initial porosity of random packing - tu delft

Initial Porosity of Random Packing

Computer simulation of grain rearrangement

L.J.H. Alberts


Computer simulation of grain rearrangement

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College van Promoties,

in het openbaar te verdedigen

op woensdag 12 oktober 2005 te 10.30 uur

door Luc Jan Hendrik ALBERTS

mijnbouwkundig ingenieur geboren te Heerenveen.

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. Salomon Kroonenberg Samenstelling promotiecommissie: Rector Magnificus voorzitter Prof. dr. S.B. Kroonenberg Technische Universiteit Delft, promotor Dr. G.J. Weltje Technische Universiteit Delft, toegevoegd

promotor Prof. ir. C.P.J.W. van Kruijsdijk Technische Universiteit Delft Prof. dr. S.M. Luthi Technische Universiteit Delft Prof. dr. ir. S.M. Hassanizadeh Universiteit Utrecht Dr. D.K. Dysthe Universitetet i Oslo Dr. M.R. Giles Shell Research Rijswijk Prof. dr. ir. P.K. Currie Technische Universiteit Delft, reservelid This research was funded by the Delft Interfacultary Research Centre (DIOC- 3) of Delft University of Technology. ISBN-10: 9090199772 ISBN-13: 9789090199771 Copyright © 2005 by Luc J.H. Alberts, Department of Geotechnology, Delft University of Technology All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilised in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without the prior written permission of the author.

Aan mijn moeder

The cure for boredom is curiosity. There is no cure for curiosity. – Dorothy Parker

Acknowledgements A research project that takes so many years to complete requires assistance and support from the people around you. Especially a topic that is new and not directly related to other work at the department cannot be brought to a successful end, unless the working environment is good. Pioneering can be very satisfactory at times, but it also brings the risk of being isolated. I was in the fortunate circumstance that I had many great people around, for which I am thankful. There are several people, who I would like to mention in particular. Salle Kroonenberg was my promotor, and if it weren’t for his enthusiasm, I would possibly not have done a PhD research. I will always remember the speech he gave when I received my MSc degree. Salle provided the time and freedom that is needed for creativity and own initiatives, and the motivation to grab the opportunities as they arrived. It was always a pleasure to show him the latest results and to discuss about it. A great source of inspiration was Gert-Jan Weltje, my daily supervisor. If you are short of ideas, visit Gert-Jan and you will return with ten good ideas. Also, his thorough and timely revisions of all manuscripts and abstracts are greatly appreciated. I couldn’t have wished for a better supervisor.


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All colleagues and friends at the department aided to create an enjoyable working atmosphere. Humour and wit caused many a laugh, and made the office never a boring place to go to. Joep, Bob, Klaas, Remco, Irina, Kees, Marit, Israel, Jose, Jelle, Gert-Jan, Rory, Jan-Kees, Rick, Maaike, Alexey and Jon: thank you for this great time. Room 2.53 deserves a special mention. In the seven years I occupied this room, I’ve outlived the furniture and the photo panel on the wall, and it nearly became my second home (even though it was gradually transformed into our lunch room). Although I thought I already knew much about computers, I learned a lot from Kees Geel. I wonder how I would have been able to visualise my packings otherwise. Thanks Kees, and I’m glad I could teach even you a thing or two. In the darkest hours of my project, the meanest of all programming bugs struck (a minus sign!!!). Luckily Jelle de Haan was willing to help me out. His demonstration of debugging skills helped me also during the rest of my research, which is greatly appreciated. I wish to acknowledge Peter Kavanagh and Gary Nolan for providing me with the source code of their model, which formed the basis for my model, and John Finney for providing me with the position data of his experimental sphere pack. A substantial part of this thesis originates from the work I did at PGP in Oslo for three months. It provided me with an entirely different view on how science can also be done. The long and fruitful discussions with Dag, Lina, Sean, Anders and Renaud contributed significantly to my understanding of the physics of granular media, and I am grateful for that. To Lina, I hope that your thesis will benefit just as much from our exchange of thoughts and data as mine. I would also like to thank everyone else from PGP for the great times during the wine seminars, the dinner and drinks in town afterwards, the parties and all other activities. Evelyn, thank you for your kindness and hospitality. I appreciate it very much that I have been able to meet so many friendly Norwegian people through you. Altogether it made my stay in Oslo a worthwhile experience. Of course, a life outside the university is just as important. Thanks to all my friends there have been plenty occasions to distract my thoughts from work and enjoy life. Thank you, Jelle, Hilde, Marten, Sacha, Gerard, Mila, Bart, Chris, Danny, Bas en Matthijs!

Acknowledgements

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I would like to thank my father and Anjenet for their continuous support and interest in my work. Finally, I’d like to say a special thanks to my mother, to John and my brother. For being there when I needed it the most. For the love, support and encouragement. For the place to come home, which seems to be taken for granted, but never is.


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Contents

Acknowlegdements .................................................................................... i

Table of contents ...................................................................................... v

Summary ................................................................................................... ix

Samenvatting .......................................................................................... xiii

1 General Introduction ......................................................................... 1 1.1 Background 1.2 Problem definition 1.3 Objectives and approach 1.4 Thesis outline

2 Compaction of Sandy Sediments and The Evolution of ............... 7

Porosity 2.1 Introduction 2.2 Definitions of relevant terminology 2.3 Components of compaction and porosity loss

2.3.1 Grain rearrangement 2.3.2 Grain deformation and brittle failure 2.3.3 Pressure solution 2.3.4 Cementation 2.3.5 Relative importance of the diagenetic processes


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2.4 Effect of material and mineral composition 2.5 Overview of quantitative models

2.5.1 Empirical models 2.5.2 Process models 2.5.3 Rock reconstruction 2.5.4 Numerical modelling of granular media

2.6 Discussion

3 Simulation of Grain Rearrangement during Compaction .......... 37 3.1 Introduction 3.2 The rearrangement model

3.2.1 Particle-pack representation 3.2.2 Numerical simulation scheme 3.2.3 The force-displacement law 3.2.4 Stability and rolling

3.3 Optimisation 3.4 Implementation 3.5 Simulation of Grain Rearrangement 3.6 Discussion

4 Analysis of Mono Sized Particle Systems ...................................... 61

4.1 Introduction 4.2 Available packings

4.2.1 Regular packings 4.2.2 Steel-ball packings: Finney pack and packings of others 4.2.3 Play-Doh packings 4.2.4 Rampage packings 4.2.5 Other factor affecting packing variations

4.3 Statistical analysis of packings 4.3.1 Porosity 4.3.2 Radial distribution function 4.3.3 Contacts: coordination and contact angles

4.4 Assessment of boundary effects 4.5 Influence of gravity on simulated packings 4.6 Conclusions

5 The Relation between Grain Size Distribution and Porosity ..... 89

5.1 Introduction 5.2 Previous work on particle mixtures 5.3 Methodology for porosity determination 5.4 Porosities of binary and ternary mixtures

5.4.1 Binary mixtures 5.4.2 Ternary mixtures

5.5 Sorting and skewness effect

Contents

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5.5.1 Sorting 5.5.2 Skewness

5.6 Conclusions

6 Synthesis ........................................................................................ 113

References ............................................................................................. 119

Curriculum Vitae .................................................................................. 133

List of Publications .............................................................................. 135


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Summary The initial porosity of clastic sediments is poorly defined. In spite of this, it is an important parameter in many models that describe the diagenetic processes taking place during the burial of sediments and which are responsible for the transition from sand to sandstone. Diagenetic models are of importance to predict the sub-seismic heterogeneity of reservoir rock. Also, initial porosity is an important parameter for decompaction routines to reconstruct the burial history of rock used to determine the maturation of oil source rock. Measurement of initial porosity is usually difficult, because unconsolidated sediments are easily disturbed during sampling and because sediments close to the surface already have been subjected to varying degrees of compaction. Neither is it possible to observe the processes that take place during compaction, since these take place over geological time scales. Laboratory experiments do not allow us to accurately mimic these processes due to the relatively short time span available. For these reasons, no analytical methods exist to quantify the relation between the grain-size distribution, grain shape and the (initial) porosity. Therefore, these parameters are ignored in many models that describe porosity loss, despite the knowledge that they have a large influence on the heterogeneities inside a sand body.


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In this thesis an object-based simulation model is presented that is used to improve our insight into the relation between the parameters of the grain-size distribution, the initial porosity of sandy sediments, and the evolution of porosity decrease during the initial phase of compaction. In Chapter 2 the different processes that take place during diagenesis in sediments are introduced and an overview is given of the different types of models that have been applied to describe porosity loss. In the past much research has been focussed on recognition of the different processes taking place during compaction and cementation (grain rearrangement, grain deformation, pressure solution and cementation), which led to much debate about the relative importance of these processes. The interaction of these processes has received much less attention. It has become clear, however, that none of the processes can be neglected. The majority of the models have been limited to empirically determined porosity-depth trends, such as simple exponential functions that use initial porosity as a starting point. A promising modelling technique, that approaches the problem on a grain scale using numerical simulations, has only occasionally been applied within the geosciences. The concept of the model, the algorithms and the implementation are described in Chapter 3. The initial condition of the model involves a unit cell with periodic boundary conditions that have to minimise the influence of the boundaries on the grains. The cell is partially filled with spheres of arbitrary sizes, which are positioned randomly. Elastic interparticle forces decrease the overlap between the spheres, after which the spheres move, drop or roll under the influence of gravity and the remaining interparticle forces, until a gravitationally stable configuration develops. Further compaction is stimulated by a small downward force applied to the top of the packing, and a potential agitation of the packing. In this way the model simulates a realistic trajectory of porosity loss. The sphere packs that can be simulated with the model are analysed and compared to several different types of packing in Chapter 4. For this analysis data were available from various experimental sphere packs. Statistical methods involving the porosities, the numbers of contacts between the spheres, and the mutual distances between the spheres, indicated that the structure of the dense packed simulated sphere assemblages show large agreement with the structure of experimental disordered packings of solid spheres. A close inspection of the wall effects of the cell proved that the use of periodic boundaries effectively minimises the wall effects. In Chapter 5 the relations between the parameters of the grain-size distribution and porosity are investigated using the model. A large number of

Summary

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simulations have been carried out for binary and ternary mixtures of sphere of size ratio 1:2:4. The spheres were mixed in different proportions, and each mixture was simulated until a fully stable packing was reached with minimum overlap of spheres. Sorting coefficient and skewness of the grain-size distribution were determined for each packing and plotted versus the obtained porosity. Although the variations of porosity values are relatively large, it can be concluded from the obtained trends that the lowest porosities are obtained for mixtures with poor sorting in combination with a slightly positive skewness (a slightly larger volume percentage of large spheres). The most important conclusion of the research is given in Chapter 6. Two different types of simulation illustrate that the model is capable of simulating all different disordered packings in a process-based manner. The ability to model all kinds of packing offers many opportunities for further research. It enables us for example to study the influence of different methods of sediment deposition on the properties of rock (e.g. porosity). The simulated packings of different grain assemblages can also be used to analyse the effect of the parameters of the grain-size distribution on flow through porous media. Aside from this, the model can easily be adapted to simulate also the other diagenetic processes, which creates further opportunities to increase our insight into the controls on heterogeneity in reservoir rock.


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Samenvatting De initiële porositeit van klastische sedimenten is maar matig gedefinieerd. Desondanks vormt deze eigenschap een belangrijke parameter in veel modellen die de diagenese van zand tot zandsteen tijdens de begraving van het sediment beschrijven. Deze modellen zijn belangrijk om de sub-seismische heterogeniteit van reservoir gesteenten te voorspellen. Tevens vormt de initiële porositeit een belangrijke parameter tijdens de reconstructie van de begravingsgeschiedenis van gesteenten aan de hand van decompactie routines, waarmee de maturatie van olievormend gesteente bepaald kan worden. Meting van initiële porositeit van sediment is doorgaans moeilijk, omdat de ongeconsolideerde sedimenten eenvoudig verstoord worden tijdens het monsteren en deze sedimenten dicht onder de oppervlakte al verschillende gradaties van compactie kunnen hebben ondergaan. Evenmin is het mogelijk de processen die plaatsvinden gedurende compactie te observeren, omdat deze plaatsvinden op een geologische tijdschaal. Aan de hand van laboratorium experimenten is men ook slechts deels in staat om deze processen na te bootsen, vanwege de relatief korte tijdsduur die hiervoor beschikbaar is. Om deze reden bestaan er nog geen analytische methoden om de relatie tussen korrelgrootte verdeling, korrelvorm en (initiële) porositeit te


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beschrijven. Vandaar dat deze parameters in veel modellen die porositeitsverlies beschrijven grotendeels genegeerd worden, ondanks dat bekend is dat deze een grote invloed hebben op de heterogeniteit binnen een zandlichaam. In dit proefschrift wordt een object-gebaseerd simulatie model gepresenteerd dat een beter inzicht moet verschaffen in de invloed van deze parameters op de initiële porositeit van zandige sedimenten en het verloop van porositeitsverlies gedurende het eerste compactie traject. In Hoofdstuk 2 worden de verschillende processen die plaatsvinden gedurende de diagenese van sedimenten geintroduceerd en wordt een overzicht gegeven van de verschillende typen modellen die gebruikt worden om porositeitsverlies te beschrijven. Veel onderzoek in het verleden is gericht geweest op het herkennen van de verschillende compactie en cementatie processen (korrel herschikking, korrel deformatie, pressure solution, en cementatie), hetgeen heeft geleid tot veel debat over het relatieve belang van deze processen. De onderlinge interactie van deze processen heeft daarbij echter een ondergeschikte rol gekregen. Het is echter duidelijk dat geen van deze processen genegeerd kan worden. De meerderheid van de modellen is beperkt tot empirisch bepaalde porositeit-diepte trends. Deze modellen zijn vaak herleid tot eenvoudige exponentiële functies met initiële porositeit als beginwaarde. Een veelbelovende modelleertechniek, die het probleem op korrelniveau benadert door middel van numerieke simulaties, is slechts mondjesmaat toegepast binnen de geowetenschappen. De werking van het model, de algorithmen en de implementatie, wordt beschreven in Hoofdstuk 3. Als uitgangssituatie voor het model wordt gebruik gemaakt van een eenheidscel met periodieke randvoorwaarden, die de invloed van de wanden van de cel moeten beperken. De cel wordt gedeeltelijk gevuld met bollen van gewenste grootte, die willekeurig gepositioneerd worden. De overlap wordt door middel van elastische krachtwerking tussen de bollen grotendeels geminimaliseerd, waarna onder invloed van zwaartekracht en de overgebleven krachten tussen de deeltje, de bollen verplaatsen, rollen, of vallen, tot een gravitationeel stabiele stapeling ontstaat. Verdere compactie wordt gestimuleerd door een lichte neerwaartse druk aan de bovenkant van de stapeling en eventueel agitatie van de stapeling. Zodoende wordt een realistisch traject van porositeitsverlies gesimuleerd. De bolstapelingen die gesimuleerd worden met het model, worden geanalyseerd en vergeleken met verschillende typen bolstapelingen in Hoofdstuk 4. Voor deze analyse waren gegevens verkregen van verschillende experimentele stapelingen. Statische methoden die de porositeiten, de aantallen contacten tussen bollen, en de onderlinge afstanden tussen bollen

Samenvatting

xv

beschouwen hebben aangetoond dat de structuur van de dichtste gesimuleerde stapelingen grote overeenkomsten vertonen met de structuur van experimentele ongeordende stapelingen van harde bollen. Een nadere beschouwing van de invloed van de randen van de cel laat zien dat door het gebruik van periodieke celgrenzen deze invloed effectief geminimaliseerd wordt. In Hoofdstuk 5 worden de relaties tussen de parameters van de korrelgrootteverdeling en porositeit onderzocht met behulp van het model. Een groot aantal simulaties is uitgevoerd voor mengsels van bollen van twee of drie verschillende groottes (ratio 1:2:4). Deze bolverzamelingen zijn gemengd in verschillende verhoudingen, waarbij elk mengsel gesimuleerd is met het model tot een stabiele stapeling ontstaat met minimale overlapping van bollen. Voor ieder mengsel zijn de sorteringscoëfficiënt (sorting) en de scheefheid (skewness) van de verdeling bepaald en uitgezet tegen de verkregen porositeit. Hoewel de variatie van porositeitswaarden relatief groot is, kan uit de trend geconcludeerd worden dat de laagste porositeits waarden verkregen worden voor verdelingen met een slechte sortering in combinatie met een licht overwicht van grote bollen in de verzameling (licht positieve scheefheid van de verdeling). De belangrijkste conclusie van het onderzoek wordt gegeven in Hoofdstuk 6. Aan de hand van twee verschillende soorten simulatieruns wordt aangetoond dat het model in staat is om alle verschillende wanordelijke stapelingstypen op procesmatige wijze te simuleren. Het vermogen om allerhande soorten stapelingen te verkrijgen met behulp van het model biedt vele mogelijkheden voor verder onderzoek. Zo kan bijvoorbeeld de invloed van verschillende afzettingsmechanismen van sediment op de gesteente-eigenschappen (zoals porositeit) bestudeerd worden. Ook kunnen gesimuleerde stapelingen van verschillende korrelmengsels gebruikt worden om het effect van de parameters van de korrelgrootteverdeling op stroming door de poreuze media te analyseren. Het model zelf biedt daarnaast nog alle ruimte voor uitbreidingen met de overige diagenetische processen, hetgeen mogelijkheden schept om het inzicht te vergroten omtrent de mechanismen die de heterogeniteit van reservoirgesteenten bepalen.


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1 General Introduction 1.1 Background Sandstone is consolidated sedimentary rock composed of predominantly sand-sized particles and it is defined to consist mostly of quartz grains, although other constituents such as feldspar, rock fragments and mica are also very common. The size of the grains ranges by definition between 0.0625 mm and 2 mm (between 4 phi and –1 phi on a logarithmic scale with base 2). This definition still allows for a large variety of rocks with different properties and characteristics. A key issue that has as yet proven very difficult to solve is to fully characterise sandstone using the scientific method (Griffiths, 1967). Griffiths proposed that a population of minerals forming a rock is characterised by the properties of its elements, and discerned five fundamental properties:

• kinds and proportions of its elements (mineral particles); • sizes of its elements; • their shapes; • their orientation; • and the packing of the population.


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Besides these fundamental properties of rock there are also derived properties (Griffiths, 1967). Porosity and density of a rock are considered derived properties that can be determined quantitatively, whereas bedding or stratification is a qualitative derived property. Derived property means that knowledge of its fundamental properties is necessary to understand it. Most research on sandstone during the last century has focused on derived properties such as porosity. Yet, porosity is still poorly understood, because knowledge of the relation between the porosity and the fundamental properties is insufficient. Interest in sandstone was raised in particular after oil started being produced in large quantities. Most of the earliest oil fields were found in sandstone reservoirs and in many parts of the world sandstone is still the dominant reservoir lithology (Selley, 1998; Gluyas & Swarbrick, 2004). With increasing demand for oil in the 1950s, a lot of knowledge has been gathered about environments in which typical reservoir sands have been deposited (Selley, 1985). The interaction of processes involved in the distribution of sediments in the various depositional systems is now fairly well understood, on account of many data gathered in outcrop and recent sedimentary environments. Much less is still known about the interaction of diagenetic processes, which take place after the sediments have been buried, and gradually transform loose sediments into solid rock. However, these processes eventually determine the porosity of the reservoir rock. 1.2 Problem definition Porosity is the percentage of the volume of voids in a volume of rock. It may be subdivided into primary and secondary porosity, or into intergranular and intragranular porosity. For sandstones the primary porosity represents the pore space that exists between particles after deposition; secondary porosity is formed during later stages of diagenesis as a result of dissolution of solid material. Intergranular porosity refers to the pores between the grains, and intragranular porosity to the pores within the grains. In discussions about porosity one should take note, however, of the type of porosity under consideration. This is dependent on the type of pores that the rock contains (Fig. 1.1). For the recovery of oil or gas, pores need to be in communication with other pores to enable the flow of fluid towards the well bore. The pore space that is interconnected is the effective porosity. In most sandstones, the majority of the pores are in communication with one another, so that we use the term porosity also where effective porosity is meant.

General Introduction

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Figure 1.1 Schematic diagram illustrating the three basic pore-types, which define the effective porosity open for flow (From Selley, 1985).

Porosity is considered to be one of the most important properties of hydrocarbon reservoirs, as the amount of hydrocarbons it can contain depends largely on the porosity and porosity has a large influence on the permeability of a reservoir (Panda & Lake, 1994). Besides the indication that porosity gives about the total reserves in the reservoir, the variability of this parameter also expresses the heterogeneity of the reservoir. Heterogeneity enhances the flow of fluids through certain parts of the reservoir and obstructs flow through other parts and thus determines the amount of recoverable reserves. A major problem is however that no direct way exists to measure porosity over large areas (at reservoir scale) with sufficient accuracy. Geological reservoir models are based upon seismic data cubes of great volume but usually insufficient spatial resolution on the one hand, and well and core data with excellent spatial resolution, but very limited coverage on the other hand (Fig. 1.2). The method usually adopted to fill this data gap is to interpolate between wells using more or less advanced geostatistical methods and trends from seismic attribute analyses (Doyen, 1988; Wolf et al., 1994; Dubrule, 2000). Even though these methods have improved considerably over the years through use of multiple seismic attributes and neural networks (Schultz et al., 1994a, b; Ronen et al., 1994; Trappe and Hellmich, 2000), they are often limited by the seismic resolution, as illustrated by poor correlations between predictions and measurements at well locations (Pramanik et al., 2004).


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Figure 1.2 Methods of data acquisition and their spatial resolution. Although advances in technology may increase the spatial resolution of seismic data, it is doubtful that it will reach the level of core data and well log data.

Another way to fill the data gap is to generate viable subsurface architectural patterns through numerical simulation of sedimentary processes. Modelling of stratigraphic records allows for translation of geological scenarios to realistic layer geometries and stacked sediment patterns. Many of such simulation models have become increasingly detailed, in spatial scales ranging from bed scale to grain scale (Paola, 2000; Storms, 2003). Porosity will also affect the simulated stratal architecture, because the thickness of the beds is directly related to the porosity. A better insight into the relation between grain size parameters and porosity results also in more realistic bed geometry. The translation from grain size data to (initial) porosity is, however, still open to improvements. Beard & Weyl (1973) presented an excellent quantitative, but empirical study on the effects of sediment properties on porosity, but theoretical studies of the relation between grain size distribution and porosity do not exist. Not only the translation of sediment mixtures to porosity at the surface, but also the evolution of porosity during burial is often heavily simplified. Many

General Introduction

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compaction models consider one or more processes, but the majority of these is predominantly analytical and regards only few parameters. One of the most popular models is an exponential porosity decrease with depth. The initial or depositional porosity is the starting point for many compaction models. The lack of an analytical relation between the grain size distribution of the sediment and the initial porosity is reflected as absence of grain size distribution as a parameter in those models. 1.3 Objectives and approach The objective of this study is to investigate the relationship between the grain size distribution and porosity, and thus contributing to decreasing the data gap that exists in reservoir modelling. A numerical model has been designed that can simulate the compactional behaviour in sediment during and after deposition. The focus for studying compaction is on simulation of grain rearrangement, and diagenetic processes such as pressure solution and cementation will be ignored. The model is based on sphere-packing models, developed in powder technology to predict physical properties of realistic granular materials. This implies that all individual spheres are traced, and are subject to a complex interaction with surrounding particles. 1.4 Thesis outline Compaction of particle populations is a strongly multi-disciplinary research topic. Compaction and its underlying mechanisms have been researched in many disciplines such as physics, chemistry, geotechnique, and mining engineering. To avoid unnecessary misconceptions, especially for readers who are not entirely familiar with geologic terminology, Chapter 2 will start off by giving some definitions of relevant terms. Subsequently, a compact overview of relevant literature is provided from a geologic point of view. After a description in Chapter 2 of published results on the different processes that are involved in diagenesis, the different modelling approaches are discussed. These discussions lead to the introduction of the model, placing it in the proper context of earlier work. Chapter 3 focuses entirely on the numerical model, named RAMPAGE, which can best be defined as a micro-scale grain rearrangement simulation scheme. The model workflow, boundary conditions, assumptions and the physical foundations are described. Different results and visualization modes are


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presented to illustrate how the model functions (for example its state of equilibrium), and how this may be interpreted in geological terms. Since simulation models, such as RAMPAGE, are commonly designed to reproduce reality to a high degree, they contain a large number of parameters that can be adjusted by the user. As a consequence, such models are much harder to test than abstract analytical models, which involve only a small number of free parameters (Paola, 2000). In Chapters 4 and 5, the RAMPAGE model is tested in two different ways. The structure of the simulated particle packings is analysed in Chapter 4, and compared with results of similar analyses of experimental particle packings. In addition, effects of the boundary conditions and some elementary choices of the model components are evaluated. The model can simulate rearrangement of almost any size distribution of particles (although the necessary computing resources will normally create a physical boundary). It is, nevertheless, important to consider what the resulting porosity values represent, with respect to porosities of real, experimental packings. In Chapter 5, the porosities of mono-sized, binary and ternary mixtures are presented and compared to binary and ternary diagrams obtained from laboratory measurements and idealized models. Also, in this chapter the effect of sorting and skewness of the grain size distribution on porosity is assessed.

