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

ii 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

v 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

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Table 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

Page

Figure 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

viii Figure A1. 4. Velocities distribution at Cambao station during a low discharge period............54 Figure A1. 5. Velocities distribution at Nariño station during a high discharge period..............54 Figure A1. 6. Velocities distribution at Nariño station during a low discharge period...............55 Figure A2. 1. Calibrated balance and tank for solid volume estimation......................................57 Figure A2. 2. Weighing of the wooden mannequin by using the balance...................................58 Figure A2. 3. Irregular object submersion...................................................................................58 Figure A2. 4. Mannequin’s submersion in an overflow tank. .....................................................59 Figure A3. 1. Mid-section of the Teusaca River, Cabaña Gauging Station.................................61 Figure A3. 2. Teusaca River........................................................................................................62 Figure A3. 3. Schematic diagram showing the configuration of the reach of study at Teusacá River.............................................................................................................................................63 Figure A3. 4. Measurements of flow velocity by using a flow meter at spaced positions..........64 Figure A3. 5. Object’s pattern of movement along the surface. Teusacá River..........................68 Figure A3. 6. Object’s submerged pattern of movement at Teusacá River.................................69 Figure A4. 1. Experimental stretch, Magdalena Medio region. Circle shows the stretch’s total length............................................................................................................................................71 Figure A4. 2. Mannequin’s releasing in the Magdalena River....................................................72 Figure A4. 3. Mannequin getting trapped into an eddy...............................................................72 Figure A4. 4. Release cross-section of the Magdalena River at the Variante Bridge Station. X axis: river width, Y axis: Elevation..............................................................................................73 Figure A6. 1. Data entry chart......................................................................................................83 Figure A6. 2. Output interface.....................................................................................................83 Figure A6. 3. Output diagram showing distance travelled vs. object’s mean and minimum travel times and flow mean travel time. ...............................................................................................84

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ABBREVIATIONS A

Area

ADD

Accumulated degree days

ANOVA

Analysis of variance

CI

Confidence interval

CIDH

Interamerican Commission for Human Rights

D

Distance

Diamb

Trunk diameter

FMT

Flow mean travel time

FV

Flow velocity

FD

Flotation depth

IDEAM

Institute of Hydrology, Meteorology and Environmental Studies of Colombia

hrs.

hours

K

Constant

Kb

Mass degradation constant

kms

kilometres

Lbmax

Maximum feasible object length

m

meters

Mbo

Object mass

min.

minutes

OMT

Object mean travel time time

OMinT

Object minimum travel time time

ORM

Object residual mass

OV

Object velocity

PMI

Post mortem interval

PMSI

Post mortem submersion interval

ρ

Density

rb

Density (for the output computer model)

r

Pearson’s correlation coefficient

Q

Discharge

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x RM1

Experimental stretch Variante Bridge – Girardot’s Gas Pipe (10.7kms)

RM2

Complete modelled stretch Variante Bridge – Puerto Berrío City (339kms)

SD

Standard deviation about the mean

Sg

Specific gravity

Sig

Significance

t

Time

Te

Water temperature

TrapF

Trapping factor

UNAL

National University of Colombia

V

Velocity

V

Volume

W

Weight

x1

Object initial position

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ABSTRACT There are many problems that arise in studies of bodies disposed in moving waters, as these remains do not only decompose, but are also subject to transport, disarticulation, and dispersion.

In such cases, computer modelling has proven to be an invaluable tool to understand both trajectories of bodies in former cases, as well as the prediction of body flow patterns.

In this paper, a one-dimensional hydraulic model has been coupled with an object transport model in order to predict both drift trajectories as well as distances travelled in a specific time interval. Transport is modelled taking into account buoyant, hydrostatic, and dynamic forces, calculated by using velocity, discharge, and depth computed in a numerical hydraulic model.

Results and information from previous research studies were incorporated into the modelling framework to represent the transport of bodies with different densities and specific gravities.

The model was calibrated by means of physical experiments carried out in the Teusacá and Magdalena rivers (Colombia). These experiments provided detailed hydraulic data, as well as objects’ travel times. This information was used to calibrate and validate the numerical model.

The calibrated model has been applied to a 350kms stretch of the Magdalena River in Colombia’s Magdalena Medio region, in order to simulate objects’ transport and predict their location after being disposed into the river at a certain time. Travel times recorded ranged from 23.7hrs (≅1day) to 307.6hrs (≅12.8) days, and a maximum mass loss of 35kg was documented in bodies that completed the 350kms river stretch.

The study concludes that the main extrinsic variables affecting the objects’ movement rate were discharge, initial position of the object into the river, and trapping factor; all of these variables were either catalysts for increasing or reducing velocity. Density is the main intrinsic factor affecting the pattern of movement and distance travelled during a specific time interval along a river, given its influence in the determination of the object’s position in the water column.

The coupled method developed in this study is a promising computational tool for simulating the drift of objects similar to the human body along a river system. However, further verifications are still needed for complex body composition and motion, where the calculation of changes in body volume resulting from inhaled water and mass loss due to disarticulation continue to represent a considerable challenge.

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CHAPTER 1 INTRODUCTION 1.1 Geographical and Political Context of the Study Colombia 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 area The 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,

2 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 Justification The modus operandi for concealing the death bodies by throwing them into rivers poses specific logistical difficulties for judicial investigators in terms of both the recovery and the identification of the victims, especially given the lack of taphonomic studies of non-terrestrial deposition in Colombia and elsewhere (Gómez-López and Patiño-Umaña, 2007).

Both the unique legal and social context of the missing in Colombia challenge the forensic sciences to develop scientific methods for improving the techniques of searching for human remains within the State’s territory, especially in those places in which the searches are more difficult to be carried out, such as rivers.

Because of the highly complex nature of the dynamics underlying fluvial behaviour, the use of prediction models to infer patterns of transport that allow to describe likely scenarios in which victims can be found becomes necessary. However, the success for the development of this kind of tools depends on the amount and quality of information that can be used to make the predictions, especially that regarding the behaviour of the rivers. In Colombia this information is mainly provided by the IDEAM although the Laboratory of Hydraulics of the National University of Colombia (UNAL) has made important contributions to a current data base of flow gaugings, and hydraulic characteristics of Colombian rivers.

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1.4

Aims and objectives

Due to the advantages that controlled simulations provide for the observation of the transport and behaviour of bodies into the rivers, the aim of this project is to produce a predictive transport model of objects along the Magdalena River, Magdalena Medio region, from the section of the Variante Bridge, near the town of Girardot, to the town of Puerto Berrío, 350kms stretch (Figure 1.1), through the observation of the behaviour of objects in a simulated computing environment.

This model should predict the location of the objects disposed into the river at certain time intervals, and conversely, allow the determination of potential points of their entry. In both circumstances, the success of the model will be evaluated on the identification and quantification of the factors that are most likely to affect the transport. To produce such a model, extensive investigation with reference to drift trajectories of bodies disposed in flowing water, the rivers’ physical behaviour and their relationship with the masses they may contain have to be conducted.

1.4.1. Objectives 1

Identify the intrinsic variables that influence the movement of bodies in the Magdalena River with especial concern given to body density and weight.

2

Simulate different flow situations and their interaction with a moving body, and identify the extrinsic variables that determine its transport in the Magdalena River.

3

Develop a modelling framework for body transport for the Magdalena River.

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Figure 1. 1. Magdalena River Section Variante Bridge – Puerto Berrío © Guatame-García, 2007

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CHAPTER 2 LITERATURE REVIEW

2.1 Human Decomposition in Water Environments and Postmortem Interval Estimations Human remains disposed in water environments are subject to specific intrinsic and extrinsic variables that can make the process of decay highly variable (Lyman, 1994). Intrinsic events are related to the semi-regular sequence of soft tissue degeneration that often concludes with skeletonization and the subsequent modification of bone (Haglund and Sorg, 1997); the extrinsic ones refer to transport by moving waters, scattering, and consumption by predators (Haglund and Sorg, 2002).

