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An investigation into preserving spatially-distinct pore systemsin multi-component rocks using a fossiliferous limestoneexample

Citation for published version:Jiang, Z, Couples, GD, Lewis, MH & Mangione, A 2018, 'An investigation into preserving spatially-distinctpore systems in multi-component rocks using a fossiliferous limestone example', Computers andGeosciences, vol. 116, pp. 1–11. https://doi.org/10.1016/j.cageo.2018.04.004

Digital Object Identifier (DOI):10.1016/j.cageo.2018.04.004

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Accepted Manuscript

An investigation into preserving spatially-distinct pore systems in multi-componentrocks using a fossiliferous limestone example

Zeyun Jiang, Gary D. Couples, Helen Lewis, Alessandro Mangione

PII: S0098-3004(17)30811-7

DOI: 10.1016/j.cageo.2018.04.004

Reference: CAGEO 4116

To appear in: Computers and Geosciences

Received Date: 28 July 2017

Revised Date: 4 April 2018

Accepted Date: 6 April 2018

Please cite this article as: Jiang, Z., Couples, G.D., Lewis, H., Mangione, A., An investigation intopreserving spatially-distinct pore systems in multi-component rocks using a fossiliferous limestoneexample, Computers and Geosciences (2018), doi: 10.1016/j.cageo.2018.04.004.

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An Investigation into Preserving Spatially-distinct Pore Systems in Multi-Component Rocks 1

using a Fossiliferous Limestone Example 2

3

Zeyun Jianga,b,*, Gary D. Couplesa, Helen Lewisa, Alessandro Mangionea1 4

a Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh, EH144AS, UK 5 b School of sciences, Southwest Petroleum University, Chengdu, 610500, China 6 *

Corresponding author, ZJ6@hw.ac.uk, Phone: +44 (0) 131 451 3808, Fax: +44 (0)131 451 3127 7

8

Abstract 9

Limestones containing abundant disc-shaped fossil Nummulites can form significant hydrocarbon 10

reservoirs but they have a distinctly heterogeneous distribution of pore shapes, sizes and 11

connectivities, which make it particularly difficult to calculate petrophysical properties and 12

consequent flow outcomes. The severity of the problem rests on the wide length-scale range from the 13

millimetre scale of the fossil’s pore space to the micron scale of rock matrix pores. This work 14

develops a technique to incorporate multi-scale void systems into a pore network, which is used to 15

calculate the petrophysical properties for subsequent flow simulations at different stages in the 16

limestone’s petrophysical evolution. While rock pore size, shape and connectivity can be determined, 17

with varying levels of fidelity, using techniques such as X-ray computed tomography (CT) or 18

scanning electron microscopy (SEM), this work represents a more challenging class where the rock of 19

interest is insufficiently sampled or, as here, has been overprinted by extensive chemical diagenesis. 20

The main challenge is integrating multi-scale void structures derived from both SEM and CT images, 21

into a single model or a pore-scale network while still honouring the nature of the connections across 22

these length scales. Pore network flow simulations are used to illustrate the technique but of equal 23

importance, to demonstrate how supportable earlier-stage petrophysical property distributions can be 24

used to assess the viability of several potential geological event sequences. The results of our flow 25

simulations on generated models highlight the requirement for correct determination of the dominant 26

pore scales (one plus of nm, µm, mm, cm), the spatial correlation and the cross-scale connections. 27

Key words: Carbonates; Nummulites; Multi-scale pore network; Two-phase flow 28

29

1. Introduction 30

Over the last decade, a considerable body of work has focused on using pore-scale images of rocks to 31

derive quantitative estimates of emergent physical properties, such as those associated with fluid flow 32

(see Blunt et al. 2013 for a comprehensive review). The vast majority of approaches use three-33

dimensional (3D) digital rock models, at a small scale, derived either through tomographic methods 34

(e.g. Vinegar et al. 1987, Flannery et al. 1987; Ketcham and Carlson, 2001; Arns et al. 2005), or as 35

reconstructed from 2D images (e.g. Øren and Bakke, 2003; Okabe et al. 2004; Wu et al. 2006; 36

Kopf et al. 2007; Chen and Wang, 2010; Tahmasebi et al. 2012). The tomographic grey-level images 37

are further segmented into two phases, solid components and the pore space, although in some studies, 38

fluid phase distributions are also determined (Pak et al. 2015). These provide 3D solid-pore (binary) 39

1 Authorship statement:

Dr Zeyun Jiang, PhD, wrote the manuscript, clarified the ideas, developed the methodology, and carried out the analysis on

pore network simulations;

Prof Gary D. Couples, PhD, pioneered at the ideas and the framework with many in-depth interpretations of numerical

results;

Dr Helen Lewis, PhD, conceived the ideas, steered the development and was regularly instrumental in guidance;

Dr Alessandro Mangione, PhD, provided Nummulites templates and basic geological knowledge plus quantified

characteristics of Nummulitic rocks.