2 Compaction of Sandy

Sediments and the Evolution of Porosity

2.1 Introduction Compaction has taken a prominent place in today’s research. A large amount of literature dealing with this topic has become available in a broad range of research areas over the last century. The topic does not only cover compaction of sediments or soil, but also compaction of granular materials like powders, rubble, construction material, pills, etc. The aims of investigation are as diverse as the areas of research, because compaction affects for instance the strength of a granular material, the volume of voids (which in turn may affect fluid flow through it), the packing of the particles (which can be of economic interest e.g. for packing materials or transport of rubble), or even the stability of structures placed on top of an unconsolidated soil. Geological interest in compaction is largely driven by exploration and production of hydrocarbons. The porosity of a rock provides information about the amount of hydrocarbons in place and is often a good measure for the permeability of the rock. Both the decrease of porosity and the decrease of layer thickness with depth are of importance to determine the burial history (maximum depth of burial) and thus the maturation of source rocks. Seismic velocity is also dependent on the porosity of beds, which is important for


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time-depth conversion of seismic surveys. Another motivation for compaction studies is given by the effects of pore fluids during abnormal compaction rates, which can cause serious hazards in overpressured zones during drilling, while on the other hand overpressure can also have positive economic effects on hydrocarbon reservoirs, especially if higher porosity is preserved during early cementation. An important driving force for research nowadays is the assessment of the reservoir quality, which plays a more and more important role in increasingly deeper reservoirs. This explains why currently, besides the scientific goal of fundamental understanding, much of the research in this discipline is predominantly focussed on deeper diagenetic processes such as pressure solution and cementation. Considering the long history of research, one might expect that compaction is well understood nowadays. This is, however, far from true. Most effort in prediction of porosity has been data-driven. Borehole measurements have been used to construct porosity-depth curves, which in turn have led to a range of empirical equations that describe the relations between porosity and depth, or rather, overburden stresses. Thin sections are being used to describe the mineralogy of reservoir rocks and much research has been devoted to mineralogical alterations, cementation and other chemical reactions in these rocks. Models of compaction are often generalised and constrained to reservoir scales, whereas the small-scale models are mostly reconstructions based on thin-section data with a limited predictive value for primary porosities. However, it is believed that a large part of the variability of porosity and permeability within a reservoir can be explained or even predicted through modelling of the processes at a small scale (Milliken, 2001). In this chapter we give a background for the model by defining the relevant terms and give an overview of significant literature in this field. The overview is predominantly focussed on compaction of sandy sediments, although other materials will be touched upon. 2.2 Definitions of relevant terminology The broadest definition of diagenesis covers all the physical, chemical and biological processes that modify the sediments during burial. Diagenesis is considered to start at the moment that sediment is deposited and buried by other sediments. It continues until high pressure and temperature drastically change the structure and mineralogical composition of the rock, which is referred to as metamorphism. Sometimes the term diagenesis is only used for modifications to the rock at larger depths, excluding mechanical compaction

Compaction of Sandy Sediments and the Evolution of Porosity

9

from its definition (Djéran-Maigre et al., 1998; Worden et al., 2000), whereas other workers prefer a division into diagenesis and catagenesis (or epigenesis) with the former comprising all processes that take place from deposition until lithification and the latter the processes from diagenesis until metamorphism (Chilingar et al., 1979; Buryakovski et al., 1991); a division that seems especially appropriate for carbonates. Hence, we use here the broadest definition comprising mechanical and chemical compaction as well as cementation, but excluding biological and weathering processes at the surface. Compaction is the reduction of the bulk volume of sediment or sedimentary rock. Bulk volume (Vb) encompasses the volume of the pore space (Vp) and the volume of the solid material (Vs) as presented in Eq. 2.1: b p sV V V= + (2.1)

Compaction can occur due to overburden loading (vertically) and due to loading as a result of tectonic stresses (laterally). Often the tectonic stresses are neglected and compaction is considered predominantly one dimensionally as a decrease of bed thickness. The bulk volume reduction for sandy sediments is often represented as -or rather considered equal to- porosity loss. These assumptions are commonly acceptable for the early stages of compaction, but will fail as soon as the solid matter is added or removed from the system by cementation or dissolution, as can be deduced from Eq. 2.1. Nevertheless, the reduction of pore volume during compaction is significant and can only be accomplished by expulsion of pore fluids. Especially in areas of high burial rates, the rate of expulsion is critically dependent on the permeability of the porous medium. If the permeability is too low to dewater the sediment under the increasing overburden pressure an overpressured zone will build up and both porosity and fluid pressure will become anomalously high. Overpressured zones can cause serious hazards during drilling. The exact definition of compaction is very important during decompaction routines for the reconstruction of the burial history, where accurate reconstructions of sediment thickness through geological time are needed (Giles et al., 1998; Perrier & Quiblier, 1974; Guidish et al., 1985; Ungerer et al., 1990). A narrower definition is adopted by soil engineers, who use the term compaction to describe the increase in bulk specific weight as a result of applied mechanical or hydraulic means, such as vibrating, loading or wetting (Allen & Chilingarian, 1976). Compaction can be subdivided into mechanical and chemical compaction. Mechanical compaction involves rearrangement of the grains due to reorientation, slippage, rotation, elastic and plastic deformation of grains and


10

fracturing of brittle grains. Chemical compaction or pressure solution is the dissolution of grains at the points of contacts under influence of high pressure, often in combination with precipitation of the dissolved material on free surfaces of adjacent grains where pressure is lower. Although mechanical and chemical compaction are often investigated separately, the two mechanisms are commonly working in combination to decrease volume. Removal of a small edge of a quartz grain by pressure solution can permit the grain to move into a much tighter packing position (Füchtbauer, 1967; Wilson & McBride, 1988; McBride et al., 1991; see also Fig. 2.1).

Figure 2.1 Only a small volume of the grain has to be removed by pressure solution (A) to permit a large porosity decrease through grain rearrangement (B). After Füchtbauer (1967).

Consolidation is the acquisition of structural coherence of sediments owing to reduction of volume, induration, cementation, etc. (Allen & Chilingarian, 1976). Again this is a broader definition than its meaning in the field of soil mechanics, where the term consolidation is used for the reduction of volume of a porous material by the expulsion of pore water upon the application of a load. A term that is directly related is unconsolidated sediments, which applies to all sediments that have not obtained structural coherence. The transformation of loose or unconsolidated sediments into solid rock is also often referred to as lithification. Cementation is precipitation of minerals in the pore space. These minerals can be clays, carbonates or silica (quartz minerals). The role of the flow of pore


11

fluid through the rocks is very important. If the flux is reasonably high, solids can be supplied in solution to the system and precipitation will cause a net porosity decrease without any loss in bulk volume. Early cementation of sands may sometimes strengthen the packing of the grains and can retard compaction significantly, or even halt it completely.

2.3 Components of compaction and porosity loss A number of processes contribute to a decrease of porosity of sands and sandstones in sedimentary basins (Wilson & Stanton, 1994; Giles et al., 1998). The processes are schematically illustrated in Figure 2.2.

Figure 2.2 Diagram of diagenetic processes, subdivided into different categories of processes.

These different controls on porosity loss are not all equally significant and are usually subdivided into the following four classes of processes: grain rearrangement, grain deformation and breakage, (pressure) dissolution and cementation. In this chapter this subdivision is also followed. In the next paragraphs a short overview is given on the mechanisms of each diagenetic process, followed by an overview of contributions that discuss the relative importance of the different mechanisms.


12

2.3.1 Grain rearrangement Grain rearrangement is the first diagenetic process to take place after sediment has been deposited. Immediately after settling from suspension, grains will move, drop, slip or rotate into a stable position. Relatively few workers have studied this process that is arguably one of the most important components of compaction. Grain rearrangement is often described as repacking of grains into a closer-fitting system. The system starts as a loose packing of grains immediately after deposition, in which even pores with a size greater than the grain size of the rock may occur regularly (Atkins & McBride, 1992). Porosity measurements by Pryor (1973) and Atkins & McBride (1992) on naturally deposited sands in river, beach and dune environments gave average porosity values ranging between 41% and 52%, and these values are known to be even higher (up to 80%) for carbonates and shales (Giles, 1996). The variability in measured porosity values is partly explained by differences in the sorting coefficient. Beard & Weyl (1973) artificially packed natural sands of fixed grain size and sorting, and measured porosity and permeability. Although the porosity values they reported were slightly lower than values reported for naturally deposited sands, their experiments showed a strong dependence of porosity on grain sorting, while grain size has hardly any effect on porosity but affects permeability, because porosity is dimensionless, whereas permeability is not. The observation that porosities reported from laboratory grain packs are generally lower than depositional porosities obtained from natural samples can be attributed to a higher degree of stabilisation of the laboratory sand packs (i.e., more gravitationally stable grains, less oversized pores). Usually such experiments are also carried out to determine the tightest possible packings of those specific grain populations (Palmer & Barton, 1987; Wilson & Stanton, 1994). Several sub-processes can induce rearrangement of the grain framework:

- Slight seismic tremors or vibrations in the subsurface can open spaces where grains are being pushed into or they can destroy existing bridges of particles.

- Existing contacts between grains fail as a function of increasing effective stress (overburden pressure minus pore fluid pressure), which results in slippage of individual grains accompanied by a movement of the grains into a new (perhaps temporary) stable position.

- Even geological time may influence the structure of a grain framework. De Waal (1986) recognised a relation between the strain rate and the lifetime of grain contact points as a result of loading. At high strain rates, the contacts exist only for a short time, resulting in a lower frictional


13

coefficient between grains. Once loading ceases, compaction must continue until the frictional force between grains is sufficient to withstand the overburden load. (De Waal, 1986; Giles, 1996).

- As a result of pressure solution, additional rearrangement can occur when small protruding volumes are dissolved as illustrated in Figure 2.1.

Figure 2.3 A) Photomicrograph (plane polarized light) of sandstone sample showing poor definition of grains and cement; B) Cathodoluminescence micrograph of same field of view, illustrating clear grain-cement relationships and porosity (P). Different grain contacts can be distinguished: floating (F), tangential (T), long (L) and concavo-convex (C). From Barker & Kopp (1991).

In thin sections rearrangement is usually associated with the presence of grain contacts. Different types of grains and grain contacts can be identified: floating grains (no contacts); tangential contacts (point contacts); long contacts (embayed); concavo-convex and sutured (or serrated) contacts (Fig. 2.3). Taylor (1950) illustrated that the presence of three or more contacts per grain in a thin section gives a strong indication that grain rearrangement has taken place. It needs to be noted that the number of contacts per grain in thin sections is the apparent number of contacts; floating grains have no contacts in 2-D, but may still have a point contact outside the thin section plane. The average number of contacts per grain counted in thin sections was later defined by Pettijohn et al. (1972) as Contacts Index (CI) and used by others as indicator of grain rearrangement (Wilson & McBride, 1988; McBride et al. 1991). A large number of long, concavo-convex and sutured contacts indicates that pressure solution was a relatively important factor of porosity decrease. No measurements of the number of contacts in three dimensions on natural sands and sandstone are known. Artificial packings of equal spheres are known to possess between 6 and 7 contacts per sphere after rearrangement (Bernal & Mason, 1960; Aste et al., 2005). Similar packings have often been


14

used as analogues for granular materials to analyse characteristics of these assemblies (Scott, 1960; Finney, 1970; Cumberland & Crawford, 1987). It has, however, proven to be extremely difficult to minimize the boundary effects of such arrangements. The results obtained from this type of experimental data apply only to examples where granular materials are placed in containers or other constrained environments. 2.3.2 Grain deformation and brittle failure The petrographic composition and the properties of grains play an important role in whether or not grain deformation or grain failure will take place. In “clean” quartzose sandstones the grains will not plastically deform and grain deformation is not an important mechanism for porosity reduction. If the percentage of ductile grains increases, grain deformation becomes a very important component of porosity loss upon application of overburden stress. At high contents of ductile material, grain deformation can even completely erase all porosity (Pittman & Larese, 1991). The ductile grains commonly found in sediment mixtures are clays and micas, whereas the brittle components are quartz and feldspars. For prediction of occurrence of grain deformation in a specific area, it is important to know the constituents of the sediments in that area. To predict the mineralogy of a sediment mixture, the provenance of sediments in the area of interest should be taken into account (Weltje & von Eynatten, 2004). Brittle fracturing of grains is usually not considered to be a very important component of compaction. It requires the presence of relatively large grains and is often related to areas of intensive tectonic deformation (Wilson & Stanton, 1994). It is also noted that dissolution of soluble cements may increase stresses at contacts, which can result in fracturing of grains. A recent experimental study on failure (or yield point) of loose sands during compression also indicated that fracturing of grains is highly dependent on grain size, grain shape, grain-size distribution and mineralogy. Sands with coarser and angular grains had lower yield stresses than chert-rich sands or very fine-grained sands. As an example, at about 2 km of overburden at hydrostatic pore pressure, very fine to fine-grained sands may have nearly 10% higher porosities than the very coarse grained sands (Chuhan et al., 2003). 2.3.3 Pressure solution Pressure solution takes place as sandstones undergo burial and the overburden stress concentrates at the points of contact between grains. These


15

stressed locations of quartz are more soluble than unstressed regions of the same grain. Dissolved material diffuses away from the contact and precipitates on unstressed surfaces of nearby grains as quartz overgrowths (Houseknecht, 1984; Wilson & Stanton, 1994) or can be transported away in solution. The process of pressure solution is associated with two mechanisms of porosity reduction. The first mechanism is the dissolution of quartz at grain contacts, which causes the grains to move closer to each other. The second mechanism is precipitation of the dissolved material, which leads to additional pore-space reduction. Figure 2.4 shows four possible scenarios for pressure solution between two grains with equal or different solubility. The two types of porosity reduction can also easily be derived from this diagram.

Figure 2.4 Schematic diagram to illustrate four types of contacts resulting from pressure solution. The grains (or detrital quartz) are displayed in grey, quartz cement is indicated with stippling, and dotted lines indicate the original grain boundaries. (a) Contact with equal dissolution of involved grains; (b) Contact with equal dissolution along a sutured boundary; (c) Contact with unequal dissolution of involved grain; and (d) Contact with dissolution of only one grain. (After Houseknecht, 1984)

The extent of pressure solution shows a clear relationship with several factors such as mean grain size, thermal maturity and clay content. A smaller mean


16

grain size appears to be beneficial for dissolution of quartz, whereas large volumes of quartz cement were encountered in samples of larger mean grain size (Houseknecht, 1984). These relations were derived from observations in thin sections, where the fine-grained sandstones were observed to have more contacts per grain than coarse-grained sandstones. The coarse-grained sandstones retained higher pore volumes, and thus more available surface area for precipitation of cements. Among sands of equal grain size, more intergranular pressure solution had occurred in areas of higher thermal maturity. As a result sands in these areas had lower porosities. The presence of clayey material at the grain contact is believed to catalyse the process of intergranular pressure solution, by enabling pressure solution to operate at lower pressures and temperatures (Wilson & Stanton, 1994). Rittenhouse (1971) analysed the porosity loss from pressure solution geometrically for several regular packings. He found that a 30° rotated orthorhombic packing would lead to a maximum porosity loss by precipitation of dissolved material relative to porosity loss by solution, and considered this also to be the maximum for any sand with well-rounded and well-sorted grains with the assumption of all strain being in the vertical plain. Sibley & Blatt (1976) argued that even larger porosity losses occur when strain is equal in all directions. They remarked that neither their model nor Rittenhouse’s model is valid for natural sands, since most strain in regions of intense pressure solution is in the horizontal plane. Discrepancies from a uniform size distribution and perfect sphericity contribute even more to porosity loss from pressure solution. 2.3.4 Cementation Precipitation of (mostly) foreign material in the available pore space is termed cementation. Cementation is generally not considered to play a very important role during the first few kilometres of burial. Contrary to other mechanisms of porosity reduction, cementation does not decrease the bulk volume of the porous media. Neither does it depend strongly on depth, although pressure and especially temperature may enhance cementation, but cementation strongly depends on the ions that are present in the pore fluids. The precipitated material can consist of a broad range of minerals, but most common are quartz, clay (kaolinite, chlorite, smectite, illite) and carbonate cements. The dominant cement type in quartzose sandstones is quartz cement. Authigenic quartz precipitates in the form of overgrowths, causing predominantly porosity loss, but is less disadvantageous for permeability than cements that occupy the pore or the pore throats. The source of silica for cementation is much debated and often believed to be supplied by pressure solution, although research by Sibley & Blatt (1976) indicated that pressure


17

solution could only account for one third of the quartz cement and that other sources such as illitisation of smectitic clays, expulsion of pore water from shales, dissolution of siliceous fossils and volcanic glass are more important. Although cementation decreases the porosity of the rock, early cementation may slow down or even halt mechanical compaction by increasing strength and rigidity of the grain structure, and thus prevent rearrangement, grain deformation and pressure solution. A good example of this effect is the study by Cavazza & Dahl (1990); in a single sandstone formation they found both early-cemented samples with loose-packed framework grains and samples without early cementation that showed densely packed framework grains, occurrence of pressure solution and squeezed ductile grains. Extensive reviews of research on cementation are documented for instance by Wilson & Stanton (1994) and Bjørlykke et al. (1989). 2.3.5 Relative importance of the diagenetic processes Several workers have assessed the relative importance of the different mechanisms that reduce porosity. Houseknecht (1987) developed a technique to examine the respective roles of compaction and cementation. His method is based on the assumption that compaction permanently reduces the volumes between the grains by reducing the bulk volume, while cementation reduces the intergranular porosity without reduction of the bulk volume. This implies that the precipitated material is supplied from other (nearby) formations. Using point counting, performed on a cathodoluminescence microscope to distinguish between detrital grains and cements (see Fig. 2.3 for an example), he determined both the intergranular porosity and the intergranular volume (a term introduced by Weller, 1959, which indicates the sum of the pore volume and the cement volume, and is synonymous to minus-cement porosity) and derived the volume of cement. The quantities of intergranular volumes and cement volumes could then be plotted on a simple graph, from which he could easily determine the diagenetic events that were most relevant to porosity reduction. Houseknecht demonstrated this approach with data from two sandstone reservoirs and showed that in both reservoirs compaction led to a more significant reduction of intergranular porosity than cementation. Although his model has some shortcomings due to oversimplification, as Pate (1989) and Ehrenberg (1989) later pointed out, it is a very effective method to quickly evaluate the relative importance of compaction and cementation. Lundegard (1992) gathered a large database and plotted these data in a diagram using an improved version of the method suggested by Houseknecht (1987) taking into account the comments by Pate (1989) and Ehrenberg (1989). Lundegard concluded that porosity loss in the majority of sandstones was dominated by compaction.


18

Analyses comparable to the one by Houseknecht (1987) were carried out by few others (Wilson & McBride, 1988; McBride et al., 1991) and they also concluded that compaction was the dominant cause of porosity reduction. Besides the relative importance of compaction and cementation, they established in thin section analyses the amount of porosity loss due to grain rearrangement, ductile grain deformation and pressure solution, albeit somewhat subjectively. The volume percentage of the rock where grains overlap at concavo-convex and sutured boundaries (see Fig. 2.3-B and Fig. 2.4) was related to the pore volume lost by pressure solution. The volume percentage of the rock that was pore space prior to straining rock fragments and bending of micas was related to the pore volume lost by grain deformation. The remainder of the decrease of minus-cement porosity was associated with grain rearrangement. In both studies they found that twice as much porosity was lost by grain rearrangement than by pressure solution and ductile grain deformation, although they admit that the impact of pressure solution may be larger because it can initiate additional rearrangement as illustrated in Figure 2.1. Palmer & Barton (1987) conducted an experimental study on uncemented quartzose natural sands and sandstones, which they disaggregated and recompacted using a device that applied lateral vibrations during sedimentation. In contrast to the aforementioned studies, they conclude that grain rearrangement is only responsible for a limited porosity reduction, which proceeds rapidly after deposition. They suggest that increasing depth and time have little effect on the porosity afterwards. Disaggregation and recompaction of the natural sands yield similar porosities as those of the undisturbed samples observed near the surface, but the authors consider the laboratory fabrics to have achieved higher stabilities. By cancelling out other fundamental processes that primarily control porosity reduction, Palmer & Barton strongly favour pressure solution as the dominant process of porosity decrease. Sibley & Blatt (1976) were amongst the first to demonstrate the effectiveness of cathodoluminescence techniques to distinguish between quartz cement (authigenic quartz) and detrital quartz (see Fig. 2.3). In an area that was believed to be the site of extensive pressure solution, they showed that less pressure solution was taking place than previously believed, but argued that only little pressure solution was necessary to achieve considerable mechanical porosity reduction. Pittman and Larese (1991) found experimentally that a large fraction of ductile lithic fragments can diminish virtually all porosity purely by plastic grain deformation.


