computer simulation for drift trajectories of objects in the magdalena river, colombia
DESCRIPTION
The coupled method developed in this study is a promising computational tool for simulating the drift of objects similar to the human body along a river system.TRANSCRIPT
UNIVERSITY OF CENTRAL LANCASHIRE
DEPARTMENT OF FORENSIC AND INVESTIGATIVE SCIENCES
COMPUTER SIMULATION FOR DRIFT TRAJECTORIES
OF OBJECTS IN THE MAGDALENA RIVER, COLOMBIA
A.C. GUATAME-GARCIA
SUPERVISORS
LUIS CAMACHO, PhD.
TAL SIMMONS, PhD.
A DISSERTATION SUBMITTED AS PART OF THE REQUIREMENT
FOR
MSc FORENSIC ANTHROPOLOGY
SEPTEMBER 2007
I confirm that this Report is all my own work and that all references and quotations from both
primary and secondary sources have been fully identified and properly acknowledged in
footnotes and bibliography.
Signed ……………………………………. ………… Date…………………………
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ACKNOWLEDGEMENTS
Many people deserve my gratitude for their personal and academic support along this
process.
First, I want to thank my parents who, in spite of the distance, have always provided me
with unwavering support and encouraged me to pursue my goals. I also am grateful to my
brother for keeping me full of energy, to my sister for her constant advice for both the trivial
and the complex, and to Diego for his companionship and love.
This project would not have been possible without the knowledge of Dr. Luis A. Camacho
who provided the scientific assistance for developing the numerical-computer model: for this, I
owe him my thanks. I am also grateful to Dr. Tal Simmons for supervising all stages of this
project.
As this project progressed, the number of people who provided me with help has grown; to
all of them, I express my gratitude and appreciation.
Finally, thanks to God, to whom I owe all my successes and happiness.
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TABLE OF CONTENTS
CHAPTER 1 ................................................................................................................... 1
INTRODUCTION ............................................................................................................ 1
1.1 Geographical and Political Context of the Study .............................................................................................. 1
1.2 Outline of the subject area .................................................................................................................................. 1
1.3 Justification .......................................................................................................................................................... 2
1.4 Aims and objectives ............................................................................................................................................ 3
CHAPTER 2 ................................................................................................................... 5
LITERATURE REVIEW ................................................................................................. 5
2.1 Human Decomposition in Water Environments and Postmortem Interval Estimations ............................. 5
2.2 Natural Open-Channels ...................................................................................................................................... 7
2.3 Theory of Transport in Open-Channels ......................................................................................................... 10
2.4 Position and Transport of Bodies in Water .................................................................................................... 12
2.5 Hydrodynamics’ Modelling .............................................................................................................................. 15
CHAPTER 3 ................................................................................................................. 17
MATERIALS AND METHODS .................................................................................... 17
3.1 Object's drift test 1 ............................................................................................................................................ 18
3.2 Object's drift test 2 ............................................................................................................................................ 18
3.3 Model Implementation for the Magdalena River .......................................................................................... 19
3.4 Statistical Analysis of Data ............................................................................................................................... 19
CHAPTER 4 ................................................................................................................. 21
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RESULTS ..................................................................................................................... 21
4.1 Model calibration results .................................................................................................................................. 21
4.2 Simulation results and statistical analyses ...................................................................................................... 22
CHAPTER 5 ................................................................................................................. 37
DISCUSSION ............................................................................................................... 37
5.1 General observations from physical experimentation at the Teusacá and Magdalena Rivers ................. 37
5.2 External factors affecting the drift of objects along the Magdalena River ................................................. 37
5.3 Flotation effects on the object’s rate of movement ........................................................................................ 39
5.4 Intrinsic factors affecting the object’s rate of movement .............................................................................. 41
5.5 Study implications and limitations .................................................................................................................. 42
5.6 Discussion summary ........................................................................................................................................ 42
CONCLUSIONS ........................................................................................................... 44
REFERENCES ............................................................................................................. 46
APPENDIX 1 ................................................................................................................ 51
APPENDIX 2 ................................................................................................................ 55
APPENDIX 3 ................................................................................................................ 60
APPENDIX 4 ................................................................................................................ 70
APPENDIX 5 ................................................................................................................ 75
APPENDIX 6 ................................................................................................................ 81
APPENDIX 7 ................................................................................................................ 86
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LIST OF TABLES
Table PageTable 2. 1. Body density and percent of fat in Adult males (Krzywicki and Chinn, 1967:307)..12
Table 4. 1. Object’s experiment data..........................................................................................22
Table 4. 2. Sub-sections for the Variante Bridge- Gas Pipe stretch of the Magdalena River......22
Table 4. 3. Object’s observed vs. predicted travel times and velocities for each subsection......22
Table 4. 4. Sub-sections for the Variante Bridge- Puerto Berrío stretch.....................................23
Table 4. 5. Descriptive statistics for RM1....................................................................................24
Table 4. 6. Descriptive statistics for RM2....................................................................................24
Table 4. 7. Pearson’s correlations for RM1.................................................................................25
Table 4. 8. Pearson’s correlations for RM2.................................................................................26
Table A2. 1. Volume and density estimations for wooden objects..............................................60
Table A3. 1. Flow Gauging data gathered from Teusacá experiment.........................................65
Table A4. 1. Data obtained from the mannequin drift experiment..............................................75
Table A6. 1. Variables and values used as data entry on the computer model............................82
Table A6. 2. Table showing the example results.........................................................................85
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LIST OF FIGURES
Figure PageFigure 1. 1. Magdalena River Section Variante Bridge – Puerto Berrío.......................................4
Figure 2.1. Temperatures for the Magdalena River, section Variante Bridge-Arrancaplumas. .. .6
Figure 2. 2. Channel section shapes for open-channels. ...............................................................8
Figure 2. 3. Velocities’ distribution in rectangular channels ........................................................9
Figure 2. 4. Velocities’ distribution in trapezoidal channels ........................................................9
Figure 2. 5. Three dimensional flow in a straight river channel with double spiral flow ..........10
Figure 2. 6. Flow in a bend..........................................................................................................10
Figure 2. 7. Archimedes’ Principle .............................................................................................12
Figure 2. 8. Floating position of fresh remains............................................................................14
Figure 3. 1. Magdalena River profile...........................................................................................18
Figure 4. 1. Differences on object mean time due to changes on river discharge.......................28
Figure 4. 2. Differences on object mean time due to changes on object initial position.............29
Figure 4. 3. Differences on object mean time due to changes on trapping factor. ......................29
Figure 4. 4. Differences on object mean time due to changes on water temperature .................30
Figure 4. 5. Differences on object velocity due to changes on river discharge...........................31
Figure 4. 6. Differences on object velocity due to changes on object initial position.................31
Figure 4. 7. Differences on object velocity due to changes on trapping factor. ..........................32
Figure 4. 8. Differences on object velocity due to changes on water temperature .....................32
Figure 4. 9. Difference on object mean time due to flotation depth. ..........................................33
Figure 4. 10. Difference of object’s velocity due to flotation depth............................................34
Figure 4. 11. Differences on object mean time due to changes on object density.......................35
Figure 4. 12. Differences on object’s velocity due to changes on object density........................35
Figure 4. 13. Differences on object mean time due to weight.....................................................36
Figure 4. 14. Differences on object’ velocity due to weight.......................................................37
Figure A1. 1. Velocities distribution at Arrancaplumas station during a high discharge period 52
Figure A1. 2. Velocities distribution at Arrancaplumas station during a low discharge period 53
Figure A1. 3. Velocities distribution at Cambao station during a high discharge period...........53
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Figure A1. 4. Velocities distribution at Cambao station during a low discharge period............54
Figure A1. 5. Velocities distribution at Nariño station during a high discharge period..............54
Figure A1. 6. Velocities distribution at Nariño station during a low discharge period...............55
Figure A2. 1. Calibrated balance and tank for solid volume estimation......................................57
Figure A2. 2. Weighing of the wooden mannequin by using the balance...................................58
Figure A2. 3. Irregular object submersion...................................................................................58
Figure A2. 4. Mannequin’s submersion in an overflow tank. .....................................................59
Figure A3. 1. Mid-section of the Teusaca River, Cabaña Gauging Station.................................61
Figure A3. 2. Teusaca River........................................................................................................62
Figure A3. 3. Schematic diagram showing the configuration of the reach of study at Teusacá
River.............................................................................................................................................63
Figure A3. 4. Measurements of flow velocity by using a flow meter at spaced positions..........64
Figure A3. 5. Object’s pattern of movement along the surface. Teusacá River..........................68
Figure A3. 6. Object’s submerged pattern of movement at Teusacá River.................................69
Figure A4. 1. Experimental stretch, Magdalena Medio region. Circle shows the stretch’s total
length............................................................................................................................................71
Figure A4. 2. Mannequin’s releasing in the Magdalena River....................................................72
Figure A4. 3. Mannequin getting trapped into an eddy...............................................................72
Figure A4. 4. Release cross-section of the Magdalena River at the Variante Bridge Station. X
axis: river width, Y axis: Elevation..............................................................................................73
Figure A6. 1. Data entry chart......................................................................................................83
Figure A6. 2. Output interface.....................................................................................................83
Figure A6. 3. Output diagram showing distance travelled vs. object’s mean and minimum travel
times and flow mean travel time. ...............................................................................................84
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ABBREVIATIONS
A Area
ADD Accumulated degree days
ANOVA Analysis of variance
CI Confidence interval
CIDH Interamerican Commission for Human Rights
D Distance
Diamb Trunk diameter
FMT Flow mean travel time
FV Flow velocity
FD Flotation depth
IDEAM Institute of Hydrology, Meteorology and Environmental Studies of Colombia
hrs. hours
K Constant
Kb Mass degradation constant
kms kilometres
Lbmax Maximum feasible object length
m meters
Mbo Object mass
min. minutes
OMT Object mean travel time time
OMinT Object minimum travel time time
ORM Object residual mass
OV Object velocity
PMI Post mortem interval
PMSI Post mortem submersion interval
ρ Density
rb Density (for the output computer model)
r Pearson’s correlation coefficient
Q Discharge
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RM1 Experimental stretch Variante Bridge – Girardot’s Gas Pipe (10.7kms)
RM2 Complete modelled stretch Variante Bridge – Puerto Berrío City (339kms)
SD Standard deviation about the mean
Sg Specific gravity
Sig Significance
t Time
Te Water temperature
TrapF Trapping factor
UNAL National University of Colombia
V Velocity
V Volume
W Weight
x1 Object initial position
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ABSTRACT
There are many problems that arise in studies of bodies disposed in moving waters, as these
remains do not only decompose, but are also subject to transport, disarticulation, and dispersion.
In such cases, computer modelling has proven to be an invaluable tool to understand both
trajectories of bodies in former cases, as well as the prediction of body flow patterns.
In this paper, a one-dimensional hydraulic model has been coupled with an object transport
model in order to predict both drift trajectories as well as distances travelled in a specific time
interval. Transport is modelled taking into account buoyant, hydrostatic, and dynamic forces,
calculated by using velocity, discharge, and depth computed in a numerical hydraulic model.
Results and information from previous research studies were incorporated into the modelling
framework to represent the transport of bodies with different densities and specific gravities.
The model was calibrated by means of physical experiments carried out in the Teusacá and
Magdalena rivers (Colombia). These experiments provided detailed hydraulic data, as well as
objects’ travel times. This information was used to calibrate and validate the numerical model.
The calibrated model has been applied to a 350kms stretch of the Magdalena River in
Colombia’s Magdalena Medio region, in order to simulate objects’ transport and predict their
location after being disposed into the river at a certain time. Travel times recorded ranged from
23.7hrs (≅1day) to 307.6hrs (≅12.8) days, and a maximum mass loss of 35kg was documented
in bodies that completed the 350kms river stretch.
The study concludes that the main extrinsic variables affecting the objects’ movement rate
were discharge, initial position of the object into the river, and trapping factor; all of these
variables were either catalysts for increasing or reducing velocity. Density is the main intrinsic
factor affecting the pattern of movement and distance travelled during a specific time interval
along a river, given its influence in the determination of the object’s position in the water
column.
The coupled method developed in this study is a promising computational tool for
simulating the drift of objects similar to the human body along a river system. However, further
verifications are still needed for complex body composition and motion, where the calculation
of changes in body volume resulting from inhaled water and mass loss due to disarticulation
continue to represent a considerable challenge.
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CHAPTER 1
INTRODUCTION
1.1 Geographical and Political Context of the StudyColombia is a country located in the northwestern region of South America. One of the
main rivers of the country is The Magdalena, which runs 1612kms from south to north
(Martínez, 1990). The central region of the river, known locally as the “Magdalena Medio” is a
strategic location because it allows the communication between the north, the centre and the
south of the country (Acevedo, 1981).
During the late 1940s and the early 1950s, an intense conflict between partisan groups gave
rise to the formation of different guerrilla groups. These groups have occupied vast zones across
the country developing an insurgent 'armed revolution' mainly financed by illegal drug trade,
extortion and kidnapping (Rabasa and Chalk, 2001). Thus, during the 1980s and the 1990s
many paramilitary groups were formed in order to defend their economical and political
interests. Paramilitary forces also operate the illegal drug trade and in many regions were aided
by traditional agrarian elites (Avilés, 2006).
Since the 1980s the Magdalena Medio has been one of the most important zones of
paramilitary “limpieza”, which refers to the cleansing of guerrillas and anyone remotely
suspected of sympathies with the insurgents (Taussig, 2005). The brutality and terror in the
paramilitary war can be observed in the pattern of their massacres and selective and systematic
homicides of the civil population: torture, killing, dismembering and throwing of the victims’
bodies into the closest rivers (CIDH, 2005; Brittain, 2006).
