MODELLING AND NUMERICAL SIMULATION OF GROUNDWATER FLOW …dlib.lib.cas.cz/5282/1/2007_55_3_burger_168.pdf · Modelling and numerical simulation of groundwater flow in the reparian

J. Hydrol. Hydromech., 55, 2007, 3, 168–184

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MODELLING AND NUMERICAL SIMULATION OF GROUNDWATER FLOW IN THE REPARIAN ALLUVIAL AQUIFER FRANTIŠEK BURGER Institute of Hydrology, Slovak Academy of Sciences, Račianska 75, 831 02 Bratislava, Slovak Republic; mailto:[email protected]

The objective of this study is to make a conceptual and numerical model of the groundwater flow system which will improve the understanding of the groundwater cycle in the area of the Čenkov Valley, Slovakia. Extreme deficits of atmospheric precipitation and thereof resulting periods of low water flows and dis-charges could very negative impact the water management. Increasing water consumption in the future will be the most critical in strong and intensive dry periods. Almost every climatic zone could suffer from drought, although its features could considerably vary from region to region. The study is handling with creating, calibration and verification of numerical model of groundwater flow in the reparian alluvial aqui-fer of the Čenkov Valley in south-east part of the Danubian lowland for minimal anthropogenic disrupted natural conditions in the past and quasi-steady deficit water regime of the area. The conceptual model is based on data from earlier studies in the area complemented with data collected in the field. Results of model solutions are presented in the study – groundwater level, filtration velocity vectors, groundwater paths by particle tracking and water budget of study area. Created numerical model could be used for simu-lation of underground dam function, which belongs to the types of artificial recharge of reparian alluvial aquifer management, and also for creating prognostic scenarios concerning expected climatic changes. Ad-ditional future work may include adding a solute transport model to the flow model. KEY WORDS: Stream–Aquifer Model, Model Calibration/Verification, MODFLOW, Water Regime, Natural State.

František Burger: MODELOVANIE A NUMERICKÁ SIMULÁCIA PRÚDENIA PODZEMNEJ VODY V PRIRIEČNOM ALUVIÁLNOM HYDROGEOLOGICKOM KOLEKTORE. Vodohosp. Čas., 55, 2007, 3; 33 lit., 14 obr., 2 tab.

Cieľom predloženej štúdie je vytvorenie koncepčného a numerického modelu systému prúdenia

podzemnej vody na území Čenkovskej nivy na Slovensku. Extrémne deficity atmosférických zrážok a z toho vyplývajúce obdobia nízkych vodných stavov a prietokov môžu vplývať na vodné hospodárstvo veľmi negatívne. Zvýšená spotreba vody bude v budúcnosti najkritickejšia práve počas drasticky suchých periód. Takmer každá klimatická zóna môže trpieť suchom, hoci jeho charakteristiky sa môžu od regiónu po región značne líšiť. Štúdia sa zaoberá tvorbou, kalibráciou a verifikáciou numerického modelu prúdenia podzemnej vody v pririečnom hydrogeologickom kolektore Čenkovskej nivy v juhovýchodnej časti Podu-najskej roviny, za minimálne antropogénne narušených prírodných podmienok v minulosti a kvázi ustáleného deficitného vodného režimu územia. Koncepčný model je založený na údajoch z predošlých štúdií doplnených o údaje zhromaždené v teréne. V štúdii sú prezentované výsledky modelového riešenia – poloha hladiny podzemnej vody, vektory filtračnej rýchlosti, smery prúdenia podzemnej vody prostredníct-vom trasovania pohybu častíc a vodná bilancia územia. Vytvorený numerický model môže byť využitý na simuláciu funkcie podzemnej priehrady, ktorá patrí medzi typy umelého nasycovania pririečneho aluviál-neho kolektora a tiež pri tvorbe prognostických scenárov, zaoberajúcich sa klimatickými zmenami. Doplnková budúca štúdia sa môže venovať pripojeniu transportného modelu chemických látok k prezen-tovanému tokovému modelu.

KĽÚČOVÉ SLOVÁ: model povrchový tok–kolektor, kalibrácia a verifikácia modelu, MODFLOW, vodný režim, prírodý stav.

Modelling and numerical simulation of groundwater flow in the reparian alluvial aquifer

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Introduction

The solution of groundwater flow features as-sessment, which are due to later anthropogenic encroachments in the area ranked as almost natural, is going out from the evaluation of former ground-water regime based on observations in the nature, knowledge of geological structure of the area and hydrogeologic conditions, which is serving as a base to the water level regime assessment and the assessment of the main groundwater flow direc-tions.

Various experimental methods were used in the past for solving the problem e.g. analogy methods and physical models. The works of e.g. Hálek (1956), Duba (1968), Lukner and Šestakov (1975), Pricket (1975), Mucha and Šestakov (1987), and others were dedicated to analogy models. Mathe-matical models started to be used with computer approach; however these models are facing certain limits by the solution of practical problems (Kosorin, 1999). Rash evolution of numerical methods for modelling of groundwater flow and transport of substances was noted in latest period, these methods are described in many documents e.g. Huyakorn and Pinder, 1983; Kazda, 1983; Bear and Verruijt, 1987; Mucha and Šestakov (1987); Goode, 1990; Konikow, Goode, Honberger, 1996; Kipp, Konikow, Hornberger, 1998; Har-baugh, Banta, Hill, Mc Donald, 2000; Heberton, Russell, Konikow, Hornberger, 2000; Hill, Banta, Harbaugh, Anderman, 2000; Merritt and Konikow, 2000; Anderson and Woessner, 1992; Anderson and Pricket, 1992; Kovařík, 2000; Chiang and Kinzelbach, 2000; Čelková, 2003, Šoltész and Baroková, 2003–2004, Novák and Majerčák, 2004, Štekauerová, Šútor, Mikulec, 2005 and others.

