AUTHORS

Matthew D. Jackson � Center for Petro-leum Studies, Department of Earth Science andEngineering, Imperial College London, PrinceConsort Road, London SW7 2BP, UnitedKingdom; mattjack@imperial.ac.uk

Matthew Jackson is a lecturer in petroleumreservoir engineering at Imperial College Lon-don. His research interests include the impactof geological heterogeneity on flow duringhydrocarbon recovery, the development of newreservoir and outcrop modeling techniques,and the optimal application of smart-well tech-nology. He is a graduate in physics from Im-perial College London and holds a Ph.D. ingeological fluid mechanics from the Universityof Liverpool.

Shuji Yoshida � Center for PetroleumStudies, Department of Earth Science andEngineering, Imperial College London, PrinceConsort Road, London SW7 2BP, United King-dom; present address: Department of Geo-logical Sciences, University of Texas at Austin,Austin, Texas; shuji@mail.utexas.edu

Shuji Yoshida has a B.Eng. degree (miningengineering) from Kyushu University, Japan, anM.Sc. degree (petroleum geology) from Impe-rial College London, United Kingdom, and aPh.D. (basin analysis) from Toronto University,Canada. He has held industry and postdoctoralpositions in reservoir characterization (ImperialCollege), coastal geomorphology (Royal Hollo-way, University of London), and hydrocarbonexploration (University of Wyoming). Hismain interest is shallow-marine and nonmarinesedimentology.

Ann H. Muggeridge � Center for Petro-leum Studies, Department of Earth Scienceand Engineering, Imperial College London,Prince Consort Road, London SW7 2BP,United Kingdom

Ann Muggeridge is a senior lecturer in petro-leum engineering at Imperial College. Her re-search interests include modeling the influenceof reservoir heterogeneities on oil recoveryand upscaling. Before joining Imperial College,she worked in simulation technical supportat Scientific Software Inc. (United Kingdom)and as a research reservoir engineer at BP.

Three-dimensional reservoircharacterization and flowsimulation of heterolithictidal sandstonesMatthew D. Jackson, Shuji Yoshida, Ann H. Muggeridge,and Howard D. Johnson

ABSTRACT

Tidal sandstone reservoirs contain significant intervals of hydrocarbon-

bearing heterolithic facies, characterized by the presence of tide-

generated sedimentary structures such as flaser, wavy, and lenticular

bedding (millimeter to centimeter sand-mud alternations). We have

characterized the reservoir properties (sandstone connectivity, ef-

fective permeability, and displacement efficiency) of these facies

using three-dimensional (3-D) models reconstructed directly from

large rock specimens. The models are significantly larger than a core

plug, but smaller than a typical reservoir model grid block. We find

that the key control on reservoir quality is the connectivity and

continuity of the sandstone and mudstone layers. If the sandstone

layers form a connected network, they are likely to be productive

even at low values of net-to-gross (about 0.3–0.5). This may explain

why the productivity of low net-to-gross, heterolithic tidal sand-

stones is commonly underestimated or overlooked. Connectivity

is the dominant control on the transition between productive (pay)

and nonproductive (nonpay) heterolithic facies. However, connec-

tivity is difficult to characterize because core plugs sampled from

the subsurface are too small to capture connectivity, whereas two-

dimensional outcrop measurements can significantly underestimate

the true 3-D value. Our results suggest that core-plug measurements

of permeability and displacement efficiency are unlikely to yield

representative values at the scale of a reservoir model grid block

because the connectivity and continuity of sandstone and mudstone

layers varies significantly with length scale.

AAPG Bulletin, v. 89, no. 4 (April 2005), pp. 507–528 507

Copyright #2005. The American Association of Petroleum Geologists. All rights reserved.

Manuscript received March 12, 2004; provisional acceptance June 24, 2004; revised manuscriptreceived November 1, 2004; final acceptance November 23, 2004.

DOI:10.1306/11230404036

INTRODUCTION

Hydrocarbon reservoirs formed in tide-influenced environments

commonly contain significant intervals of heterolithic sandstones,

which are characterized by the presence of complex millimeter-

to centimeter-scale intercalations of sandstone and mudstone (e.g.,

Maguregui and Tyler, 1991; Dixon et al., 1997; Marjanac and Steel,

1997; Robertson, 1997; Gupta and Johnson, 2001; Martinius et al.,

2001). These small-scale intercalations are highly variable both

laterally and vertically and commonly reflect diurnal and/or semi-

diurnal variations in depositional energy during the tidal cycle (e.g.,

Reineck and Wunderlich, 1968; Dalrymple, 1992). Heterolithic

intervals frequently host significant hydrocarbon reserves, yet their

reservoir properties (such as porosity, permeability, capillary pres-

sure, and mobile oil saturation) are highly variable and difficult to

predict (e.g., Norris and Lewis, 1991; Martinius et al., 2001). More-

over, it is not clear how they should best be represented in either

static or dynamic reservoir models (Jackson et al., 2003). In the

absence of core, they are not easy to identify in the subsurface as

productive zones because of their low resistivity and low contrast

pay characteristics (Darling and Sneider, 1993). The aim of this

paper is to characterize the reservoir properties of heterolithic

tidal sandstones and assess their impact on fluid flow.

The objectives of the paper are fourfold. The first is to present

a methodology for reconstructing the complex, three-dimensional

(3-D) architecture of the sandstone-mudstone intercalations with-

in heterolithic tidal intervals directly from outcrop data. Existing

geological models are based either on two-dimensional (2-D) out-

crop data, or on modern analogs or laboratory experiments, in which

the preservation potential is uncertain (e.g., Reineck and Wunder-

lich, 1968; Terwindt and Breusers, 1971, 1982; Reineck and Singh,

1980; Terwindt, 1981; Oost and Baas, 1994). Consequently, little

quantitative 3-D data are available. Yet, as we demonstrate in this

paper, to properly characterize the reservoir properties of hetero-

lithic tidal sandstones, it is essential to have a good understanding of

their internal 3-D architecture. To achieve this, we use serial sec-

tioning techniques, in conjunction with surface mapping software,

to reconstruct this architecture directly from large rock specimens

(blocks measuring about 0.5 � 0.5 � 0.3 m [1.6 � 1.6 � 1 ft])

collected from outcrop. The resulting models are a close repre-

sentation of the rock specimens and are not based on synthetic data

or geostatistical modeling techniques.

The second objective is to calculate numerically the connec-

tivity of the sandstone intercalations, the effective permeability, and

the displacement efficiency of each rock specimen model. We cal-

culate the sandstone connectivity because this is likely to be the

dominant control on the transition between hydrocarbon-bearing

(reservoir) and nonhydrocarbon-bearing (nonreservoir) rock (King,

1990; King et al., 2001). Connectivity also controls the effective

permeability and displacement efficiency. However, information on

sandstone connectivity within heterolithic intervals is not available

Howard D. Johnson � Center for Petro-leum Studies, Department of Earth Scienceand Engineering, Imperial College London,Prince Consort Road, London SW7 2BP, UnitedKingdom

Howard Johnson holds the Shell Chair of Pe-troleum Geology at Imperial College London.His main interests are in clastic sedimentology,sequence stratigraphy, reservoir characteriza-tion, and basin studies. Previously, he spent15 years with Shell working in research, explo-ration and development geology, and petro-leum engineering. He holds a B.Sc. degree ingeology from the University of Liverpool and aD.Phil. in sedimentology from the Universityof Oxford.