1.2 Outline of the subject areaThe main concern of this research project is the analysis of drift trajectories of objects that
have been deposited in fluvial environments through computational experimentation. Transport
of bodies by waters is a taphonomic process worthy of study since it produces complex patterns
of dispersion and deposition of human remains. Experimental taphonomic research, also called
actualistic research, has been used to observe particular processes through model building,
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where the key independent variables are controlled and observation of the independent variables
is carefully systematized (Haglund and Sorg, 2002).
In general terms, flow models provide very close approximations to real situations
associated with natural currents such as rivers, and hence, they can be extended to represent the
behaviour of masses moving within them. In the forensic context, the information obtained
from these models may allow the setting of geographic limits for victim searches, the
determination of the potential origin of the remains, and the confirmation and correlation of
information regarding time and place of entry into the water and/or sightings of remains along
the river’s trajectory (Ebbesmeyer and Haglund, 2002; Haglund and Sorg, 2002).
1.3 JustificationThe modus operandi for concealing the death bodies by throwing them into rivers poses
specific logistical difficulties for judicial investigators in terms of both the recovery and the
identification of the victims, especially given the lack of taphonomic studies of non-terrestrial
deposition in Colombia and elsewhere (Gómez-López and Patiño-Umaña, 2007).
Both the unique legal and social context of the missing in Colombia challenge the forensic
sciences to develop scientific methods for improving the techniques of searching for human
remains within the State’s territory, especially in those places in which the searches are more
difficult to be carried out, such as rivers.
Because of the highly complex nature of the dynamics underlying fluvial behaviour, the use
of prediction models to infer patterns of transport that allow to describe likely scenarios in
which victims can be found becomes necessary. However, the success for the development of
this kind of tools depends on the amount and quality of information that can be used to make
the predictions, especially that regarding the behaviour of the rivers. In Colombia this
information is mainly provided by the IDEAM although the Laboratory of Hydraulics of the
National University of Colombia (UNAL) has made important contributions to a current data
base of flow gaugings, and hydraulic characteristics of Colombian rivers.
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1.4 Aims and objectivesDue to the advantages that controlled simulations provide for the observation of the
transport and behaviour of bodies into the rivers, the aim of this project is to produce a
predictive transport model of objects along the Magdalena River, Magdalena Medio region,
from the section of the Variante Bridge, near the town of Girardot, to the town of Puerto Berrío,
350kms stretch (Figure 1.1), through the observation of the behaviour of objects in a simulated
computing environment.
This model should predict the location of the objects disposed into the river at certain time
intervals, and conversely, allow the determination of potential points of their entry. In both
circumstances, the success of the model will be evaluated on the identification and
quantification of the factors that are most likely to affect the transport. To produce such a
model, extensive investigation with reference to drift trajectories of bodies disposed in flowing
water, the rivers’ physical behaviour and their relationship with the masses they may contain
have to be conducted.
1.4.1. Objectives
1 Identify the intrinsic variables that influence the movement of bodies in the
Magdalena River with especial concern given to body density and weight.
2 Simulate different flow situations and their interaction with a moving body, and
identify the extrinsic variables that determine its transport in the Magdalena River.
3 Develop a modelling framework for body transport for the Magdalena River.
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Figure 1. 1. Magdalena River Section Variante Bridge – Puerto Berrío
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CHAPTER 2
LITERATURE REVIEW
2.1 Human Decomposition in Water Environments and Postmortem Interval Estimations
Human remains disposed in water environments are subject to specific intrinsic and
extrinsic variables that can make the process of decay highly variable (Lyman, 1994). Intrinsic
events are related to the semi-regular sequence of soft tissue degeneration that often concludes
with skeletonization and the subsequent modification of bone (Haglund and Sorg, 1997); the
extrinsic ones refer to transport by moving waters, scattering, and consumption by predators
(Haglund and Sorg, 2002).
Studies of decomposition in water environments refer to the PMSI, which attests the
amount of time a corpse has been exposed to water (Haskell et al, 1989) that sometimes is
equivalent to the PMI. Since decomposition in water may exhibit different rates according to
specific microenvironment biochemical and physical characteristics (Brewer, 2005), estimation
of PMSI has to be treat more carefully than traditional estimations of PMI. However, Nawrocki
et al. (1997) have stated that if the features of a specific water environment are known,
relationships between decomposition, transport, and time since death can be established.
Techniques for PMI estimations are based on the relationship between the process of
decomposition and the time a body takes to reach a certain decompositional stage. Then, PMI
estimations are generally based on the assessment of these stages of decay through qualitative
observation (Mann et al, 1990; Galloway, 1997) or the quantification of the process by either
measuring mass loss (Payne, 1965; Payne and King, 1968) or accumulated degree days ADD
(Vass et al., 1992; Megyesi et al., 2005).
The accuracy of expressing decomposition as directly related to weight loss has been tested
by Adlam (2004), who stated that biomass removal has a strong correlation with decomposition
score, which is in turn related to the environmental temperature, and therefore is a validated
indicator for the advancement of decomposition.
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Because as well as in land the main factor in aquatic decomposition is temperature,
decomposition can be quantified taking into account both ADD (thermal time) (Haefner et al,
2004) and mass loss. In these cases, average daily temperature can be replaced by the average
water temperature.
Water temperatures in rivers are determined by air temperature, wind speed, cloud cover,
and the relative humidity of the atmosphere (Chapra, 1997). In the Magdalena River, water
temperature is also affected by changes in altitude (Universidad de los Andes-ACUAGYR,
2004). For the Magdalena Medio, temperatures oscillate between 22 and 290C (Figure 2.1).
Figure 2.1. Temperatures for the Magdalena River, section Variante Bridge-Arrancaplumas. Red lines correspond to September 9 -11/2004 at low discharge, and blue lines correspond to September 28-30/2004 ay high discharge. Squares indicate first day of sampling, triangles second day of sampling, and dots third day of sampling. Adapted from: Universidad de los Andes – ACUAGYR, 2005:51, Modelación de la Calidad del Agua del Río Magdalena y Caracterización de las Aguas Lluvias y Residuales de Girardot.
Human bodies may undergo different processes of decomposition in marine, lacustrine or
riverine environments given their dissimilar chemical composition (salinity, pH, and CO2
concentrations), temperatures, depth and movement (Nittrouer et al., 1995; Sorg et al., 1997).
Regarding movement, different transport patterns have been identified in marine and fluvial
environments since the variables affecting the motion of water can vary from the tidal
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atmospheric oceanic behaviour (Ebbesmeyer and Haglund, 2002; Carniel et al., 2002) to the
channel driven waters of a river (Dilen, 1984; Nawrocki et al., 1997). The later are called
natural open-channels in the hydraulic literature and are explained below.
2.2 Natural Open-ChannelsA natural Open-Channel is a water flow confined to a channel whose characteristics
changes over time and space and become more complex as all the variables are interdependent
one from another (Chow, 1988). These systems are affected by several extrinsic factors that
make their behaviour unsteady and difficult to predict. Rivers and streams are typical natural
open-channel systems.
The shapes of a natural channel are very irregular and many often vary in shape from a
parabola to a trapezium (Chow, 1988). The parabola is used as approximation of medium and
small natural channel sections; the trapezium commonly refers to old river channels. (Figure
2.2).
Figure 2. 2. Channel section shapes for open-channels (Adapted from Chow, 1998:21, Open-Channel
Hydraulics, New York: McGraw-Hill).
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Flow velocity and discharge are two important variables that contribute to fully understand
the behaviour of a natural open-channel. Velocity is expressed by
tD=V /
Where D is the distance travelled by an object, and t the time the object reach to get the given
distance.
Because of the open surface and the friction along the channel walls, the velocities of a
channel are not homogeneously distributed over the whole section. The maximum velocity
often occurs underneath the surface at 0.05 to 0.25 of the total depth; closer to the banks this
maximum is reached at a deepest point. Actual velocities near the bottom are lower and those
near the surface are higher (Boaz and Behrensmeyer, 1976); figures 2.3 and 2.4 show the
general distribution of velocities for rectangular and trapezoidal channels. Appendix 1 shows
the distribution of velocities for the Magdalena River.
Figure 2. 3. Velocities’ distribution in rectangular channels (Adapted from Chow, 1998:24, Open-Channel Hydraulics, New York: McGraw-Hill).
Figure 2. 4. Velocities’ distribution in trapezoidal channels (Adapted from Chow, 1998:25, Open-Channel Hydraulics, New York: McGraw-Hill).
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This distribution is also affected by factors such as the channel roughness and bends
(Bishop and Prosser, 2001). For example, in a wide, fast and shallow current, the maximum
velocity is reached at the surface; in a bend, the velocity increases in its convex side due to the
centrifuge action of the flow (Chow, 1988).
In straight prismatic channels the flow is three-dimensional and produces a spiral
movement. The movement is characterised by a double spiral which allows the equilibrium of
forces at each side of the channel (Dilen, 1993). The model includes one spiral at each side of
the central line where the water level is higher (Figure 2.5.) (Chow, 1988).
Figure 2. 5. Diagram showing the three dimensional flow in a straight river channel with double spiral flow (Dilen, 1984:1029).
Because of the velocity structure of a stream, and especially in streams flowing over low
gradients with easily eroded banks, straight channels will eventually erode into meandering
channels (Novak, 2001).Velocities at the meanders also change due to their curve geometry. In
the meanders, the depth gradually increases to a maximum downstream of the apex of the bend;
it is characterised by spiral flow and triangular sections with the maximum depth and velocity at
the concave bank, and maximum sediment transport at the convex bank and the talweg (line of
maximum depth) deviating from the river centreline (Novak, 2001)(Figure 2.6).
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Figure 2. 6. Diagram showing the flow in a bend (Novak, 2001:299).
On the other hand, discharge refers to the volume of water passing through a given cross-
section of a river at a given period of time (Smith and Stopp, 1979). For any flow the discharge
Q in a channel section is expressed by
AV=Q ∗
where V is the mean velocity and A is the cross section area of the perpendicular flow in its
direction. As water in a river moves downstream, the discharge is affected by the size, depth
and shape of the channel, the slope of the riverbed, the smoothness or roughness of the banks
and river bottom, and by bends in the river's channel (Dilen, 1984), and its variability also
depends upon the amount of water being delivered from precipitation, evaporation, throughflow
and channel networks (Smith and Stopp, 1979).
2.3 Theory of Transport in Open-ChannelsHanson (1980) has shown that the behaviour of bones in artificial and natural currents is to
some extent predictable and that bones act as sedimentary particles that follow the general
principles of hydraulics. This author also suggests that the area of sediment transport fully
developed by the hydraulic engineering field is compatible with fluvial taphonomy, and hence,
the basic equations of fluid flow and resistance forces can be applied to these problems.
In general terms, the movement of any particle deposited in water follows the Archimedes'
Principle which states that any object floating upon or submerged in a fluid is buoyed upward
by a force equal to the weight of the displaced fluid (Donoghue and Minniguerode, 1977). Any
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submerged object is subject to a greater pressure force on its lower surface than on its upper
surface, creating a tendency for the object to rise (Figure 2.7).
a. b.Figure 2. 7. Archimedes’ Principle. a) Forces acting over a completely submerged object. b) Forces acting over a partially submerged object. F= Force; P = Pressure; A= Area; y = Depth, mg = Weight; FB = Buoyant Force.
Elements deposited into flowing water have some transport patterns that are time and
distance dependant defined by a velocity Vt in the downstream direction (Hanson, 1980). These
elements are controlled by hydraulic factors such as depth, width, stream velocity, shear stress
and sediment load, which in turn are related to discharge (Chow et al., 1988). Other extrinsic
physical factors in the fluvial realm such as channel geometry and lateral migration rate,
aggradation or degradation of the channel and floodplain, bed forms, and clast size also
influence rates of transport and patterns of movement (Hanson, 1980).
Another physical law applied to the hydraulics is the principle of motion resulting from the
relationship between the object's mass and its velocity vector. Newton originally formulated the
laws of motion in terms of this property which he called “quantity of motion” and which is now
called linear momentum (Lea and Burke, 1997). The law of conservation of momentum is a
fundamental law of nature, and it states that the total momentum of a closed system of objects
which has no interactions with external agents is constant. The principle of continuity is closely
related to the principle of momentum and states that a fluid mass is neither created nor
destroyed during its flow (Lea and Burke, 1997). In fluid dynamics the continuity equation is an
equation of conservation of mass.
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2.4 Position and Transport of Bodies in WaterDifferent water environments may differ radically with respect to temperature, depth,
salinity, oxygenation, or discharge. Dependent upon these factors, bodies entering water may
initially float or sink, possibly to surface later, or remain submerged, or even be buried by
accumulating sediment (Haglund and Sorg, 2002).
Density (ρ) and specific gravity (Sg) are useful concepts for predicting whether a body will
float or sink. Density is defined as the ratio of weight W to the volume V , and specific gravity
as the density of a body to the weight and volume of the reference standard water. A body with
a density and/or specific gravity of greater than 1.000 will sink in freshwater, while one with a
specific gravity of 1.000 or less will float in freshwater (Donoghue and Minniguerode, 1977).
In human bodies, density mainly depends on the amount of body fat and the gases
concentrated in the lungs and those produced by the gastrointestinal system (Krzywicki and
Chinn, 1967). In the latter study, 173 living males ranged from 17 to 69 years old were studied
to obtain body densities and percentages of body fat through the water displacement method.
The study observed a progressive decline of the mean body density with age as well as a
gradual increase in body fat, which are independent of body weight (Table 2.1). The subjects
with the highest mean body weight described the lowest mean body density, which reflects a
high percent of body fat (Krzywicki and Chinn, 1967).