Studies of decomposition in water environments refer to the PMSI, which attests the amount of time a corpse has been exposed to water (Haskell et al, 1989) that sometimes is equivalent to the PMI. Since decomposition in water may exhibit different rates according to specific microenvironment biochemical and physical characteristics (Brewer, 2005), estimation of PMSI has to be treat more carefully than traditional estimations of PMI. However, Nawrocki et al. (1997) have stated that if the features of a specific water environment are known, relationships between decomposition, transport, and time since death can be established.

Techniques for PMI estimations are based on the relationship between the process of decomposition and the time a body takes to reach a certain decompositional stage. Then, PMI estimations are generally based on the assessment of these stages of decay through qualitative observation (Mann et al, 1990; Galloway, 1997) or the quantification of the process by either measuring mass loss (Payne, 1965; Payne and King, 1968) or accumulated degree days ADD (Vass et al., 1992; Megyesi et al., 2005).

The accuracy of expressing decomposition as directly related to weight loss has been tested

by Adlam (2004), who stated that biomass removal has a strong correlation with decomposition score, which is in turn related to the environmental temperature, and therefore is a validated indicator for the advancement of decomposition.

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6 Because as well as in land the main factor in aquatic decomposition is temperature, decomposition can be quantified taking into account both ADD (thermal time) (Haefner et al, 2004) and mass loss. In these cases, average daily temperature can be replaced by the average water temperature.

Water temperatures in rivers are determined by air temperature, wind speed, cloud cover, and the relative humidity of the atmosphere (Chapra, 1997). In the Magdalena River, water temperature is also affected by changes in altitude (Universidad de los Andes-ACUAGYR, 2004). For the Magdalena Medio, temperatures oscillate between 22 and 290C (Figure 2.1).

Figure 2.1. Temperatures for the Magdalena River, section Variante Bridge-Arrancaplumas. Red lines correspond to September 9 -11/2004 at low discharge, and blue lines correspond to September 28-30/2004 ay high discharge. Squares indicate first day of sampling, triangles second day of sampling, and dots third day of sampling. Adapted from: Universidad de los Andes – ACUAGYR, 2005:51, Modelación de la Calidad del Agua del Río Magdalena y Caracterización de las Aguas Lluvias y Residuales de Girardot.

Human bodies may undergo different processes of decomposition in marine, lacustrine or riverine environments given their dissimilar chemical composition (salinity, pH, and CO2 concentrations), temperatures, depth and movement (Nittrouer et al., 1995; Sorg et al., 1997).

Regarding movement, different transport patterns have been identified in marine and fluvial environments since the variables affecting the motion of water can vary from the tidal

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7 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-Channels A natural Open-Channel is a water flow confined to a channel whose characteristics changes over time and space and become more complex as all the variables are interdependent one from another (Chow, 1988). These systems are affected by several extrinsic factors that make their behaviour unsteady and difficult to predict. Rivers and streams are typical natural open-channel systems.

The shapes of a natural channel are very irregular and many often vary in shape from a parabola to a trapezium (Chow, 1988). The parabola is used as approximation of medium and small natural channel sections; the trapezium commonly refers to old river channels. (Figure 2.2).

Figure 2. 2. Channel section shapes for open-channels (Adapted from Chow, 1998:21, Open-Channel Hydraulics, New York: McGraw-Hill).

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8 Flow velocity and discharge are two important variables that contribute to fully understand the behaviour of a natural open-channel. Velocity is expressed by V = D/t Where D is the distance travelled by an object, and t the time the object reach to get the given distance.

Because of the open surface and the friction along the channel walls, the velocities of a channel are not homogeneously distributed over the whole section. The maximum velocity often occurs underneath the surface at 0.05 to 0.25 of the total depth; closer to the banks this maximum is reached at a deepest point. Actual velocities near the bottom are lower and those near the surface are higher (Boaz and Behrensmeyer, 1976); figures 2.3 and 2.4 show the general distribution of velocities for rectangular and trapezoidal channels. Appendix 1 shows the distribution of velocities for the Magdalena River.

Figure 2. 3. Velocities’ distribution in rectangular channels (Adapted from Chow, 1998:24, Open-Channel Hydraulics, New York: McGraw-Hill).

Figure 2. 4. Velocities’ distribution in trapezoidal channels (Adapted from Chow, 1998:25, Open-Channel Hydraulics, New York: McGraw-Hill).

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9 This distribution is also affected by factors such as the channel roughness and bends (Bishop and Prosser, 2001). For example, in a wide, fast and shallow current, the maximum velocity is reached at the surface; in a bend, the velocity increases in its convex side due to the centrifuge action of the flow (Chow, 1988).

In straight prismatic channels the flow is three-dimensional and produces a spiral movement. The movement is characterised by a double spiral which allows the equilibrium of forces at each side of the channel (Dilen, 1993). The model includes one spiral at each side of the central line where the water level is higher (Figure 2.5.) (Chow, 1988).

Figure 2. 5. Diagram showing the three dimensional flow in a straight river channel with double spiral flow (Dilen, 1984:1029).

Because of the velocity structure of a stream, and especially in streams flowing over low gradients with easily eroded banks, straight channels will eventually erode into meandering channels (Novak, 2001).Velocities at the meanders also change due to their curve geometry. In the meanders, the depth gradually increases to a maximum downstream of the apex of the bend; it is characterised by spiral flow and triangular sections with the maximum depth and velocity at the concave bank, and maximum sediment transport at the convex bank and the talweg (line of maximum depth) deviating from the river centreline (Novak, 2001)(Figure 2.6).

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Figure 2. 6. Diagram showing the flow in a bend (Novak, 2001:299).

On the other hand, discharge refers to the volume of water passing through a given crosssection 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 Q=V ∗ A 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-Channels Hanson (1980) has shown that the behaviour of bones in artificial and natural currents is to some extent predictable and that bones act as sedimentary particles that follow the general principles of hydraulics. This author also suggests that the area of sediment transport fully developed by the hydraulic engineering field is compatible with fluvial taphonomy, and hence, the basic equations of fluid flow and resistance forces can be applied to these problems.

In general terms, the movement of any particle deposited in water follows the Archimedes' Principle which states that any object floating upon or submerged in a fluid is buoyed upward by a force equal to the weight of the displaced fluid (Donoghue and Minniguerode, 1977). Any

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11 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; F B = Buoyant Force.

Elements deposited into flowing water have some transport patterns that are time and distance dependant defined by a velocity Vt in the downstream direction (Hanson, 1980). These elements are controlled by hydraulic factors such as depth, width, stream velocity, shear stress and sediment load, which in turn are related to discharge (Chow et al., 1988). Other extrinsic physical factors in the fluvial realm such as channel geometry and lateral migration rate, aggradation or degradation of the channel and floodplain, bed forms, and clast size also influence rates of transport and patterns of movement (Hanson, 1980).

Another physical law applied to the hydraulics is the principle of motion resulting from the relationship between the object's mass and its velocity vector. Newton originally formulated the laws of motion in terms of this property which he called “quantity of motion” and which is now called linear momentum (Lea and Burke, 1997). The law of conservation of momentum is a fundamental law of nature, and it states that the total momentum of a closed system of objects which has no interactions with external agents is constant. The principle of continuity is closely related to the principle of momentum and states that a fluid mass is neither created nor destroyed during its flow (Lea and Burke, 1997). In fluid dynamics the continuity equation is an equation of conservation of mass.