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calculating fluid flow through this solid-pore structure (see Bultreys et al. 2016 for a broad review) 41

operate directly on the pore space model (e.g. Chen and Doolen, 1998; Kang et al. 2006; Ma et al. 42

2010; Raeini et al. 2012), while others simplify the pore space into a pore network model (e.g. 43

Lindquist et al. 2000; Jiang et al. 2007; Dong and Blunt, 2009). 44

Techniques to characterise, replicate or quantify the pore (void) distribution mainly focus on using an 45

actual rock specimen as a template for deriving a digital rock model. Some combination of X-ray CT, 46

SEM, and optical petrographic data is typically utilised to generate 3D models from which the pore-47

space models are extracted (e.g. Prodanovic et al. 2014; Hebert et al. 2015; Soulaine et al. 2016; Lin 48

et al. 2016). However, for some rocks, the pore-size range is so large that a single image cannot 49

capture the whole pore distribution. Consequently, there is an insufficient number, abundance and 50

frequency of fractal properties (e.g. pore size, pore tortuosity) present across this large range of length 51

scales, so even the fractal models (e.g. Xu 2015) are not useful in this case. Different sampled 52

volumes, and potentially different sampling techniques can be used to create digital rock models at 53

each scale, but the authors do not know of any published approach to combine those models into a 54

single unified model that is suitable at the highest resolution required by the size of the smallest pores 55

resolved, and from which the pore system could be extracted and used in the normal way. 56

Such separation of imaging scales, and then the extraction of pore systems at each required scale, has 57

been mainly used in two ways worldwide: pore network integration and pore model generation, 58

corresponding to the two workflows shown in Fig. 1. The multiscale pore network integrated using 59

the approach of Jiang et al. (2013) is capable of honouring the pore structure spanning multiple orders 60

of magnitudes, e.g. from µm to mm, and so can predict petrophysical properties (e.g. porosity, 61

permeability, capillary pressure) accurately. Aiming to reduce the two-scale network size significantly, 62

Bultreys et al. (2015) introduce virtual links that are assigned with pre-computed properties to 63

represent local microporous regions, into a macropore network. To tackle spatial distribution of 64

micropores, meanwhile, a deterministic approach is proposed in Prodanovic et al. (2014) and 65

Mehmani et al. (2015) by populating the microporosity with a pore network shrunk by a given factor, 66

which can be estimated by experiments such as Mercury Injection Capillary Pressure (MICP). Both 67

validation and good results are reported by these authors, however the multiscale network technique is 68

unsuited to materials where the different-sized pore systems occur in distinct spatial regions, and 69

where preservation of that spatial arrangement is considered to be important. This paper addresses that 70

situation, describing a new pore model generation method (as illustrated in Fig. 1 as operation (3) that 71

combines original models (a) and (d) into model (f)) that allows the characteristics of multi-scale pore 72

systems to be retained and to be represented in appropriate spatial arrangements. We illustrate the first 73

version of this method by using it to assess the flow behaviours of a dolomitised, fossiliferous 74

limestone in which the spatially-distinct fossils have partially- or fully-preserved voids that occur in a 75

very precise arrangement, being remnants of the living chambers of the animal, that can be orders of 76

magnitude larger than the sizes of pores in the surrounding rock matrix. 77

In the limestone sample used for the case study, two distinct pore systems exist. One is the residual 78

void space of the fossil tests of Nummulites (Fig. 2), consisting of chambers and connections that are 79

distributed in a spiral pattern inside the bounding surface, with individual fossils varying from mm to 80

cm in size. The other pore space is the inter-crystalline pores in the surrounding fine-grained rock 81

matrix, with pores ranging from 1-10 microns in diameter. These matrix pores have a spatially 82

random distribution, which is very different from the ordered voids of the Nummulites spirals in terms 83

of length-scale, shape and spatial location. Nummulites can have chambers up to 10s of millimetres in 84

diameter, significantly larger than the average matrix pore radii of 10s microns. With current 85

topographies, it is not possible to acquire images that are able to simultaneously describe both 86

Nummulites void structures and matrix pore system, because this would require the creation of a 87

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2001; Arns et al. 2005, Garing et al. 2014; Lin et al. 2016; Hebert et al. 2015). 89

Therefore, it is necessary here to introduce a method so that different pore-space components and 90

multi-scale pore structures can all be preserved in the resulting model with correct spatial 91

arrangements. This method will produce realistic models for extremely complex pore structures, 92

allowing us to perform single- and two-phase flow simulations effectively. However, is such a 93

complex model necessary both here and in the general case? In this paper, we show that the 94

preservation of spatially-distinct pore systems does make a difference in the calculated flow properties, 95

so the answer is that there is a big value in progressing towards the additional complexity when the 96

multi-scale pore space contains spatially distinct regions. 97

In this paper, we present a new methodology that models the pore systems that have two or more 98

components, each with very different pore sizes located in different regions, identified through: (1) 99

extracting templates of pore-solid structures from images at differing length scales, potentially using 100

different tools (e.g. SEM, CT), and all acquired from potentially different sub-samples (e.g. plugs, 101

sub-plugs or fragments) of a single rock, with the aim being to capture the real pore morphology in 102

each identified component; (2) performing transformations, if required, on the extracted templates; (3) 103

enhancing our pore network extraction so as to allow a better voxel representation of a composite pore 104

system. A Nummulitic limestone rock is selected because it matches the requirements listed above for 105

interception purposes. In this case study we demonstrate here the fossils’ images were needed in their 106

pre-dolomitisation state but the available images, from the diagenetically altered rock, showed too 107

many components of diagenesis to be used unadjusted. So, the shapes and sizes of the pore system 108

seen in the tomographic images were used to control the generation of fossil representations using the 109

observed geometric characteristics, and were subsequently checked against published descriptions of 110

similar Nummulites species from the fossil record. The methodology certainly has potential 111

applications for other multi-component and complex rocks whether the pore system components are 112

described from existing samples or their analogues or, as here, by a constructed complex geometric 113

shape that is informed by component images but is not generated directly from them. As implied 114

above, the method is designed to work equally well for representing the earlier stages of now altered 115

rocks or for determining how changes in, for example the distribution, density or size of a component 116

affects its petrophysical properties. The methodology is also designed to test if and where the spatial 117

distribution into separated pore networks, here the Nummulites fossils and the rock matrix matter. 118