Table 2. 1. Body density and percent of fat in Adult males (Krzywicki and Chinn, 1967:307)Age Group
NBody
Weight Density % Fat17-19 9 71.9 ± 14.4 1.060 ± 0.016 19.6 ± 7.020-24 35 73.6 ± 7.5 1.060 ± 0.013 19.5 ± 5.525-29 29 76.8 ± 14.0 1.053 ± 0.017 22.6 ± 7.330-34 15 85.8 ± 17.6 1.044 ± 0.013 26.3 ± 6.135-39 13 76.2 ± 10.6 1.043 ± 0.012 26.9 ± 3.640-44 25 75.4 ± 11.1 1.042 ± 0.012 27.1 ± 5.545-49 24 76.2 ± 10.0 1.038 ± 0.010 29.3 ± 4.550-54 12 75.5 ± 10.1 1.032 ± 0.026 32.8 ± 9.155-59 4 79.0 ± 10.3 1.031 ± 0.021 32.5 ± 4.860-64 5 69.7 ± 7.5 1.026 ± 0.010 34.7 ± 4.565-69 2 68.6 ± 2.1 1.017 ± 0.001 38.7 ± 0.6
Total 173
In Donoghue and Minniguerode’s (1977) study, specific gravities corrected for the lungs at
residual volumes ranged from 1.021 to 1.097, which means that a recent dead human body will
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sink. This phenomenon is also affected due to the absorption of water into the circulation
system which results in an abrupt increase in blood volume (Boyle et al, 1997).
Because the specific weight of the human body is very close to that of water, small
variations in specific gravity have considerable effect in whether a body will sink or float
(Donoghue and Minniguerode, 1977). While on the surface, the body will float with the head
and limbs hanging down beneath the surface (Figure 2.8); this position in the water results in
the head and limbs exhibiting more lividity than observed on the trunk (Spitz and Fisher, 1993;
Rodriguez, 1997)
Figure 2. 8. Floating position of fresh remains (Haglund and Sorg, 2002:205)
Having sunk to the bottom, the body will remain there until putrefactive gas formation in
the chest and abdomen decreases the specific gravity of the body and creates sufficient
buoyancy to allow it to rise to the surface and float (Donoghue and Minniguerode, 1977). Once
decomposition gases are released, bodies/remains will become submerged again (Sorg et al.,
1997); this changing position in the water allows differential access to agents of destruction,
and also alters the sequence of soft tissue decay and disarticulation of bones or body units
(Haglund and Sorg, 2002). As the decay progresses, most of the exposed flesh is lost, while the
submerged flesh remains intact; During this stage, the head, shoulders, abdomen, and legs
frequently become separated (Boyle et al., 1997). This process of decay becomes more rapid as
the water temperature increases.
As Nawrocki et al. (1997) have pointed out, when bodies become disarticulated, it is
necessary to take into account that densities and buoyancy vary in different regions of the body
and that not all will react to the currents in the same way. Boaz and Behrensmeyer (1976)
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support this statement since their study of transport of bones have established that density is the
most useful variable in predicting whether a skeletal part will move or not and what its velocity
will be once it does move. Disarticulated body units and skeletal elements may be moved by
deeper currents and come to rest on bottom substrates. At that point they may be moved along
the substrate or be silted over (Haglund, 1993).
Boaz and Behrensmeyer (1976) utilised an artificial recirculating flume to observe the
movement of bones in moving water to establish the pattern of hominid assemblages’
formation. They divided the bones in lag and transport groups according to the their hydraulic
behaviour, and as has been mentioned above, concluded that density is the most important
variable for determining the movement of bones into the water, and that shape and conservation
of bones is also important although no-quantifiable at that time. Hanson (1980) utilised the
same recirculating flume and also made river experiments with the aim to test mathematical
transport hypotheses for several mammals’ bones.
Dilen (1984) used mannequins floating in the Chattahoochee River, Atlanta, to examine the
movement of a body floating downstream on the surface and to determine the flow patterns of
surface currents through bends in river. The study concluded that objects floating downstream
tend to stay near the bank they were dropped at, and that submerged bodies orient parallel with
the direction of the flow and tend to resist downstream movement near the banks due to the low
current velocity. However, this pattern is case-specific and the author recommended carrying
out similar experiments in rivers with different characteristics to test his own results.
Ebbesmeyer and Haglund (2002) used the Hydraulic Model of Puget Sound, Seattle, to
demonstrate trajectories of floating objects by ocean waves that lead to the identification of the
time required for arrival at selected locations. This study used a physical scale replica which did
not utilise numerical predictive models. The results of the simulation did not show with
complete reliability where a floating object would travel under all environmental conditions, but
provided useful information with regard to probable trajectories the body could travel and areas
that the body is not likely to travel.
On the other hand, Carniel et al. (2002) used a computer simulation based on atmospherical
prediction models applied for the Mediterranean coast of the Ligurian Sea. In this study,
trajectories of floating bodies were simulated to infer the location of discovery of bodies that
have been drifted by surface ocean currents, based on the case of a woman who disappeared in
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the city of Portofino (Italy) and found in the Isle du Levante (France) 14 days after. This study
did not take into account submerged bodies and the model was not able to simulate accurately
the behaviour of waves near the line coast; however. The model showed a very approximate
trajectory of drift since the results were consistent with the broad area in which the body was
finally found and not somewhere else.
2.5 Hydrodynamics’ ModellingHydrological modelling systems can be subdivided into physical and mathematical models.
The first types include scale models that represent the real system in a reduced scale.
Mathematical models represent the system by numerical reasoning that describes the system
through a group of equations of input and output variables (Chow et al., 1988).
In the forensic arena some hydraulic physical simulations and numerical models have been
developed (Dilen, 1984; Ebbesmeyer and Haglund, 2002; Carniel et al., 2002). These studies
constitute an advantage with regard to previous studies derived from retrospective case files’
analysis (Brewer, 2005; Megyesi et al., 2005; Heaton, 2006) or actualistic forensic cases (Dix,
1987; Nawrocki et al., 1997; Kahana et al., 1999) in which recovery of few body parts and a
limited number of cases hinder the comprehension of the whole transport process.
In Colombia several computer simulations of the Magdalena River have been developed to
model water quality and solute transport (Universidad de los Andes – ACUAGYR, 2005); to
model hydrodynamics in flood prediction applications (Lombana, 2003); and to model solute
transport (Camacho et al., 2003; Camacho and DiazGranados, 2003). Thanks to these studies,
an important corpus of data of the behaviour of the river obtained from field work is currently
available and different extensions and numerical methods tested. Consequently, the most
predictive-accurate methods according with the particular aims of each project have been
identified and refined.
One of the methods that has been used to model the Magdalena River is the distributed
hydraulic model, which considers the hydraulic process as a phenomenon that occurs in several
points of space and defines variables simultaneously as functions of space and time (Chow et
al., 1988). For simulating the transport of human bodies this approximation is appropriate given
that the movement of bodies suspended in water occurs in three dimensions (Haglund, 1993)
and is temporally dependent.
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Since the water flow in a river or stream acts as a distributed process in which discharge,
depth and velocity vary in the space along the channel, the distributed model generates an
accurate approximation to the real behaviour of the river. However, the application of the
principles of momentum and continuity for this kind of system is only possible under very
simplified conditions, resulting in one-dimensional or bi-dimensional models (Chow et al.,
1988). The numerical schemes are discussed elsewhere (Fread 1985, 1993; Chow et al., 1988)
and will not be explained in detail herein, as it is not the purpose of this study.
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CHAPTER 3
MATERIALS AND METHODS
Both physical and mathematical methods were used to identify the drift of objects
throughout the 350kms of the Magdalena River section studied (Figure 3.1). The methods
follow the laws of physics and theory of flow dynamics explained above in Chapter Two.
Mathematical models must be calibrated to a particular site by comparing observed and
predicted water surface levels and/or discharges (Camacho and Lees, 1998). Two preliminary
experiments at the Teusacá and Magdalena rivers were performed; relevant information such as
orientation of the object and flood effects on the object's movement was derived and used to
calibrate the computer model. Afterwards, the numerical hydraulic distributed model developed
by Camacho and Lees (1998) was adjusted and implemented for the Magdalena River, thus
providing an accurate representation of the real flow pattern of the river section and object’s
movement.
972 tests were performed and statistically analysed in order to identify the correlations
between the variables used by the model, and to assess the truthfulness or falsehood of the
hypotheses proposed.
Figure 3. 1. Magdalena River profile. Shaded section shows the modelled route (350kms).
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3.1 Object's drift test 1Two sets of physical experiments in a straight section of a uniform flow stream were
devised to test the submerged and surface movement of objects in a river (See Appendices 2
and 3 for details). First, an object yielding a density of 0.75gr/cm3 was used to examine the way
in which a body floating in the surface moves downstream. Second, a similar object yielding a
density of 1.02gr/cm3 was used to observe the submerged movement of the object while drifting
downstream.
The stream flow discharge was measured using the velocity area method. The time the
object took to reach the end of the section was recorded and average travel times for each of the
runs were established. Due to the orientation of the element relative to the current direction at
each test, movement patterns of the object near the river banks and at the centre of the channel
were identified.
3.2 Object's drift test 2A physical experiment was carried out in the Magdalena River for object’s velocity
calibration, and pattern of surface movement testing (See Appendix 4 for details). The river
section extended from the La Variante Bridge to the gas pipe of the town of Girardot, a distance
of 10.7kms downstream. This section has a mean width of 164.47m width, a longitudinal slope
of 0.0070483, and a mean flow discharge of 1200m3/s (Uniandes-ACUAGYR, 2005).
The experiment was devised to calibrate the velocity of a wooden mannequin as it moved
downstream, as well as to test the surface movement in an area characterised by several bends,
eddies, and shrubs. The mannequin was manufactured taking into account human body
proportions reported by Krzywicki and Chinn (1967). The mannequin weighted 45kg, with a
centre of gravity located near the abdomen, and a bodily density of 0.98gr/cm3.
The mannequin was released at the centre of the current, without specific orientation
relative to the current direction. The mannequin floated freely, approximately 85% of the
volume submerged. Five control points were previously established to identify the time the
body took to reach each subsection, and the average travel time along the whole section was
calculated. The mannequin’s route downstream was drawn, photographed, and video recorded
from a boat.
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The average travel time along the whole section was calculated, as well as the ratio flow
velocity to body velocity. The river flow discharge during the experiment was obtained using
the stage-discharge relationship of Nariño flow gauging station. The measured discharge during
the experiment was 900m3/s.
3.3 Model Implementation for the Magdalena RiverThe objects’ drift modelling and calibration is fully explained in Appendix 5. After
calibrating the computer model by using real data obtained from the physical experiments, 972
tests were performed to observe the predicted movement of objects in the Magdalena River.
The first set of experiments (n=486) was implemented for the experimental stretch Variante
Bridge – Girardot’s Gas Pipe (10.7kms), and the second one (n=486) was carried out for the
complete section of study Variante Bridge – Puerto Berrío (350kms).
The 972 tests composed the total universe of combinations of the 5 external and 4 internal
variables used by the computer model (see Appendix 6).
3.4 Statistical Analysis of DataThe following hypotheses were tested through statistical analysis:
Hypothesis A:
H1: The extrinsic variables considered by the model significantly affect the movement of
objects in the Magdalena River (travel times and velocity).
H0: The extrinsic variables considered by the model do not affect the transport of objects
throughout the river.
Hypothesis B:
H1: Floating objects move faster than the submerged ones
H0: Floating and submerged objects undergo the same velocities
Hypothesis C:
H1: As object density increases, the drift of the object along the river decreases.
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H0: The drift of the object along the river either increases or stays the same while the object
density increases.
Hypothesis D:
H1: Density is the main factor affecting whether a body will float or sink.
H0: Density is not the only factor affecting floating or sinking.
Hypothesis E:
H1: Heavier bodies move faster than the lighter ones.
H0: Heavy and light bodies undergo the same travel times.
Data manipulation and statistical analyses were carried out using the computer software
package SPSS v. 14.0. The Pearson’s r correlation coefficient was calculated for every pair of
variables to identify the significant relationships at α=0.01 and α=0.05 levels.
The pairs revealing a significant correlation were subject to an Analysis of Variance
(ANOVA), which makes the following assumptions:
1. Samples are randomly drawn, and/or conditions assigned randomly.
2. Data are scale.
3. The data in each population are normally distributed.
4. The variability in each population is similar.
The last was tested using Levene’s test for Homogeneity of Error of Variance. This sets the
hypothesis that error variance between the two groups is equal. A non-significant result (i.e.
p>0.05) indicates that this is indeed the case, and the ANOVA can proceed. Where Levene’s
test showed homogeneity of variance, the significance level was set at α=0.10 in order to
increase the statistical power of the analysis; in cases where Levene’s test showed error
variance to differ significantly (p<0.05), the significance level was set at the traditional level of
α=0.05. Where two independent variables showed to affect a single dependent variable, a two-
way ANOVA was performed to identify the significance of each independent variable and the
occurrence of an interaction effect. These tests were also restricted to the Levene’s test for
Homogeneity of Error of Variance.
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CHAPTER 4
RESULTS
This section will discuss the predicted drift by the computer model. In addition, the results
from the 952 tests undertaken by using the computer model will be summarised and statistically
analysed according to the aforementioned hypotheses of the study.
4.1 Model calibration resultsThe model was calibrated taking into account the pattern of drift observed in the second
physical experiment and the predicted movement by the computer model, for both the same
river section and object’s characteristics (Table 4.1). The section was divided into six sub-
sections (Table 4.2) and both observed and predicted travel times and velocities were compared
(Table 4.3). Table 4. 1. Object’s experiment data
Table 4. 2. Sub-sections for the Variante Bridge- Gas Pipe stretch of the Magdalena River.
Number Length (kms) Reach’s name1 1.125 Variante Bridge – Sumapaz River2 3.35 Sumapaz River – Flow derivation structure Girardot City3 5.5 Flow derivation structure – Bogota River4 6.65 Bogota River – Ospina Bridge5 7.65 Ospina Bridge – Ferrocarril Bridge6 10.7 Ferrocarril Bridge – Gas Pipe
Table 4. 3. Object’s observed vs. predicted travel times and velocities for each river subsection
Reach Observed hours
Predicted hours
Observed Ob. Vel
Predicted Ob. Vel
1 0.12 0.11 2.68 2.782 0.47 0.51 1.51 1.773 0.72 0.64 2.34 2.364 0.90 0.84 2.18 2.065 0.98 0.83 2.78 2.306 1.13 1.04 3.60 3.18
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Q (m3/s)
Water temp. (0C)
Object Mass (kg)
Density (g/cm3)
Trunk Dia-meter (m)
Body Length
(m)
K De-grad.