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2.4 Position and Transport of Bodies in Water Different 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 Body N Weight Density % Fat 17-19 9 71.9 ± 14.4 1.060 ± 0.016 19.6 ± 7.0 20-24 35 73.6 ± 7.5 1.060 ± 0.013 19.5 ± 5.5 25-29 29 76.8 ± 14.0 1.053 ± 0.017 22.6 ± 7.3 30-34 15 85.8 ± 17.6 1.044 ± 0.013 26.3 ± 6.1 35-39 13 76.2 ± 10.6 1.043 ± 0.012 26.9 ± 3.6 40-44 25 75.4 ± 11.1 1.042 ± 0.012 27.1 ± 5.5 45-49 24 76.2 ± 10.0 1.038 ± 0.010 29.3 ± 4.5 50-54 12 75.5 ± 10.1 1.032 ± 0.026 32.8 ± 9.1 55-59 4 79.0 ± 10.3 1.031 ± 0.021 32.5 ± 4.8 60-64 5 69.7 ± 7.5 1.026 ± 0.010 34.7 ± 4.5 65-69 2 68.6 ± 2.1 1.017 ± 0.001 38.7 ± 0.6 Total 173

In Donoghue and Minniguerode’s (1977) study, specific gravities corrected for the lungs at residual volumes ranged from 1.021 to 1.097, which means that a recent dead human body will

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13 sink. This phenomenon is also affected due to the absorption of water into the circulation system which results in an abrupt increase in blood volume (Boyle et al, 1997).

Because the specific weight of the human body is very close to that of water, small variations in specific gravity have considerable effect in whether a body will sink or float (Donoghue and Minniguerode, 1977). While on the surface, the body will float with the head and limbs hanging down beneath the surface (Figure 2.8); this position in the water results in the head and limbs exhibiting more lividity than observed on the trunk (Spitz and Fisher, 1993; Rodriguez, 1997)

Figure 2. 8. Floating position of fresh remains (Haglund and Sorg, 2002:205)

Having sunk to the bottom, the body will remain there until putrefactive gas formation in the chest and abdomen decreases the specific gravity of the body and creates sufficient buoyancy to allow it to rise to the surface and float (Donoghue and Minniguerode, 1977). Once decomposition gases are released, bodies/remains will become submerged again (Sorg et al., 1997); this changing position in the water allows differential access to agents of destruction, and also alters the sequence of soft tissue decay and disarticulation of bones or body units (Haglund and Sorg, 2002). As the decay progresses, most of the exposed flesh is lost, while the submerged flesh remains intact; During this stage, the head, shoulders, abdomen, and legs frequently become separated (Boyle et al., 1997). This process of decay becomes more rapid as the water temperature increases.

As Nawrocki et al. (1997) have pointed out, when bodies become disarticulated, it is necessary to take into account that densities and buoyancy vary in different regions of the body and that not all will react to the currents in the same way. Boaz and Behrensmeyer (1976)

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14 support this statement since their study of transport of bones have established that density is the most useful variable in predicting whether a skeletal part will move or not and what its velocity will be once it does move. Disarticulated body units and skeletal elements may be moved by deeper currents and come to rest on bottom substrates. At that point they may be moved along the substrate or be silted over (Haglund, 1993).

Boaz and Behrensmeyer (1976) utilised an artificial recirculating flume to observe the movement of bones in moving water to establish the pattern of hominid assemblages’ formation. They divided the bones in lag and transport groups according to the their hydraulic behaviour, and as has been mentioned above, concluded that density is the most important variable for determining the movement of bones into the water, and that shape and conservation of bones is also important although no-quantifiable at that time. Hanson (1980) utilised the same recirculating flume and also made river experiments with the aim to test mathematical transport hypotheses for several mammals’ bones.

Dilen (1984) used mannequins floating in the Chattahoochee River, Atlanta, to examine the movement of a body floating downstream on the surface and to determine the flow patterns of surface currents through bends in river. The study concluded that objects floating downstream tend to stay near the bank they were dropped at, and that submerged bodies orient parallel with the direction of the flow and tend to resist downstream movement near the banks due to the low current velocity. However, this pattern is case-specific and the author recommended carrying out similar experiments in rivers with different characteristics to test his own results.

Ebbesmeyer and Haglund (2002) used the Hydraulic Model of Puget Sound, Seattle, to demonstrate trajectories of floating objects by ocean waves that lead to the identification of the time required for arrival at selected locations. This study used a physical scale replica which did not utilise numerical predictive models. The results of the simulation did not show with complete reliability where a floating object would travel under all environmental conditions, but provided useful information with regard to probable trajectories the body could travel and areas that the body is not likely to travel.

On the other hand, Carniel et al. (2002) used a computer simulation based on atmospherical prediction models applied for the Mediterranean coast of the Ligurian Sea. In this study, trajectories of floating bodies were simulated to infer the location of discovery of bodies that have been drifted by surface ocean currents, based on the case of a woman who disappeared in

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15 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’ Modelling Hydrological 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. © Guatame-García, 2007

16 Since the water flow in a river or stream acts as a distributed process in which discharge, depth and velocity vary in the space along the channel, the distributed model generates an accurate approximation to the real behaviour of the river. However, the application of the principles of momentum and continuity for this kind of system is only possible under very simplified conditions, resulting in one-dimensional or bi-dimensional models (Chow et al., 1988). The numerical schemes are discussed elsewhere (Fread 1985, 1993; Chow et al., 1988) and will not be explained in detail herein, as it is not the purpose of this study.

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17

CHAPTER 3 MATERIALS AND METHODS Both physical and mathematical methods were used to identify the drift of objects throughout the 350kms of the Magdalena River section studied (Figure 3.1). The methods follow the laws of physics and theory of flow dynamics explained above in Chapter Two. Mathematical models must be calibrated to a particular site by comparing observed and predicted water surface levels and/or discharges (Camacho and Lees, 1998). Two preliminary experiments at the Teusacá and Magdalena rivers were performed; relevant information such as orientation of the object and flood effects on the object's movement was derived and used to calibrate the computer model. Afterwards, the numerical hydraulic distributed model developed

by Camacho and Lees (1998) was adjusted and implemented for the Magdalena River, thus providing an accurate representation of the real flow pattern of the river section and object’s movement.

972 tests were performed and statistically analysed in order to identify the correlations between the variables used by the model, and to assess the truthfulness or falsehood of the hypotheses proposed.

Figure 3. 1. Magdalena River profile. Shaded section shows the modelled route (350kms).

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18

3.1 Object's drift test 1 Two 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/cm 3 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 2 A physical experiment was carried out in the Magdalena River for object’s velocity calibration, and pattern of surface movement testing (See Appendix 4 for details). The river section extended from the La Variante Bridge to the gas pipe of the town of Girardot, a distance of 10.7kms downstream. This section has a mean width of 164.47m width, a longitudinal slope of 0.0070483, and a mean flow discharge of 1200m3/s (Uniandes-ACUAGYR, 2005).

The experiment was devised to calibrate the velocity of a wooden mannequin as it moved downstream, as well as to test the surface movement in an area characterised by several bends, eddies, and shrubs. The mannequin was manufactured taking into account human body proportions reported by Krzywicki and Chinn (1967). The mannequin weighted 45kg, with a centre of gravity located near the abdomen, and a bodily density of 0.98gr/cm3.

The mannequin was released at the centre of the current, without specific orientation relative to the current direction. The mannequin floated freely, approximately 85% of the volume submerged. Five control points were previously established to identify the time the body took to reach each subsection, and the average travel time along the whole section was calculated. The mannequin’s route downstream was drawn, photographed, and video recorded from a boat.

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19 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 River The 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 Data The following hypotheses were tested through statistical analysis: Hypothesis A: H1: The extrinsic variables considered by the model significantly affect the movement of objects in the Magdalena River (travel times and velocity).

H0: The extrinsic variables considered by the model do not affect the transport of objects throughout the river.

Hypothesis B: H1: Floating objects move faster than the submerged ones H0: Floating and submerged objects undergo the same velocities Hypothesis C: H1: As object density increases, the drift of the object along the river decreases.

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20 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 twoway ANOVA was performed to identify the significance of each independent variable and the occurrence of an interaction effect. These tests were also restricted to the Levene’s test for Homogeneity of Error of Variance.

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21

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 results The 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 subsections (Table 4.2) and both observed and predicted travel times and velocities were compared (Table 4.3).

Table 4. 1. Object’s experiment data Q Water temp. Object Mass (m3/s) (0C) (kg) 900

30

Density (g/cm3)

Trunk Diameter (m)

0.98

0.23

45

Body Length (m) 1.5

K Degrad.

Init. po- Trapping sition factor

0

0.6

0.4

Table 4. 2. Sub-sections for the Variante Bridge- Gas Pipe stretch of the Magdalena River.

