Reservoir characterisation and upscaling

0 downloads 0 Views 17MB Size Report
European 3-D Reservoir Modeling Conference (Stavanger Norway), 16-17 April. ...... Sedimentary rocks lend themselves naturally for a hierarchical descriptive ...
Reservoir characterisation and upscaling From core to reservoir

Daniel MIKEŠ

1

Acknowledgements I acknowledge the following people and institutions for their contribution to this research at Delft University of Technology. The Beek funds sponsored the project. Koen Weber and Rick Donselaar provided space. Robert Lafite and Jean-Paul Dupont provided time. With Hans Bruining, Paul van Lingen, Alies Bartelds, Cor van Kruijskijk, Hans Batenburg and Patrick Corbett we held long scientific discussions that built the bridge between geology and engineering. Sonny Buter, Johan ten Veen, Renate Ruitenberg and Cindy Assen assisted in the field. Alexander Wiefkers and Wouter van de Waal worked on the probe-permeameter, Niels Nouwens and Omar Barzandji on the flow cell simulations. The Meades family offered their home during the fieldwork, Kentucky Geological Survey supplied outcrop data, Jacques Hagoort supplied Stars and Imex and Karl-Heinz Wolff a personal computer. Rudy Ephraim assisted with petrography and image analysis, André Hoving performed the technical design of the probe-permeameter. Leo Vogt built the probe-permeameter, sampled all cores and photographed them. Boudewijn de Haas critically read the manuscript and Salomon Kroonenberg, Kees Geel and Gert-Jan Weltje pre-reviewed chapters. Patrick Corbett and various anonymi reviewed the chapters. Margit Schmöltz assisted with publishing. All other colleagues at Universities of Delft and Rouen contributed in some way or another, a.o. Saskia Blom, Guido Bracco-Gartner, Pilar Clemente, Allard Martinius, Nicolas Massei, Valérie Mesnage, Buu-Long Nguyen, Jean-Marie Stam. Friends and family provided moral support.

2

Contents Acknowledgements ....................................................................... 2 Summary ....................................................................................... 5 Introduction .................................................................................. 6

Part I Model Construction ............................................ 9 1 Standard facies models ............................................................. 9 Summary ..................................................................................................................9 1.1 Introduction ........................................................................................................9 1.2 Method .............................................................................................................12 1.3 Geological model .............................................................................................13 1.4 Reservoir model ...............................................................................................21 1.5 Example............................................................................................................23 1.6 Results ..............................................................................................................25 1.7 Discussion/Conclusions ...................................................................................25 1.8 Nomenclature ...................................................................................................26 1.9 More standard models ......................................................................................29

2 Standard bed models .............................................................. 38 Summary ................................................................................................................38 2.1 Introduction ......................................................................................................38 2.2 Method .............................................................................................................41 2.3 Dune bedding ...................................................................................................42 2.4 Flow cells .........................................................................................................49 2.5 Flow cell modelling..........................................................................................55 2.6 Discussion/conclusions ....................................................................................62 2.7 Nomenclature ...................................................................................................64

3 Upscaling .................................................................................. 65 Summary ................................................................................................................65 3.1 Introduction ......................................................................................................65 3.2 Method .............................................................................................................67 3.3 Discussion/conclusions ....................................................................................83 3.4 Nomenclature ...................................................................................................85

3

Part II Permeability sampling .................................... 86 4 Sampling ................................................................................... 86 Summary ................................................................................................................86 4.1 Introduction ......................................................................................................86 4.2 Method .............................................................................................................89 4.3 Discussion/conclusions ..................................................................................107 4.4 Recommendations ..........................................................................................108 4.5 Nomenclature .................................................................................................108

5 Probe permeameter............................................................... 110 Summary ..............................................................................................................110 5.1 Introduction ....................................................................................................110 5.2 Experimental Set-Up ......................................................................................112 5.3 Probe ..............................................................................................................112 5.4 Physical model ...............................................................................................116 5.5 Theoretical model...........................................................................................119 5.6 Results and discussion....................................................................................123 5.7 Conclusions ....................................................................................................129 5.8 Nomenclature .................................................................................................129

Part III Appendices ................................................... 131 6 Outcrop studies...................................................................... 131 6.1 Introduction ....................................................................................................131 6.2 Pikeville delta .................................................................................................131 6.3 Baronia Delta .................................................................................................144 6.4 Nomenclature .................................................................................................147

7 Core studies............................................................................ 154 7.1 Introduction ....................................................................................................154 7.2 Pikeville outcrop ............................................................................................154 7.3 Baronia outcrop ..............................................................................................164 7.4 Ardross outcrop ..............................................................................................164 7.5 Ruurloo core ...................................................................................................164

References ................................................................................. 172 Discussion .................................................................................. 181 Conclusion ................................................................................. 182 Index .......................................................................................... 183 4

Summary This work presents a hierarchical and geologically constrained deterministic approach to incorporate small-scale heterogeneities into reservoir flow simulators. We use a hierarchical structure to encompass all scales from the lamina to an entire depositional system. For the geological models under consideration we propose a five-scale hierarchy with sedimentary structure and subfacies as key elements. The five scales are the lamina, sedimentary structure or bed, subfacies, facies, and depositional system. We use the term subfacies to denote a rock type, i.e. a body of rock with internally consistent sedimentary properties. The term facies denotes an assemblage of subfacies. The depositional system may be thought of as a sequence, i.e. an assemblage of genetically related deposits formed during periods between “catastrophic” events such as an abrupt sea level rise. Although sedimentary architecture is very complicated, the hierarchical approach lends itself naturally to implementation of heterogeneity in a reservoir simulator. For this reason we distinguish flow units, flow cells and grid cells. Flow units are subfacies schematised in a rectangular structure. The building block of the flow unit is a particular type of flow cell. Flow cells are periodic unit cells (PUC) that contain one or more sedimentary structures. The advantage of the assumed periodicity is that PUC‟s can be used to obtain average flow characteristics over the representative elementary volume (REV) of the flow unit. Such an REV consists of a large collection of PUC‟s. For flow simulations on the oil field scale we divide the flow units into rectangular grid cells that contain the flow characteristics of the flow cell. The simulations based on SPE 9th comparative program show the importance of incorporating small-scale heterogeneities. The work is divided into three parts: Chapters 1 to 3 cover the hierarchical descriptive model; Chapters 4 and 5 highlight pertinent aspects of data acquisition and measurement strategy. Appendices A and B show results and data. In Chapter 1 eight standard facies models are defined and used as a template for the geologic model, i.e. coastal facies (lagoon, lagoonal estuary, estuary), fluvial facies (braid, meander, stable) and deltaic facies (Gilbert type, mouthbar type). The facies model is converted into an assemblage of flow units. In Chapter 2 nine bedding types are schematised, i.e. massive bedding, graded bedding, horizontal bedding, longitudinal dune bedding, planar dune bedding, trough dune bedding, hummocky cross-stratification, low angle lamination and ripple bedding. The flow cell consists of one or more of these bedding types. In Chapter 3 an integrated upscaling procedure is presented and applied to a specific example, based on the SPE 9th comparative study. This example shows the straighforward application of the approach and the importance of incorporating small-scale heterogeneities into reservoir simulators. In Chapter 4 a measurement strategy is proposed, that is closely related to the small-scale heterogeneity and the sampling instrument. The bedding geometries must be obtained from outcrops. The common occurrence of repetitive bedding patterns is used to optimise the measurement strategy. 5

In Chapter 5 a detailed study of the use of pressure depletion probe permeameters is presented. It is shown that probe permeameters can obtain good estimates of small-scale permeability and a value for the inertia factor, but that they are only suitable for fresh cores. Chapter 6 contains the outcrop studies of a mouthbar-type delta and of a Gilbert-type delta. The geological structure is described in terms of the hierarchical geological model used in our upscaling approach. Chapter 7 presents four data sets of probe permeability measurements. They comprise the two above-mentioned fieldwork areas, an outcrop sample, and a well core. Each is shown to pose specific problems to acquisition of representative permeability measurements.

Introduction Reservoirs are heterogeneous at all scales. An adequate representation of the heterogeneity of the reservoir is a basic requirement for reliable oil production forecasts with flow simulators. Current methods to obtain a reservoir heterogeneity model can be classified into stochastic, process-based and deterministic. In stochastic modelling, uncertainty in the input data is quantified by averaging over multiple realisations. Stochastic models can represent spatial variability and correlation through semivariograms. These statistics are often not directly related to the geological structures and often disregard the fine scale detail (Abbaszadeh 1995, Alabert 1992, Dubrule 1998, Volpi 1995). Unlike stochastic models, process-based models can represent geological heterogeneity in a realistic way. The basic problem facing process-based geological modellers is the dissipative nature of sedimentary systems, which poses fundamental problems to conditioning (Watney et al., 1999). This technology can not yet be used to build fully conditioned 3-D models of reservoir architecture, but the future looks promising. There is a vast literature on deterministic geological models, but a fully integrated approach from a geological description to a reservoir model that can be incorporated in reservoir simulators is lacking. Such an integrated approach evidently includes a measurement strategy to obtain the relevant parameters. Therefore our objective here is to find a geologically based reservoir model that is sufficiently comprehensive to be used in the reservoir simulation practice. It turns out that a hierarchical procedure forms a natural way to accomplish this. From the geological point of view the sedimentary structures on the small scale are well known. There is now a coherent classification for dunes (Ashley 1990) and dune bedding (Allen 1963). Dimensions for dunes and dune bedding can be inferred from literature (Allen 1973, Bridge 1997). Dunes come in many varieties and dimensions, and occur in a wide range of depositional environments (Collinson and Thompson 1989, Reineck 1960). Studies elaborate on dune dimensions, dune characteristics against hydrodynamic conditions, dune migration, and internal build-up (Allen 1963, Allen 1973, Ashley 1990, Collinson and Thompson 1989, Reineck 1960) and by numerical modelling (Rubin 1987). Also, the dune bedding that results from a particular dune type is well-understood (Ashley 1990, Reineck 1960, Rubin 1987). Pickup et al. (1994) focus on the small scale and give extensive information for numerous sedimentary environments including dimensions, bedding type, permeability distribution, trends, and contrasts. They come up with a 6

geologic atlas and they use 'geopseudos' to scale up the effective flow properties of sedimentary structures. A number of authors present a more or less integrated approach. Although each deals with several upscaling steps, none gives the link to reservoir simulation. Yinan (1984) and Peihua (1986) give an elaborate model of a specific genetic unit i.e. point bar, which contains all bedding types, internal and external boundaries and vertical and lateral trends. Ringrose et al. (1996A) integrate small-scale heterogeneities into an upscaling procedure incorporating all types of internal structures of cross-beds, their lamina types and boundary characteristics. Kjønsvik et al. (1994) incorporate several scales into an upscaling procedure giving flow unit dispositions, the average parameters and their boundary characteristics, acknowledging the small-scale characteristics. Aigner et al. (1999) present a methodology that uses the “genetically driven” hierarchy of heterogeneities and the facts that units are repetitive and heterogeneities occur on several scales. The approach presented here builds on the work of Weber and Van Geuns (1990), which is further elaborated by Mijnssen (1991). Weber and Van Geuns propose three types of geometrical build up. They distinguish layer cake, jigsaw puzzle, and labyrinth reservoirs. Mijnssen uses the flow unit as a building block of deltaic structures. In other words he offers a set of genetic units with tentative dimensions, sedimentary structures, grain size, sorting, permeability, porosity and gamma ray logs for each of his „typical‟ deltas. In this work, the hierarchical concept is added to this approach, which proves to be the natural way to incorporate geological structures into reservoir simulation models. Chapter 1 illustrates the large-scale part of the approach by defining eight standard facies models that are used as a template for the geologic model. These eight facies do not cover all sedimentary environments, but serve as a demonstration of the procedure. The standard facies defined are coasts (lagoon, lagoonal estuary, estuary), rivers (braid, meander, stable) and deltas (Gilbert type, mouth bar type) as the basis of the geological model. (Litho)facies denotes an association of rock types, (litho)subfacies means rock type. A standard facies model is a morphological model that contains all subfacies of the system. A flow unit is a rectangularized version of the subfacies. Chapter 2 shows the small scale part of the procedure by schematizing nine bedding types i.e. massive bedding, graded bedding, horizontal bedding, longitudinal bedding, planar dune bedding, trough dune bedding, Hummocky cross-stratification, low angle lamination and ripple bedding. The flow cell consists of one or more of the bedding types. Chapter 3 presents the integrated approach, which goes from geological modelling to flow simulation. It is demonstrated how Periodic unit cells (PUC) are used to obtain average flow characteristics for the representative elementary volume (REV) of the flow unit. Flow simulations on the oil field scale use the thus obtained flow characteristics. A specific example, based on the SPE 9th comparative study is presented in detail. This example shows the importance of incorporation of small-scale heterogeneities, because they can cause trapping of a considerable fraction of the Oil Initially In Place. Chapter 4 proposes a measurement strategy that is closely related with the small-scale heterogeneity and the sampling instrument. The properties of the bedding structures must be obtained from outcrops. One of the central issues in a measurement program is the optimization of the measurement strategy. The new aspect introduced here is to use the geological understanding of 7

the heterogeneity to confine the measurements to representative parts of the entire structure. The measurements comprise length and height of cross-bed, the foreset dip, thickness and permeability of the coarse, fine foreset and the bottomset. Chapter 5 presents a detailed study on the use of pressure depletion probe permeameters. It is shown that probe permeameters can obtain good estimates of small-scale permeability and a value for the inertia factor can be obtained. However, it is found that the probe permeameter is only suitable for freshly obtained cores. For outcrops grain size and porosity analysis can give an estimate of permeability, by the application of Carman-Kozeny relation. The measurements are an essential element in the deterministic geologically based upscaling procedure. Chapter 6 gives the descriptions of the two main fieldwork studies that comprise an example of a mouthbar-type delta and of a Gilbert-type delta. The studies focus on the description of the geological structure in terms of the hierarchical geological model that is the objective in this upscaling approach. Chapter 7 presents the cases of four sites on which probe permeability had been intended. They comprise the two afore-mentioned fieldword areas, an outcrop sample, and a well core. All of them pose problems for permeability measurement.

8

Part I Model Construction 1 Standard facies models Summary The construction of reservoir models is frustrated by the fact that core and well cover only a fraction of the reservoir volume and it is therefore difficult to determine features like facies shape, size and -distribution, inter- and intra facies boundaries and lateral trends from them. These features are, however, critical to fluid flow and they should necessarily be incorporated in the reservoir model and we therefore propose to systematically describe geometry and distribution of facies. To this end we make use of “standard facies models” that a priori contain all elements and boundaries of facies for a number of typical depositional environments.

1.1 Introduction Geological models are commonly based on traditional interpretative techniques, i.e. facies analysis on geometries and sedimentary structures in outcrops. Combined with knowledge from present sedimentary systems this is a powerful tool to reconstruct general sedimentary setting and spatial information. One can identify sedimentary elements, i.e. facies, and construct their spatial distribution yielding so-called facies models. Conventional reservoir models based on wells generally focus on sand and mud distribution yielding rather simplistic geological models with respect to traditional geological models. We also observe that these models lack important information required for flow simulation, e.g. facies distribution, inter and intra-facies boundaries, small-scale heterogeneity (sedimentary bedding) and intra-facies trends. Hence, as a model of a reservoir these models are inadequate. Therefore we propose a systematic procedure to describe sedimentary rocks in a way that all hydraulic properties and heterogeneity levels be incorporated. To this end we make use of “standard facies models” that include all facies, their spatial distribution, shape, flow boundaries and bedding types. Subsequently we demonstrate how to transform this model into a reservoir model consisting of flow units. The procedure provides for 2 important aspects: (1) A robust geometrical model of the geology including all hydraulic elements. (2) The ensuing model is ready to be used for reservoir flow simulation. This chapter covers the large-scale descriptive part of an upscaling procedure that is presented in Figure 1.1 (Chapter 3; Mikeš & Geel, 2006, Mikeš & Bruining, 2006 and Mikeš, 2006). A number of review papers gives a concise overiew of some of the latest upscaling procedures (Ewing, 1997, Pickup & Stephen, 2000, Christie, 2001, Moulton et al., 1998). Although some of these methods are mathematically rather sophisticated, they do not explicitly incorporate true geological structures nor can they easily be incorporated in reservoir simulators. In this paper we present the large-scale descriptive part of an upscaling procedure that does (Figure 1.1), i.e. a 9

geologically based deterministic reservoir model that is sufficiently comprehensive to be used in reservoir simulation practice.

Figure 1.1. Schematic representation of the 4-step upscaling procedure of this work. Step 1: Geological and reservoir model construction. Step 2: Relative permeability and capillary pressure calculation. Step 3: Micro simulation on flow cell. Step 4: Macro simulation on reservoir.

10

Our approach is inspired on work of Weber & van Geuns (1990) that identified three different types of geometrical build-ups of reservoirs, reflecting their facies distribution that is related to their depositional environment. Mijnssen (1991) extended this approach to the geometrical build-up of facies, reflecting their bedding type related to hydrodynamic conditions. Mijnssen (1991) offers a set of genetic units with tentative dimensions, sedimentary structures, grain size, sorting, permeability, porosity and gamma ray logs for each of his „typical‟ deltas. The objective of this work is to present a systematic procedure to construct a geological model and transform it into a reservoir model, whilst incorporating all levels of heterogeneity within. The procedure builds on a six-scale hierarchy that conforms to naturally occurring levels of heterogeneity viz. lamina → bed → facies → facies association/parasequence → systems tract/parasequence set → sequence (Table 1.1). The key element in the procedure for the reservoir model is the flow unit (facies). Table 1.1. Six-scale hierarchy for facies and reservoir models in this study. Geological key elements are facies and bed. Reservoir key elements are flow unit and flow cell. Reservoir units Geological units Facies example Sequence

River valley

Parasequence set

Alluvial plain

Facies association/Parasequence

River

Flow-Unit

Facies

Meander belt

Flow-Cell

Bed

Trough bed

Lamina

Foreset

A common definition of facies, a body of rock with similar sedimentary properties, we read as “consisting of a repetition of one (cross)bed”. And the one of flow unit, a body of rock with similar hydraulic properties, we read as “consisting of a repetition of one flow cell”. The parasequence is a repetitive facies association that is either progradational, aggradational, or retrogradational. A parasequence set or systems tract is also either progradational, aggradational, or retrogradational. This is where generality stops and specificity starts, i.e. the succession and dimensions of parasequences, systems tracts and sequences are to be determined for every specific case. The sequence stratigraphic methodology has been designed for and is therefore only applicable to coastal/shallow marine and deltaic environments, i.e. environments that occupy both coast and shelf and are controlled by marine signals. For environments that are strictly continental, i.e. rivers or intracontinental lacustrine systems, sedimentary sequences are controlled by climatic and/or tectonic signals and therefore traditional sequence stratigraphy does not apply. In this paper we demonstrate our systematic facies and flow unit description on two facies: Gilbert-type and mouthbar-type delta. The remaining standard facies types are presented in the appendix. We chose deltas, because they are the most complicated in terms of internal build-up and because both river and coast can compose parts of it. On the two delta types we follow the procedure of constructing a reservoir model. The construction of the bed model and sampling strategy are presented in Chapters 2 and 4 (Mikeš & Bruining, 2006 and Mikeš, 2006). 11

We started this work with the intention to extend the works of Weber & van Geuns (1990) for the large scale and Mijnssen (1991) for the small scale. We thought to fill-in one of the large-scale models of Weber & van Geuns (1990) with some of the genetic types of Mijnssen (1991). However, this proved to be not too evident. Weber & van Geuns (1990) neither relate facies to depositional environment nor do they provide a specific association and spatial distribution of facies. Mijnssen (1991) neither relates bedding with facies or depositional environment nor does he provide a specific association and spatial distribution of laminae. Neither of them provides spatial distribution of boundaries. We understood that simple extension of these works wouldn‟t suffice and that we needed a new approach. We realised that every depositional environment has a typical facies distribution and decided to select a number of typical depositional environments. To construct their typical facies distribution and determine the bedding type for each facies and include all boundaries. Hence, once the depositional environment of a reservoir be recognised, we can “a priori” construct its geological model, including all elements typical of it. The rest is assignment of values to boundaries and beds.

1.2 Method We provide a systematic construction of a reservoir model, based on two key elements: 1) Flow unit (facies) and 2) Flow cell (bed). Thereto we pose two statements: 1) A depositional system has a typical set of sedimentary facies, 2) A facies consists of a repetition of one bedding type. This allows us to represent a depositional system by a conceptual “standard facies model” in which the spatial distribution of facies is predefined. This model can easily be transformed into a (geometrically simplified) flow unit model. The characteristics of each flow unit (i.e. dimensions, boundary characteristics, and flow cell characteristics) can be acquired from the reservoir deterministically (Mikeš, 2006) or from a data set. Even with lack of data, this methodology supplies a reservoir model that represents all heterogeneity types and scales, enabling natural integration into an upscaling procedure (see also Mikeš, Barzandji, Bruining & Geel, 2001). For deltas we use the classification of Postma (1990), for coasts the one of Leeder (1982), for rivers the one of Miall (1992). From these classifications we select eight facies types as being most typical for subaqueous reservoirs. Gilbert and mouthbar type for deltas (shallow varieties), braid, meander and stable/anastomosed type for rivers. Lagoon, lagoonal estuary, and estuary for shores. For each of these environments we design a "standard facies model". We then transform facies into flow units and hence the geological model into a reservoir model. The value of this approach is that the reservoir model preserves the spatial distribution of facies. It is especially this spatial distribution that controls flow on a regional scale. Moreover, a small number of flow units represents characteristics to hydraulic flow. This makes the method an efficient tool to quickly model fluid flow through a reservoir, enabling routine modelling and sensitivity analysis.

12

1.3 Geological model 1.3.1 Facies models A delta has four distinct areas: delta plain, delta front, delta slope and prodelta (Nemec, 1990). The delta plain consists of an alluvial system; the upper-delta plain is governed by fluvial processes only, the lower-delta plain by both fluvial and marine processes, i.e. river, tides and weather (waves and wind). Distributary channels can be any of the fluvial river types (braid, meander, stable/anastomosed). Inter-channel areas can be marshes/swamps, lakes, evaporite flats, mudflats, chenier plains, beach ridges, bays, aeolian dunes/ergs. The delta front distributary mouthbars (or shortly mouthbars) form the active growth-front of a delta and are governed by fluvial and marine influences. The appearance (morphology, architecture, and lithology) of the mouthbar will depend on many factors, viz.: effluent type, depth ratio and weather- and tidal action. Coastal and shallow marine processes influence the delta front interdistributaries (or shortly inter-mouthbar). The delta slope occurs only if a delta sits at the shelf margin. It is situated below the storm wave base, only fed by gravity flows (debris flows, slumps, and mass flows). The prodelta is in fact a shallow submarine fan, receiving sediment only from suspension out of the riverine flume and occasionally by storm surges (turbidity currents). The architecture of deltaic systems may be extremely complex. Figure 1.2 shows geometric appearances of delta plain, delta front and entire delta, respectively. Table 1.2 presents a list of controlling processes together with features of deltaic elements for the macro, meso and micro scale. Neither list nor figure are exhaustive and are to be considered as a conceptual overview that may be helpful when describing a deltaic system. Most deposition of a delta takes place at the river mouths by a uniform front, mouth bars, or tidal flats. Off shore, sediment is redistributed through wave, wind, and tidal action. The interaction of river and weather together with basin setting dictates the morphology of a delta. In the short term, this morphology remains more or less constant. In the long term, however, allocyclic and autocyclic controls (tectonics, eustasy, climate, and lobe switching) will vary, modifying the delta morphology. As a result, the tri-dimensional evolution of the system will either exhibit a regressive, aggrading, prograding or backstepping structure (Figure 1.2).

1.3.2 Delta types Postma (1990) distinguishes 12 prototypes for so-called “river-dominated deltas”, which are based on a unique combination of distributary system, depth ratio and effluent type. It should be noted that his classification would yield 36 delta types if extended to “wave-“ and “tide-dominated deltas”. From his 12 prototypes we filtered out four, i.e. Gilbert-type delta and mouthbar-type delta for both shallow- and deep water setting (Figure 1.3), from which we only discuss the shallowwater varieties.

13

Figure 1.2. Geometrical appearances of delta plain, delta front and entire delta. Delta plain: channel pattern and channel type. Delta front: mouthbar type and river pattern. Entire delta: delta profile, migration sense and delta plan. Recognition of the geometric distribution of elements is vital for the regional flow paths of fluid flow.

14

Table 1.2. Characteristic properties of a delta for three scales: Parasequence, facies association and facies/bed scales. These items provide a geometric framework for the geological model. They incorporate important properties for fluid flow that are to be incorporated in the reservoir model. Parasequence

Basin

Delta

Position

Depth

Setting

Shelf shelf margin

Shallow Deep

Gilbert-type Mouthbar-type

Size

Shape

Growth

Large small

Lobate Cuspate Elongate/estuarine

Regressive Aggrading Prograding Backstepping

Facies association River type Delta Plain

Delta Front

Supply

Grain size

alluvial fan braidplain meander anastomosed

bed-load suspended-load

gravel, sand sand sand/mud mud

Effluent type

Mouthbar type

Process

inertia friction buoyancy

mid-ground Gilbert-type long narrow beach-spit tidal flat

River Waves Tides

Facies/Bed

Facies

Delta plain

Delta front

Delta slope

Prodelta

Distributary Interdistributary beach floodplain

Mouthbar Clinoform Coast

Offshore Slope

Hemipelagic

Distributary

SubFacies

hor.lam. dune-bdg

Interdistr. Floodplain

Mouthbar

Shore

Offshore

Dune-bdg Hor.lam. hor.lam. Ripple-bdg Convolute bdg ripple-bdg ripple-bdg HCS Low-ang. x-bdg Slumps hor.lam. hor.lam. low-ang. x-bdg HCS Turbidites

15

Figure 1.3. Schematic delta models: A) Shallow-water Gilbert delta. B) Deep-water Gilbert delta. C) Shallow-water mouthbar delta. D) Deep-water mouthbar delta. D.P.= delta plain; D.F.= delta front; D.S.= delta slope; P.D.= prodelta; B.= basin. (from the 12 prototypes for river-dominated deltas in Postmas (1990)). These four typical deltas demonstrate the variety of internal architecture of deltas.

1.3.3 Gilbert-type delta The mouthbars of a Gilbert-type delta join to form a closed front, the active depositional area being the entire delta front. The resulting structure is a single delta lobe, prograding in the shape of large-scale clinoforms (Figure 1.4A). The delta plain typically consists of a subaqueous channel with subaqueous levees and sand flats, filled with planar- and trough cross-bedded coarse sediments, horizontal bedding and superimposed ripple lamination. The delta front is a prograding Gilbert-type mouthbar; its upper part filled by large-scale planar and trough cross-bedded coarse sediments, its lower part by horizontally bedded finer sediments, slumps, convolute bedding, and graded beds. The prodelta consists of horizontal lamination with the finest sediments of the sequence. The grain size varies from one delta to another but always decreases from delta plain to prodelta. 16

The typical build-up of a Gilbert-type delta is rather simple (Figure 1.4A). On the whole one parasequence is coarsening-up with intercalated sand and mud in the lower part and sand in the upper part. The lower generally consists of horizontal bedding and/or horizontal lamination, the upper of graded beds and/or large-scale dune bedding. The progradation of the mouthbar/delta front shows as accretion surfaces called clinoforms.

Figure 1.4. Schematic model of shallow-water deltas. A) Gilbert-type delta (Beaumont and Foster, 1987). B) Mouthbar-type delta (Shannon and Dahl, 1976). They show the contrast of simple build up of a Gilbert delta front and complex build up of a mouthbar delta front.

1.3.4 Mouthbar-type delta In contrast to a Gilbert-type delta, the mouthbars of a mouthbar-type delta form separate active fronts with passive areas in-between. The resulting structure is again a single delta lobe, within which the mouthbar deposits occupy elongated lenses separated by the inter-mouthbar fines (Figure 1.4B). For a “river-dominated delta”, the delta plain consists of channels with graded bedding, large- to medium-scale trough-dune bedding topped with ripple lamination and/or horizontal lamination. The delta plain inter-channels have horizontally laminated fines with an occasional crevasse that has horizontal bedding in the lower part; graded bedding and trough-dune bedding in the upper. The delta front is made up of mouthbars and delta front inter-mouthbars. The prodelta consists of offshore fines with horizontal lamination. The mouthbars themselves consist of topset (bar top), foreset (bar front) and bottomset (probar) Terminology for delta and mouthbar might be confusing due to the fact that mouthbars partially coincide with delta front (bar front) and prodelta (probar). The typical build-up of a Mouthbar-type delta is rather complex (Figure 1.4B). The typical parasequence for mouthbar-type deltas is a function of locally dominant controlling processes (Figure 1.5). Typically the bar top consists of bar back and bar crest with scour and fill of the channel. The bar front is filled with large-scale planar and trough dune bedding in its proximal part, horizontal bedding in its distal part. The probar is made up of horizontal lamination, again the grain size decreases from bar top to probar. Inter-mouthbars compose a common shoreline profile laterally fed by the mouthbars.

17

Figure 1.5. Delta sequences of deltas in the Upper-Cretaceous Funvegan Formation, Alberta (Bhattacharya and Walker, 1992). Delta front successions for socalled A) fluvial-dominated, B) wave-influenced, and C) wave-dominated deltas; proximal prodelta is equivalent to bar front. D) Interchannel bay fill in a riverdominated delta. These sequences show natural variability at bedding scale. Nonetheless, each facies consists of one bedding type or a succession of two bedding types.

Figure 1.6 shows a cross-section through a typical Pikeville delta mouthbar with its type sections for each element. Each of the higher elements overrides the lower one, hence preserving the vertical succession. An exception to this rule is the bar crest that can in cases be eroded by the prograding bar back facies. Figure 1.7 shows an outcrop of a Pikeville mouthbar. Synsedimentary deformation structures evidence the unstable situation of sediments during deposition, due to rapid sedimentation. Figure 1.8 shows a detail of a mouthbar succession and several of its elements.