19

2.4 Effect of material and mineral composition The source area of sediments and the depositional environment determine the mineralogical composition and the size distribution of the sediment mixture subject to compaction (Weltje & Von Eynatten, 2004). When studying diagenesis it is important to consider the material that is affected, since this will have a large influence on porosity evolution. The parameters of the grain size distribution are known to affect porosity. Beard & Weyl (1973) found empirical relations between the sorting coefficient of sediment and the porosity. Tickell et al. (1933) found that porosity also varies with the skewness of the size distribution. Various researches in related disciplines that regard granular media have shown porosity trends for binary and ternary mixtures (Furnas, 1929; Koltermann & Gorelick, 1995; Yu & Standish, 1987) that are dependent on the volumes of the large and fine fractions and the ratio between the diameter sizes of the large and the fine grains. Sohn & Moreland (1968) showed that these trends are similar for binary mixtures of continuous size distributions. In Chapter 5 the effects of grain size distribution on porosity will be discussed and analysed in more detail. Clean sandstones commonly have initial porosities around 45% and are able to retain their porosity very well during burial. Shales, on the other hand, have depositional porosities in the range 75-80%, that will fall to 45% within a few tens of meters of burial (Burst, 1969) and eventually to very low porosities at larger depths. Carbonates can behave very unpredictably during burial (Giles, 1996). Some carbonates have a very reactive nature, like for instance aragonite-rich carbonates in the shallow subsurface. Major modifications to the pore network and lithification can already occur in the shallow subsurface, which enable some carbonates to resist compaction until high effective stresses are reached. The presence of specific minerals in sandstone can also have a huge influence on the behaviour of the sediment mixture during burial. The effect of lithic grains, investigated by Pittman & Larese (1991), can cause a major porosity loss through grain deformation, if the lithic fragments are ductile. The presence of clays or carbonates can induce cementation at a later stage during burial (Bjørlykke et al., 1989), and especially clays can be responsible for major destruction of permeability of sandstone reservoirs. Tri-axial experiments (variable vertical and lateral stresses and variable pore pressure) on unconsolidated materials by several workers (De Boer, 1975; Giles, 1996) have also indicated that the rate of compaction is a function of the mineralogy of the rock. They found that clay-rich lithologies loose porosity at a faster rate than quartz-rich lithologies. Mixtures of sand and clay result in


20

a faster rate of porosity loss as well, with a minimal porosity/depth envelope at the limit between the clayey sand and the sandy shale domains (Revil et al., 2002). It is also noteworthy that lithic sands containing large quantities of ductile grains not only loose porosity at a faster rate, but also show a completely different compaction behaviour than most other compositions, for which an essentially linear relation between the natural logarithm of porosity and vertical effective stress is observed (Giles, 1996). 2.5 Overview of quantitative models A large number of models has been developed to describe diagenesis, either in terms of bulk volume reduction, porosity loss or decrease of layer thickness. Most of these models are limited to certain applications, materials or topographic regions, and possess assumptions and boundary conditions specifically tailored to their own needs. A complete predictive deterministic compaction model is not readily available. The lack of such a universal model is not surprising, if one considers the large number of variables that influence the composition of the sediments and the changes to the sediment package during diagenesis. An overview of the main variables by Giles (1996) comprises:

- mineralogy of the rock (provenance) - sorting of sediments (initial porosity) - vertical effective stress (overburden pressure – pore fluid pressure) - type of pore fluid - temperature - loading rate - time (creep) - maximum burial depth (irreversibility of porosity loss) - cementation

2.5.1 Empirical models Where the driving forces of a process are not easily recognized, or not well understood in relation to the dependent variable, very simple models are often used that aim to predict the unknown variable directly from a single, and often independent, variable. A well-known example is the correlation between porosity and depth of burial, which do not have a direct dependence,


21

but often show good linear or exponential correlations. This type of approach is widely used, particularly for mechanical compaction. Porosity-depth curves A compilation by Giles (1996) of porosity-depth models is shown in Figures 2.5, 2.6 and 2.7 for sandstones, shales and carbonates respectively. One can readily observe that the plots are hardly comparable, which demonstrates that standard curves do not exist. The curves are based on rocks of different ages, compositions, from areas of different thermal histories, etc., and the data are often obtained from different sources, such as logs, cores, thin sections, etc. Nevertheless, models like these have the advantage that they are easy to construct, as they only require regression techniques and a limited data set. Also, they are fairly simple to apply, since only a few parameters are needed to predict the approximate porosity at a certain depth. However, as a rule these models are often only applicable to specific lithologies, or a specific sedimentary basin, while the different conditions in other lithologies or basins affect the porosity evolution in ways not accounted for in the models. So, before they can be used as predictive tools they need to be calibrated to local conditions.

Figure 2.5 A compilation of porosity-depth trends for sandstones. The grey area is a visual aid to show the range of possible porosity-depth pairs. After Giles (1996). 1, 6: Galloway, 1974; 2, 3, 5, 7, 8: Giles, 1996; 9: Loucks et al., 1979; 10, 12: Scherer, 1987; 11: Baldwin & Butler, 1985; 13: Sclater & Christie, 1980; 14: Falvey & Deighton, 1982.


22

Figure 2.6 A compilation of porosity-depth trends for shales. The grey area is a visual aid to show the range of possible porosity-depth pairs. After Giles (1996). 1: Athy, 1930; 2: Hosoi, 1963; 3: Meade, 1966; 4: Hedberg, 1936; 5: Magara, 1968; 6: Weller, 1959; 7, 8, 11, 14: Giles, 1996; 9: Proshlyakov, 1960; 10: Foster & Whalen, 1966; 12, 16: Dzevanshir et al., 1986; 13: Ham, 1966; 15: Sclater & Christie, 1980; 17: Falvey & Deighton, 1982; 18: Dickinson, 1953.

Figure 2.7 A compilation of porosity-depth trends for carbonates. The grey area is a visual aid to show the range of possible porosity-depth pairs. After Giles (1996). 1: Limestone. Royden & Keen, 1980; 2: Dolomite. Schmoker, 1984; 3: 75-100% Limestone. Schmoker, 1984; 4: Mouldic limestone, Borneo. Giles, 1996; 5: Chalk. Sclater & Christie, 1980; 6: Dolomite. Schmoker, 1984.


23

Correlation of porosity to depth is commonly successful, because pressure, temperature and time frequently exhibit a high degree of correlation with depth (Byrnes, 1994). Moreover, while the influence of pressure associated with mechanical compaction decreases with depth, the influence of temperature expressed by pressure solution and cementation increases with depth (Fig. 2.8). The downside is, however, that with increasing depth the correlation with other variables often decreases. Also, there is no correlation between depth and composition. This causes an increasing scatter of porosity values with depth, complicating the predictive value of these correlations (Fig. 2.5). The most accurate porosity-depth relations have been obtained for shales (Byrnes, 1994), as can also be seen by the relatively narrower grey-coloured band in Figure 2.6. Byrnes suggests that this may be attributed to the relatively minor role of chemical diagenesis in shales, which is a result of their low permeability. The significant role of mechanical compaction, which reduces porosity to very low values, can be directly related to pressure, and hence is highly correlated with depth.

Figure 2.8 While the relative influence of pressure on porosity generally decreases with depth, the influence of temperature increases. Hence, a good (linear) correlation is usually observed between porosity and depth.

Porosity loss equations Based on the porosity-depth curves a range of equations has been produced. Linear equations of the form 0 azφ φ= + (2.2)


24

with φ0 equal to the average porosity at the surface, a a constant, and z the burial depth, are straightforward, but the use of this type of equations is of course limited to shallow depths since negative porosities will be predicted at larger depths. These equations have nevertheless been successfully applied to restricted intervals (e.g. Galloway, 1984; Loucks et al., 1979). The prediction of negative porosity values is naturally overcome with an exponential form. The basic equation was first advocated by Athy (1930) and has since become known as Athy’s Law, 0

bzeφ φ −= (2.3)

where φ0 is again the average porosity at the surface, b a constant, and z is the burial depth. Often this equation is slightly modified using effective stress rather than depth as the driving force, which generally appears to fit porosity data extremely well (Giles, 1996), as long as the vertical effective stress σveff equals the maximum stress reached at maximum burial:

0

veff

ceσ

φ φ−

= (2.4)

This equation is the solution of the differential equation dd cφ

σ φ= − , which

assumes that under identical loads, a sediment with high initial porosity will compact more than a sediment with low initial porosity (Bahr et al., 2001). If the pressure gradients for fluid and overburden are more or less constant, Eq. 2.4 can be rewritten as a function of depth. Combined with the assumptions of no excess pore pressure and a constant grain density, Bahr et al. (2001) derived the following equation:

( )

( )1

( )s w

s w

cg z

cg z

ez

e k

ρ ρ

ρ ρφ− −

− −=+

(2.5)

where 1 (1 (0)) / (0)k φ φ= − , thus related to the porosity at the surface,

ρs, ρw is grain density and water density respectively, g is gravity acceleration, z is depth, and c is a constant. Equation 2.5 shows a linear behaviour at shallow depths and an exponential behaviour at greater burial depths. Bahr et al. (2001) demonstrate that this


25

exponential behaviour accurately fits a large worldwide collection of porosity data from compacted shales, silts and sandstones. In the field of (pharmaceutical) powder research two independently derived empirical relations have been used, namely the compaction equations of Heckel (1961) and Kawakita & Ludde (1970-1971). In powder metallurgy, the Heckel equation is most popular, since Heckel used metal powders for his work, and such powders all compact in the same manner (plastic deformation) (Heckel, 1961; Denny, 2002). The Kawakita equation is presented by Kawakita & Ludde (1970-1971) as follows:

0

0 1V V abP

CV bP−

= =+

⎛ ⎞⎜ ⎟⎝ ⎠

(2.6)

with: C = degree of volume reduction V0 = initial apparent volume V = powder volume under applied pressure P = applied pressure a,b = characteristic constants The Heckel equation (Heckel, 1961) was derived from the consideration that reduction in porosity obeys a first-order type of reaction with applied pressure (P):

d

KdP

φ φ− = (2.7)

which on integration gives

0

1 1ln ln KP

φ φ= + (2.8)

where 0φ is the porosity at P = 0 and K is a constant, which Heckel empirically

found to equal 01 (3 )K σ= with σ0 being the yield strength of the

compacted material. Denny (2002) showed that, after adding some minor modifications to the Heckel equation, the Kawakita equation represents a special case of the Heckel equation. He converted the volume terms in Eq. 2.6 into porosities:


26

0 0

0 1V V

Vφ φ

φ− −

=−

(2.9)

which, after rearranging and taking logarithms across, becomes:

[ ]00

1 1ln ln ln 1 (1 )b Pφ

φ φ= + + − (2.10)

In cases that 0(1 )b Pφ− is small, Eq. 2.10 can be written as:

00

1 1ln ln (1 )b Pφ

φ φ= + − (2.11)

which is of the same form as Eq. 2.8. Continuing this line of reasoning, it can be seen that Eq. 2.8 is also quite similar to Athy’s Law in Eq. 2.4, which can be rewritten by taking logarithms across as:

0

1 1ln ln effv

c

σ

φ φ= + (2.12)

Byrnes (1994) classified published diagenetic models into effect-oriented and process-oriented. The porosity-loss models discussed above typically fall in the first category. All these models describe a more or less similar effect under application of pressure, and thus have been captured in a series of essentially identical equations. 2.5.2 Process models Besides the empirical, effect-oriented models that were treated in the previous section, several process-oriented models have been developed. A large share of such theoretically derived models with a diagenetic context have been developed in geochemistry. These models typically involve physical compaction routines, thermodynamics, and reaction kinetics. Computer models in this field have reached a rather advanced stage, which allows investigators to perform fairly realistic experiments with control over a large number of variables. (Wood, 1994). Fewer examples are available for the other processes of compaction.


27

Lander & Walderhaug (1999) demonstrated the advantage of process-oriented modelling over effect-oriented modelling. Their model approach may be considered to be a hybrid model according to the classification of Byrnes (1994), because they describe compaction as a function of the effective stress using an exponential relation, but use a precipitation-rate-controlled process model for quartz cementation, based on Walderhaug (1994, 1996). Their model enabled them to provide probabilistic predictions of reservoir quality and to assess the effects of different burial histories (effective stress history, thermal history, timing of fluid overpressure development, cementation) on reservoir quality for sandstones with similar initial compositions and textures. A very complete analytical model for porosity loss of a sample consisting of various minerals was derived by Giles (1996) and presented in the following form:

, 0 2(1 )5

0

*,

1

,

1(1 )

3 (1 ) [ ]1

3 ( ) 3 2

( ) ( , )

1

m veffv r

veff veff ref d i hRTv

J

veff sj p j ej j pj jj

j effsm j j j j

s

t

m V D idVk e

K K V hr

n ddTV n

V t dt dt

σ

φ

φφ

σ σ φ δφ πρ

σ ε ε ζ ε γ

σϑ β

−

=

⎛ ⎞⎜ ⎟⎝ ⎠

∂− =

− ∂

−− − − − −

− ℘ − ℘ − +

∂− −

∂

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

⎛ ⎞⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠

∑

, ,1

,max

( )j J

w e w pj

veff veff

R R

σ σ

=

=− −

=

⎧ ⎫⎨ ⎬⎟⎩ ⎭∑

(2.13)

with: φ = porosity

t = time σveff(,max) = (maximum) vertical effective stress K(φ) = elastic modulus 3K0 = σ*/ε* , where σ* is a material constant and ε* is the limit

strain kv = viscosity coefficient V = bulk volume m = fraction of grains in favourable position for pressure solution (1-φ)Vref = solid volume in reference volume Vref


28

δ = thickness of absorbed water layer Dd,i = diffusion coefficient for species i [i]0 = concentration of species i in the bulk pore fluid at some

pressure Pf ρ = density of mineral h = thickness of dissolved grain r = grain radius vm = molar volume of the mineral R = gas constant T = temperature J = number of types of mineral ε = strain εp

* = strain at which plastic flow is initiated εp = plastic strain rate ζ = volume fraction of plastic grains γ = constant of proportionality ℘s,℘e = Heavyside functions, defining the period over which plastic

deformation occurs Vs = total solid volume Vsm = molar volume of solid component n = number of moles of component

ϑ = expansitivity of mineral β = compressibility of mineral Rw,e = rate of water escape from the sediment Rw,p = rate of water production from dehydration reactions per

unit solid volume In words, this impressive equation can be summarized as:

Rate of Porosity Loss = Rate of Elastic Strain + Rate of Mechanical Creep + Rate of Pressure Solution + Rate of Plastic Deformation of Ductile Grains – Rate of Change of Solid Volume.

Although the equation looks very complicated, essentially it is of the same form as Eqs. 2.4 and 2.8, having only now basically a stress term:

(1 ) ( )ftφ φ σ∂

− = −∂

(2.14)


29

The major difference, however, is that every process that may reduce porosity is accounted for, which adds the possibility to quantify the effects of the different processes. After integration, which requires the initial porosity as boundary condition, this equation demonstrates that besides time and temperature, porosity loss depends heavily on facies and initial composition, because these factors have a large influence on the initial porosity. Since facies and initial composition also govern the elastic constants and plastic deformation (as suggested by Eq. 2.13), these data should be available in a complete porosity model. Unfortunately, an extensive model such as this one will in practice be quite laborious to use. The formula cannot be solved analytically, and for a numerical solution a detailed burial history of the rock is needed. In combination with the aforementioned compositional data and the remaining parameters of the equation, the data set has to be rather substantial. 2.5.3 Rock reconstruction An entirely different approach to modelling of compaction and/or porosity loss, yet also process-based, is numerical simulation of object-based particle packings, also known as discrete element models (DEM). Workers in fields such as physics, powder technology and geotechnique have adopted this approach (Cundall & Strack, 1979; Nolan & Kavanagh, 1992), but it has found little follow-up within the geological sciences so far. A group of models that has been applied within geosciences and which is closely related to the DEMs, are the reconstruction models of realistic porous media. These models mainly differ in their method of construction. The generation of granular media does not necessarily need to be realistic for this type of models, and the purpose of the construction is mainly focused on acquiring a realistic end product. Such a reproduced medium can then be further used as a boundary topology for solving standing problems concerning flow through porous media, electric conductivity and mechanical strength of granular assemblages. In this paragraph an overview of different rock reconstruction models is given, while the DEM-type models are treated in the next paragraph. One of the most simple theoretical models was designed by Bradley (1980). It is based on regular packings of equal spheres and although this can hardly be considered to represent realistic rock, it was used to approximate relationships between the porosity and permeability, formation factors and conductivity. The porosity of the packs was reduced by coating the spheres with additional material (onion-skin or growth model). This growth model was adopted by Roberts & Schwartz (1985) and Schwartz & Kimminau (1987) to analyse the electric conductivity for ordered and disordered


30

packings of equal spheres. To model the disordered packings they used the Bernal distribution (the coordinates of the particle centres of a dense random packing) (Bernal, 1960). Their findings were consistent with experimental data and showed that preferential growth of the grains towards the pore throats led to an increasing conductivity with decreasing porosity in comparison with uniform growth. Alternatively, a random packing of equal spheres that was constructed in the laboratory by Finney (1970), was used by several workers (Bryant et al., 1993; Cade et al., 1994; Bryant & Pallatt, 1996) to generate a realistic network of spheres, pores and throats for modelling of diagenetic processes. Tacher et al. (1997) generated heterogeneous and anisotropic packed beds in 2-D and 3-D by placing grains inside a field of random initial points using a density function representative of the largest grains. The process starts with the largest grain and adjusts size and location of the remaining grains to minimise porosity. A disadvantage of the method is that the grain size distribution cannot be directly specified. It can however easily be extended to generate packings of grains with arbitrary shapes. Yao et al. (1997) reconstructed random numerical samples from porosity and an autocorrelation function obtained from thin section data. These samples were constructed as a set of elementary cubes, with each cube representing either pore space or solid material. The computed values for permeability and formation factor were comparable to the values of the initial sandstone samples. The reconstruction models discussed above do not incorporate a sedimentation method. The positions of the grains are either determined by regular lattices, measured experimental packs, or random methods. An algorithm that simulates simple sedimentation was designed by Visscher & Bolsterli (1972). The Visscher-Bolsterli algorithm consists of sequentially dropping spheres of arbitrary sizes onto an xy-plane, until the sphere comes to rest on the floor or in a stable position on three other spheres. Schwartz & Banavar (1989) investigated the transport properties of granular packings that were initially generated with the Visscher-Bolsterli algorithm and the obtained system was then changed using the growth model to control porosity. This combination yields complex granular media without wall effects, since these were eliminated through the employment of periodic boundaries. Another model using sequential deposition of particles was proposed by Coelho et al. (1997), who applied the model to non-spherical particles. Each particle settled after moving under the action of steepest descent and conjugate gradient methods. Each increment of movement that lowers the body’s centre of gravity without penetration into other grains was granted. Pilotti (1998, 2000) later refined this algorithm by solving the law of motion for every falling grain and by simulating the interaction with already settled grains. While the simulation of deposition was carried out with perfect


31

spheres, the shape of the particle could be adapted after the packing was generated. Probably one of the most complete rock reconstruction models is the rule-based process model by Bakke & Øren (1997). With an integrated numerical computer model of sedimentation, compaction and cementation, they were able to generate realistic homogeneous and heterogeneous sandstone models. Grain-size data were obtained from thin sections, and individual grains with sizes randomly selected from a distribution curve were consecutively deposited in a stable position. The deposited pack was subsequently subjected to linear compaction until a representative intergranular porosity was reached. Diagenesis was further modelled by a uniform increase of radii to simulate quartz cement overgrowth, and by random precipitation of clay coatings on the grain surfaces. For the latter process the initially object-based model had to be voxelised. From the generated model the pore network could be extracted and used as input for flow simulators. The predicted drainage capillary pressure and relative permeabilities were in good agreement with experimental data. In many of the reconstruction models presented above, the amount of compaction or grain coating that the grains undergo is determined by a target porosity value. This categorises these models as rock reconstruction models rather than compaction models. Unfortunately, it makes this type of models not very appropriate to use as porosity prediction tools. Yet, a more or less similar approach can very well be adopted to construct such a model. In the next paragraph a process-based method of modelling granular media is reviewed. 2.5.4 Numerical modelling of granular media Cundall & Strack (1979) were amongst the first to apply an object-based numerical modelling technique to granular media and verified the resulting force fields with experimental data on photo-elastic discs from De Josselin de Jong & Verruijt (1969). Based on fair qualitative agreement, Cundall & Strack (1979) concluded that their discrete element method was a valid tool for fundamental research into the behaviour of granular assemblages. The DEM has been successfully applied afterwards in many other disciplines working with granular materials such as (geo)physics, material science, powder technology, and geotechnique. Much of this research has been focussed on the strength or compressibility of such assemblages. The opportunities of the DEM to study the influence of the different parameters on porosity decrease during compaction have been left unnoticed. This is remarkable, because aside from permeability, porosity is generally considered to be one of the more important parameters of reservoir sands.


32

A number of possible reasons why such numerical simulations of discrete elements have not been adopted within geological research might be:

- The complex mechanical and physical interactions of a large number of bodies require considerable computational resources.

- Many successful applications of the DEM consider only a small number of particles, and are often limited to two-dimensional problems. Practical geological problems are usually three-dimensional in nature (for instance porosity prediction or fluid flow through reservoirs).

- Geological processes take place over long timescales. A discrete numerical model behaving in a strictly physical manner usually needs to incorporate discretised changes at very small time steps.

- The hard boundaries of a simulated container are likely to affect the results of the obtained assembly. In several other disciplines these boundaries are often part of the problem. Geological granular media are however deposited in layered form and therefore have a much longer lateral continuity than a vertical continuity.

Figure 2.9 The average number of contacting spheres per sphere (mean coordination number) versus the porosity defines the boundaries of random packing of uniform spheres (modified from Nolan & Kavanagh, 1992).


33

Nolan & Kavanagh (1992, 1993) developed a rather intuitive model. Instead of applying physical laws, they used a rule-based approach to simulate random packing of hard spheres for gravitationally stable lattices. Spheres with either equal sizes or radii obeying a lognormal distribution were randomly placed in a box in an overlapping state and gradually transformed into an overlap-free and stable lattice. With their model they claimed to be able to generate all known types of random packing (Fig. 2.9). An interesting aspect of their model is that it is able to simulate packings that contain bridges, a prerequisite for forming oversized pores. This advantage follows mainly from their choice of initialisation that permits overlapping particles. Other possible ways to construct initial states can be either a) reconstructions of real granular media through measurement (micro-tomography), or, as was discussed in the previous paragraph, b) sequential dropping of particles at a random lateral location that are subsequently allowed to slide into a local low in a stable position, or c) random positioning followed by radial growth to prevent overlaps. The model which will be presented in Chapter 3 can be placed somewhere between the physical but complex DEM, advocated first by Cundall & Strack (1979) and the rule-based model of Nolan & Kavanagh (1992, 1993). 2.6 Discussion In the previous section a brief overview of the broad range of models that describe the transformation of sediment into rock was given. It is believed that the selection of models discussed above, is a reasonable representation of the developments within the field of geological sciences. The interplay of the mechanisms causing porosity loss is possibly the most difficult characteristic of diagenesis. All models describe either the net result from one or two varying parameters with burial, or they treat the different processes separately. The tendency to study only a single process has caused several investigators to focus the debate on the question which of the processes is the most dominant control on porosity reduction, rather than on the interaction of the different processes. Most workers have more or less agreed that every process can be of significance. The one thing that can be taken for certain is that the diagenetic change starts off with grain rearrangement. While several workers concluded that in many samples grain rearrangement was a very important factor of porosity decrease, only Palmer & Barton (1987) concluded that occurrence of grain rearrangement was purely restricted to the near surface. However, they


34

based their conclusions solely on the inability to reproduce porosities of the samples in the laboratory. In their reasoning they disregarded the effect of geologic time on stabilisation of the packing, as was shown by De Waal (1984). Therefore, already from the start it could not be expected that comparable porosities could be obtained from a mechanical compaction experiment in such a short period. After examination of the different models, it is evident that the general trend of porosity loss during diagenesis, and compaction in particular, can be described in quite a satisfactory way. However, the degree of small-scale variability and heterogeneity of sedimentary rock, which affects both porosity and permeability values, is poorly covered by the published models. As Giles (1996) pointed out after finalizing his attempt to obtain the universal analytical porosity loss equation, much of the heterogeneity of sandstone is related to the composition of the sediment mixture. While his formula accounted for the effects of the separate mineralogical compounds, the full solution requires the availability of a sediment sample for analysis, and the ability to measure the necessary parameters for each material (see Eq. 2.13). A second, and perhaps equally important source of variability in reservoir sands is the initial porosity of the sediment. Practically all the porosity loss models that have been developed use initial porosity as a boundary condition, and very often this parameter is estimated to have a value of somewhere between 40 and 50 percent. When only an approximation of the porosity at a certain depth or after a certain burial history is required, a precise estimation of the initial porosity will not be of primary importance. If, however, better accuracy is desired, a misjudgement of the initial porosity of sediment may cause a substantial error at depth. It is remarkable that the initial porosity of sediments has received only little attention (Beard & Weyl, 1973; Pryor, 1973; Atkins & McBride, 1992). While it is known that the grain size distribution, and in particular the sorting and the skewness of the distribution, are of importance, there is no readily available model that determines the (initial) porosity as a function of these parameters. Equally important, there is no generally accepted unique definition of initial porosity. The way forward in diagenetic modelling needs to be sought in pore-scale simulation of the different processes. Models like the reconstructed granular media have already proven their value for further analyses of transport phenomena. The methods of construction of these packings are too simplistic, however, for the prediction of porosity loss. For instance, sedimentation algorithms that describe consecutive settling of spheres prevent the construction of bridging pairs of spheres.


35

The use of DEMs seems inevitable, despite the objections against the process-based numerical schemes listed in the previous paragraph. This implies that efficient algorithms are required to incorporate collective settling of particles in 3-D and that a system with periodic boundaries should be implemented.


