clear responses only at relatively short dis- tances from the river, especially under semi-confined conditions. For the phreatic part of the aquifer, the response was difficult to detect beyond a distance of about 9 km because of the strong dampening of the river-induced pressure wave, and probably the super- imposed effect of the surface recharge.

However, when the lag times corresponding to the highest correlation coefficient were plotted against the distance from the river Danube, it could be demon- strated that a defined functional relationship existed. In other words, physical laws which govern propaga- tion of a high river stage into the aquifer had an effect on the heads. In the phreatic part the effect could be detected up to at least a distance of 8800 m.

In an attempt to predict the propagation of river stage rises, two equations have been selected because of their simplicity, one for confined conditions (Leenen, 1994) and one for phreatic conditions (Edelman, 1947).

For the part of the aquifer where the equation for confined conditions has been used, the calculated results matched the observed ones only for river stage rises not higher than the river bank. For over- bank flows the calculated values were lower than the measured ones. This can, at least partly, be attributed to the complex geometry of the flood plain. Uncertain- ties in the determination of the hydraulic parameters of the cover layer and the aquifer may play a role too.

In the phreatic part of the aquifer the calculated heads were generally higher than what was observed. Relatively small floods measured at the closest gauging station had no observed effects on the groundwater, whilst the calculations predicted some groundwater head rise. It can be assumed that the

Z Vekerdy, A.M.J. Me!/erink/Journal q/Hvdrologv 205 (1998) 112 125 123

small dams across the upstream end of the river branches prohibited the smaller floods from entering these branches. Larger floods had smaller observed effects than predicted by the calculations that are attributed to the delay of river stage rise in the branches near the wells. The possibly low river bed conductance can explain why calculated rises are too high.

The one-dimensional hydraulic analysis has shown that, apart from local variations of transmissivity, accurate data about the surface hydraulic configura- tion are needed. The analysis indicated what type of additional field data are needed, and showed that there can be little confidence in calibration results of a three-dimensional groundwater model if those data are not available.

Acknowledgements

We thank Mr J.C. van Dam for remembering and retrieving the work of Edelman used in this study and for his critical attitude in our application of the method. Mr D. Kovfics attended for the appli- cation of Leenen's work. The support of Mr O. Starosolsky, Director General of the Water Resources Research Centre (VITUKI, Budapest) and Mr Gy. Jakus, Director of the North-Transdanubian Water Authority (I~DUVIZIG, Gy6r) is acknowledged, in providing access to their data sets. We are grateful for the constructive comments of the two reviewers of this paper: Mr M.A. Sophocleous and an anony- mous referee.

Appendix A

This appendix contains the most important steps of the derivations of the one-dimensional groundwater flow equations.

Appendix A. 1 Semi-confined condition

For an aquifer covered by a semi-pervious layer and bordered by a river with no river bed resistance [Fig. l(c)], the heads can be calculated from the one-dimensional flow equation, as a function of the river stage and the distance from the river in a

section perpendicular to the river (Leenen, 1994):

Se 04, 4, 024,= - - 0 (A1)

T Ot Tecl Ox 2

and the boundary conditions, assuming linear rise of the river stage:

x = 0 4,=Hpt (t-<tp) tp

04, X -----~ oe : - - - - * 0

0x

where Se is the elastic storage coefficient or stora- tivity of the aquifer, T is the aquifer transmissivity, 4, is the head in the aquifer, t is the time, tp is the duration of the rising phase of the river stage, c'cl is the resistance of the cover layer, x is the distance from the river, Hp is the full rise of the river stage. The resistance of the cover layer can be calculated from the thickness of the cover layer (dcl) and its hydraulic conductivity (Kz,d) by:

dcl Cc~ = (12)

Kz, cl

It can be proved by a dimensional analysis of Eq. (1) that the process is determined by the following characteristic time (tu) and length (l.) units:

tu = Secd (A3)

l u = ~ (14)

It is possible to rewrite Eq. (1) in a dimensionless form taking:

, X t * = t X = - - ; --" 4 , * = 4, l u tp ' Hp

and defining:

SeCcl tu tp tp

Eq. (1) in dimensionless form is:

~ °Ot** + 4,. 024, * - Ox~=O (15)

The boundary conditions:

x . = 0 : 4, .=t . (t. ~< 1)

04,. X, --* ac : - - - - - ~ 0

0x,

124 z. Vekerdv, A.M.J. Meijerink/Journal o]" Hydrolog?. ' 205 (1998) 112-125

Leenen (1994) gives an approximate analyt ical solu- tion of Eq. (5):

1 0 , = < t , - ~/3x,) e (16 )

Appendix A.2 Phreat ic conditions

Assuming a vertical river bank, an initial horizon- tal g roundwater table and no surface recharge, the one-d imens iona l groundwater flow in an unconf ined aquifer with a water body at one of its boundar ies can be described by:

00 T 0z4' = 0 (A7) at ix Ox 2

and the boundary condi t ions are:

t : 0 ; 4 ,=0 ( 0 < - x < - ~ )

x : o < q~=0 ( 0 < - t - < ~ )

x = 0 ; 0 = H p ( 0 - t - < w )

where T is the aquifer t ransmissivi ty, 4' is the head in the aquifer (the da tum is the head at t = 0), t is the time, x is the distance from the r iver bank and IX is the specific yield of the aquifer. The Dupuit assump- tions must be met:

1. the equipotential lines are vertical; 2. the slope of the groundwater table is equivalent to

the hydraulic gradient and is invariant with depth.

Ede lman (1947) gave analyt ical solutions for Eq. (7) to calculate the head rise in a phreatic aquifer. The most important steps are as follows. He intro-

duced:

T ~- = --t (A8)

IX

Subst i tut ing ~- into Eq. (7):

a2~ a~ - 0 ( A 9 )

0x 2 OT

Solut ions o f the fol lowing form satisfy Eq. (9):

C~ : bTn/2/'(U)

where b is a constant and

x u = - - (A10)

andJ (u ) has to satisfy the fol lowing equation:

f " (u ) + 2u f ' (u ) - 2 n f (u) : 0 (A 1 1)

Ede lman (1947) gives solut ions for the cases n = 0,1,2 and 3. Case n = 2 describes a l inearly rising head in the river (H=c~t, so o~ = H/t):

4~ = o~t~(u) (A12)

where fz (u ) is a function of u. Ede lman (1947) gives the analytical form off2(u):

.f2 (u) = ~ u e - u2 - (2u 2 + 1 )( 1 - E(u)) (A 13)

where

E ( u ) = e - " du (A14) 0

Edelman (1947) provides the corresponding numer i - cal values of u and fz(u) in tabular form too.

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