Number 1 2 3 4 5 6

Length (kms) 1.125 3.35 5.5 6.65 7.65 10.7

Reach’s name Variante Bridge – Sumapaz River Sumapaz River – Flow derivation structure Girardot City Flow derivation structure – Bogota River Bogota River – Ospina Bridge Ospina Bridge – Ferrocarril Bridge Ferrocarril Bridge – Gas Pipe

Table 4. 3. Object’s observed vs. predicted travel times and velocities for each river subsection

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Reach

Observed hours

Predicted hours

Observed Ob. Vel

Predicted Ob. Vel

1 2 3 4 5 6

0.12 0.47 0.72 0.90 0.98 1.13

0.11 0.51 0.64 0.84 0.83 1.04

2.68 1.51 2.34 2.18 2.78 3.60

2.78 1.77 2.36 2.06 2.30 3.18

22 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 R =1− 2

∑ ( ot − pt ) ∑ ( ot − ot ) i

2

i

2

i

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 analyses Two 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 River 2 3.35 Sumapaz River – Flow derivation structure Girardot City 3 5.5 Flow derivation structure – Bogota River 4 6.65 Bogota River – Ospina Bridge 5 7.65 Ospina Bridge – Ferrocarril Bridge 6 10.7 Ferrocarril Bridge – Gas Pipe 7 19.05 Gas Pipe – Coello 8 36.3 Coello- Nariño 9 97.67 Nariño – Ambalema

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23 10 11 12 13 14 15 16

141.1 163.17 195.67 133.17 238.17 295.67 338.97

Ambalema – Sabandija Sabandija – Arrancaplumas Arrancaplumas – Puerto Salgar Puerto Salgar – Río Negro Río Negro – La Miel La Miel – Nare 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):

Object mean time O. minimum time Object velocity Flotation depth Object mass Valid N (listwise)

Object mean time O. minimum time Object velocity Flotation depth Object mass Valid N (listwise)

Table 4. 5. Descriptive statistics for RM1 N Minimum Maximum Mean 486 0.726080 7.297416 2.42817316 486 0.497315 4.998230 1.66313229 486 0.4784 4.5167 1.843278 486 0.210 8.947 3.49331 486 48.527 99.894 74.51462 486

Table 4. 6. Descriptive statistics for RM2 N Minimum Maximum Mean 486 34.617580 307.648266 108.53195772 486 23.710671 210.717991 74.33695731 486 0.2683 2.2889 0.978099 486 0.210 3.705 1.47422 486 14.171 95.084 57.31390

Std. Deviation 1.451654078 0.994283628 0.9187313 3.493398 20.303513

Std. Deviation 62.791330679 43.007760738 0.4717623 1.346912 18.991889

486

In the RM1 set of tests, the minimum mean object travel time was yielded by the test number T268 (fastest object, t=0.72hr; 43.2min), where data entry was Q= 2170.9, water temperature = 300C, object mass=50kg, density=0.9g/cm3, initial position: 0.6, trapping factor= 0.5. The minimum object travel time was yielded by the same test (t=0.49hr; 24.9min), as well as the highest object’s velocity (V= 4.5167m/s). The observed residual mass in this test was 49.85kg.

The maximum mean travel time for this set of experiments (slowest object, t=7.2974hr; ≅ 1/3 days) was yielded by the test T231, where data entry was Q= 444.7, water temperature = 220C, object mass=50kg, density=1.06.9g/cm3, initial position= 0.9, trapping factor= 1.5. The

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24 maximum minimum1 object travel time was yielded by the same test (t=4.99823hr; ≅1/5 days), as well as the lowest object’s velocity (V= 0.4784m/s); the observed residual mass in this test was 49.48kg. The test number T444 yielded very similar results: maximum mean travel time t=7.2973hr; ≅1/3 days; maximum minimum object travel time t=4.99819hr; ≅1/5 day); object’s velocity V= 0.4784m/s; and residual mass = 48.52kg. The data entry for this test was Q= 444.7, water temperature = 300C, object mass=50kg, density=1.06.9g/cm3, initial position: 0.9, and trapping factor= 1.5.

The ratio object velocity/flow velocity for the fastest object (T268) was estimated as 1.2; for the lowest object (T231), the ratio was calculated as 0.2.

As well as in RM1, the minimum mean object travel time in RM2 (fastest object) was yielded by the test number T268 (t=34.61hrs; 1.4 days). Also, the minimum object travel time was yielded by the test T268 (t=23.71hr; ≅1day), which yielded the highest object’s velocity too (V= 2.28m/s). The observed residual mass in this test was 43.8kg.

The maximum mean travel time for RM2 was also yielded by the test T231 (t=307.6482hrs; ≅12.8 days), which also described the maximum minimum object travel time (t=210.71791hrs; ≅8.7days), and the lowest object’s velocity (V= 0.2683m/s). The observed residual mass in this test was 32.19kg. The test T444 was very close to these results, yielding a mean object travel time of 307.6439hrs, a minimum travel time of 210.7150hrs, and a velocity of 0.2684m/s. The observed residual mass in this test was 14.17kg, which corresponds to the lowest residual mass of the total universe of tests that compose this set.

The ratio object velocity/flow velocity for the fastest object (T268) was estimated as 1.1; for the lowest object (T231), the ratio was calculated as 0.2.

In summary, RM1 travel times ranged between 0.72 and 7.29 hours, and RM2 between 1.4 and 12.8 days. For RM1 the mass loss ratio was calculated as 0.2kg/hr in T268, 0.1kg/h in T231, and also 0.2kg/hr in T444. For RM2 the mass loss ratio was calculated as 0.17kg/hr in T268, 0.05kg/h in T231, and 0.11kg/hr in T444.

1

The “maximum minimum” corresponds to the highest value identified for the object minimum travel time predicted by the computer model.

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25 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 RM1 Kb x1 TrapF OMT FMT Q -.357 Te 1.000 Rb .218 X1 .650 TrapF .342 OMT -.357 .218 .650 .342 .371 FMT .371 FV -.364 OV .396 -.264 -.547 -.419 -.389 FD .254 .932 .102 -.253 Correlation is significant at the 0.01 level (2-tailed).

Q

rb

FV

-.364

.397 .255

OV .396 -.264 -.547 -.419 -.844 -.389 .397 -.164

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 3

The variables are shown in order of the strength of the correlation from left to right.

Relationships marked as (+) mean a positive correlation and those marked as (-), an inverse correlation.

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26

Q Q Te Rb X1 TrapF OMT FMT FV OV FD ORM

-.330

.386 .286 .147

Table 4. 8. Pearson’s correlations for RM2 Rb Kb X1 TrapF OMT -.330 1.000 .169 .672 .353 .169 .672 .353 .343 -.339 -.191 -.565 -.433 .910 -.316 -.316 -.298 -.157 -.428 Te

FMT

FV

OV .386 -.191 -.565 -.433

.343

-.339 -.384 .389

-.384 -.281 -.149

.389 .285 .148

.385

Correlation is significant at the 0.01 level (2-tailed).

As well as in RM2, the highest correlations observed lie on the pairs water temperaturemass degradation factor (r=1.0), and flotation depth-density (r=0.91).

The other significant correlations found below the r=0.8 cutoff are listed below4: -Object mean time: initial position (+): object residual mass (-): trapping factor (+): flow mean time (+): flow velocity (-): discharge (-): density (+).

- Object velocity: initial position (-): trapping factor (-): flow velocity (+): discharge (+): object residual mass (+): flow mean time (-): density (-).

- Flotation depth: density (+): flow velocity (+): discharge (+): flow velocity time (-): flow mean time (+).

- Residual mass: Object mean time (-): object velocity (+): density (-): water temperature (-): initial position (-): trapping factor (-): flow mean time (-); flow velocity (+): discharge (+).

4.2.3. Analysis of variance (ANOVA) Since the pattern of transport of elements deposited into flowing water is time and distance dependant (Hanson, 1980), the dependent variables that have been chosen as descriptors of the object’s rate of movement are object mean time and object velocity.

