Acta hydrotechnica 37/67 (2024), Ljubljana                                                                                       Open Access Journal 
ISSN 1581-0267                       Odprtodostopna revija 
173 
UDK/UDC: 519.87:627.13(282)(497.6) Prejeto/Received: 29.01.2025 
Izvirni znanstveni članek – Original scientific paper Sprejeto/Accepted: 04.03.2025 
DOI: 10.15292/acta.hydro.2024.10 Objavljeno na spletu/Published online: 24.03.2025 
ESTIMATING THE DEFORMATION OF A RIVER BED USING DIMENSIONLESS 
GEOMETRIC PARAMETERS AND MATHEMATICAL MODEL  
OCENA DEFORMACIJE REČNE STRUGE Z UPORABO BREZDIMENZIJSKIH 
GEOMETRIJSKIH PARAMETROV IN MATEMATIČNEGA MODELA 
Nerma Lazović1, Mario Krzyk2 
1Faculty of Civil Engineering, University of Sarajevo, Patriotske lige 30, 71000 Sarajevo, Bosnia and Herzegovina 
2Fakulteta za gradbeništvo in geodezijo, Univerza v Ljubljani, Jamova cesta 2, 1000 Ljubljana, Slovenia 
Abstract 
The river channel cut into an alluvial substrate constantly adapts to changes in water flow and sediment 
transport. This paper presents a new approach to determine the magnitude of channel deformation based on 
the definition of dimensionless spatial parameters of channel deformation. The new approach’s applicability 
is evaluated by experimental studies on the Željeznica River in the Sarajevo Field in central Bosnia and 
Herzegovina. The morphological changes of the channel over a period of ten years were analyzed using 
dimensionless parameters of the channel geometry. A numerical analysis of changes in the channel of the 
Željeznica River was carried out using a mathematical model and a selected equation for sediment transport. 
Sensitivity analysis of the model parameters for channel deformation and statistical reliability analysis of the 
numerical model showed a good agreement between the modeled and observed values of channel deformation 
parameters during the analyzed period. 
Keywords: river morphology, dimensionless geometric parameters, river regime, sediment transport, channel 
deformation, HEC-RAS, Željeznica River, model reliability. 
Izvleček 
Struga reke, vrezana v aluvialno podlago, se nenehno prilagaja spremembam vodnega toka in transporta plavin. 
V prispevku je predstavljen nov pristop k določanju velikosti deformacije struge, ki temelji na opredelitvi 
brezdimenzijskih prostorskih parametrov deformacije struge. Ocena uporabnosti novega pristopa temelji na 
eksperimentalnih študijah na reki Željeznici na območju Sarajevskega polja v osrednjem delu Bosne in 
Hercegovine. Morfološke spremembe struge v desetletnem obdobju so analizirane na podlagi 
brezdimenzijskih parametrov geometrije struge. S pomočjo matematičnega modela in izbrane enačbe za 
transport plavin je izvedena numerična analiza sprememb v strugi reke Željeznice. Analiza občutljivosti 
parametrov modela deformacije rečne struge in analiza statistične zanesljivosti numeričnega modela sta 
 
2 Stik / Correspondence: mario.krzyk@fgg.uni-lj.si  
© Lazović N., Krzyk M.; This is an open-access article distributed under the terms of the Creative Commons Attribution – 
NonCommercial – ShareAlike 4.0 Licence. 
© Lazović N., Krzyk M.; Vsebina tega članka se sme uporabljati v skladu s pogoji licence Creative Commons Priznanje avtorstva – 
Nekomercialno – Deljenje pod enakimi pogoji 4.0.  
 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
174 
pokazali dobro ujemanje med modeliranimi in opazovanimi vrednostmi deformacije rečne struge v 
analiziranem obdobju. 
Ključne besede: rečna morfologija, brezdimenzijski geometrijski parametri, rečni režim, transport plavin, 
deformacija struge, HEC-RAS, reka Željeznica, zanesljivost modela. 
 
