Initial Porosity of Random
Packing
Computer simulation of grain rearrangement
Initial Porosity of Random Packing
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.
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.
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!
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.
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 ... 7Porosity
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
2.4 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
5.5.1 Sorting 5.5.2 Skewness 5.6 Conclusions 6 Synthesis ... 113 References ... 119 Curriculum Vitae ... 133 List of Publications ... 135
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.
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
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.
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
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
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.
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;
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.
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).
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
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
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
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
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 s
V =V + (2.1) V
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
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
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.
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
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
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
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
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
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.
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.
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
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,
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.
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.
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 az
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
bz
e
φ 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 c e σ φ φ − = (2.4)
This equation is the solution of the differential equation dd c
φ
σ = − , which φ
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 e z e k ρ ρ ρ ρ φ = − − − − + (2.5)
where k1= −(1 φ(0)) / (0)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
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 1 V V abP C V 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 K dP φ φ − = (2.7)
which on integration gives
0
1 1
ln ln KP
φ = φ + (2.8)
whereφ0 is the porosity at P = 0 and K is a constant, which Heckel empirically
found to equal K =1 (3 )σ0 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: