computer simulation for drift trajectories of objects in the magdalena river, colombia

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

© Guatame-García, 2007

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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


x

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.


2

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.


3

Figure 1. 1. Magdalena River Section Variante Bridge – Puerto Berrío


4

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.


5

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


6

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).


7

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).


8

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).


9

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


10

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.


11

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


12

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)


13

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


14

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.


15

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.


16

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).


17

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.


18

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.


19

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.


20

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


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


22

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


23

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.


24

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.


25

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.


26

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.


27

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.


28

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.


29

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.


30

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.


31

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.


32

Figure 4. 9. Difference on object mean time due to flotation depth.

Figure 4. 10. Difference of object’s velocity due to flotation depth.


33

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.


34

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.


35

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.


36

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


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


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.


39

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.


40

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.


41

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


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.


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


44

example of the use of technological tools in the investigation of forensic cases and the

interdisciplinary collaboration.


45

REFERENCES

Acevedo, L. (1981). El Río Grande de la Magdalena: apuntes sobre su historia, su geografía y

sus problemas. Bogotá: Banco de la República.

Adlam, R. (2004). “The Effect of Repeated Physical Disturbance on Soft Tissue

Decomposition” Dissertation submitted for Msc/PGDip Forensic and Biological

Anthropology.

Avilés, W. (2006). Global Capitalism, Democracy and Civil Military Relations in Colombia.

Albany: State University of New York Press.

Bishop, V. and R. Prosser (2001). Landform Systems. 2nd Ed. London: Collins Educational.

Boyle, S., A. Galloway and R. Mason (1997). “Human Aquatic Taphonomy in the Monterey

Bay Area” In: Forensic Taphonomy: The Postmortem fate of Human Remains. Eds. W.

Haglund and M. Sorg, Ch. 35, pp 553-558, Boca Raton: CRC Press.

Brewer, V.L. (2005). “Taphonomic Changes and Drift Trajectories in the River Thames”

Dissertation submitted for MSc/PGDip Forensic and Biological Anthropology.

Brittain, J. (2006). “Human Rights and the Colombian Government: An analysis of state-based

atrocities toward non-combatants” New Politics 10(4) [On line]. Available from:

http://www.wpunj.edu/newpol/issue40/Brittain40.htm [Accessed 21 March 2006].

Camacho, L and M. Lees (1998). Implementation of a Preissman Scheme Solver for the

Solution of the One-dimensional Saint-Venant Equations. Technical Document Version 1.0.

Imperial College of Science Technology and Medicine. Environmental and Water

Resources Engineering Section. London.

Camacho, L. and M. DiazGranados (2003). Metodología para la Obtención de un Modelo

Predictivo de Transporte de Solutos y Calidad de Agua en Ríos – Caso Río Bogotá. Paper

presented in the Seminario Internacional Hidroinformática en la Gestión Integrada de los

Recursos Hídricos. Universidad del Valle – Instituto Cinara.

Carniel, S., M. Sclavo, L. Kantha and S. Monthi (2002). “Tracking the drift of a human body in

the coastal ocean using numerical prediction models of the oceanic, atmospheric and wave

conditions” Science and Justice 42:143-151.

Chapra, S. (1997). Surface Water Quality Model. New York: McGraw-Hill.

Chow, V (1988). Open-Channel Hydraulics. New York: McGraw-Hill.


46

http://www.polodemocratico.net/article.php3?id_article=692

Chow, V., D. Maidment, and L. Mays (1988). Applied Hydrology. New York: McGraw-Hill.

CIDH (2005) Sentencia Caso de la Masacre de Mapiripán vs. Colombia, September 15.

Cotton, G., A. Aufderheide, and V. Goldschmidt (1987). “Preservation of Human Tissue

Immersed for Five Years in Fresh Water of Known Temperatura” Journal of Forensic

Sciences 32 (4):1125-1130.

Dilen, D. (1984). “The Motion of Floating and Submerged Objects in the Chattahoochee River,

Atlanta, GA” Journal of Forensic Sciences 29(4):1027-1037.

Dix, J. (1987). “Missouri’s Lakes and the Disposal of Homicide Victims” Journal of Forensic

Sciences 32(3): 806–809.

Donoghue E. and S. Minniguerode (1977). “Human Body Buoyancy: A Study of 98 Men”

Journal of Forensic Sciences 22(3):573-578.

Ebbesmeyer, J. and W. Haglund (1994). “Drift Trajectories of a Floating Human Body

Simulated in a Hydraulic Model of Puget Sound” Journal of Forensic Sciences 39(1):231-

240.

Fread, D. (1985). “Channel Routing” In: Hydrological Forecasting. Eds. Anderson, M., and T.

Burt, Chichester: John Willey and Sons:. pp.XX.

____ (1993) “Flood Routing”. In: Handbook pof Hydrology. Maidment, D. (ed.), pp. 946-951,

New York: McGraw-Hill

Galloway, A. (1997). “The Process of Decomposition: A Model from the Arizona-Sonoran

Desert” In: Forensic Taphonomy: The Postmortem fate of Human Remains. Eds. W.

Haglund and M. Sorg, Ch. 8, pp 139-150, Boca Raton: CRC Press.

Gómez-López, AM and A. Patiño-Umaña (2007). “Who is Missing? Problems in the

Application of Forensic Archaeology and Anthropology in Colombia’s Conflict”

Forensic Archaeology and Human Rights, Ed. Ferllini R., Springfield: Charles C. Thomas

Press.

Hanson, B. (1980). “Fluvial Taphonomic Processes: Models and Experiments” In: Fossils in

the Making. Vertebrate Taphonomy and Paleoecology. Eds. Behrensmeyer, A. and A. Hill.