The aim of the task to be solved in this article is to create numerical model for steady groundwater flow in reparian alluvial aquifer of the Čenkov Val-ley, its calibration, verification and obtaining the results by simulation, that is groundwater level, filtration velocity vectors, groundwater paths by particle tracking and water budget. Assumption is long-term minimal anthropogenic disrupted natural conditions of the study area. As the date of the simulation was chosen the day of 29 Sept. 1954, because of steady state of deficit water regime of study area, and also of the Danube low stage and because of existence of solving similar task by other methods in the past (Duba, 1964, 1968) and thereby availability of data needed for modelling and simulation.

Description of study area

The Čenkov Valley is situated in the eastern part of the Danubian lowland, west apart from Štúrovo town. It is fluvial plain of the Danube, which bor-ders in the south on 23 kilometers long river section between rkm 1722 and 1745 and in the north on arc stretching terrace platform, on its the boundary are the Moča village, the Búč village, the Júrsky Chlm village, the Mužla village and the Obid village. The fluvial plain is from 2.5 km up to 6 km wide and has overall area circa 66 km2. Its surface is flat. The heights of terrain vary from 106.0 up to 108.0 m a.s.l. The lowest situated section under terrace is on height level ca. 105.0 m a.s.l. and the highest situ-ated section in the Čenkov wood is on the middle of the area 108.0 – 110.0 m a.s.l. (Fig. 1). Evaluation of natural conditions – status before 29. September, 1954 Climate

From climatic point of view the study area be-longs to warm locality in the scope of south-eastern part of the Danubian lowland, warm and dry cli-matic zone with soft winter. First temperature char-acteristic, yearly air temperature average shows markedly that the south of Slovakia is the warmest locality of the republic. The average 10.4 oC at Štúrovo is convincing. Uniformity of moisture con-ditions is clear already from the yearly relative air moisture average, which varies from 74 – 81 % and is the lowest in lowest parts of the Danubian low-land (Štúrovo 74 %, Komárno 75 %).

In the territory of the West-Slovakian district, which is the most productive agricultural locality, the precipitation has significant importance. The centre of this locality is the Danubian lowland, which is indeed the warmest but also driest locality. Its hydrologic conditions show significant contrasts in particular years. Although this locality is passed by the Danube, big European river, which is ac-creted here by conspicuous Slovak rivers: the Váh River, the Nitra River, the Hron River and the Ipeľ River, district often suffers of absence of water, therefore agricultural cultures are supported by artificial irrigation. In long-term observed series the lowest annual precipitation totals vary in terms of 300 – 400 mm and minimum monthly precipitation totals in particular months do not even reach (ex-cept July) 5 mm precipitation, whereupon signifi-cant dry periods are more often in summer half-

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year than in winter half-year. The lowest July pre-cipitation totals do not drop under 10 mm (Fig. 2 and Fig. 3).

On the other hand wet (precipitation) periods are lasting here mostly 18 – 20 days, their appearance is relative more rarely than the appearance of dry periods and it occurs mostly in spring and autumn periods. The highest annual precipitation totals could reach 900 mm, even in singular cases up to 1000 mm of precipitation.

Wind conditions are important climatic charac-teristic, because they considerably affect the course of meteorological elements such as air temperature, evaporation, snow cover, the occurrence of fogs and others. Territory of the West-Slovakian district presents complicated orographic entity. Particular mountain chains cause in various ways on wind velocity and wind direction in order to total air pressure distribution in various atmospheric situa-tions (Kolektív, 1968).

Location of study area

Fig. 1. Location map of the Čenkov Valley. Obr. 1. Zemepisná mapa Čenkovskej nivy.

Fig. 2. Average monthly precipitation totals in mm (period 1931–1960). Obr. 2. Priemerné mesačné úhrny zrážok v mm (obdobie r. 1931–1960).


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Fig. 3. Average yearly, winter and summer precipitation totals in mm (period 1931–1960). Obr. 3. Priemerné ročné, zimné a letné úhrny zrážok v mm (obdobie r. 1931–1960). Geology and hydrogeology

The Danube fluvial plain in observed river sec-tion is built by sediment deposits of the Danube River, their thickness varies irregularly between 5 – 12 m, the most frequent thicknesses are between 6 – 9 m. In soil layers there dominate gravel and sand, which are in the highest part covered by allu-vial loams. Gravel – sandy filling of the Danube fluvial plain’s bed in this section belongs to würm, the cover of sandy loam is holocene. It is possible to observe their partial subtilization in longitudinal profiles of gravel’s sand alluvia (Fig. 4, 5 and 6) in the Danube direction, although the appearance of heavier gravel’s layers is possible in the whole pro-file. The left edge of the Danube’s fluvial plain is lined by markedly terrace step with relative height approximately 15 m and base 3 m above the Da-nube water level. Absolute height of the base is around 110 m a.s.l., it is slowly descending from the Chotin village to the Štúrovo town. Vaškovský (1977) refers this terrace to trias (Figs 7 and 8).