In Section 2 the main points of the geological system of the case-study are provided together with the 119

use of the new methodology. Section 3 outlines the structural and network modelling methodology for 120

the Nummulitic limestone. Section 4 details the impact of pore features and distributions in terms of 121

different parameters, such as Nummulites density, size, cementation, corrosion and rock matrix, on 122

single- and two-phase flow. 123

2. Geological Setting 124

The dolomitised Nummulitic limestone is of Eocene age and now forms a hydrocarbon reservoir 125

approximately 2km deep off the north shore of North Africa. Originally deposited in shallow water 126

just off a series of nearshore islands (Beavington-Penny et al., 2008), it originally consisted of a 127

calcium carbonate mud containing abundant Nummulites fossils, which are approximately oblate 128

resembling a disc, with diameters varying from a few mm to a few cm. The deposited Nummulites 129

often contained partial mud infill though some remained empty at this stage. During burial the 130

calcium carbonate mud, a micrite, was very prone to crystal form change resulting in a general 131

alteration from a needle-dominant form to a more equant form. Crystal and pore size is believed to be 132

in the order of a few microns to tens of microns. During burial, at approximately 1500m depth and at 133

approximately 30 m.y. before present (Mangione, 2016; Mangione et al. 2017) the micrite was 134

replaced by dolomite by addition of calcium ions to the molecular structure. This burial 135

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necessarily any calcium minerals contained within them, but core samples of this rock strongly 137

suggest that dolomitisation of the matrix was complete in this reservoir. Mangione (2016) also 138

identifies and illustrates the sequence of less significant diagenetic events, such as calcite cementation, 139

which add complexity to the geological evolution but do not add any fundamentally different 140

elements to the problem. Mangione et al. (2017), used the physical and petrophysical properties of the 141

Nummultic limestone immediately before dolomitisation created using this method, together with the 142

basin evolution derived by basin modelling, to simulate the dolomitisation process and indicate the 143

shape, extent and drivers of the dolomitisation. In this work the emphasis is on the methodology to (1) 144

generate suitable digital models of pore systems where there is a large range of pore network 145

characteristics and the networks have distinct identities, and particularly test if the spatial identities of 146

the different components need to be preserved, and (2) to incorporate network alteration techniques to 147

accommodate past, or potentially future, alterations in the pore systems of these components. Section 148

4 addresses the consequences for petrophysical properties of the various matrix and fossil types. 149

The methodology developed was designed to accommodate the very different pore systems, in both 150

pore size groups, here varying from microns to millimetres, and in distinct spatial regions, here 151

represented by the fossils and the matrix. The pore system created must also be able to accommodate 152

both very structured and unstructured pore arrangements. The methodology also needs to be capable 153

of using images of the pore-rock system derived from 3D representations such as from X-ray 154

tomography or SEM as well as to develop equation-driven representations of regular geometric shapes, 155

together with techniques to modify images to represent inferred prior or future diagenetic states. 156

3. Methodology 157

The methodology is composed of the following steps: 158

1) Choosing rock matrix model (see Fig. 1(a)); 159

2) Obtaining templates (see Fig. 1(d) and below) of spatially-distinct pore systems; 160

3) Integrating (see Fig. (3)) rock matrix and templates into a pore model; 161

4) Extracting (see Fig. 1(4)) a pore network from the resultant model; 162

5) Conducting (see Fig. 1(7)) network simulation of single- and two-phase flow. 163

In the model generation, the actual structural characteristics of a rock matrix might be observed using 164

2D images such as BSE images or 3D X-ray tomographic reconstructions, or by representing for 165

example now overprinted material with selected analogue images. To match the morphological 166

aspects of rock matrix, we can stochastically reconstruct a 3D model (Fig. 3(a)), or choose an 167

appropriate image (Fig. 1(a)) from an existing database of CT images. So, the chosen rock matrix 168

model is equivalent to the targeting structures in terms of pore geometry and topology. The details are 169

discussed below. Where relevant we use the dolomitised Nummulitic limestone as the example: study 170

petrophysical results are provided in Section 4. 171

For a spatially-distinct pore system, i.e. a pore structure that differs significantly from its background 172

in both spatial distribution and in length scales, the detailed digital model called a template, can be 173

extracted (see Fig. 1(d)) from a CT image that contains the component, here a Nummulites, or be 174

created (see Fig. 3(b)) mathematically by, for a Nummulites, the Archimedean spiral equation 175

The next step is to superpose the Nummulites onto a rock matrix, replacing the voxels of the chosen 176

matrix model with those of the superposed model, to generate digital rock models (Fig. 1(f)). After 177

extracting the pore network for the resultant model, our workflow culminates in the calculation of 178

petrophysical properties (e.g. Fig. 1(i,j)). 179

3.1 Rock Matrix 180

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In the chosen Nummulitic limestone example the rock matrix is observed in thin section and 181

BSE to be micro-porous and somewhat heterogeneous. The matrix pore size ranges from 182

approximately one to several hundred µm (Mangione et al. 2014). That multi-scale matrix 183

could be represented by a multi-scale pore network (see Fig. 1(h)), but for the present 184

purpose, we just need a voxellated model to represent the whole rock. For simplicity, we 185

chose to use a homogeneous matrix model and we do not attempt to represent exactly the 186

details of the real matrix material. 187

3.2 Nummulites Template 188

Nummulites have a test (or shell) shape that varies from fairly flat to sub-globular, with the peripheral 189

edge ranging from sharp to rounded (Racey 1992). Nummulites are composed of plani-spirally coiled 190

chambers separated by septa (walls), and they are bilaterally symmetrical about the equatorial plane 191