1.3.5 Standard facies models For the geological models under consideration we propose a six-scale hierarchy with sedimentary structure and facies as key elements. The six elements are the lamina, bed, facies, facies association/parasequence, parasequence set/systems tract, and sequence. We use the word facies to denote a rock type, i.e. a body of rock with similar sedimentary properties which we read as a repetition of one type of sedimentary structure (or a particular aggregate of sedimentary structures). We focus on the build-up of facies within a parasequence and not on the build-up of parasequence sets. First we interpret the facies of the reservoir and choose its corresponding standard facies model. Instead of fitting a facies model onto the data, we define a facies model and search for the data required. An assumption is of course, that ancient and recent sedimentary environments are comparable. Figure 1.9 shows the standard facies models for our two delta types and their boundary characteristics. The continuous delta front of the Gilbert delta makes a uniform build up, with coarsening-up and fining from apex to outline, i.e. coarsening-up on the whole and within parasequences, lateral fining towards the distal parts within every parasequence. The discontinuous delta front of the mouthbar, on the other hand, combined with lateral lobe switching makes both vertical as well as lateral stacking of coarsening-up and fining-out successions.

18

Figure 1.6. Mouthbar elements. A typical Pikeville delta mouthbar consists of four elements: A) probar, B) bar front, C) bar crest, and D) bar back. Shown are their typical sedimentary logs. The bar crest might be eroded by the prograding bar back and hence not be preserved.

Figure 1.7. An outcrop of the Pikeville delta exhibiting a mouthbar succession. The picture shows the increasing thickness of layers towards the top and the synsedimentary deformation of layers.

On this scale the important boundaries are parasequence boundaries and facies boundaries. We assume the parasequence boundaries to act as barriers and the facies boundaries as conductors. We argue that a parasequence boundary inherently represents some time of non-deposition and/or reworking enough to consolidate the surface and render it significantly less permeable than the underlying facies. We argue that a facies boundary within a parasequence is a gradual transition from one lithology to another that doesn‟t limit flow. These assumptions would require testing for every case, but it‟s difficult to obtain permeability values from such a boundary.

19

Figure 1.8. Outcrops of the Pikeville Formation showing a mouthbar succession and details of its elements.

20

Figure 1.9. Standard facies models for shallow-water deltas. A) Gilbert-type. B) Mouthbar-type. Also shown are their stratigraphic sequence model and boundary model for one parasequence. These models contain vital geometric information, i.e. distribution of units and boundaries.

1.4 Reservoir model 1.4.1 Flow unit models Although the geometry of natural systems may be very complicated, a hierarchical approach enables simplification and implementation of heterogeneity into a reservoir simulator. For this reason we distinguish flow units, flow cells and grid cells. Flow units are in our case rectangular structures building up a facies. A flow unit is build up by a repetition of one type of flow cell. The flow unit has the effective or upscaled properties of a flow cell. These effective properties are obtained by flow simulation on the flow cell (Mikeš & Bruining, 2006). Essentially the crossbed corresponds to the flow cell and the facies to the flow unit (Table 1.1), but there are some differences. Facies have an irregular shape (Figure 1.1), whereas flow units in our case have a rectangular shape. A crossbed and foreset-laminae have an irregular shape and may 21

vary along a facies, whereas a flow cell in our case has a rectangular shape and laminae are planes. The flow cell has average properties of crossbeds within the flow unit. If a facies consists of a number of zones with different bedding types, it is to be split up into the same number of sub-facies. If variation is systematic, the flow cell itself can be divided into zones with different characteristics. For example, a consistent vertical sequence throughout the facies or vertical grading within the crossbeds or clay drapes as bottomsets (Mikeš et al., 2006). A flow unit is thus characterised by its internal properties on the one hand, and by those of its boundaries on the other. Here we find another difference between facies and flow unit. A geologically important boundary might have no effect on flow or vice versa. Since a boundary determines the exchange of fluids between two flow units, both its permeability and its lateral extent are important and particularly so because the permeability of these boundaries is difficult to sample. However, boundaries show certain facies specific characteristics, which can be taken into account. Continuous clay layers, for instance, such as maximum flooding surface, partial clay drapes over point bars in a meander belt (Peihua, 1986) or partial bottomsets in 3-D dunes. Figure 1.10 shows the flow-unit models for our two deltaic examples. For both delta types they consist of a single lobe with three large-scale elements, i.e. delta plain, delta front and prodelta. In the figure, the three major elements are taken apart. The Gilbert-type delta is ready as it is, but the mouthbar-type delta is internally made up of sub-elements. The delta plain holds a complete fluvial system and the delta front alternates mouthbars with inter-mouthbars. The choice to use superfacies and facies or facies and sub-facies is academic.

Figure 1.10. Flow unit models for shallow-water deltaic reservoirs. A) Gilbert-type. B) Mouthbar-type. Facies association, facies, flow-unit model and boundaries for one parasequence are shown. The elements of the flow unit model are those of the standard facies model in Figure 1.9, but facies are simplified to rectilinear geometries and boundaries to planes.

22

Figure 1.11. Hypothetical Reservoir model for one parasequence of Gilbert-type and mouthbar-type delta (shallow varieties). The model and its elements are reduced to rectangular blocks, conserving vertical and lateral distribution of facies.

1.4.2 Standard flow unit models The standard flow units for shallow-water Gilbert-type and mouthbar-type deltas (Figure 1.11) are geometrically simplified versions of the above flow-unit models (Figure 1.10). The resulting model is readily usable for a flow simulation study, given that the properties of flow units are yet to be upscaled from the flow cells (Mikeš et al, 2006 and Mikeš & Bruining, 2006) and the flow units have to be split-up into grid blocks. In effect, one flow unit will consist of several grid blocks. Although in this case all gridblocks are rectangular and of equal size neither should necessarily be the case.

1.5 Example As an example we use the Baronia Delta. The delta is roughly 12 km. wide and 12 km. long and its thickness is approximately 200 m. The delta formed a progradational wedge into one of the foreland basins of the southern Pyrenees. The external outline (wedge shape) is that of an estuary that experienced strong tidal influence. Nonetheless the internal structure (clinoforms) is that of a Gilbert-type delta, with topset, foreset, and bottomset. Although we name it estuarine delta it has all characteristics of a "shallow-water, Gilbert-type delta", with a „classic‟ Gilbert-type profile and type B/C feeder system according to Postmas 23

classification (Postma, 1990) for river-dominated deltas. The overall structure is a thickening-up parasequence set, internally consisting of seven parasequences, which are thickening-up on their own account. We interpret this to be shallowing-up successions in a progradational delta. One of the parasequences has been studied in detail (Figure 1.12), which in turn consists of three units with discretely increasing bed thickness from base to top of the parasequence, each having a typical crossbed type (Figure 1.13): Unit 1) Crossbeds with distinct bottomsets (silt/clay) that completely separate subsequent crossbeds and occasional mud drapes on the foresets. Unit 2) Crossbeds with distinct bottomsets, but no mud drapes on the foresets. Unit 3) Crossbeds with occasional mud drapes on the foresets, but no bottomsets.

Figure 1.12. An outcrop of the Baronia Formation showing parasequence number 6 consisting of a delta progradation exhibiting a succession of three units, discretely increasing in bed thickness from base to top.

Figure 1.13. The typical beds for each of the three units in the Baronia delta succession of Figure 1.12.

The three major minerals are quartz, glauconite, and kaolinite, with abundances of 15-70, 20-35, and 10-35%, for units 1, 2 and 3 respectively. Since calcite cement fills all pores, the primary porosity was calculated using image analysis on the calcite cement, giving 9.1-16.6%. All the succession consists of trough-dune bedding. The troughs are supposedly formed by linguoid 3-D dunes and their thickness, width, and length are 0.2-2, 1-20, and 15-40 m., for units 1, 2 and 3 respectively. The thickness of coarse, fine, and bottomsets ranges: 0.8-6, 0.05-3, and 0.1-20 cm. for 24

units 1, 2 and 3 respectively. The foresets dip 20 to 30° and most of the foreset dip directions fall within N350E to N060E. Coarse foresets consist of mL-mU sand, fine foresets and bottomsets of silt and clay. The bottomsets are very pronounced and form „pseudo-beds‟ of up to 20 cm. in thickness. The strong contrast between coarse and fine foreset and the presence of wave ripples and bi-directional current ripples on palaeotroughs indicate a tidal influence. Generally, the larger crossbeds have thicker coarse foresets and thinner fine- and bottomsets than the smaller crossbeds. The larger crossbeds also tend to have their foresets cut at the top and tangential toesets, whereas the smaller beds tend to have topsets and angular toesets. The resulting geological model is shown in Figure 1.14. The system represents a funnel shape, with seven consecutive parasequences. Parasequence 6 has been studied in detail and consists of three thickening-up successions. Each succession has one typical crossbed. The dominant bedding types for successions 1, 2, and 3 are crossbed type 1, 2, and 3, respectively. The resulting reservoir model is shown in Figure 1.15. The outlines are rectilinearised, but the overall geometry is conserved. We chose to represent every sub-succession by only one flow cell type with average values although there is a trend within each successsion.

1.6 Results This study shows that a hierarchical and geologically constrained deterministic approach is appropriate to incorporate all heterogeneity scales, i.e. from lamina to depositional system, into reservoir flow simulators. It presents a procedure that is straightforward and intuitively applicable to characterisation and modelling of reservoirs. It also shows that reservoir models can very well be conceptualised through the use of rigid standard models with facies and crossbed as key elements for the geological model and flow unit and flow cell for the reservoir model. The strength of this approach is the characterisation of the reservoir with a small number of flow units, whilst preserving the spatial distribution of these.

1.7 Discussion/Conclusions Quantitative geometrical data are best acquired from the reservoir from wells, cores, and seismic lines. If not available an estimate of minimum and maximum values of parameters has to be made. In the modelling we then take minimum and maximum values from available data. We propose parasequence boundaries to be barriers and facies boundaries to be conductors to flow. We argue that a parasequence boundary inherently represents some time of non-deposition and/or reworking enough to consolidate the surface and render it significantly less permeable than the underlying facies. We argue that a facies boundary within a parasequence is a gradual transition from one lithology to another that doesn‟t limit flow. These assumptions would require testing for every case, but it‟s difficult to obtain permeability values from such a boundary. Our approach can be applied to a reservoir at any stage of exploration or production. One can use it with relatively few data and simple methods and subsequently add data and more sophisticated methods to availability of data, will and time. The idea of this approach is to create a 25

straightforward procedure that works for any depositional system and for any level of sophistication. This approach presents some aspects that make it a valid and original method. From this work the following conclusions can be drawn: Standard facies models can serve as a template for reservoir models. Flow unit and facies are key elements of the reservoir and geological model, respectively. Reservoir models can be represented by a small number of units, whilst preserving all levels of heterogeneity, the spatial distribution of facies/flow units and their hydraulic flow properties. All elements (facies) and boundaries for a specific depositional environment are universal, whilst facies geometries are specific. Facies are similar and bedding is repetitive, hence 1 bedding type per facies is sufficient.

1.8 Nomenclature T W L t w l

= = = = = =

Thickness Width Length thickness of crossbed width of crossbed length of crossbed

tB tF tC DB DF DC

= = = = = = =

foreset angle dip thickness of bottomset thickness of fine foreset thickness of coarse foreset mean grainsize of bottomset mean grainsize of fine foreset mean grainsize of coarse foreset

B

=

mean porosity of bottomset

F

=

mean porosity of fine foreset

C

=

mean porosity of coarse foreset

26

Figure 1.14. Geological model for the Baronia delta. A) Facies association with outline, lateral cross-section and longitudinal cross-section. B) Facies with longitudinal cross-section and lateral cross-section. C) Bed with bed types 1, 2 and 3. Black triangles indicate thickening beds towards the broader end of the triangle. Arrows point in the direction of grain coarsening. This syntectonic estuarine/deltaic sedimentary system might have been entirely continental. In that case sealevel might not have played a role and (para)sequences reflect tectonic and/or climatic variability.

27

Figure 1.15. Reservoir model for the Baronia delta. A) Facies association with outline, lateral cross-section and longitudinal cross-section. B) Facies with longitudinal cross-section and lateral cross-section. C) Bed with bed types 1, 2 and 3. This is a rectilinearised version of the geological model in Figure 1.14 without trends in the units.

28

1.9 More standard models The construction and application of standard facies and flow unit models has been extensively discussed in the body of this paper for deltaic models. In this appendix we present the remaining six facies models deployed in this work, i.e. shorelines (lagoon, lagoonal estuary, estuary) and rivers (braid, meander, stable). We make no mention of bedding features here, they have to be incorporated once the facies and flow unit model established (Mikeš et al, 2006 and Mikeš, 2006). Figure 1.16 presents hypothetical parasequence build-ups of the three major depositional systems in this work (delta, shore, and river). Figure 1.17 gives a schematic of the entire depositional system from alluvial fan to submarine fan and possible geometric appearances of their elements.

Figure 1.16. Hypothetical parasequence successions for shorelines, rivers, and deltas. A) Coastal/shallow marine. B) Fluvial. C) Deltaic. For marine settings, parasequences would be controlled by sealevel fluctuations. For entirely continental settings, however, they would relate to tectonic and/or climatic fluctuations.

29

Figure 1.17. Possible appearances of a depositional system, from alluvial fan to submarine fan. A) Schematic of depositional system including deposition on the continental shelf, slope, and abyssal plain. B) Appearances of the facies indicated with numbers 1 to 5 in Figure 1.15A. This figure gives an idea of the variability of facies within a depositional system. Their recognition is an asset for reconstruction of a reservoir model.

1.9.1 Shorelines (lagoon, lagoonal estuary, estuary) For shorelines we selected three typical types: lagoon, lagoonal estuary, and estuary (Figure 1.18). In fact, lagoon and estuary are end-members of a continuous spectrum, a lagoon having a barrier island and no input from the land, an estuary having no barrier island and strong input from the land. A lagoonal estuary is an intermediate situation with partial barrier islands and input from 30

land. For all three, the strength of tidal and wave energy modifies their appearance. Possible successions for lagoon and estuary are shown in Figure 1.19. For the lagoon, the vertical succession is identical to the lateral succession from shore to land and for the estuary the succession depends on the position in the estuary.

Figure 1.18. Schematic models of shorelines with moderate wave energy (Leeder, 1982). A) Estuary: note absence of barrier islands and presence of tidal current ridges. B) Lagoonal estuary: note barriers, tidal inlets, and ebb-tidal deltas. C) Lagoon: note abundant washovers and occasional flood-tidal deltas inshore from rare tidal inlets. This Figure 1.shows three discrete members on a continuous range that‟s controlled by the weight of wave energy, tectonic activity and climate.