Hydrogeological conditions of the terrace were proofed only by a few boreholes, after which hy-draulic conductivity of gravels varies from 6,6.10-5 m s-1 (the Chotín village) up to 2.0.10-3 m s-1 (the Štúrovo town – the Nana village). Groundwater recharge happens entirely from precipitation in locations where permeable blown sands or loamy sands and sandy loams are located in hanger. Groundwater from the terrace is drained on its edge to the lower step, partly on its contacts comes up to the surface and it is taken away by drainage chan-

nels. It is shown in some locations by tiny outflow, and their yield reaches maximum tenths of l s-1. River alluvia are strongly watered in almost whole vertical profile. Groundwater level in the alluvia is mainly influenced by the surface stream of the Da-nube River and on other side also by water seeping down from adjacent terrace and by precipitation. Groundwater depths at high water levels were smaller than 1,0 m, in locations moderate elevated by blown sands were the depths about 2 – 3 m. Groundwater level local ascended up to the little permeable to impermeable hanger and gained aqui-fer pressure regime. Groundwater flow direction according to bilateral relation of the Danube water level and groundwater level was either to the area or to the Danube. The beginning of the ascending of groundwater levels after the Hydrometeorologi-cal institute appraisal of regime observation felt already to winter months, where the influence of precipitation was registered. But it was only grad-ual and it went on very smoothly to the maximums period of the Danube – May, June. Afterwards con-sistent decrease occurs and lasts to the winter re-vival. Hydrology

By the hydrological characteristic of study area and description of surface flows in objective time it is necessary to concentrate on the Danube River, which has here first- rated importance. Slovak Da-nube River reach belongs to the upper part of mid-dle part of the river. Danube is keeping its alpine

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1

1

22

Hydrogeological profile: Fig. 4. Localization of hydrogeological profiles. Obr. 4. Polohopis hydrogeologických profilov.

Fig. 5. Hydrogeological profile 1-1 (400x exceeded). Comments: 1 – young Pleistocene blown sands, 2 – medium to smooth sands, 3 – sandy gravels to rough sands with gravel, 4 – downhill loamy sediments along upper terrace step, 5 – dusty to loamy sands, eventually dusty – sandy loams, 9 – marking of the tertiary base surface, 10 – groundwater level on 29 Sept. 1954, 11 – the highest groundwater level in years 1954 – 1956 (after Duba, 1968). Obr. 5. Hydrogeologický profil 1-1 (400x prevýšený). Vysvetlivky: 1 – mladopleistocénne naviate piesky, 2 – stredné až jemné piesky, 3 – piesočnaté štrky až hrubé piesky so štrkom, 4 – svahové hlinité sedimenty pozdĺž vyššieho terasového stupňa, 5 – pra-chovité až hlinité piesky, poprípade prachovito-piesočnaté hliny, 9 – označenie povrchu treťohorného podkladu, 10 – hladina podzemnej vody dňa 29. 9. 1954, 11 – najvyššia hladina podzemnej vody za r. 1954 – 1956 (podľa Duba, 1968).


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Fig. 6. Hydrogeological profile 2-2 (400x exceeded), (after Duba, 1968). Obr. 6. Hydrogeologický profil 2-2 (400x prevýšený), (podľa Duba, 1968).

Fig. 7. Aquifer of Čenkov Valley – 3-D view from south-west. Obr. 7. Hydrogeologický kolektor Čenkovskej nivy – 3-D pohľad od juhozápadu. character in Slovak reach, in its upper part it has considerable slope around 0.4 ‰, it is flowing in its own alluvia and it is creating multiple system of river arms. Water stages are first of all depending on water supply from the Alps. Maximum water stage reaches the Danube in June at the time of alpine snow and glacier melting. From June it comes to permanent decrease and minimum water stages are reached in December and January. The Danube water stage on 29. Sept. 1954 in rkm 1742.9 (Radvaň nad Dunajom) was 105.20 m a.s.l. (Fig. 9).

Other surface flows in study area are rather small and short and their discharges are low. Maximum is in spring months, in summer is their discharge con-siderably decreased. Such streams are Modrianský potok (creek) (from Veľká Dolina), its left-hand side tributary Vojnický potok (creek), and Mužliansky potok (creek). Main channels: the Obidský, the Búčsky, the Kraviansky and the Krížny channel belong to the system, as well as amount of side drainage channels without any name.

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NE NW

Obr. 8. Boundary of Čenkov Valley aquifer from the north by higher old-würm terrace step: 1 – tertiary base surface, 2 – sandy gravels to rough sands with facia gravel from river bed, 3 – smooth to medium sands of littoral sandbank’s facia, 4 – alluvia loam cover layer, 5 – downhill sediments along the foot of the terrace step, 6 – blown loess and sandy loess (after Duba, 1968). Obr. 8. Ohraničenie hydrogeologického kolektora Čenkovskej nivy od severu vyšším starowürmským terasovým stupňom: 1 – povrch treťohorného podložia, 2 – piesočnaté štrky až hrubé piesky so štrkom fácie riečneho dna, 3 – jemné až stredné piesky fácie príbrežných plytčín, 4 – pokryvná vrstva náplavových hlín, 5 – svahové sedimenty pozdĺž úpätia terasového stupňa, 6 – naviate spraše a piesočnaté spraše (podľa Duba, 1968).

Fig. 9. The Danube water level time series (Duba, 1968). Obr. 9. Časový priebeh hladiny Dunaja (Duba, 1968). Methods and material Modelling Mathematical model of groundwater flow

Three-dimensional groundwater flow of constant density through porous earth material may be de-scribed by partial differential equation:

,

xx yy

zz s

h hK Kx x y y

h hK W Sz z t

∂ ∂ ∂ ∂∂ ∂ ∂ ∂

∂ ∂ ∂∂ ∂ ∂

⎛ ⎞⎛ ⎞ + +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞+ − =⎜ ⎟⎝ ⎠

(1)

where


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x, y, z – Cartesian coordinates in the direction of main axis of hydraulic conductivity Kxx, Kyy, Kzz, Kxx, Kyy, Kz – values of hydraulic conductivity in the direction of axis of Cartesian coordinates x, y, z, about which is assumed that they are parallel with major axis of hydraulic conductivity [L T-1], h – the piezometric pressure head [L], W – a volumetric flux per unit volume, which represents sources and (or) sinks of water [T-1], Ss – the specific storage of the porous material [L

-1],

t – time [T]. In general, Ss, Kxx, Kyy, and Kzz may be the

functions of space (Ss = Ss (x, y, z) and Kxx = Kxx (x, y, z), etc. and h and W could be the functions of space and time (h = h (x, y, z, t), W = W(x, y, z, t)) thus means Eq. (1) describes groundwater flow for unsteady conditions in a heterogeneous and anisot-ropic medium, provided the principal axes of hy-draulic conductivity are aligned with the coordinate directions.