36

3 Simulation of Grain

Rearrangement during Compaction

3.1 Introduction In the previous chapter it has been shown that irrespective of the complexity of a compaction model, it must contain at least the initial porosity as a parameter. Despite this strong dependence on initial conditions, only few workers have studied the controls on initial porosity of a mixture of particles of different sizes. Measurements by Pryor (1973) indicated a large natural variation of porosity values, partly correlated by Beard & Weyl (1973) to variations in grain shape and sediment sorting. Often, generally accepted average initial sandstone porosities of about 45 to 50 percent are used as input to such models. This may suffice for many purposes, but when predictions of porosities and permeabilities of heterogeneous reservoir sands are required, it is less desirable to neglect heterogeneities that have already been present from the start (Lander & Walderhaug, 1999; Milliken, 2001). In this chapter a numerical model is presented, designed to predict the porosity of a population of spheres of arbitrary size distribution. The model realistically rearranges the particles into denser configurations under the influence of gravity. It is able to produce packings ranging from loose disordered packings to dense disordered packings with and without bridging (Tory et al., 1968). Although the model has been designed to treat samples of various sizes, the examples in this chapter will be restricted to mono-sized

Initial Porosity of Random Packings

38

packings. Packings consisting of multi-sized spheres will be dealt with in Chapter 5. 3.2 The rearrangement model The proposed model simulates the movements of a number of individual particles under the influence of gravity and an applied pressure at the top of the assemblage. The randomly positioned population of particles starts with a rather high initial porosity in a state of disequilibrium and the particles are rearranged to approach a state of equilibrium. Unrealistic boundary effects are avoided through the presence of periodic boundaries. The model has been named RAMPAGE, initially as an acronym for Random Packing Generator, but since the model has been developed into an increasingly mature and complex computer program the acronym was changed to ReArrangement Model for Packed Grain Ensembles. In this paragraph the outline of the model is described. First the simplifications and requirements for the model are treated, after which the general workflow of the numerical simulation is presented. Important mechanisms of the model, such as the transition from force to movements, will be explained in some detail. 3.2.1 Particle-pack representation A model is a simplification of reality in order to help us understand the complexity of reality better. Consequently our model also includes several assumptions and simplifications. Two of the most important choices will be motivated below. Particle shape The influence of the shape on porosity and permeability is generally believed to be of secondary importance compared to grain size and sorting (Beard & Weyl, 1973). The shape of a particle can be described by its sphericity and roundness. The first is a measure of the resemblance of a particle to a sphere; the second refers to the sharpness of edges or corners and is not related to the sphericity of a particle (Rittenhouse, 1943; Powers, 1953). In both cases it can be imagined that with deviations from the ideal state (i.e. a sphere or perfect roundness) the porosity increases, since more elongated particles or more angular particles are more likely to bridge pore space (Fraser, 1935). On the other hand, recent studies by Williams & Philipse (2003) and Donev et al. (2004) indicated that particles with a slightly cylindrical shape and

Simulation of Grain Rearrangement during Compaction

39

ellipsoids with size ratios resembling M&M’s candies (oblate ellipsoids) produce closer packings than perfect spheres. Despite the influence of irregular shapes on porosity, we decided to use perfect spheres for our model. The most important consideration for this choice is the simple geometric description that can be used for this shape, namely the radius and the x-, y- and z-position of the centre point. The simple fact that with these two parameters every point on the surface of a particle can be easily determined, avoids unnecessary complications during calculations. Unit cell boundaries Sediment is deposited in laminae with very large width-thickness ratios (e.g. 102-103). Computer simulations of simultaneous movements of individual particles require many computing resources and thus have to be limited to a sample of particles placed in a restricted volume. In order to construct a representative model, it makes, however, little sense to confine the configuration to a box with hard, straight boundaries, introducing wall effects. Therefore it was decided to use periodic boundaries, despite the disadvantage that this will complicate calculations, since a single particle can influence the packing in different areas of the box (Fig. 3.1).

Figure 3.1 Plan view illustration of the unit cell indicating the consequences of periodic boundaries: a sphere can cross resp. a) one; b) two; or c) three boundaries.

The bottom of the cell is chosen to act in a similar manner as a cushion. If a sphere touches the bottom it experiences only a small counteracting force. The farther the sphere protrudes into the bottom layer, the larger the force will be. A top layer placed at the highest coordinate of the pack similarly prevents spheres from popping upwards.


40

Figure 3.2 Flow chart of the RAMPAGE-model.

3.2.2 Numerical simulation scheme Figure 3.2 shows a schematic diagram of the calculation steps, which are followed during execution of the RAMPAGE-program. These steps are described below:

1) The first step is the initialisation of the unit cell, which comprises the construction and placement of a specified number of spheres, either read from file or stochastically positioned (Fig. 3.3). The centres of


41

individual spheres are randomly placed inside the box, i.e. positions of other spheres are not taken into consideration. This implies that spheres may overlap. Overlap between spheres is the source of interparticle forces, which are the driving forces for rearrangement of particles in conjunction with gravitation. Overlap of two neighbouring spheres means that both spheres are in contact. Gravitational effects (dropping and rolling) are ignored until the excessive overlap that may be present at the beginning of the simulation is removed.

Figure 3.3 Initial placement of 999 equally sized spheres in the unit cell. The cell has an initial fill of circa 50 percent. Note the overlapping and floating spheres.

2) After initialisation global properties of the pack are calculated, such as porosity, mean coordination number (= average number of contacts per sphere), fraction of gravitationally stable spheres (further referred to as stability), mean overlap (= average overlap per contact), etc.

3) Based on user-specified criteria, it is decided whether the simulation stops or continues. This can be after a fixed number of iterations, or for example at a desired degree of stability and mean overlap below a certain threshold.


42

4) On continuation, the packing can be either expanded or compressed vertically to simulate respectively shaking or tapping. The average translation of the spheres due to this mechanism should be less than the displacement resulting from interparticular forces and gravity.

5) The most important step is the search for contacting spheres. For every sphere it is determined which spheres are in contact with it. This means that for every pair of spheres the distance between their centres needs to be calculated. If the distance is smaller than the sum of the radii, the spheres are in contact and the volume that overlaps is used to calculate the resulting elastic force. We will elaborate on this mechanism in the next paragraph.

6) After all forces have been calculated the net force acting on every sphere is determined and the associated movement of each sphere is computed. Displacement of the spheres takes place simultaneously, after which the gravitational stability can be determined for each sphere.

7) If a sphere is in a gravitationally unstable configuration it will start to move downward. This can be either through dropping a small increment when no other contacting spheres are positioned below its centre (a “floating” sphere) or by rolling when one or more spheres are positioned below its centre.

3.2.3 The force-displacement law The resultant force exerted on the bodies of the spheres determines the movement of spheres. According to Newton’s Law the force F [M L T-2] will result in acceleration a [L T-2] inversely related to the body’s mass m [M]:

F

am

= (3.1)

Interpreting Eq. (3.1) from a geologic perspective, with time steps that are much larger than the duration that the force is applied, leads to the assumption that the distance of particle movement as a result of the force will be proportional to the acceleration: dx a∝ (3.2) The movement of the particles will be largest when the mass is small, it is therefore important to scale the displacement to the smallest grains present in the packing. By combining equations (3.1) and (3.2) we can scale the maximum movement of the particles and relate it to the movement of other


43

particles, but this requires more information about the maximum contact-force on the smallest particle. In the model contacts are represented as overlaps. If the distance between the centres of two spheres is smaller than the sum of their radii, there is a certain amount of overlap between the spheres. The overlap is assumed to be comparable to elastic compression, similar to a spring, and as a result the spheres contain an amount of elastic potential energy. For springs, the force F as a result of compression over a distance x is F kx= , where k is the stiffness of the spring. Contrary to springs, however, we deal with a compressed volume instead of a distance. For three-dimensional problems we can use Hooke's Law: Eσ ε= (3.3) where σ is the stress [M L-1 T-2], ε the compression [-], and E the elasticity modulus [M L-1 T-2]. When two spheres have an overlap, the intersection of the spheres is a circular plane, or contact plane. Because the stress, resulting from the compression, is the force acting on the area of the intersection, or contact area Ac [L2], we can also write: c cF E Aε= (3.4)

The volume of the overlap Vo [L

3] is the compressed volume, which has the shape of a lens. We simplify this by replacing the lens by a cylinder of the same volume with area Ac and length Vo / Ac. With the sum of the diameters of both spheres as the original length and the length of the overlap cylinder as the compression distance, we can write the compression as:

( )1 2

/2

o cl V Al r r

ε Δ= =

+ (3.5)

where r1 and r2 are the radii of the two spheres. Substitution in Eq. (3.4) gives:

( )1 22

oc

EVF

r r=

+ (3.6)

When we consider a pack consisting of grains of the same material, we can assume the elasticity modulus to be constant. As a consequence, under


44

conditions of constant overlap volume, a smaller sphere will cause a larger contact force. Knowing that the overlap volume can never be larger than the volume of the smallest sphere, the contact force will always be smaller than:

( ) ( )

3 22 2

1 2 1 2

2 23 3 1c

E r E rF

r r r rπ π

< =+ +

(3.7)

where r2 is the radius of the smallest sphere. When we substitute the ratio of radii by R and dimension the smallest sphere to r2 = 1, we get (see also Fig. 3.4):

( ) { }1 21 1 31 where and 1F k R k E Rπ−< + = ≥ (3.8)

Figure 3.4 Maximum contact force versus the diameter ratio.

If we rewrite Eq. (3.1) in terms of r and ρ using 343m V rρ πρ= = and

combine this with Eq. (3.2) we get:

{ }42 33

2

with and F

dx a dx kk r

λ πλρ= = = (3.9)

During the simulation a constant maximum displacement is used, to prevent extremely large displacements for small particles. This maximum


45

displacement determines the value of constant k2 in Eq. (3.9) for every iteration, which is applied to calculate the displacement for each sphere as a function of the resultant force on the sphere. The limitation of the movement of the spheres is also a way to dissipate energy in the system. Without dissipation the assemblage would not be able to converge to a state of lower energy, and particles would continue to vibrate.

3.2.4 Stability and rolling

Figure 3.5 Stability determination for a sphere with centre point M (plan view), on contact points A, B and C with three supporting spheres. All points are projected on the same horizontal plane. The angles are sorted such that αA < αB < αC.

Stability When a sphere is not in a gravitationally stable position, it will either drop or roll a small increment. Stability and rotation are geometrically defined. A sphere is considered stable if it has at least three contacting spheres and the plane through the centre points of those spheres lies below its centre point. Besides this requirement, it is also necessary that the contacting spheres are evenly spaced around the centre of the sphere (Fig. 3.5), in such way that the following equations are valid:


46

C B

B A

C A

α α πα α πα α π

− <

− <

− >

⎧⎪⎨⎪⎩

(3.10)

Since our model does not account for shear forces, jammed particles might still be able to drop or slide. To compensate for this effect, all overlaps (i.e. normal forces) of a sphere should be below a fixed threshold value before it will be allowed to drop or roll. Even though such a particle may be stuck for a number of iterations, it will not be counted as a stable sphere. Rolling Obviously, for a sphere to roll it needs to have at least one contacting sphere below its midpoint or else it will drop. In the situation with only one contacting sphere located below its centre point, the contact point between both spheres will function as point of rotation. Rotation becomes more complicated when the sphere has more than one contact. Depending on the location of the contact-points the rotational movement is directed perpendicular to the axis that connects the two most important contact-points. We suggest those are the lowest contact (C1) and the contact most opposite to the lowest contact (C2, see Fig. 3.6), because the lowest contact is located most central underneath the sphere and thus will obstruct downward movement the most. If the angle α between the centre point (M) and both contacts is smaller than ½ π or larger than 1½ π, then the rotation point will be C1, else the rotation point is located on the axis C1-C2 at point R as shown in Figure 3.6.


47

Figure 3.6 Procedure to find the point of rotation (R) for sphere with midpoint M with C1 representing the lowest contact point and C2 the contact point most opposite to C1. If C2 is located in the shaded area point C1 is taken as rotation point. (The upper part of the figure is a plan view representation.)

Rotation will then take place by a small increment, scaled in accordance with the overall movement per iteration within the pack. 3.3 Optimisation The most time-consuming step of the simulation is to find the contacting spheres for each single sphere. The least efficient way to do this is by calculating the distance between every pair of spheres, and if necessary correcting for the periodic boundaries. For an assemblage of N spheres this

would mean more than 20.5N tests per iteration. A combination of methods has been used to optimise this process. Firstly, all spheres are projected onto the bottom of the unit cell. The cell floor is gridded in a pattern of squares with a side length of about twice the maximum radius.


48

These squares will be referred to as tiles. An example of a tiling is shown in Figure 3.7.

Figure 3.7 Plan view of a tiling example for a fictitious unit cell with size l x l. Tiles

are sized t x t (with max2t r≈ ). Each tile knows the exact position of its surrounding

tiles even with respect to the periodic boundaries, as indicated for the shaded tile.

Each tile holds the memory addresses of the spheres located within its x- and y-coordinates. To find the contacts of a sphere only the distances to spheres mapped on the same tile and spheres mapped on surrounding tiles need to be tested. Vectors to correct for periodic boundaries are stored in the tiles along the border and facilitate the distance calculations. When a sphere crosses a tile boundary during movement, the mappings are updated in the two involved

tiles. This method alone saves an order of tests per iteration (530.5N tests).

If tiles contain addresses of a very large number of spheres, the process may benefit from further optimisation. A method to improve performance very effectively for packings inside a unit cell with a large height or for packings with particles with large diameter differences is similar to the method proposed by Nolan & Kavanagh (1992). This method encompasses storing all the spheres within a specified distance of a sphere’s surface into an array.


49

When a sphere has moved beyond half that distance this array needs to be updated. 3.4 Implementation The program is implemented in C/C++ using the Microsoft Foundation Class Library (MFC) and runs on a normal PC under Windows. This language is chosen predominantly for its flexibility with system resources and use of memory addresses (pointers/references) in the code. C++ allows for definition of objects, implemented as classes, each with their own functions, operators (+, -, *, etc.) and variables. The use of classes also promotes well-organized and structured programming. MFC encapsulates the native Windows API and assists in creating a Windows-based application. Although this might have a negative effect on the overall performance of a program, it adds many possibilities such as a familiar user interface and options to save, load or pause ongoing simulations. The five main classes, which were implemented for the RAMPAGE program, are summarized in Figure 3.8. Other implemented classes are a) the standard Document/View classes that handle the Windows-view and menu commands, b) the Dialogue boxes, c) supporting classes for vector- and matrix arithmetic and d) thread-classes that capture and handle windows messages (D. Filipp, 1999). As soon as a simulation is started, objects of the classes shown in Figure 3.8 are constructed. The main iterative loop is defined in the class that also contains the threading procedure that captures and handles Windows messages, which prevents a system lock during simulation. It further contains the array of spheres and the tiling grid. The tiling encapsulates the tiles, which store pointers to the mapped spheres. The spheres in return store the pointer of the associated tile, keep track of a mapping of pointers of contacting spheres and contain the forces resulting from their contacts. At any time during simulation output files can be written in two different formats. A VRML-file contains the location and radius of each sphere and can be either viewed with a standard VRML viewer, or post-processed with specifically designed software. A log of packing properties can be saved as a comma-separated value file (CSV), which can be further processed in any desired plotting software.


50

Figure 3.8 Implementation structure of RAMPAGE: The five main classes and their properties.

Furthermore, possibilities are implemented to run a large number of simulations in batch mode, although modifications in the code might be necessary for advanced batch simulations.


51

On a system with a 2.7 GHz Pentium-4 processor the model calculates typically over 40000 iterations per hour for a lattice of approximately 1000 spheres (see Fig. 3.2 for the work flow of a single iteration). 3.5 Simulation of grain rearrangement General model behaviour The behaviour of the model is shown in Figures 3.9, 3.10 and 3.11 for a population of 1000 spheres of equal size. During this simulation the spheres will only relax under the influence of gravity and a minimal downward force exerted by the top layer. No dilation is applied. The initial lattice is characterized by a predominantly unstable configuration (Fig. 3.9-A), large, but scattered forces (Fig. 3.10-A) and high porosities, few contacts per sphere and a large mean overlap (Fig. 3.11). After the start of the simulation five different phases following the initialisation can be recognized, and are indicated in Figure 3.11 with dashed lines. 0) Initialisation: random positioning of 1000 spheres in the unit cell (~50% porosity). 1) The first phase of the simulation may be regarded as a surrealistic state resulting from the initialisation. During this phase the extremely large overlaps are reduced. The effects of gravity are not yet considered, because particularly in multi-sized sphere mixtures smaller particles will erase their overlaps faster due to the restricted maximum displacement, thus being able to fall down much earlier than large particles. This would inevitably lead to a sort of undesired segregation (inverse grading), which would have no physical meaning. Generally, this first phase passes rather quickly, taking only a few hundred iterations (Figs. 3.9-B, 3.10-B, 3.11-1). During this period porosity and mean coordination number remain the same, while overlap decreases linearly. 2) Dropping of unstable spheres dominates the second phase. At first this will have no influence on porosity and will only slightly increase the mean coordination number. The decrease of overlap accelerates with unstable spheres dropping out of clusters. Densification of the packing will start at the bottom of the pack, creating new space to drop and roll into. The average porosity of the pack will deteriorate quickly after the densification has reached the top of the packing, and the number of contacts per sphere will increase significantly.


52

Figure 3.9 A series of 3D visualizations of the positions of the spheres during a simulation run. The packing contains 1000 spheres and started with an initial fill of about 50 percent (A). Figures B-H illustrate relaxation and compaction of the packing due to grain rearrangement, with porosities after resp. 250, 1000, 2000, 10 000, 100 000, 1 000 000 and 2 000 000 iterations of 51.9, 51.7, 48.1, 46.4, 44.2, 40.8 and 38.9 percent respectively.


53

Figure 3.10 3-D Diagrams of the interparticular forces during a simulation, clearly showing the relaxation of the packing. Forces are shown as a tube between the centre-points of the involved spheres, larger magnitudes are represented as a darker shade of grey. Only forces with magnitude above a fixed threshold value are shown. Sphere centres are represented as tiny dots and are in exactly the same configuration as in Fig. 3.9-A to 3.9-H.


54

Figure 3.11 Plots of the evolution of properties during simulation: a) Porosity, b) Mean coordination number and c) Mean overlap (as fraction of the radius). Dotted lines divide five distinctive phases of simulation (see text).


55

A gravitationally stable packing will come into existence after a few thousand iterations and the force network will start to be streamlined into continuous diagonals (Figs. 3.9-C, 3.9-D, 3.10-C, 3.10-D, 3.11-2). 3) The streamlined force paths facilitate the dispersion of the forces. Hence, the third phase is characterized by a further decrease of overlaps and relaxation of the force network (Fig. 3.10-E). The porosity and mean coordination number of the packing remain rather constant during this phase (Fig. 3.11-3). The porosity of 46 percent reached at the end of this phase represents the “undisturbed” depositional porosity of this assemblage (i.e., at surface conditions). 4) At the onset of the fourth phase a sudden increase of mean overlap is noticed (Fig. 3.11-4), which is believed to be the effect of a growing impact of the top boundary that exerts a small downward force on the packing. Initially this was added to the model to prevent the formation of piles of spheres on top of the packing, and to destroy existing irregularities at the top surface that result from the random nature of the starting condition. However, during ongoing simulation it also stimulates further rearrangement, resembling the effect of the overburden at the stage of early burial. The observation that the increase of the mean overlap coincides with an instant drop of porosity of approximately one percent and an increase of the average number of contacts per sphere (Fig. 3.11) indicates that at this moment a larger number of spheres experiences the downward pressure, which results in a simultaneous downward displacement of the spheres. 5) During the last phase of the simulation the influence of the overburden continues to be felt. Intervals of increased mean overlap are followed by a slight increase of the mean coordination number and further particle rearrangement that results in decreasing porosities (Fig. 3.11-5). All major forces have been dispersed throughout the lattice, which is illustrated as the near-equilibrium systems in Figure 3.10-F and 3.10-G. Addition of external energy to the system from the overburden will dissipate rapidly as long as the spheres are able to move into tighter packing and porosity continues to decrease. The few forces that remain visible at the bottom have a very short lifetime and result from the rebound effect of the bottom boundary. Towards the end of the simulation individual interparticle forces tend to increase again (Fig. 3.10-H), but net forces become smaller and further decrease of porosity slows down. Although it cannot readily be inferred from Figure 3.11-A, one can expect a halt of porosity loss with extended simulation, when the force exerted by the overburden equals the reaction force from the packing.


56

An extended simulation of nearly 10 million iterations (250 hours) is shown in Figure 3.12-A. This illustration demonstrates the asymptotic deviation from the logarithmic porosity decrease at a porosity of approximately 39 percent after roughly one million iterations. Because of the particularly lengthy calculation times required to visualize this effect, only a few of such extensive runs have been carried out, all showing the same asymptotic effect. Nearly all other simulations are stopped after about one or two million iterations, while prolongation of a simulation hardly gives any additional data or insights, besides the confirmation that equilibrium has been achieved. The porosity of 39 percent is in good agreement with the minimum porosities that Pryor (1973) measured for well-sorted beach sands, and also a little below the range of porosities that Beard & Weyl (1973) obtained from their experiments with very-well and extremely-well sorted sands that were artificially mixed and packed in distilled water (i.e., Beard & Weyl recorded porosity values between 39.8 percent and 41.8 percent for sorting coefficients between 1.1 and 1.2). Palmer & Barton (1987) collected samples of uncemented, well-sorted quartzose sands, which experienced only a relatively small depth of burial. They recorded the porosities, estimated the ages of the samples, and recompacted the disaggregated grains. The youngest sands, which have supposedly been buried less than 100 metres, have porosities ranging from 39 to 48 percent. Also, the range of porosities for the remoulded sands is within one percentile of 39 percent. All these examples suggest that the RAMPAGE model produces packings that closely resemble recently deposited well-sorted sands and shallow uncemented, well-sorted sandstones with limited compaction resulting from early grain rearrangement.

Figure 3.12 Three examples of different modes of simulation: (a) merely relaxation without stimulation by means of shaking or tapping; (b) stimulated by tapping at the top at states of low mean overlap (< 0.01); and (c) stimulated by tapping at the top at states of moderate mean overlap (< 0.1).


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Stimulated simulation through dilation of the pack The model of Nolan & Kavanagh (1992) was remarkable in its ability to produce a broad range of packing configurations. The loosest lattices (random loose packing) were characterized by low mean coordination numbers and high porosities, another extreme packing configuration was characterized by low mean coordination numbers and low porosities (random bridged dense packing) and their densest lattices (random dense packing) had high mean coordination numbers and low porosities. They reached these different states not through the simulation of an entire trajectory starting from a loose state, but instead used an initial pack with an initial fill near the desired packing state. During construction of a very dense packing state, the large initial overlaps provide enough energy required to reach a bridge-free gravitationally stable configuration. Packings simulated with the RAMPAGE model as shown in Figs. 3.9, 3.11 and 3.12-A have a mean coordination number of approximately five contacts per sphere. Compared with experimental packings like those from Bernal & Mason (1960) and Finney (1970), which have coordination numbers of about seven contacts per sphere, this is rather low. Experimental packings, even those with natural sands (Beard & Weyl, 1973), are usually shaken, vibrated, tapped or otherwise stimulated to obtain the densest possible packing within a reasonably short time. In order to make simulated RAMPAGE packings comparable to those experimental packings, a dilation mechanism is used to supply energy to the packing, and which can be either enabled or disabled. This mechanism is a very simple multiplier (c) that is applied to the heights (z) of all particles in the pack at a specified minimum relaxation interval (n), so we get: 1i iz cz+ = ,

after which n iterations are run before the next dilation is allowed to take place. Secondary criteria can also be applied to determine when such dilations may take place, such as a minimum stability of the pack that needs to be fulfilled and/or a maximum allowed mean overlap in the simulated packings. To construct packings with the high densities such as the vibrated experimental packings, it is most effective to use a multiplier c < 1, and typically in the vicinity of unity to simulate a steady relaxation. The effectiveness of the multiplier can be understood by the type of supplied energy that is associated with the multiplier. If the multiplier is larger than one, spheres will rise in the packing and get a higher potential energy. If the multiplier is smaller than one, the spheres will be slightly pressed inside each other and the elastic energy within the packing is increased. Since potential energy is damped most with the chosen implementation, whereas elastic energy remains stored in the overlap volumes, the multiplier will be chosen around c = 0.9999.