Init. po-sition
Trapping factor
900 30 45 0.98 0.23 1.5 0 0.6 0.4
21
The Nash-Sutcliffe model efficiency coefficient was calculated to asses the predictive power
of the model regarding object travel times and velocity. The coefficient is defined as
( )( )∑
∑−
−− 2
22 1
otot
ptot=R
i
ii
where ot is the observed object travel time and pti the predicted object travel time.
Nash-Sutcliffe efficiencies can range from -∞ to 1. An efficiency of 1 (R2=1) corresponds
to a perfect match of modelled data to the observed data; an efficiency of 0 (R2=0) indicates that
the model predictions are as accurate as the mean of the observed data. In conclusion, the closer
the model efficiency is to 1, the more accurate the model is.
The comparison between both the data obtained from the physical experiment and the
predicted values by the computer model, yielded a R2= 0.94 for object travel times, and R2=0.8
for object’s velocity. Thus, the coefficient reveals that the numerical results obtained from the
model are in good agreement with the experimental ones.
4.2 Simulation results and statistical analysesTwo sets of tests were run by using the computer model. The first one (n=486) was applied
to the short experimental stretch, whose sub-sections were mentioned above in Table 4.2. The
second one was applied to the complete modelled stretch which runs about 339kms from the
Variante Bridge to the city of Puerto Berrío. This stretch is subdivided into sixteen sub-sections
(Table 4.4) of those the first six correspond to the experimental stretch.
Each test was numbered as T1, T2, T3, etc. and its respective entry values recorded. Since
the model produces the eight aforementioned outcomes (see Appendix 7), only the last row was
taken into account for statistical analysis.
Table 4. 4. Sub-sections for the Variante Bridge- Puerto Berrío stretch.Number Length (kms) Reach’s name
1 1.125 Variante Bridge – Sumapaz River2 3.35 Sumapaz River – Flow derivation structure Girardot City3 5.5 Flow derivation structure – Bogota River4 6.65 Bogota River – Ospina Bridge5 7.65 Ospina Bridge – Ferrocarril Bridge6 10.7 Ferrocarril Bridge – Gas Pipe7 19.05 Gas Pipe – Coello8 36.3 Coello- Nariño9 97.67 Nariño – Ambalema
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10 141.1 Ambalema – Sabandija11 163.17 Sabandija – Arrancaplumas12 195.67 Arrancaplumas – Puerto Salgar13 133.17 Puerto Salgar – Río Negro14 238.17 Río Negro – La Miel15 295.67 La Miel – Nare16 338.97 Nare _ Puerto Berrío
4.2.1. Descriptive statistics
Descriptive statistics of each set of tests yielded the following results (Tables 4.5 and 4.6):
Table 4. 5. Descriptive statistics for RM1N Minimum Maximum Mean Std. Deviation
Object mean time 486 0.726080 7.297416 2.42817316 1.451654078O. minimum time 486 0.497315 4.998230 1.66313229 0.994283628Object velocity 486 0.4784 4.5167 1.843278 0.9187313Flotation depth 486 0.210 8.947 3.49331 3.493398
Object mass 486 48.527 99.894 74.51462 20.303513Valid N (listwise) 486
Table 4. 6. Descriptive statistics for RM2N Minimum Maximum Mean Std. Deviation
Object mean time 486 34.617580 307.648266 108.53195772 62.791330679O. minimum time 486 23.710671 210.717991 74.33695731 43.007760738Object velocity 486 0.2683 2.2889 0.978099 0.4717623Flotation depth 486 0.210 3.705 1.47422 1.346912
Object mass 486 14.171 95.084 57.31390 18.991889Valid N (list-
wise) 486
In the RM1 set of tests, the minimum mean object travel time was yielded by the test
number T268 (fastest object, t=0.72hr; 43.2min), where data entry was Q= 2170.9, water
temperature = 300C, object mass=50kg, density=0.9g/cm3, initial position: 0.6, trapping factor=
0.5. The minimum object travel time was yielded by the same test (t=0.49hr; 24.9min), as well
as the highest object’s velocity (V= 4.5167m/s). The observed residual mass in this test was
49.85kg.
The maximum mean travel time for this set of experiments (slowest object, t=7.2974hr; ≅
1/3 days) was yielded by the test T231, where data entry was Q= 444.7, water temperature =
220C, object mass=50kg, density=1.06.9g/cm3, initial position= 0.9, trapping factor= 1.5. The
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maximum minimum1 object travel time was yielded by the same test (t=4.99823hr; ≅1/5 days),
as well as the lowest object’s velocity (V= 0.4784m/s); the observed residual mass in this test
was 49.48kg. The test number T444 yielded very similar results: maximum mean travel time
t=7.2973hr; ≅1/3 days; maximum minimum object travel time t=4.99819hr; ≅1/5 day); object’s
velocity V= 0.4784m/s; and residual mass = 48.52kg. The data entry for this test was Q= 444.7,
water temperature = 300C, object mass=50kg, density=1.06.9g/cm3, initial position: 0.9, and
trapping factor= 1.5.
The ratio object velocity/flow velocity for the fastest object (T268) was estimated as 1.2;
for the lowest object (T231), the ratio was calculated as 0.2.
As well as in RM1, the minimum mean object travel time in RM2 (fastest object) was
yielded by the test number T268 (t=34.61hrs; 1.4 days). Also, the minimum object travel time
was yielded by the test T268 (t=23.71hr; ≅1day), which yielded the highest object’s velocity too
(V= 2.28m/s). The observed residual mass in this test was 43.8kg.
The maximum mean travel time for RM2 was also yielded by the test T231
(t=307.6482hrs; ≅12.8 days), which also described the maximum minimum object travel time
(t=210.71791hrs; ≅8.7days), and the lowest object’s velocity (V= 0.2683m/s). The observed
residual mass in this test was 32.19kg. The test T444 was very close to these results, yielding a
mean object travel time of 307.6439hrs, a minimum travel time of 210.7150hrs, and a velocity
of 0.2684m/s. The observed residual mass in this test was 14.17kg, which corresponds to the
lowest residual mass of the total universe of tests that compose this set.
The ratio object velocity/flow velocity for the fastest object (T268) was estimated as 1.1;
for the lowest object (T231), the ratio was calculated as 0.2.
In summary, RM1 travel times ranged between 0.72 and 7.29 hours, and RM2 between 1.4
and 12.8 days. For RM1 the mass loss ratio was calculated as 0.2kg/hr in T268, 0.1kg/h in
T231, and also 0.2kg/hr in T444. For RM2 the mass loss ratio was calculated as 0.17kg/hr in
T268, 0.05kg/h in T231, and 0.11kg/hr in T444.
1 The “maximum minimum” corresponds to the highest value identified for the object minimum travel time predicted by the computer model.
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4.2.2. Correlation
Pearson’s r correlation coefficient was calculated for every pair of variables within each set
of tests. Tables 4.7 and 4.8 show the pairs that expressed a significant correlation. Correlations
determined by its definition (i.e. discharge, flow velocity) were not analysed. Object minimum
time was also omitted since its behaviour completely matches the variable object mean time.
Table 4. 7. Pearson’s correlations for RM1Q rb Kb x1 TrapF OMT FMT FV OV
Q -.357 .396Te 1.000Rb .218 -.264X1 .650 -.547
TrapF .342 -.419OMT -.357 .218 .650 .342 .371 -.364 -.844FMT .371 -.389FV -.364 .397OV .396 -.264 -.547 -.419 -.389 .397FD .254 .932 .102 -.253 .255 -.164
Correlation is significant at the 0.01 level (2-tailed).
For RM1, the highest correlations observed lie on the pairs water temperature-mass
degradation factor (r=1.0), and flotation depth-density (r=0.93). The positive correlation means
a proportional correlation (as x increases, y also increases).
Correlations under 0.8 are considered as significant but no main correlations. These kinds of
correlations may be interpreted as: a) a direct effect of one variable to another in a small
proportion, or b) the transitive effect of one variable through another. Correlations regarding the
output data of RM1 can be listed as2:
-Object mean time: initial position3 (+): flow mean time (+): flow velocity (-): discharge (-):
trapping factor (+): density (+): flotation depth (-).
- Object velocity: initial position (-): trapping factor (-): flow velocity (+): discharge (+):
flow mean time (-): density (-); flotation depth (-).
- Flotation depth: density (+): flow velocity (+): discharge (+): flow velocity (-): flow mean
time (+).
2 The variables are shown in order of the strength of the correlation from left to right.3 Relationships marked as (+) mean a positive correlation and those marked as (-), an inverse correlation.
© Guatame-García, 2007
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Table 4. 8. Pearson’s correlations for RM2Q Te Rb Kb X1 TrapF OMT FMT FV OV
Q -.330 .386Te 1.000Rb .169 -.191X1 .672 -.565
TrapF .353 -.433OMT -.330 .169 .672 .353 .343 -.339FMT .343 -.384FV -.339 .389OV .386 -.191 -.565 -.433 -.384 .389FD .286 .910 -.281 .285
ORM .147 -.316 -.316 -.298 -.157 -.428 -.149 .148 .385Correlation is significant at the 0.01 level (2-tailed).
As well as in RM2, the highest correlations observed lie on the pairs water temperature-
mass degradation factor (r=1.0), and flotation depth-density (r=0.91).
The other significant correlations found below the r=0.8 cutoff are listed below4:
-Object mean time: initial position (+): object residual mass (-): trapping factor (+): flow
mean time (+): flow velocity (-): discharge (-): density (+).
- Object velocity: initial position (-): trapping factor (-): flow velocity (+): discharge (+):
object residual mass (+): flow mean time (-): density (-).
- Flotation depth: density (+): flow velocity (+): discharge (+): flow velocity time (-): flow
mean time (+).
- Residual mass: Object mean time (-): object velocity (+): density (-): water temperature
(-): initial position (-): trapping factor (-): flow mean time (-); flow velocity (+): discharge
(+).
4.2.3. Analysis of variance (ANOVA)
Since the pattern of transport of elements deposited into flowing water is time and distance
dependant (Hanson, 1980), the dependent variables that have been chosen as descriptors of the
object’s rate of movement are object mean time and object velocity.
For either one or two-way ANOVA, where Levene’s test for error of variance indicated a
non-homogeneity of the data, the significance of the tests was set at α=0.05; if homogeneity of
4 The variables are shown in order of the strength of the correlation from left to right.
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variance was identified, significance was set at α=0.01. The tests contained both RM1 and RM2
(n=972).
4.2.3.1. Extrinsic variables affecting the objects’ rate of movement (Hypothesis A)
Object mean time
- Discharge and water temperature: The significance of the two-way ANOVA was set at
α=0.05. The results show discharge to highly affect the object mean time (F=102755.3,
p=0.000), but temperature (F=1.920, p=0.260) and the interaction of discharge and temperature
(F=0.000, p=1.000) showed no significance in their effects.
- Initial position and discharge: The significance of the two-way ANOVA was set at
α=0.05. The results show a significant effect of discharge on object’s mean time (F=8.913,
p=0.034); the effect of initial position on mean time is also significant (F= 50.702, p=0.01).
However, the interaction of discharge and initial position on object’s mean time had not a
significant effect (F= 1.637, p=0.163).
- Initial position and trapping factor: The significance of the two-way ANOVA was set at
α=0.05. The results show a significant effect of initial position on object’s mean time
(F=48.014, p=0.02); the effect of trapping factor on mean time is also significant (F= 9.035,
p=0.033). However, the interaction of initial position and trapping factor on object’s mean time
did not show to have a significant effect (F= 1.733, p=0.140).
These results indicate that object’s mean travel time is highly affected by these extrinsic
variables, excluding water temperature (Figures 4.1 to 4.4). However, there is not sufficient
statistical evidence for establishing any interaction effect of the variables studied on object
mean travel time.
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Figure 4. 1. Differences on object mean time due to changes on river discharge
Figure 4. 2. Differences on object mean time due to changes on object initial position.
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Figure 4. 3. Differences on object mean time due to changes on trapping factor.
Figure 4. 4. Differences on object mean time due to changes on water temperature. Note that water temperature does not produce any means difference.
© Guatame-García, 2007
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Object velocity
-Discharge and water temperature: The significance of the two-way ANOVA was set at
α=0.05. The results show discharge to have a very highly effect on object velocity (F=400375.5,
p=0.000), but temperature (F=1.936, p=0.258) and the interaction of discharge and temperature
(F=0.001, p=1.000) showed no significant interaction effect.
- Initial position and discharge: The significance of the two-way ANOVA was set at
α=0.05. The results show a significant effect of discharge on object’s velocity (F=15.438,
p=0.013); the effect of initial position on object velocity is also significant (F= 53.243, p=0.01).
In addition, the interaction of discharge and initial position shows to highly affect object
velocity (F= 6.190, p=0.000).
- Initial position and trapping factor: The significance of the two-way ANOVA was set at
α=0.05. The results show a significant effect of initial position on object’s velocity (F=45.093,
p=0.02); the effect of trapping factor on object’s velocity is also significant (F= 15.429,
p=0.013). The interaction of initial position and trapping factor on object’s velocity is
significant (F= 7.617, p=0.000).
As well as object mean time, object velocity is highly affected by discharge, initial
position, and trapping factor (Figures 4.5 to 4.8). Conversely to object mean time, two
interactions were identified to have a significant effect on object’s velocity: initial position-
discharge and initial position-trapping factor.
Figure 4. 5. Differences on object velocity due to changes on river discharge.
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Figure 4. 6. Differences on object velocity due to changes on object initial position.
Figure 4. 7. Differences on object velocity due to changes on trapping factor.