Eq. (1), together with specification of flow and (or) head conditions on aquifer boundaries and specification of initial head conditions, creates mathematical model of groundwater flow. A solu-tion of Eq. (1) in an analytical sense, is an algebraic formula, which indicates h (x, y, z, t) so that when the derivatives of h with respect to space and time are substituted into Eq. (1), the equation and its initial and boundary conditions are satisfied.

Besides very simple systems it is possible to reach analytical solution of Eq. (1) only rarely, therefore it is necessary to use numerical methods for solution. One of the methods is the finite differ-ence method, where the continuous system of Eq. (1) is substituted by the finite set of discrete points in the space and time, and the partial derivatives are substituted by terms calculated from the differences in head values at these points. Such approach leads to the system of linear algebraic differential equa-tions. Values of head in specific points in time are obtained by their solution. These values represent approximation of the time-variable distribution of piezometric head, which could have been obtained by analytical solution of Eq. (1). Three-dimensional modular model of groundwater flow ”MODFLOW”

The finite difference model originally published by McDonald and Harbaugh (1988), in the form of later modifications and addendums, and its modular

computer program was utilized by the solution of mentioned task. Modular structure consists of the "main program" and a series of independent sub-routines called "modules". The explanation of physical and mathematical concept, on which the model is based, and explanation how the modules are implemented into the structure of computer program is in detail listed in mentioned work. Groundwater flow in hydrogeological groundwater body is simulated by the use of finite difference block-central method. The solution of systems of simultaneous linear equations is possible to obtain by various methods. Conceptual groundwater modelling Definition of model’s boundaries

Northern to western boundary of the modeled area is chosen regarding to demarcation of hydro-geological groundwater body of Čenkov Valley from the northern side by higher old-würm terrace step. Southern to eastern border is created by the Danube River (Fig. 1). Transformation of the hydrogeological system – vertical schematization

Topography of the surface terrain was processed into the digital form from maps with the scale 1 : 10 000. Density of height points (spacing 50 x 50 m) corresponds to the amount of cells in computa-tional grid, whereas every midpoint of the cell sur-face in the 1st layer represents one point from the map of surface terrain topography (Fig. 10). For purposes of modelling, upper described hydro-geological system was transformed into three-layers system, which consists of upper covering soil layer (1st layer), of medium to soft sand layer (2nd layer) and of sandy gravel to rough sands with gravel layer (3rd layer) (Fig. 7).

Type of layers was selected for modelling as fol-lows: 1st layer: unconfined, 2nd layer: confined/unconfined (transmissivity = = const), 3rd layer: confined/unconfined (transmissivity = = const). Thickness of particular layers was defined from geological and hydrogeological data from the sur-vey. Surface of tertiary base is considered as im-permeable.

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Fig.10. 3-D visualization of the surface terrain Čenkov Valley. Obr. 10. 3-D vizualizácia povrchu terénu Čenkovskej nivy. Discretization in the space and time

Hydrogeological system is divided into the mesh of blocks called cells, the locations of which are described in terms of rows, columns, and layers. Within each cell there is a point called a “node” at which head is to be calculated. Regarding to Carte-sian system of coordinates, points along rows are parallel with x-axis, points along columns are paral-lel with y-axis and points along vertical are parallel with z-axis. Conceptually nodes represent prisms from porous material, in which hydraulic features are constant, and so whichever value connected with a node refers for the whole cell.

Footprint dimensions are picked so that whole area of the Čenkov Valley with smooth overlay is covered. Dimensions of cells are: ∆x = ∆y = 50 m. Geometry of the model is: 22.5 km x 6.5 km, (130 rows and 450 columns); 3 layers. Whole three-layers model consists of 175 500 cells. Grid orien-tation was picked in the direction of the general groundwater flow and coordinate axis x, y, z are approximately parallel to the main hydraulic con-ductivity axis.

Groundwater flow has always in certain measure character of unsteady flow. It results from natural conditions of recharge and drainage of groundwa-ter. However if the recharge and drainage ground-water conditions are changing in the time slightly, flow is quasi-steady and practically represents cer-tain boundary status. From the modelling target point of view a steady status of groundwater flow was considered to the date 29. Sept. 1954. Filtration parameters of the aquifer

Following filtration parameters were necessary for the modelling of this case: horizontal hydraulic conductivity, transmissivity, vertical hydraulic con-

ductivity, effective porosity and coefficient of ver-tical leakage.

Horizontal hydraulic conductivity of the groundwater body was obtained from the results of the hydropedological and hydrogeological survey in the study area. Data from pumping tests in probes and boreholes were globally processed by the means of interpolation method of kriging. Values varies from 7.48.10-7 m s-1 to 3.99.10-3 m s-1.

Transmissivity of the layers was calculated as a multiple of horizontal hydraulic conductivity and thickness of the layer.

Vertical hydraulic conductivity. By the model-ling applications the usual ratio of the horizontal to vertical hydraulic conductivity is from 1 to 10 (Anderson and Woessner, 1992). For the first and second layer ratio 1.0 was selected in compliance with results of the field research and for the third layer the ratio 2.0 according to Benetin´s work (Benetin, 1961). Effective porosity is the feature of aquifer to receive and send out fluid in order to hydrostatic pressure in the layer and to the ground-water level. Quantitatively it is expressed by coeffi-cient of flexible storage and coefficient of free wa-ter level storage. The value of the coefficient of the free water level storage depends on hydraulic con-ductivity and also on grain size distribution of sedi-ments and varies around 0.05 up to 0.15 for loamy sands, 0.15 for soft granulated to dusty sands, 0.19 for soft granulated sands, 0.22 for medium granulated sands and 0.24 for rough granulated sands, gravels etc. Estimated values of flexible storage for unit volume of the groundwater body are stated in work of Mucha and Šestakov (1987).