ACKNOWLEDGEMENTS

Funding for this work is gratefully acknowledgedfrom Amoco (now BP), BP, Fortum PetroleumA/S (now Agip-ENI), Norske Conoco A /S (nowConocoPhillips), Saga Petroleum A /S (nowNorsk Hydro), and Statoil. The Norwegian Pe-troleum Directorate is thanked for coordinatingthis Forum for Reservoir Characterization andEngineering project. We are grateful to MidlandValley Exploration for providing the 3DMove1

surface mapping software with which the rockspecimen architectures were reconstructed,Roxar for providing the IRAP RMS1 reservoirmodeling software with which the rock speci-men models were gridded and visualized, andStreamsim Technologies for providing the3DSL1 software with which the streamlinesimulations were performed. Special thanks toPeter King at Imperial College London, AlistairJones at BP, and Arve Næss, Phil Ringrose, andAllard Martinius at Statoil, for their helpful dis-cussions. Finally, this project could not havebeen achieved without the help of Mark Buckleyin collecting and sectioning the rock specimenmodels and John Dennis in the Imperial Col-lege Rock Mechanics Laboratory and his in-valuable expertise in rock sectioning.

508 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

from subsurface data. We calculate the effective per-

meability and displacement efficiency because both are

normally estimated using conventional core-plug mea-

surements (about 2–3 cm [0.78–1.18 in.] in diameter

and 3–7 cm [1.18–2.75 in.] in length), yet core-plug

measurements of heterolithic sandstones are unlikely

to be representative at the scale of a reservoir model

grid block (about 10–500 � 10–500 � 0.1–5 m [33–

1600 � 33–1600 � 0.3–16 ft]) (Jackson et al., 2003).

The rock specimen models help to bridge this scale gap;

they sample about 104 cm3 (610 in.3) of rock, which,

although less than that represented by a reservoir model

grid block (>109 cm3; >6.1 � 107 in.3), is significantly

greater than that sampled by a core plug (about 30 cm3

[1.83 in.3]). The models therefore provide quantita-

tive information on reservoir properties at a scale that

would otherwise be inaccessible.

The third objective is to compare sand body con-

nectivity in 3-D and 2-D to determine whether 2-D

models provide useful estimates of reservoir properties

in heterolithic tidal sandstones. Most previous studies

have used 2-D models derived from outcrop or sub-

surface data to characterize the reservoir properties

of different bedform types (e.g., Pryor, 1973; Yoko-

yama and Lake, 1981; Kortekaas, 1985; Weber, 1986;

Goggin et al., 1988, 1992; Corbett and Jensen, 1991,

1993; Jacobsen and Rendall, 1991; Corbett et al., 1992;

Hartkamp-Bakker, 1993; Lake and Malik, 1993; Mc-

Dougall and Sorbie, 1993; Prosser and Maskall, 1993;

Ringrose et al., 1993; Huang et al., 1995, 1996). Only

a minority have used 3-D models, and these were syn-

thetic models of bedforms deposited in the fluvial

(Jones et al., 1995), shallow-marine (Kjønsvik et al.,

1994), and tidal (Ringrose et al., 2003) environments.

Norris and Lewis (1991) used 2-D, outcrop-derived

models to characterize the reservoir properties of het-

erolithic tidal sandstones, but no previous studies have

used 3-D outcrop models.

The fourth objective is to suggest an explanation as

to why intervals of low resistivity and low contrast pay

in the tidal environment commonly yield large vol-

umes of hydrocarbons despite their low net to gross.

There is significant production from such intervals in

the North Sea; for example, the Lower Jurassic Tilje

Formation of the Heidrun and Smørbukk (part of

Asgard) fields (e.g., Whitley, 1992; Olsen et al., 1999;

Martinius et al., 2001), the Lower Jurassic Cook For-

mation of the Gullfaks field (e.g., Marjanac and Steel,

1997), and the Middle Jurassic Beryl Formation (Bruce

Group) of the Beryl and Bruce fields (e.g., Robertson,

1997; Dixon et al., 1997). Large volumes are also pro-

duced from heterolithic tidal intervals within the Eo-

cene Misoa Formation in the Lagunillas and LL-652

fields, Venezuela (e.g., Maguregui and Tyler, 1991),

and in Colombia, Libya, and parts of the Cretaceous

Western Interior basin in North America. Consequent-

ly, understanding and predicting when they will be

productive is of great importance.

We investigate two rock specimens: one that rep-

resents a high net-to-gross (sandstone rich) hetero-

lithic interval and is characterized by the presence of

flaser-wavy bedding, and one that represents a low net-

to-gross heterolithic interval and is characterized by the

presence of lenticular-wavy bedding. The classification

of flaser-, wavy-, and lenticular-bedding types follows

Reineck and Wunderlich (1968).

GEOLOGICAL SETTING

The heterolithic tidal deposits selected for this study

were taken from the Lower Cretaceous Barnes High

Sandstone Member (Vectis Formation, Wealden Group)

in the southwest of the Isle of Wight, United Kingdom,

where it is exposed along the coastal cliffs between

Shepherd’s Chine and Cowleaze Chine (Figure 1). In

this area, the member is about 6 m (20 ft) thick and

comprises several vertically stacked, coarsening-upward

facies successions containing a wide variety of tidal sed-

imentary structures. It is interpreted as a composite tidal-

bar complex deposited in the inner, tide-dominated part

of a mixed-energy estuary, similar to the modern Gironde

estuary in northwest France (Yoshida et al., 2001).

The Barnes High Sandstone Member occurs in the

basal part of the Vectis Formation, which is an 80-m

(260-ft)-thick, mud-dominated succession with abun-

dant fresh- and brackish-water fauna. The formation

records the first brackish transgression above the terres-

trial Wessex Formation, which contains about 500 m

(1600 ft) of flood-plain shales and channel sandstone

bodies. The Vectis Formation is erosively overlain by

the open-marine Atherfield Clay Formation of the

Lower Greensand Group (Figure 2) (Ruffel and Wach,

1991, 1998; Wach and Ruffel, 1991). The Barnes High

Sandstone Member serves as an excellent outcrop ana-

log for many tidal sandstone reservoirs, such as the

Lower Jurassic Cook Formation in the Gullfaks field,

northern North Sea (Gupta and Johnson, 2001) and the

Lower Jurassic Tilje Formation in the Heidrun field,

offshore Norway (Martinius et al., 2001).

This study focused on heterolithic facies in the

lowermost coarsening-upward succession (Figure 1).