Figure 1.19. Schematic models of shoreline sequences. A) Lagoon (Selley, 1996), B) Tidal bar (Reinson, 1992), C) Tidal estuarine point bar (Reinson, 1992), D) Fluvial point bar (Reinson, 1992). It shows the strong variability of coastal/lagoonal sequences and the abundance of clay laminae.

Figure 1.20 shows the standard facies models for lagoon and estuary. The models of lagoon and lagoonal estuary are similar, only that the barrier for the lagoonal estuary is discontinuous. Again the sequence boundaries are barriers, the facies boundaries are conductors or baffles to flow. Figure 1.21 shows the simplified facies models, their facies, flow units, and boundaries for lagoon and lagoonal estuary. The lagoon consists of a simple lateral succession of facies. The lagoonal estuary has in fact the same succession, with addition of ebb and flood tidal deltas and channels in the lagoon. Figure 1.22 shows standard facies and flow unit models for lagoon, lagoonal estuary, and 31

estuary and only the shoreface is represented by inclined laminae. The estuarine model is more complex than the other two, in both global geometry and composition of facies.

Figure 1.20. Standard facies models for shorelines. A) Lagoon. B) Estuary. The geometrical distribution of these shoreline models is rather simple, since the facies are laterally uniform.

Figure 1.21. Facies models and flow-unit models for shoreline reservoirs. A) Lagoon. B) Estuary. Facies, subfacies, flow-units and boundaries are shown for one parasequence. The models contain geometrically simple rectilinear elements.

32

Figure 1.22. A possible geological and reservoir model for one parasequence of shoreline reservoirs. A) Lagoon. B) Lagoonal estuary. C) Estuary. Although all ellements are rectangular, the spatial distribution of elements is conserved. They show increasing complexity fom lagoon to estuary for three discrete elements of a continuous range of settings. A similar range might be proposed for an estuary via estuarine delta to delta. It is a demonstration of non-triviality of geological classification.

33

1.9.2 Rivers (braid, meander, stable) For rivers we selected three typical river types, viz. braid, meander, stable river (Figure 1.23). Braid rivers have essentially a broad channel belt with flow overall or in channels between bars for respectively high and low river-runoff. Meander rivers have a migrating sinuous channel path in a meander belt and floodplains at both sides. Stable rivers have one or several anastomosed channels that do not change position. Figure 1.24 shows schematic successions for the three river types. The braid river has overall thinning-up and bar and channel deposits. The meander river has a thinningup pointbar succession and channel fill and the stable river has thinning-up of channel and interchannel deposits.

Figure 1.23. Schematic models of rivers (Galloway and Hobday, 1983). A) Low-sinuosity, braid river. B) High-sinuosity, meander river. C) Stable, anastomosed river. They show main elements and internal build up.

Figure 1.24. Schematic fluvial sequences (Galloway and Hobday, 1983). A) Braid river. B) Meander river. C) Stable river. They indicate typical successions for the three river types. The complexity indicates the sensitivity of the periodic unit cell and its properties to facies analysis.

34

Figure 1.25 shows the standard facies models and boundaries of the three river types. A braid river consists of channels and bars, a meander river of channel, meander belt (several pointbars), crevasses and floodplains. A stable river consists of channels, inter-channels and floodplains. For the braid river only the outer limits are barriers to flow, all internal boundaries are either conductors or baffles. For the meander river, the meander belt boundaries, channel limits and upper part of pointbar boundaries are barriers and all other facies and pointbar boundaries are either conductors or baffles to flow. For the stable river the channel boundaries are barriers to flow. Figure 1.26 gives simplified facies models, their facies, flow units and boundaries for the three river types. Figure 1.27 shows the standard flow unit models for the three river types again with the same facies elements.

Figure 1.25. Standard facies models for rivers. A) Braid river. B) Meander river. C) Stable river. They show typical build up of sequences and distribution of facies and boundaries. Although prone to strong natural variability they contain most elements.

35

Figure 1.26. Facies models and flow-unit models for fluvial reservoirs. A) Braid river. B) Meander river. C) Stable river. Facies, subfacies, flow-units and boundaries are shown for one parasequence. The models are rectilinear simplifications of facies models.

36

Figure 1.27. A possible reservoir model for one parasequence of fluvial reservoirs. A) Braid river. B) Meander river. C) Stable river. Although all ellements are rectangular, the spatial distribution of elements is conserved.

37

2 Standard bed models Summary Small-scale heterogeneities like dune bedding hamper fluid flow and cause oil to be capillary trapped at the intersections of coarser to finer lamina. These effects are fairly well established, but less well established is the method to construct an adequate flow cell model containing critical hydraulic properties like bed shape and –size, lamina shape, -angle and boundaries. Nor is it trivial how to obtain those features from well and core. We therefore propose to systematically describe geometry and distribution of beds. To this end we make use of standard flow cell models that a priori contain all elements and boundaries of beds for a number of bedding types.

2.1 Introduction Geometry and sedimentary structure (bedding) are the key elements for geologists to interpret sedimentary facies. With information like lamina shape, -angle, -thickness, -grain size and -trends a geologist deduces hydrodynamic regime, current, depth, position, sedimentary facies and depositional environment at the time of deposition. Combined with studies on present sedimentary systems, outcrops, lab- and computer studies this is an efficient tool to reconstruct geometry and distribution of bedding and facies. In reservoir studies we observe that, when the same tool is used on wells and cores, the result is less adequate. Wells and cores lack sufficient lateral coverage to interpret lateral extension of bedding and facies. Seismic lines, on the other hand, lack resolution to identify bedding or facies. Hence, reservoir models lack the geological variability and detail of geological models. As a consequence, geometry and distribution of bedding and facies are lost in the reservoir model. In water-wet rock the interfacial energy is less between water and rock than between oil and rock. In order to minimise its interfacial energy the porous medium prefers water in the finer grained (low permeable) part and oil in the coarser grained (high permeable) part. In laminated structures the oil will hence be trapped in the coarse grained laminae against boundaries with finegrained laminae. With many laminae present, especially when inclined, part of the oil may be trapped and thus not produced. The amount of trapped oil depends on lamina thickness, permeability contrast, interfacial energy, fluid flow rate, and viscosity. Estimates of the trapped fraction of the Oil Initially In Place vary from 10% to 60% (Kortekaas, 1985, Hartkamp-Bakker, 1991, and van Lingen, 1998). It is therefore evident that fine-scale heterogeneities may play an important role in oil recovery and must be taken into account for oil recovery predictions. We observe that bedding models used for upscaling are often chosen inappropriately and the data inadequately represent their hydraulic properties. Hence we can conclude that flow cell and reservoir model lack crucial information required for hydraulic flow modelling, i.e. bed shape, bed size, inter and intra-bed boundaries, lamina shape, lamina thickness, lamina trends, grain size and trends. As a consequence the ensuing reservoir model is inadequate. 38

Therefore we propose a systematic procedure to describe sedimentary rocks in a way that all hydraulic properties and heterogeneity levels be incorporated. To this end we make use of “standard bed models”, that include bedding types, spatial distribution, shape, flow boundaries and lamina types. Subsequently we demonstrate how to transform a bed model into a flow cell model. The procedure provides for 2 important aspects: (1) A robust geometrical model of the bedding including all hydraulic elements. (2) The ensuing model is ready to be used for upscaling. A common definition of bedding, a body of rock with similar sedimentary properties, we read as “consisting of a repetition of two laminae”. And the one of flow cell, a body of rock with similar hydraulic properties, we read as “consisting of a repetition of two laminae”. In reservoir fluid flow simulators grid block size tends to exceed lamina size by orders of magnitude, hence to incorporate the effect of bedding, upscaling is required. We propose to perform this upscaling via two-step numerical flow modelling, i.e. first on the flow cells and then on the reservoir (Mikeš et al., 2006). This work covers the small-scale descriptive part of an upscaling procedure that is presented in Figure 2.1 (Chapter 3; Mikeš et al, 2006, Mikeš & Geel, 2006, and Mikeš, 2006). The procedure builds on a six-scale hierarchy, which conforms to naturally occurring levels of heterogeneity, viz. lamina

bed

facies

facies association/parasequence

systems

tract/parasequence set sequence (Table 2.1). The key element in the procedure for the flow cell model is the lamina, which by our definition is homogeneous and isotropic. We hence intentionally disregard intralamina trends. There are several requirements to characterise dune bedding for use in numerical modelling. (1) A consistent classification for dunes and dune bedding. (2) The relation between dune features and sedimentary environment. (3) The relation between dunes and their resulting dune bedding. (4) A flow cell model for every bedding type. The latter are to be called standard flow cells in this work. Since the unit cell of a sub-facies may consist of one ore more bedding types the flow cell may be an aggregate of one or more of these standard flow cells. Dunes come in many varieties and dimensions, and occur in a wide range of depositional environments (Reineck, 1960 and Collinson & Thompson, 1989). Studies elaborate on dune dimensions (Allen, 1963 and Ashley, 1990) dune characteristics against hydrodynamic conditions (Reineck & Singh, 1980), dune migration (Reineck, 1960 and Jopling, 1966) and internal build-up (Allen, 1973, Reineck & Singh, 1980, Hunter, 1985, and Collinson & Thompson, 1989). There is also a variety of dune bedding types, which have been intensively studied in the lab (Reineck, 1960 and Jopling, 1966) in the field (Reineck, 1960, Boersma, 1967, and Ashley, 1990) and by numerical modelling (Rubin, 1987). The dune bedding that results from a particular dune type is fairly well understood (Reineck & Singh, 1980 and Rubin, 1987) and there is a coherent classification for dunes (Ashley, 1990) and dune bedding (Allen, 1963). Dimensions for dunes and dune bedding can be inferred from literature (Allen, 1973 and Bridge, 1997).

39

Figure 2.1. Schematic representation of the 4-step upscaling procedure of this work. Step 1: Geological and reservoir model construction. Step 2: Relative permeability and capillary pressure calculation. Step 3: Micro simulation on flow cell. Step 4: Macro simulation on reservoir.

40

Table 2.1. Six-scale hierarchy of heterogeneity levels for facies and reservoir models in this study. Geological key elements are facies and bed. Reservoir key elements are flow unit and flow cell. Reservoir units Geological units Facies example Sequence

River valley

Parasequence set

Alluvial plain

Facies association/Parasequence

River

Flow-Unit

Facies

Meander belt

Flow-Cell

Bed

Trough bed

Lamina

Foreset

In summary, all knowledge to characterise flow cells is there. The contribution of this work is therefore to simplify and synthesise the relating state-of-the-art and to develop a systematic description of a flow cell including all its elements. The ensuing procedure lends itself to more sophistication whenever required. We started this work with the intention to provide a typical bedding type to each facies in a reservoir model. We thought that this would be a simple appointment of bedding type to facies using one of genetic units from Mijnssen (1991) into one of reservoir models from Weber & van Geuns (1990). We realised, however, that both approaches were to generic and that things were not so easy. We therefore constructed a new approach to reservoir description and we realised that we needed a new approach to description of bedding also. Existing studies do not provide bedding models related to facies and depositional environment. They do not provide bedding associations within a facies. They do not provide facies-specific internal build-up of bedding, i.e. tri-dimensional bed shape and dimensions, bottomset configuration, foreset angle and -thickness. Moreover, they do not provide accurate internal lamina properties. We realised that simple extension of these works wouldn‟t suffice and that we needed a new approach. We decided to select a number of typical bedding types, construct their typical bedding distribution and determine the lamina types for each bedding and include all boundaries. Hence, once the bedding type of a facies be recognised, we can “a priori” construct its bedding model, including all elements typical of it. The rest is assignment of values to laminae following a rigid sampling strategy (Mikeš, 2006).

2.2 Method We define nine bedding types to represent most typical bedding occurring in nature and define a “standard flow cell” for each. We describe this flow cell by its length L, width W, height H, inclination of foresets , bottomset thickness tb, bottomset permeability kb, coarse/fine foreset thickness tc,tf, coarse/fine foreset permeability kc,kf. We thus describe the flow cell with 10 parameters: L,W,H, ,tc,tb,tf,kc,kf,kb. We assign relative permeability and capillary pressure to each lamina according to van Lingen (1998) and perform steady state flow simulation on each flow cell to test sensitivity to the parameters. Capillary trapping occurs on two levels namely bed scale and lamina scale (Nouwens, 41

1997 and van Lingen, 1998). Critical parameters (Nouwens, 1997) for both bed and lamina scale are permeability and permeability contrast. For the bed scale they are also bed length and bottomset thickness. For the lamina scale they are also lamina angle and coarse/fine foreset thickness. The shape of the bed is insignificant.

2.3 Dune bedding 2.3.1 Classification and definitions of dunes and dune bedding Dunes can be either subaqueous or subaerial and either small, medium, large or very large (Table 2.2) and either bi- or tri-dimensional. Bi-dimensional dunes have a laterally continuous crest and form dune trains, whereas tri-dimensional dunes consist of an arrangement of separated dunes. Accordingly, bi-dimensional crossbeds are laterally continuous exhibiting planar bedding in a crosssection perpendicular to paleocurrent, whereas tri-dimensional crossbeds are laterally discontinuous and exhibit dune bedding in a cross-section perpendicular to paleocurrent. Rubin (1987) finds that in principle bi-dimensional and tri-dimensional bedforms yield bi-dimensional and tri-dimensional crossbeds respectively, although flow conditions can impede this straightforward relation e.g. oblique migration of bi-dimensional dunes yields tri-dimensional crossbeds.

Table 2.2. Bedding hierarchy. Five classes of bedforms and dune bedding, with their ranges of length and height. Length categories for bedforms are from Ashley (1990). Height h of the bed is related to its length L‟ by H‟ = 0.0677·l‟ 0.8098 (Flemming, 1988). Length L of crossbeds from Allen (1973) (L‟/2 < L < 4·L‟). Height of crossbeds from Bridge (1997) (0.4·H‟ < H < H‟). Widths of bedforms are not given in literature. Widths for crossbeds in literature range from 5 < W/H< 15; lengths from 10 < L/H < 25 (Hartkamp-Bakker, 1991; Weber 1968), but values of L/H = 80 (Ashley, 1990) have been observed. See for further definitions Figure 2.1. To differentiate between bedforms and ripples we use an accent for bedforms. BEDFORMS (Ashley, 1990) RIPPLES

DUNES small

medium

large

very large

L‟ (Ashley, 1990)

0.04 - 0.6 m

0.6 - 5 m

5 - 10 m

10 - 100 m

100 - 1.000 m

H‟ (Flemming, 1988)

0.003 - 0.075 m

0.075 - 0.4 m

0.4 - 0.75 m

0.75 - 5 m

5 - 30 m

CROSS-BEDS

(Conform Ashley, 1990) RIPPLE LAMINATION

DUNE-BEDDING small

medium

large

very large

L (Allen, 1973)

0.02 - 2.4 m

0.3 - 20 m

2.5 - 40 m

5 - 400 m

50 - 4.000 m

H (Bridge, 1997)

0.0012 - 0.075 m

0.03 - 0.4 m

0.16 - 0.75 m

0.3 - 5 m

2 - 30 m

Bedform types fall into six categories, i.e.: straight crested, sinuous, linguoid, lunate, asymmetrical wave and Hummocks (Figure 2.2). The resulting beds can be planar or trough cross42

bedded in both longitudinal (parallel to paleocurrent) as well as transverse (perpendicular to paleocurrent) cross-section. Notice, that although very different in bedform, sinuous, linguoid and lunate dunes yield similar bedding. Height is often the only readily measurable dimension of crossbeds; length and width have to be estimated using an equation (Table 2.2) or ratios observed in outcrop for specific facies (Weber, 1968, Allen, 1973, and Hartkamp-Bakker, 1991).