Vertical leakage is required in the case of multi-ple layers groundwater body and represents the resistance to the water leakage at adjacent layers.


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Used modules

Model is divided into modules, which enable to study specific hydrogeological characteristics of the area. Modules are grouped into packages. Each package handles with specific outline of simulated hydrogeological system, or with a specific method of solution of linear equations describing the flow-ing system. In presented case were utilized follow-ing modules: basic package, block centred flow package, output control, evapotranspiration, re-charge, general-head boundary, and streamflow-routing packages. Basic package

The Basic package should always be utilised and input for it must always be defined. The basic package handles those aspects that define the groundwater system. Block centred flow package

The Block centred flow package is also required for all simulations. It computes the conductance components of the finite-difference equation, which determine flow between adjancent cells and the rate of movement of water to and from storage. Output control

The output control option is actually not an indi-vidual package but rather an optional segment of the basic package Head and drawdown by layer and time step and the overrall volumetric budget can be stored or printed. Evapotranspiration package

The Evapotranspiration package simulates the ef-fects of plant transpiration and direct evaporation in removing water from the saturated groundwater regime. In this case initial input data from Duba (1964, 1968) works were utilized. Recharge package

This package is utilized to simulate space-time distribution of the water inflow from the surface terrain to the groundwater level. Natural recharge enters the groundwater system from surface and so it is applied into the first layer of the model. Similar as in the case of evapotranspiration initial input data from Duba (1964, 1968) works were also util-ized in this case.

General-head boundary The General-head boundary package is used to

simulate head-dependent flow boundaries (Cauchy boundary conditions). By the means of package inflow of outland water into groundwater from behind the boundary of study area is simulated and that from the north from the edge of old-würm ter-race step. It is applied only to the second layer of the model. Input data were obtained from hydro-geological survey works stored in Geofond Brati-slava, and from the Duba (1964) publication. Streamflow-routing package

The Streamflow-routing package is used to ac-count for the amount of flow in streams and to simulate the interaction between surface streams and groundwater. Only three creeks, flowing through the study area and lager drainage canals, were taken into account. Their parameters and char-acteristics, which were needed as entering data into the model, were obtained by the means of particular survey and research, and also by the help of archive materials stored in the Geofond Bratislava and State melioration direction Bratislava. Solvers

To calculate heads in each cell in the finite-difference grid, Modflow prepares one finite differ-ence equation for each cell, expressing the relation-ship between the head at a node and heads at each of the six adjancent nodes at the end of a time step. Because each equation may involve up to seven unknown values of head, and because the set of unknown head values changes from one equation to the next through the grid, the equations for the en-tire grid must be solved simultaneously at each time step. The best results from utilized methods for solution of the system of the equations (Direct So-lution-DE45, Preconditioned Conjugate-Gradient (PCG2), Strongly Implicit Procedure-SIP, Slice-Successive Over Relaxation (SSOR)) give us the method called Preconditioned Conjugate-Gradient 2 (PCG2), (Hill, 1990). Calibration and verification of the model

After creating the numerical model of the area by means of creating responding initial input files it is necessary to calibrate the model. Calibration is a process, when initial input model parameters are adjusted until output (dependent) model parameters at most approach to the values measured in the

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terrain. Calibration of the model is an inverse model process, i.e. the problem of parameters esti-mation is an inverse problem. Calibration of the model or inverse model process could be performed either repetitively – on manual basis by the way trial - error or by the way of using a special com-puter program. Calibration was executed by the means of special computer program PEST with manual tuning of some zones.

Calibration results for the status up to date 29. Sept. 1954 are listed in Tab. 1. Difference be-tween measured groundwater stages in sounds from the Hydrometeorological institute’s observing net-work and calculated groundwater stages has a maximum value 0.17 m and the regression coeffi-cient has a value near to one, what refers to high

correspondence of calculated results with measured results. Bigger differences are only in two purpose boreholes, which were constructed and temporary observed in the scope of creating the project of the Gabčíkovo system of hydraulic structures on the Danube (Duba, 1968) and they are appended lower credit.

Calibrated model of steady groundwater flow was verified at the low water stage in the Danube up to date 7. Aug. 2002 and at high water stage in the Danube up to date 7. May 2000. For both cases a very good accordance of measured and calculated groundwater levels was reached (see Burger, 2005a), 2005b)).

T a b l e 1. Results of the model calibration. T a b u ľ k a 1. Výsledky kalibrácie modelu; 1 – sonda-číslo, 2 – vypočítané piezometrické výšky [m n.m. ], 3 – namerané piezometrické výšky [m n.m. ], 4 – rozdiel [m], 5 – účelové sondy.

Borehole-number1) Calculated heads2) [m a.s.l.]