(Racey 1992, 2001). The initial, central chamber is spherical and the second chamber is generally 192

kidney shaped. The chambers and separating walls are covered by a shell with one or a few small 193

apertures linking to the surrounding rock matrix. The shell material is not solid (as seen in Fig. 4), but 194

possesses nano-tubules. 195

Nummulites preserved in rocks have generally been altered physically and chemically during 196

geological time, and their current structure can differ from the initial. Therefore, the analysis 197

workflow described here includes options to assess the variations in calculated properties related both 198

to differences between individual Nummulites, and to differences in geo-historical events. A precise 199

way to obtain a Nummulites model is to extract it directly from an X-ray CT image. Fig. 5(a) shows an 200

image slice containing Nummulites, and Fig. 5(b, c, d) shows three extracted Nummulites. 201

Nummulites extraction from an image is as follows (see Fig. 6): 202

(1) Determine Nummulites boundary; 203

(2) Mark out boundaries of individual Nummulites; 204

(3) Identify different components of Nummulites. 205

Unfortunately, the very small-scale details of the fossils’ preserved pore systems are not clearly 206

captured via the X-ray tomography because of their lower resolutions and the low grey-level contrast 207

between chambers and nearby walls. Thus, the focus must be to choose an appropriate segmentation 208

algorithm that is sufficiently good (at least, visually) to distinguish the obvious voids and the (mostly) 209

solid parts of Nummulites. Through intensive testing, we identify three filters with ImageJ 210

(https://imagej.net) including the mean, median, minimum/maximum filters, which prove suitable for 211

extracting individual Nummulites that contains four components (Fig. 6) through the above steps. 212

The potentially-open boundary areas (see Fig. 6) could play a major role in providing pore 213

connectivity between the internal void space and the surrounding rock matrix; that is, the matrix pores 214

outside of Nummulites can be connected to the interior voids through the open areas, which are places 215

where the initial boundary of the test has been corroded or otherwise diagenetically altered. That is 216

the open boundary areas could potentially be crucial for fluid/gas flow through, in or out of a 217

Nummulites. For a single Nummulites, the chambers and the open boundary areas form the void space, 218

and the closed boundary areas and chamber walls constitute the solid parts. The nano-tubules shown 219

in Fig. 4 that initially passed through the chamber walls may or may not be preserved after burial and 220

diagenesis. Here, they are not considered because they are far too small to include in the eventual 221

composite model whose voxel resolution is far too coarse to capture such small features. They are a 222

topic of interest for subsequent developments (e.g. Fig. 1(h)). 223

Having a limited number of extracted Nummulites templates, we need to further transform them to 224

capture more diagenetic effects (for instance of corrosion, cementation or precipitation) noted from 225

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Nummulites being available a feasible way to make Nummulites models (Fig. 3(b)) is to define the 227

basic geometry (Fig. 2 and Fig. 7) by the Archimedean spiral equations: 228

�� = 2�� = ���� = �� � . (1) 229

The void space can be modelled by a sequence of chambers along an Archimedean spiral curve (see 230

Fig. 2). The intersection segments on the spiral curve, which could be considered as the central lines 231

of the walls separating the chambers, can be given with a fixed distance (e.g. 2π2) by a unit vector 232

(−�� � + ��, ��� + � �). We fill in each chamber by a series of balls of different radii in 233

decreasing order away from each chamber’s centre, as shown in Fig. 7(a), and the corresponding 234

model is generated by adding a certain degree of randomness to choosing ball radii and locations. We 235

notice that the void space displays a changing trend with smaller touching and tighter chambers in the 236

spiral centre, but bigger and scattered chambers outwards. Fig. 7(b) shows the chamber void space, 237

and the central spiral curves highlight and correlate the chamber positions. 238

3.3 Nummulitic Limestone Model 239

With a set of Nummulites templates of varying sizes and diagenetic states, and two rock matrix 240

models, the templates can be superposed onto the physical space previously occupied by rock matrix 241

voxels, thus forming a Nummulitic limestone model. The Nummulites used can be as extracted from 242

the images, or resulting from a series of transformations. Possible transformations consist of 243

alterations in size, in orientation and in void space and can be accompanied by changes in the overall 244

volume or density, and by adjusting the rock matrix volume. The corresponding parameters used in 245

building the composite model are: density, size, orientation (angle), void enlarging/shrinking 246

coefficient, boundary open/close switch, volumetric and pore enlarging/shrinking coefficient for the 247

rock matrix, plus intersection switch. These are explained below. 248

Density 249

Density is defined as the number of Nummulites per volume, and is used to adjust the Nummulites 250

fraction in a model. An approximation for density can be determined from thin section images by 251

manually identifying and counting Nummulites (Fig. 6(a)). 252

Size 253

The size of a Nummulites is defined by its three dimensions, e.g. 154×149×43 voxels shown in Fig. 254

5(b). An example Nummulites might need to be resized as a whole in order to generate Nummulites 255

with the required features. For example, for a rock matrix model of 2 µm per voxel, the Nummulites 256

with a resolution of 8.05 µm needs to be enlarged 8.05/2 ≈ 4.025 (i.e. the size coefficient) times. 257

These magnified Nummulites are then superposed on the rock matrix to obtain a model with a uniform 258

resolution of 2 µm. To reduce the deviation, we first enlarge or shrink a Nummulites by rounding 259

down to an integer (e.g. 4) from the floating size coefficient (e.g. 4.025) and then interpolate a number 260

of rows, columns or layers equidistantly, in X-, Y- or Z-direction. Alternately, a mathematically-261

derived model (Fig. 7) of Nummulites can be generated at any chosen resolution, using the 262

Archimedean spiral approach. 263

Orientation 264

Nummulites can present a certain pattern of orientations in rocks such as being dominantly horizontal, 265

or at some angle to the horizontal. To specify orientation, we sequentially rotate the templates (Fig. 266