For either one or two-way ANOVA, where Levene’s test for error of variance indicated a non-homogeneity of the data, the significance of the tests was set at α=0.05; if homogeneity of

4

The variables are shown in order of the strength of the correlation from left to right.

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27 variance was identified, significance was set at α=0.01. The tests contained both RM1 and RM2 (n=972).

4.2.3.1. Extrinsic variables affecting the objects’ rate of movement (Hypothesis A) Object mean time - Discharge and water temperature: The significance of the two-way ANOVA was set at α=0.05. The results show discharge to highly affect the object mean time (F=102755.3, p=0.000), but temperature (F=1.920, p=0.260) and the interaction of discharge and temperature (F=0.000, p=1.000) showed no significance in their effects.

- Initial position and discharge: The significance of the two-way ANOVA was set at α=0.05. The results show a significant effect of discharge on object’s mean time (F=8.913, p=0.034); the effect of initial position on mean time is also significant (F= 50.702, p=0.01).

However, the interaction of discharge and initial position on object’s mean time had not a significant effect (F= 1.637, p=0.163).

- Initial position and trapping factor: The significance of the two-way ANOVA was set at α=0.05. The results show a significant effect of initial position on object’s mean time (F=48.014, p=0.02); the effect of trapping factor on mean time is also significant (F= 9.035, p=0.033). However, the interaction of initial position and trapping factor on object’s mean time did not show to have a significant effect (F= 1.733, p=0.140).

These results indicate that object’s mean travel time is highly affected by these extrinsic variables, excluding water temperature (Figures 4.1 to 4.4). However, there is not sufficient statistical evidence for establishing any interaction effect of the variables studied on object mean travel time.

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28

Figure 4. 1. Differences on object mean time due to changes on river discharge

Figure 4. 2. Differences on object mean time due to changes on object initial position.

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29

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.

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30 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 positiondischarge and initial position-trapping factor.

Figure 4. 5. Differences on object velocity due to changes on river discharge.

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31

Figure 4. 6. Differences on object velocity due to changes on object initial position.

Figure 4. 7. Differences on object velocity due to changes on trapping factor.

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32

Figure 4. 8. Differences on object velocity due to changes on water temperature. Note that water temperature does not produce any means difference.

In summary, statistical analysis of the external variables used by the model reveals that only water temperature does not affect the object’s rate of movement (mean time and velocity).

The results allow to reject the null hypothesis and to accept the alternative hypothesis as true.

4.2.3.2. Comparison of the rate of movement of objects moving either submerged or floating (Hypothesis B) Object mean time The significance of the one-way ANOVA was set at α=0.01. The flotation depth, which determines whether the body moves either along the bottom or at the surface, have a significant effect on the object’s mean travel time at the 1% level (F=10.252, p=0.000) (Figure 4.9).

Object velocity The significance of the one-way ANOVA was set at α=0.01. The flotation depth have a significant effect on the object’s velocity at the 1% level (F=3.253, p=0.000) (Figure 4.10) The results allow to reject the null hypothesis and to accept the alternative hypothesis as true.

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33

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

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

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34 4.2.3.3. Significance of density over objects’ rate of movement (Hypotheses C and D) Object mean time The significance of the one-way ANOVA was set at α=0.05. The results show object density to have a significant effect on object mean time (F=6.050, p=0.014) (Figure 4.11).

Object velocity The significance of the one-way ANOVA was set at α=0.05. The results show object density to have a significant effect on the object’s velocity (F=38.754, p=0.000) (Figure 4.12) The results allow to reject the null hypothesis and to accept the alternative hypothesis as true. Also, the high Pearson’s correlation (r=-0.91) supports a) the preponderance of body density on the definition of the object’s flotation depth, and b) the inverse relationship between object’s rate of movement and density.

Figure 4. 11. Differences on object mean time due to changes on object density.

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35

Figure 4. 12. Differences on object’s velocity due to changes on object density.

4.2.3.4. Comparison of the rate of movement of objects heavier and lighter objects (Hypothesis E) Object mean time The significance of the one-way ANOVA was set at α=0.01. The results show a no significant effect of the object mass on the object’s mean time (F=0.03, p=0.997) (Figure 4.13).

Object velocity The significance of the one-way ANOVA was set at α=0.01. The results show a no significant effect of the object mass on the object’s velocity (F=0.020, p=0.980) (Figure 4.14).

The results allow accepting the null hypothesis as true.

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36

Figure 4. 13. Differences on object mean time due to weight. Note that weight does not produce any means difference.

Figure 4. 14. Differences on object’ velocity due to weight. Note that weight does not produce any means difference.

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37

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 River Computational calculations were undertaken considering five extrinsic independent variables: discharge, water temperature, object initial position, trapping factor and mass degradation constant. Analysis of results of RM1 and RM2 showed that the minimum and maximum travel times and velocities were reached by tests in which the data entered was similar in terms of discharge, object initial position, and trapping factor. Highest discharge (2170.9) produced the lowest travel times and highest object’s velocities, meanwhile the highest values in object initial position (0.9) and trapping factor (1.5) derived on the highest travel

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38 times and lowest object’s velocities. These results are consistent with the ANOVA tests, which demonstrate that these three variables have a significant influence on the object’s rate of movement Discharge, understood as the longitudinal velocity of the flow passing through a particular cross section (Smith and Stopp, 1979), produces an effect on travel time and object’s velocity that can be explained by the sequence: high discharge ⇒ high flow velocity which derives on high flow velocity ⇒ high object velocity and then high object velocity ⇒ low travel time However, since the velocity in a river cross-section decreases at the banks and the at bottom due to the friction produced along the channel walls and the river bed (Chow, 1998), objects disposed at the centre of the channel (0.6) described the highest velocities, whereas those disposed at the borders (0.9) described the lowest velocities.

The calculated ratios object velocity/flow velocity indicate that under ideal circumstances (no obstruction) the object may travel slightly faster than the mean flow velocity, and that the minimum object velocities are equivalent to half-mean flow velocity.

The effect produced by the trapping factor must be understood as an abstraction of the likelihood of an object to be trapped by substances that are not directly related to the hydraulic behaviour of the river such as shrubs and debris. The trapping factor variable acts as a “hold mechanism” that allows the simulation of situations in which the body gets trapped by eddies such as those observed in the experiment carried out at the Magdalena River. Consequently, its increment will considerably reduce the rate of movement. Since trapping factor is an abstract variable, its value resides on the capability of interpreting the external conditions of the river (vegetation, intrusive structures) as a ratio of trapping.

The relationship identified by the two-way ANOVA between discharge-initial position with regard to object’s velocity is explained as a “lag” effect. It means that in spite of supposing that an object will describe high velocity due to the high discharge of the flow, the velocity will be slightly reduced by the decreasing velocity at the borders. In the same way, the relationship

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39 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 movement The model does not consider flotation depth as a pre-defined variable but it is calculated from the mathematical reasoning of the buoyancy and hydrostatic forces (i.e. specific gravity) that act once the body is deposited into a fluid.

The one-way ANOVA revealed a very significant effect of flotation depth on the object’s rate of movement (p=0.000). It is consistent with the theory that object travel times and

5

Note that mass degradation factor and residual mass are different variables. The first one is the input variable defined as the ratio of mass loss at specific water temperatures; the second one is the amount of mass left after certain period of time.

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40 velocities depend, among others, on the vertical coordinates that indicate the position at which the object moves downstream.

Looking at figures 4.9 and 4.10, it can be observed that there is an actual difference on travel times and velocities as a result of changes in the flotation depth; however, they do not meet the decreasing scale with depth found in trapezoidal channels. This indicates that the channel displays an irregular geometric configuration that makes the velocities irregularly distributed. However, given that the model structure relies on the knowledge of the geometric configuration of the river at spaced cross-sections, the model is capable of identifying the velocities at each point of the vertical column, allowing the prediction of the object’s velocity at the calculated flotation depth.

In addition to the irregular distribution of flow velocities, high object’s velocities near to the water surface can be explained by the low gravity force displayed by a buoyant object. This kind of drift has only a longitudinal direction that reduces the resistance force, and then increases the object’s velocity.