Figure 2.2. Schematics of crossbed types and their dune bedding in longitudinal and lateral cross-section (Davis, 1985; Reineck, 1960). A: Straight-crested. B: Sinuous. C: Linguoid. D: Lunate. E: Asymmetrical wave. F: Hummocky Cross-Stratification (HCS).

In accordance with Ashley (1990), we use ripples for the smallest bedforms and dunes for all the larger ones. A dune/ripple consists of several distinct regions (Figure 2.3). Transport takes place over its stoss-side, deposition on its lee-side. There can be three types of laminae in a crossbed, viz. coarse-, fine foreset and bottomset. The foreset results from deposition on the leeface by avalanching, the bottomset from fall-out in the trough (at the toe of the foreset) out of suspension, the topset from fall-out on the stoss-side out of suspension. The foresets can dip 20 to 35°, but in most of the cases dip between 20 and 30° (Collinson & Thompson, 1989). 43

Figure 2.3. Dune features. A dune has three possible lamina types: foreset, bottomset and topset laminae that deposit on stoss side, slipface, and trough, resp.

The existing bedding types are: massive bedding, graded bedding, horizontal bedding, lateral accretion, planar dune bedding, trough dune bedding, hummocky cross stratification (HCS), low angle lamination and ripple bedding. Lateral accretion is a particular case, because the beds themselves are inclined transverse to the current. A crossbed is defined as a layer with internal cross-lamination i.e. inclined lamination (Figure 2.4). The thickness of a lamina varies strongly i.e. between 1 mm. and 10 cm. Several crossbeds bound by a bedding surface are a crossbed set (Figure 2.4). Compilation of the classifications of Ashley (1990) and Allen (1963) for lengths and the use of the formulae of Flemming (1988) and Bridge (1997) for the heights enable us to obtain one hierarchical classification for dunes and dune-bedding (Table 2.2). We disregard ripple-lamination and focus only on dune-bedding for two reasons: (1) Our interest is in those laminae that can be measured with a probe-permeameter (laminae in ripple lamination are too thin); (2) Dune bedding is more important in terms of reservoir properties (ripple lamination hardly allows for flow). Bedding planes, that are a surface of erosion, non-deposition, or abrupt change in character, bound the beds. These boundaries are visible because of a change in composition, texture, or internal structure. Heterogeneity in crossbeds manifests itself in two ways, either as (1) Alternation of coarse- and fine laminae, or as (2) Grading within laminae. Alternation of fine and coarse laminae is due to (a) Intermittent avalanching (Reineck, 1960) caused by the building-up of grains at the crest before sliding down, or (b) Velocity changes during a tidal cycle, causing avalanching at times of strong current and fall-out from suspension at times of weak current. Grading within laminae is due to (i) Differential trajectories and sorting within the heavy-fluid layer (Jopling, 1966), or (ii) Segregation into flow zones (Reineck, 1960), coarse grains on the foreset, suspension fall-out on the bottomset, or (iii) Differential shear within avalanches (Reineck, 1960), larger grains move up and farther away. 44

Figure 2.4. Schematics of the three main dune bedding types and their hierarchy, modified from McKee & Weir (1953). A) Crossbed sets for the three main dune bedding types. B) Simple dune bedding. C) Planar dune bedding. D) Trough dune bedding. Internal build up, however, may present a wide variety of alternatives.

Hydrodynamic conditions thus dictate shape and heterogeneity of laminae. The above variations may give the following varieties. Lamina shape can be either 1) straight, 2) concave i.e. tangential towards the bed base, or 3) concavo-convex i.e. tangential towards both bed top and bed base. The heterogeneity of foresets can be either bimodal i.e. alternating coarser and finer laminae or unimodal i.e. repetition of identical laminae that have internal grading. Figure 2.5 tentatively presents some typical flow regimes and their resulting lamina characteristics. Figure 2.5A shows hydrodynamic zones around a dune. Figure 2.5B,C refer to a high/low -energy river, respectively. Figure 2.5D,E refers to a high/low-energy distributary channel, respectively, with strong tidal influence. Figure 2.5F refers to a bi-directional tidal regime. Figure 2.5G refers to a high-energy distributary channel with weak tidal influence. To the left of the profiles, the current direction i.e. unidirectional/bi-directional, current velocity i.e. unimodal/bimodal, and current strength i.e. nil/weak/strong are indicated. Schematic curves near the downstream side represent the velocity profile through time with the x-axis corresponding to velocity zero, the scale is relative. The area fill indicates the main settling grain size i.e. black = mud, small dots = finer sand, large dots = coarser sand. To the left of the profiles, the avalanche grading i.e. weak/strong, avalanche continuity i.e. intermittent/continuous, and tidal influence i.e. absent-very strong are also indicated.

45

Figure 2.5. Tentative schematics of internal build up of bedforms with respect to flow regime. A) Hydrodynamic zones around a dune. B) Strong river current. C) Weak river current. D) Strong river with strong tidal influence. E) Weak river with strong tidal influence. F) Estuarine or wadden channel. G) Strong river with weak tidal influence. Current and grain flow features are indicated to the left. The curves to the right represent velocity vs. time. Shading indicates grain size of settling sediment.

46

Figure 2.5B: In a high-energy river channel the current velocity is uniform; the bedload grains bypass the crest without accumulating to settle in the flow-shadow. Avalanching is continuous; the foresets dip gently and are concave. Grading is subtle. Figure 2.5C: In a low-energy river channel the current velocity is also uniform; but here the bedload grains accumulate at the dune crest, before avalanching down the lee-slope. Avalanching is intermittent, the foresets are steep and straight. Grading is coarsening-up and possibly coarsening from brink-point to toe-point. Mud settles at the toe of slope and in the trough forming bottomsets. Figure 2.5D: In a high-energy river channel affected by tidal influence the current velocity is periodical and unidirectional, varying between high- and low energy. Foresets deposit in continuous avalanching when the current is strong like in Figure 2.5B. Mud settles from suspension when the current is weaker. Avalanching is continuous, foresets dip gently, and are concave. Figure 2.5E: In a low-energy river channel affected by tidal influence, the current velocity is also periodical, yet unidirectional, varying between low- and even lower energy. During stronger current foresets are deposited by intermittent avalanching as in Figure 2.5C, during weaker current mud settles from suspension. Foresets are steep and alternate with concave bottomsets. Figure 2.5F: In an estuarine or wadden deposit proximal to the sea the current velocity is not only periodical, but bi-directional as well. Depending on the strength of the current, the foresets are of the types described in Figure 2.5B or Figure 2.5C; commonly the current in one direction is stronger than in the other. The result is an alternation of sets of coarser and finer laminae in both directions, separated by mud layers, formed by settling out from suspension during slack water. Figure 2.5G: In a an estuarine or wadden deposit distal to the sea the current velocity is periodic, yet unidirectional. Coarser foresets deposit at times of strong current like in Figure 2.5B, being steeper, concave, with hardly any grading. Finer foresets deposit at times of weaker current, like in Figure 2.5C, dipping gentler. They are straight and exhibit strong grading. The combination of processes yields a large number of lamina features, which fall into five groups that characterise the heterogeneities of a crossbed (Figure 2.6). These groups are lamina shape (straight; concave; concavo-convex), lamina variation (sand; coarse/fine-sand; sand/mud) layer continuity (continuous; semi continuous; bottomset), climbing type (supercritical; critical; subcritical), and lamina structure (coarsening-up; lateral coarsening). The appearance of the three main lamina features corresponds with their controlling processes (Figure 2.7). Beneath the profiles the lamina features are indicated: lamina shape (straight; concave), lamina variation (unimodal; bimodal), continuity (continuous; discontinuous); to the right their corresponding controlling processes: velocity strength (low; high), process (river; tides), avalanching (continuous; intermittent). Thus, angular and tangential laminae correspond to weak and strong current strength, respectively. Unimodal and bimodal laminae correspond to river or tide dominance, respectively. Continuous and discontinuous bottomset correspond to continuous and intermittent avalanching, respectively.

47

Figure 2.6. Schematic overview of lamina structures: 1) Lamina shape. 2) Lamina variation. 3) Mud layer continuity. 4) Climbing type. 5) Lamina structure. These show possible geometries and heterogeneities within a crossbed, recognition of which is vital to a flow cell model.

Figure 2.7. Tentative schematic of lamina structure as a function of depositional process. Primary processes: 1) Velocity: weak/strong/zero. 2) Process: river/tides. 3) Avalanching: continuous/intermittent. Laminae structure: a) Shape i.e. angular/tangential. b) Variation i.e. uniform/binary. c) Continuity i.e. continuous/discontinuous. These six examples present type heterogeneity models. The resulting lamina structure would be a function of those three processes and structures. These structures incorporate important features for fluid flow.

48

In Figure 2.7 the most common of the lamina appearances of Figure 2.6 are shown. Moreover, concave and concavo-convex are essentially identical to a fluid simulation, because the bottomset is commonly much less permeable than the topset. Since the top of one flow cell corresponds to the base of another, there is no need to represent the topset. Similarly, coarse/fine sand and sand/mud alternations have different values, but the same geometry. Semicontinuous foresets are rather rare. Supercritical- and critical climbing occur mostly in ripples. For now, we disregard intra-lamina texture in our models, since we consider a lamina homogeneous and isotropic.

2.4 Flow cells 2.4.1 Flow cell types We use the flow cell to represent bedding in a numerical flow simulation. Figure 2.8 shows this flow-cell and its properties. The model has a rectangular shape and an infill of three kinds of laminae; the coarse- (c) and fine foreset (f), which alternate and are inclined to the bed base, and the bottomset (b).

Figure 2.8. The flow cell and its properties: β = inclination of flow cell, α = inclination of laminae, L/W/H = length/width/height of flow cell, tc,f,b = thickness of laminae, kc,f,b = permeability of laminae. The orientation of the long axis of the crossbed can be obtained from dipmeter logs or borehole imaging logs. The geometry of a crossbed is hence characterised with 9 parameters and the crossbeds 1-phase permeability with 3 parameters.

The properties partly relate to the geometry of these elements i.e. inclination of flow cell β, inclination of laminae α, length/width/height of flow cell L/W/H, thickness of laminae tc,f,b and for the other part to the permeabilities of laminae kc,f,b. The inclination/sedimentary bedding is taken horizontal (β=0), assuming flow to be parallel to the sedimentary bedding plane. For the bidimensional numerical flow simulation we assign W=1 without loss of generality. This leaves us with 9 free parameters: L,H,α,tc,tf,tb,kc,kf,kb.

49

In practice, only a limited set of typical flow cells occurs (Figure 2.9), which we call “standard flow cell models”. At the level of a flow cell, the only difference between planar- and trough dune bedding is the presence of a „bottomset‟ enveloping the trough dune model.

Figure 2.9. Standard flow cell models for nine typical bedding types. Lamina shapes are rectilinearised, conserving average dip. Massive bedding is homogeneous, whereas graded bedding may have permeability increasing upward or downward. HCS = Hummocky Cross Stratification. Depending on their complexity, the flow cell models are described by 9 parameters or less.

2.4.2 Bed properties The bed properties (L, H) determine the geometry of the flow-cell model. The height (H) of the bed is readily available from a core. Length (L) of the bed is to be assigned using the formulae from Table 2.2 or from H:L-ratios in literature (Table 2.2) for specific facies (Weber, 1968 and Flemming, 1988) or drawn from a distribution function.

2.4.3 Boundary properties The fine-grained envelope of a crossbed is commonly called the bottomset. It is in reality a superposition of multiple bottomsets that together form one mud layer. The position and continuity of the mud layer depend on the nature of the dune. Figure 2.10 tentatively shows the position of mud settling (read: bottomset) for straight crested, sinuous, linguoid, and lunate dunes. Straight-

50

crested and sinuous dunes yield planar dune bedding (Figure 2.10A,B) and linguoid and lunate dunes yield trough dune bedding (Figure 2.10C,D).

Figure 2.10. Possible bedforms (dunes) and their supposed bottomset configuration indicated in grey. A) Straight-crested. B) Sinuous. C) Linguoid. D) Lunate. For so-called two-dimensional dunes (A,B) the resulting composed bottomset is laterally continuous and of constant thickness, whereas for so-called threedimensional dunes (C,D) the resulting composed bottomset configuration is discontinuous and of varying thickness.

As the dune migrates, each new bottomset partially overlaps its predecessor, forming a stack of bottomsets. The bottomset for straight crested and sinuous dunes is laterally continuous and has the same thickness everywhere (Figure 2.11C), whereas the bottomset of linguoid and lunate dunes can be either continuous or discontinuous and its thickness varies (Figure 2.11B). Now, there are three possibilities as to the bed envelope: (1) No bottomset (Figure 2.11A); there is no bottomset, thus there is no mud layer enveloping the crossbed: (2) Discontinuous bottomset (Figure 2.11B); only part of the bed base has bottomset properties yielding two alternatives: a) Non-touching: bedding planes consist only partially of bottomsets; b) Overlapping: bedding planes consist entirely of bottomsets and the sides of the flow cells consist partially of a bottomset. (3) Continuous bottomset (Figure 2.11C); the bottomset encloses the flow cell entirely. If we consider one flow-cell, its boundary configuration is one of the alternatives in Figure 2.12 that correspond to Figure 2.11, completed with the two possibilities for planar dune bedding. Figure 2.11 shows possible cross-bedded sequences, whereas Figure 2.12 shows only one flow-cell for each of the possibilities. Case 1 is an „open box‟ without bottomsets, thus yielding no capillary trapping on the bed scale. Case 3 is a „closed box‟ enclosed by bottomsets, yielding capillary trapping on the bed scale for both horizontal and vertical flow. Case 2a is partially closed to the base, yielding only retardation of vertical flow. Case 2b is closed to the base and partially closed to the sides, yielding capillary trapping on the bed scale for vertical flow and retardation for horizontal flow. Capillary trapping on lamina scale occurs for all scenarios. 51

Figure 2.11. Three bottomset scenarios and their corresponding flow model for a bedset: A) Absent. B) Discontinuous. C) Continuous. If discontinuous it can be either non-touching or overlapping. In all cases the composed bottomset thins towards the sides, because the individual bottomset narrows towards the sides (Figure 2.10). The bottomset configuration is constructed from imaginary migration of linguoid/lunate dunes (Figure 2.10). The difference between overlapping and continuous bottomset is that the overlapping does not reach over the full height of the bed.