Jackson et al. 509

Figure 1. The rock specimens used in this study were collected from Lower Cretaceous estuarine deposits on the Isle of Wight insouthern England. (a) The rock blocks were obtained from outcrops of the Barnes High Sandstone Member (Lower Cretaceous VectisFormation) along the southwestern coast of the Isle of Wight, between Barnes High (BH) and Cowleaze Chine (CC). (b) A dip-oriented stratigraphic section of the uppermost Wessex Formation and the lower Vectis Formation, between Barnes High andCowleaze Chine, shows that the Barnes High Sandstone Member comprises three vertically stacked tidal bars (tidal bars 1–3). Eachtidal bar is characterized by a coarsening-upward facies trend, and together, they form a composite coarsening-upward and sandierupward succession. Deposition occurred in the axis of a tidally influenced estuarine-valley system. The rock specimens studied in thispaper were collected from the lower part of tidal bar 1 in Cowleaze Chine, immediately above a composite stratigraphic surfacecomprising a sequence boundary and transgressive surface (SB/TS) (modified from Yoshida et al., 2001). (c) Outcrop photograph ofthe lower part of tidal bar 1 in Cowleaze Chine, showing the coarsening-upward facies succession. Light colors denote sandstone;dark colors denote mudstone. In detail, the coarsening-upward profile is characterized by three distinct intervals: (1) a basal mud-dominated zone with extensive lenticular bedding and minor wavy bedding (lithofacies assemblage C); (2) sand-rich zone dominatedby ripple cross-lamination, laterally extensive mud layers, and lenticular to wavy bedding (lithofacies assemblage B); and (3) a thickerbedded, sand-dominated zone with trough cross-bedding, current-ripple cross-lamination, wavy to flaser bedding, and mud drapes ofvariable extent, which occur along foresets, toesets, and set boundaries (lithofacies assemblage A). Detailed outcrop facies mapsshow that lithofacies assemblage C passes laterally and up depositional dip into lithofacies assemblage B, which in turn passeslaterally and up depositional dip into lithofacies assemblage A. Details of the lithofacies assemblages and their environmentalsignificance are outlined in Yoshida et al. (2001). Rock specimen 1 corresponds to lithofacies assemblage A, and rock specimen 2corresponds to lithofacies assemblage B.

510 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

To characterize the complex internal 3-D architecture

of these facies, we collected two large rock specimens

(blocks) measuring about 0.5 � 0.5 � 0.3 m (1.6 �1.6 � 1 ft) and described them in detail. Specimen 1 is

sandstone rich and corresponds to lithofacies assem-

blage A in Figure 1. It contains discontinuous mud

flasers and more continuous wavy mud drapes along

the foresets and toesets of low-angle cross-beds. The

mud drapes are commonly bifurcating, particularly at

the toesets of cross-beds, and both single and double

mud drapes are observed. Specimen 2 is mudstone

rich and corresponds to lithofacies assemblage B in

Figure 1. It contains abundant lenticular bedding with

subordinate wavy bedding. Both samples contain

negligible bioturbation, although specimen 2 contains

slightly more than specimen 1. Hence, the continuity

of the sandstone and mudstone layers mainly repre-

sents primary depositional processes.

MODEL CONSTRUCTION

Serial Sectioning and Reconstruction

To reconstruct the complex internal 3-D architecture of

the rock specimens, they were serially sectioned. Prior to

sectioning, the specimens were treated with a water-

resistant compound to minimize damage caused by the

blade lubricant, then set in plaster to minimize damage

caused by the blade cuts. Each specimen was sectioned

at about 20-mm (0.78-in.) intervals, with the section

spacing determined accurately using inclined lines inter-

secting at a known angle scored on the base of each

specimen (Figure 3a). For reconstruction, each section

face was photographed, the photographs were digitized,

and the lithologies on each were identified as either

sandstone or mudstone (mainly silty mudstone with

uniform grain size). The boundaries between sandstone

and mudstone layers were then traced (Figure 3b). By

manually correlating these lithological boundaries be-

tween each 2-D section, the 3-D architecture of the

bounding surfaces separating each sandstone and mud-

stone layer was reconstructed (Figure 3c). The following

rules were applied to correlating the surfaces to ensure

the results were suitable for 3-D reconstruction and

gridding:

1. Sandstone and mudstone layers must alternate and

be separated by a traced bounding surface.

2. All surfaces and layers are continuous across the

specimen.

3. Where a sandstone or mudstone layer pinches out,

the layer is still defined but has zero thickness (i.e.,

the upper and lower bounding surfaces coincide).

Figure 2. Simplified vertical stratigraphic section through the Lower Cretaceous of the Isle of Wight, based on Ruffell and Wach(1991, 1998) with our interpretation of depositional environments (Yoshida et al., 2001).

Jackson et al. 511

Figure 3. Reconstruction and visualization of the rock specimens. (a) Serial sectioning. The spacing of the sections was calculatedaccurately using inclined lines intersecting at a known angle (a) scored on the base of each specimen. By measuring the spacing ofthe lines (b) at each section, the distance (d) of the section from the origin could be calculated and from that the spacing of eachsection. (b) Tracing of the sandstone and mudstone boundaries. Section photographs from specimen 2. Light colors denotesandstone; dark colors denote mudstone. (c) Surface correlation. Bounding surfaces between sandstone and mudstone layers werecorrelated manually between the 2-D sections in an iterative process. (d) Surface reconstruction. The correlated bounding surfaceswere reconstructed in 3-D using surface mapping software. Selected surfaces from specimen 2 are shown. (e) Rendering andvisualization. The reconstructed rock specimen models. Light colors denote sandstone; dark colors denote mudstone.

512 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

4. The bounding surfaces cannot cross.

5. All surfaces are vertically monotonic (no overhang-

ing surfaces).

Ambiguities in correlation between sections oc-

curred occasionally when sandstone and mudstone lay-

ers pinched out. Miscorrelations were identified

primarily by excessive vertical relief of the bounding

surfaces across a particular section, and these were re-

fined iteratively. The minimum sandstone or mud-

stone layer thickness resolved was 1 mm; thinner layers

were neglected or pinched out.

Once the surfaces had been successfully correlat-

ed, they were reconstructed in 3-D from the 2-D dig-

ital boundary tracings using surface mapping software

(Figure 3d). The reconstruction of specimen 2 was

complicated by mild bioturbation of the sandstone

and mudstone layers, which caused slight disruption

of the bounding surfaces. During reconstruction, this

effect was smoothed so that the reconstructed sur-

faces reflect their original depositional architecture.

Consequently, the sandstone and mudstone layers in

the reconstructed model are slightly cleaner than in

the original rock specimen. Finally, the volume be-

tween each pair of surfaces was designated either

sandstone or mudstone, and the models were rendered

for visualization (Figure 3e). The reconstructed model

of specimen 1 measures 44 � 28 � 14 cm (17 � 11 �5.5 in.); that of specimen 2 measures 38 � 32 � 12 cm

(15 � 12.5 � 4.7 in.).