Examining the Pearson’s correlation coefficients, it can be stated that in spite of the significance of the effect of flotation depth on the object’s travel time and velocity, variables such as initial position and trapping factor have higher influence on the rate of movement. Thus, these variables cannot be single-handled, but their influence must be analysed as a coalescence of interrelations in which the impact is to some extent predictable.

It is worth mentioning that the calculated flotation depth remains the same along the whole of the object’s route, and hence, it does not describe the movement of the body in the water column. According to Cotton et al. (1987) a human body disposed in water initially sinks, and may resurface depending on water temperature. With fairly warm water temperatures, a body can be expected to surface within a few days, whereas with cold or near-freezing water temperatures, resurfacing of the body can be delayed for several weeks to several months.

However, because the model is not able to simulate these changes in flotation depth, the significance of the relationship between flotation depth and water temperature cannot be established at this time.

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41

5.4 Intrinsic factors affecting the object’s rate of movement A 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 effect of this variable on the object’s rate of movement. This finding is in agreement with Boaz and Behrensmeyer (1976) who also proved that there is not a correlation between weight and rate of movement. A possible explanation to this observable fact is that weight is acting more as one of the variables defining density than one factor affecting movement itself.

The residual mass appears to be mainly correlated to the object mean time (r=-0.428), and lesser to the water temperature (r=-0.316). This correlation was expected as the residual mass results from the mass degradation constant which is time and temperature dependent.

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42

5.5 Study implications and limitations Although 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 summary A 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

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43 main intrinsic and extrinsic factors affecting the rate of movement of objects which composition was similar to the human body.

Discharge, object initial position, and trapping factor were established as the most important extrinsic factors affecting the objects’ travel times and velocities. Body density, affected in turn by body volume (mass, length and diameter), was identified as the solely intrinsic factor affecting the rate of movement, since it determines the body position in the water column.

The model must be adjusted according to the analysis of its limitations, but despite those shortcomings, is a reliable tool for predicting probable trajectories that an object similar to the human body may travel, and the travel times it could take to reach specific locations.

The experiments carried out at the Teusacá and Magdalena Rivers allowed the observation of the pattern of movement of objects moving downstream either floating or submerged during specific time intervals. This initial experimentation supports the Nawrocki et al. (1997) statement which suggests that if the features of a specific water environment are known, relationships between decomposition, transport, and time since death can be established.

Although these experiments mainly refer to transport and submersion interval, they constitute a preliminary proposal for further experimentation.

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44

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

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45 example of the use of technological tools in the investigation of forensic cases and the interdisciplinary collaboration.

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46

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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.

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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.

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

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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):231240.

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 Application of Forensic Archaeology and

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Haglund, W. (1993). “Disappearance of Soft Tissue and Disarticulation of Human Remains from Water Environments” Journal of Forensic Sciences 38(4):806-815.

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48 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.

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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):305310 Lea, S. and J. Burke (1997). Physics: The Nature of Things. Minnesota: West Publishing Company.

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49 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.

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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.

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50 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 1 DISRIBUTION OF VELOCITIES FOR THE MAGDALENA RIVER MAGDALENA MEDIO REGION

The distribution of velocities at Arrancaplumas, Cambao, and Nariño stations is shown.

Note that in all cases higher velocities are described near to the water surface (0.2 of the total depth).

Figure A1. 1. Velocities distribution at Arrancaplumas station during a high discharge period.

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52

Figure A1. 2. Velocities distribution at Arrancaplumas station during a low discharge period.

Figure A1. 3. Velocities distribution at Cambao station during a high discharge period.

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Figure A1. 4. Velocities distribution at Cambao station during a low discharge period.

Figure A1. 5. Velocities distribution at Nariño station during a high discharge period.

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

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55

APPENDIX 2 LABORATORY REPORT WOODEN OBJECTS’ DENSITY ESTIMATION

Summary Two laboratory sessions for volume and density estimation of wooden objects were carried out at the Hydraulic Laboratory of the National University of Colombia (UNAL).The objects would be used for physical experimentation at the Teusacá and Magdalena Rivers.

The estimations combined the water submersion technique based on the Archimedes’ principle for objects volume calculation, and the numerical estimation of density. Results were used as reference for drift observations.

Participants Researchers: Ana Carolina Guatame-Garcia, BSc; Luis Alejandro Camacho, PhD. Assistant: Luz Adriana Guatame-Garcia BSc. Geology student, UNAL. Introduction Archimedes' principle stated that when a body is fully or partly immersed in a liquid, that body experiences an upward force (buoyant force) equal to the weight of the displaced liquid.

The displaced liquid is that volume of liquid equal to the volume of the body below the water surface.

Density is given by the relationship between mass and volume of an object. Direct determination of density can be made by means of mass and volume measurement. Mass is determined by weighting, and volume can be estimated based on the Archimedes’ principle.

Instruments •

A calibrated balance with platform

•

Two water tanks

•

One measure tape

•

Stick tape

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56 •

Thermometer

•

Three catch buckets

•

A straight plastic tube (pipette) Methods Volume calculation Estimation of volume was attempted through the overflow method, which consists on

measuring of the volume of water displaced by an object submerged in the liquid. Calculations were made for: -

An object of irregular shape composed by three wood sticks and an iron bar

-

Wooden human-like mannequin

1. Objects’ weighing Objects were weighed by a calibrated balance with platform (Figures A2.1 and A2.2). Both dry and wet weights were registered for comparison.

Figure A2. 1. Calibrated balance and tank for solid volume estimation.

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57

Figure A2. 2. Weighing of the wooden mannequin by using the balance.

2. Irregular object volume estimation A small circular tank was filled with water until it weighted 168kg and the water level was 38.5cm. Water level was marked at the tank walls with stick tape, and the level recorded. The object was introduced into the water until it was fully submerged (Figure A2.3) and the weight of the water with the object, and the water level, were measured.

Figure A2. 3. Irregular object submersion

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58 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. © Guatame-García, 2007

59 Table A2. 1. Volume and density estimations for wooden objects Irregular Object Wooden Mannequin Dry weight 21kg 43.8kg Wet weight 22kg 45kg Volume of water displaced 21647ml 48882ml Object’s Density 1.02gr/cm3 0.98gr/cm3 0 Water temperature 16 160 3 Water density 0.998gr/cm 0.998gr/cm3 Specific gravity 1.02 0.98

In spite of the small dimensions of the tanks, wooded objects tend to remain afloat since the wood’s density is lower than that of the water. However, observed results indicate that the iron bar helps for increasing density, and then, it makes the object to slightly sink.

These observed and calculated values would be used as reference for tracking the drift of both the irregular wooden object in the Teusacá River, and the mannequin in the Magdalena River.

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APPENDIX 3 LABORATORY REPORT OBJECT’S DRIFT TEST 1

Summary Two sets of experiments in a straight section of the Teusacá River (Cabaña Gauging Station, Figure A3.1) were carried out to identify the pattern of transport of submerged and floating objects. A flow gauge was performed to obtain measurements of velocity and discharge. The identified path of movement and its relationship with discharge were used to calibrate the numerical model developed in this research.

Figure A3. 1. Mid-section of the Teusaca River, Cabaña Gauging Station.

Participants Researchers: Ana Carolina Guatame-Garcia, BSc; Luis Alejandro Camacho, PhD.

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61 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.

© Guatame-García, 2007

62 Instruments •

Flow meter: Device composed by a vertical axis and a rotor which, introduced in the water 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 measured 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 crosssection. 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

© Guatame-García, 2007

63 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. © Guatame-García, 2007

64 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 PlotŠ 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).

