**3. Methods and material**

#### **3.1 Modelling**

58 Studies on Water Management Issues

a markedly terraced step with relative height approximately 15 m and base 3 m above the Danube 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. Hydrogeological conditions of the terrace were proofed only by a few boreholes, after which hydraulic conductivity of gravel varies from 6.6E-05 m.s-1 (the Chotín village) up to 2.0E-03 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 contact as it comes up to the surface and it is taken away by the drainage channels. The ground-water level in the alluvia is mainly influenced by the surface stream of the Danube River and then on other side by water seeping down from an adjacent terrace and through precipitation. The ground-water flow direction according to bilateral relation of the Danube water level and

Fig. 4. Hydrogeological profile along the Danube bank (400x exceeded), (Duba, 1964)

Through the hydrological characteristics of the study area and the description of surface flows in an objective time it is necessary to concentrate on the Danube River, which has here first- rated importance. Slovak Danube river reach belongs to the upper part of middle part of the river. Danube is keeping its alpine character in Slovak reach, in its upper part it has considerable slope around 0.4%o, it is flowing in its own alluvia and it is creating multiple systems of river arms. Water stages are first of all dependent on the 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 September 1954 in RK 1742.9 (Radvaň nad Dunajom) was 105.20 m a.s.l. Other surface flows in study area are rather small and short and their discharges are low. The maximum occurs in spring months, and in summer their discharge is considerably 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 Búčský, the Kraviansky and the Krížny channel belong to the system, as well as the large amount of side drainage channels

ground-water level was either to the aquifer or to the Danube.

**2.1.2.3 Hydrology** 

without any name.

#### **3.1.1 Mathematical model of groundwater flow**

Three-dimensional groundwater flow of constant density through porous earth material may be described by a partial differential equation (McDonald, M.G. & Harbaugh A.W., 1988):

$$\frac{\partial}{\partial \mathbf{x}} \left( \mathbf{K}\_{xx} \frac{\partial \mathbf{h}}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( \mathbf{K}\_{yy} \frac{\partial \mathbf{h}}{\partial y} \right) + \frac{\partial}{\partial z} \left( \mathbf{K}\_{zz} \frac{\partial \mathbf{h}}{\partial z} \right) - \mathcal{W} = \mathbf{S}\_s \frac{\partial \mathbf{h}}{\partial \mathbf{t}} \tag{1}$$

Where

*x*, *y*, z are Cartesian coordinates in the direction of main axis of hydraulic conductivity *Kxx*, *Kyy*, *Kzz*, *Kxx*, *Kyy*, *Kzz* are the values of hydraulic conductivity in the direction of the axis of Cartesian coordinates *x*, *y*, *z*, which are assumed that they are parallel with major axis of hydraulic conductivity [L T-1], *h* piezometric pressure head [L], *W* volumetric flux per unit volume, which represents sources and (or) sinks of water [T-1], *Ss* specific storage of the porous material [L-1], and *t* time [T]. In general, *Ss*, *Kxx*, *Kyy*, and *Kzz* may be the functions of space (*Ss = Ss* (*x*, *y*, *z*) a *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*)) which means that equation (1) describes groundwater flow for unsteady conditions in a heterogeneous and anisotropic medium, provided that the principal axes of hydraulic conductivity are aligned with the coordinate directions. Equation (1), together with the specification of flow and (or) head conditions on aquifer boundaries and specification of initial head conditions, creates a mathematical model of groundwater flow. A solution of equation (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 equation (1), the equation and its initial and boundary conditions are satisfied. Besides these very simple systems, it is possible to reach an analytical solution of equation (1) only rarely, so therefore it is necessary to use numerical methods for solution. One of the methods is the finite difference method, where the continuous system of equations (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 an approach leads to the system of linear algebraic differential equations. Values of head in specific points in time are obtained by their solutions. These values represent approximation of the time-variable distribution of piezometric head, which could have been obtained by analytical solution of equation (1).

#### **3.1.2 Three-dimensional modular model of groundwater flow" MODFLOW"**

The finite difference model originally published by McDonald & Harbaugh (1988), in the form of later modifications and addendums, and its modular computer program was utilized by the solution of the mentioned task. The modular structure consists of the "main program" and a series of independent subroutines called "modules". The explanation of physical and mathematical concepts, on which the model is based, and an explanation on how the modules are implemented into the structure of computer program, is listed in detail in the mentioned work. Ground-water flow in hydrogeological ground-water body is

Change of Groundwater Flow Characteristics After Construction of the

**3.1.3.3 Discretization in the space and time** 

**3.1.3.4 Filtration parameters of the aquifer** 

**3.1.4 Calibration and verification of the model** 

cells are:

Δ*x* = Δ

September 1954.

Waterworks System Protective Measures on the Danube River – A Case Study in Slovakia 61

Hydrogeological systems are divided into a mesh of blocks called cells, the locations of which are described in terms of rows, columns, and layers. Footprint dimensions are picked so that the whole area of the Čenkov plain is covered with a smooth overlay. Dimensions of

columns); 3 layers. Grid orientation was picked in the direction of the general groundwater flow and coordinate axis *x*, *y*, *z* are approximately parallel to the main hydraulic conductivity axis. Groundwater flow has always had a certain measure of unsteady flow. This results from natural conditions of recharge and drainage of groundwater. However, if the recharge and drainage groundwater conditions are changing in the time slightly, the flow is quasi-steady and practically represents certain boundary status. From the modelling target point of view a steady status of groundwater flow was considered to the date 29

Following filtration parameters were necessary for the modelling of this case: horizontal hydraulic conductivity, transmissivity, vertical hydraulic conductivity, effective porosity and coefficient of vertical 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 vary from 7.48E-07 m s-1 to 3.99E-03 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 modelling applications the usual ratio of the horizontal to vertical hydraulic conductivity is from 1 to 10 (Anderson & 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. *Effective porosity* is the feature of an aquifer to receive and to send out fluid in order to build hydrostatic pressure in the layer and through to the groundwater level. Quantitatively it is expressed by the coefficient of flexible storage and coefficient of free water level storage. The value of the coefficient of the free water level storage depends on hydraulic conductivity and also on grain size distribution of sediments 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 the work of Mucha & Šestakov (1987). *Vertical leakage* is required in the case of multiple layers groundwater body and represents the resistance to the water leakage at adjacent layers.

Calibration is a process, when the initial input model parameters are adjusted until output (dependent) model parameters at most approach the values measured in the terrain. Calibration of the model is an inverse-model process, i.e. the problem of parameter estimation is an inverse problem. Calibration of the model or the inverse model process could be performed either repetitively, either on a manual basis by way of trial and error, or by using a special computer program. The calibration was executed by the means of the special computer program PEST with manual tuning of some zones. Calibration results for

*y* = 50 m. Geometry of the model is: 22.5 km x 6.5 km, (130 rows and 450

simulated by the use of a finite difference block-central method. The solution of systems of simultaneous linear equations is possible to obtain by various methods.