Grid Construction

To quantitatively investigate the reservoir properties

of the reconstructed rock specimen models, they must

be represented on a grid. The grid must preserve the

complex 3-D depositional architecture of the sand-

stone and mudstone layers, without requiring such a

large number of grid blocks that permeability calcu-

lations and flow simulations become too computa-

tionally expensive. We used an adaptive cornerpoint

gridding scheme in which variations in layer architec-

ture are represented by variations in grid architecture.

The grid has vertical pillars with a constant spacing in

the x and y (horizontal) directions; between each

bounding surface, one or more grid layers are defined

in the z (vertical) direction (Figure 4a). The geometry

of the upper and lower grid layers between each pair

of bounding surfaces is governed by the geometry of

the surfaces, and intermediate grid layers are distrib-

uted proportionately (Figure 4b). The geometry of the

grid is therefore governed by the geometry of the

bounding surfaces, ensuring geological integrity and

preserving their curvature. Where sandstone or mud-

stone layers pinch out, the bounding surfaces coincide,

and the blocks in between have zero thickness. These

pinched-out blocks are set to be inactive in flow sim-

ulations and bridged using non-neighbor connections,

so they do not act as barriers to flow (Figure 4c). This

approach minimizes the number of grid blocks re-

quired to represent the complex architecture of the

layers. The advantage of defining all surfaces and layers

to be continuous across a specimen is that the under-

lying structure of the grid is regular, although blocks

may be pinched out. This allows the grids to be used in

conventional flow simulators. The model of specimen 1

was represented on a grid containing 134 � 84 � 76

grid blocks in the x, y (horizontal), and z (vertical)

directions, respectively, with a total of 855,456, from

which 622,200 were active. The grid blocks measured

3.3 mm (0.13 in.) in the x and y directions and from

zero (pinched out) to 5 mm (0.19 in.) in the z di-

rection. The model of specimen 2 was represented on

a grid containing 114 � 96 � 76 grid blocks, with a

total of 831,744, from which 703,908 were active.

The grid blocks measured 3.3 mm (0.13 in.) in the xand y directions and from zero (pinched out) to 3 mm

(0.11 in.) in the z direction. Each sandstone and mud-

stone layer contained four grid layers.

Figure 4. Adaptive distribution of simulation grid layers tocapture sandstone-mudstone layer architecture. (a) A minimumof four grid layers are defined to represent each sandstone andmudstone layer; (b) the geometry of the upper and lower gridlayers is governed by the geometry of the bedding surfaces, andthe intermediate grid layers are distributed proportionately;(c) most reservoir simulators require an underlying rectangularmatrix on which to solve the flow equations. Where sandstonelayers pinch out, the cells are set inactive and bridged verticallyusing non-neighbor connections.

Jackson et al. 513

CHARACTERIZATION OFRESERVOIR PROPERTIES

Visualization

The reconstructed models can be easily displayed and

dissected on a computer, allowing qualitative analysis

of the architecture and connectivity of the sandstone

and mudstone layers. Vertical sections through spec-

imen 1 reveal large variations in lateral continuity and

vertical connectivity of the mud drapes (Figure 5). The

drapes are commonly bifurcated; they connect verti-

cally forming double mud drapes (Reineck and Wun-

derlich, 1968; Visser, 1980). Some isolated flasers have

a conventional concave-up form where they fill rip-

ple troughs, but others have a convex-up form where

they lie over ripple crests but do not extend into the

adjacent troughs. Sections through specimen 2 reveal

even larger variations in the continuity and connectivity

of the sandstone layers (Figure 5). Several generations

of sandstone lenses frequently stack up and are con-

nected vertically, but many also appear to be isolated

in mudstone.

If a core-plug-sized sample was taken from spec-

imen 2, the lateral continuity of sandstone and mud-

stone layers would be higher at the scale of a core

plug than at the scale of the whole model (Figure 6).

In a layer-parallel sample, the sandstone layers ex-

tend over the entire length of the sample; in a layer-

perpendicular sample, the mudstone layers extend over

the entire width of the sample (Figure 6). However,

in the rock specimen, the sandstone and mudstone

layers are laterally discontinuous (Figures 3, 5, 6).

This scale-dependent variation in layer continuity sug-

gests that core-plug measurements will not sample a

representative volume in heterolithic tidal sandstones

(see also Jackson et al., 2003). We discuss this issue

and its implications for upscaling in a section below.

Visualizing individual sandstone and mudstone

layers in specimen 2 reveals how lateral sandstone and

Figure 5. Orthogonal sections through the two rock specimen models, showing large variations in sandstone connectivity andmudstone continuity. Images (a) and (b) show specimen 1 (lithofacies assemblage A) with mud drapes of variable extent preservedin trough cross-bedded and ripple cross-laminated sandstone layers along toesets, foresets, and formsets (simplified here as flaser-wavy bedding). Images (c) and (d) show specimen 2 (lithofacies assemblage B), with more laterally continuous mud drapes overcontinuous to discontinuous rippled sandstone layers and lenses (simplified here as lenticular-wavy bedding). Light colors denotesandstone; dark colors denote mudstone.

514 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

mudstone layer continuity varies with the fraction of

sandstone (net-to-gross) (Figure 7). This specimen was

taken from a thin fining- and muddier upward interval

of lithofacies assemblage B, superimposed on the over-

all coarsening-upward trend (Figure 1c). The decrease

in sandstone fraction toward the top of the specimen is

accompanied by a decrease in lateral sandstone con-

nectivity and an increase in lateral mudstone continu-

ity. Hence, the sandstone lenses appear isolated near

the top of the model but become connected toward the

base. These qualitative estimates of sandstone connec-

tivity are augmented by quantitative calculations in the

next section.

Sandstone Connectivity

For each rock specimen model, we calculated both the

total fraction of sandstone (the net to gross) and the

fraction of connected sandstone. We defined the latter

in a given model to be that which is connected to each

pair of opposing faces in each of the x, y (horizontal),

and z (vertical) directions (Figure 8). This definition of

connectivity is consistent with our definition of ef-

fective permeability, which is also measured between

opposing faces, and yields a connectivity tensor. For

both connectivity and permeability tensors, we calcu-

lated only the diagonal components. Our definition of

connectivity omits (1) sandstone entirely surrounded

by mudstone, (2) sandstone connected to only one

face, and (3) sandstone connected to two faces but not

to two opposing faces. However, it includes sandstone

that forms a dead end but is nevertheless connected to

both opposing faces (Figure 8). This definition is rea-

sonable if we assume that the specimen models are

representative samples, and that flow occurs primarily

between opposing faces.

In addition to calculating sandstone connectivity

in each of the 3-D rock specimen models, we also

calculated connectivity in a selection of vertical 2-D

sections. This allowed us to investigate quantitatively

whether 2-D measurements of connectivity are rep-

resentative of the true 3-D value. If 2-D measurements

do not properly capture connectivity, then 2-D out-

crop models will not properly capture other reservoir

properties such as effective permeability and displace-

ment efficiency. In each model, 10 vertical sections were

taken, 5 in each of the x and y directions and the con-

nected fraction of sandstone calculated in both the

horizontal (x or y) and vertical (z) directions.