Measured heads3)

[m a.s.l.] Difference4)

[m] Moča-6024 106.80 106.74 0.06 Kravany-7422 105.07 105.15 –0.08 Kravany-6026 105.09 104.91 0.17 Čenkov-7426 105.09 105.21 –0.12 Obid-7438 104.21 104.32 –0.11 Kendeleš-514 104.34 104.36 –0.02 Mária-519 105.85 105.79 0.06 Deravý chrbát-7432 104.43 104.48 –0.05 Langov majer-520 105.58 105.43 0.15 Purpose boreholes5) Búčska lúka-9 104.54 104.91 –0.37 Obid-S1 104.33 104.11 0.22

Sumary output

Regression statistics Multiple R 0.989895204 R Square 0.979892516 Adjusted R Square 0.854892516 Standard Error 0.110284044 Observations 9

Results and discussion

Data about groundwater fluctuation has shown that the basic factors which are influencing groundwater level changes are atmospheric precipi-tation, bank filtration from the Danube River and underground inflow from the upper terrace direc-tion north-west and on the northern edge of the area, then evapotranspiration, underground outflow to the Danube (draining effect of the Danube) and

outflow through drainage canals. Presented solution for groundwater regime represents response to the certain below average up to low water stage of the Danube (up to date 29. Sept. 1954), which regard-ing to former water stages it is possible to consider as almost steady (Fig. 11). On the Fig. 12 we can see, that the drainage effect of the Danube is from rkm 1745 to rkm 1733 and from rkm 1727 to rkm 1722, this presents together 17 km bank filtration drainage function of the river. Bank filtration re-


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charge is from rkm 1733 to rkm 1727 that is 6 km bank length of the Danube. The line of direct drain-age influence of the Danube goes from the Moča village along the edge of the terrace step to the Búč village, where it turns to the south approximately to the Mária farmstead whence it continues the south-ern-east direction to the river in the Čenkov local-ity. Second drainage area is bounded from the south from the Obid canal river mouth to the Danube up to the Štúrovo town and from the north by the “Pod kopanicami” drainage canal. The feeding effect of the river is decreased by the drainage of lower parts of the Mužliansky creek and the Obid canal. Re-maining part of the study area is drained by the system of drainage (the Obid, Krížny and Búčsky canal). At the modeled low water stage of the Da-nube it doesn’t come in the area to the confined groundwater flow as it was expected (see Duba, 1964) at high water stages in the river. On the Fig. 13 there is 3D visualization of the whole groundwa-ter body with velocity vectors and on the Fig. 14 there is 3-D visualization of the groundwater body with the pathlines of particle tracking in the flowing field. In the second layer of the groundwater body there are the highest filtration velocities along the Obid canal and Mužliansky creek and at their river mouths, also along the bank of the Danube (drain-age) around rkm 1742. Similary in the third layer there are the highest filtration velocities along open streams and along the Danube bank from rkm 1744 to rkm 1742 and around rkm 1729. The range of

filtration velocity values in the whole groundwater body varies from 0.0 m s-1 to max. 1.19E-03 m s-1. Particle tracking is used for the tracing of the groundwater flow directions that means for creating flow lines by the means of tracing of the infinitely small imaginary elements movement situated in the flow field. Computer codes for the particle tracking are regarding to the groundwater flow models post-processors, because they accept the distribution of piezometric heads from the model and they use it for the calculation of the velocity distribution. In the reach from the Moča village to the Čenkov set-tlement groundwater flows into the Danube except the northern part in which groundwater flows to the drainage canals. From the Obid canal mouth to the Danube River (rkm 1727) up to the Štúrovo town groundwater flow direction is again into the Da-nube.

Volume water budget for the whole model at the end of simulation is calculated in order to control the results. It indicates acceptation of numerical solution. Numerical solution technology is based on system of equation solved by the model by main-taining the continuity equation for each cell. Conti-nuity has to be preserved also for the total model inflow and outflow or subregion of the layer. The difference in water budget of the study area is 0,00 % (Tab. 2). The difference should be in ideal case smaller than 0,1 %, what is fulfilled. In general error up to 1 % is accepted (Konikow and Brede-hoeft, 1978).

Fig. 11. 3-D visualization – groundwater table isosurfaces (layout every 0.25 m) – view from the south. Obr. 11. 3-D vizualizácia – izoplochy hladiny podzemnej vody (klad po 0,25 m) – pohľad od juhu.

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Fig. 12. Simulated steady head distribution and flowlines in the second model layer. Obr. 12. Hydroizohypsy hladiny podzemnej vody a trajektórie pohybu častíc v 2. modelovej vrstve.

Fig. 13. 3-D visualization – filtration velocity vectors – view from the south. Obr. 13. 3-D vizualizácia – vektory filtračnej pórovej rýchlosti – pohľad od juhu.

Fig. 14. 3-D visualization – trajectory of particles movement – view from the south. Obr. 14. 3-D vizualizácia – trajektórie pohybu častíc – pohľad od juhu.


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T a b l e 2. Water budget of the whole model domain. Flows are considered "IN" if they are entering a model domain. The unit of the flows is [L3 T-1]. T a b u ľ k a 2. Vodná bilancia celej modelovej oblasti. Toky vstupujúce do modelovej oblasti sú označené „IN“. Jednotka toku je [L3 T-1]. 1 – tokový člen, 2 – konštantná piezometrická výška, 3 – prítok z povrchu, 4 – evapotranspirácia, 5 – cudzí hraničný prítok, 6 – priesak z (do) povrchových tokov, 7 – suma, 8 – diskrepancia.

Flow term1) IN OUT IN-OUT

Constant head2) 1.5719140E-02 6.6387028E-02 –5.0667889E-02 Recharge3) 1.5193854E-01 0.0000000E+00 1.5193854E-01 ET4) 0.0000000E+00 4.2560022E-02 –4.2560022E-02 Head dep bounds5) 1.0380716E-03 4.5224038E-05 9.9284749E-04 Stream leakage6) 1.7055156E-02 7.6753564E-02 –5.9698410E-02

Sum7) 1.8575092E-01 1.8574584E-01 5.0812960E-06

Discrepancy [%]8) 0.00

Conclusions

The aim of this contribution was to developed a model to simulate groundwater flow with MOD-FLOW for the near natural situation. This means the simulation without the large-scale abstractions of groundwater, huge pumping stations, seepage canals and undeground impermeable walls along the bank of the Danube River. One of the problems faced in the Čenkov Valley is lack of understanding of what the natural hydrological regime would have been before large-scale human interventions started in the mid 1980´s. This study investigates the near natural situation in the Čenkov Valley by using simulation modelling. The most important conclu-sions from simulations are as follows: − Groundwater level varies from 103.26 m a.s.l. to