58

Figures 3.12-B and 3.12-C show two examples of stimulated simulations with a dilation factor of c = 0.9999 with different rules to determine when dilation takes place. The first example illustrates the porosity decrease for a pack that is “tapped” when at least 99% of the spheres are gravitationally stable and the mean overlap is smaller than 1% of the radii. The simulation approaches equilibrium at a porosity of 34% and a mean coordination number of roughly 7. These values are in good agreement with porosities and mean coordination numbers obtained from the few available experimental packs (i.e. Bernal & Mason, 1960; Finney, 1970). It also corresponds very well with the minimum porosities of well-sorted sands measured by Palmer & Barton (1987), which are believed to have been buried no more than 780 metres, and rapidly reached a porosity floor of 34 percent at 300 metres depth. They contended, however, that the reduction of porosity in their samples could only be partially explained by mechanical compaction and assumed pressure solution to be the dominant process of porosity decrease, since recompaction of the disaggregated sands led to minimum values of only 39%. However, their discussion does not incorporate the influence of geologic time scales and entirely rules out the possibility for interplay of processes such as pressure solution and grain rearrangement due to both an increase of overburden pressure and seismic disturbances at geological time scales. Because a slight compression was applied to the packing during this example’s simulation, the small (elastic) overlaps are preserved throughout the simulation. This retains the energy in the packing that can initiate renewed movements as soon as a little space is available. The slightly closer configuration of spheres that results from the tiny overlaps may well be compared to the effects of pressure solution, and consequently can yield some additional opportunities for rearrangement to occur. Hence, the porosities that are obtained with this type of simulations are slightly below the values of around 36.4 percent that are generally measured for random dense packings (Finney, 1970). This example demonstrates nevertheless that porosities of less than 39 percent can be reached with grain rearrangement as the sole or at least dominant process of porosity reduction. The second example shows the porosity decrease when the packing is tapped at an average stability of 90% of the spheres and a mean overlap of less than 10% of the radii. One might consider this to represent a packing of slightly elastically deformed particles. After a prolonged simulation it approaches an asymptotic line at 29% porosity and a mean coordination number of 8. Although there is no published evidence on experimental packs, the additional porosity decrease due to deformation is in good agreement with observations of porosity decrease in ductile grain packings (Pitman & Larese, 1991; Wilson & Stanton, 1994).


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3.6 Discussion The numerical model described in this chapter is capable of simulating an entire rearrangement trajectory from an unconsolidated and disordered lattice of spheres into a densely packed assemblage, both with and without bridging of particles, within a reasonable computing time. In contrast to the model of Nolan & Kavanagh (1992, 1993), our model does not require an initialisation with a high fill to reach a state of dense packing. The periodic boundaries also create a pack that is more representative of layered granular media than boxed packings with fixed boundaries. Furthermore, the model differs considerably from the physically based DEMs derived from the Cundall & Strack model (1979), through the use of a simplified approach of time-dependent processes. The introduction of a maximum displacement per iteration justifies comparisons with geologically produced packings. The discretisation of movement provides a damping mechanism to prevent eternal vibration of the particles (energy dissipation). The comparison of simulated porosity trajectories and porosities of RAMPAGE packings with laboratory experiments and measurements of well-sorted natural sands show a remarkable agreement. It suggests that the model can simulate a broad range of gravitationally stable packings with porosities similar to well-sorted, natural sands found near the surface (cf. Pryor (1973) and Palmer & Barton (1987)). Estimates of age and estimated maximum depths of burial were also provided for the samples of the latter workers. Wilson & Stanton (1994) plotted the porosity data of Palmer & Barton versus estimated depth of burial and found a logarithmic relation that clearly indicated that occurrence of porosities larger than 39 percent was limited to depths shallower than about 20 metres. This implies that the bulk of porosity reduction resulting from grain rearrangement takes place near the surface. It is therefore questionable whether averaged porosity data gathered near the surface are representative for initial porosities in (analytical) compaction models. The RAMPAGE model distinguishes itself from other numerical models in its ability to mimic transitions between gravitationally stable packings as a result of grain rearrangement. The different properties of the grain population during the evolution of the packing can be assessed throughout the simulation and separate phases can be recognized. The first gravitationally stable packs of mono-sized spheres appear rather rapidly after the start of the simulation at porosities of about 45%. In well-sorted natural sands, such porosities are in many cases amongst the highest reported values, unless a large number of oversized pores are present (Atkins & McBride, 1992).


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Starting from these first stable packs, a spectrum of stable packings can be generated and analysed until the densest packings with porosities of approximately 39% are reached. Although Palmer & Barton (1987) postulate that these are the lowest porosities that can result from grain rearrangement, and that further compaction is induced by other processes (in particular pressure solution), we believe that grain rearrangement remains an important process on geologic time scales. It is responsible for porosity decrease during the early stages of burial, and can still contribute to compaction in conjunction with complementary processes that trigger small-scale movement of particles. The contribution of grain rearrangement may, however, diminish slowly, and will become more and more dependant on the degree of lithification of the sands. This hypothesis is supported by the model’s ability to rearrange into packings with porosities down to 34 percent, and by experimental packings such as those from Finney (1970), which record a porosity of 36%. In the above discussion, all comparisons were based on averaged porosity data for mono-sized sphere populations and well-sorted sands. The fact that poor sorting causes lower porosities has been well established (Beard & Weyl, 1973). Yet only few quantitative analyses or models have been published on this relation. Our model opens up opportunities to analyse this relation for all possible mixtures of grains of different sizes. However, before we can apply our model to such compositions, it seems appropriate to calibrate our simulated packings to experimental data, based on structural characteristics of the packings.

4 Analysis of Mono Sized Particle Systems

4.1 Introduction A model requires thorough validation before it can be applied. Only when the model holds for certain conditions, it can create new insights about the simulated processes, by giving information that could not be obtained by experiments. To check the validity of the model the results are compared to field data and experimental data. Compaction is difficult to observe in situ, since the diagenetic processes are imperceptibly slow and most of the processes only take place at great depth under extreme conditions. We can nevertheless use the result under the assumption that compaction is an irreversible process. However, any attempt to analyse a sample in situ requires disturbance of the sample’s state of stress. As a result, one can only compare two different samples at different times or locations and assume that the differences in compositions of the samples are irrelevant. As an alternative, we can study the separate processes of compaction using experimental set-ups in laboratories. This way one can control a large number of variables to investigate the influence of the property of interest. Many workers in diverse research fields have studied packings of spheres and their properties. It would be nearly impossible to give a complete summary of the work that has been carried out, but relevant standards on experimental random packings and the analysis of those packs were set by Scott (1960;


62

1962), Bernal & Mason (1960), Scott & Kilgour (1969) and Finney (1970). A thorough overview on the topic of the packing of spheres was given by Cumberland and Crawford (1987). In this chapter, packings generated with the RAMPAGE-model are analysed and compared with similar analyses of experimental packings that were made available by workers of the University of Oslo, analyses of a random close packing made by Finney (1970) and analyses of regular packings (crystal lattices). Before going into detail, the regular packings, the available experimental packings and the simulated packings will be described, as well as the differences between them, as those differences will undoubtedly lead to variations in the results. Also, assumptions and simplifications in the model, which might be expected to cause deviations from the experimental packings, are mentioned. Several statistical methods have been applied to analyse the different packings and the effects of the boundaries are evaluated. Finally the influence of gravity on the simulated packings will be analysed and assessed, by the same methods that were used to compare simulated packing with experimental packings. 4.2 Available packings The packings, which were available for analysis, are subdivided into four groups. The group of regular packings form the basic type of packing of particles and will be discussed first. The second group consists of the Finney-pack and other experimental packings that have been discussed in literature. We will refer to these packings by using the name of the worker, and use the term Steel-ball packing as a reference to this entire group. The third group of packings are the experimental packs measured at the University of Oslo by Uri et al. (2004a; 2004b). Since the material used for the particles of the pack is PLAY-DOH, these packings will be referred to as Play-Doh packings. The last group of packings are the results of the simulations with RAMPAGE and will further be referred to as RAMPAGE packings. 4.2.1 Regular packings Regular packing arrangements exist of well-ordered rows and layers in crystallographic patterns. Using layers with rows ordered at either 60° or 90°, five basic packing types can be constructed, namely cubic, orthorhombic,

Analysis of Mono Sized Particles Systems

63

tetragonal-sphenoidal and rhombohedral; the last can be subdivided into pyramidal and hexagonal arrangements (e.g. Graton & Fraser, 1935; Cumberland & Crawford, 1987). The packings are illustrated in Figure 4.1. For this analysis a model consisting of 1000 spheres (10 layers of 10 rows of 10 spheres) was constructed for each regular packing. Because several properties can be easily calculated for regular packings, this will give us the opportunity to derive the effect of the boundaries and -where possible- correct for them.

Figure 4.1 Five basic regular packing arrangements, characterised by the smallest possible number of spheres to represent the packing. The rectangular and rhombic shapes indicate the unit cell for each packing.


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4.2.2 Steel-ball packings: Finney pack and packings of others The Finney pack is a random packing consisting of 7994 accurately measured solid steel balls (Finney, 1970; Bernal et al, 1970) with a diameter of ¼ inch. It is probably one of the largest and most accurate experimentally constructed random close packings without notable boundary effects. The coordinates of 7935 spheres in this packing were made available for the purpose of this study (Finney, pers. comm.) Of other studies of interest (Scott, 1960, 1962; Bernal & Mason, 1960), which used similar packings consisting of between 1000 and 20 000 balls, only the published properties were used for comparison. All these packings are constructed in a more or less similar manner. The steel balls were poured into a container and shaken or tapped quickly to reduce the volume, until no further reduction in volume was observed. Scott (1960) also produced “loose” packings by tipping over the container horizontally and then slowly returning the container to its vertical position 4.2.3 Play-Doh packings The packings constructed at the University of Oslo were made with the purpose to study the compaction of ductile grains (Uri et al, 2004a; 2004b). Spherical grains with a diameter of 8.8mm (±0.2mm) were made (by hand) using PLAY-DOH (by Hasbro International Inc., available in toy shops). PLAY-DOH is a patented modelling compound largely similar to modelling clay, but for which the material properties are not made available by the distributor. It is however very ductile and thus easily plastically deformed. The particles were poured into five small cylinders (3.2 cm diameter, ~150 spheres) and one large cylinder (~12.5-13 cm, ~1000 spheres). The positions of the grains were recorded with a position measurement apparatus (MICROSCRIBE) to a resolution of ca. 0.5 mm. The measurements were done for each cylinder at a different stage of compaction. Unlike the steel-ball packings, the purpose of this research was to record the changes in the packings due to compaction, and as a consequence these ensembles could not be corrected for boundary effects. 4.2.4 Rampage packings Disordered mono-sized grain populations can exist in several different packing states. The extreme packing states have been defined as bridged loose packing, bridged dense packing and bridge-free dense packing by Nolan & Kavanagh (1992). These states differ predominantly in the degree of compaction and the relative presence of bridges. A bridge can be formed,


65

when two or more spheres attempt to drop into a single gap simultaneously, and become poised against each other (Fig. 4.2). The presence of bridges causes a significant decrease of the average number of contacts per sphere.

Figure 4.2 Illustration of bridging of spheres in 2-D. The two shaded spheres formed a bridge while moving into the same gap.

Simulated packings of 500 to 2000 spheres can quickly be constructed using the RAMPAGE model. Some larger packings (up to 5000 spheres) are generated only for a rather loose state, because of the long simulation time needed to construct them. The model is dimensionless and the radius of spheres in mono-sized sphere packs is set equal to 1 unit. The methodology for constructing these packings is described in Chapter 3, and was summarized by Alberts & Weltje (2001). In short, the model works as follows: The spheres (defined by centre coordinates and radius) are randomly positioned inside a box up to a 50% fill, irrespective of possible overlap with other spheres. Overlapping spheres are pushed apart to reduce the overlap distance and as soon as the maximum overlap is below a specified threshold the spheres can drop and roll as a result of gravity. A combination of gravity and elastic interparticle contact forces are responsible for further rearrangement until gravitational stability under minimal overlap conditions has been reached. A small vertical dilation or compression can be imposed upon the packing to simulate shaking or tapping, which can either create or destroy bridges. Horizontal boundary effects are avoided through the use of periodic boundary conditions in the horizontal directions. Top and bottom boundaries act as a soft cushion, and the effects of those boundaries can be reduced by increasing the height of the unit box. For this analysis mostly packings with a size of 1000 spheres were used.


66

4.2.5 Other factors affecting packing variations A comparison between the groups of packings above will only be useful if we remain aware of the differences between the packs. The construction of the packs will in itself be responsible for differences (Macrae & Gray, 1961). While the steel-ball packings were actively treated to create an ideal close packing and the Play-Doh packings were just poured into a cylinder to observe plastic deformation, the standard RAMPAGE packings are created in a fashion simulating loose deposition followed by ongoing rearrangement, which can eventually approach a close packing. Addition of a tapping mechanism is needed to simulate the ideal close packings, such as the steel-ball packings. Other deviations will undoubtedly arise from the differences between the boundary effects of each packing group and the number of particles used for each packing. This will especially influence the porosity values of the Play-Doh packings for instance, which have not been corrected for boundary effects. However, the expected loose packed structure and ductility of the grains make these packings an interesting reference for other analyses, which cannot be supplied by the other experimental packs. 4.3 Statistical analysis of packings 4.3.1 Porosity The ratios between the volume of the voids or pore volume (Vp), the volume of the solid material (VS) and bulk volume (Vb = Vp + Vs) can be specified in several ways. Porosity (φ) relates the pore volume to the bulk volume (Vp / Vb) and is thus mostly used when interests lie in porous media, whereas packing density (ρp) relates the solids volume to the bulk volume (Vs / Vb) and is commonly used in physical research of granular matter. From the definitions above, it directly follows that porosity relates to packing density as φ = 1 - ρp. Porosity or packing density is traditionally a very important first indicator for the state of a granular medium (Scott, 1960; Cumberland & Crawford, 1987; Bezrukov et al, 2001; Lundegard, 1992). Even though there are many other variables that influence the porosity, such as sorting and shape (Beard & Weyl, 1973), it hints at the structure of a packing and the degree of compaction it has undergone. For a random close packing of equal spheres the limiting density has empirically been established at 0.6366 ± 0.0008 (Scott & Kilgour, 1969) corresponding to 36.34% porosity, and for random loose packings the limiting density was determined to be about 0.60 (Scott,


67

1960) or 40% porosity. Pryor (1973) took samples of recent sands at various dune and beach locations, and measured porosities ranging from 39% to 56% with averages well over 45%. Porosities of artificial packings of well-sorted natural sands by Beard & Weyl (1973) averaged 39% for well-sorted sands to 42% for extremely well sorted sands. They commented that these data were probably representative of minimum porosities for unconsolidated sands in most depositional environments following mechanical rearrangement under near-surface conditions. Table 4.1 shows the porosities for the different packings. In the first column the uncorrected measurements are listed. For the regular packing and the Play-Doh packings those values are based on calculations of the volumes. The porosity of the large Play-Doh packing could not be calculated, as not all positions of the grains in the cylinder were recorded, and the measured grains did not form an entirely cylindrical shape. The small cylinder packings show an expected decrease of porosity in accordance with the duration of compaction due to plastic deformation before measurement. The nature of the Play-Doh packings seems to have changed slowly from a loose packing to a denser packing. However, because grain deformation during compaction plays a more important role in the decrease of porosity than grain rearrangement, the topology of the packing of these denser packings can be expected to remain more or less similar to the looser packings. Unfortunately, one can hardly draw reliable conclusions from these figures, as the population of spheres is very small and the values will show significant overestimation due to the dominance of boundary effects. The RAMPAGE packings with a mean overlap of less than 0.5% of the diameter size and gravitational stability cover a range of packings of between 500-5000 spheres. For each number of spheres, ten different simulations have been run and average porosities ranging from 44.1% to 44.9% were obtained (see also Table 4.3), with individual porosities ranging from 43.3% to 45.9%. Although the spread in values is not as large as that in the measurements of Pryor (1973), the obtained variability and the agreement in average values give sufficient reason to believe that the type of packing for these models are comparable to the packing type of recently deposited sands. The bridged dense RAMPAGE packing is obtained after a prolonged run of a 1000-sphere packing. Small perturbations disturb the equilibrium, which causes the particles to rearrange to a denser packing. Because of the long simulation time for these runs, only two of these packings were created. The porosity of 38.5% is comparable to the 39% porosity of well-sorted sands after rearrangement according to the study of Beard & Weyl (1973), but is significantly lower than the porosity of extremely well sorted sands (42%).


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Table 4.1 Porosities of mono-sized regular, steel-ball, Play-Doh, and RAMPAGE packings. The measured/calculated porosities are the rough measurements or calculations without corrections for boundary effects. A post-processing program has been used for porosities in the last column. This software can calculate the porosity for any region within the packing, the tabulated values represent the porosity in the region within one diameter from the boundaries for the steel-ball packings and the RAMPAGE packings and the largest box that fits in the cylinders for the Play-Doh packings, within 90% confidence limits.

Porosity (%)

Packing measured/ calculated

post-processing software

Regular packings: Cubic 47.64 47.40 ± 2.56 Orthorhombic 39.54 39.70 ± 1.24 Tetragonal-Sphenoidal 30.19 30.48 ± 0.89 Rhombohedral-Pyramidal 25.95 25.94 ± 0.27 Rhombohedral-Hexagonal 25.95 26.19 ± 0.60 Steel-ball packings: Finney (1970) 36.34 36.383 ± 0.03 Scott (1960) dense 39-411) n/a Scott (1960) loose 42-461) n/a Play-Doh packings: Large Cylinder n/a 29.90 ± 0.67 Small Cylinder 1 (0h) 45.9 36.29 ± 2.01 Small Cylinder 2 (40h) 43.3 36.25 ± 1.49 Small Cylinder 3 (75h) 43.7 34.16 ± 1.30 Small Cylinder 4 (145h) 41.8 31.76 ± 2.39 Small Cylinder 5 (305h) 41.4 34.70 ± 0.97 Rampage packings: Bridged Loose (Stable @ < 0.5% mean overlap)

44.6 ± 0.142) 44.57 ± 0.09

Bridged Dense 38.5 38.13 ± 0.21 Dense 34.3 33.94 ± 0.18

1) Range of values determined for packings of different numbers of spheres.

2) Averaged over 70 simulations.

Similarly, the dense RAMPAGE packing, which is obtained after a run with simulation of vertical tapping, has a porosity of 34.3 percent against 36.3 percent for dense packings obtained from experiments by Scott (1960) and


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Finney (1970). This suggests that RAMPAGE is able to create slightly denser packings than what can be created with experimental set-ups, which can nearly entirely be explained by the overlaps that are allowed in the packings. The last column of Table 4.1 lists the porosity values obtained from calculations using post-processing software. This software determines the bounding box of the packing, based on the locations of the spheres, and calculates the porosity for a smaller box with the walls at a specified distance (rim) from the bounding box. The rim needs a default value of one sphere diameter to correct for boundary effects. However, for a small number of spheres in a packing the porosity values are very sensitive to the size of the

box. Therefore the average porosity (φ ) is calculated over a number of

porosity calculations (n) covering a range of rim values between one diameter and two diameters. The average porosity obeys the following equation:

( ) 0

0

1

1( )

n

inr i rφ φ

=+= + Δ∑ (4.1)

with:

D

rn

Δ =

where D is the sphere diameter size and ( )rφ the calculated porosity as a

function of rim r. The accuracy of this procedure is demonstrated by porosities for the regular packings (Table 4.1). All porosities are accurately estimated, taking the small amount of 1000 spheres into consideration. For the Finney pack, consisting of eight times as many spheres, a nearly identical porosity has been estimated. The RAMPAGE packings were also analysed using the post-processing program. Probably due to the reason explained above (sensitivity to small population sizes), the observed variation in values was slightly higher than the variations in porosities obtained from the RAMPAGE model itself. Application of post-processing to the Play-Doh packings gives thus a much better indication for the true porosity values. As one would expect, the variability is rather high, especially for the small cylinders, as a result of the small number of grains, but nevertheless the small porosities show that grain deformation has a very large effect on porosity decrease in these highly ductile grain assemblages.


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4.3.2 Radial distribution function A standard tool to study the structure of a packing is the radial distribution function (Scott, 1962), which is often also referred to as the pair correlation function. It shows a histogram of the distribution of distances between sphere centres (in all directions), normalised for the volume of scope. The radial distribution function g(r) is defined as follows:

2

( , )( )

4N r r dr

g rr drπρ+

= (4.2)

with: r radial distance in diameters dr increment size N(r,r + dr) average number of centres between radial distance r

and r + dr ρ average number density Figure 4.3 shows the radial distribution for the Finney packing and the RAMPAGE packings. The radial distribution function for the Play-Doh packings is given in Figure 4.4. A comparison between the radial distribution function for the Finney packing and the regular packings is shown in Figure 4.5. The most prominent feature in these plots is the large peak at the radial distance of one sphere diameter, which obviously represents the contacts between neighbouring spheres. It is, however, notable that the first peaks for the Play-Doh packings (Fig. 4.4) span a much broader range of distances than the packings of Figure 4.3. This is plainly related to the deformation of the ductile Play-Doh grains and unfortunately this character will also be reflected in later peaks, hiding any further structural information contained in those peaks. Especially the second peak is of interest, while the occurrence of a split peak explains much about the geometrical arrangement with the second-nearest neighbour, which was well explained by Finney (1970). Closer examination of the regular packings (Fig. 4.5) indicates that the location of the highs of the second peaks suggests a degree of regularity.


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Figure 4.3 The radial distribution function g(r) versus the radial distance expressed in units of sphere diameter for A) the Finney pack and B) the different RAMPAGE packings after resp. 10 000 (loose), 1 000 000 (bridged dense) and 750 000 iterations (dense). The average number of spheres was measured in intervals of 0.025 of the sphere diameter. The RAMPAGE packings were corrected for the boundary effects caused by the top and the bottom.


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Figure 4.4 The radial distribution function for the Play-Doh packings, with A) the large cylinder and B) the small cylinders measured after 0, 40, 75, 145 and 305 hours respectively. Again the average number of spheres is measured in intervals of 0.025 of the sphere diameter.


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Figure 4.5 One-to-one comparisons between the radial distribution functions of the regular packings and the Finney packing.

The second high of the split peak at a distance of two diameters is caused by the trivial linear arrangement of spheres (‘three in a row’) and also explains the steep drop after the radial distance of two sphere diameters. The first high of the split peak has a larger deviation, which is mainly caused by two regular arrangements; the first case with a triangular layer results in a second nearest neighbour at the distance of 1.732, while packing of triangular layers results in a second nearest neighbour at a distance of 1.633 (Fig. 4.6). It is not surprising that these two peaks are the most dominantly represented distances in the hexagonal packing. With this consideration in mind, it is interesting to observe the growth of these peaks for the RAMPAGE packings in Figure 4.3-B. The loose packing appears to have no clear split peak, but continuation of rearrangement causes the development of a split peak. The first high of the split peak shows the highest sensitivity to ordering; when no tapping is applied to compact the packing this peak will show up, but not as distinct as when the packing has been tapped. A degree of order exists in both cases, given the presence of the third and even a slight fourth peak, but existence of bridges in the untapped packing is apparently responsible for a less well defined first high of the split peak.


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Figure 4.6 Distances to the second nearest neighbour in number of diameters.