We found that the total sandstone fraction (net to

gross) in each specimen model ranges from 0.935 in

specimen 1 to 0.445 in specimen 2 (Figure 9). In spec-

imen 1, the connected fraction of sandstone is 1 (all

the sandstone is connected) in both the horizontal and

vertical directions. In specimen 2, the connected frac-

tion is 0.94 in the x direction and 0.75 in the y di-

rection. This is perhaps surprisingly high, given their

2-D appearance (Figures 3, 5, 6). The nonconnected

fraction is mostly limited to several isolated sand-

stone bodies toward the top of the model. Connectiv-

ity is lower in the y direction because a sandstone body

toward the base of the model is connected to only one

face (Figure 10). The vertical sandstone connectivity is

zero because one mudstone layer extends completely

across the specimen.

For both models, the 2-D connectivity is consis-

tently lower than the 3-D connectivity. For specimen 1,

the 2-D horizontal connectivity is still high but is

commonly less than 1, because in 2-D, some sandstone

Figure 6. Core-plug-sized sam-ples from specimen 2 (lithofa-cies assemblage B). Sandstoneand mudstone layer continuity ishigher at the core-plug scalethan at the rock specimen modelscale. Consequently, core-plugmeasurements tend to overes-timate horizontal permeabilityand underestimate vertical per-meability (see Figure 11 andassociated text). Each samplemeasures 8 � 2 � 2 cm (3.14 �0.78 � 0.78 in.).

Jackson et al. 515

bodies are entirely enclosed by mudstone (e.g., the

front face shown in Figure 3e). The 2-D vertical con-

nectivity is also high in most sections but is zero in

some where one or more mudstone layers extend com-

pletely across the specimen (e.g., left-hand face in

Figure 3e). For specimen 2, the 2-D horizontal con-

nectivity is significantly lower than the 3-D value,

because in all sections, there are sandstone bodies that

connect only with one face or that are entirely enclosed

by mudstone. In many sections, none of the sandstone

bodies are connected horizontally (cf. Figures 3e, 6;

note that no sandstone bodies are connected across the

front face). The vertical connectivity is zero as observed

in 3-D.

For both models, the average of the 2-D connec-

tivity measurements is lower than the true 3-D value,

albeit by a very small amount in the horizontal direc-

tion in specimen 1. In specimen 2, the 3-D connec-

tivity is lower in the y direction than in the x direction,

yet 2-D connectivity measurements suggest the oppo-

site. These observations suggest that estimates of con-

nectivity (and hence, other reservoir properties such

as permeability and displacement efficiency) obtained

from 2-D outcrop studies, no matter how extensive,

are not representative of the true 3-D values.

The results are similar to those of King (1990), who

investigated the connectivity of overlapping sandstone

bodies using percolation theory. He found that there

Figure 7. Individual sandstone and mudstone layers in specimen 2. This specimen was taken from a thin fining- and muddier upwardsuccession in lithofacies assemblage B, imposed on the overall coarsening-upward and sandier upward succession (Figure 1c).Consequently, the net to gross decreases upward through the specimen resulting in a decrease in sandstone layer connectivity and anincrease in mudstone layer continuity. Images (a) and (b) show sandstone and mudstone layers toward the top of the specimen;images (c) and (d) show sandstone and mudstone layers toward the base of the specimen.

516 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

is a distinct threshold in sandstone fraction below

which there is no connectivity. At low sandstone frac-

tions, the bodies are isolated; as more bodies are add-

ed, they begin to overlap and form clusters. Eventual-

ly, at the threshold sandstone fraction, a cluster spans

the region, and connectivity is established. This thresh-

old is termed the ‘‘percolation threshold.’’ For a con-

ceptual infinite system containing isotropic (square)

sand bodies, King (1990) found that the percolation

threshold occurs at a sandstone fraction of 0.276 ±

0.0004 in 3-D and 0.668 ± 0.003 in 2-D. In general,

King (1990) observed that connectivity is lower in 2-D

than in 3-D.

Qualitatively, we observe similar behavior in the

rock specimen models. For example, in layers with a

low sandstone content, individual sandstone bodies

can be resolved, and these sandstone bodies are like-

ly to be isolated (Figure 7a). In layers with a higher

sandstone content, the bodies overlap to form a con-

nected sheet (Figure 7c). Furthermore, the measured

Figure 8. Illustration of our definition of connected sandstone. Two-dimensional elliptical sandstone bodies are shown in a blackmudstone background. In this example, connectivity is measured between the left-hand and right-hand faces. We define ‘‘connected’’sandstone to be that which is connected to each pair of opposing faces. This is shown in white. Our definition omits (1) sandstoneentirely surrounded by mudstone, (2) sandstone connected to only one face, and (3) sandstone connected to two faces but not to twoopposing faces. These are shown in shades of gray. Note that the connectivity between upper and lower faces is zero.

Figure 9. Total fraction of sandstone in each 3-D rock specimen model (identical to net to gross; denoted t) and the fraction of thissandstone that is connected between opposing model faces in each of the x and y (horizontal) and z (vertical) directions. Sandstoneconnectivity is high despite the 2-D appearance of the rock specimens. Also shown for comparison is the (arithmetic) average of 10measurements of total and connected sandstone fraction on a selection of 2-D sections. The 2-D measurements consistentlyunderestimate the true 3-D connectivity, although the difference is too small to be resolved in this figure for specimen 1 in thehorizontal direction.

Jackson et al. 517

connectivity is generally lower in 2-D than in 3-D

(Figure 9). However, quantitatively, our results are

different. For example, we find zero vertical connec-

tivity in specimen 2, although the sandstone fraction

is significantly larger than the percolation threshold

predicted by King (1990) (Figure 9). This is partly

because we need to account for the anisotropy in the

dimensions of the sandstone bodies, which are gen-

erally much longer in the horizontal (x and y) direction

than they are in the vertical (z) direction. The impor-

tant ratio is that of the system size to the sandstone

body length in each direction. If this is not isotropic,

then the connectivity must be suitably rescaled, and

the percolation threshold will change (King, 1990). Un-

fortunately, the sandstone body length in the rock

specimen models is very poorly constrained and may

be highly variable. Moreover, both the sandstone body

length and the length of a sandstone cluster may be

significantly larger than the model length. In either

case, there will be finite-size effects that cause sig-

nificant variability in the observed percolation thres-

hold (King, 1990). As yet, the percolation threshold in

heterolithic tidal sandstones is poorly understood. Es-

timates based on synthetic models (Ringrose et al.,

2003) suggest that the horizontal threshold is similar

to the value of 0.28 obtained by King (1990), but the

vertical threshold is larger (about 0.5). This is caused

by the anisotropy in the dimensions of the sandstone

bodies.

Effective Permeability

We investigated quantitatively the impact of scale-

dependent sandstone and mudstone layer continuity

on permeability by calculating the effective perme-

ability of both the rock specimen models and a selec-

tion of core-plug-sized samples taken from models.