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5(b, c, d)) around X-, Y- and Z-axis at angles of θx, θy, and θz sequentially, and re-cast that model into 267

the target voxel arrangement. 268

Void Enlarging/Shrinking Coefficient 269

The corrosion of Nummulites could be considered as the chamber walls (see Fig. 6) being partly or 270

completely dissolved, becoming voids. In contrast, cementation can infill a fraction of the chamber 271

spaces with minerals precipitated during geological history. Corrosion adds volume to the void space 272

while cementation reduces it. To implement these operations, a distance map (Fig. 8) is introduced to 273

define the exact part to be converted from void to solid, or vice versa. The value for each void voxel is 274

defined as the distance to its closest solid voxel, but with zero values for every solid voxel. Setting a 275

threshold ξ, all void voxels of distance ≤ ξ are converted into additional wall voxels, and all 276

Nummulites wall voxels of distance > ξ are converted into additional void voxels, respectively, for the 277

simulation of cementation and corrosion, respectively. 278

Boundary Open/Close 279

As discussed previously, the open/closed boundary area is one component of a Nummulites (Fig. 6). 280

Nummulites originally deposited in carbonate muds would always be connected to the outside through 281

their apertures but these could become closed during later cementation. Additionally, Nummulites 282

could connect with the outside matrix after a part of shell is corroded as open boundary areas during 283

diagenesis, or through breakage during transport before sedimentation or during compaction with 284

burial. To mimic these alterations, we can open or close a fraction of the boundary areas of each 285

Nummulites, where the locations and areas can be selected from measurements, or chosen randomly. 286

Volumetric Enlarging/Shrinking 287

In many cases a rock matrix model as a whole needs to be enlarged or shrunk to match the resolution 288

associated with the extracted Nummulites. This can be mimicked simply by replicating the rock matrix 289

model in X-, Y- and/or Z-direction and cascading the multiple parts symmetrically; see the replicated 290

pattern of rock matrix in Fig. 1(f) as an example. 291

Intersection 292

From Fig. 9, we observe that Nummulites might sometimes touch or intersect each other, where 293

material has been lost. For each Nummulites that is to be added into the model, we randomly locate its 294

centre and then address its overlap with others. Within the region surrounding a Nummulites, the 295

matrix pore-solid structure will be replaced completely by the Nummulites, so the matrix pores make 296

connection to Nummulites interior only through boundary open areas. To avoid overlap, we count the 297

number of voxels that overlap with other Nummulites, and once there are no overlaps, we stop this 298

process and select the Nummulites. Alternately, we can also choose the template with, for example, 299

the smallest number of overlapping voxels, if that is the target. 300

3.4 Pore Network Simulation 301

All two-phase flow simulations are conducted using the network simulator developed by Ryazanov et 302

al. (2009), on the pore networks that are extracted from the rock models using the method of Jiang et 303

al. (2007). The pore networks are assumed to be strongly water-wet and initially are water-saturated. 304

The displacement is dominated by capillary pressure. To simulate the drainage relative permeability, a 305

receding contact angle is assumed to be zero everywhere, and no water trapping is considered for 306

simplicity. 307

4. A case-study 308

4.1 Data preparation for the Nummulitic limestone 309

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Tahmasebi and Sahimi (2012), followed by the optimization by Chen and Wang (2010)) to 311

reconstruct a rock matrix model (Fig. 10(b)) of 4003 voxels, with a resolution of 2.05 µm, called 312

model A. To create an alternate matrix model, called model B, we selected an image (Fig. 10(d)) of 313

10003 voxels, with a resolution of 3.003 µm from a database of CT images. The selection was made to 314

match the pore structure determined from the square region in the 2D image shown in Fig. 10(c), so 315

that the selected image (Fig. 10(d)) has the similar pore geometry and the same pore connectivity (see 316

Fig.10) as the reconstructed model. Model A is homogeneous but model B is heterogeneous although 317

both are isotropic. 318

From Fig. 10 and Fig. 11, we compare the two matrix models, and the three Nummulites templates 319

(Fig. 5(b, c, d)), in terms of pore size distribution (PSD) and pore connectivity function (PCF, Jiang et 320

al. 2007). Model A has tiny and consistently-sized pores (4.88 ~ 11.81 µm), with the volumetric 321

fraction of the sub-pore space of radii < 8 µm being more than 90%, whereas model B is composed of 322

larger and inconsistently-sized pores ranging from 6.10 ~ 84.27 µm. The three templates all have 323

similar distributions and the original void chamber sizes fall between the pore sizes of model A and B. 324

Note that the Nummulites chamber sizes can be modified through resizing, corrosion and cementation, 325

using image manipulation methods such as those described in Jiang et al. (2007). 326

When the void-pore space is taken into account, model A has a specific Euler number χ of -2.7×105 327

mm-3, which indicates that the pore connectivity is much better than that of model B (χ = 2.3×104 mm-328

3). An explanation of this is that only a small number of pores in model B contribute to the pore 329

connectivity. Regarding a Nummulites, the void connectivity mainly comes from the corrosion or 330

cementation, because the chambers are always disconnected from each other at the resolution of 8.05 331

µm. Hence, it is not meaningful to compare individual Nummulites as such. 332

To investigate the impact of the void-pore structures on fluid flow, we designed and generated four 333

groups of models shown in Table 1. In these models, Nummulites can be either the original templates 334

given in Fig. 5 or ones that are altered by transformations. The models are coded in this way: the first 335

two groups (i.e. AN) and the second two groups (i.e. BN) are built up based on model A and B, 336

respectively, and N represents Nummulites. The embedded Nummulites can be either corroded or 337

cemented, indicated by a suffix of e (expanding) or s (shrinking) in model codes. 338