1. Introduction 
Investigating the deformation of river channels, 
understanding the processes that influence 
deformation and assessing the stability of river 
banks have long been areas of research for 
geologists and engineers. Classical contributions 
laid the foundation for today's studies on and 
practices regarding river channel deformation and 
management. The first investigations began in the 
second half of the nineteenth century (Gilbert, 1877; 
du Boys, 1879; Davis, 1889; Lindley, 1919; Shields, 
1936; Lane, 1937; Mackin, 1948; Leopold and 
Maddock, 1953; Schumm, 1969) (Singh, 2003). 
The theories of hydraulic geometry, which describe 
the relationships between river channel features and 
water flow, were significantly advanced by the work 
of Leopold and Maddock (Buffington, 2012). These 
theories help predict how changes in water flow and 
sediment load affect the geometric characteristics of 
river channels. The integration of geomorphology 
and hydraulic engineering has been crucial to the 
understanding of river processes. This 
interdisciplinary approach has become widely 
accepted in modern river engineering. 
Rivers change their channel geometry (longitudinal 
profile, cross-sections and channel shape in 
horizontal projection) depending on the geological, 
hydrologic and hydraulic characteristics of the 
watercourses and the catchment area, as well as on 
the characteristics of the sediments (volume, 
material properties of the sediments, granulometric 
properties). Natural and anthropogenic factors 
influence fluvial processes and riparian morphology 
(Figure 1). Flow regime and sediment quality, along 
with valley characteristics, are among the most 
important natural factors determining boundary 
conditions. (Ibisate et al., 2011). Natural processes 
can proceed extremely slowly and be practically 
imperceptible to humans (Minh Hai et al., 2019, 
Simon, 1989). In contrast, human activities such as 
damming rivers, building reservoirs, constructing 
dikes, and dredging waterways can significantly and 
rapidly affect natural processes and trends. In these 
cases, the time scale for river adaptation is reduced 
(Minh Hai et al., 2019, Rinaldi and Simon, 1998, 
Zheng et al., 2019). The interaction between these 
parameters or processes leads to new river dynamics 
and forms (Lane, 1955). 
The precipitation regime, the frequency and 
intensity of precipitation, and, in particular, extreme 
events and periods of drought determine runoff 
formation and erosion capacity, and thus sediment 
input and transport. The type and density of 
vegetation, soil type and land use depend on 
climatic conditions, which influence runoff 
dynamics and sediment movement and transport 
(Ibisate et al., 2011). 
Natural streams that are cut into the alluvial bedrock 
undergo the following processes: deposition of 
bedload and suspended sediments at low discharge 
and destabilisation of the riverbed and the deposited 
suspended sediments by their reintegration into the 
flow at high discharge. The local hydrodynamic 
conditions of the stream and the characteristics of 
the sediments can cause various combinations of 
processes. Depending on the stream’s current 
hydrologic condition, one of these processes may be 
more pronounced than the others, but always with 
the additional influence of other parameters that 
make analyzing the processes even more difficult. 
Over time, the methodology for calculating the 
parameters that characterize the sediment transport 
has evolved. From the first, rather crude predictions 
based on empirical and semi-empirical equations, 
mathematical models are now being developed that 
should be able to predict the concentration of 
suspended sediments and the balance of load in 
space and time using basic differential equations. 
Recently, hydro-morphodynamic processes in river 
channels have been studied in detail using 
numerical models (Krzyk, 1997, Martínez-Aranda, 
2019, Mahdizadeh and Sharifi, 2021, Stipić et al., 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
175 
2022, Peng et al., 2022, Chang et al., 2024). 
Considering that the changes in the physical 
properties of sediments due to mechanical forces are 
negligible, sediments behave like conservative 
materials. However, they should be considered as 
non-conservative materials because their 
concentration varies spatially and temporally, 
depending on the conditions for the deposition or 
flushing of the material from the bottom and its 
displacement (Krzyk, 1997).  
There are various calculation and measurement 
methods for determining sediment transport. 
Despite the availability of mathematical models, 
calculation and measurement devices, it is still 
difficult to accurately determine the amount of 
transported material and the changes in the 
configuration of a channel’s bed. The calibration of 
larger and more accurate mathematical models also 
requires a range of input data that can only be 
obtained through more complex and accurate 
measurements. The development of suitable 
measurement techniques is therefore also important. 
Furthermore, the experience of other researchers 
with similar problems shows that the process of 
calibrating mathematical models is very time-
consuming. In some cases, this process has taken 
more than ten years. Given the quality of the 
available measurement data on the hydrodynamic 
characteristics of flow, sediment transport, turbidity 
and bedload changes, it makes sense to select the 
complexity of a mathematical model. This makes it 
possible to avoid the use of parameters for which 
there is no validated data but for which more 
complex mathematical models are required. 
Many researchers have endeavored to equate the 
morphological responses of rivers with the 
corresponding driving factors. Minh Hai et al. 
(2019) established an empirical model by 
evaluating the spatiotemporal long-term variations 
of morphological characteristics of the lower 
reaches of the Tedori River in Japan. The results 
showed that sediment removal and dam 
construction strongly influenced the morphological 
features of the lower Tedori River and that the river 
channel’s vertical adjustment went through five 
phases.
 