We did this because qualitative observations suggest

that layer continuity is higher at the core-plug scale

than the specimen model scale (Figure 6). The effec-

tive permeability was calculated numerically in the

x, y (horizontal), and z (vertical) directions, using the

sealed side pressure solver of Warren and Price (1961).

We assume that flow in a given rock specimen model

occurs between each pair of opposing faces, in each of

the x, y (horizontal), and z (vertical) directions, and

that there is no flow across the other faces. This is the

numerical equivalent of measuring the permeability

of the rock specimens in the laboratory, and the

boundary conditions are consistent with those used to

calculate the sandstone connectivity. For simplicity,

the permeabilities of the sandstone (k s) and mudstone

(km) lithologies within each rock specimen model

were assumed to be uniform and isotropic, and per-

meability values were assigned on the basis of lithology.

For each sample, effective permeability values (keff)

were obtained for the x, y (horizontal), and z (vertical)

directions and expressed in terms of a dimensionless

(normalized) permeability kn for a given direction

kn ¼ ðkeff � kmÞ=ðks � kmÞ

Expressed in this way, the normalized effective per-

meability values for a given bedding type are inde-

pendent of the dimensional permeability values (k s

and km) used to populate the model; instead, they

depend only on the sandstone/mudstone permeability

contrast

Rk ¼ ks=km

Figure 10. Specimen 2 display-ing only those sandstone bodiesthat are isolated by mudstonewhen connectivity is measuredbetween the front and back faces(i.e., in the y direction). The wire-frame denotes the outline of themodel; the connected sandstoneand all of the mudstone has beenremoved. A systematic upwarddecrease in the connectivity ofsand bodies is associated withthe fining-upward and muddierupward succession of the spec-imen (cf. Figure 7).

518 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

A normalized permeability of unity corresponds to

an effective permeability equal to the sandstone per-

meability (k s), whereas a normalized permeability of

zero corresponds to an effective permeability equal to

the mudstone permeability (km). The normalized ef-

fective permeability of each rock specimen model was

calculated for a range of sandstone/mudstone perme-

ability ratios, which are representative of those mea-

sured in analogous subsurface facies using a miniper-

meameter on slabbed cores (e.g., Martinius et al., 1999).

As expected, the permeability of the low net-to-

gross specimen 2 is significantly lower than that of

specimen 1 (Figure 11). For both specimen models,

the normalized permeability in all three (x, y, and z)directions decreases with increasing permeability con-

trast (Rk). However, the decrease is more significant

for specimen 2 than for specimen 1 and for the vertical

permeability (z) than for the horizontal permeability

(x and y). Anisotropy in the x and y directions is small.

Permeability values are lower in specimen 2 because

flow must follow more tortuous paths within the sand-

stone or through the low-permeability mudstone. The

vertical permeability in specimen 2 is particularly low

because one mudstone layer extends completely across

the specimen.

In the subsurface, permeability values are typi-

cally measured from conventional core plugs and then

upscaled to the scale of a reservoir model grid block using

averaging techniques (e.g., Haldorsen, 1986; Jackson

et al., 2003). Horizontal (or layer-parallel) permeability

measurements are commonly upscaled using an arith-

metic average, whereas vertical (or layer-perpendicular)

permeability measurements are upscaled using a geo-

metric or harmonic average (e.g., Weber and van Geuns,

1990). The problem in these heterolithic facies is that

the continuity of the sandstone and mudstone layers is

different at different length scales. Consequently, the

arithmetic average of numerous horizontal core-plug

measurements tends to overestimate the true value of

the host rock specimen model (Figure 11) because the

sandstone layers are more laterally continuous at the

scale of a core plug than at the scale of a rock specimen

(Figure 6). Conversely, the geometric or harmonic

average of vertical core-plug measurements tends to

significantly underestimate the true vertical perme-

ability of the rock specimen (Figure 11) because the

mudstone layers are more laterally continuous at the

scale of a vertical core plug (Figure 6).

These results suggest that using averaging tech-

niques to upscale permeability measurements in het-

erolithic tidal facies will introduce an error, regardless

of the technique used, because the continuity of the

sandstone and mudstone layers varies significantly

with length scale. Jackson et al. (2003) suggested that

this error is likely to be minimized in heterolithic tidal

facies if the geometric mean is used to upscale both

horizontal and vertical core-plug permeability mea-

surements. However, we find that both the geometric

and harmonic means significantly underestimate ver-

tical permeability, particularly in the low net-to-gross

specimen 2; the most suitable scheme for upscaling

core-plug measurements of vertical permeability would

appear to be the arithmetic mean. This directly con-

tradicts conventional thinking.

The dimensionless results shown in Figure 11 may

be expressed in dimensional terms for given sandstone

and mudstone permeabilities using the equation

keff ¼ knðks � kmÞ þ km

For example, a sandstone permeability of k s =

100 md and a mudstone permeability of km = 0.1 md

yields the ratio Rk = 103. The corresponding dimen-

sionless horizontal permeability for specimen 1 of kn =

0.9 is read from Figure 11. Substituting these values

into the above equation yields a dimensional horizon-

tal permeability for specimen 1 of about 90 md. The

same process yields a vertical permeability of about

11 md. For specimen 2, the horizontal permeability

is about 28 md, and the vertical permeability is about

0.37 md. If the mudstone permeability is lower, with

km = 0.001 md (Rk = 105), the vertical permeability of

specimen 1 falls to about 8 md, and the vertical per-

meability of specimen 2 falls to about 0.025 md.

Displacement Efficiency

The displacement efficiency in each rock specimen

model was characterized using streamline simulation

techniques (Batycky et al., 1997). We simulated flow

in a given rock specimen model between each pair of

opposing faces, in each of the x, y (horizontal), and z(vertical) directions, and assumed that there was no

flow across the other faces. These boundary condi-

tions are consistent with those used to calculate both

the connectivity and the effective permeability. We

assumed that flow was single phase and injected tracer

with an initial concentration of 1 over one boundary

in each simulation. Diffusion of the tracer was zero.

Mudstone layers had zero porosity and permeability,

so flow occurred only through the sandstone layers.

Tracer simulations provide a reasonable estimate of

Jackson et al. 519

the volume of sandstone likely to be swept during a

multiphase displacement (such as water displacing oil)

and are computationally tractable on these complex

3-D models.

Displacement efficiency was characterized by cal-

culating the fraction of sandstone swept by tracer for

flow in the x, y (horizontal), and z (vertical) directions

after 2, 10, and 50 pore volumes (PV; one PV is the

total volume of the pore space within a model) of

tracer injected. The higher the sweep, the less tor-

tuous the fluid flow paths, and the more efficiently oil

will be recovered.