107.96 m a.s.l. Its depth under the terrain varies from 0.50 m in the area of Čenkov wood to 5.70 m by the Danube;

− Groundwater flow velocities in the second layer of aquifer vary in the scope from 0.00 m s-1 to 1.19E-03 m s-1 and in third layer from 0.00 m s-1 to 9.94E-05 m s-1;

− Trajectory of particles movement show to largely drainage function of the Danube, as well as the creeks and drainage channels in the study area of Čenkov Valley;

− Obtained results (data, maps, pictures) will serve as a reference basis for numerical simulations of the underground dam creation, which belongs to the management types of artificial supplies of reparian alluvial aquifer, as well as for prognos-tic scenarios of water regime evolution in the ar-eas regarding to climatic changes scenarios in this region.

Acknowledgement. Authors would like to express thanks to the Grant Agency of Slovak Academy of Sciences VEGA for the financial support from pro-

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Received 26. January 2007

Scientific paper accepted 20. March 2007 MODELOVANIE A NUMERICKÁ SIMULÁCIA PRÚDENIA PODZEMNEJ VODY V PRIRIEČNOM ALUVIÁLNOM HYDROGEOLOGICKOM KOLEKTORE František Burger

Príspevok obsahuje tvorbu koncepčného a numeric-kého modelu systému prúdenia podzemnej vody a jeho simulácie na území Čenkovskej nivy na Slovensku za minimálne antropogénne narušených prírodných pod-mienok a kvázi ustáleného deficitného vodného režimu územia. Dátum simulácie bol vybraný ku dňu 29. 9. 1954 z dôvodov ustálenosti deficitného vodného režimu záujmového územia, ako aj nízkeho vodného stavu Du-naja a existencie riešenia príbuznej problematiky inými metódami v minulosti (Duba, 1964, 1968) a tým aj do-stupnosti údajov potrebných pre modelovanie. Pri riešení problematiky bol použitý model konečných diferencií a jeho modulárny počítačový program, pôvodne publiko-vaný McDonaldom a Harbaughom (1988), v tvare ne-skorších modifikácií a doplnkov.

Severná až západná hranica modelovanej oblasti je zvolená vzhľadom na ohraničenie hydrogeologického kolektora nivy od severu vyšším starowürmským teraso-vým stupňom. Južnú až východnú hranicu tvorí rieka Dunaj (obr. 2). Topografia povrchu terénu bola pomocou máp mierky 1 : 10 000 spracovaná do digitálneho tvaru. Hustota výškových bodov (spon 50 x 50 m) odpovedá množstvu buniek výpočtovej siete, pričom každý stred povrchu bunky v 1. vrstve reprezentuje jeden bod mapy topografie povrchu terénu (obr. 10). Vyššie opísaný hydrogeologický systém bol pre potreby modelovania pretransformovaný do trojvrstvového systému, ktorý je zložený z vrchnej pokryvnej pôdnej vrstvy (1. vrstva), vrstvy stredných až jemných pieskov (2. vrstva) a z vrst-vy piesočnatých štrkov až hrubých pieskov so štrkom


183

(3. vrstva), (obr. 7). Orientácia siete bola zvolená v sme-re generálneho toku podzemnej vody, pričom súradnico-vé osi x, y, z sú približne paralelné s hlavnými osami hy-draulickej vodivosti.

Prúdenie podzemných vôd má vždy v určitej miere charakter neustáleného prúdenia. Vyplýva to z prírod-ných podmienok napájania a odvádzania podzemných vôd. Ak sa však podmienky napájania a odvádzania podzemných vôd menia v čase nepatrne, prúdenie je kvázi ustálené a prakticky reprezentuje určitý hranič- ný stav.

Pre modelovanie tohto prípadu boli potrebné nasledu-júce filtračné parametre: horizontálna hydraulická vodi-vosť, koeficient prietočnosti, vertikálna hydraulická vodivosť, efektívna pórovitosť a koeficient vertikálnej netesnosti.

Model je rozdelený na moduly, ktoré dovoľujú študo-vať špecifické hydrologické charakteristiky oblasti. Mo-duly sú zgrupované do balíkov. Každý balík sa zaoberá špecifickou črtou simulovaného hydrologického systé-mu, alebo špecifickou metódou riešenia lineárnych rov-níc opisujúcich tokový system. V prezentovanom prípa-de boli použité nasledujúce moduly – tokové členy: základný balík, blokovo-centrálny tokový balík, výstup-ný kontrolný balík, evapotranspirácia, prítok vody do hladiny podzemnej vody, hraničná piezometrická výška a systém otvorených tokov.

Najlepšie výsledky z použitých metód riešenia sústa-vy rovníc dala metóda riešenia nazývaná Preconditioned Conjugate-Gradient 2 (PCG2).

Kalibrácia bola vykonaná pomocou špeciálneho počí-tačového programu PEST s manuálnym doladením nie-ktorých zón. Kalibračné výsledky pre stav ku dňu 29. 9. 1954 sú uvedené v tab. 1. Rozdiel medzi nameranými stavmi hladín podzemnej vody v sondách pozorovacej siete SHMÚ a vypočítanými stavmi hladín má maximál-nu hodnotu 0,17 m a koeficient regresie má hodnotu blízku jednotke, čo poukazuje na vysokú zhodu výsled-kov vypočítaných s nameranými.