 
Figure 1: The interaction of the main factors on the morphology of the riverbed (Ibisate et al., 2011). 
Slika 1: Medsebojni vpliv glavnih dejavnikov na morfologijo rečne struge (Ibisate et al., 2011).  
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
176 
Tang et al (Tang et al., 2021) analyzed the long-term 
(1977–2006) riverbed deformation in the Lower 
Atchafalaya River, the largest tributary of the 
Mississippi River. The riverbed is degraded in its 
upper part, which is mainly influenced by the 
frequency of floods. The riverbed grows larger 
mainly in the last 35 km of its course, although less 
sediment is delivered there. Zheng et al. (2019) used 
data from four decades (1976–2015) of systematic 
surveys of water discharge, sediment load, cross-
sectional profiles, and water surface elevation. They 
showed that the backwater zone influences the 
development and discharge of the Qingshuigou 
Channel, a recent bulge in the Yellow River delta. 
Rahman et al. (2024) analyzed the characteristics of 
riverbed deformation, bank erosion, and channel 
changes in the lower Sangu River basin using a 
depth-averaged 2-D flow model in which suspended 
sediment transport dominates and flow 
characteristics are influenced by active tides. These 
studies have clarified the characteristics and 
mechanisms of morphodynamics in the lower 
Sangu catchment, particularly the potential of the 
combination of tides and floods to enhance riverbed 
deformation and associated bank shifts. Ma et al. 
(2022) investigated the morphological evolution 
characteristics based on annual comprehensive 
cross-sectional data and water and sediment data 
from the Xianyang, Lintong, and Huaxian 
hydrological stations in the lower Weihe River from 
2006 to 2018, as well as the response to water and 
sediment fluctuations. The results show that large 
floods impact the channel’s cross-sectional 
morphology. The fluctuations in the meander 
elevation of the main channel in the lower Weihe 
River are closely correlated with changes in the 
inflowing sediment coefficient. In addition, human 
activities such as river development projects can 
also have a direct impact on the cross-sectional 
morphology. 
Information on historical morphological changes is 
useful for understanding the evolutionary trends and 
condition of rivers. The integration of remote 
sensing and GIS techniques has contributed 
significantly to investigations into rivers’ 
morphological parameters by analyzing geospatial 
data (Beyene et al., 2023, Ibitoye, 2021, Momin et 
al., 2022). Langat et al. (2020) characterized the 
planform changes and floodplain dynamics of the 
Tana River system in Kenya between 1975 and 
2017 using an integrated methodological 
framework of remote sensing and geographic 
information system (GIS). The river has undergone 
significant changes in its channel morphology over 
this period of 42 years, indicating a very active and 
meandering nature with a generally increasing 
tortuosity in the downstream direction. After the 
river regulation, which took place between 1981 and 
1988, the annual lateral movement of the river 
decreased considerably. Andualem et al. (2024) 
developed a new framework for studying changes in 
overall channel morphology by using very high-
resolution aerial imagery and a LiDAR-derived 
digital elevation model (DEM). By digitizing 
channel boundaries with ArcGIS Pro 3.0 and 
analyzing various morphological parameters as well 
as erosion and deposition patterns, they investigated 
the impact of urban expansion and infrastructure 
development on channel adjustments. Đorđević et 
al. (2023) investigated the relationship between the 
observed decline in water level and the incision of 
the riverbed recorded at the gauging stations along 
a length of around 300 km on the Middle Danube. 
The study found that in the absence of systematic 
monitoring of sediment transport, regularly 
surveyed cross-sections of the navigable river can 
be used to estimate the long-term averaged sediment 
transport, i.e. to infer the average rates of channel 
aggradation and degradation along the reach. 
Considering the increasing frequency of flood 
events due to climate change and the exposure of 
watercourses to anthropogenic activities, 
understanding the magnitude of morphological 
changes in river channels is of great importance for 
effective water management and an invaluable tool 
for decision makers in the water management 
process. 
The paper presents new dimensionless parameters 
for the geometric characteristics of the river channel 
using the example of the Željeznica River in the 
Sarajevo Field region in Bosnia and Herzegovina. 
Due to the nature of the Želježnica River, which has 
a certain torrential character, its high discharges 
threaten the surrounding settlements with flooding. 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
177 
The Želježnica River is an important source for 
filling the underground reservoirs used for 
Sarajevo’s water supply. Therefore, it is very 
important to understand the morphological changes 
of its riverbed. 
 
2. Analysis of the changes in riverbed 
geometry 
2.1 Riverbed spatial deformation 
The analysis of morphological changes in river 
channels involves a comprehensive investigation of 
various parameters that influence channel 
dynamics. Parametric analysis focuses on 
quantifying aspects such as channel depth, width, 
and cross-sectional area to understand the riverbed’s 
evolution over time. Deepening of the channel, 
often caused by increased flow velocities or 
sediment transport, can lead to significant changes 
in cross-sectional area and influence hydraulic 
conditions. In addition, changes in the river’s 
horizontal alignment, such as meander migration or 
bank erosion, are crucial for assessing the river 
system’s stability. Topographic surveys are 
becoming increasingly affordable and are possible 
with higher spatial resolution and over larger spatial 
extents. The digital elevation models (DEMs) 
created from such surveys are used to create DEM 
of Difference (DoD) maps and to estimate the net 
change in storage conditions for morphological 
sediment budgets. 
 