Figure 11. Effective permeability calculated for each model and for a selection of core-plug-sized samples taken from eachmodel. (a) Normalized horizontal (x and y) permeability (k n) as a function of sandstone-mudstone permeability contrast (R k ) foreach rock specimen. (b) Normalized arithmetic average of 500 core-plug-sized measurements of horizontal (x and y) perme-ability (k n) as a function of sandstone-mudstone permeability contrast (R k) for each rock specimen. (c) Normalized vertical (z)permeability (k n) as a function of sandstone-mudstone permeability contrast (R k) for each rock specimen model. (d ) Normalizedarithmetic, geometric, and harmonic averages of 500 core-plug-sized measurements of vertical (z) permeability (k n) as a functionof sandstone-mudstone permeability contrast (R k ) for each rock specimen. Each core-plug sample measures 5 � 2 � 2 cm (2 �0.78 � 0.78 in.).

520 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

In specimen 1, the fraction of sandstone swept by

tracer is high for flow in the horizontal (x and y)

direction (Figure 12). After only 2 PV of tracer in-

jected, the fraction swept is greater than 0.98 (i.e.,

greater than 98%); after 50 PV injected, the fraction

swept is greater than 0.99. For flow in the vertical

direction, the fraction of swept sandstone is slightly

lower, 0.83 after 2 PV injected, rising to 0.97 after

50 PV injected. The presence of the mudstone drapes

does not have a significant impact on the sweep ef-

ficiency (Figure 13).

In specimen 2, the fraction of sandstone swept is

lower than in specimen 1 and depends more strongly

on the direction of flow. In the x direction, the frac-

tion is 0.92 after 2 PV injected, rising to 0.94 after

50 PV injected; in the y direction, the fraction is 0.67,

rising to 0.74 after 50 PV injected (Figure 12). How-

ever, when compared with the connected fraction of

sandstone, which is 0.94 in the x direction and 0.75 in

the y direction (Figure 10), it is clear that virtually all

of the connected sandstone has been swept by tracer

(Figure 13). However, there is no flow in the vertical

(z) direction because one no-flow mudstone layer ex-

tends completely across the model.

Excepting only this case, after 50 PV of tracer

has passed through either rock specimen, virtually all

Figure 12. Fraction of sandstone invaded by tracer for each rock specimen and for each flow direction. The higher the fraction ofsandstone invaded, the higher the sweep efficiency of a displacement process such as water-flooding. Flow in the x direction wasfrom left to right as viewed in Figure 6; flow in the y direction was from front to back; flow in the z direction was from base to top.The total length of each bar denotes the fraction of sandstone invaded after 50 pore volumes (PV) of tracer injected; the gray sectionof each bar denotes the fraction of sandstone pore space invaded after 10 PV of tracer injected, whereas the black section denotesthe fraction of sandstone invaded after 2 PV of tracer injected. Note that none of the pore space was invaded by tracer for specimen 2with flow in the vertical (z) direction because one no-flow mudstone layer extends completely across the specimen.

Figure 13. Tracer saturation in (a) specimen 1 and (b) specimen 2 after 0.5 pore volumes (PV) injected (that is, soon after adisplacing fluid has invaded each model). Tracer flow was from left to right. The presence of mudstone drapes in specimen 1 has notsignificantly disrupted the flow of tracer. Connected sandstone layers in specimen 2 are well swept except for occasional dead ends;these will be swept later in the displacement. Isolated sandstones remain unswept.

Jackson et al. 521

(>97%) of the connected sandstone has been swept.

Comparable volumes of rock in a reservoir may have

several hundred pore volumes of fluid pass through

them, depending on their location and the large-scale

movement of the displacing fluid. Consequently, re-

gardless of the bedding structure, essentially all of the

connected sandstone in these heterolithic sandstones

is likely to be swept by a displacing fluid such as water,

if sufficient volumes flow through. However, the effi-

ciency with which this fluid displaces the oil from the

pores will depend on other physical processes not cap-

tured by these tracer simulations.

DISCUSSION

Importance of Sandstone Connectivity

We have found that the key control on effective per-

meability and displacement efficiency in these hetero-

lithic tidal sandstones is the continuity and connec-

tivity of both sandstone and mudstone layers. This

depends strongly on the net to gross. At low net-to-

gross values (likely to be <0.28; King, 1990), the sand-

stone bodies will not form a connected network, and

the effective permeability of the facies will approach

that of the mudstone. Such isolated sandstones will be

nonproductive, so these intervals will not contribute to

net pay. As the net to gross increases, the sandstone

connectivity and effective permeability will remain low,

until at some threshold net to gross (the percolation

threshold about 0.28; King, 1990), the sandstone bodies

overlap and form a connected network. The effective

horizontal permeability of the facies will then increase

greatly. Sandstone bodies that are connected will be

efficiently swept by a displacing fluid such as water, and

the facies will be productive. Above the percolation

threshold, as the net to gross increases, the effective

horizontal permeability of the facies will approach that

of the sandstone, and the mudstone layers will have

a smaller effect on the effective vertical permeability

and on the sweep efficiency.

Consequently, high net-to-gross (>0.9), flaser-

wavy-bedded facies will be of good quality, with high

horizontal permeabilities but rather lower vertical per-

meabilities. The presence of discontinuous mud drapes

will not cause significant bypassing of oil. Lower net-

to-gross (<0.5), wavy-lenticular-bedded facies will be

of significantly lower quality but are still likely to be

productive unless the net to gross falls below the per-

colation threshold. The value of this threshold is poorly

constrained but is likely to be less than 0.5 and greater

than or similar to 0.28 (King, 1990; Ringrose et al.,

2003). These intervals represent one type of low

resistivity and low contrast pay (Darling and Sneider,

1993), and our results help to explain why their pro-

ductivity is commonly underestimated or even over-

looked (Gupta and Johnson, 2001). Despite their rel-

atively low net to gross, the sandstone layers may be

well connected, yielding high horizontal permeabil-

ities and sweep efficiencies. Nevertheless, their ver-

tical permeability will be very low.

Our results may also explain the hydrocarbon

shows observed in cored intervals of low net-to-gross,

lenticular- and wavy-bedded heterolithic tidal facies

from the Tilje Formation in the Heidrun field, off-

shore Norway (Martinius et al., 2001). These inter-

vals are volumetrically significant, and the apparently

discontinuous (in 2-D) sandstone lenses and ribbons

are hydrocarbon bearing. There has been debate as to

whether the hydrocarbons migrated into these sand

bodies through microfractures, or because they are

connected in 3-D. Our results strongly suggest that

the sand bodies will be connected, so there is no need

to invoke the presence of microfractures as a hydro-

carbon migration path.

Upscaling and the Representative Element Volume

The rock specimens from which the models were re-

constructed sample only a small volume of the flaser-,

wavy-, and lenticular-bedded facies observed in the

Barnes High Sandstone Member. It is therefore likely

that they do not sample a representative element

volume (REV; Bear, 1972) of the facies. We inves-

tigated whether this is the case by calculating 2-D

correlelograms in the horizontal (x–y) and vertical

(x–z and y–z) planes in each model (Chatfield, 1989)

(Figures 14, 15). These show the correlation between

sample values as a function of sample separation.