The radial distribution for the small cylinders of Play-Doh grains in Figure 4.4-B needs to be considered with care. The function drops almost immediately to values below 1, which is caused by the walls of the cylinder. However, the more or less similar signature of these plots indicates that during the time span of 305 hours the structural arrangement of the spheres remains the same, suggesting that grain rearrangement is not an important factor during compaction of these ensembles. 4.3.2 Contacts: coordination and contact angles Another important characteristic of packings is the frequency distribution of the number of contacts between the particles. The average number of contacts per sphere (or mean coordination number) gives an impression about the connected network of the packing, the closeness of the packing (Bernal & Mason, 1960) and the occurrence of bridging structures within the pack (Nolan & Kavanagh, 1992). The distribution of coordination numbers also contains information on the occurrences of large holes or very close packed clusters (Bernal & Mason, 1960) and thus the degree of disorder in the pack. Examination of the distribution of contact angles may provide information on, for example, the topology of the packing and the direction of forces in the network. To find the contacts for the Finney-pack the region of 10% of the sphere radius around each sphere is searched for the presence of neighbouring spheres. This may lead to a small overestimation of the coordination number (see also Figure 4.3-A). However, the spheres near the boundaries will have a lower number of observed contacts. The contacts of the RAMPAGE packings are, by definition, the spheres for which the distance between both centres is


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smaller than the sum of the radii. The contacts for the Play-Doh packings were laboriously counted and recorded (Uri, pers.comm.). An impression of the mean coordination number is presented in Table 4.2; a breakdown of the number of contacts per sphere into a histogram is shown Figure 4.7.

Table 4.2 The mean coordination number, or the average number of contacts per particle, calculated for each pack. The first column presents the calculations for all spheres with 3 or more contacts; the second column is calculated for all spheres with 4 or more contacts. The real values are those by definition for the regular packing and were available for the RAMPAGE packings directly from the simulation program. The corrected values are derived from the trend of the differences between the real and calculated values from the second column.

Mean coordination number Packing

3 and more 4 and more corrected real Cubic 5.4 5.57 5.72 6 Orthorhombic 7.02 7.04 7.79 8 Tetragonal-sphenoidal 8.64 8.66 10.04 10 Rhombohedral-Pyramidal 10.1 10.11 11.91 12 Rhombohedral-Hexagonal 10.26 10.27 12.11 12 Finney pack 6.36 6.52 7.05 Play-Doh large 7.04 7.06 7.81 Play-Doh small 1 5.28 5.56 5.71 Play-Doh small 2 5.26 5.43 5.53 Play-Doh small 3 5.42 5.72 5.93 Play-Doh small 4 5.54 5.71 5.91 Rampage loose 4.8 5.16 5.16 5.12 Rampage bridged dense 4.9 5.29 5.34 5.16 Rampage dense (tapped) 6.31 6.52 7.05 6.91

In this analysis, the number of contacts per sphere for the experimental packings is affected by the presence of boundaries. As a result the values will be influenced by an overrepresentation of spheres with apparently few contacts. To give a reasonable approximation of the mean coordination number, the spheres with two or fewer contacts are left out as a fast correction for boundary effects, since spheres cannot be in a stable configuration with


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only one or two contacting spheres, implying that these spheres must be located near the boundaries. Theoretically a sphere can have only three contacts and still exist in a stable position, because it is possible that a sphere is overarched by bridging spheres. Therefore, a column is presented showing the mean coordination number calculated for spheres with three or more contacts as well as a column with four or more contacts. The mean coordination numbers for the untapped RAMPAGE packings are automatically corrected for wall effects during the calculations and therefore considered the actual (real) mean coordination numbers. Comparison between the mean coordination numbers for these RAMPAGE packings and the post-processed calculations show that there may be only a few occurrences of this hypothetical effect of overarching.

Figure 4.7 Contact histograms showing the fractional distribution of the number of contacting spheres per sphere for A) the Finney packing vs. calculated contacts of the dense RAMPAGE packing; B) RAMPAGE packings; C) the large Play-Doh packing and D) the Play-Doh packings from the small cylinders.


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Not surprisingly, according to Table 4.2 the closest disordered packings are the Finney pack, the tapped RAMPAGE packing and the large Play-Doh-pack, and the histograms of their coordination numbers show a high degree of similarity (Fig. 4.7-A, 4.7-B and 4.7-C). The untapped RAMPAGE packings are quite loose. The Play-Doh packings from the small cylinders appear to resemble the untapped RAMPAGE packing (Fig 4.7-B and 4.7-D), but probably suffer too much from the small size of the population, with a large fraction being located near the boundaries, to allow for a reliable direct comparison. The contact angle is defined as the angle between two spheres, which are in contact, and the horizontal plane. It provides information on the structure of the packing and direction of the forces. Of course, the extreme cases are again illustrated by the regular packing, which all have only two directions present; one of those is always horizontal, indicating the layering, while the other angle indicates the manner of stacking of the layers. Cubic shows of course the largest angle at 0.5π, and Rhombohedral-Pyramidal has the smallest angle at 0.25π. The contact angles for the disordered packings are shown in Figure 4.8. For the Play-Doh packing only one cylinder was used, because the lengths of the radii had to be accurately adjusted to reconstruct the correct number of contacts for each sphere. Several interesting features can be seen in these plots. According to Figure 4.8-A, the RAMPAGE packings have the highest percentage of contacts in a near horizontal plane. The Play-Doh packing has on the other hand a rather low percentage of grains in a horizontal arrangement. The reason for this difference might be twofold: the horizontal space in the cylinder was too limited for Play-Doh grains to fall next to each other, or the absence of friction in the RAMPAGE model might lead to an excessive amount of sliding. In Figure 4.8-B can be seen that the Play-Doh packing has a relatively lower number of low-angle contacts over the entire sphere shell. This implies that friction is playing a very important factor in these experiments, which explains also the relatively higher percentage of high-angle contacts in Figure 4.8-A. The RAMPAGE packings are relatively closer to the cumulative distribution of the Finney-pack (Fig. 4.8-B) and neglecting friction in the model seems to affect predominantly the spheres at lower contact angles.


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Figure 4.8 Contact angles for the different packings. The angle is in radians from the horizontal plane. A) Histogram of angles in percentage of the total number of contacts. B) Cumulative distribution of contact angles.


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4.4 Assessment of boundary effects Even though RAMPAGE uses periodic boundaries in the horizontal directions, the top and bottom boundaries still might cause deviations in the obtained values. Scott (1960) showed in his paper a method to extrapolate his experimentally measured values for a limited number of balls to a value for an infinite amount of balls in a packing, and thus corrected the measured values for the boundary effects. This procedure was repeated here for the RAMPAGE packings at gravitational stability and minimal overlap (Table 4.3 and Figure 4.9). The porosities for 10 random simulations with a packing of N spheres were averaged for each value of N. From Table 4.3 it can be readily concluded that the variations between simulations become smaller and that in general the average porosity decreases for an increasing number of spheres. The slightly higher porosity value for 2500 spheres is probably a random effect caused by the limited number of simulations.

Table 4.3 Results of repeated RAMPAGE simulations for a varying number of spheres N at a stop criterion of 99% stability and <0.5% mean overlap of diameter. N-1/3 is the value used for the plot in Figures 4.9 and 4.10 and represents the inverse of the length (in number of spheres) of the edge of the sampled cube. The last column represents the 90% confidence level.

N N-1/3 Average porosity

Standard deviation 0.9 CL

500 0.125992 0.448764 0.007544 0.005397 1000 0.1 0.447112 0.006883 0.004924 1500 0.087358 0.446759 0.006039 0.00432 2000 0.07937 0.446164 0.005101 0.003649 2500 0.073681 0.447075 0.004223 0.003021 3000 0.069336 0.444593 0.004573 0.003272 5000 0.05848 0.441538 0.003229 0.00231


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Figure 4.9 Porosities averaged over 10 RAMPAGE simulations. Porosities are plotted versus N-1/3, with N being the number of spheres. The porosity indicated at the axis is the extrapolated average porosity for an infinite number of spheres. The error bars correspond to the 90% confidence interval of the mean.

Scott eliminated the peripheral error by plotting the packing density against the reciprocal of the radius of the vessel. Similarly, we can plot the porosity against the reciprocal of the length of a side. In case of a cubic container this will also be proportional to N-1/3, which similarly gives the porosity for an infinite volume when it is extrapolated to N-1/3 = 0. As can be seen in Figure 4.9, the porosity is only slightly affected by the boundaries. When the 90% confidence limits are taken into consideration, the maximum corrected porosity will be 44.3% and the minimum 42.7%. This implies that the present influence of the boundaries is only very small and is most likely related to the top and bottom boundaries. This proves clearly that the use of periodic boundary conditions in horizontal direction avoids the boundary effects, which would be present with a rigid wall. Application of the procedure shown in Figure 4.9 to the porosities obtained from the post-processing software gave a similar range of averages (44.2-45.1%) and the plot indicates now even more clearly the absence of boundary effects (Fig. 4.10).


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Figure 4.10 Averaged porosities over 10 simulations by RAMPAGE, as in Figure 4.9, but with values obtained from post-processing software.

In a similar procedure to derive the porosity of an infinite space by changing the number of spheres (and thus affecting the size of a cubic unit cell), the influence of the height of the unit cell on the simulated porosity can be derived. For this purpose 10 x 10 packings were simulated, all consisting of 1000 spheres and a square intersection of the unit cell, but with varying height. Figure 4.11-A indicates that if the height of the pack is too small, the porosities will be significantly overestimated, and implies that the height of the packing is required to be at least 17 radial units high (= 0.06 inversed height). Higher packings do not seem to change the porosity of the simulation. A negative effect of increasing height is shown in Figure 4.11-B, where the number of iterations before the packing is in a stable configuration is plotted versus the inverse of the height. It appears that higher packings require longer simulations, likely resulting from instabilities lower in the pack, which propagate upwards throughout the lattice during the simulation. It is therefore inferred that a unit cell with a height of 20 radial units is representative for a layered sediment with a homogenous size distribution. Post-processing of simulated packings somewhat decreases overestimation of the average porosities as is illustrated in Figure 4.11-C. It is, however, only a marginal improvement, which does not alter the above conclusions.


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Figure 4.11 Influence of height (in number of radii) on: A) porosity; B) length of simulation; and C) porosity obtained from post-processing. Values are averaged over 10 simulations. Error bars indicate the 90% confidence limits.


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4.5 Influence of gravity on simulated packings With the previous analysis of different types of packings, a good reference is available to study the effect of gravitational forces on the structure of the packing. In the simulation model RAMPAGE, the magnitude of gravitational force in relation to the interparticle force can be easily modified to become a more dominant or either less dominant factor during grain rearrangement. This ratio will further be referred to as gravity ratio G. It is important to investigate the variations of this ratio to find the optimal value for efficient rearrangement. The results of these variations must also be compared and evaluated with the results of the experimental packings to prevent the simulation of unrealistic packings. A higher gravity ratio leads inevitably to larger vertical distances that particles travel in a smaller number of calculation steps. Hypothetically, an increase or decrease of downward movement per iteration can be regarded as the easiest adjustable control on the time duration that each iteration represents. It is, however, rather unpredictable how the model will behave with a large downward force when particles start to interact. For this analysis, one initial lattice was generated and has been subjected to five different magnitudes of the gravity ratio. Besides the default ratio of G = 1, gravity ratios of G = 0.1, G = 0.316, G = 3.16 and G = 10 were used. The effects of these variations on the various properties of the pack are illustrated in Figure 4.12. The influence of the gravity ratio on the porosity decrease during the simulation is shown in Figure 4.12-A. A noteworthy attribute on this diagram is that the largest gravity ratio of G = 10 causes the slowest loss of porosity during the simulation. After 100 000 iterations a gravity ratio of 10 results in a porosity decrease from 52% to 46%, whereas a gravity ratio of 1 causes a decrease from 52% to nearly 42%. This seems counterintuitive, but can be understood as the result of particles being pushed too tightly together, causing dominant vertical forces and preventing significant lateral displacements. The gravity ratio of 1 shows the earliest and fastest decrease along with the simulation with a gravity ratio of 0.316. A gravity ratio of 3.16 will eventually also result in an end porosity comparable to those of G = 1 and G = 0.316, but the fastest loss of porosity starts a few thousand iterations later than the others. This is believed to be the phase where the majority of the spheres is dropping. Apparently, a higher downward force seems to hamper the initiation of this phase. The explanation for this is that the maximum move distance is related to the high downward force, and that movements related to interparticle forces are proportionally scaled to relatively small distances. As a result, removal of the large initial overlaps entails many more iterations. A very small gravitational pull seems to extend the duration of the


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Figure 4.12 Change of a) porosity; b) mean coordination number; and c) mean overlap during simulation, for five different magnitudes of gravity ratio G.


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phase with collective particle drops as is shown by the graph for G = 0.1 (see Fig 4.12-A). The evolution of the mean coordination number (Fig. 4.12-B) basically follows the same trends for the different gravitations as the porosity trends. The mean overlap is as a result of a larger gravity ratio considerably higher at the start of the simulations (Fig. 4.12-C). For the three lower values of the gravity ratio the mean overlap rapidly evolves to equivalent values, for higher gravity ratios this appears to require a considerably larger number of iterations. For the highest gravity ratio the mean overlap tends to stay a small increment above mean overlap for the other simulations. The structure of the packings seems also hardly influenced by variable gravitational magnitudes, with the only exception being the simulation with gravity ratio of 10 as can be inferred from Figure 4.13. It shows that the second peak at a distance of two diameters has only one peak in contrast to the development of a split peak for the simulations with gravity ratios of 0.1 and 1.

Figure 4.13 Radial distribution function after simulation of 100 000 iterations for three different magnitudes of the gravity ratio.


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Figure 4.14 Contact angle distribution after simulation of 100 000 iterations for five different magnitudes of the gravity ratio.

The deviations of the contact angle distributions are all within a limited interval (Fig. 4.14) and do not indicate a large discrepancy in the structures of the packings resulting from a varying influence of gravity ratio. These results of the five simulations with different gravity ratios imply that the choice of the gravity ratio does not seriously affect the structure of the packings when the gravity ratio is smaller than 10, but influences the number of iterations before a representative initial porosity has been reached. The most efficient simulations are obtained with a gravity ratio between 0.316 and 1. For these values the phase with collectively dropping particles begins earliest and proceeds most rapidly. 4.6 Conclusions

• The dense packings that are generated with the RAMPAGE model are consistent with experimental particle packs in both packing structure and spatial statistical variables such as porosity and mean coordination number.


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• The ductile Play-Doh packings are structurally most similar to loose RAMPAGE packings. This suggests that the ductile packings hardly undergo grain rearrangement due to friction.

• The use of periodic boundaries in the RAMPAGE model effectively

eliminates the influence of the boundaries.

• The height of the unit cell does not affect the porosity of the packings, provided that it is higher than a minimum of 17 radial units (20 is recommended).

• The gravity ratio G hardly influences the structure of the simulated

packings, unless the gravity ratio is increased by one or more orders of magnitude. The most efficient simulations are obtained with a gravity ratio of approximately 1.

• Most importantly, the above conclusions imply that RAMPAGE

packings are good models for a broad range of realistic natural packings, ranging from loose to dense systems. In addition, the RAMPAGE model is also capable of simulating the entire trajectory between these two extremes.


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5 The Relation between

Grain Size Distribution and Porosity

5.1 Introduction Clastic sediments typically consist of grains of different sizes. Differences in grain diameter influence packing behaviour and efficiency, and consequently affect the porosity of sediments. It has been well established that porosity decreases when the sorting becomes poorer (Beard & Weyl, 1973), and that it also varies with skewness (Tickell, et al., 1933), but a generally accepted quantitative relation has never been derived. Research on the relation between grain size distribution and porosity has been limited to analyses of mixtures of two or three grain sizes. This allows for a description of the problem in terms of the size ratio and proportions of the grain sizes. Since different authors have used different definitions for these units, we define the size ratios within mixtures here as the larger diameter divided by the smaller diameter, and use the volumetric fraction of the entire volume for the larger grain size population to indicate the proportions. In this chapter the effects of grain size parameters are studied with the use of the numerical simulation program RAMPAGE. The results of the model are compared with experimental data and earlier models, of which an overview is given in the next section. In the remainder of the chapter the results of binary and ternary packings simulated with the RAMPAGE model are presented. The effects of commonly


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used parameters such as mean grain size, sorting and skewness on porosity are evaluated. 5.2 Previous work on particle mixtures Experiments of packings of binary and ternary mixtures have already been reported as early as the 1930s (Furnas, 1929, see Fig. 5.1; Westman & Hugill, 1930; Tickell et al., 1933) and the observed behaviour has been well explained by Fraser (1935) as follows.

Figure 5.1 Porosities of binary mixtures (after Furnas, 1929).

If a comparatively large sphere is introduced into an assemblage of small spheres, the porosity will be affected in two ways: (1) the large sphere completely occupies the volume with solid material, which would otherwise be filled with smaller spheres and the pores between them thereby decreasing the porosity of the pack, and (2) it disturbs the packing of the smaller spheres, which results in a looser packing of the smaller spheres in the vicinity of the large sphere. The negative effect on porosity of the first influence grows stronger if the size ratio increases, whereas the positive effect of the latter influence decreases at larger size ratios. When the proportion of large spheres increases, the porosity will drop rather uniformly as long as the proportion of small spheres is sufficient to keep the large spheres separated (matrix-supported packing). If the proportion becomes too large, two alternatives are possible, depending on the size ratio of the binary mixture: (a) the small spheres fit inside the pores between

The Relation between Grain Size Distribution and Porosity

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packed large spheres, or (b) the small sphere is unable to occupy the pore space between large spheres. In the latter case, both sphere populations will interfere with each other’s optimal packing configuration and consequently the porosity will be higher than the optimal porosity minimum. On the other hand, a pore inside a tetrahedron of three large spheres and a small sphere will be smaller than the pore inside a tetrahedron of four large spheres, so that the overall porosity of the packing will remain below the porosity of an assembly of uniform spheres. If the diameter of the small spheres is smaller than the average pore size of large spheres, the large spheres will increasingly control the structure of the assembly. When the large spheres become self-supporting and the structure is independent of the small spheres, the small spheres will only occupy the pore space. The degree of “filling” of the pore space will decrease with an increase of the large sphere population, and as a result the porosity increases again (Fraser, 1935; Marion et al., 1992; Koltermann & Gorelick, 1995, in Figure 5.2). Conversely, if small spheres are introduced into a packing of large spheres, the same reasoning can be applied in the inverse direction.

Figure 5.2 Ideal packing model, illustrated for (A) a binary mixture of natural sediments and (B) its associated predicted porosity curve. The porosity minimum occurs when the volume of the fines is equal to the porosity of the coarse-grained component. After Marion et al. (1992) and Koltermann & Gorelick (1995).

Addition of a third intermediate sized population decreases the magnitude of the porosity range, which followed from experimental data from Westman &


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Hugill (1930) and Tickell et al. (1933), as shown in Figure 5.3. Tickell extended this also further to more complex mixtures and concluded that in such a case the porosities vary even less.

Figure 5.3 Ternary diagram showing porosity contours for ternary mixtures of Ottawa Silica sands (–24+28 Mesh = 0.7 mm – 0.59 mm; -42+48 Mesh = 0.42 mm – 0.297 mm; -80+100 Mesh = 0.177 mm – 0.149 mm). After Tickell et al. (1933).

The results of experiments with ternary mixtures have invariably shown the occurrence of the minimum porosity at, or very near to, the binary packing of the largest and the smallest grain sizes, i.e. one of the edges of the ternary diagram. Any other size of particles will only increase the porosity, which is a tendency that persists to even very small contents of the two extreme sizes in the mixture. This results in lines of equal porosity radiating towards the medium size sphere corner. Standish & Borger (1979) noted that this theory also forms the underlying mechanism for the additional trough (see Fig. 5.4) that was observed in the experiments by Ridgway & Tarbuck (1968) and suggested that such troughs should be present in all ternary porosity diagrams, but that errors or the choice of mixture compositions mask their existence. It may well be, that the occurrence of such a trough is caused by a very small difference between the medium and small sphere sizes. There have, however, been no other studies that support this hypothesis.


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Figure 5.4 Porosities of ternary mixtures of spheres; the dotted line indicates the axis of the trough. After Ridgway & Tarbuck (1968).

One might expect the existence of very dense ternary packings in which the medium-sized spheres fit inside the pores of the largest component, and the smallest component fits inside the remaining pores. Such a packing is known as the fractal Apollonian packing (Roualt, 1999; Hecht, 2000), which is an ideal packed assembly. However, Hecht (2004) illustrated in later work that, as soon as such size distributions are slightly randomised, the porosity increases significantly. Many real particle systems are of continuously distributed sizes. Such systems have often distributions that resemble either Gaussian or lognormal distributions by weight. Sohn & Moreland (1968) conducted experiments on such continuous size distributions and reported that an increasing width of distribution, equivalent to poorer sorting, caused lower porosities. More importantly, they found that binary mixtures of continuous distributions showed a porosity trend similar to binary mixtures of uniform sizes, although porosities were somewhat higher for larger size ratios. Again, these higher porosities can be attributed to the introduction of other (intermediate) particle sizes that interfere with the packing of two grain sizes of the size ratio. These results are important because provenance studies (Weltje & Prins, 2003) indicate that many sediments consist of mixed end-members that follow relatively simple, unimodal size distributions. The possibilities of predicting the properties of such mixtures on the basis of models consisting of only two or three uniform sizes would be a significant improvement over existing (empirical) methods.


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Besides this, it is also believed by several workers that the results of (experimental) binary mixtures can be used to predict the porosity and permeability of multi-component mixtures (Standish & Borger, 1979; Yu & Standish, 1987).

Figure 5.5 Comparison between measurements (points) and Yu & Standish’ model predictions (lines) for various size ratios. (After Yu & Standish, 1987).

Many existing models for both binary and ternary packings are based on the ideal packing model for binary mixtures (Furnas, 1929; Cumberland & Crawford, 1987; Yu & Standish, 1987; Koltermann & Gorelick, 1995; Fig. 5.2). The ideal packing model is based on the concept of filling the pores between the large spheres with infinitely small spheres (limiting case). The volume of small spheres that can occupy the pore space is however restricted to the initial porosity of the packing of large spheres. The other volume fractions are regarded from an initial packing of small spheres, which is then “solidified” by combining small spheres to form larger spheres. This solidification can also be continued until a packing exists of large spheres with the interstices of the large spheres completely filled with small spheres. However, the ideal packing model underpredicts the porosity considerably, especially in the region where the minimum porosity is expected, and particularly for smaller diameter ratios (Koltermann & Gorelick, 1995). The reason for this is that only one packing is being considered at a time, while in


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Figure 5.6 Ternary mixture porosities. (Size ratios presented as DM/DS – DL/DM – DL/DS, with DL = large diameter, DM = medium diameter, DS = small diameter) (a) Measurements from Ridgway & Tarbuck (1968), size ratios: 1.33 – 1.67 – 2.22; (b) model predictions for same size ratios from Yu & Standish’ (1987) model; (c) measurements from Standish & Collins (1983), size ratios: 1.85 – 1.97 – 3.65; (d) model predictions for same size ratios from Yu & Standish’ (1987) model; (e) measurements from Jeschar et al. (1975), size ratios: 2 – 2 – 4; (f) model predictions for same size ratios from Yu & Standish’ (1987) model. (After Yu & Standish, 1987).


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reality the two packings disturb each other’s arrangement. Modified models have been developed by Yu & Standish (1987) and Koltermann & Gorelick (1995), and produce quite satisfactory results (Fig. 5.5). Models for ternary mixtures, on the other hand, which are derived from the ideal packing model and its extensions (Westman & Hugill, 1930; Dexter & Tanner, 1971; Yu & Standish, 1987), do not resemble the experimental data very well (Fig. 5.6). The predicted isopores (lines of equal porosity) are straight and angular, whereas the experimental isopores are curved. Furthermore, the earlier models (Westman & Hugill, 1930; Tickell et al., 1933) show minima towards the middle of the ternary diagram, while theory and experimental data have never convincingly shown proof of such behaviour. Making the assumption that all spheres touch their neighbours, Dodds (1980) followed another, statistical, and potentially more promising approach. He divided the structure of a random packing of spheres into tetrahedral subunits, and determined the frequency distribution of these subunits from the sphere size distribution. Next, he calculated the porosities of all possible tetrahedra for various binary and ternary mixtures. From this he statistically determined the porosity diagrams for these binary and ternary mixtures, which show a qualitative realistic behaviour for different compositions. The porosities were, however, not extended to realistic values for packings, and the model can therefore unfortunately not directly be compared with experimental data. 5.3 Methodology for porosity determination The RAMPAGE model has been used to construct both binary and ternary packings with an average of approximately 1000 spheres. The same stopping criterion was used for all simulations, namely a mean overlap of less than 1% of the diameter and at least 99% of the spheres in a gravitationally stable position. This will stop the simulation in a rather early phase of grain rearrangement, but it is assumed that this generates packings that are representative of sediments at surface conditions, and secondly it enables us to run a larger number of simulations to obtain a large and usable data set. 68 Simulations of binary mixtures were completed with a size ratio of 2; and 68 simulations of binary mixtures with a size ratio of 4. For the ternary diagrams, 37 simulations were run with diameter size ratios of DM/DS = 2, DL/DM = 2, and DL/DS = 4. In all these simulations the volume fractions of large, medium and small spheres were varied. In addition, 10 simulations with a uniform size distribution were used for the monodisperse variant in the


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diagrams (i.e. 100% solid volume of large particles, 100% of small particles and 100% of medium sized particles). The final porosities of the binary mixtures as calculated by the model for each simulation are plotted in Figure 5.7. The compositions of the ternary mixtures are plotted in the diagram of Figure 5.8.