2.2 Numerical modelling 
The goal of river modelling is to predict a river’s 
hydraulic behavior over a wide range of hydraulic 
conditions. Water flow and sediment transport in a 
natural river are part of a very complex natural 
process that is unstable, irregular, and highly 
turbulent, with characteristics that are constantly 
changing and varying in time and space. The 
hydraulic roughness of the river channel’s wetted 
contour also varies, and there is a multiphase 
composition to the flow (biphasic: mixture of water 
and sediment, or triphasic: mixture of water, 
sediment, and air). Due to the geometric 
characteristics, the solution to the equations 
describing the flow of water and substances can 
often be simplified in practice. The most common 
simplifications are the two-dimensional (2D) 
approach, where the problem is considered in the 
vertical or horizontal plane (as a width- or depth-
averaged model), and the one-dimensional (1D) 
approach (Krzyk, 1997, Krzyk and Četina, 2003).  
Erosion processes in rivers are 3D phenomena and 
their intensity depends on the water's local flow 
velocities, which are homogeneous neither in the 
lateral nor in the vertical direction. The question is 
how this inhomogeneity affects the results of 
hydrodynamic and sediment transport models. 
Laboratory experiments were carried out at the 
Hydraulic Engineering Institute of the University of 
Innsbruck and then simulated with numerical 1D, 
2D, and 3D models (Glock et al., 2019) to obtain 
measurement data from the laboratory experiment. 
A standardised, relative comparison of the models 
shows that, after successful calibration to measured 
water levels, the corresponding 2D/1D and 3D/1D 
ratios are almost unity, while the bed shear stresses 
in the 3D models are about 62-86% of the simulated 
1D values and 90-100% in the case of 2D/1D. In the 
studies of morphological changes of the Željeznica 
river channel, a separate calibration procedure was 
performed for the hydrodynamic parameters and 
sediment transport. For this reason, the (1D) 
hydraulic mathematical model used can be justified. 
This approach has been proven for decades in 
numerical modelling of water and sediment flow for 
cases where the length of the reach is 20 times or 
more the channel width and where the transverse 
variations in flow velocity and water depth are not 
significant (Spasojević, 1996). 
Deformation of river channels is a spatially three-
dimensional phenomenon whose intensity largely 
depends on local velocities that are not 
homogeneous in either the transverse or vertical 
direction. However, since the main aim of this study 
was to develop a simple method that facilitates 
analysis of the general trend of riverbed 
deformation, which could be applied to a wider area 
and based on available data, the 1D hydraulic model 
was chosen. 
In hydraulic terms, applying a 1D model is justified 
on longer river sections where the cross-section 
length is 20 or more times greater than the channel 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
178 
width and where transverse changes in flow velocity 
and water depth are not significant (Spasojević, 
1996). 
The flow velocity is the fundamental parameter for 
mass transport in the form of a solution or solid 
particles. Therefore, precise knowledge of the 
hydrodynamic parameters such as water velocity 
and water depth is essential. Only when the 
hydrodynamic situation has been accurately 
calculated and confirmed by measurements can the 
modelling of mass transport begin. Based on the 
measurements, we need to determine the actual 
values of the coefficients in the transport equation 
(turbulent diffusion coefficients) that describe the 
movement of sediments, as it is often difficult to 
determine the order of these coefficients’ magnitude 
from the literature alone (Krzyk, 1997). 
In the analyzed example of the Željeznica River 
section, a one-dimensional approach was used to 
solve the problem due to the river’s geometric 
characteristics with a significantly greater length of 
the section compared to the width of the riverbed 
and the depth of the river. In linear (1D) hydraulic 
calculation models, the space is limited to the main 
axis of the river and the variation of the relevant 
parameters in the orthogonal direction to the river 
axis (i.e. in the river’s cross-sectional profile) is 
ignored. The calculation results are determined in 
each cross-sectional profile of the river: water level, 
mean profile velocity, etc. Considering the higher 
flow resistance due to vegetation in inundations 
compared to the main riverbed, the concept of a 
complex riverbed cross-section is normally used for 
hydraulic calculation in line flow models. This 
concept means that the calculation is performed 
with three adjacent 1D flows: in the basic riverbed, 
to the left, and to the right inundation. 
The numerical modelling was carried out using the 
HEC-RAS model (Chang, 1988). Two modules of 
the HEC-RAS model were used: the hydrodynamic 
module (HD) and the transport module (ST). In the 
HD module, the equations of one-dimensional (1D) 
unsteady flow in a non-prismatic channel, which 
describe the laws of conservation of mass and 
momentum (Saint-Venant equations), are solved 
numerically using an implicit finite difference 
scheme. The Preissmann scheme was used to 
calculate the wave propagation in a watercourse. 
The ST module uses the results of the HD module 
as a basis. The flow duration represents the duration 
of each constant flow value, which is further divided 
into calculation time steps, where for each time step 
the hydraulic volumes and sediment properties are 
calculated for each cross-section. Although the flow 
is constant over the time step of the flow duration, 
HEC-RAS calculates the channel geometry and 
hydrodynamic parameters after each time step. If 
the geometry changes rapidly and sudden erosion or 
sediment deposition occurs, the model may become 
unstable and the calculation time step must be 
shortened. The calculation time step is divided into 
calculations of the sediment transport capacity 
according to the granulometric properties of the 
sediments in the active layer. In one calculation 
step, the hydraulic properties and the transport 
potential are constant, but the transport capacities 
for different sediment grains vary, which leads to 
changes in the geometry of the cross-sections. The 
HEC-RAS software allows the user to select an 
empirical equation to calculate the sediment 
transport capacity. The equations available are 
Ackers–White, Engelund–Hansen, Laursen 
(Copeland), Meyer-Peter and Muller, Toffaletti, and 
Yang and Wilcock (Chang, 1988). 
The mathematical modeling of the general 
deformation of the riverbed is based on the 
numerical solution of the system of four equations 
(1)–(4): the continuity and momentum equation, 
continuity equation for the sediment and selected 
equations for sediment transport with the given 
initial and boundary conditions (Chang, 1988). 
𝜕𝑍
𝜕𝑡
+
1
𝐵
𝜕𝑄
𝜕𝑥
= 0          (1) 
𝜕𝑄
𝜕𝑡
+
𝜕
𝜕𝑥
(
𝑄2
𝐴
) + 𝑔𝐴
𝜕𝑍
𝜕𝑥
+ 𝑔 𝐴 𝐼𝑒 = 0          (2) 
(1 − 𝜆0)
𝜕𝐴𝑑
𝜕𝑡
+
𝜕𝑄𝑣
𝜕𝑥
= 0          (3) 
𝑞𝑣 = 𝑞𝑣(𝑍, 𝑄)          (4) 
The equations of this system are linked by the 
dependent variables Z(x,t) and Q(x,t) and the 
elevations of the bed Zd(x,t), which are implicit in 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
179 
the geometry of the profile and the surface Ad(x,t), 
where: 
Z(x,t) and Q(x,t) are the water level and the 
discharge in space and time, 
A(x,t) is the cross-sectional area, and 
Ad(x,t) is the cross-sectional deformation area. 
The change in the area (Ad) in the time interval Δt is 
obtained by solving the continuity equation for the 
sediment: 
(∆𝐴𝑑)𝑖 = 
= −
∆𝑡
1 − 𝜆0
·
[
𝑄𝑣
𝑘 + 𝑄𝑣
𝑘+1
2
]
𝑖
− [
𝑄𝑣
𝑘 + 𝑄𝑣
𝑘+1
2
]
𝑖−𝑖
∆𝑥𝑖−1
 