52

Figure 2.12. Basic flow-cell models for planar- and trough dune bedding for an individual bed. Planar dune bedding either has no or a continuous bottomset. Trough dune bedding either has no, a discontinuous or a continuous bottomset. The discontinuous bottomset can be either non-touching or overlapping. In this Figure 2.it is well seen that the overlapping bottomset breaches only half of the bed height, whereas the continuous bottomset breaches the entire height. When there is no bottomset planar and trough dune bedding are essentially identical to fluid flow.

A core of a through cross-bedded sandstone that contains a series of crossbeds cuts each bottomset at a random location. Within one facies the crossbeds are similar, therefore these points might be seen as if randomly distributed on one bottomset. They will thus yield the permeability distribution of the bottomset of one flow cell. Hence, if they are less permeable than the foresets, we have type A of Figure 2.11. If half of them have low permeability, we can take type B of Figure 2.11 and if they all have low permeability, type C of Figure 2.11 is valid. Also, borehole imaging tools can reveal the nature of bottomsets (Figure 2.13; Mercadier & Livera, 1993, Williams & Soek, 1993, and Knight, 2002).

53

Figure 2.13. Trough dune bedding as seen by the SHDT tool. A) A cross-bedded reservoir section with a bore hole. F is foreset, BL is bottomset. Arrow indicates the paleo-depositional flow direction. B) The unfolded bore hole wall with correlations of the vertical SHDT sections (Hartkamp-Bakker, 1991). Such images provide valuable geometric information of dune bedding that improves flow cell construction.

2.4.4 Lamina properties The lamina properties (α,tc,f,kc,f) describe the characteristics of coarse- and fine foreset. Foreset laminae can have many appearances that depend on the hydrodynamic conditions around the dune. All characteristics of a lamina, i.e. its shape, dip, permeability contrast and bottomset relate to grain size, grain texture, sorting, mud presence and flow velocity at the time of its formation. We consider the permeability of foresets to be bimodal and lognormally distributed, although in some cases e.g. fluvial channel crossbeds it can be unimodal. Furthermore, we take the foresets to be straight i.e. uncurved in either direction and of constant dip α. To avoid complex flow cell models, other properties can be incorporated in an alternative way. Figure 2.14 shows some examples of how a complex lamina configuration can be represented by a simpler configuration that yields similar fluid flow behaviour. Straight laminae without bottomsets can be represented as a flow cell with foresets only. To account for the effect of decreasing inclination and lamina thinning for concave- and concavo-convex laminae, the flow cell can be divided into two regions: an upper with dune bedding and a lower with horizontal lamination. Thickness tc,f and dip α of the foresets can readily be obtained from a core or borehole imaging logs (Weber, 1968, Flemming, 1988, Mercadier & Livera, 1993, Williams & Soek, 1993). Permeability (kc,f) has to come from probe-permeameter or from tri-dimensional directional measurements on cubes using a Hassler-sleeve permeameter.

54

Figure 2.14. Simplification of the three possible lamina shapes of a crossbed into a flow-unit model: A) Straight. B) Concave. C) Concavo-convex. Decreasing dip and thinning of laminae towards the bed base for B and C is represented by a set of thin horizontal laminae. This shows alternatives to represent the effect on flow of a complex geometry through a reduced number of grid cells.

2.5 Flow cell modelling 2.5.1 Upscaling procedure We construct the flow cell through a systematic description of crossbeds (Figure 2.8) that is the small-scale part (Chapter 2; Mikeš & Bruining, 2006) of step 1 in our integrative upscaling procedure (Chapter 3; Mikeš et al, 2006). The large-scale part has been covered earlier (Chapter 1; Mikeš & Geel, 2006). We define both flow unit and flow cell as a body of rock with similar hydraulic properties, which we read for the flow unit as “a repetition of one flow cell” and for the flow cell as “a repetition of two laminae”. The procedure uses two assumptions for the laminae of a crossbed: (1) the permeability distribution within a crossbed is trimodal i.e. coarse-, fine-, bottomset; (2) The permeability distribution of each lamina type in a crossbed is lognormally distributed. Table 2.1 shows the hierarchy we propose for geologic elements. We consider six scales: lamina, bed, facies, facies association/parasequence, parasequence set/systems tract and sequence, which need not all be represented in one reservoir. Numerical simulation is carried out in two steps, micro55

simulation on the flow cells and macro-simulation on the flow units (Mikeš et al, 2006). Therefore flow cell and flow unit are key elements in this procedure. For each scale we define a representative elementary volume REV (Bear, 1972) or periodic unit cell PUC (van Lingen, 1998) in case that such a REV is periodic. We use the averaged properties within the REV such as relative permeability and capillary pressure to describe the properties in a numerical grid for flow simulation in a domain consisting of grid blocks, which are considerably larger than the REV. For us the following two REVs are relevant: (1) The lamina. We consider the lamina to be the REV for the flow cell and the averaging volume the lamina volume. We obtain permeability on a core by use of a probe-permeameter (van de Waal et al., 1998) or averaging of cubical samples in a Hassler-sleeve permeameter (Bartelds et al., 1996). Alternatively, we might estimate permeability from image analysis on thin sections with specific surface, porosity, cementation, and sorting (Ruzyla, 1986 and Bartelds et al., 1996). We then calculate for the laminae relative permeabilities with Brooks-Corey equations (Brooks & Corey, 1966) and capillary pressure with Buckley-Leverett-J equation (Leverett, 1941). (2) The flow cell. We consider the flow cell to be the PUC for the flow unit and the averaging volume the flow cell. Periodicity implies that averaged properties of one flow cell represent average properties of the entire flow unit. We subject the flow cell to numerical steady-state flow simulation and use capillary pressure PC and 1- and 2-phase permeability of the laminae Kc,f,b and Krel|c,f,b as input of the flow cell (Mikeš et al., 2001). We use the output of the smallscale simulation, PC,FC, KFC and Krel|FC as input for the flow unit in the large-scale simulation, PC,FU, KFU and Krel|FU (Mikeš et al., 2006).

2.5.2 Baronia Delta We use the Baronia Delta as an example of huw to construct flow cell models. The Baronia Delta can be considered a Gilbert type Delta consisting of seven coarsening-up and thickening-up successions/parasequences. We studied parasequence number 6 in detail. Figure 2.15 shows the sedimentary log and the curves for bed characteristics we obtained, i.e. an average value for each meter of the section for the following parameters: width W and length L of the troughs; thickness of the bed Tbed,sd; thickness of the bed base Tbed,cy; thickness of the troughs Ttrough,sd; thickness of the trough base Ttrough,cy; thickness of the coarse foreset Tforeset,sd; thickness of the fine foreset Tforeset,cy. The parasequence shows a distinction in three units for which the bed thickness increases discretely. Figure 2.16 shows parasequence number 6 and its 3 units and the PDF functions for W and L. Figure 2.17 shows the typical bedding of the three units of this parasequence and the PDF functions of the lamina parameters. Bed thickness refers to thickness of the bed set, whereas trough thickness indicates the individual troughs.

56

Figure 2.15. Sedimentary section of the Baronia Delta, parasequence number 6 and the values for bed and lamina parameters. Each data point denotes the estimated average value for one meter of section. The discontinuous lines indicate the three units. Notably the boundary of units 2 and 3 shows a significant jump in values.

57

Figure 2.16. Photograph of parasequence number 6 of the Baronia Delta with the three units indicated with lines and numbers. The graphs show PDF functions for trough width and length. The difference between units 2 and 3 is obvious, whereas the difference between units 1 and 2 is less subtle. It‟s interesting to see, that unit 1 has narrowest troughs, unit 3 widest and unit 2 has a bimodal distribution. Trough lengths could only be discerned in unit 3.

58

Figure 2.17. Typical bedding of each of the 3 units of parasequence 6 of Figure 2.16. Bed numbers correspond to unit numbers. Bed 1 has bottomsets and occasional clay drapes on the foresets. Bed 2 has bottomsets, but no clay drapes on the foresets. Bed 3 has occasional clay drapes on the foresets, but no bottomsets. The graphs show the PDF functions for lamina properties. Bed thickness increases most clearly from unit 1 to 3. The other parameters are less indicative of a clear trend.

2.5.3 Numerical flow simulation The capillary pressure function of a porous medium can be expressed as a conjunction of capillary tubes as in Figure 2.18A. In a heterogeneous (layered or cross-bedded) medium, each of the layers would have a different capillary pressure curve as in Figure 2.18B. Producing oil from such a medium would increase water saturation and decrease capillary pressure. Each layer would theoretically reach a minimum capillary pressure and hence maximum water saturation dictated by the residual oil saturaton for that layer, 1-Sor in Figure 2.18B. Capillary equilibrium, however, would preclude the capillary pressure to pass below the minimum of the finest layer present. Hence, if a bottomset present, SwC would stick at Sw*CB and SwF would stick at Sw*FB; this would be called bed trapping. If no bottomset present, SwC would stick at Sw*Cf and SwF would reach 1-Sor; 59

this would be called lamina trapping. Hence bed trapping occurs for both coarse- and fine laminae and lamina trapping only for coarse laminae.

Figure 2.18. Capillary pressure curves of porous media. A) Capillary tube model to construct a capillary pressure curve for a porous medium. Capillary pressure decreases with increasing capillary radius (Nouwens, 1997). B) Capillary pressure curves for the three lamina types in a cross-bedded rock, i.e. bottomset B, fine foreset F and coarse foreset C. Individual curves for these layers yield identical end saturations, but different capillary end pressures. If all three are present in a rock, capillary equilibrium precludes the capillary pressure to drop below the capillary end pressure of the bottomset, hence trapping end water saturations of coarse and fine foreset to Sw*CB and Sw*FB. The amount of capillary trapped oil in the layers is then (1Sor)-Sw* (Nouwens, 1997).

Even if a bottomset present, the orientation of flow through a crossbed has an effect on the type of capillary trapping that occurs. Flow perpendicular to the laminae will result in bed trapping for coarse- and fine laminae as well as lamina trapping in the coarse laminae (Figure 2.19A). Flow parallel to the laminae will result in bed trapping for coarse and fine laminae, but no lamina trapping in the coarse laminae (Figure 2.19B). Moreover, part of the trapped oil might escape from coarse to fine layer (Figure 2.19C). The size of the bed in the orientation of flow would determine how much of the oil is trapped on the bedding scale. Hence, a large crossbed would yield less bed trapping than a small crossbed, if the characteristics of the coarse and fine layers be equal (Figure 2.20).

Figure 2.19. The effect of flow orientation to trapping in laminae (Nouwens, 1997). A) Flow parallel to laminae yields bed trapping in both coarse and fine laminae and lamina trapping in each coarse layer against its fine neighbour. B) Flow perpendicular to laminae yields bed trapping in the coarse and fine layers, but no lamina trapping in the coarse layers. C) Part of the trapped oil in the coarse layers might escape to the fine layers.

60

Figure 2.20. The effect of bed size to the amount of capillary trapped oil on the bed scale, i.e. against the bottomset, showing only the coarse layers (After Nouwens, 1997). A) In a large crossbed, bed trapping occurs only in part of the bed. B) In a small crossbed bed trapping occurs in all of the crossbed. C) Water saturation profile. The length at which water saturation reaches its maximum relates directly to the end of capillary bed trapping in figures A and B.

In order to assess the sensitivity of fluid flow to the various parameters we performed a series of two-phase steady-state numerical flow simulations on a cross-bedded flow cell (Nouwens, 1997). The results show that the model shape has hardly any influence so that a rectangular flow cell adequately represents a trough-shaped crossbed. What is critical to bed-scale trapping is whether the bottomset is continuous or not, i.e. whether it reaches up to the uppermost down-stream corner of the crossbed. If it does, the bed is “closed” by the bottomset and trapping in fine foresets occurs. If it is open, even if slightly, the bed is “open” and trapping in fine foresets does not occur. The size of the hole then determines the retardation of the flow, the smaller the hole, the more flow is retarded. Critical parameters for bed-scale trapping show to be bed length and height, thickness of bottomset, permeability of bottomset, and permeability contrast of fine foreset to bottomset. Critical parameters for lamina-scale trapping show to be length of foresets, lamina angle, thickness of fine foresets, relative volume of coarse foresets, permeability of fine foreset, and permeability contrast of fine to coarse foreset. Considering a crossbed in three dimensions, the orientation of flow with respect to the bed is crucial. For flow in the plane of, but perpendicular to paleocurrent, the case is that of flow parallel to the layers, hence yielding bed trapping in coarse and fine foreset and no lamina trapping (Figure 2.21A). For flow parallel to paleocurrent the case is that of (more or less) perpendicular to laminae, hence yielding bed trapping in coarse and fine laminae (Figure 2.21B) and lamina trapping in coarse laminae (Figure 2.21C), which decreases with decreasing lamina dip. An overview of the amount of capillary trapping is given in Figure 2.22 for both transverse and longitudinal flow.

61

Figure 2.21. Types of trapping that occur with relation to the orientation of flow through a cross-bedded rock (van Lingen, 1998). A) Transverse flow compares to flow parallel to laminae and yields reduced bed trapping in coarse laminae. B) Longitudinal flow compares to flow perpendicular to laminae and yields bed trapping in coarse and fine laminae and C) Lamina trapping in coarse laminae.

On the whole, the amount of produced oil depends on a number of parameters that are summarised in Figure 2.23. Large-scale heterogeneity affects sweep efficiency and small-scale heterogeneity affects displacement efficiency. Together with the position with respect to injection and production as well as the position in time in the production history determine the amount of oil produced and trapped.

2.6 Discussion/conclusions In our approach we have intentionally made several assumptions to simplify the model. At a later stage, these should deserve some closer examination to establish their effect on fluid flow. We assume a crossbed to have three types of laminae, coarser- and finer-grained foreset and bottomset. In nature things can be more complex, e.g. when mud drapes partly reach up onto the foresets, hence creating a third “foreset type” within the crossbed. We assume the lamina to be homogeneous and isotropic. In nature neither of these need be true. Laminae exhibit grain size trends, causing heterogeneity. Flattened grains can be arranged, causing anisotropy. We assume a flow unit to have one flow cell only. However, there can be trends within one flow unit, which might oblige us to use sub-facies. 62

The direction of flow with respect to lamina configuration has an important effect on flow behaviour and the resulting capillary trapped oil (van Lingen, 1998). To this end we might obtain a permeability tensor from tri-dimensional flow cell simulation. This study yield us the fundamental information required to construct flow cells and allows to draw the following conclusions: Critical parameters for bed-scale trapping are bed length and height, thickness of bottomset, permeability of bottomset, and permeability contrast of fine foreset to bottomset. Critical parameters for lamina-scale trapping are length of foresets, lamina angle, thickness of fine foresets, relative volume of coarse foresets, permeability of fine foreset and permeability contrast of fine to coarse foreset. The flow cell can effectively characterise bedding by a rectangular shape with 10 parameters, viz. length L, width W, height H, inclination of foresets α, bottomset thickness db, bottomset permeability kb, coarse/fine foreset thickness tc,tf, coarse/fine foreset permeability kc,kf. The flow cell can straightforwardly incorporate the effects of small-scale permeability heterogeneity, like in dune bedding, into upscaling procedures.