Figure 5.7 Raw simulated porosity data of binary mixtures.

The porosities of the 10 simulations with mono-sized spheres range between 43.5% and 45%, with an average of 44.6%. The locations and radii of the spheres of each packing are stored and available for further analysis. A higher density of simulations with a specific initial volume fraction of large spheres was chosen when the variability of the final porosities for these percentages was high. For the diameter size ratio of 4 the initial percentage of large spheres was restricted to about 40% of the volume, since smaller percentages require a total of more than 1000 spheres to obtain a representative distribution of small and large spheres, which would also require a considerable amount of extra calculation time per simulation. For the same reason, there is an underrepresentation of compositions with a large volume fraction of small spheres in the ternary diagram of Figure 5.8. Fortunately, this will not significantly influence the final diagram, because the largest porosity variations are expected to be around 70% volume fraction of large spheres.


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Figure 5.8 Compositions of ternary mixtures that are used for the construction of the ternary diagrams for the bottom and the middle slice. The different points are plotted according to their respective volume fractions of large (L), medium (M) and fine (F) particles.

The porosities and volume percentages in Figure 5.7 are not the best estimates of the model output, for the following two reasons: (a) Segregation within the packing during rearrangement causes small spheres to drop to the bottom more rapidly than large spheres, which results in overprediction of the volume percentage of large spheres in the calculation window (inverse grading). The calculation window lies one maximum radius distance above the lowest sphere coordinate; and (b) The rather early stage of rearrangement at the moment that the stop criterion is satisfied results possibly in very irregular top surfaces, accompanied by extremely large voidage areas. As a consequence, porosities can be severely overestimated in such cases. Recalculation of porosities and volume fractions of size classes was therefore carried out in three equal horizontal slices through the unit cell following the procedure described and illustrated in Figure 5.9.


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Figure 5.9 Procedure of recalculation of the porosity and volume fractions of size classes. Of any sampled container the calculations should be inside the shaded volume at a distance of the maximum radius size from its boundaries to correct for holes left by particles just outside the container (above). Splitting the container in three slices reduces the number of particles inside the shaded volume. Four containers are placed next to each other to increase the number of particles by exploiting the periodic boundaries (below).


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5.4 Porosities of binary and ternary mixtures 5.4.1 Binary mixtures The corrected porosities of the binary mixtures are presented in Figures 5.10 and 5.11. The porosity values for the different slices show a clear segregation. Such a segregation of an initial mixture of spheres that initially has been randomly distributed causes several changes in the arrangement, when it is assumed that a haphazard placement results in a more or less even distribution of properties throughout the pack. The lowest slice contains a larger volume fraction of small particles, while the top slice gets a relatively high representation of large spheres. This is illustrated by the highest porosity values being plotted for the top slices towards the right of the figures in the upper plot of Figures 5.10 and 5.11. Furthermore, due to the downward pressure from spheres higher up in the packing, the lowest slice has achieved a denser state of packing than the middle and –especially– the top slice, as can also be noted from the trend lines, that were obtained by polynomial regression. The points show an abundance of scatter in all of the plots of Figures 5.10 and 5.11, which indicates that porosity values vary considerably within a single slice. The scatter in the lower plots is, however, clearly bounded by two extremes that are particularly conspicuous for size ratio 4: (a) the porosity is lower than the maximum porosities for the monodisperse packing and (b) the porosity is higher than porosities that would be predicted with the ideal packing model. This also immediately demonstrates why the ideal packing model is not very useful for predicting initial porosities of random packings. Models like these give a good indication of the minimum porosities that may be found, but give a poor impression of the actual observed initial porosities. Experimental data for binary mixtures generally fit much better to the modified ideal packing models, because these experimental packings have been carefully generated through tapping and shaking until a dense packing state has been reached. The considerable increase of the squared correlation coefficient for size ratio 4 from 0.5027 in the middle slice to 0.6255 lower in the packing, suggests that the trend becomes more pronounced and less variable. This implies that the size distribution has a serious effect on the amount of porosity decrease already during the early stages of grain rearrangement at the surface and during very shallow burial.


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Figure 5.10 Porosities of the simulated binary mixtures with size ratio 2 for: A) the entire packing and the individual horizontal top slice through the packing; and B) the middle and bottom slices and a trend line for each slice. The fractional packing model of Yu & Standish (1987) is used to indicate the theoretical porosity minimum.


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Figure 5.11 Porosities of the simulated binary mixtures with size ratio 4 for: A) the entire packing and the individual horizontal top slice through the packing; and B) the middle and bottom slices and a trend line for each slice. The fractional packing model of Yu & Standish (1987) is used to indicate the theoretical porosity minimum.


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5.4.2 Ternary mixtures The porosity data of the ternary mixtures were plotted in ternary diagrams. Besides the ternary mixture data, also the porosities of binary mixtures were used for the sides of the diagram and the mono-size data were used for the corner points. Three different methods were chosen, for different reasons:

• Kriging was used because it obeys the actual data best while interpolating. It relies on the spatial correlation between the data points of the entire data set to determine how much weight has to be given to each data point. This leads generally to more irregular surfaces, where isolated data points with larger deviations will be clearly visible. The kriged diagrams for the bottom slice, the middle slice and both slices together are presented in Figure 5.12.

• Inverse square distance was used to obtain a smoother diagram and to reduce the effect of the notorious outliers. These diagrams are possibly more useful for the comparisons with the experimental data in Figure 5.6. The diagrams produced with inverse square distance for the bottom slice, the middle slice and both slices together are presented in Figure 5.13.

• The polynomial regression of the data using a cubic surface was carried out on the data to obtain an averaged surface (trend surface analysis), similar to the regression lines for the binary mixtures in Figures 5.10 and 5.11, which can be described by a simple equation. The diagrams for the polynomial regression of the bottom slice porosities, the middle slice porosities and the porosities of both slices together are presented in Figure 5.14.


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Figure 5.12 Ternary porosity diagrams constructed with Kriging from the porosity values of the simulated ternary mixtures over: (a) Bottom slice; (b) Middle slice; and (c) Bottom and Middle slices.


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Figure 5.13 Ternary porosity diagrams constructed by Inversed Square Distance, from the porosity values of the simulated ternary mixtures over: (a) Bottom slice; (b) Middle slice; and (c) Bottom and Middle slices.


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Figure 5.14 Ternary porosity diagrams constructed with Polynomial Regression (cubic surface definition) from the porosity values of the simulated ternary mixtures over: (a) Bottom slice; (b) Middle slice; and (c) Bottom and Middle slices.


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In all the diagrams the lowest porosities are at approximately 70 vol% of large spheres, and 30 vol% of small spheres, and isopores radiate outwards from this composition on the diagram to the highest porosity values at the uniform size compositions at the corner points. Similar to the observations in the binary diagrams, the porosities of the bottom slice are the lowest in all the diagrams (Figs. 5.12-A, 5.13-A and 5.14-A). For the middle slice the kriged diagram in particular, but also the inverse distance diagram (Figs. 5.12-B and 5.13-B) show rather distinctive irregularities. This is most likely an effect of larger variation in porosity values in the middle slices, that disappears as the spheres are further rearranged into a denser packing, as the lowest slice has a much more regular surface in those figures. The experimental data from Standish & Collins (1983) and Jeschar et al. (1975) in Figure 5.6 have size ratios comparable to the simulated data. The porosities that were found by Standish & Collins (1983) are also within the same range as the simulated porosities for the bottom slice, although the porosities of the mono sized packings are approximately 4% higher for the simulated packings. A trough comparable to the one in the ternary diagram of Ridgway & Tarbuck’s data (Figs. 5.4, 5.6-A and possibly as modelled in 5.6-B), which would support the theory of Standish & Borger (1979) that all ternary diagrams should contain one, is not evident in any of our simulated diagrams. 5.5 Sorting and skewness effect 5.5.1 Sorting While the negative effect of poor sorting on porosity has been well established in petroleum geology, an analytical explanation has never been provided. Earlier works on porosities of binary and ternary mixtures, on the other hand, have never been translated into parameters of the grain size distribution, such as sorting and skewness. The use of ternary diagrams may increase insight into the effect of those parameters on porosity. The sorting can be calculated as the standard deviation of the mean grain size of a particle distribution. All calculations in this paragraph use the logarithmic

phi-scale for grain sizes: 2 log(1/ )phi D= with D being the diameter in millimetres. The exact calculation then becomes:

2 2

1 1

( )n n

std i i i ii i

S w phi w phi= =

= −∑ ∑ (5.1)


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with n number of size classes wi weight fraction of size class i Since sieving is normally the procedure to obtain the grain size distribution of a sediment, the mean grain size, sorting and skewness are logically determined by weight fractions. Figure 5.15 illustrates the differences for mean grain size and sorting between frequency distribution and weight distribution. On the diagrams for the frequency distribution the effect of the large number of small particles dominates the plots, whereas the weight distribution shows a symmetric distribution of the properties.

Figure 5.15 Mean grain size (a) and sorting (b) of the ternary diagrams calculated with frequency distribution of the samples (left) and the weight distribution of the samples (right).

Figure 5.16 shows the plots of porosity vs. sorting for the Beard & Weyl data and the simulated RAMPAGE data. Both exhibit a broad range of porosity values where the minimum values are observed, although the largest


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variations are observed for the RAMPAGE data. This may partly be contributed to the small degree of compaction that these samples have been subjected to, but this is clearly not the only reason. The regression line derived from the experimental data from Beard & Weyl (1973) predicts for example a porosity of 34% around a sorting coefficient of 1.0, while Beard & Weyl’s data range between 31-36%. The values of the RAMPAGE packings range in a much broader band from values much lower than 30% to values reaching over 40%, which may be an effect of the limited number of grain in the RAMPAGE simulations.

Figure 5.16 Porosity vs. sorting of the RAMPAGE simulations and the Beard & Weyl (1973) experiments. The regression line (P = porosity; S=sorting) is based on data of Beard & Weyl, and fits their data very well for a=3.76 and b=–0.76.

Comparison of Figure 5.15-B with the ternary diagrams of Figure 5.14 shows that the sorting of the weight distribution looks very much alike, but that there is a clear shift between the diagrams. This implies that another characteristic of the distribution must play an important role to determine the low porosity values. It has to be noted that all Beard & Weyl (1973) experiments were carried out with sediment weight distributions at zero


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skewness. In the next paragraph, the effect of the skewness on porosity will be assessed. 5.5.2 Skewness The skewness characterises the degree of asymmetry of a distribution. A negative skewness indicates that a relatively larger volume of small spheres is present in the sample. The skewness is the third order deviation of the size distribution and is calculated as follows:

3

1

ni

ii std

phi phiSk w

S=

−=

⎛ ⎞⎜ ⎟⎝ ⎠

∑ (5.2)

where phi is the mean grain size.

Figure 5.17 Skewness of the distribution for the ternary diagrams based on frequency distributions (left) and weight distributions (right).

The skewness of the ternary mixtures in this chapter is shown in Figure 5.17. The skewness for the frequency distribution is strongly negative all across the diagram. The skewness for the weight distribution has a more or less point-symmetrical character near the vertex of medium-sized grains.


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Figure 5.18 Porosity vs. skewness of the weight size distribution.

The absence of a clear trend for the porosity with skewness in Figure 5.18 can also be easily deduced if the ternary diagrams for the skewness in Figures 5.17 are compared to the porosity trends in Figure 5.14. It follows from these diagrams that the lowest porosities are obtained for compositions with a slight positive skewness in the weight size distribution. Combining the trends for sorting and observations for the binary mixtures indicates that the lowest porosities will be obtained for the most poorly sorted samples with a slightly positive skewness and the largest diameter ratios. 5.6 Conclusions The RAMPAGE model has been used successfully to simulate a large number of packings consisting of various mixtures of two or three grain sizes. The data were used to generate binary and ternary diagrams that show comparable trends to diagrams obtained from experimental data. These porosity trends are in better agreement with porosity values that might be expected from natural mixtures than those that are predicted with ideal packing models and their derivatives, which are more suitable for predicting the minimum porosity values that may be expected for such mixtures. The effects of sorting and skewness of the weight distribution on the porosity of the mixtures were evaluated and indicate a slight trend of decreasing


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porosity with poorer sorting, while the skewness shows the lowest porosities for a slight positive skewness. There is, however, a strong non-uniqueness in both parameters separately, so compositions need to fulfil both conditions to achieve a low porosity of the packs. It needs to be pointed out that all these results for ternary mixtures are obtained for sphere populations with a single size ratio. Further simulations with other size ratios are required to confirm these findings for grain-size distributions with even poorer sorting.

6 Synthesis In the preceding chapters a model to simulate collective particle rearrangement was presented. The structure of the resulting particle packs was successfully compared with other known packings and initial porosities of mixtures of spheres of different sizes were presented in ternary diagrams. It is shown that the model is capable of simulating realistic heterogeneous granular media, ranging from loose packings that underwent only initial stabilisation to fully rearranged dense packings. This offers a solid basis to use the model for a variety of applications and further research. A number of these opportunities are briefly described in this synthesis. First, the most important feature of the model is graphically represented and the definition of disordered sphere packings is reassessed. Packing State A good overview of the state of mono-sized sphere packings is obtained when the mean coordination numbers are plotted versus porosity (Fig. 6.1). The regular (crystalline) packings show a linear alignment, indicating a maximum porosity for each mean coordination number. The disordered packings remain well below that imaginary line, with all the Play-Doh packings tending to have the lowest porosities in relation to their mean coordination numbers, most likely because of the deformation of the ductile grains.


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Nolan & Kavanagh (1992) defined a subspace into which all their simulated mono-sized packings fitted (see Fig. 2.9), however, the position of this plane in Figure 6.1 (illustrated as a shaded area) appears to be shifted a bit towards lower mean coordination numbers, since both the simulated dense Rampage packing as well as the Finney pack have higher mean coordination numbers and fall outside this region. The dense rampage packing has however a lower porosity as well, which is an effect of the allowance of overlap during rearrangement. Even though the overlaps are generally very small, they cause on average a thin shell of the size of the mean overlap around every sphere to occupy the pore space. Secondly, the allowance of overlaps may have enabled the spheres to “wriggle” through tight gaps, where solid steel-balls would not be able to go through (although this is probably just a minor effect). The first effect can be geometrically corrected, by recalculating the porosity for spheres of the size of the initial diameter minus the mean overlap.

Figure 6.1 The mean coordination number plotted versus porosity, showing the state of the different packings that were treated in Chapter 4. The shaded area is the subspace of the random packings according to Nolan & Kavanagh (1992). See Figures 2.9 and 6.2 for the names of the bounding packing states.

To correct porosities for overlap for mono-sized packings, the following equation is used:

3

3

( )1 (1 )

i i

ir r

i

rrδ

δφ φ−

−= − − (6.1)

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115

where ri initial radius size

irφ porosity calculated with initial radius size

δ mean overlap

ir δφ − porosity corrected for overlap

Two simulation runs for mono-sized spheres have been corrected and are plotted in Figure 6.2. One simulation started as a loose disordered packing that was rearranged under the influence of gravity without additional compaction stimulation, until a bridged disordered close packing was reached. A second simulation was tapped during rearrangement until a disordered close packing was reached with the same properties as the Finney pack. All packing states falling between these two trajectories can be easily constructed by initiating tapping at a later stage during the unstimulated simulation.

Figure 6.2 Subspace of natural disordered packings, encapsulating all possible packings that can be simulated by the Rampage model, which is illustrated by the full simulation trajectories for rearrangement with and without tapping. The dashed lines define the boundaries of the subspace.

Nolan & Kavanagh (1992) included a fourth packing state in their definition of the subspace of random packings, namely the bridge-free random loose


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packing. However, they were unable to generate this packing state with their model. We believe that a bridge-free loose packing, which approaches a crystalline cubic packing, cannot exist in a stable configuration and therefore cannot be produced by rearrangement in a gravity field. Figure 6.2 shows the redefined subspace of natural disordered packings: a triangular area with disordered loose packing, bridged disordered close packing and disordered close packing at the three vertices. Opportunities The most obvious opportunity of the model is to use the generated packings for further analysis. Obtained packings of spheres with arbitrary grain-size distributions can be used for various studies on properties of disordered granular media. The pore network can be extracted and used for research on flow properties, such as the prediction of permeability and multi-phase flow through porous media (cf. Bakke & Øren, 1997). Such analyses will certainly benefit of a few further logical advancements of the model, like incorporation of advanced physical algorithms for pressure solution and/or grain deformation and eventually also cementation. This latter process can also be quickly added in either a simple or more advanced manner on any obtained packing. Another advancement that deserves attention is the effect of particles shape on porosity and permeability. The main challenge for using particles of other shapes will be the search for neighbouring particles that are in contact. Redefinition of the model for ellipsoidal particles will be a flexible method to approximate a large range of realistic grains, but involves additional rearrangement mechanics such as reorientation of elongated particles to be added to the model. Analysis of the statistical characteristics of the Finney and Rampage packings indicates that grain rearrangement creates a larger degree of order. The increase of regular arrangements inside these packings is expressed by the split peak of the radial distribution functions (Scott, 1962). Comparisons of disordered close packings with the soft PlayDoh packings suggest that grain rearrangement forms a minor process during compaction of ductile materials. It is thus an interesting exercise to examine the compaction behaviour of the model, if the rearrangement step involving rolling of spheres is removed from the simulation. It is not unlikely that this will result in a radial distribution function that exhibits a fingerprint similar to the ductile packings. A method that may possibly improve quantification of the degree of structural regularity of disordered packings can be found in a procedure that resembles the approach that Dodds (1980) applied to predict porosities of binary mixtures of spheres by clustering all possible tetrahedra formed by combining

Synthesis

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grains of the two different grain sizes. Such clustering could also be adopted to the different crystalline packings up to a distance of two diameter sizes. It can then be attempted to unmix the radial distribution function of disordered packings into the fingerprints of the radial distribution of the crystalline packings. This would possibly provide a measure for the degree of crystallinity of packing, which could aid the quantification of the degree of grain rearrangement of the assemblage.

Figure 6.3 Three packings of a binary mixture of 1000 spheres with size ratio 4 and a volume percentage of 70% large spheres clearly showing inverse grading.

From the simulations that were run to construct the ternary diagrams, it was clear that grain rearrangement of particle mixtures of multiple grain sizes seems to produce inverse grading (an upward coarsening size profile) from the initial uniform distribution of sizes. An example of inverse grading for a few binary packings is visualised in Figure 6.3 by plotting the average diameter size versus the vertical position in the pack. Inverse grading within beds is usually considered uncommon, but it has been found in nearly every type of sedimentary rock (Fisk & Costa, 1974). The simulated packings are however only a few grain diameters thick, which is more comparable to the thickness of laminae (Allen, 1984). At a lamina-scale, inverse grading is commonly observed in highly concentrated flows of sand-


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size and larger particles. The development of inverse grading like for instance in beach foreshore laminations and slip-face foreset strata was believed to be controlled by dispersive pressure created by grain collisions (Sallenger, 1979). This behaviour is simulated by the model. Many laminae show however normal grading. Paola et al. (1989) produced parallel laminations -all of which had normal grading- in laboratory experiments of turbulent sediment transport over flat beds. Using high-speed photography they were able to observe the deposition processes. The normal grading results from a surge of high-concentration bed load that decelerates and is deposited within a few tenths of a second. In the following seconds the irregular surface is then filled with fine grains. To be also able to simulate normal grading samples using the RAMPAGE program, it would be necessary to investigate the response of the model on different initial gradings during simulation. A procedure following the events during deposition described by Paola et al. (1989) could be a useful starting point to generate such laminae. A stack of different laminae generated at separate time intervals would be interesting to study the effect of different depositional patterns on porosity and eventually permeability. As a concluding remark it can be stated that the model offers numerous opportunities to study the relationship between the parameters of the grain-size distribution and the porosity of clastic sediments, although more simulations are definitely required to fully quantify this relationship.

References Alberts, L.J.H. and Weltje, G.J. (2001) Predicting initial porosity as a function of grain-size distribution from simulations of random sphere packs. In: Proc. 6th Annual Conference of the International Association for Mathematical Geology. Cancun, Mexico. Allen, J.R.L. (1984) Sedimentary structures, their character and physical basis. Vol 1. Elsevier, Amsterdam, 593pp. Allen, D.R. and Chilingarian, G.V. (1976) Mechanics of sand compaction, In: Chilingarian, G.V. and Wolf, K.H. (eds), Compaction of coarse-grained sediments I, Elsevier, Amsterdam, 43-77. Aste, T., Saadatfar, M. and Senden, T.J. (2005) The geometrical structure of disordered sphere packings. ArXiv:cond-mat/0502016 v1. Athy, L.F. (1930) Density, porosity, and compaction of sedimentary rocks. The American Association of Petroleum Geologists Bulletin 14, 1-24.


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Atkins, J.E. and McBride, E.F. (1992) Porosity and packing of Holocene river, dune, and beach sands. The American Association of Petroleum Geologists Bulletin 76, 339-355. Bahr, D.B., Hutton, E.W., Syvitski, J.P.M. and Pratson, L.F. (2001) Exponential approximations to compacted sediment porosity profiles. Computers & Geosciences 27, 691-700. Bakke, S. and Øren, P.-E. (1997) 3-D Pore-scale modelling of sandstones and flow simulations in the pore networks. SPE Journal 2, 136-149. Baldwin, B. and Butler, C.O. (1985) Compaction curves. The American Association of Petroleum Geologists Bulletin 69, 622-626. Barker, C.E. and Kopp, O.C. (1991) Luminescence microscopy and spectroscopy; qualitative and quantitative applications. SEPM Short Course 25, 195pp. Beard, D.C. and Weyl, P.K. (1973) Influence of texture on porosity and permeability of unconsolidated sand. The American Association of Petroleum Geologists Bulletin 57, 349-369. Bernal, J.D. (1960) Geometry of the structure of monatomic liquids. Nature 185, 68. Bernal, J.D. and Mason, J. (1960) Co-ordination of randomly packed spheres. Nature 188, 910-911. Bernal, J.D., Cherry, I.A., Finney, J.L. and Knight, K.R. (1970) An optical machine for measuring sphere coordinates in random packings. Journal of Physics E: Scientific Instruments 3, 388-390. Bezrukov, A., Stoyan, D. and Bargieł, M. (2001) Spatial statistics for simulated packings of spheres. Image Analysis and Stereology 20, 203-206. Bjørlykke, K., Ramm, M. and Saigal, G.C. (1989) Sandstone diagenesis and porosity modification during basin evolution. Geologische Rundschau 78, 243-268. Boer, R.B. de (1975) Influence of pore solutions on rock strength. Ph.D. Thesis, University of Utrecht, Utrecht. Bradley, J.S. (1980) Fluid and electrical formation conductivity factors calculated for a spherical-grain onion-skin model. Log Analyst 21, 24-32.