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Figure 4. 8. Differences on object velocity due to changes on water temperature. Note that water temperature does not produce any means difference.
In summary, statistical analysis of the external variables used by the model reveals that
only water temperature does not affect the object’s rate of movement (mean time and velocity).
The results allow to reject the null hypothesis and to accept the alternative hypothesis as true.
4.2.3.2. Comparison of the rate of movement of objects moving either submerged or
floating (Hypothesis B)
Object mean time
The significance of the one-way ANOVA was set at α=0.01. The flotation depth, which
determines whether the body moves either along the bottom or at the surface, have a significant
effect on the object’s mean travel time at the 1% level (F=10.252, p=0.000) (Figure 4.9).
Object velocity
The significance of the one-way ANOVA was set at α=0.01. The flotation depth have a
significant effect on the object’s velocity at the 1% level (F=3.253, p=0.000) (Figure 4.10)
The results allow to reject the null hypothesis and to accept the alternative hypothesis as
true.
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Figure 4. 9. Difference on object mean time due to flotation depth.
Figure 4. 10. Difference of object’s velocity due to flotation depth.
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4.2.3.3. Significance of density over objects’ rate of movement (Hypotheses C and D)
Object mean time
The significance of the one-way ANOVA was set at α=0.05. The results show object
density to have a significant effect on object mean time (F=6.050, p=0.014) (Figure 4.11).
Object velocity
The significance of the one-way ANOVA was set at α=0.05. The results show object
density to have a significant effect on the object’s velocity (F=38.754, p=0.000) (Figure 4.12)
The results allow to reject the null hypothesis and to accept the alternative hypothesis as
true. Also, the high Pearson’s correlation (r=-0.91) supports a) the preponderance of body
density on the definition of the object’s flotation depth, and b) the inverse relationship between
object’s rate of movement and density.
Figure 4. 11. Differences on object mean time due to changes on object density.
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Figure 4. 12. Differences on object’s velocity due to changes on object density.
4.2.3.4. Comparison of the rate of movement of objects heavier and lighter objects (Hypothesis
E)
Object mean time
The significance of the one-way ANOVA was set at α=0.01. The results show a no
significant effect of the object mass on the object’s mean time (F=0.03, p=0.997) (Figure 4.13).
Object velocity
The significance of the one-way ANOVA was set at α=0.01. The results show a no
significant effect of the object mass on the object’s velocity (F=0.020, p=0.980) (Figure 4.14).
The results allow accepting the null hypothesis as true.
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Figure 4. 13. Differences on object mean time due to weight. Note that weight does not produce any means difference.
Figure 4. 14. Differences on object’ velocity due to weight. Note that weight does not produce any means difference.
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CHAPTER 5
DISCUSSION
This chapter summarises and explains the main findings of this research, and discusses
them with regard to the theory presented earlier in Chapter Two. In addition, implications of the
study and recommendations for future work are considered.
5.1 General observations from physical experimentation at the Teusacá and Magdalena Rivers
Objects disposed at high flow velocity sections were observed to travel downstream with
the main flow. Along bends, the objects typically followed a path close to the external river
banks. In most of the cases, the objects moved after the bends with the main surface flow from
the external river bank in the direction towards the opposite bank.
The presence of debris and snags in the river banks altered the direction and velocity of the
surface flow, producing whirls and eddies where the floating bodies got trapped, reducing their
effective longitudinal velocity. The bodies were forced to get back into the main flow because
of the tangential force of the water at the eddies’ borders.
Objects drifting downstream display a circular pattern of movement as result of the driving
forces exerted by the water on different sections of the objects’ surface. This pattern is altered
by the presence of debris and shrubs at the river banks, which invert the direction of the circular
movement, also reducing the objects’ longitudinal velocity.
5.2 External factors affecting the drift of objects along the Magdalena RiverComputational calculations were undertaken considering five extrinsic independent
variables: discharge, water temperature, object initial position, trapping factor and mass
degradation constant. Analysis of results of RM1 and RM2 showed that the minimum and
maximum travel times and velocities were reached by tests in which the data entered was
similar in terms of discharge, object initial position, and trapping factor. Highest discharge
(2170.9) produced the lowest travel times and highest object’s velocities, meanwhile the highest
values in object initial position (0.9) and trapping factor (1.5) derived on the highest travel
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37
times and lowest object’s velocities. These results are consistent with the ANOVA tests, which
demonstrate that these three variables have a significant influence on the object’s rate of
movement
Discharge, understood as the longitudinal velocity of the flow passing through a particular
cross section (Smith and Stopp, 1979), produces an effect on travel time and object’s velocity
that can be explained by the sequence:
high discharge ⇒ high flow velocity
which derives on
high flow velocity ⇒ high object velocity
and then
high object velocity ⇒ low travel time
However, since the velocity in a river cross-section decreases at the banks and the at
bottom due to the friction produced along the channel walls and the river bed (Chow, 1998),
objects disposed at the centre of the channel (0.6) described the highest velocities, whereas
those disposed at the borders (0.9) described the lowest velocities.
The calculated ratios object velocity/flow velocity indicate that under ideal circumstances
(no obstruction) the object may travel slightly faster than the mean flow velocity, and that the
minimum object velocities are equivalent to half-mean flow velocity.
The effect produced by the trapping factor must be understood as an abstraction of the
likelihood of an object to be trapped by substances that are not directly related to the hydraulic
behaviour of the river such as shrubs and debris. The trapping factor variable acts as a “hold
mechanism” that allows the simulation of situations in which the body gets trapped by eddies
such as those observed in the experiment carried out at the Magdalena River. Consequently, its
increment will considerably reduce the rate of movement. Since trapping factor is an abstract
variable, its value resides on the capability of interpreting the external conditions of the river
(vegetation, intrusive structures) as a ratio of trapping.
The relationship identified by the two-way ANOVA between discharge-initial position with
regard to object’s velocity is explained as a “lag” effect. It means that in spite of supposing that
an object will describe high velocity due to the high discharge of the flow, the velocity will be
slightly reduced by the decreasing velocity at the borders. In the same way, the relationship
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38
amongst initial position and trapping factor can be explained as a “double-lag” effect when both
of variables increase. Conversely, when both variables decrease it is a catalyst for increasing
velocity.
In both, RM1 and RM2, water temperature appeared not to have any significant effect on
the object’s rate of movement. However, the correlation between water temperature and mass
degradation factor5 (r= 1) appeared to be almost exclusive. This relationship is explained by the
model’s assumption that an increment on temperature will produce a higher degree of
degradation of organic material. This assumption arises from previous studies (see Chapter 1,
measuring decomposition) that state that temperature is the main factor affecting the rate of soft
tissue decay. Consequently, mass decomposition was defined as a ratio of weight loss acting as
function of temperature and time.
Concerning the relationship between water temperature and residual mass, the correlation
analysis for RM1 does not display any significant relationship between variables. The most
likely explanation is that to have a statistically considerable weight loss, a minimum time is
required that is not reached by this set of tests, since the maximum travel time was identified as
7.29hrs, an interval that is unlikely to describe high rates of mass loss. Conversely to RM1,
RM2 demonstrates a high but not unique correlation between water temperature and residual
mass (r=-0.316). This can be explained by the longer time intervals, which allow the
observation of soft tissue decomposition over extended periods of time at different temperatures
i.e. T232 and T444 yielded the same travel times but their residual masses significantly differed
because T232 was run at 220C while T444 was run at 300C.
5.3 Flotation effects on the object’s rate of movementThe model does not consider flotation depth as a pre-defined variable but it is calculated
from the mathematical reasoning of the buoyancy and hydrostatic forces (i.e. specific gravity)
that act once the body is deposited into a fluid.
The one-way ANOVA revealed a very significant effect of flotation depth on the object’s
rate of movement (p=0.000). It is consistent with the theory that object travel times and
5 Note that mass degradation factor and residual mass are different variables. The first one is the input variable defined as the ratio of mass loss at specific water temperatures; the second one is the amount of mass left after certain period of time.
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velocities depend, among others, on the vertical coordinates that indicate the position at which
the object moves downstream.
Looking at figures 4.9 and 4.10, it can be observed that there is an actual difference on
travel times and velocities as a result of changes in the flotation depth; however, they do not
meet the decreasing scale with depth found in trapezoidal channels. This indicates that the
channel displays an irregular geometric configuration that makes the velocities irregularly
distributed. However, given that the model structure relies on the knowledge of the geometric
configuration of the river at spaced cross-sections, the model is capable of identifying the
velocities at each point of the vertical column, allowing the prediction of the object’s velocity at
the calculated flotation depth.
In addition to the irregular distribution of flow velocities, high object’s velocities near to
the water surface can be explained by the low gravity force displayed by a buoyant object. This
kind of drift has only a longitudinal direction that reduces the resistance force, and then
increases the object’s velocity.
Examining the Pearson’s correlation coefficients, it can be stated that in spite of the
significance of the effect of flotation depth on the object’s travel time and velocity, variables
such as initial position and trapping factor have higher influence on the rate of movement. Thus,
these variables cannot be single-handled, but their influence must be analysed as a coalescence
of interrelations in which the impact is to some extent predictable.
It is worth mentioning that the calculated flotation depth remains the same along the whole
of the object’s route, and hence, it does not describe the movement of the body in the water
column. According to Cotton et al. (1987) a human body disposed in water initially sinks, and
may resurface depending on water temperature. With fairly warm water temperatures, a body
can be expected to surface within a few days, whereas with cold or near-freezing water
temperatures, resurfacing of the body can be delayed for several weeks to several months.
However, because the model is not able to simulate these changes in flotation depth, the
significance of the relationship between flotation depth and water temperature cannot be
established at this time.
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5.4 Intrinsic factors affecting the object’s rate of movementA preliminary hypothesis based on Donoghue and Minniguerode (1977) and Boaz and
Behrensmeyer (1976), stated that body density is correlated with the average rate of movement.
The one-way ANOVA proved this alternative hypothesis yielding a significance of p=0.014 for
object mean travel time and p=0.000 for object’s velocity.
Also, the strong proportional relationship between density and flotation depth (r=0.93)
demonstrated the direct effect of density on the vertical location of the body and its “transitive”
effect on the object’s rate of movement. Nevertheless, density appears not to be the only factor
affecting whether a body will float or sink, although it seems to have a high influence.
When the data introduced into the model slightly varied the body volume with respect to
the body mass, results significantly differ in terms of flotation depth. This phenomenon obeys to
the fact that the human density is very close to that of the water, and small variations in body
density, defined as the ratio of mass to the volume, will critically affect body buoyancy
(Donoghue and Minniguerode, 1977). It demonstrates the fragile equilibrium of the buoyant
forces at densities very close to 1, which is the case of the human body.
While decomposing, putrefactive gas formation in the chest and abdomen decreases the
density of the body and creates sufficient buoyancy to allow it to rise to the surface and float
(Donoghue and Minniguerode, 1977). However, the model is not able to simulate the changes
on the body’s density, and the pattern of both vertical and longitudinal movement cannot be
studied at this time.
With regard to the body weight, ANOVA tests proved that there is not any significant ef-
fect of this variable on the object’s rate of movement. This finding is in agreement with Boaz
and Behrensmeyer (1976) who also proved that there is not a correlation between weight and
rate of movement. A possible explanation to this observable fact is that weight is acting more as
one of the variables defining density than one factor affecting movement itself.
The residual mass appears to be mainly correlated to the object mean time (r=-0.428), and
lesser to the water temperature (r=-0.316). This correlation was expected as the residual mass
results from the mass degradation constant which is time and temperature dependent.
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5.5 Study implications and limitationsAlthough the simulated results are in good agreement with the physical observations and
hydraulic theory, the model does not show with complete reliability how a human body will
travel throughout the river for a number of reasons.
First, although the mannequin used for the calibration satisfied the proportions of a human
body (height, weight and diameter), the material from what it was made (wood) just allowed the
observation of floating trajectories. In addition, once partially submerged, the mannequin
underwent some changes in weight and density due to the wood’s absorption of water that could
not be accurately measured.
Second, human body shape, changes in density as the body decomposes, and
disarticulation, were not represented in the modelled environment since those simulations
require: a) knowledge on the geometry of human body parts; b) information about body parts’
densities; c) knowledge on the changes in body density at particular stages of decomposition;
and d) the quantification of the process of disarticulation either as a ratio of mass loss, or as a
cutoff of ADD that must be yielded for a body part to become separated from the trunk. In
general terms, the aforementioned variables must be quantified to be derived into a
mathematical function that would be able to be introduced into the computer model.
Third, even though the model takes into account five extrinsic and four intrinsic variables
that have shown to distinctly affect the rate movement of objects drifting along the Magdalena
River, there are several other variables that may also affect this pattern, for instance, the
increasing on river discharge due to the rain seasons or the floatation of the body because of the
trapping of air into the clothes.
Fourth, the mass degradation factor was calculated based on the degradation of organic
waste, and did not take into account the measuring of decomposition in ADD’s. Thus, it is
suggested that future work will include measurements of ADD for calculating the ratio of mass
loss, which also will include the process of disarticulation.
5.6 Discussion summaryA computer model was used to simulate the drift of bodies along 338kms of the Magdalena
River, at the Magdalena Medio region. The 972 tests performed allowed the identification of the
© Guatame-García, 2007
42
main intrinsic and extrinsic factors affecting the rate of movement of objects which composition
was similar to the human body.
Discharge, object initial position, and trapping factor were established as the most
important extrinsic factors affecting the objects’ travel times and velocities. Body density,
affected in turn by body volume (mass, length and diameter), was identified as the solely
intrinsic factor affecting the rate of movement, since it determines the body position in the
water column.
The model must be adjusted according to the analysis of its limitations, but despite those
shortcomings, is a reliable tool for predicting probable trajectories that an object similar to the
human body may travel, and the travel times it could take to reach specific locations.
The experiments carried out at the Teusacá and Magdalena Rivers allowed the observation
of the pattern of movement of objects moving downstream either floating or submerged during
specific time intervals. This initial experimentation supports the Nawrocki et al. (1997)
statement which suggests that if the features of a specific water environment are known,
relationships between decomposition, transport, and time since death can be established.
Although these experiments mainly refer to transport and submersion interval, they constitute a
preliminary proposal for further experimentation.
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43
CONCLUSIONS
In this paper a one-dimensional hydraulic model has been coupled with an object transport
model in order to predict the object’s drift trajectories and distance travelled with time. The
transport of objects was modelled taking into account buoyant, hydrostatic and dynamic forces
calculated by using velocity, discharge and depth computed by the numerical hydraulic model.
Results and information from previous research studies were incorporated into the modelling
framework to represent the transport of living and dead human bodies with different densities
and specific gravities. Nevertheless, basic assumptions and simplifications were made in the
derivation of the simulation, and therefore its application is certainly constrained to particular
conditions and its accuracy must be carefully interpreted.
Thanks to the numerical model, intrinsic and extrinsic variables affecting the pattern of
transport of objects along the Magdalena River were controlled and assessed. This assessment
led to an identification of the factors that may become critical at the time of evaluating forensic
cases with regard to the calculation of submersion intervals and the prediction of points of entry
and recovery.
The model allowed the calculation of the mean and minimum times a particular body
requires to arrive at certain locations under specific environmental conditions. Consequently, in
Colombia this model can help to satisfy the need of investigating crimes which have used the
‘river-concealing’ technique to impede the location and recovery of human remains.
The coupled method developed in this study is a promising computational tool to simulate
the drift of objects similar to the human body along a river system. The tests have supported the
accuracy of the present model with regard to the prediction of body velocity, mean travel time,
and flotation depth. However, further verifications are still needed for complex body
composition and motion, during which calculations of the object’s pattern of movement in the
water column, changes in body volume resulting from inhaled water, and mass loss due to
disarticulation represent a considerable challenge.
The model can be applied to any river from which discharges and geometrical constitution
at several cross-sections are known.
This research allows the understanding of the application of computer modelling in the
analysis of processes of transport concerning bodies disposed in water systems. It is also an
© Guatame-García, 2007
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example of the use of technological tools in the investigation of forensic cases and the
interdisciplinary collaboration.
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Vass, A., W. Bass, J. Wolt, J. Foss, J. Ammons (1992). “Time Since Death Determinations of
Human Cadavers Using Soil Solution” Journal of Forensic Sciences 37(5):1236-1253.
Velandia, C. (2005). “Obtención de la constante de reaireación y modelación de la calidad del
agua de un río de montaña colombiano (Río Teusacá)” Dissertation submitted for BSc.
Environmental Engineering.
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APPENDIX 1DISRIBUTION OF VELOCITIES FOR THE MAGDALENA RIVER
MAGDALENA MEDIO REGION
The distribution of velocities at Arrancaplumas, Cambao, and Nariño stations is shown.
Note that in all cases higher velocities are described near to the water surface (0.2 of the total
depth).
Figure A1. 1. Velocities distribution at Arrancaplumas station during a high discharge period.
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Figure A1. 2. Velocities distribution at Arrancaplumas station during a low discharge period.
Figure A1. 3. Velocities distribution at Cambao station during a high discharge period.
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Figure A1. 4. Velocities distribution at Cambao station during a low discharge period.
Figure A1. 5. Velocities distribution at Nariño station during a high discharge period.
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Figure A1. 6. Velocities distribution at Nariño station during a low discharge period.
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APPENDIX 2LABORATORY REPORT
WOODEN OBJECTS’ DENSITY ESTIMATION
Summary
Two laboratory sessions for volume and density estimation of wooden objects were carried
out at the Hydraulic Laboratory of the National University of Colombia (UNAL).The objects
would be used for physical experimentation at the Teusacá and Magdalena Rivers.
The estimations combined the water submersion technique based on the Archimedes’
principle for objects volume calculation, and the numerical estimation of density. Results were
used as reference for drift observations.
Participants
Researchers: Ana Carolina Guatame-Garcia, BSc; Luis Alejandro Camacho, PhD.
Assistant: Luz Adriana Guatame-Garcia BSc. Geology student, UNAL.
Introduction
Archimedes' principle stated that when a body is fully or partly immersed in a liquid, that
body experiences an upward force (buoyant force) equal to the weight of the displaced liquid.
The displaced liquid is that volume of liquid equal to the volume of the body below the water
surface.
Density is given by the relationship between mass and volume of an object. Direct
determination of density can be made by means of mass and volume measurement. Mass is
determined by weighting, and volume can be estimated based on the Archimedes’ principle.
Instruments
• A calibrated balance with platform
• Two water tanks
• One measure tape
• Stick tape
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• Thermometer
• Three catch buckets
• A straight plastic tube (pipette)
Methods
Volume calculation
Estimation of volume was attempted through the overflow method, which consists on
measuring of the volume of water displaced by an object submerged in the liquid. Calculations
were made for:
- An object of irregular shape composed by three wood sticks and an iron bar
- Wooden human-like mannequin
1. Objects’ weighing
Objects were weighed by a calibrated balance with platform (Figures A2.1 and A2.2). Both
dry and wet weights were registered for comparison.
Figure A2. 1. Calibrated balance and tank for solid volume estimation.
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Figure A2. 2. Weighing of the wooden mannequin by using the balance.
2. Irregular object volume estimation
A small circular tank was filled with water until it weighted 168kg and the water level was
38.5cm. Water level was marked at the tank walls with stick tape, and the level recorded. The
object was introduced into the water until it was fully submerged (Figure A2.3) and the weight
of the water with the object, and the water level, were measured.
Figure A2. 3. Irregular object submersion
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Afterwards, the displaced water was taken out of the tank through a lateral valve, and the
content disposed in several buckets until the water level went down to the first reported level.
Water contained into the buckets was measured by using a pipette.
3. Wooden mannequin volume estimation
A big overflow rectangular tank was filled with water until it reached the border of the
tank’s side spout; the mannequin was introduced into the water afterwards (Figure A2.4). The
overflowed water was caught by several buckets located underneath the tank. Once the buckets
got filled, the water was placed into another tank to be weighed by the balance. Results were
recorded.
Since the mannequin was not completely submerged, an external force was used to fully
submerge the object and the water displaced caught and measured.
Figure A2. 4. Mannequin’s submersion in an overflow tank.
4. Water temperature
Water temperature was measured by using a thermometer introduced into the water for 60
seconds.
Results
Measurements and numerical estimations are presented in table A2.1.
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Table A2. 1. Volume and density estimations for wooden objectsIrregular Object Wooden Mannequin
Dry weight 21kg 43.8kgWet weight 22kg 45kg
Volume of water displaced 21647ml 48882mlObject’s Density 1.02gr/cm3 0.98gr/cm3
Water temperature 160 160
Water density 0.998gr/cm3 0.998gr/cm3
Specific gravity 1.02 0.98
In spite of the small dimensions of the tanks, wooded objects tend to remain afloat since the
wood’s density is lower than that of the water. However, observed results indicate that the iron
bar helps for increasing density, and then, it makes the object to slightly sink.
These observed and calculated values would be used as reference for tracking the drift of
both the irregular wooden object in the Teusacá River, and the mannequin in the Magdalena
River.
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APPENDIX 3LABORATORY REPORT
OBJECT’S DRIFT TEST 1
Summary
Two sets of experiments in a straight section of the Teusacá River (Cabaña Gauging
Station, Figure A3.1) were carried out to identify the pattern of transport of submerged and
floating objects. A flow gauge was performed to obtain measurements of velocity and
discharge. The identified path of movement and its relationship with discharge were used to
calibrate the numerical model developed in this research.
Figure A3. 1. Mid-section of the Teusaca River, Cabaña Gauging Station.
Participants
Researchers: Ana Carolina Guatame-Garcia, BSc; Luis Alejandro Camacho, PhD.
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Assistants: Luz Adriana Guatame-Garcia and Mauricio Cantor. BSc. Geology and MSc.
Hydraulic Resources students, UNAL.
Introduction
The Teusacá River (Figure A3.2) is a tributary of the Bogotá River, which is in turn,
tributary of the Magdalena River. The river is located in the Cundinamarca Department, central
region of Colombia. The reach of study located at Cabaña Station (120 51’ 51” North, 650, 30’,
3” West) yielded 47m length, 0.48m average depth, and 7m average width.
Figure A3. 2. Teusaca River. Red box indicates the location where the experiments were performed. (Velandia, 2005: 47)
The section starts after a pronounced bend and is characterised by the presence of several
shrubs and debris in both the left and right banks, where the centre of the reach is the thickest
area (Figure A3.3). Along this reach, seven runs were devised to test the movement of a
wooden object both floating and submerged. The differences in the pattern of movement were
produced due to the variation in the object’s density, as was explained before in Appendix 1.
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Instruments
• Flow meter: Device composed by a vertical axis and a rotor which, introduced in the wa-
ter measures the velocity of flow at a particular point. The number of revolutions of the
rotor is obtained by an electric circuit through a contact chamber; the electrical impulses
produce a signal which registers a unit in a counting device. Intervals of time are meas-
ured by an automatic timing device (Herschy, 1998).
• Digital Photo Camera
• Stopwatch
• Wooden object
• Iron bars
Figure A3. 3. Schematic diagram showing the configuration of the reach of study at Teusacá River.
Methods
Discharge calculation
Discharge was calculated through the velocity area method. This method is based on the
principle Q = V*A (Herschy, 1998).
The area of the cross-section was determined from soundings at each 50cm over the cross-
section. Velocity was measured by using a flow meter introduced into the water flow path. The
flow meter counted the revolutions of the rotor during 30 seconds time interval at points 0.2.,
0.6 and 0.8 of depth from the water surface, at spaced vertical positions from bank to bank
(Figure A3.4).
Discharge was obtained as
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Q= Σ Ai Vi
First test
An object weighting 20kg (ρ = 0.75g/cm3) composed by four pieces of wood was used to
examine the way in which a body floating in the surface move downstream. The object was
released four times, one at the left and right banks of the stream and two at the centre of the
channel, and allowed to float freely. The object floated with about 70% of the volume
submerged. Orientation of the element relative to the current direction was changed for each of
the runs located at the centre of the channel.
The movement was hand drawn and video recorded as the object drifted downstream, and
later plotted in a personal computer using GNU Plot© data plotting programme for Linux.
Figure A3. 4. Measurements of flow velocity by using a flow meter at spaced positions. The figure shows the measurements taken at 0.2 of the total depth of the section.
Second test
A piece of wood weighting 5kg was removed from the object used in the first test, and
replaced by an iron bar of the same weight for increasing density (ρ = 1.02); thus the submerged
movement of the object was observed.
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The object was released three times, one at each bank of the stream and one at the centre.
The object drifted downstream suspended by the flow with 100% of the volume submerged;
however, complete sinking of this object was not produced.
The time to reach the end of the section was recorded by using a stopwatch, which yielded
the average travel times for each of the runs.
The movement while submerged was drawn and video recorded as the object drifted
downstream, and later plotted in a personal computer using GNU Plot© data plotting
programme for Linux.
Results
Through the velocity area method, discharge for the reach of study was estimated as
1.97m3/s, with a cross sectional mean velocity of 0.53m/s (Table A3.1). No changes at the stage
were perceived, which means the discharge remained steady along the whole experiment. The
lowest cross sectional velocity, V = 0.149m/s, was yielded at the right river bank (y = 0.52m,
distance from the left border = 6.50m), and the fastest, V = 0.778m/s, was yielded at the centre
of the channel (y = 0.51m; distance from the left border = 3.50m). Nevertheless, the deepest
points of the river were reached between the 4 and 4.50m from the left river bank.
The fastest flow at 0.2 (surface), V = 0.629m/s, was identified at 4m from the left bank, and
the fastest flow at 0.6 (middle depth), V = 0.851m/s, was identified at 3.5m from the same bank.
These observations demonstrate the symmetric geometric configuration of the river that
behaves as a trapezoidal channel, in which the maximum velocities are reached at the centre of
the channel at some distance from the water surface, and the lowest ones at the river banks (See
figure 2.4, Chapter Two).