(5) 
The type of deformation is determined by the sign 
of the calculated change: 
(∆𝐴𝑑)𝑖 > 0 sedimentation and 
(∆𝐴𝑑)𝑖 < 0 erosion. 
In the mathematical model used, the system of 
equations is solved step by step: first, the 
hydrodynamic equations are solved to obtain the 
water depths and velocities of the analyzed flow, 
and then the sediment transport is calculated in all 
profiles on the same time scale using the chosen 
formula Qv = Qv (Z, Q). Finally, the Exner equation 
is applied to determine the change in profile 
geometry through the values of ΔAd (or Δzd). The 
results obtained describe the propagation of the 
water waves and the "sediment waves". 
The results of the numerical calculations carried out 
with the mathematical model were used as the basis 
for analyzing the morphological changes of the river 
channel in the lower reaches of the Željeznica River. 
The deepening and widening of the riverbed were 
identified as changes to the riverbed in the section 
under consideration.  
 
3. Case study 
3.1 Location 
On the lower course of the Željeznica River in the 
Sarajevo area, which is characterized by an unstable 
and changing riverbed, studies were carried out on 
the riverbed’s deformation. The 10-year period 
from 2009 to 2019 was analyzed. The Sarajevo 
Field, located at the foot of the Igman and 
Bjelašnica mountains in central Bosnia and 
Herzegovina (Figure 2), plays an important role in 
Sarajevo’s water supply. The Željeznica River is the 
main source enriching the underground water 
reservoir. Morphological changes in the Željeznica 
River influence the amount of water infiltrated into 
the ground. Vertical movements of the riverbed 
have the greatest impact (Spasojević, 1996, Lazović 
et al., 2023). 
 
3.2 Hydrologic data 
The closest gauging station to the Željeznica section 
under consideration is the Ilidža gauging station, 
which is located 200 m downstream from the area 
under investigation. Hourly water level 
observations were carried out at the Ilidža gauging 
station in the period from 2009 to 2019. The hourly 
discharge values were determined based on the 
established flow curves. These data were used for 
statistical data processing, in which the maximum, 
minimum, average daily and monthly discharges, 
and the flow duration curve were determined. 
The average annual discharge of the Željeznica 
River at the Ilidža gaging station is 8.65 m3/s for the 
period 2009–2019. The maximum discharge was 
measured on 01 December 2010 at 8 p.m. and 
amounted to 272.50 m3/s. 
 
3.3 Geodetic data 
Geodetic surveys of the channel of the Željeznica 
River in the considered section were carried out in 
2009, 2018, and 2019. The geodetic surveys were 
carried out along cross-sections, which were located 
in the same places each time. 36 cross-sections of 
the channel were created for the considered section 
with a maximum distance of about 60 m between 
them (Figure 3). 
The morphological changes of the Željeznica 
riverbed in the area and period under consideration 
were analyzed by superimposing the same cross-
sections from different years. With the help of 
AutoCAD, CAD – Earth and Civil 3D tools, 
information on the changes of the channel was 
obtained in all three spatial directions: in the 
horizontal plane, on the cross-sections, and on the 
longitudinal profile of the channel. 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
180 
 
Figure 2: Location of experimental investigations area. 
Slika 2: Lokacija območja eksperimentalnih raziskav. 
The data on the configuration of the river channel at 
different time intervals form the basis for the 
evaluation of its deformation. The configuration of 
the Željeznica riverbed in the considered section 
was digitally represented by the Digital Terrain 
Model (DTM). It was created based on a 
Triangulated Irregular Network (TIN) created from 
the results of geodetic measurements. In order to 
assess the deformation of the Željeznica riverbed 
between 2009 and 2019, raster-based 1 m resolution 
digital terrain models (DTM) were created using 
surveyed geodetic points obtained from the riverbed 
terrain, which are shown in Figure 4. The AutoCAD 
– Civil 3D tool was used to process and analyze the 
changes in river morphology. 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
181 
 
Figure 3: The section of the Željeznica 
River under investigation for the years 
2009 (blue) and 2019 (red) with defined 
cross-sections. 
Slika 3: Obravnavani odsek reke 
Željeznice za leti 2009 (modra) in 2019 
(rdeča) z označenimi prečnimi profili. 
  
Figure 4: Digital terrain models (DTM) of the Željeznica riverbed for 2009 and 2019. 
Slika 4: Digitalna modela terena (DTM) struge reke Željeznice za leti 2009 in 2019. 
3.4 Analysis of the spatial deformation of cross-
section 
Hydraulic geometry refers to the interrelationship 
between rivers’ water discharge, sediment 
discharge, stream width, depth, velocity, and 
planform (Chang, 1988). In order to observe the 
morphological changes to the channel over time, 
three dimensionless parameters were defined to 
measure the deformation: 1. the change in channel 
bed height as deepening/increase in relation to the 
water depth in the channel (∆z/h), 2. the change in 
cross-sectional area of the channel (A1/A2), and 3. 
the horizontal displacement of the channel axis in 
relation to the channel width (∆L/B) (Lazović et al., 
2023). The definition and graphical description of 
Bottom level from – 
to (m asl) 
Colour 
500.00 501.50  
501.50 503.00  
503.00 504.50  
504.50 506.00  
506.00 507.50  
507.50 509.00  
509.00 510.50  
510.50 512.00  
512.00 513.50  
513.50 514.00  
514.00 516.50  
516.50 518.00  
518.00 519.50  
519.50 521.00  
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
182 
the spatial deformation elements of the river 
channel are shown in Figures 5a, 5b, and 5c. 
Figure 5a: Rate of vertical movement of a riverbed 
bottom (∆z/h). 
Slika 5a: Mera vertikalnega premika dna struge 
(∆z/h). 
Figure 5b: Rate of change of the cross-sectional 
area of a channel (A1/A2). 
Slika 5b: Mera spremembe ploščine prečnega 
prereza struge (A1/A2). 
 