We found that the principal axes of the cor-

relelograms are not aligned with the grid axes, and

specimen 1 has different principal axes to specimen 2

(Figures 14, 15). This gives us confidence that the

serial sectioning and subsequent model reconstruction

have not added undue bias to the data. However, it

is not clear whether the different orientation of the

principal axes reflects our arbitrary choice of grid axes

during sectioning and reconstruction or a change in

conditions during deposition. Both specimens show

periodicity in the z direction (Figure 16), reflecting

the alternations between sandstone and mudstone.

522 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

However, the wavelength of this periodicity is much

larger in specimen 1 (approximately 1.5 cm [0.59 in.])

than in specimen 2 (approximately 1 cm [0.39 in.]),

which reflects differences in the thickness of the

layers.

In both models, the sandstone and mudstone lay-

ers are positively correlated in the horizontal direc-

tion up to the boundary of the models (the correlation

is always greater than zero in Figure 16a). This is

evidence of statistically significant correlation of the

sandstone and mudstone layers beyond the bound-

aries of the model. Moreover, the analysis of sand-

stone connectivity described earlier demonstrated that

one mudstone layer extends completely across speci-

men 2. However, 2-D outcrop observations suggest

that mudstone layers in wavy-bedded intervals are not

continuous at length scales greater than that sampled

by the specimen. Typically, they will be truncated by

Figure 14. Two-dimensional correlelogram in the horizontal (x–y) plane, for (a, b) specimen 1 and (c, d) specimen 2. Plots (a) and(c) show the correlelogram surfaces in 3-D; note the spike in each surface where the correlation reaches 1 at the center of the model.Plots (c) and (d) show the same data but in contour form; correlation values greater than 0.3 have been eliminated, so variationsaway from the central spike can be more clearly resolved. A positive value between 0 and 1 at a given distance indicates that pointsseparated by that distance are correlated. The closer the value is to unity, the more closely values are correlated. A value of zeromeans that the points are uncorrelated. A negative value indicates a negative correlation; the more negative the number, the greaterthe anticorrelation.

Jackson et al. 523

Figure 15. Vertical 2-D correlelograms for specimen 1: (a) in the x–z plane and (b) in the y–z plane; and for specimen 2: (c) in thex–z plane and (d) in the y–z plane. Correlation values greater than 0.3 have been eliminated, so variations away from the centralspike can be more clearly resolved (cf. Figure 14).

524 3-D Reservoir Characterization and Flow Simulation of Heterolithic Tidal Sandstones

adjacent sandstone layers. Together, these observa-

tions suggest that the models do not sample a repre-

sentative volume of the facies. If the models do not

sample a representative volume, then the reservoir

characteristics we measure will be specific to the model

and its boundary conditions instead of being generally

representative of the facies (Renard and de Marsily,

1997). Nevertheless, the models provide valuable in-

formation about the 3-D distribution of sandstone

and mudstone layers and their effect on flow, which is

not available from 2-D outcrop observations. Moreover,

they sample a much greater volume (about 104 cm3

[610 in.3]) than a core plug (about 30 cm3 [1.83 in.3]),

so they provide information that is not available from

the subsurface. However, we recognize that the sub-

surface identification and characterization of thinly

bedded, heterolithic tidal facies may be best achieved

through a combination of core and borehole image data

(e.g., Mercadier and Livera, 1993; Goodall et al., 1998;

Prosser et al., 1999). This improves confidence in dis-

tinguishing different types of heterolithic facies (e.g.,

tidal flats, distal lower shoreface, distal mouth bar,

deepwater levees, outer fan fringe facies, etc.). Hence,

it also helps select the most appropriate outcrop ana-

log data for 3-D modeling purposes. Indeed, borehole

image data could be used more extensively to help

acquire representative outcrop data for quantifying res-

ervoir properties at the scale of our rock specimen

models in other reservoir systems.

Ideally, we would reconstruct several larger rock

specimens to investigate representative volumes and

lateral and vertical variations within facies. These mod-

els may also be used to derive effective (upscaled) flow

properties for use in reservoir-scale models (e.g., Hal-

dorsen, 1986; Kjønsvik et al., 1994; Pickup et al., 1994;

Jones et al., 1995; Jackson et al., 2003). We will recon-

struct further specimens in the future, although their

size is limited by logistical considerations. However,

flaser, wavy, and lenticular bedding may also be inves-

tigated using a recently developed, process-oriented

mathematical code, which generates 3-D models of the

bedforms by simulating small-scale tidal sedimenta-

tion processes (Wen et al., 1998; Ringrose et al., 2003)

The models generated by this code may be calibrated

against our real 3-D rock data and the code used to

construct a variety of model architectures at larger sam-

ple volumes.

CONCLUSIONS

1. The key control on effective permeability and dis-

placement efficiency in heterolithic tidal intervals is

the continuity and connectivity of sandstone and

mudstone layers.

2. If the sandstone layers are disconnected (isolated by

mudstone), then the effective permeability will ap-

proach that of the mudstone, and the heterolithic

interval is unlikely to be productive. However, if

the sandstone layers form a connected network, then

the effective permeability will be significantly higher,

and they may be efficiently swept by a displacing

fluid such as water, provided that sufficient vol-

umes flow through the interval. Hence, the facies

interval is likely to be productive.

Figure 16. (a) Horizontal profiles along the center lines of the2-D horizontal correlelograms for specimen 1 and (b) specimen 2;(b) vertical profiles along the center lines of the 2-D verticalcorrelelogram for specimen 1 and specimen 2. The periodicity inthe vertical profiles reflects the alternations between sandstoneand mudstone layers.

Jackson et al. 525

3. Sandstone connectivity is the dominant control on

the transition between productive (pay) and non-

productive (nonpay) rock. There is a threshold net

to gross at which the sandstone layers become con-

nected in 3-D. As yet, this threshold is poorly

understood in heterolithic tidal sandstones. Esti-

mates based on synthetic models suggest that the

horizontal threshold is about 0.28, but the vertical

threshold is rather larger at about 0.5.

4. Measurements of sandstone connectivity in 2-D un-

derestimate the true 3-D value. Consequently, both

individual and averaged 2-D connectivity measure-

ments are poor indicators of 3-D connectivity. If

2-D measurements do not properly capture con-

nectivity, then 2-D models will not properly capture

other reservoir properties such as effective perme-

ability and displacement efficiency. The results of

2-D outcrop studies should therefore be applied

with caution.

5. Core-plug measurements of permeability, no mat-

ter how numerous, are unlikely to yield represen-

tative values at the scale of a reservoir model grid

block because the connectivity and continuity of

sandstone and mudstone layers varies significantly

with length scale. Upscaling vertical permeability

measurements in moderate to low net-to-gross fa-

cies using the geometric or harmonic mean may

lead to a significant underestimate.

6. Intervals of low to moderate net-to-gross, hetero-

lithic tidal sandstones represent one type of low re-

sistivity and low contrast pay. Their productivity is

commonly underestimated or even overlooked. This

is likely to be because despite their low net to gross,

the sandstone layers are well connected in 3-D,

yielding relatively high horizontal permeabilities and

sweep efficiencies but low vertical permeabilities.

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