Kalibrovaný model ustáleného prúdenia podzemnej vody bol verifikovaný pre nízky stav hladiny vody v Dunaji ku dňu 7. 8. 2002 aj pre vysoký stav Dunaja ku dňu 7. 5. 2000. Pre obidva prípady bola dosiahnutá veľmi dobrá zhoda nameraných a vypočítaných výšok hladiny podzemnej vody (pozri Burger, 2005a, 2005b).

Údaje o fluktuácii hladiny podzemnej vody ukázali, že základnými činiteľmi, ovplyvňujúcimi režim zmeny hladiny sú: atmosferické zrážky, brehový filtračný prie-tok z Dunaja a podzemný prítok z vyššej terasy na SZ a severnom okraji oblasti, ďalej evapotranspirácia, pod-zemný odtok do Dunaja (odvodňovací, drenážny účinok Dunaja) a odtok odvodňovacími kanálmi. Prezentované riešenie režimu podzemnej vody predstavuje odozvu na určitý podpriemerný, až nízky stav hladiny vody v Dunaji (ku dňu 29. 9. 1954), ktorý vzhľadom na pred-chádzajúce stavy možno pokladať za takmer ustálený (obr. 11). Na obr. 12 vidieť, že drenážny účinok Dunaja sa prejavuje od rkm 1745 po rkm 1733 a od rkm 1727 po

rkm 1722, čo predstavuje spolu 17 km brehovej filtrač-nej odvodňovacej funkcie rieky. Brehové filtračné napá-janie sa prejavuje od rkm 1733 po rkm 1727, čo je 6 km dĺžky brehu Dunaja. Čiara priameho drenážneho vplyvu Dunaja ide od Moče po hrane vyššieho terasového stup-ňa až po obec Búč, kde sa stáča na juh zhruba po usad-losť Mária, odkiaľ pokračuje juhovýchodným smerom po rieku v lokalite Čenkova. Druhá drénovaná oblasť je ohraničená od juhu od zaústenia Obidského kanála do Dunaja po Štúrovo a od severu záchytným kanálom „Pod kopanicami”. Napájací účinok rieky je znižovaný od-vodňovaním dolnými časťami Mužlianskeho potoka a Obidského kanála. Ostávajúcu plochu záujmového úze-mia odvodňuje sústava odvodňovacích kanálov (Obid-ský, Krížny a Búčsky kanál). Za modelovaného nízkeho vodného stavu Dunaja nedochádza v území k prúdeniu podzemnej vody s napätou hladinou, ako sa to predpo-kladalo (pozri Duba, 1964) pri vysokých stavoch hladiny vody v rieke. Na obr. 13 je 3-D vizualizácia celého ko-lektora s vektormi filtračnej pórovej rýchlosti a na obr. 14 je 3-D vizualizácia kolektora s trajektóriami pohybu častíc v prúdovom poli. V druhej vrstve kolektora sú najvyššie filtračné rýchlosti pozdĺž Obidského kanála a Mužlianskeho potoka a pri ich zaústení, taktiež pozdĺž brehu Dunaja (drénovanie) okolo rkm 1742. Podobne v tretej vrstve sú najvyššie filtračné rýchlosti pozdĺž otvo-rených tokov a pozdĺž brehu Dunaja od rkm 1744 po rkm 1742 a okolo rkm 1729. Rozsah hodnôt filtračnej rýchlosti v celom kolektore je od 0,0 m s-1 do max. 1,19E-03 m s-1. Trasovanie pohybu častíc sa používa na trasovanie smerov prúdenia podzemnej vody, čiže na vytvorenie prúdových čiar pomocou trasovania pohybu nekonečne malých imaginárnych častíc umiestnených v prúdovom poli. Trajektórie pohybu častíc poukazujú na prevažne drenážnu funkciu Dunaja, ako aj potokov a odvodňovacích kanálov na záujmovom území Čenkov-skej nivy. V úseku od obce Moča až po osadu Čenkov prúdi podzemná voda do Dunaja okrem severnej časti, z ktorej prúdi smerom do odvodňovacích kanálov. Od zaústenia Obidského kanála do Dunaja (rkm 1727) až po Štúrovo prúd podzemnej vody smeruje znovu do Dunaja.

Za účelom kontroly výsledkov simulácie sa počíta ob-jemová vodná bilancia pre celý model na konci výpočto-vej doby. Indikuje akceptovateľnosť numerického rieše-nia. Technika numerického riešenia je založená na sys-téme rovníc riešených modelom pri zachovaní rovnice kontinuity pre každú bunku. Kontinuita musí byť zacho-vaná aj pre celkový tok dnu a von z modelu, alebo sub-regiónu vrstvy. Rozdiel vo vodnej bilancii záujmového územia je 0,00 % (tab. 2). Rozdiel by mal byť v ideál-nom prípade menší ako 0,1 %, čo je splnené. Všeobecne sa akceptuje chyba do 1 % (Konikow a Bredehoeft, 1978).

Získané výsledky budú slúžiť ako referenčná báza pre numerické simulácie vytvorenia podzemnej priehrady, ktorá patrí medzi typy manažmentu umelého napájania pririečneho hydrogeologického kolektora, ako aj pre prognostické scenáre vývoja vodného režimu záujmové-

F. Burger

184

ho územia vzhľadom na scenáre klimatických zmien v tomto regióne. Zoznam symbolov x, y, z – kartézske súradnice v smere hlavných osí hydrau-

lickej vodivosti Kxx, Kyy, Kzz, Kxx, Kyy, Kzz – hydraulická vodivosť v smere osí kartézskych

súradníc x, y, z, o ktorých sa predpokladá, že sú paralelné s hlavnými osami hydraulickej vodivosti [ L T-1],

h – piezometrická výška [L], W – objemový tok cez jednotkový objem, ktorý reprezentuje

zdroje a (alebo) prepady vody [T-1], Ss – súčiniteľ zásobnosti pórovitého materiálu [L

-1], t – čas [T].

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