4.2 Structure Impact 339

To assess the impact of the spatially-organised void-pore structure, we compare basic petrophysical 340

properties such as porosity, permeability and relative permeabilities. During drainage, oil pressure is 341

gradually increased at the inlet, and water pressure maintains a constant value at the outlet. Initially, 342

oil replaces water in the network elements of the largest pores connected to the inlet when the 343

required entry pressure is reached; and then more and more network elements associated with smaller 344

pores along the invasion fronts are occupied by oil until all available network elements are filled with 345

oil. The results are given in Fig. 12 for a series of oil-water drainage simulations. Note that the 346

irreducible water saturation is almost zero, as invasion of the smallest network elements occurs at 347

very high oil pressures, which can often not be achieved in reality. Additionally, in each model only 348

the percolating pore space is taken into consideration. 349

Rock matrix 350

When no Nummulites are embedded in models, such as for ANe0 and BNe0, the flow properties are 351

completely determined by the matrix pore structure. Model B has much larger pores than Model A 352

(25.79 µm vs 5.06 µm, on average), but lower porosity (11.3% vs 20.8%). But the pore connectivities 353

show the opposite effect on flow mobility – many more flow paths exist in model A than in model B. 354

As a result of the combination of pore sizes and connectivity, similar permeabilites (78.81 mD vs 355

121.82 mD) are obtained. However, the relative permeability curves demonstrate different trends (see 356

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respectively, which means that model B favours oil to move rather than water, even though the whole 358

system is assumed to be water-wet. 359

Nummulites Density 360

As Nummulites density increases, the porosity and permeability changes (Fig. 12(a, c, e, g)), as shown 361

in Table 2, for model ANe, ANs, BNe and BNs. Regarding relative permeability, models with prefix 362

AN are more sensitive to the density than are BN models, and ANs have increasing Kro (decreasing 363

Krw) and cross-over points moving to the right by > 0.1 water saturation. In comparison, BNs have no 364

notable change or motion. Through structure analysis, we note that the pore-void structures are 365

dominated by the overlapping and intersection of void chambers, and the increase in density leads to 366

higher connectivity. We also observe that as more Nummulites are added, the influence of rock matrix 367

decreases because major flow paths in the matrix are replaced by void chambers within highly 368

corroded Nummulites or blocked by heavily cemented Nummulites. 369

Corrosion and Cementation 370

Note that both porosity and permeability decease in ANs and BNs models, as more-cemented 371

Nummulites are present. The reason is that the multiple cemented Nummulites have deteriorated the 372

pore connectivity in comparison to either model A or model B, even though the cemented Nummulites 373

have much larger void chambers than the matrix pores. 374

Spatially-Preserved and Mixed Models 375

Here, we create two network models, Ahomo (Fig. 1(g)) and Bhomo (Fig. 1(h)), in which we 376

generated a rock model (Fig. 1(f)) to preserve the spatial arrangement of Nummulites (Fig. 1(d)) and 377

surrounding rock matrix (Fig. 1(a)) using our new methodology (Fig. 1(3)), and have also created a 378

stochastic network (Fig. 1(h)) that combines all the statistical information (see Fig. 1(c)), using a two-379

scale network model, where the information is obtained from two pore networks (Fig. 1(b, e)). An 380

image of 3003 voxels (2.75 µm per voxel) was resized into a model (Fig. 1(a)) of 1023 voxels and so 381

has an altered resolution of ~8.05 µm. The Nummulites (Fig. 1(d)) is obtained by transforming the one 382

shown in Fig. 5(c) with open boundary areas and highly corroded void chambers. 383

We simulate (Fig. 1(7)) for the two network models (Fig. 1(g, h)) both single - and two-phase flow. 384

Key network properties are listed in Table 3, and the simulated capillary pressure and relative 385

permeability curves are also given in Fig. 1(i) and (j). The results indicate that the two workflows 386

shown in Fig. 1 behave differently, although they can provide similar pore networks and 387

permeabilities. This comparison demonstrates the necessity of improving the multi-scale network 388

model (Jiang et al. 2013), aiming to also preserve the spatial characteristics in order to deal with 389

extremely heterogeneous carbonate rocks, including one or more of fossiliferous materials, those with 390

micro-fractures, and those with macro-/ and micro-pores. 391

Now we can summarise the main observations from these numerical experiments. 392

1. Corroded Nummulites increase both porosity and permeability if Nummulites chambers are 393

bigger than nearby pores in the rock matrix. Otherwise they continue to act as baffles to fluid 394

flow; 395

2. Partly cemented Nummulites are more likely be a negative influence on single-phase flow, i.e. 396

reducing fluid flow pathways (lower permeability); 397

3. Models containing the spatially preserved pore systems can develop a higher degree of 398

diagenetic change than that which can develop in a matrix-only model, even though both 399

models, and their original rocks, are subject to the same diagenetic event. As a consequence, 400

models with spatially-preserved pore-systems display a much higher uncertainty in estimates 401

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models. 403

404

5. Conclusions 405

We have introduced a new methodology to create rock models that contain spatially-distinct pore-406

system components. We apply this method to the example of a fossiliferous limestone, where the void 407

spaces in the fossils are very different to the pore system in the surrounding rock matrix. The resulting 408

models permit us to examine the single-and multi-phase flow impacts of variations in the pore 409

structure of the fossil components associated with cementation or chemical corrosion and the density 410

of fossils, along with how these characteristics give different answers with different matrix types. It is 411

clear that the extreme size-gap and the various spatial organizations of the pore/void spaces in 412