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Table A3. 1. Flow Gauging data gathered from Teusacá experiment.DATE June 23rd
2007
STATION Total Width (m) 7
TimeStart 9:20 Discharge 1.970 m3/s
Final 11:20 Velocity 0.583 m/s
TEUSACÁ RIVER - CABAÑA GAUGING ST. Stage
Start 0.71 m Mean Area 3.380 m2
Final 0.71 m Mean Depth 0.483 m
D from the
border (mt)
DEPTH (m)
Turns
N
VELOCITIES (m/s) SECTION
MD (m) PW (m)
SP (m2)
TD GD (No) T (s) N/T VP VMV VM
Partial Q (m3/s)
0.000 0.28 0 0 0 0.000 0.145 0.36 0.500 0.180 0.0261
0.50 0.44 0.2 28 30 0.933 0.237 0.6 36 30 1.200 0.305 0.290 0.8 37 30 1.233 0.314 0.352 0.445 0.50 0.223 0.0783
1.00 0.45 0.2 39 30 1.300 0.331 0.6 50 30 1.667 0.425 0.414 0.8 56 30 1.867 0.476 0.494 0.46 0.50 0.230 0.1136
1.50 0.47 0.2 57 30 1.900 0.484 0.6 73 30 2.433 0.621 0.574 0.8 67 30 2.233 0.570 0.632 0.475 0.50 0.238 0.1500
2.00 0.48 0.2 72 30 2.400 0.612 0.6 85 30 2.833 0.723 0.689 0.8 82 30 2.733 0.698 0.723 0.48 0.50 0.240 0.1736
2.50 0.48 0.2 73 30 2.433 0.621 0.6 95 30 3.167 0.809 0.757 0.8 93 30 3.100 0.792 0.764 0.495 0.50 0.248 0.1891
3.00 0.51 0.2 70 30 2.333 0.595 0.6 96 30 3.200 0.817 0.770 0.8 100 30 3.333 0.851 0.777 0.51 0.50 0.255 0.1981
3.50 0.51 0.2 69 30 2.300 0.587 0.6 100 30 3.333 0.851 0.783 0.8 99 30 3.300 0.843 0.778 0.52 0.50 0.260 0.2022
4.00 0.53 0.2 74 30 2.467 0.629 0.6 94 30 3.133 0.800 0.772 0.8 101 30 3.367 0.860 0.771 0.53 0.50 0.265 0.2044
4.50 0.53 0.2 72 30 2.400 0.612 0.6 96 30 3.200 0.817 0.770
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0.8 98 30 3.267 0.834 0.721 0.515 0.50 0.258 0.1857
5.00 0.50 0.2 55 30 1.833 0.467 0.6 85 30 2.833 0.723 0.672 0.8 91 30 3.033 0.775 0.649 0.505 0.50 0.253 0.1638
5.50 0.51 0.2 59 30 1.967 0.501 0.6 76 30 2.533 0.647 0.625 0.8 83 30 2.767 0.706 0.577 0.505 0.50 0.253 0.1457
6.00 0.50 0.2 40 30 1.333 0.339 0.6 68 30 2.267 0.578 0.529 0.8 73 30 2.433 0.621 0.414 0.51 0.50 0.255 0.1056
6.50 0.52 0.2 38 30 1.267 0.322 0.6 35 30 1.167 0.297 0.299 0.8 33 30 1.100 0.280 0.149 0.45 0.50 0.225 0.0336
7.00 0.38 0.2 0 30 0.000 0.000 0.6 0 30 0.000 0.000 0.000 0.8 0 30 0.000 0.000 Sum 1.970
Figure A3.5 shows the pattern of movement identified for the object moving at the water
surface. The object was released four times just after the bend at which the reach started, and
yielded different travel times ranging between 60 and 73 seconds, being 62.5s the average travel
time, and 0.75m/s the average object’s velocity.
Entry 1 was located at the right river bank and oriented in the same direction of the flow.
The object initially moved towards the opposite bank due to the characteristic spiral flow of
meanders that generates the maximum velocities at the concave bank, and the transport of
substances towards the convex bank (see figure 2.6, Chapter Two). Once oriented to the left
bank the object crashed against the shrubs and this movement drifted the object towards the
right bank. The object never left the middle section of the channel, and yielded the shorter time
interval to reach the section (fastest velocity = 0.78m/s).
Entries 2 and 3 were placed at the centre of the channel. The first one was oriented in the
same direction of the flow, and the second one perpendicular to it. Both of them stayed at the
middle section of the channel, although entry two experienced a mild bending movement along
the travel segment. Entry three undergone a straight movement, but travel times did not differ
substantially. Hence, orientation of the object to the current direction seems to affect the kind of
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movement the object will undergo while drifting downstream but does not affect the velocity of
the object.
Entry four was located at the left river bank where lowest velocities and discharge were
identified. The object was not able to get the fastest flow and moved along the same bank
throughout the whole section. Consequently, this run experienced the longer time interval to
reach the travel segment.
Figure A3. 5. Object’s pattern of movement along the surface. Teusacá River. Arrow head indicates the object’s orientation.
Due to the elongated-like shape of the object, a circular pattern of movement was produced.
It results from the water velocity variation at each micro-section of the object, and the torque-
like effect it produces.
Figure A3.6 shows second set of experiments devised with the object suspended by the
flow with 100% of its volume submerged. Due to the location of the object in the water column,
interpretations can be derived from its relationship with the flow observed at 0.6 of the total
depth.
The object was released three times just after the bend at which the reach started, and
yielded different travel times ranging between 58 and 63 seconds, being 64.3s the average travel
time, and 0.73m/s the average object’s velocity.
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Figure A3. 6. Object’s submerged pattern of movement at Teusacá River. Arrow head indicates the object’s orientation.
In this case, entry one was located at the right river bank. As well as occurred in entry one
at the first set of experiments, the object started its movement drifting towards the opposite river
bank due to the deviated direction of the current resulting from the bend located just before the
point at which the reach started. The curve followed by the object this time tended to be more
dramatic since the object was drifted at a level in which discharge and velocity yielded their
highest values. In spite of this, the object took a longer time interval to reach the section in
comparison to entry one at the first set of experiments (68s vs. 60s). This fact can be explained
by the increased density of the object and the sharp curve movement it did once it reaches the
opposite bank.
Entry 2 was released perpendicular to the flow direction at the centre of the channel and
underwent a very similar pattern of movement to entry 2 at the first set of tests, although this
time the time interval was shorter than the first one (58s vs. 63s). These variations would be
associated with the different orientation of the object at the time of entry. The object did not
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move out of the central section of the channel and was transported by the fastest current
producing a circular movement.
The object at entry three was also placed perpendicular to the flow direction. The releasing
biased towards the left river bank close to the central section, however. The object remained at
this banks for approximately 15m and latter trapped by the fastest flow crossing to the central
section of the channel. This fact explains why it registered a longer time interval in comparison
to entry 2.
In both sets of experiments the object’s velocity was over the reach’s mean velocity. This
phenomenon can be explained as result of the general tendency of the object to follow the
fastest area of the section. Nevertheless, the object’s velocity is always less than the fastest
cross sectional velocity reported by the channel.
Conclusions
The aforementioned observations support that:
1. Once trapped by the fastest flow, objects stay at that section of the channel until an ex-
ternal factor comes into the travel. In this case, the fastest flow was identified at the
centre of the channel and most of the times the object drifted downstream following this
section.
2. Objects underwent a circular pattern of movement as result of the driving force exerted
by the moving water. This pattern is altered by the presence of debris and shrubs at the
river banks. These obstacles sometimes invert the direction of the circular movement or
may trap the object for some time until the current is strong enough to get the object off
of the obstacle.
3. Increased density affects the velocity of transport. Objects that experience submerged
movement tend to drift slower because of the different flow velocities in the water
column.
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APPENDIX 4LABORATORY REPORT
OBJECT’S DRIFT TEST 2
Summary
An experiment for tracking the drift of a wooden mannequin in a 10.7km stretch of the
Magdalena River was carried out. Travel times to reach each of the six subsections along the
stretch were recorded in order to obtain the object transport velocity and to calibrate the object
tranport model. Pattern of movement regarding longitudinal sections of the river, meanders, and
influence of shrubs and debris, and external structures was observed.
Participants
Researchers: Ana Carolina Guatame-Garcia, BSc; Luis Alejandro Camacho, PhD.
Assistants: Luz Adriana Guatame-Garcia BSc. Geology student, UNAL. Carlos J. Guatame. ,
Ricardo González, MSc. Water Resources student, UNAL
Introduction
The Magdalena River is 1612km long and drains a 257.438km2 basin. It is the largest
fluvial system in Colombia and originates in the Andean Cordillera at an elevation of 3300m
(Restrepo, 2001).
The studied river stretch extended from the Variante Bridge to the gas pipe of the town of
Girardot, a distance of 10.7km downstream (Figure A4.1). This section has a mean width of
164.47m width, a longitudinal slope of 0.0070483, and a mean flow discharge of 1200m3/s
(Uniandes-ACUAGYR, 2005).
The stretch is characterised by four main bends located at the pumping station of Girardot
Town, the Bogotá River’s mouth, and Chicalá and Honda discharges, at which strong eddies are
produced. The river banks are also distinguished at some sections by their thick vegetation and
the releasing of debris and shrubs into the water.
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Figure A4. 1. Experimental stretch, Magdalena Medio region. Circle shows the stretch’s total length.
Instruments
• Wooden mannequin
• Motor boat
• Video Camera
• Digital Photo Camera
• Stopwatch
• Rope
© Guatame-García, 2007
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Methods
A wooden mannequin weighing 45kg that yielded a density of 0.98gr/cm3 was released at
the Variante Bridge, a known hydrological control point of the Magdalena River. The body was
disposed at the centre of the flow without specific orientation relative to the current direction
(Figure A4.2).
Figure A4. 2. Mannequin’s releasing in the Magdalena River
The mannequin floated freely with about 85% of the volume submerged. Six control points
were previously established to measure the time the body took to reach each subsection.
Tracking of the mannequin was drawn, photographed and video recorded from a boat as the
body drifted downstream (Figure A4.3).
Figure A4. 3. Mannequin getting trapped into an eddy.
© Guatame-García, 2007
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The average travel time along the whole section was calculated, as well as the ratio flow
velocity to body velocity. The river flow discharge during the experiment was obtained using
the stage-discharge relationship of Nariño flow gauging station. The measured discharge was
900 m3/s.
Results
The mannequin was released at the middle section of the channel, slightly biased towards
the left bank. It starts moving towards the middle-right portion of the channel, at which the
deepest points of the river at this section are found (Figure A4.4).
Figure A4. 4. Release cross-section of the Magdalena River at the Variante Bridge Station. X axis: river width, Y axis: Elevation.
Along the bends, the mannequin followed a path close to the external river banks. In most
of the cases, the object moved after the bend with the main surface flow from the external river
bank in the direction towards the opposite bank. When the object was again located at the centre
of the channel, it travelled with the main flow.
The object travel time was compared to the distance travelled, to obtain the object’s
velocity at each subsection. The highest velocity was obtained at the Ospina Bridge –
Ferrocarril Bridge subsection (V = 3.97m/s). Through this straight section, the object tended to
move at the centre of the channel with the main flow and was not affected by external obstacles.
The lowest velocities were reached at the Sumapaz River – Girardot’s gas pipe (V= 1.77m/s),
and Bogotá River – Ospina Bridge (V=1.73m/s) subsections. These results answer the fact that
© Guatame-García, 2007
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in both sections the object got trapped by some eddies produced by the structures located at the
east river bank. Summary of the experiment results are shown in table A4.1.
Table A4. 1. Data obtained from the mannequin drift experiment.
Reach Discharge Length Depth Cross-
sectionalMean Flow
Max Flow
Observed Object
Observed Object Vob/Vfl
(m3/s) (m) (m)Area* (m2)
Velocity (m/s)
Velocity (m/s)
Travel Time (s)
Velocity (m/s)
Variante Bridge - Sumapaz River 950 1125 9.34 340.16 2.79 4.11 420 2.68 0.96
Sumapaz River - Flow Derivation Structure Girardot city 800 2225 7.02 385.71 2.07 2.70 1260 1.77 0.85Flow Derivation Structure- Bogota River 900 2150 7.02 385.18 2.34 3.04 910.2 2.36 1.01Bogotá River - Ospina Bridge 900 1150 5.54 420.91 2.14 3.14 666 1.73 0.81Ospina Bridge - Ferrocarril Bridge 968 1000 6.20 494.86 1.96 2.88 252 3.97 2.03Ferrocarril Bridge - Gas Pipe 968 3050 6.41 345.40 2.80 4.12 582 5.24 1.87
Total or Average 914 10700 2.41 4090.2 2.62 1.09
Partial Conclusions
- The mannequin tended to be drifted by the faster flows until external factors such as
debris, shrubs, or other structures, obstructed the object’s path. In this case, the faster
flow was located at the centre-right section of the channel due to the additional dis-
charges produced by the confluence of the Sumapaz and Bogotá Rivers, and the river
geometric configuration (Figure A4.4)
- Bends also alter the path of movement since they moved the object towards the convex
bank. This pattern is similar to that observed in the experiment carried out in the
Teusacá River.
- The mannequin moved downstream following a circular pattern resulting from the dif-
fering forces exerted by the moving water on different sections of the mannequin’s sur-
face. Trapping of the body by external factors altered this movement reducing its longit-
udinal velocity due to the discontinued movement of the object.
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APPENDIX 5OBJECT DRIF MODELLING AND CALIBRATION
By Dr. Luis Camacho
Water temperature, as well as suspended and dissolved solids concentrations, affects
water density. One of the more useful relationships relating density to temperature, dissolved
solids, and suspended solids is (Martin and McCutcheon, 1999),
sssTw ρρρρ ∆+∆+= (1)
where, ρw is the water density (kg m-3), ρT is the density of pure water as a function of
temperature, and sρ∆ and ssρ∆ are the changes in density due to dissolved and suspended
solids, respectively. Density is best calculated as a function of temperature using the Thiesen-
Scheel-Diesselhorst equation (Martin and McCutcheon, 1999),
( )( ) ( )
−+⋅
+−= 29863.312963.682.508929
9414.28811000 TT
TTρ (2)
where, T is water temperature in °C. In turn, the most generally accepted relationship between
dissolved solids or salinity and density is,
(3)
where, CSL is salinity in g of salt per kg of seawater or ppt (parts per thousand) written as %, and
T is temperature in °C. The effect of suspended solids concentrations on density can be
calculated as,
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5.12643
49
37253
108314.4
)106546.1100277.11072466.5(
)103875.5
102467.8106438.7100899.4824493.0(
SL
SL
SL
s
C
CTT
CT
TTT
−
−−−
−
−−−
×+
⋅×−×+×−+
⋅×+
×−×+×−=∆ ρ
75
31011 −⋅
−=∆
SGCssssρ (4)
where, Css is the suspended solids concentration (gm-3 or mg/L), and SG is the specific gravity
of the suspended solids.
The volume of an object, ∀ b, of initial mass Mbo and density ρb is,
b
bob
Mρ
81.9⋅=∀ (5)
Assuming that the object is mainly of a cylindrical shape, its length Lb will be related to its
diameter, Db and volume by,
24
bD
Lb ⋅∀=
π (6)
Due to hydrostatic and buoyancy forces the submerged depth, yb, in water of density ρw of such
a cylindrical object will approximately be,
bbw
bob LD
My⋅⋅⋅
⋅=81.9
81.9ρ (7)
If yb is less than Db the object will move of float at a depth yfl equal to,
bfl yyy −= (8)
where, y is the total hydraulic flow depth. Otherwise, if yb computed by means of Eq. (7) results
to be greater than Db, the object will sunk and remain or move close to the river bed,
bfl Dy ⋅= 05.1 (9)
The previous theory could be used to compute the submerged depth, and the depth at which
approximately a human body of cylindrical abdomen shape will move along a river. As the
cylindrical shape is just an approximation, it should be noted that the length computed by Eq.