London: University of Chicago Press. 156-181.

Haglund, W. (1993). “Disappearance of Soft Tissue and Disarticulation of Human Remains

from Water Environments” Journal of Forensic Sciences 38(4):806-815.


47

Haglund, W. and M. Sorg (1997). “Method and theory of forensic taphonomic research” In

Forensic Taphonomy: The Postmortem Fate of Human Remains, Eds. W. Haglund and M.

Sorg, Ch. 1, pp. 13-26, Boca Raton: CRC Press.

Haglund, W. and M. Sorg (eds) (2002). “Human Remains in Water Environments” in Advances

in Forensic Taphonomy: Method, Theory and Archaeological Perspectives. Ch. 10, pp.

201-218, Boca Raton: CRC Press.

Haskell, N. H., D. McShaffrey, D. Hawley, R. Williams, and J. Pless (1989). “Use of Aquatic

Insects in Determining Submersion Interval” Journal of Forensic Sciences 34(3):622-632.

Heaton, V. (2006). “The Decomposition of Human Remains Recovered from the Rover Clyde,

Scotland: A Comparative Study of UK Fluvial Systems” Dissertation submitted for

MSc/PGDip Forensic and Biological Anthropology.

Herschy, R. (1998). Hydrometry: Principles and Practices. New York: John Wiley and Sons.

Kahana, T., J. Almog, J. Levy, E. Shmeltzer, Y. Spier, and J. Hiss (1999). “Marine taphonomy:

adipocere formation in a series of bodies recovered from a single shipwreck” Journal of

Forensic Sciences 44(5):897–901.

Krzywicki, H. and K. Chinn (1967). “Human Body Density and Fat of an Adult Population as

Measured by Water Displacement” The American Journal of Clinical Nutrition 20 (4):305-

310

Lea, S. and J. Burke (1997). Physics: The Nature of Things. Minnesota: West Publishing

Company.

Lombana, C (2003) “Modelo de Alarma Integrado de Flujo y Transporte de Contaminantes.

Aplicación al tramo Palermo-Puerto Berrío en el Río Magdalena” Dissertation submitted

for BSc. Civil Engineering.

Lyman, R. (1994). Vertebrate Taphonomy. Cambridge: Cambridge University Press.

Mann, R., W. Bass and L. Meadows (1990). “Time since Death and Decomposition of the

Human Body: Variables and Observations in Case and Experimental Field Studies”

Journal of Forensic Sciences 35(1):103-111.

Martínez, L (1990). Datos geográficos de Colombia. Bogotá: Instituto Geográfico Agustín

Codazzi.


48

Megyesi, M., S. Nawrocki and N. Haskell (2005). “Using Accumulated Degree-Days to

Estimate the Postmortem Interval from Decomposed Human Remains” Journal of Forensic

Sciences 50(3):618-626.

Nawrocki, S., J. Pless, D.A. Hawley and S.A. Wagner (1997). “Fluvial Transport of Human

Crania” In: Forensic Taphonomy: The Postmortem Fate of Human Remains. Eds. W.

Haglund and M. Sorg, Ch. 34, pp. 529-552. Boca Raton: CRC Press.

Nittrouer, C., J. Brunskill and A. Figueiredo (1995). “Importance of tropical coastal

environments” Geo-Marine Letters 15(3-4): 121-126.

Novak, P. (2001). Hydraulic Structures. London: Spon Press.

Payne, J. (1965). “A Summer Carrion Study of the Baby Pig Sus Crofa Linnaeus” Ecology

46(5): 592-602.

Payne, J. and E. King (1968). “Arthropod Succession and Decomposition of Buried Pigs”

Nature 219:1180-1181.

Rabasa A. and P. Chalk (2001). Colombian Labyrinth: The Synergy of Drugs and Insurgency

and Its Implications for Regional Stability. Pittsburgh: RAND Corporation.

Restrepo, J (2001) “Physical process in the San Juan River delta and comparisons to the

Magdalena River, Colombia”. Dissertation submitted for PhD.

Rodriguez, W. (1997) “Decomposition of buried and submerged bodies” In: Forensic

Taphonomy: The Postmortem fate of Human Remains. Eds. W. Haglund and M. Sorg, Ch.

29, pp 559-565, Boca Raton: CRC Press.

Smith, D., and P. Stopp (1979). The River Basin – An Introduction to the Study of Hydrology.

Cambridge: Cambridge University Press.

Sorg, M., J. Dearborn, E. Monahan, H. Ryan, K. Sweeney, and E. David (1997). “Forensic

Taphonomy in Marine Contexts” In: Forensic Taphonomy: The Postmortem fate of Human

Remains. Eds. W. Haglund and M. Sorg, Ch. 37, pp. 567-604, Boca Raton: CRC Press.

Spitz, W. and R. Fisher (1980). Medicolegal Investigation of Death. Springfield: Charles C

Thomas.

Taussig, M. (2005). Law in a Lawless Land: Diary of a Limpieza in Colombia. Chicago:

University of Chicago Press.

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.


49

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.


51

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.


52

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.


53

Figure A1. 6. Velocities distribution at Nariño station during a low discharge period.


54

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


55

• 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.


56

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


57

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.


58

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.


59


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



60

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.


61

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


62

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.


63

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).


64

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


65

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


66

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.


67

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


68

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.


69


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


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.


70

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


71

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.


72

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


73

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.


74

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,


24

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


76

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)


77

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


78

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.


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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computer simulation for drift trajectories of objects in the magdalena river, colombia

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