References

121

Bryant, S.L., Cade, C.A. and Mellor, D.W. (1993) Permeability prediction from geologic models. The American Association of Petroleum Geologists Bulletin 77, 1338-1350. Bryant, S.L. and Pallatt, N. (1996) Predicting formation factor and resistivity index in simple sandstones. Journal of Petroleum Science and Engineering 15, 169-179. Burst, J.F. (1969) Diagenesis of Gulf Coast clayey sediments and its possible relation to petroleum migration. The American Association of Petroleum Geologists Bulletin 53, 73-93. Buryakovskiy, L.A., Djevanshir, R.D. and Chilingarian, G.V. (1991) Mathematical simulation of sediment compaction. Journal of Petroleum Science and Engineering 5, 151-161. Byrnes, A.P. (1994) Empirical methods of reservoir quality prediction. In: Wilson, M.D. (ed) Reservoir Quality Assessment and Prediction in Clastic Rocks. SEPM Short Course 30, 9-21. Cade, C.L., Evans, I.J. and Bryant, S.L. (1994) Analysis of permeability controls: a new approach. Clay Minerals 29, 491-501. Cavazza, W. and Dahl, J. (1990) Note on the temporal relationships between sandstone compaction and precipitation of authigenic minerals. Sedimentary Geology 69, 37-44. Chilingar, G.V., Bissell, H.J. and Wolf, K.H. (1979) Diagenesis of carbonate sediments and epigenesis (or catagenesis) of limestones. In: Larsen, G. and Chilingar, G.V. (eds), Diagenesis in Sediments and Sedimentary Rocks. Developments in Sedimentology 25A, Elsevier, Amsterdam, 247-422. Chuhan, F.A., Kjeldstad, A., Bjørlykke, K. and Høeg, K. (2003) Experimental compression of loose sands: relevance to porosity reduction during burial in sedimentary basins. Canadian Geotechnical Journal 40, 995-1011. Coelho, D., Thovert, J.-F. and Adler, P.-M. (1997) Geometrical and transport properties of random packings of spheres and aspherical particles. Physical Review E 55, 1959-1978. Cumberland, D.J. and Crawford, R.J. (1987) The packing of particles. Amsterdam, Elsevier, 148pp.


122

Cundall, P.A. and Strack, O.D.L. (1979) A discrete numerical model for granular assemblies. Géotechnique 29, 47-65. Denny, P.J. (2002) Compaction equations: a comparison of the Heckel and Kawakita equations. Powder Technology 127, 162-172. Dexter, A.R. and Tanner, D.W. (1971) Packing of ternary mixtures of spheres. Nature-Physical Science 230, 177-179. Dickinson, G. (1953) Geological aspects of abnormal reservoir pressures in Gulf Coast Louisiana. The American Association of Petroleum Geologists Bulletin 37, 410-432. Djéran-Maigre, I., Tessier, D., Grunberger, D., Velde, B. and Vasseur, G. (1998) Evolution of microstructures and of macroscopic properties of some clays during experimental compaction. Marine and Petroleum Geology 15, 109-128. Dodds, J.A. (1980) The porosity and contact points in multicomponent random sphere packings calculated by a simple statistical geometric model. Journal of Colloid and Interface Science 77, 317-327. Donev, A., Cisse, I., Sachs, D., Variano, E., Stillinger, F.H., Connelly, R., Torquato, S. and Chaikin, P.M. (2004) Improving the density of jammed disordered packings using ellipsoids. Science 303, 990-993. Doyen, P.M. (1988) Porosity from seismic data—A geostatistical approach. Geophysics, 53, 1263-1275. Dubrule, O. (2000) Geostatistics in Petroleum geology. AAPG Distinguished Lecture Course Notes, 4-52. Dzevanshir, R.D., Buryakovskiy, L.A. and Chilingarian, G.V. (1986) Simple quantitative evaluation of porosity of argillaceous sediments at various depths of burial. Sedimentary Geology 46, 169-173. Ehrenberg, S.N. (1989) Assessing the relative importance of compaction processes and cementation to reduction of porosity in sandstones: Discussion; Compaction and porosity evolution of Pliocene Sandstones, Ventura Basin, California: Discussion. The American Association of Petroleum Geologists Bulletin 73, 1274-1276.

References

123

Falvey, D.A. and Deighton, I. (1982) Recent advances in burial and thermal geohistory analysis. Australian Petroleum Exploration Association Journal 22, 65-81. Filipp, D. (1999) CThread - a worker thread wrapper class. (http://www.codeproject.com/threads/cthread.asp) Finney, J.L. (1970) Random packing and the structure of simple liquids: I. The geometry of random close packing. Proceedings of the Royal Society of London 319, 479-493. Fisk, L.H. and Costa, J.E. (1974) Inverse grading as stratigraphic evidence of large floods: Comment & Reply. Geology 2, 613-615. Foster, J.B. and Whalen, H.E. (1966) Estimation of formation pressures from electrical surveys – offshore Louisiana. Journal of Petroleum Technology 18, 165-171. Fraser, H.J. (1935) Experimental study of the porosity and permeability of clastic sediments. Journal of Geology 43, 910-1010. Füchtbauer, H. (1967) Influence of different types of diagenesis on sandstone porosity. Seventh World Petroleum Congress Proceedings, v.2, New York, Elsevier, 353-369. Furnas, C.C. (1929) Flow of gases through beds of broken solids. U.S. Bureau of Mines Bulletin 130, 144pp. Galloway, W.E. (1974) Deposition and the diagenetic alteration of sandstone in northeast Pacific arc-related basins. Implications for graywacke genesis. Bulletin of the Geological Society of America 85, 379-390. Galloway, W.E. (1984) Hydrogeologic regimes of sandstone diagenesis. In: MacDonald, D.A. and Surdam, R.C. (Eds.) Clastic diagenesis. American Association of Petroleum Geologists Memoir 37, 3-14. Giles, M.R. (1996) Diagenesis – a quantitative perspective. Kluwer Academic Publishers, Dordrecht, 435pp. Giles, M.R., Indrelid, S.L. and James, D.M.D. (1998) Compaction – the great unknown in basin modelling. In: Düppenbecker, S.J. and Iliffe, J.E. (eds) Basin Modelling: Practice and Progress. Geological Society, London, Special Publications 141, 15-43.


124

Gluyas, J.G. and Swarbrick, R.E. (2004) Petroleum geoscience. Blackwell Science Ltd., Oxford, United Kingdom, 359pp. Graton, L.C. and Fraser, H.J. (1935) Systematic packing of spheres – with particular relation to porosity and permeability. Journal of Geology 43, 785-909. Griffiths, J.C. (1967) Scientific method in analysis of sediments. McGraw-Hill, Inc., United States of America, 508pp. Guidish, T.M., Kendall, C.G.St.C., Lerche, I., Toth, D.J. and Yarzab, R.F. (1985) Basin evaluation using burial history calculations: an overview. The American Association of Petroleum Geologists Bulletin 69, 92-105. Ham, H.H. (1966) New charts help estimate formation pressures. Oil and Gas Journal 64, 58-63. Hecht, C.A. (2000) Appolonian packing and fractal shape of grains improving geomechanical properties in engineering geology. Pure and Applied Geophysics 157, 487- 504. Hecht, C.A. (2004) Geomechanical models for clastic grain packing. Pure and Applied Physics 161, 331-349. Heckel, R.W. (1961) Density-pressure relationships in powder compaction. Transactions of the Metallurgical Society of AIME 221, 671-675. Hedberg, H.D. (1936) Gravitation compaction of clays and shales. American Journal of Science 231, 241-287. Hosoi, H. (1963a) First migration of petroleum in Akita and Yamagata Prefectures. Japanese Association of Mineralogists, Petrologists, and Economic Geologists 49, 43-55. Hosoi, H. (1963b) First migration of petroleum in Akita and Yamagata Prefectures. Japanese Association of Mineralogists, Petrologists, and Economic Geologists 49, 101-114. Houseknecht, D.W. (1984) Influence of grain size and temperature on intergranular pressure solution, quartz cementation and porosity in a quartzose sandstone. Journal of Sedimentary Petrology 54, 348-361.

References

125

Houseknecht, D.W. (1987) Assessing the relative importance of compaction processes and cementation to reduction of porosity in sandstones. The American Association of Petroleum Geologists Bulletin 71, 633-642. Jeschar, R., Pötke, W., Petersen, V. and Polthier, K. (1975), In: Standish, N. (ed), Blast Furnace Aerodynamics, Aust. I.M.M. Press, Wollongong, New South Wales, 136-147. Josselin de Jong, G. de, and Verruijt, A. (1969) Etude photo-élastique d’un empilement de disques. Cah. Grpe fr. Etud. Rhéol. 2, 73-86. Kawakita, K. and Ludde, K.-H. (1970-1971) Some considerations on powder compression equations. Powder Technology 4, 61-68. Koltermann, C.E. and Gorelick, S.M. (1995) Fractional packing model for hydraulic conductivity derived from sediment mixtures. Water Resources Research 31, 3283-3297. Lander, R.H. and Walderhaug, O. (1999) Predicting porosity through simulating sandstone compaction and quartz cementation. The American Association of Petroleum Geologists Bulletin 83, 433-449. Loucks, R.G., Dodge, M.M. and Galloway, W.E. (1979) Sandstone consolidation analysis to delineate areas of high-quality reservoirs suitable for production of geopressured geothermal energy along the Texas Gulf Coast. Bureau of Economic Geology, University of Texas, 98pp. Lundegard, P.G. (1992) Sandstone porosity loss – a “big picture” view of the importance of compaction. Journal of Sedimentary Petrology 62, 250-260. Macrae, J.C. and Gray, W.A. (1961) Significance of the properties of materials in the packing of real spherical particles. British Journal of Applied Physics 12, 164-172. Magara, K. (1968) Compaction and migration of fluids in Miocene mudstone, Nagaoka Plain, Japan. The American Association of Petroleum Geologists Bulletin 52, 2466-2501. Marion, D., Nur, A., Yin, H. and Han, D. (1992) Compressional velocity and porosity in sand-clay mixtures. Geophysics 57, 554-563. McBride, E.F., Diggs, T.N. and Wilson, J.C. (1991) Compaction of Wilcox and Carrizo Sandstones (Paleocene-Eocene) to 4420 m, Texas Gulf Coast. Journal of Sedimentary Petrology 61, 73-85.


126

McGeary, R.K. (1961) Mechanical packing of spherical particles. Journal of the American Ceramic Society 44, 513-522. Meade, R.H. (1966) Factors influencing the early stages of compaction of clays and sands – review. Journal of Sedimentary Petrology 36, 1085-1101. Milliken, K.L. (2001) Diagenetic heterogeneity in sandstone at the outcrop scale, Breathitt Formation (Pennsylvanian), eastern Kentucky. The American Association of Petroleum Geologists Bulletin 85, 795-815. Nolan, G.T. and Kavanagh, P.E. (1992) Computer simulation of random packing of hard spheres. Powder Technology 72, 149-155. Nolan, G.T. and Kavanagh, P.E. (1993) Computer simulation of random packings of spheres with log-normal distributions. Powder Technology 76, 309-316. Palmer, S.N. and Barton, M.E. (1987) Porosity reduction, microfabric and resultant lithification in UK uncemented sands. In: Marshall, J.D. (ed) Diagenesis of Sedimentary Sequences. Geological Society Special Publication 36, 29-40. Panda, M.N. & Lake, L.W. (1994) Estimation of single-phase permeability from parameters of particle-size distribution. The American Association of Petroleum Geologists Bulletin, 78, 1028-1039. Paola, C. (2000) Quantitative models of sedimentary basin filling. Sedimentology 47, 121-178. Paola, C., Wiele, S.M. and Reinhart, M.A. (1989) Upper-regime parallel lamination as the result of turbulent sediment transport and low-amplitude bed forms. Sedimentology 36, 47-59. Pate, C.R. (1989) Assessing the relative importance of compaction processes and cementation to reduction of porosity in sandstones: Discussion. The American Association of Petroleum Geologists Bulletin 73, 1270-1273. Perrier, R. and Quiblier, J. (1974) Thickness changes in sedimentary layers during compaction history; methods for quantitative evaluation. The American Association of Petroleum Geologists Bulletin 58, 507-520. Pettijohn, F.J., Potter, P.E. and Siever, R. (1972) Sand and sandstone. New York, Springer-Verlag, 618pp.

References

127

Pilotti, M. (1998) Generation of realistic porous media by grains sedimentation. Transport in Porous Media 33, 257-278. Pilotti, M. (2000) Reconstruction of clastic porous media. Transport in Porous Media 41, 359-364. Pittman, E.D. and Larese, R.E. (1991) Compaction of lithic sands: Experimental results and applications. The American Association of Petroleum Geologists Bulletin 75, 1279-1299. Powers, M.C. (1953) A new roundness scale for sedimentary particles. Journal of Sedimentary Petrology 23, 117-119. Pramanik, A.G., Singh, V., Vig, R., Srivastava, A.K. and Tiwary, D.N. (2004) Estimation of effective porosity using geostatistics and multiattribute transforms: A case study. Geophysics, 69, 352-372 Proshlyakov, B.K. (1960) Reservoir properties of rocks as functions of their depth and lithology. Geologiia Nefti I Gaza 12, 24-29. Pryor, W.A. (1973) Permeability-porosity patterns and variations in some Holocene sand bodies. The American Association of Petroleum Geologists Bulletin 57, 162-189. Revil, A., Grauls, D. and Brévart, O. (2002) Mechanical compaction of sand/clay mixtures. Journal of Geophysical Research – Solid Earth 107 (B11), 2293. Ridgway, K. and Tarbuck, K.J. (1968) Particulate mixture bulk densities. Chemical and Process Engineering 49, 103- . Rittenhouse, G. (1943) A visual method of estimating two-dimensional sphericity. Journal of Sedimentary Petrology 13, 79-81. Rittenhouse, G. (1971) Pore-space reduction by solution and cementation. The American Association of Petroleum Geologists Bulletin 55, 80-91. Roberts, J.N. and Schwartz, L.M. (1985) Grain consolidation and electrical conductivity in porous media. Physical Review B 31, 5990-5997. Ronen, S., Schultz, P. S., Hattori, M. and Corbett, C. (1994) Seismic guided estimation of log properties, Part 2: Using artificial neural networks for nonlinear attribute calibration. The Leading Edge, 13, 674-678.


128

Rouault, Y. (1999) The construction of an appolonian packing. Powder Technology 102, 274-280. Royden, L. and Keen, C.E. (1980) Rifting process and thermal evolution of the continental margin of Eastern Canada determined from subsidence curves. Earth and Planetary Science Letters 51, 343-361. Scherer, M. (1987) Parameters influencing porosity in sandstones: A model for sandstone porosity prediction. The American Association of Petroleum Geologists Bulletin 71, 485-491. Schmoker, J.W. (1984) Empirical relation between carbonate porosity and thermal maturity - an approach to regional porosity prediction. The American Association of Petroleum Geologists Bulletin 68, 1697-1703. Schultz, P.S., Ronen, S., Hattori, M. and Corbett, C. (1994a) Seismic guided estimation of log properties, Part 1: A data-driven interpretation technology. The Leading Edge, 13, 305-315. Schultz, P.S., Ronen, S., Hattori, M., Mantran, P. and Corbett, C. (1994b) Seismic guided estimation of log properties, Part 3: A controlled study. The Leading Edge, 13, 770-776. Schwartz, L.M. and Banavar, J.R. (1989) Transport properties of disordered continuum systems. Physical Review B 39, 11965-11970. Schwartz, L.M. and Kimminau, S. (1987) Analysis of electrical conduction in the grain consolidation model. Geophysics 52, 1402-1411. Sclater, J.G. and Christie, P.A.F. (1980) Continental stretching: An explanation of the post Mid-Cretaceous subsidence of the Central North Sea Basin. Journal of Geophysical Research 85, 3711-3739. Scott, G.D. (1960) Packing of equal spheres. Nature 188, 908-909. Scott, G.D. (1962) Radial distribution of the random close packing of equal spheres. Nature 194, 956-957. Scott, G.D. and Kilgour, D.M. (1969) The density of random close packing of spheres. Journal of Physics D: Applied Physics 2, 863-866. Sallenger Jr., A.H. (1979) Inverse grading and hydraulic equivalence in grain-flow deposits. Journal of Sedimentary Petrology 49, 553-562.

References

129

Selley, R.C. (1985) Ancient sedimentary environments. 3rd Edition. University Press, Cambridge, Great Britain, 317pp. Selley, R.C. (1998) Elements of petroleum geology. 2nd Edition. Academic Press, San Diego, California, 470pp. Sibley, D.F. and Blatt, H. (1976) Intergranular pressure solution and cementation of the Tuscarora orthoquartzite. Journal of Sedimentary Petrology 46, 881-896. Sohn, H.Y. and Moreland, C. (1968) The effect of particle size distribution on packing density. The Canadian Journal of Chemical Engineering 46, 162-167. Standish, N. and Borger, D.E. (1979) The porosity of particulate mixtures. Powder Technology 22, 121-125. Standish, N. and Collins, D.N. (1983) The permeability of ternary particular mixtures for laminar-flow. Powder Technology 36, 55-60. Storms, J.E.A. (2003) Event-based stratigraphic simulation of wave-dominated shallow-marine environments. Marine Geology, 199, 83-100. Tacher, L., Perrochet, P. and Parriaux, A. (1997) Generation of granular media. Transport in Porous Media 26, 99-107. Taylor, J.M. (1950) Pore space reduction in sandstones. The American Association of Petroleum Geologists Bulletin 34, 710-716. Tickell, F.G., Mechem, O.E. and McCurdy, R.C. (1933) Some studies on the porosity and permeability of rocks. Transactions of the American Institute of Mining and Metallurgic Engineering (Petroleum Division) 103, 250-260. Tory, E.M., Cochrane, N.A. and Waddell, S.R. (1968) Anisotropy in simulated random packing of equal spheres. Nature, 220, 1023. Trappe, H. and Hellmich, C. (2000) Using neural networks to predict porosity thickness from 3D seismic. First Break, 18, 377-384. Ungerer, P., Burrus, J., Doligez, B., Chénet, P.Y. and Bessis, F. (1990) Basin evaluation by integrated two-dimensional modeling of heat transfer, fluid flow, hydrocarbon generation and migration. The American Association of Petroleum Geologists Bulletin 74, 309-335.


130

Uri, L., Dysthe, D.K. and Feder, J. (2004a) Complex behaviour and structure of ductile granular material. Geophysical Research Abstracts 6, 06356. Uri, L., Alberts, L., Dysthe, D.K. and Feder, J. (2004b) Structure of compacting ductile grain ensembles. Geophysical Research Abstracts 6, 06379. Visscher, W.M. and Bolsterli, M. (1972) Random packing of equal and unequal spheres in two and three dimensions. Nature 239, 504-507. Waal, J.A. de (1986) On the rate type compaction behaviour of sandstone reservoir rock. Ph.D. Thesis, Technische Hogeschool Delft, Delft, 166pp. Walderhaug, O. (1994) Precipitation rates for quartz cement in sandstones determined by fluid-inclusion microthermometry and temperature-history modeling. Journal of Sedimentary Research A64, 324-333. Walderhaug, O. (1996) Kinetic modeling of quartz cementation and porosity loss in deeply buried sandstone reservoirs. The American Association of Petroleum Geologists Bulletin 80, 731-745. Weller, J.M. (1959) Compaction of sediments. The American Association of Petroleum Geologists Bulletin 43, 273-310. Weltje, G.J. and Eynatten, H. von (2004) Quantitative provenance analysis of sediments: review and outlook. Sedimentary Geology 171, 1-11. Weltje, G.J. and Prins, M.A. (2003) Muddled or mixed? Inferring palaeoclimate from size distributions of deep-sea clastics. Sedimentary Geology 162, 39-62. Westman, A.E.R. and Hugill, H.R. (1930) The packing of particles. Journal of the American Ceramic Society 13, 767-779. Williams, S.R. and Philipse, A.P. (2003) Random packings of spheres and spherocylinders simulated by mechanical contraction. Physical Review E, 67, 051301. Wilson, J.C. and McBride, E.F. (1988) Compaction and porosity evolution of Pliocene Sandstone, Ventura Basin, California. The American Association of Petroleum Geologists Bulletin 72, 664-681.

References

131

Wilson, M.D. and Stanton, P.T. (1994) Diagenetic mechanisms of porosity and permeability reduction and enhancement. In: Wilson, M.D. (ed) Reservoir Quality Assessment and Prediction in Clastic Rocks. SEPM Short Course 30, 59-119. Wolf, D.J., Withers, K.D. and Burman, M.D. (1994) Integration of well and seismic data using geostatistics. AAPG Computer Applications in Geology, 177-199. Wood, J.R. (1994) Geochemical models. In: Wilson, M.D. (ed) Reservoir Quality Assessment and Prediction in Clastic Rocks. SEPM Short Course 30, 23-40. Worden, R.H., Mayall, M. and Evans, I.J. (2000) The effect of ductile-lithic sand grains and quartz cement on porosity and permeability in Oligocene and Lower Miocene clastics, South China Sea: Prediction of reservoir quality. The American Association of Petroleum Geologists Bulletin 84, 345-359. Yao, J., Thovert, J.-F., Adler, P.-M., Tsakiroglou, C.D., Burganos, V.N., Payatakes, A.C., Moulu, J.-C. and Kalaydjian, F. (1997) Characterization, reconstruction and transport properties of Vosges Sandstones. Revue de l’Institut Français du Pétrole 52, 3-21. Yu, A.B. and Standish, N (1987) Porosity calculations of multi-component mixtures of spherical particles. Powder Technnology 52, 233-241.


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Curriculum Vitae Luc Alberts was born in Heerenveen, the Netherlands on November 10, 1974, but grew up in Borne. In 1993 he graduated from secondary school, Twickel College in Hengelo. He studied Mining and Petroleum Engineering at the Delft University of Technology. He graduated in 2000 on the characterisation of the Rijn Field, an oil field in the West Netherlands Basin, The Netherlands. For this study he combined seismic data and well data with novel modelling techniques to unravel the reservoir architecture of the field. After his study he prolonged his stay in Delft and started a PhD at the Department of Applied Earth Sciences of the Faculty of Civil Engineering and Geosciences, where he developed the numerical simulation model Rampage. In 2003 he visited the Norwegian Centre of Excellence “Physics of Geological Processes” for three months to test his simulation results with experimental data. Besides his PhD work, he was also involved in the organisation of the Aarderijkskunde Olympiade 2003 in Delft, a national geography contest for high school students, for which he organised the field exercise. Currently he is working for Shell International Exploration & Production B.V. in Rijswijk.


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List of publications Alberts, L.J.H. and Weltje, G.J. (in prep.) A numerical micro-scale model for simulation of grain rearrangement. Uri, L., Alberts, L., Feder, J. and Dysthe, D.K. (in prep.) Structure of packings of plastically deforming grains. Uri, L., Alberts, L.J.H., Dysthe, D.K. and Feder, J. (2004) Structure of compacting ductile grain ensembles. EGU General Assembly, Nice, France. Geophysical Research Abstracts 6. Alberts, L.J.H., Geel, C.R. and Klasen, J.J. (2003) Reservoir characterisation using process-response simulations: the Lower Cretaceous Rijn Field, West Netherlands Basin. Netherlands Journal of Geosciences 82: 313-324 Alberts, L.J.H. (2003) Porosity prediction through micro-scale simulation of grain rearrangement. Symposium analogue and numerical forward modelling of sedimentary systems, Utrecht, The Netherlands.


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Alberts, L.J.H. and Weltje, G.J. (2002) A dynamic sphere-packing model for prediction of compaction behaviour of sand-sized sediments. 6e Nederlands Aardwetenschappelijk Congres, Veldhoven, The Netherlands. Alberts, L.J.H. and Weltje, G.J. (2001) Predicting initial porosity as a function of grain-size distribution from simulations of random sphere packs. Extended abstract, 6th annual meeting IAMG, Cancun, Mexico. Alberts, L.J.H. and Geel, C.R. (2001) Reservoir architecture of a barrier-type oil field explained by process-response modelling. Extended abstract, 63rd EAGE Conference, Amsterdam, The Netherlands. Alberts, L.J.H. and Geel, C.R. (2000) Towards integrating well data, seismics and process-response simulations in a barrier-type reservoir: The Rijn Field in the Lower Cretaceous of the West-Netherlands Basin. 5e Nederlands Aardwetenschappelijk Congres, Verldhoven, The Netherlands.

initial porosity of random packing - tu delft

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