Nummulitic limestones causes not only pore channels to have high tortuosity but also pore 413

connectivity to occur at multiple levels. Our Nummulitic models highlight the requirement for correct 414

determination of the dominant pore scales (nm, µm, mm or cm), the spatial correlation and the cross-415

scale connections, aiming to create multiscale networks capable of fully capturing the heterogeneous 416

structures in carbonate rocks with larger volume whenever possible. They provide a method of 417

representing the distinct pore components of a rock as either spatially distinctive or otherwise and 418

permit a comparison of the consequences of this decision. They also allow representation of the pore 419

systems from rock-derived images or from equation-driven numerical representations, together with 420

tools to alter either one to accommodate observed, expected or speculative burial, deformational or 421

diagenetic alterations. The methodology is flexible enough to be used for other multi-component 422

rocks or sediments. 423

Acknowledgements 424

This work is financially supported by Petrobras and BG Group (now Shell) and the NSF Grant of 425

China (no. 61572007). Dr Jim Buckman of Heriot-Watt University and Dr Stephen A Hall of the 426

University of Lund are thanked for providing 2D and 3D images used in this work. 427

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TABLES TABLES TABLES TABLES

Number of

Nummulites 1 2 5

ANe

ANs

BNe

BNs

Model code

characters

A - Model A; B - Model B; N - Nummulites;

e - expanded/corroded; s - shrunk/cemented

Table 1. A set of resultant models by embedding a number of Nummulites into matrix models. The model

in the 3rd row and 4th column is coded as ANs5, for instance, denoting that it is generated by embedding 5

cemented Nummulites into the matrix model A. For each resultant model, only one slice is shown in the 2nd,

3rd and 4th columns but a 3D view is given in the 4th column, where the colours represent different

components (see Fig. 6).

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Property

Models

With no Nummulites With 10 Nummulites

Porosity (%) Permeability (mD) Porosity (%) Permeability (mD)

ANe 20.88 78.81 41.38 873.61

ANs 20.88 78.81 5.46 0.0

BNe 11.3 121.81 29.71 74.54

BNs 11.3 121.81 5.44 0.0

Table 2. Petrophysical properties changes as Nummulites density increase.

Table 3. Comparison of our two methods – spatially-preserved (Ahomo) and integrated network (Bhomo)

models.

Models Nodes Bonds Porosity (%) Permeability (mD)

Rock matrix (Fig. 1(a, b))

(1023 voxels, 2.75 µm/voxel) 1339 2480 25.27 1865.72

Nummulites (Fig. 1(d, e))

(5103 voxels, 8.05 µm/voxel) 1348 2450 10.01 0.253

Ahomo (Fig. 1(g)) 294451 517836 35.38 3460.08

Bhomo (Fig. 1(h)) 242674 571729 31.87 3627.85

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FIGURESFIGURESFIGURESFIGURES

Fig.1. Two workflows moving from pore voxel representations to petrophysical properties: multi-scale

pore network and model generation. The former developed by Jiang et al. (2013) mainly consists of

extracting (1, 2) pore networks (b, e) from pore voxel representations (a, d), computing (5) statistical

information (c), and generating (6) multi-scale pore network (h). The latter, first presented in this paper,

involves the generation (3) of a complex pore model (f) based on various information obtained from one or

more of a series of images ((a, d), e.g. CT, SEM) and the extraction (4) of a pore network (g) from the pore

Spatially preserved network Two-scale stochastic

(f) (g) (h)

(4(4(4(4

(a) (b)

(1(1(1(1

(d) (e)

(2(2(2(2

(c)

(5)

(7)(7)(7)(7)

(3) CT image – rock matrix

A Nummulites template Extracted Nummulites void network

A spatially preserved model

Petrophysical properties – porosity, absolute permeability, capillary pressure, relative permeability ….

(6)

0

5000

10000

15000

0 0.2 0.4 0.6 0.8 1

Ca

pilla

ry p

ress

ure

(P

a)

Ca

pilla

ry p

ress

ure

(P

a)

Ca

pilla

ry p

ress

ure

(P

a)

Ca

pilla

ry p

ress

ure

(P

a)

Water saturationWater saturationWater saturationWater saturation

(i)Spatially preserved network

Two-scale stochastic network

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1Re

lati

ve p

erm

ea

bili

tyR

ela

tive

pe

rme

ab

ility

Re

lati

ve p

erm

ea

bili

tyR

ela

tive

pe

rme

ab

ility

Water saturationWater saturationWater saturationWater saturation

(j) Krw - spatially preserved

Kro - spatially preserved

Krw - Two-scale stochstic

Kro - Two-scale stochastic

-3,000

-2,000

-1,000

0

1,000

5 125

Sp

eci

fic

Nu

mb

er

nu

mb

er

(mm

-3)

Minimal pore/vpod radius (µm)

Pore Connectivity Function

Nummulites Rock matrix

-20

-10

0

10

5 15 25

0.00

0.01

0.02

0.03

0.04

5 25 125 625

Vo

lum

etr

ic f

req

ue

ncy

Pore/void radius (µm)

Pore Size DistributionNummulites

Rock matrix

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model using the method of Jiang et al. (2007). Both workflows culminate in the network simulation (7) of

petrophysical properties (e.g. (i) – capillary pressure, (j) – relative permeability).

Fig. 2. Schematic diagram of a Nummulites (http://vmek.oszk.hu)

(a) (b) (c) Fig. 3. Model generation of the pore system in Nummulitic rock. (a) a reconstructed rock matrix

model, excluding the black disk region, of a homogeneous background pore structure; (b) a created

Nummulites templates (model) using the Archimedean spiral equation in the region corresponding to

the black disk; (c) the pore network extracted from the resultant model by embedding the Nummulites

template in the black disk region, where network nodes are represented by balls, and network bonds

by cylinders.