 
Figure 5c: Rate of horizontal displacement of the 
riverbed axis (∆L/B). 
Slika 5c: Mera horizontalnega premika osi rečne 
struge (∆L/B). 
 
4. Numerical modelling 
After conducting field research on the 
morphological changes of the riverbed in the lower 
course of the Željeznica River, numerical 
investigations into the morphological changes to the 
riverbed were carried out. The results of the field 
investigations were used for the setup, input data, 
calibration, and verification of the numerical 
investigations. The mathematical model simulated 
the elevation changes to the riverbed bottom, i.e. the 
deepening and raising of the riverbed bottom. 
Since the field investigations showed that in the 
period from 2009 to 2019 there was a horizontal 
shift of the axis of the Željeznica River in the section 
between profiles P19 and P24, the section from 
profile P2 (0+63.60) to profile P19 (1+052.60) was 
selected for the numerical modeling of the 
morphological changes of the riverbed. In the 
section P2–P19, there were neither significant 
changes in the geometry of the cross-sections nor 
horizontal shifts in the river axis during the period 
under consideration. 
The calibration of the HEC-RAS model includes the 
calibration of the hydrodynamic model (HD) and 
the sediment transport model (ST). For the 
calibration of the hydrodynamic model, the water 
levels calculated with the HEC-RAS software were 
applied to the cross-sectional profiles of the sections 
under consideration and compared with the 
measured water levels of the automatic water level 
station in Ilidža. In addition, the level of the flood 
line Q = 102.82 m3/s was used, which was registered 
on 14 May 2014 at 21:00. The calibration parameter 
of the HD model was the Manning's roughness 
coefficient, which takes into account the friction due 
to the absolute roughness of the channel bed as well 
as a number of other factors that contribute to the 
overall friction, such as: sediment formations, 
variable channel cross-section, structures in the 
channel, vegetation, and meander formation. The 
friction coefficient used can be represented as the 
sum of the partial friction coefficients (Jovanović, 
2008): 
𝑛𝑔 = ∑(𝑛𝑔𝑖) · 𝑐𝑚                (6) 
where: 
𝑛𝑔𝑖 is the partial coefficient of friction,  
𝑐𝑚 is the coefficient of meandering. 
The HEC-RAS Version 5.0 software allows the 
modelling of water and sediment transport using the 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
183 
formulae (1)–(4). When numerical simulations were 
carried out for the various empirical equations 
chosen for the sediment transport calculations, 
considerable discrepancies were found between the 
results. The model results and the modelled channel 
bottom were compared with the measured geodetic 
data of the channel bottom. The best agreement 
between the model and the natural conditions was 
found when using the empirical Ackers-White 
equation to calculate sediment transport. The 
general form of the transport equation for the 
Ackers-White function for a single grain size is 
expressed in formulae as follows (Ackers, 1973): 
𝑋 =
𝐺𝑔𝑟𝑠𝑑𝑠
𝐷 (
𝑢∗
𝑉 )
𝑛 
𝐺𝑔𝑟 = 𝐶 (
𝐹𝑔𝑟
𝐴
− 1)
𝑚
               (7) 
 
𝐹𝑔𝑟 =
𝑢∗
𝑛𝑉1−𝑛
√𝑔(𝑠−1)𝑑[√32 log (𝛼𝑑𝑠/𝐷]
1−𝑛, 
where: 
𝑋 – sediment flux, 
𝐺𝑔𝑟 – dimensionless sediment transport rate, 
𝑠 – relative sediment density, 
𝑑𝑠 – mean particle diameter, 
𝐷 – effective depth, 
𝑢∗ – shear velocity, 
𝑉 – mean channel velocity, 
𝑛 – transition exponent, depending on sediment 
size, 
𝛼, 𝐶 – coefficient, 
𝐹𝑔𝑟 – particle mobility parameter, 
𝐴 – critical particle mobility parameter, and 
𝑚 – exponent in the general function. 
The results of the mathematical model of sediment 
transport are shown in the longitudinal section in 
Figure 6. The longitudinal sections for 2009 and 
2019 are presented, as well as the results of the 
mathematical model simulating a 10-year period.
 
Figure 6: Longitudinal profile of the deepest points of the modelled section of the Željeznica River using the 
empirical Ackers-White equation to calculate sediment transport (green) and the recorded channel point in 
2009 (blue) and 2019 (red). 
Slika 6: Vzdolžni profil modeliranega odseka reke Željeznice po najglobljih točkah prečnih profilov 
izračunanega z upoštevanjem Ackers-Whiteove enačbe za izračun transporta plavin (zelena) in posneta 
podolžna profila v letih 2009 (modra) in 2019 (rdeča).  
 