Figure 2.22. The amount of trapping expressed as a function of capillary pressure (van Lingen, 1998). A) Transverse flow can only yield bed trapping. B) Longitudinal flow can yield bed trapping and lamina trapping

63

Figure 2.23. The production efficiency expressed as the combination of sweep efficiency and displacement efficiency (van Lingen, 1998). Sweep-efficiency is dictated by large-scale heterogeneity, displacement efficiency by small-scale heterogeneity.

2.7 Nomenclature Bed base Bedform Bottomset

= = =

Capillary trapping

=

Crossbed

=

Foreset laminae Flow cell

= =

Flow unit

=

The lower boundary of the crossbed. Surface feature caused by flow (large= dune; small= ripple). The toe of the foreset when mud deposition occurs on it. Bottomset is commonly used for the lower bed boundary, which is in fact an assembly of subsequent bottomsets. the effect of oil retention due to capillary forces across an interface. Bed consisting of laminae inclined to the principal bedding surface. Internal inclined stratification of a crossbed. Body of rock with similar hydraulic properties, read here as= “consisting of a repetition of two different laminae”. Body of rock with similar hydraulic properties, read here as= “consisting of a repetition of one flow cell”. 64

3 Upscaling Summary The effect of small-scale heterogeneities on fluid flow through a reservoir is determined using one of so called mathematical upscaling methods, that as such are fairly well established. Less well established, however, are the operations that provide their input, i.e. construction of reservoir model and acquisition of data. We thereto propose systematic description of facies and bed and systematic data sampling. We upscale the ensuing model and data through a 2-step flow simulation, i.e. first on the flow cell and then on the flow unit scale. This work provides for a straightforward and easily applicable upscaling method that incorporates the effects of all heterogeneities.

3.1 Introduction Since it was discovered that small-scale heterogeneities e.g. dune bedding have an effect on fluid flow (Kortekaas, 1985) studies have addressed their effective bed permeability by a number of mathematical methods, viz. averaging, renormalization, homogenisation, and numerical flow simulation, in order to determine 1- and 2-phase permeabilities and capillary pressures. Studies also addressed geometries of facies and beds, i.e. dimensions, distributions, bedding types and the effect of clay layers. All these methods served to establish the effect of small-scale parameters on a larger scale. This procedure is commonly referred to as “upscaling”. A number of publications have addressed so called “integrated upscaling procedures”, generally referring to a procedure of constructing a reservoir model with upscaling being an integral part. Considering all publications on this subject we can conclude that all necessary techniques to construct a reservoir model have been established. Still, a satisfying “integrated” upscaling procedure has not been designed. Why is this? Let us go through all steps from rock record to reservoir flow simulation. (1) Geometrical model (large and small-scale) comprising geological description, facies analysis and facies model. (2) Data acquisition (sampling, analysis) comprising sampling, data analysis and permeability averaging in homogeneous elements. (3) Upscaling (small to large-scale) comprising flow cell model and calculation of effective flow cell permeability. (4) Reservoir flow simulation comprising petrophysical description, flow units, reservoir flow simulation. Geologists traditionally construct detailed, yet partly conceptual geological models to characterise a sedimentary environment. Reservoir geologists perform flow simulation on a naively simple reservoir model. Petrophysicists provide some theoretical basis for rock properties like porosity, permeability, wettability. Statisticians and mathematicians establish methods to calculate production of a reservoir. For the number and quality of geological tools we observe a decrease from steps 1 to 4. However, for the number and quality of mathematical tools we observe a decrease from 4 to 1. But 65

most importantly there seems to be a trend of conventional geologists focussing on step 1 and a bit on step 2 and of reservoir geologists focussing on step 4 and a bit on step 3. Hence steps 1 and 2 are well established, steps 2 and 3 less. The fundamental question of upscaling is how to incorporate small-scale structures in a largescale reservoir model, without a full-scale model at the resolution of a lamina? The answer is of course to incorporate effects of small-scale structures instead. Since geological structures are strictly hierarchical of nature, the upscaling procedure should conform to this hierarchy. A number of review papers give a concise overview of upscaling procedures (Ewing, 1997, Pickup & Stephen, 2000, Christie, 2001, Moulton et al., 1998). Traditional methods use averaging, renormalisation or homogenisation, while more modern methods use Dykstra-Parsons coefficient, pseudo functions, flow simulation, streamline, finite difference, effective medium or efficient flux. These methods are good as such. They appreciate the natural hierarchy of geological structures, repetitiveness of structures and the mathematical algorithms are robust. Then what are the problems? The problems are one or more of the following. The methods on certain elements of the geology and/or mathematics, even the ones that carry the name of integrated procedures. They do not incorporate features of geological elements explicitly viz. facies distribution, facies boundaries and bedding characteristics. They do not perform sampling and sample analysis systematically. They do not calculate relative permeability and capillary pressure of individual laminae. Hence, the reservoir model represents the reservoir inadequately and upscaling is performed on inaccurate data. And even so, most methods are too complex to be used routinely. The objective of this study is neither to invent new mathematical algorithms nor new geological techniques, but to present a procedure that focuses on the geological model. This paper presents the upscaling part of the procedure and a test on an example to demonstrate its application and efficiency. The other elements of the procedure are presented in other chapters, i.e. facies (Chapter 1; Mikeš & Geel, 2006), bed (Chapter 2; Mikeš & Bruining, 2006), sampling (Chapter 4; Mikeš, 2006). They give the scientific ground for this approach. The idea is to focus attention on geological features and their characteristics to hydraulic flow. On the one hand geometries like shape, dimension, flow boundary, spatial distribution, repetition and the like. On the other hand heterogeneities like 1-phase and 2-phase permeability, contrasts, trends, relative permeability, capillary pressure and the like. We focus on laminae, because we consider a bed the smallest heterogeneous element and a lamina the largest homogeneous element and therefore intentionally disregard intra-lamina trends. We do not consider the fundamentals of petrophysical properties like wettability, interfacial tension, adhesion, pore shape etc. The same goes for relative permeability and capillary pressure. We believe that the first priority is to construct an adequate reservoir model and assign it appropriate hydraulic flow properties. In short, we want to supply adequate data to the upscaling algorithm. To demonstrate the value of this method, we apply it to the model described in the SPE ninth comparative study (Killough, 1995). The results show that small-scale heterogeneities must be explicitly taken into account. Our procedure provides a straightforward method to accomplish this routinely in reservoir modelling. The main purpose of this work is twofold: 1) Improve the 66

transformation of geological model to reservoir model by an “a-priori” deterministic description; 2) Provide a simple and straightforward procedure to upscale reservoir heterogeneity. To this end we propose a four-step procedure (Figure 3.1): (1) Model construction: Identification of flow cells and flow units. (2) Parameter assignment: Sampling of lamina permeability and calculation of capillary pressure and relative permeability of laminae. (3) Micro-simulation: Static (or steady state) flow simulation on flow cells. (4) Macro-simulation: Assignment of the ensuing effective phase permeabilities and capillary pressures to the flow units and dynamic flow simulation on the reservoir, yielding production history. Step 1 and 2 form reservoir characterisation. Step 3 and 4 form numerical flow simulation. We started this study with the intention to sample permeability on outcrops of so-called „reservoir analogues‟. Following its more or less predefined course we encountered many difficulties. Existing probe-permeameters proved inaccurate. Existing sampling strategies and data analysis proved inappropriate. The outcrops proved to be altered. Existing geological descriptions proved inadequate. Existing upscaling procedures proved too complex. Facing so many uncertainties we decided to construct a pressure-depletion probe-permeameter and to sample only on fresh cores from producing hydrocarbon reservoirs. We constructed the instrument, calibrated it extensively on natural and artificial homogeneous and heterogeneous core plugs and it proved close to perfect (van de Waal et al, 1998). Then we confronted a problem of political nature. No oil-company was willing to borrow us a fresh reservoir core. We ended up with a ten years old core, altered just like an outcrop. Although we performed measurements on all our outcrop samples and core, we realised that none of these was representative of a true reservoir. Frustrated by our lack of data, we changed strategy. We decided to construct an upscaling procedure that would be straightforward, efficient and routinely applicable. The basis of it would be the explicit incorporation of hydraulic features of geological elements.

3.2 Method We propose a four-step procedure (Figure 3.1): (1) Systematic description of facies and beds; Assignment of flow units and flow cells and construction of geological model and reservoir model. (2) Systematic data sampling and analysis; Sampling and analysis of lamina permeability. Calculation of capillary pressure and relative permeability of laminae. (3) Systematic „upscaling‟ of lamina permeabilities; Numerical flow modelling on flow cells. Input is 1- and 2-phase permeability and capillary pressure curve of lamina. Output is the effective flow cell values of these. (4) Reservoir flow modelling; Numerical modelling on reservoir. Input is the effective flow cell 1- and 2-phase permeability and capillary pressure of flow unit. Output is the effective reservoir values of these and the production curve. Parts 1 and 2 are entirely original. Parts 3 and 4 might not be new as such, but they make the procedure complete.

67

Figure 3.1. Schematic representation of the 4-step upscaling procedure of this work. Step 1: Geological and reservoir model construction. Step 2: Relative permeability and capillary pressure calculation. Step 3: Micro simulation on flow cell. Step 4: Macro simulation on reservoir.

68

We construct the geometric model by way of a six-scale hierarchy, viz. lamina

bed

facies

facies association/parasequence systems tract/parasequence set sequence (Table 3.1), which need not all be used for every reservoir. We consider the (sedimentary) facies a body of rock with similar sedimentary properties (read: repetition of one bed), the flow unit a body of rock with similar hydraulic properties (read: repetition of one flow cell), the bed a body of rock with similar sedimentary properties (read: repetition of two laminae), and the flow cell a body of rock with similar hydraulic properties (read: repetition of two laminae). Table 3.1. Six-scale hierarchy of heterogeneity levels for facies and reservoir models in this study. Geological key elements are facies and bed. Reservoir key elements are flow unit and flow cell. Reservoir units Geological units Facies example Sequence

River valley

Parasequence set

Alluvial plain

Facies association/Parasequence

River

Flow-Unit

Facies

Meander belt

Flow-Cell

Bed

Trough bed

Lamina

Foreset

We define a Representative Elementary Volume (REV) for each heterogeneity, which may be a Periodic Unit Cell (PUC) if repetitive. According to van Lingen (1998) the lamina is the REV/PUC for the crossbed, the crossbed is the REV/PUC for the Flow Unit and the flow unit is the REV of the reservoir model. Crossbed and facies are thus key elements. Essentially the crossbed corresponds to the flow cell and the facies to the flow unit (Table 3.1), but there are some differences. Facies have an irregular shape (Figure 3.1), whereas flow units are geometrically simplified versions of facies. In the reservoir model one flow unit is build up of numerous identical grid blocks (Figure 3.1). A crossbed and foreset-laminae have an irregular shape and change along a facies, whereas a flow cell in our case has a rectangular shape and average properties of crossbeds in the flow unit and laminae are planes. If a facies consists of a number of zones with different bedding types, it is to be split up into the same number of sub-facies. If variation is systematic, the flow cell itself can be divided into zones with different characteristics. For example, a consistent vertical sequence throughout the facies or vertical grading within the crossbeds or clay drapes as bottomsets, as is the case in a pointbar. We assume, that for one bedding type (within a facies), crossbed dimensions are normally distributed, lamina thickness to be trimodal and lognormally distributed, and lamina permeability to be trimodal and lognormally distributed. We hence average intra-lamina thickness and intra-lamina permeability via simple arithmetic averaging. We use simulation on a smaller scale to obtain effective relative permeability and capillary pressure functions of the flow cell (Christie, 1996, Christie & Clifford, 1997). Set up in the x, y, z direction with no flow boundary conditions along the sides, p=1 at the inlet, p=0 at the outlet. We solve the equations and sum the fluxes. Then we calculate the effective directional relative permeabilities and capillary pressures. This upscaling method has the advantage to be easy to implement in reservoir flow simulation routine. 69

We chose for bi-dimensional flow cell simulation, but in the future we might use tri-dimensional simulation instead. Nonetheless, any of the existing procedures might do, e.g. averaging, renormalization, homogenisation, pseudoisation, streamline method, effective flux calculation. As an example we construct a hypothetical reservoir consisting of a meander river sequence. In this case we make the model naively simple to consist of three facies only, viz. channel, pointbar, and floodplain, each having one repetitive element. This is one bedding type for floodplain (horizontal bedding) and channel (trough dune bedding) and one specific succession of bedding types for the point-bar (large trough bedding at the base, small trough bedding in the middle and ripple bedding at the top). We implement our model in the model of the SPE ninth comparative study (Killough, 1995) to demonstrate the applicability of our method and to compare results. There are three important points we like to emphasise. (1) The way of incorporating small-scale heterogeneities in a reservoir model is significant for reservoir performance estimation. 2) The upscaling procedure for averaged relative permeability and capillary pressure deals with capillary trapping in foreset laminae. (3) The procedure is easy to implement in the current infrastructure of petroleum companies and institutes.

3.2.1 Geological model To demonstrate the procedure we examine a highly heterogeneous reservoir deposited in a fluvial environment. Fluvial reservoirs are known to be heterogeneous on a wide range of scales (Weber, 1986). For our purpose we generate a model in four steps using five different scales (Table 3.1). The largest scale comprises the entire reservoir (parasequence set) and consists of several parasequences (Figure 3.2A). The parasequences consist of three facies viz. floodplain, pointbar, and channel-fill (Figure 3.2B). A typical pointbar consists of a series of concentric layers, which dip down toward the outer bend of the river. Sometimes these layers are separated by thin shale streaks that extend from the top to halfway the bottom. The channel fill may contain sand in which case there is abundant trough dune bedding, or it may contain shale in a classic oxbow-lake configuration. In this case, we have chosen to model a mainly sandy channel-fill. Each facies consists of one bedding type (Figure 3.2C): horizontal bedding in the floodplain, lateral accretion in the pointbar, and trough dune bedding in the (sandy) channel fill. Each bed type consists of bottomset laminae and two types of foreset laminae, horizontal bottomset laminae are fine grained, inclined foreset laminae are medium or coarse grained. Lateral accretion bedding is in nature internally composed of trough dune bedding aligned in the direction of paleoflow. This superimposed dune bedding complicates matters severely and has therefore in the present study been omitted.

70

Figure 3.2. Geological model of the fluvial system used in this study, the first part of step 1 (Figure 3.1).

3.2.2 Reservoir model The reservoir model is a direct transformation of the geological model. Thus, the large-scale model consists of layers (Figure 3.3A) for which in this case thickness and flow unit size are given by Killough (1995). Within these layers the floodplain, pointbar, and channel fill flow units are placed (Figure 3.3B). Identical flow units compose one facies. Each flow unit consists of several identical flow-cells, the equivalent of the bedding type in the geological model (Figure 3.3C). We make an important simplification here: flow cells have the same orientation within each flow unit. Thus, lateral accretion surfaces in the pointbars and trough dune bedding in the channel fill always dip in the X-direction. As a result, each of the flow units has the same properties throughout the model.