Fig. 4. Evidence of nano-scale tubes penetrating (partially or completely) the external shell of a fossil.

Scale bar is 200 microns.

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(a) (b) (c) (d)

Fig. 5. Extracting Nummulites from 3D X-ray CT images: (a) one slice of one Nummulites 3D image; (b) a

Nummulites extracted from that image, (c) and (d) Nummulites extracted from other X-ray CT images.

(a) (b)

Fig. 6. View of the Nummulites shown in Fig. 5(b): (a) Cross showing a single slice from the reconstructed

tomographic image with Nummulites in middle left as shown by dashed lines; (b) orthoslice view of

Nummulites seen in (a) showing four identified components. Four components identified: (I) – open

boundary areas, (II) – closed boundary areas, (III) – void chambers, (IV) walls separating chambers.

(a) (b)

Fig. 7. Mathematical Nummulites model: (a) A Nummulites chamber structure and (b) the model specified

by Eq. (1) viewed from two different perspectives.

(a) (b) (c) (d)

Fig. 8. Simulating the cementation in a Nummulites: (a) a (squared) distance map coded by the rainbow

scheme – the smallest distance 0s in blue, the biggest distance in red; (b) the intial void space; (c) partly

(I)

(II)

(III)

(IV)

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cemented void space and (d) heavily cemented space. The distance colour bar is on the right.

(a) (b) (c)

(d) (e) (f)

Fig. 9. Types of Nummulitic limestones viewed in thin section with the petrographic microscope:

Uncemented chambers (a, d), partly cemented (b, e), and heavily dissolved or corroded (c, f). Green colour

in (a) and (c) denotes pore space.

(a) (b) (c) (d)

Fig. 10. Creation of a rock matrix model for the micrite and the dolomite matrixes: (a) chosen micrite

equivalent 2D training image, (b) its reconstructed matrix model A, (c) a BSE image from a thin section of

the dolomitised reservoir rock showing targeted area in red box, and (d) the matched image for (c) (model

B).

(a)

0.00

0.20

0.40

0.60

0.80

1.00

4 16 64

Fre

qu

en

cyF

req

ue

ncy

Fre

qu

en

cyF

req

ue

ncy

Logarithmic incribed radius (micron)Logarithmic incribed radius (micron)Logarithmic incribed radius (micron)Logarithmic incribed radius (micron)

Nummulites 1

Nummulites 2

Nummulites 3

Model A

Model B

-300000

-250000

-200000

-150000

-100000

-50000

0

50000

100000

1 4 16 64

Sp

eci

fic

Eu

ler

Nu

mb

er

(^S

pe

cifi

c E

ule

r N

um

be

r (^

Sp

eci

fic

Eu

ler

Nu

mb

er

(^S

pe

cifi

c E

ule

r N

um

be

r (^

-- -- 3)

3)

3)

3)

Minimal pore raius (micron)Minimal pore raius (micron)Minimal pore raius (micron)Minimal pore raius (micron)

Mode A

Model B

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(b)

Fig. 11. The PSDs (a) and PCFs (b) of matrix model A and B shown in Fig. 10(b, d). Note that (a) also

plots the PSDs of the three Nummulites shown in Fig. 5.

Fig. 12. Illustration of the impact on absolute and relative permeability during drainage for the models in

Table 1 associated with model A (see Fig. 10(b)) (a, b, c, d) and model B (see Fig. 10(d)) (e, f, g, h), of the

corrosion (a, b, e, f) and the cementation (c, d, g, h) of n Nummulites. In (b, d, f, h), Krw/o+n represents the

water/oil relative permeability for a model with n embedded Nummulites.

0

10

20

30

40

50

0

200

400

600

800

1000

0 1 2 3 4 5 6

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Pe

rme

ab

ilit

y (

mD

)

Number of corroded nummulites(a)0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1

Kr

Sw

(b)

Krw A Kro A Krw+1 Kro+1

Krw+5 Kro+5 Krw+10 Kro+10

Krw+20 Kro+20 Krw+50 Kro+50

0

10

20

30

0

20

40

60

80

100

0 1 2 3 4 5 6

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Number of cemented nummulites(c)0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1

Kr

Sw

(d)

Krw A Kro A Krw+1 Kro+1Krw+5 Kro+5 Krw+10 Kro+10Krw+20 Kro+20 Krw+50 Kro+50

0

10

20

30

40

0

25

50

75

100

125

150

0 1 2 3 4 5 6

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Number of corroded nummulites(e)0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

Kr

Sw

(f)

Krw B Kro B

Krw+1 Kro+1

Krw+5 Kro+5

Krw+10 Kro+10

Krw+20 Kro+20

0

2

4

6

8

10

12

-20

0

20

40

60

80

100

120

140

0 1 2 3 4 5 6

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Po

rosi

ty (

%)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Pe

rme

ab

ility

(m

D)

Number of cemented nummulites(g)0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1

Kr

Sw)

(h)

Krw B Kro B

Krw+1 Kro+1

Krw+5 Kro+5

Krw+10 Kro+10

Krw+20 Kro+20

MANUSCRIP

T

ACCEPTED

ACCEPTED MANUSCRIPT

MANUSCRIP

T

ACCEPTED

ACCEPTED MANUSCRIPT

Highlights

. New methodology to model spatially-distinct, multi-scale pore systems;

. Validation of the workflow to represent such multi-component pore systems in fossiliferous carbonate rocks;

. Demonstration using pre-existing flow simulation tools of the complex impact of fossil cementation and dissolution