Š Guatame-García, 2007

65 Table A3. 1. Flow Gauging data gathered from Teusacá experiment.

June 23rd 2007

DATE

Total Width (m)

STATION

Time 7

TEUSACÁ RIVER CABAÑA GAUGING ST. D from the border (mt)

Start

9:20

Discharge

1.970

m3/s

Final

11:20

Velocity

0.583

m/s

Start

0.71

m

Mean Area

3.380

m2

Final

0.71

m

Mean Depth

0.483

m

Stage

DEPTH (m)

Turns

MD (m)

N TD

GD

0.000 0.28

0.50 0.44

1.00 0.45

1.50 0.47

2.00 0.48

2.50 0.48

3.00 0.51

3.50 0.51

4.00 0.53

4.50 0.53

(No)

T (s)

0

0

N/T

VP 0

VMV

PW (m)

SP (m2)

Partial Q (m3/s)

VM

0.000

0.2

28

30

0.933

0.237

0.6

36

30

1.200

0.305

0.8

37

30

1.233

0.314

0.2

39

30

1.300

0.331

0.6

50

30

1.667

0.425

0.8

56

30

1.867

0.476

0.2

57

30

1.900

0.484

0.6

73

30

2.433

0.621

0.8

67

30

2.233

0.570

0.2

72

30

2.400

0.612

0.6

85

30

2.833

0.723

0.8

82

30

2.733

0.698

0.2

73

30

2.433

0.621

0.6

95

30

3.167

0.809

0.8

93

30

3.100

0.792

0.2

70

30

2.333

0.595

0.6

96

30

3.200

0.817

0.8

100

30

3.333

0.851

0.2

69

30

2.300

0.587

0.6

100

30

3.333

0.851

0.8

99

30

3.300

0.843

0.2

74

30

2.467

0.629

0.6

94

30

3.133

0.800

0.8

101

30

3.367

0.860

0.2

72

30

2.400

0.612

0.6

96

30

3.200

0.817

© Guatame-García, 2007

SECTION

VELOCITIES (m/s)

0.145

0.36

0.500

0.180

0.0261

0.352

0.445

0.50

0.223

0.0783

0.494

0.46

0.50

0.230

0.1136

0.632

0.475

0.50

0.238

0.1500

0.723

0.48

0.50

0.240

0.1736

0.764

0.495

0.50

0.248

0.1891

0.777

0.51

0.50

0.255

0.1981

0.778

0.52

0.50

0.260

0.2022

0.771

0.53

0.50

0.265

0.2044

0.290

0.414

0.574

0.689

0.757

0.770

0.783

0.772

0.770

66

5.00 0.50

5.50 0.51

6.00 0.50

6.50 0.52

7.00 0.38

0.8

98

30

3.267

0.834

0.2

55

30

1.833

0.467

0.6

85

30

2.833

0.723

0.8

91

30

3.033

0.775

0.2

59

30

1.967

0.501

0.6

76

30

2.533

0.647

0.8

83

30

2.767

0.706

0.2

40

30

1.333

0.339

0.6

68

30

2.267

0.578

0.8

73

30

2.433

0.621

0.2

38

30

1.267

0.322

0.6

35

30

1.167

0.297

0.8

33

30

1.100

0.280

0.2

0

30

0.000

0.000

0.6

0

30

0.000

0.000

0.8

0

30

0.000

0.000

0.721

0.515

0.50

0.258

0.1857

0.649

0.505

0.50

0.253

0.1638

0.577

0.505

0.50

0.253

0.1457

0.414

0.51

0.50

0.255

0.1056

0.149

0.45

0.50

0.225

0.0336

0.672

0.625

0.529

0.299

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 objectâ&#x20AC;&#x2122;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

Š Guatame-García, 2007

67 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 torquelike 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. © Guatame-García, 2007

68

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

© Guatame-García, 2007

69 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 external 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.

© Guatame-García, 2007

70

APPENDIX 4 LABORATORY REPORT OBJECT’S DRIFT TEST 2 Summary An experiment for tracking the drift of a wooden mannequin in a 10.7km stretch of the Magdalena River was carried out. Travel times to reach each of the six subsections along the stretch were recorded in order to obtain the object transport velocity and to calibrate the object tranport model. Pattern of movement regarding longitudinal sections of the river, meanders, and influence of shrubs and debris, and external structures was observed.

Participants Researchers: Ana Carolina Guatame-Garcia, BSc; Luis Alejandro Camacho, PhD. Assistants: Luz Adriana Guatame-Garcia BSc. Geology student, UNAL. Carlos J. Guatame. , Ricardo González, MSc. Water Resources student, UNAL Introduction The Magdalena River is 1612km long and drains a 257.438km2 basin. It is the largest fluvial system in Colombia and originates in the Andean Cordillera at an elevation of 3300m (Restrepo, 2001).

The studied river stretch extended from the Variante Bridge to the gas pipe of the town of Girardot, a distance of 10.7km downstream (Figure A4.1). This section has a mean width of 164.47m width, a longitudinal slope of 0.0070483, and a mean flow discharge of 1200m3/s (Uniandes-ACUAGYR, 2005).

The stretch is characterised by four main bends located at the pumping station of Girardot Town, the Bogotá River’s mouth, and Chicalá and Honda discharges, at which strong eddies are produced. The river banks are also distinguished at some sections by their thick vegetation and the releasing of debris and shrubs into the water.

© Guatame-García, 2007

71

Figure A4. 1. Experimental stretch, Magdalena Medio region. Circle shows the stretch’s total length.

Instruments •

Wooden mannequin

•

Motor boat

•

Video Camera

•

Digital Photo Camera

•

Stopwatch

•

Rope

© Guatame-García, 2007

72 Methods A wooden mannequin weighing 45kg that yielded a density of 0.98gr/cm3 was released at the Variante Bridge, a known hydrological control point of the Magdalena River. The body was disposed at the centre of the flow without specific orientation relative to the current direction (Figure A4.2).

Figure A4. 2. Mannequin’s releasing in the Magdalena River

The mannequin floated freely with about 85% of the volume submerged. Six control points were previously established to measure the time the body took to reach each subsection.

Tracking of the mannequin was drawn, photographed and video recorded from a boat as the body drifted downstream (Figure A4.3).

Figure A4. 3. Mannequin getting trapped into an eddy.

© Guatame-García, 2007

73 The average travel time along the whole section was calculated, as well as the ratio flow velocity to body velocity. The river flow discharge during the experiment was obtained using the stage-discharge relationship of Nariño flow gauging station. The measured discharge was 900 m3/s.

Results The mannequin was released at the middle section of the channel, slightly biased towards the left bank. It starts moving towards the middle-right portion of the channel, at which the deepest points of the river at this section are found (Figure A4.4).

Figure A4. 4. Release cross-section of the Magdalena River at the Variante Bridge Station. X axis: river width, Y axis: Elevation.

Along the bends, the mannequin followed a path close to the external river banks. In most of the cases, the object moved after the bend with the main surface flow from the external river bank in the direction towards the opposite bank. When the object was again located at the centre of the channel, it travelled with the main flow.

The object travel time was compared to the distance travelled, to obtain the object’s velocity at each subsection. The highest velocity was obtained at the Ospina Bridge – Ferrocarril Bridge subsection (V = 3.97m/s). Through this straight section, the object tended to move at the centre of the channel with the main flow and was not affected by external obstacles.

The lowest velocities were reached at the Sumapaz River – Girardot’s gas pipe (V= 1.77m/s), and Bogotá River – Ospina Bridge (V=1.73m/s) subsections. These results answer the fact that

© Guatame-García, 2007

74 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

Variante Bridge Sumapaz River Sumapaz River - Flow Derivation Structure Girardot city Flow Derivation Structure- Bogota River Bogotá River - Ospina Bridge Ospina Bridge Ferrocarril Bridge Ferrocarril Bridge Gas Pipe Total or Average

(m3/s)

(m)

(m)

Crosssectional Area* (m2)

950

1125

9.34

340.16

2.79

4.11

420

2.68

0.96

800

2225

7.02

385.71

2.07

2.70

1260

1.77

0.85

900

2150

7.02

385.18

2.34

3.04

910.2

2.36

1.01

900

1150

5.54

420.91

2.14

3.14

666

1.73

0.81

968

1000

6.20

494.86

1.96

2.88

252

3.97

2.03

968 914

3050 10700

6.41

345.40

2.80 2.41

4.12

582 4090.2

5.24 2.62

1.87 1.09

Discharge Length

Depth

Mean Flow Velocity (m/s)

Max Observed Flow Object Velocity Travel (m/s) Time (s)

Observed Object Vob/Vfl Velocity (m/s)

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 discharges 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 differing forces exerted by the moving water on different sections of the mannequin’s surface. Trapping of the body by external factors altered this movement reducing its longitudinal velocity due to the discontinued movement of the object.