 
499.5
500
500.5
501
501.5
502
502.5
503
503.5
504
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
B
o
tt
o
m
 l
ev
el
 (
m
 a
.s
.l
.)
Cross-section
2009
2019
A-W
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
184 
5. Sensitivity and reliability analysis of the 
mathematical model 
The natural process of channel deformation is 
described mathematically by a system of equations 
that describe the flow of water and sediment. As part 
of the sensitivity analysis, the parameters of the 
selected empirical Ackers-White equation for the 
calculation of the sediment transport (ST model) 
were analyzed: the coefficients n, A, m, and C of the 
selected empirical Ackers-White equations for the 
prediction of sediment transport (ST model). 
In the selected empirical Ackers-White equation, 
the coefficients n, A, m, and C, which are a function 
of the size of the mean grain size of the gravel ds, 
are determined for the calculation of bedload 
transport. By analyzing the local sensitivity of the 
coefficients n, A, m and C, it was determined which 
of the coefficients has the greatest variation for the 
different values of the mean grain size of the 
sediment, ds. 
In addition, a local sensitivity analysis of the 
empirical Ackers-White equation for the calculation 
of bedload transport to the variation of the mean 
sediment grain size ds and to the other hydraulic and 
hydromorphological parameters of the river 
specified in this equation was carried out.  
The results of the local sensitivity analysis (Figure 
7) of the coefficients n, A, m, and C of the Ackers-
White equation to the variation of the mean grain 
size ds in the range of 0.05–1 mm are as follows: 
n varies in the interval 0.23–0.96; 
A varies in the interval 0.19–0.35; 
m varies in the interval 1.96–7.42; 
C varies in the interval 0.0004–65.80. 
As can be seen from Figure 7, the coefficient C for 
different values of the mean sediment grain size ds 
shows the greatest variation compared to the other 
coefficients. Since the coefficient C in the Ackers-
White equation is directly proportional to the 
amount of bedload transport, it follows that bedload 
transport is equally 'sensitive' to a change in the 
input value of C, as can be seen in the Spider 
diagram in Figure 8. 
 
Figure 7: Results of the local sensitivity analysis – variation of the coefficients n, A, m, and C of the Ackers-
White equation for the values of the mean grain ds.  
Slika 7: Rezultati lokalne občutljivostne analize – spremembe koeficientov n, A, m in C Ackers-Whiteove 
enačbe za vrednosti srednjega zrna ds.
 
 
 
 
 
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
V
al
u
es
 o
f 
co
ef
fi
ci
en
ts
 a
, 
n
, 
A
, 
m
, 
C
Mean particle diameter ds (mm)
n
A
m
C
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
185 
6. Results and discussion 
Figure 6 shows that, along the P2–P19 section of the 
Željeznica River, the riverbed was 
lowered/deepened between 2009 and 2019. The 
largest deepening of the riverbed is observed on 
profile P6 (∆z = -0.7 m). The deepening of the river 
channel is the result of erosion processes caused by 
combined hydrological and hydraulic influences, 
especially high river flows. 
By comparing the DTMs of the river channel from 
2009 and 2019 and analyzing the results of the 
comparison, the magnitude of deformation of the 
river channel was determined at each cross-section 
of the considered section of the Željeznica River. 
Dimensionless spatial channel deformation 
parameters are introduced to comprehensively 
evaluate the geometric changes in the cross-
sectional profiles of the river channel and to 
determine the magnitude of the channel deformation 
on this basis. For the analysis, geometric 
coefficients were used to express the ratios of 
geometric changes of each cross-section. The 
vertical bed displacements (∆z/h) of each cross-
section of the considered section of the Željeznica 
River are shown in Figure 9 and modelled and 
measured deformation parameters A1/A2 of cross-
sections (P2–P19) on the Željeznica River are 
shown in Figure 10. 
It is obvious that the maximum vertical bed 
displacements are not correlated with the maximum 
changes in cross-sectional area. On average for all 
profiles, the cross-sectional area has not changed 
significantly during the period under consideration, 
but the bed of the river channel has deepened 
considerably. The horizontal displacements of the 
riverbed axis were also analyzed for the studied 
section of the Željeznica River. The results are 
shown in Figure 3. It can be observed that, in the 
cross-section profiles, where a pronounced 
horizontal displacement of the riverbed has taken 
place, a reduction in the width of the cross-section 
can be observed. This phenomenon can be 
interpreted by the fact that the river has incised a 
new channel, which requires a corresponding ability 
to erode the base material, which is achieved by the 
higher flow velocity in a narrower cross-section. 
The results of the local sensitivity analysis of 
bedload transport to changes in the river’s hydraulic 
and hydromorphological parameters show a higher 
sensitivity of sediment transport to depth and rate of 
change of water and a lower sensitivity to changes 
in the energy gradient and hydraulic radius. 
 
Figure 8: Spider diagram – results of a local sensitivity analysis of bedload transport, calculated with the 
empirical equation of Ackers-White (analytical expression) as a function of the variation of the mean particle 
diameter, the hydraulic and hydromorphological parameters of the flow. 
Slika 8: Pajkov diagram – rezultati lokalne občutljivostne analize transporta plavin, izračunane z empirično 
enačbo Ackers-White (analitični izraz) v odvisnosti od spremembe srednjega premera zrna ter hidravličnih in 
hidromorfoloških parametrov toka. 
0.00
30.00
60.00
90.00
120.00
150.00
180.00
210.00
240.00
270.00
300.00
330.00
360.00
0 10 20 30 40 50 60 70 80 90 100 110
B
ed
-l
o
ad
 t
ra
n
sp
o
rt
, 
Q
s 
[k
g
 /
s]
Input data change [%]
ds - mean grain size
I - energy gradient
R - hydraulic radius
V - mean channel velocity
h - depth
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
186 
 
Figure 9: Comparative presentation of modeled and measured deformation parameters Δz/h of cross-sections 
(P2–P19) on the Željeznica River. 
Slika 9: Primerjalni prikaz modeliranih in izmerjenih deformacijskih parametrov Δz/h prerezov (P2–P19) na 
reki Željeznici. 
 