(8) will be a bit greater than the actual body abdomen length. In that case, however, to
determine the submerged body depth, the maximum abdomen length of the actual body
multiplied by a factor of 1.12 should be considered in Eq. (7). Also note that in order to obtain
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realistic submerged depth values, the actual body diameter should be in close correspondence to
body mass and weight.
Once equilibrium has been established between dynamic and resistance forces on the
floating or sunk object, it will move downstream at the corresponding flow velocity depending
on its horizontal location at the cross section and floating depth.
Due to the horizontal and vertical velocity distributions that develop in open channels, an
object will move at a velocity given by its relative position along the channel cross-section. In a
trapezoidal channel the object will move at the mean flow cross-sectional velocity, as computed
by a one dimensional hydraulic model, if it is floating at a depth yfl = 0.6y measured from the
channel bed towards the water surface, and its relative horizontal position, x1, is either x1 =
0.22B or x1 = 0.88B (where B is the channel cross-sectional width). Therefore, in order to
compute the object transport velocity Vb at a given location along the cross-section (x1, yfl),
VKKVV vhyflxb ⋅⋅== ,1 (10)
where, Kh is a factor taking into account the horizontal velocity distribution that develops in an
open channel. It is used to compute the object flow velocity, Vx1, at a relative position x1 from
the left bank of the channel, as a function of mean flow velocity V . In turn Kv is a factor taking
into account the vertical velocity distribution that develops in an open channel. It corrects the
object transport velocity, Vfl, floating at a given depth yfl, as a function of mean flow velocity at
a floating depth of 0.6y, as computed by one dimensional hydraulic models.
Using data of thirty velocity area gauging measurements of the Magdalena River
(Uniandes-ACUAGYR, 2005), the following expressions for the correcting factors Kh and Kv
were obtained in this work:
⋅+
⋅−==
Bx
Bx
VVK x
h1
211 84.584.5 (11)
( )
( )
=
6
6
0342.0/18log
0342.0/30
log
ny
ny
K
fl
v (12)
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where, n is the Manning-n channel roughness coefficient; x1 is the fraction of the channel width
B, (x1 = 0 at the left bank and x1 = 1 at the right bank) describing the relative position where the
object is located along the cross-section; V is the mean flow velocity, as computed by a one
dimensional distributed model based on the full Saint Venant equations, or by a hydrological
model such as Muskingum-Cunge (1969); and other variables have been described above.
Equations (11) and (12) are specific for the Magdalena River after calibration of the general
equations describing velocity distributions in natural channels (Chow et al., 1988; Chiu and
Hsu, 2005). Interestingly, Eq. (10) predicts a maximum flow velocity of about Vmax = 1.47V at
the centre of the cross-section and a flow depth of 0.6y. This result is in perfect match with the
findings of (Xia, 1997) in different stretches of the Mississippi River.
The object moving downstream, can also get temporally trapped, due to recirculating
eddies and hydraulic dead zones, and by substances that are not directly related to hydraulic
factors e.g., shrubs, debris. These effective “lag or holding mechanisms” could be modelled
considering an external “trapping factor”, TF, as it is commonly performed in solute transport
applications, when the effect of transient storage or dead zone mechanisms are considered (Van
Mazijk, 1996; Camacho, 2000). Therefore, the effective object transport object velocity
downstream along a river system will be finally given by,
TF
VKKV vhb +⋅⋅=1
(13)
Therefore, the object travel time to a downstream location at a longitudinal river distance L, will
be given by,
bb V
Lt = (14)
Model CalibrationThe above modelling framework was implemented in a MATLAB (The Mathworks, Inc.)
computer program code, and coupled with a distributed hydraulic model (Camacho and Lees,
1998). The resulting object transport model was calibrated using data collected in the Object’s
drift tests 1 and 2 (see Section 3.1 and Section 3.2, and Appendices 3 and 4). The model
parameters roughness Manning-n and trapping factor, TF, where calibrated using observed data
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of travel time and object velocity. In this work, calibration, parameter uncertainty, identifiability
and sensitivity analysis of model parameters for both TS and ADZ models was investigated
using Monte-Carlo based methods. The concepts of the Generalised Likelihood Uncertainty
Estimation methodology (GLUE, Beven and Binley, 1992) are applied using the Monte-Carlo
Analysis Toolbox (MCAT, Lees and Wagener, 2000). Within the methodology, the
identifiability of model parameters were partly examined using scatter or dotty plots, where the
individual parameter values of each Monte-Carlo realization were plotted against the objective
function values (Nash determination coefficient) evaluated for the prediction results of the
observed data provided by each Monte-Carlo realization. In addition, the cumulative
distributions of 10 classes of the parameter space ranked on the likelihood measure or the
goodness of fit were plotted together, in the so-called Regional Sensitivity Analysis plot. Both
model parameters result to be sensitive parameters, as revealed by strong differences in the
cumulative distributions of each class.
The optimum model parameter values, Manning-n roughness coefficient and trapping
factor, for the Magdalena River result to be respectively 0.025 and 0.4. The calibration results
are considered very good with an overall object travel time fit in the studied stretch given by a
Nash determination coefficient of R2 = 0.95.
References
Beven, K. and A. Binley (1992). “The future of distributed models: model calibration and
uncertainty prediction” Hydrological Processes 6:279-298.
Camacho, L. (2000). “Development of a hierarchical modelling framework for solute transport
under unsteady flow conditions in rivers” PhD Dissertation, Imperial College of Science
Technology and Medicine, London.
Camacho, L. and M. Lees (1998). “Implementation of a Preissmann scheme solver for the
solution of the one-dimensional de-Saint Venant equations” Technical Report, EWRE, Civil
and Environmental Engineering Dept., Imperial College, London.
Chiu, C. S. Hsu. (2005). “Probabilistic approach to modelling of velocity distributions in fluid
flows” Journal of Hydrology 316:28-42.
Chow, V., D. Maidment, and L. Mays (1988). Applied Hydrology. New York: McGraw-Hill.
© Guatame-García, 2007
79
Cunge, J. A. (1969). “On the subject of a flood propagation computation method (Muskingum
Method)” Journal of Hydraulic Research 7:205-230.
Lees, M. and T. Wagener (2000). Monte-Carlo Analysis Tool (MCAT) v.2, User Manual, Civil
and Environmental Engineering, Imperial College of Science Technology and Medicine.
Martin, L. and S. McCutcheon (1999). Hydrodynamics and transport for water quality
modelling. London: Lewis Publishers.
Mathworks (1996) Matlab/Simulink reference guide, Natick, Mass.
Mazijk, A. Van (1996). One-dimensional approach of transport phenomena of dissolved matter
in rivers, Communications on hydraulic and geotechnical engineering, Faculty of Civil
Engineering, Delft University of Technology, Report No. 96-3.
Universidad de los Andes – ACUAGYR (2005). Modelación de la Calidad del Agua del Río
Magdalena y Caracterización de las Aguas Lluvias y Residuales de Girardot.
Xia, R. (1997). “Relation between mean and maximum velocities in a natural river” Journal of
Hydraulic Engineering 123(8)ASCE:720-722.
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APPENDIX 6COMPUTER MODEL OPERATION
For simulating the transport of bodies along the Magdalena River, the model takes into
account 5 external and 4 intrinsic variables (Table A6.1). These variables define the
environmental conditions in which the body will move, and the pattern of transport it is likely to
show. The object’s pattern of movement (model output) is described by five features: mean
travel time, minimum travel time, velocity, flotation depth, and mass loss. Besides these
features, the output also includes the predicted flow mean time and flow velocity. Table A6. 1. Variables and values used as data entry on the computer model
Variable Model pre-defined range Used values Description
EXTERNAL
River Q Undefined*
444.7
Lowest Magdalena River daily discharge reported to occur 95% of the year at Nariño station (Universidad de los Andes - ACUAGYR, 2005)
1118.9
Middle Magdalena River daily discharge reported to occur 50% of year at Nariño station (Universidad de los Andes - ACUAGYR, 2005)
2170.9
Highest Magdalena River daily discharge reported to occur 5% year at Nariño station (Universidad de los Andes - ACUAGYR, 2005)
Water temp. (0C) Undefined
22 Lowest water temperature considered for the range 20 - 300C
26 Middle water temperature considered for the range 20 - 300C
30 Highest water temperature considered for the range 20 - 300C
Init. position 0 – 10.30.060.9
Object initial position, as a fraction of the total width from the left river bank (0), where 1 indicates the opposite river bank.
K Degrad. Proportional to water temperature
0.03Mass loss ratio calculated for a human body taking into account the degradation of organic material in water at 220C
0.04 Mass degradation ratio calculated at 260C0.05 Mass degradation ratio calculated at 300C
Trapping factor
(Trap.F)0 -2
0.51
1.5
Body external trapping ratio, where 0=freely motion, 2=highly trapped.
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OBJECT
Object Mass (kg) Undefined
50 Lowest body weight75 Middle body weight100 Highest body weight
Density (g/cm3)
Proportional to body length and diameter (V )
0.9 Low density
1.06High density (Human density for young male adults according to Krzywicki and Chinn, 1967)
Trunk Diameter
(m)
Proportional to body weight
0.210.240.27
Waist diameter for male adults based on Miyatake (2005) measurements of waist circumference
Length
Maximum feasible body
length proportional to body weight
1.5 Body length calibrated from physical experimentation
1.651.8
Body length calculated as the proportion of calibrated body length to body weight
*An “Undefined” range means any value can be used according to the specific conditions of each case.
To illustrate how the model works, an example of a test is shown below:
1. Data entry: A specific value for each of the 9 variables must be established (Figure
A6.1). In this example data are:
River discharge: 850m/s
Water temperature: 220C
Object mass: 75kg
Object density: 1.07g/cm3
Trunk diameter: 0.27m
Object length: 1.7m
Mass degradation k: 0.03
Initial position: 0.85
Trapping factor: 1
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Figure A6. 1. Data entry chart.
2. Run: On the main panel, the route that will be simulated (RM1= short experimental
stretch, RM2= Complete stretch) is introduced, and the key “enter” pressed.
Immediately, results from the data previously introduced are displayed (Figure A6.2), as
well as a new window appear showing the test’s diagram (Figure A6.3).
Figure A6. 2. Output interface
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Figure A6. 3. Output diagram showing distance travelled vs. object’s mean and minimum travel times and flow mean travel time.
3. Interpretation of results: On the main panel eight columns are displayed, table A6.2
shows this example’s results.
Table A6. 2. Table showing the example resultsStretchLength (kms)
Object Mean time
(hrs)
Object min. time
(hrs)
Mean flow time (hrs)
Flow velocity
(m/s)
Object velocity
(m/s)
Flotation depth (m)
Object residual mass
(kg)1.125 0.33819 0.23163 0.12051 2.5931 0.92405 8.9346 74.9643.35 1.0544 0.72219 0.3762 2.4172 0.86296 6.6483 74.8875.5 1.7465 1.1962 0.62327 2.4172 0.86296 6.6483 74.8136.65 2.1424 1.4674 0.76486 2.2561 0.80679 5.1993 74.777.65 2.506 1.7164 0.89471 2.1394 0.76403 6.3103 74.73210.7 3.3589 2.3006 1.1994 2.7806 0.99336 6.0123 74.6419.05 6.1013 4.1789 2.183 2.3581 0.84577 3.4946 74.34836.3 12.517 8.5731 4.4838 2.0826 0.7469 3.5318 73.668
97.675 40.105 27.469 14.392 1.7207 0.61796 2.9917 70.817141.1 59.666 40.867 21.414 1.7179 0.61667 3.1524 68.862163.17 67.655 46.339 24.27 2.1469 0.76748 5.4473 68.079195.67 79.317 54.327 28.458 2.1558 0.77415 3.0232 66.952233.17 96.67 66.212 34.707 1.6667 0.60027 2.2051 65.31238.17 98.996 67.806 35.544 1.6604 0.59703 2.606 65.093295.67 124.69 85.403 44.776 1.73 0.62168 2.7951 62.743338.97 146.38 100.26 52.585 1.5404 0.55459 2.2888 60.825
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The first column shows the distance from the 0km (Variante Bridge) to the point at which
each subsection finishes; the second column indicates the accumulated mean travel time the
object takes to reach each subsection in hours; the third column shows the accumulated
minimum travel time the object is able to take to reach each subsection under ideal conditions;
the fourth column indicates the predicted flow mean travel time (the time a section of water
moving downstream takes to reach a specific point); the fifth column shows the predicted flow
velocity; the sixth column indicates the predicted object velocity; the seventh column shows the
depth at which the object is travelling in the water column; and finally, the eighth column
registers the residual object mass after being transported over a x time interval.
Regarding flotation depth, it is worth of mention that the model is not able to predict a
transitional depth (i.e. the depth at which the object is moving while sinking to the bottom), and
hence, the flotation depth either corresponds to the maximum depth of the river at each
subsection minus the body transverse length when the body composition makes it to sink (i.e.
8.9m), or to the predicted submerged area for bodies whose composition makes them to move at
the surface (i.e. 0.25m).
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APPENDIX 7COMPUTER MODEL OUTPUT
Due to its extension, Appendix 7 has been recorded in the attached CD-Rom.
The CD contains:
- RM1 and RM2 outputs (last row).
File Format: Excel (.xls)
File’s name: input_data.xls
- RM1 complete results.
File format: Matlab (.mat)
Folder: RM1_results
- RM2 complete results
File format: Matlab (.mat)
Folder: RM2_results
- RM1 diagrams
File format: Encapsulated PostScript (.eps)
Folder: RM1_diagrams
- RM2 diagrams
File format: Encapsulated PostScript (.eps)
Folder: RM2_diagrams
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