© Guatame-García, 2007

75

APPENDIX 5 OBJECT 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),

ρ w = ρ T + ∆ ρ s + ∆ ρ ss

(1)

where, ρw is the water density (kg m-3), ρT is the density of pure water as a function of temperature, and ∆ ρ s and ∆ ρ ss are the changes in density due to dissolved and suspended solids, respectively. Density is best calculated as a function of temperature using the ThiesenScheel-Diesselhorst equation (Martin and McCutcheon, 1999),  ( T + 288.9414) ( T − 3.9863) 2  ρ T = 1000 1 −  508929.2 ⋅ ( T + 68.12963) 

(2)

where, T is water temperature in °C. In turn, the most generally accepted relationship between dissolved solids or salinity and density is, ∆ ρ s = (0.824493 − 4.0899 × 10 − 3 T + 7.6438 × 10 − 5 T 2 − 8.2467 × 10 − 7 T 3 + 5.3875 × 10 − 9 T 4 ) ⋅ C SL

(3)

+ ( − 5.72466 × 10 − 3 + 1.0277 × 10 − 4 T − 1.6546 × 10 − 6 T 2 ) ⋅ C 1SL.5 2 + 4.8314 × 10 − 4 C SL

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,

© Guatame-García, 2007

76 1   −3 ∆ ρ ss = C ss  1 −  ⋅ 10 SG  

(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=

M bo ⋅ 9.81 ρb

(5)

Assuming that the object is mainly of a cylindrical shape, its length Lb will be related to its diameter, Db and volume by, Lb =

4∀ π ⋅ Db2

(6)

Due to hydrostatic and buoyancy forces the submerged depth, yb, in water of density ρw of such a cylindrical object will approximately be, yb =

M bo ⋅ 9.81 ρ w ⋅ 9.81 ⋅ Db ⋅ Lb

(7)

If yb is less than Db the object will move of float at a depth yfl equal to,

y fl = y − yb

(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, y fl = 1.05 ⋅ Db

(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

© Guatame-García, 2007

77 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),

Vb = Vx1, yfl = K h ⋅ K v ⋅ V

(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: Kh =

2 Vx1   x   x  = − 5.84 ⋅  1  + 5.84 ⋅  1    V  B  B   

30 y fl    log 6  ( n / 0.0342)   Kv =   18 y  log 6   ( n / 0.0342) 

© Guatame-García, 2007

(11)

(12)

78 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 = 1.47 V 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, Vb = K h ⋅ K v ⋅

V 1 + TF

(13)

Therefore, the object travel time to a downstream location at a longitudinal river distance L, will be given by, tb =

L Vb

(14)

Model Calibration The 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 © Guatame-García, 2007

79 of travel time and object velocity. In this work, calibration, parameter uncertainty, identifiability and sensitivity analysis of model parameters for both TS and ADZ models was investigated using Monte-Carlo based methods. The concepts of the Generalised Likelihood Uncertainty Estimation methodology (GLUE, Beven and Binley, 1992) are applied using the Monte-Carlo Analysis Toolbox (MCAT, Lees and Wagener, 2000). Within the methodology, the identifiability of model parameters were partly examined using scatter or dotty plots, where the individual parameter values of each Monte-Carlo realization were plotted against the objective function values (Nash determination coefficient) evaluated for the prediction results of the observed data provided by each Monte-Carlo realization. In addition, the cumulative distributions of 10 classes of the parameter space ranked on the likelihood measure or the goodness of fit were plotted together, in the so-called Regional Sensitivity Analysis plot. Both model parameters result to be sensitive parameters, as revealed by strong differences in the cumulative distributions of each class.

The optimum model parameter values, Manning-n roughness coefficient and trapping factor, for the Magdalena River result to be respectively 0.025 and 0.4. The calibration results are considered very good with an overall object travel time fit in the studied stretch given by a Nash determination coefficient of R2 = 0.95.

References Beven, K. and A. Binley (1992). “The future of distributed models: model calibration and uncertainty prediction” Hydrological Processes 6:279-298.

Camacho, L. (2000). “Development of a hierarchical modelling framework for solute transport under unsteady flow conditions in rivers” PhD Dissertation, Imperial College of Science Technology and Medicine, London.

Camacho, L. and M. Lees (1998). “Implementation of a Preissmann scheme solver for the solution of the one-dimensional de-Saint Venant equations” Technical Report, EWRE, Civil and Environmental Engineering Dept., Imperial College, London.

Chiu, C. S. Hsu. (2005). “Probabilistic approach to modelling of velocity distributions in fluid flows” Journal of Hydrology 316:28-42.

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

© Guatame-García, 2007

80 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 6 COMPUTER 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 Model preVariable Used values Description defined range

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)

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

EXTERNAL Water temp.

(0C)

Undefined

Init. position

0–1

K Degrad.

Proportional to water temperature

Trapping factor (Trap.F)

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0 -2

0.3 0.06 0.9 0.03 0.04 0.05 0.5 1 1.5

Object initial position, as a fraction of the total width from the left river bank (0), where 1 indicates the opposite river bank.

Mass loss ratio calculated for a human body taking into account the degradation of organic material in water at 220C Mass degradation ratio calculated at 260C Mass degradation ratio calculated at 300C Body external trapping ratio, where 0=freely motion, 2=highly trapped.

82 Object Mass (kg)

Density (g/cm3) OBJECT

Trunk Diameter (m)

Undefined Proportional to body length and diameter ( V )

Proportional to body weight

50 75 100

Lowest body weight Middle body weight Highest body weight

0.9

Low density

1.06

High density (Human density for young male adults according to Krzywicki and Chinn, 1967)

0.21 0.24 0.27

Waist diameter for male adults based on Miyatake (2005) measurements of waist circumference Body length calibrated from physical experimentation

Maximum 1.5 feasible body Length length 1.65 Body length calculated as the proportion proportional to of calibrated body length to body weight 1.8 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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83

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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84

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.

Stretch Length (kms) 1.125 3.35 5.5 6.65 7.65 10.7 19.05 36.3 97.675 141.1 163.17 195.67 233.17 238.17 295.67 338.97

Object Mean time (hrs) 0.33819 1.0544 1.7465 2.1424 2.506 3.3589 6.1013 12.517 40.105 59.666 67.655 79.317 96.67 98.996 124.69 146.38

© Guatame-García, 2007

Table A6. 2. Table showing the example results Object Flow Object Mean flow Flotation min. time velocity velocity time (hrs) depth (m) (hrs) (m/s) (m/s) 0.23163 0.12051 2.5931 0.92405 8.9346 0.72219 0.3762 2.4172 0.86296 6.6483 1.1962 0.62327 2.4172 0.86296 6.6483 1.4674 0.76486 2.2561 0.80679 5.1993 1.7164 0.89471 2.1394 0.76403 6.3103 2.3006 1.1994 2.7806 0.99336 6.0123 4.1789 2.183 2.3581 0.84577 3.4946 8.5731 4.4838 2.0826 0.7469 3.5318 27.469 14.392 1.7207 0.61796 2.9917 40.867 21.414 1.7179 0.61667 3.1524 46.339 24.27 2.1469 0.76748 5.4473 54.327 28.458 2.1558 0.77415 3.0232 66.212 34.707 1.6667 0.60027 2.2051 67.806 35.544 1.6604 0.59703 2.606 85.403 44.776 1.73 0.62168 2.7951 100.26 52.585 1.5404 0.55459 2.2888

Object residual mass (kg) 74.964 74.887 74.813 74.77 74.732 74.64 74.348 73.668 70.817 68.862 68.079 66.952 65.31 65.093 62.743 60.825

85 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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86

APPENDIX 7 COMPUTER 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

© Guatame-García, 2007