Figure 10: Comparative presentation of modeled and measured deformation parameters A1/A2 of cross-
sections (P2–P19) on the Željeznica River. 
Slika 10: Primerjalni prikaz modeliranih in izmerjenih deformacijskih parametrov A1/A2 prerezov (P2–P19) 
na reki Željeznici.
Analysis of the reliability of the calculation and 
measurement results for the ∆z/h and A1/A2 
parameters for all individual transverse profiles and 
for the entire stretch was performed using two 
statistical methods: 
- mean absolute error (MAE) and 
- the root mean square error (RMSE). 
The formulas for calculation according to these 
methods are as follows: 
-0.500
-0.400
-0.300
-0.200
-0.100
0.000
0.100
0.200
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
∆
z/
h
Cross sections P2-P19
Model
Observed
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
A
1
/A
2
Cross sections P2-P19
Model
Observed
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
187 
𝑀𝐴𝐸 =  𝑁−1 ∑(𝑋𝑚 − 𝑋𝑜)
2
𝑁
𝑖=1
 
0 < MAE < 1 
𝑅𝑀𝑆𝐸 = √𝑁−1 ∑(𝑋𝑚 − 𝑋𝑜)
2
𝑁
𝑖=1
 
0 < RMSE < 1 
where: 𝑋𝑚 – model value; 𝑋𝑜 – observed value; 𝑁 
– number of observations. 
A better model accuracy is represented by a MAE 
and RMSE value closer to 0. 
The results of the analysis by indicator are shown in 
Table 1. 
Table 1: Accuracy indicators for the mathematical 
model. 
Preglednica 1: Kazalniki točnosti matematičnega 
modela. 
Indicator 
The value for the 
parameter ∆z/h 
The value for the 
parameter A1/A2 
"MAE" 0.002 0.004 
"RMSE" 0.034 0.063 
 
7. Conclusions 
This paper presents a method for determining and 
estimating the parameters that influence the 
magnitude of spatial deformation of a river channel. 
The focus is on identifying the elements of the river 
channel’s spatial deformation based on geodetic 
data of cross-sections of the river channel and 
numerical analyses of the process of river channel 
change over the considered time period. A 
hydrodynamic 1D model and a model for sediment 
transport were used for the analysis, which enables 
the analysis of longer sections of the watercourse. 
To ensure the quality of the mathematical model 
results, the hydrodynamic model and subsequently 
the sediment transport model were calibrated. Three 
dimensionless geometric coefficients were 
introduced to allow a more transparent analysis of 
the changes in the channel’s geometric 
characteristics. The relationships between the 
dimensionless coefficients were used to estimate the 
geometric change of the river channel over time 
caused by erosion or deposition of sediments and 
the displacement of the river channel in the 
horizontal plane. The proposed approach is an 
efficient method of determining the morphological 
changes to the river channel for various time periods 
on all geodetically surveyed cross-sections. 
Numerical investigations based on simulations with 
different empirical sediment transport equations 
showed that the best agreement between the 
measured and modelled bed elevations in the 
considered river section of the Željeznica River was 
achieved when the Ackers-White equation was used 
to calculate the sediment transport. The empirical 
Ackers-White equation used to calculate the 
sediment transport represents the functional 
dependence of the hydraulic flow parameters 
(tangential stress and gradient of the energy line) 
and the sediment characteristics. The dependence 
was specified for the changing hydrodynamic 
parameters along the section under consideration 
(for different geometric properties of the cross-
sections) and for different discharges. The 
knowledge of the spatial and temporal variations of 
the tangential stress and the gradient of the energy 
line is important for the assessment of the transport 
capacity and the prediction of the morphological 
changes of the river channel. 
The calculated values of MAE and RMSE of the 
deformation model of the Željeznica River show a 
very good agreement between the measured and 
simulated values of deformation parameters, i.e. 
good model reliability. 
In order to provide information on the possibility of 
generalizing the conclusions obtained with the 
proposed method of dimensionless geometric 
parameters to other watercourses, it is necessary to 
carry out similar analyses in further studies on 
watercourses with other hydrodynamic and 
morphological characteristics. Based on the 
analysis of the dimensionless geometric parameter 
values, the numerical modelling and the comparison 
of the results with the results of the geodetic 
measurements, the most important conclusions can 
be drawn: 
Lazović N., Krzyk M.: Estimating the deformation of river bed using dimensionless geometric parameters and 
mathematical model – Ocena deformacije rečne struge z uporabo brezdimenzijskih geometrijskih parametrov in 
matematičnega modela 
Acta hydrotechnica 37/67 (2024), 173–189, Ljubljana 
 
188 
1. For the studied section of the Željeznica River, 
the Ackers-White formula for estimating the total 
sediment transport based on the energy approach 
provides the best results of the sediment transport 
simulation. 
2. The coefficient C for the different values of the 
mean sediment grain size ds shows the greatest 
variation compared to the other coefficients. Since 
the coefficient C in the Ackers-White formula is 
directly proportional to the amount of sediment 
transport, it follows that the sediment transport is 
linearly dependent on the coefficient C. 
3. The maximum vertical bottom displacements are 
not related to the maximum changes in the wet 
cross-sectional area. On average for all profiles, the 
cross-sectional area did not change significantly 
during the period under consideration, but the bed 
of the river channel deepened considerably. 
4. A decrease in the width of the cross-section can 
be observed in the cross-section profiles, where a 
pronounced horizontal displacement of the river 
course has taken place. 
 
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matematičnega modela 
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