71

Figure 3.3. Reservoir model of the fluvial system used in this study, the second part of step 1 (Figure 3.1). Note that flow unit and flow cell of lateral accretion are identical. The reason for this is, that the PUC covers the entire height of the point bar, consisting of three different bedding types (Figure 3.6).

Dimensions for the flow cells were inferred from the geological model and are listed in Table 3.2. At a smaller scale, each flow cell is made up of alternating laminae, to which different permeabilities are attributed. Typical lamina permeability values were compiled from literature and from our own data (Table 3.3). These values were simplified to the extent that we only have three permeabilities: 10 mD, 50 mD, and 200 mD. The floodplain has horizontal bedding with an alternation of 10 mD and 50 mD. The channel fill has dune bedding (alternation of 50 and 200 mD laminae) or it is shale filled (10 mD). The pointbar has three superimposed regions of equal thickness: the lower part consists of dune bedding only (alternation of 50 and 200 mD); the middle part contains dune bedding with occasional clay drapes (alternation of 50 and 200 mD and the clay drapes 10 mD). The upper part is horizontal bedding with an alternation of 10 mD and 50 mD. In this case flow cell and grid block have equal thickness.

72

Table 3.2. List of input parameters for the flow-cell models. T,W,L stand for thickness, width, length of the model; C,F,B indicate coarse-, fine grained foreset and bottomset; α denotes foreset dip; # means number. Parameter Channel Point bar Flood plain U

M

L

T (cm)

12.55

10

10

10

24

W (cm)

1

1

1

1

1

L (cm)

50

50

50

50

50

10-30

0

30

30

0

C gr.block # (-)

925

-

264

225

-

F gr.block # (-)

925

250

264

225

600

B gr.block # (-)

200

250

22

-

600

Total gridblock # (-)

2050

(°)

1600

1200

The model is a prototype of a meander river system and its geometry is mostly obtained from Galloway & Hobday (1983) and the values are guesses based on an overview of published examples. The permeability value for the bottomset might be high, but others even use values equal to foreset permeability. The channel is in nature often conserved as separated clay plugs. This would not yield a continuous channel as in Figure 3.2, but horseshoe elements of clay filled channel bends, with free passages between them as in Figure 3.4.

Figure 3.4. Three-dimensional geometries of sand bodies of braid, meander, and stable/anastomosed rivers (adapted from Galloway & Hobday 1983). The rest of the facies being clay they are impermeable and do not significantly contribute to the flow of fluids.

73

3.2.3 Micro-simulation Here we describe the procedure of obtaining upscaled phase permeability curves and capillary pressures for the three bed types constituting the reservoir. These upscaled permeabilities are derived in a three-step process, i.e. two upscaling steps and the actual simulation. First we assign Brooks-Corey relative permeability functions Leverett-J capillary functions to the high and low permeable foresets and to the bottomsetrs constituting the beds (van Lingen, 1998). (3.1) (3.2) (3.3) For convenience we use the same values as van Lingen (1998); empirical constant =0.79, connate water saturation Swc=0.0, sorting factor =1.5, porosity φ=0.3 and interfacial tension between oil and water σow =0.03 [N/m]. End point permeabilities for water and oil are taken k‟rw= 1 and k‟ro= 1. Permeabilities are summarized in Table 3.3 and would normally have been obtained from probe-permeameter measurements on cores. We would have used image analysis to estimate the sorting factor, but here we used an average value =1.5 for both foresets and bottomsetrs. In this way we completely assigned the relative permeability and capillary pressure fields for each of the three lamina types in our bedding models (Figure 3.5). The aspect that trapping is controlled by wetting state is outside the scope of this article (Huang, Ringrose, & Sorbie, 1996) and we confine interest to water wet media. Table 3.3. List of permeabilities for the flow-cell models. The permeability values for the flow cells are educated guesses based on literature. Keff is effective permeability; river indicates the model presented here; spe refers to the ninth SPE model (Killough, 1995); φ denotes porosity; ave is average; C,F,B indicate coarse-, fine grained foreset and bottomset. Parameter Channel Point bar Floodplain U

M

L

KC (mD)

200

-

200

200

-

KF (mD)

50

50

50

50

50

KB (mD)

10

10

10

-

10

Keff (mD)

87.0

34.7

Keff,model (mD)

37.0

Keff,spe (mD)

94.0

river

0.3

(-)

ave,spe

29.6

0.13

(-)

In the second step, the notions representative elementary volume (REV) and periodic unit cell (PUC) play an essential role. Following Bear (1972) we define the REV for a property as the averaging volume beyond which the average value of a property remains more or less unchanged. The periodic unit cell (PUC) is a building block of the REV that can be considered periodic for all 74

practical purposes. Here we equate the PUC to the flow cell. Because of periodicity average properties of the PUC are representative for the REV. Thus there is one flow cell for every facies, i.e. dune bedding for the channel, lateral accretion for the pointbar and horizontal bedding for the floodplain (Figure 3.2 and Figure 3.3). We perform 2D simulations on each flow cell (Figure 3.6). For a relatively small section we may assume that during reservoir production it will go through a slowly varying set of quasi steady states. This assumption is also implicit in other upscaling methods e.g. homogenisation. In accordance with previous authors, we inject and produce through two opposite faces of a rectangular block and use no-flow conditions through the other four faces. This approach ignores permeability anisotropy. We use this approach as a first example, although modern simulators can indeed handle non-diagonalised tensor permeability and cross-bedded structures are known to require full-tensor permeability (Pickup et al., 1995). The initial oil and water saturation in the flow cell are determined by assuming a given capillary pressure, Pc =107 kPa. We continue flooding until steady state is reached. The third step is the computation of relative permeability and capillary pressure from the simulation results. We use Darcy‟s law in which we substitute the flow of oil and water together with the average phase pressure difference between the injection and production side to derive the phase permeabilities with Darcy‟s law. Also, the average oil minus the average water pressure is the capillary pressure. Finally we calculate the average saturation in the flow cell. In this way, we obtain one point in the phase permeability-saturation curve and capillary pressure-saturation curve. We continue to use the same procedure with water/oil injection mixtures ranging from zero to 100% oil, with steps of 10%. In this way the upscaled phase permeabilities and capillary pressures for the channel beds, the point bar beds and the flood plain beds are obtained. We repeat the same procedure only allowing for flow in the vertical direction. In the simulation considered we have three-phase oil-water-gas flow. Hence three-phase permeabilities are required. Here we use STONE II. A somewhat better approach would be to perform 3-phase (steady state) simulations, instead of the 2-phase simulations described above. The results would, however, be history dependent and hence 3-phase upscaling is considered outside the scope of this article. If, however, we do such a study in the appraisal stage, when no free gas is liberated, a black oil simulator might be appropriate. These results are not suitable yet for implementation into a standard simulation run, because the relative permeability and capillary pressure are anisotropic. Hence we make one further simplification. We calculate the average ratio between relative permeabilities in the vertical and horizontal directions. We also calculate the 1-phase permeability anisotropy from 1-phase simulations. The product of the average ratio with the 1-phase anisotropy factor determines the effective permeability anisotropy. We use the relative permeability and capillary pressure obtained from the horizontal flow simulation.

75

3.2.4 Results for upscaled permeabilities The results from the micro-simulation are summarised in Figure 3.7 for horizontal flow and in Figure 3.8 for vertical flow. The relative permeability for oil/gas is chosen identical to the relative permeability for water/oil (Berry, Little, & Skinner, 1992).

76

Figure 3.6. The flow-cell models for Channel, Point bar, and Floodplain for STARS. On these three models the two-dimensional small-scale flow simulations have been performed. Flow is from left to right. The three flow cells are built with three lamina types: coarse foreset (200 mD), fine foreset (50 mD), and bottomset (10 mD). Inlets show individual grid cells. Shades denote permeabilities: White (200 mD), light grey (50 mD), and dark grey (10 mD).

77

Figure 3.5. Capillary-pressure- and relative-permeability curves for the three lamina types used in this study for all three flow-cells. Krw is relative permeability to water, Krow is relative permeability to oil in presence of water, Pcow is capillary pressure between oil and water.

Figure 3.7. Capillary-pressure- and relative-permeability curves for horizontal flow for the three flow-cell types. The oil-gas relative permeabilities and capillary pressures have been chosen to be identical to the water-oil relative permeabilities. Krw is relative permeability to water, Krow is relative permeability to oil in presence of water, Pcow is capillary pressure between oil and water.

Figure 3.8. The capillary-pressure- and relative-permeability curves for vertical flow for the three flow-cell types. Krw is relative permeability to water, Krow is relative permeability to oil in the presence of water, Pcow is capillary pressure between oil and water.

78

Our main observations are the following: Channel beds: With respect to the input lamina relative permeabilities we observe a substantial increase of the residual oil saturation both for horizontal and vertical flow. The higher horizontal relative water permeability at low water saturations must be attributed to the fact that at a given average water saturation of ~ Sw=0.4, the low permeable laminae are almost completely filled with water and hence are able to contribute to the relative water permeability. The vertical relative permeabilities are substantially lower, showing the additional anisotropy effect originating from the lamina structure. Note here, that channel fills are in nature often clay plugs. Point bar beds: Here the residual oil saturation appears to be affected less by cross-bed structure, proving that we cannot use a single set of relative permeabilities for all facies. Again vertical relative permeabilities are substantially lower, showing the additional anisotropy effect originating from lamina structure. Flood plain beds: Here we again observe the typical horizontal relative permeability enhancement, but this time for oil. Still the residual oil saturation is higher than for the laminae individually. We note a difference in horizontal relative permeabilities with respect to the channel and point bar. Again vertical relative permeabilities are substantially lower than horizontal ones. In summary, we observe that the relative permeability behaviour is strongly affected by the lamina distribution in channel, point bar and flood plain. The most important features are (1) Horizontal relative permeability enhancement, due to a more favourable distribution of saturations; (2) Increase in residual oil saturation; (3) Anisotropy of relative permeabilities enhancing the effective permeability anisotropy.

3.2.5 Macro-simulation The numerical simulation is run on our meander river model, using the model of the ninth SPE Comparative Solution Project (Killough, 1995) as a framework (Figure 3.9). This project concerns a black oil simulation of moderate size (9000 cells) with a high degree of heterogeneity provided by a geostatistically-based permeability field, i.e. stochastically. Here we mention the salient features of the simulation in order to highlight our deviations from this base case. Further details in particular of the fluid properties can be found in Killough (1995). The flow simulation is performed with “IMEX”. The field contains 25 somewhat randomly spaced producers and a single water injector. The dipping reservoir is divided in 24x25x15 grid blocks in rectangular coordinates without local refinement. The dimensions of the grid blocks are 300 ft both in the x- and y-directions. Cell (1,1,1) is at a depth of 9000 ft sub-sea. The remaining cells dip in the x-direction with no dip in the ydirection. In the z-direction we have 15 layers, each represented by the height of the grid block. The height of each of the layers is as given in Killough (1995). In the base case the average porosity of all layers is φ=0.1262. However, for the runs with our hypothetical reservoir model we use a constant porosity of φ=0.3. The total height of the reservoir is 359 feet. All producers were completed over a length of 56 feet in layers 2,3,4 only and the injector was completed over a length 79

Relative permeability and capillary pressure for the runs with our hypothetical reservoir model are used as derived above. There is no free gas initially in the reservoir and the oil is at the saturation pressure. PVT properties of the fluids are taken as in Killough (1995). The oil viscosity is around 1 cP, the gas viscosity around 0.015 cP and viscosities are slightly pressure dependent. We calculate the initial water saturation distribution by assuming capillary pressure gravity equilibrium and given the oil-water contact being at 9950 feet sub-sea.

3.2.6 Results for the reservoir simulation We make a comparison of runs for six different conditions (Table 3.4 and Table 3.5). First runs are made of the SPE 9th comparative program, using the stochastically generated reservoir model and all the input data of Killough (1995), whose relative permeabilities and capillary pressures are shown in Figure 3.10. However, we like to compare the situation with two effective permeability anisotropy ratios. First we use a ratio of horizontal to vertical permeability of 100 (indicated as SPE). Then we use an effective permeability ratio of 5.6 (SPEv). The other four cases concern our hypothetical reservoir model developed in this paper. We use the channel belt along strike with effective permeability anisotropy ratios of 100 (N1), and 5.6 (N1v). The last two runs concern a model with the channel belt along dip, with again anisotropy ratios of 100 (N3) and 5.6 (N3v). Figure 3.11 presents the most conspicuous results of the six runs. Shown are the cumulative oil, gas and -water productions. Admittedly the results of our reservoir model (N1, N1v, N3, N3v) with porosities φ=0.3 are difficult to compare with SPE and SPEv with average porosities of φ=0.13 (see Table 3.5), but it is important to note the differences between our different model runs. We also observe that the different anisotropy factors have a large effect on the cumulative productions from the stochastically generated reservoirs. For the hypothetical river model, there appears to be a slightly more favourable (lower) water production and less favourable oil recovery (lower) from reservoirs where the anisotropy factor is highest. This is to be expected as the injection well is injecting water in the bottom part of the reservoir, and oil is produced near the top. Whether the channel belt is along strike or along dip has little effect. This can possibly be attributed to a more or less symmetric situation with respect to the injection well and apparently a slight effect of dip in this particular situation. For the conditions shown the gas oil ratio for our hypothetical reservoir model is still approximately equal to the solution gas oil ratio indicating that we are still before free gas breakthrough. Our most important result is, however, that we have shown that facies based reservoir models can be simplified in a way that makes these models amenable to upscaling and reservoir simulation without losing the effects of the small scale. Our results present the basic concept of a procedure that can be applied using more sophisticated aspects in the future. of 207 feet in layers 11,12,13,14, and 15. The water well is located at the (24,25)-corner grid. We use the same constraints on producer and injection wells as used in the base case. Also the productivity indexes are based on a well bore radius of 0.5 ft and a drainage radius of one fifth of the grid block length.

80

Figure 3.9. Along strike (scenarios N1,N3) and along dip (scenarios N1v,N3v) flow simulation models for IMEX. Blocks denote individual grid cells. Flow in the model is updip. The model is inclined as in the 9th SPE comparative study (Killough, 1995), and grid blocks themselves are horizontal. Number, height, and average permeability of grid blocks correspond to Killough (1995). Shades indicate the three grid cells (flow units) used, i.e. white (channel), light grey (pointbar), and dark grey (floodplain).

81

Table 3.4. List of properties for the flow-unit model. Pcow is capillary pressure between oil and water, Krow is relative permeability to oil in presence of water, Krog is relative permeability to gas in presence of oil, K is permeability, Kh is effective horizontal permeability, Kv is effective vertical permeability, Tlayer is the gridblock thickness, # stands for number, river refers to our model, spe refers to the model in the SPE ninth comparative study (Killough, 1995); calc means analytically calculated; N1 is our river model with the channel belt along strike of the model. N3 is the same model with the channel belt along dip (Figure 3.9). SPE, N1, N3 have Kh/Kv = 100, SPEv, N1v, N3v have Kh/Kv = 5.6. Channel belt along strike Channel belt along dip SPE stochastic K-field N1

N1v

N3

N3v

Hspe

Vspe

Pc

Pc,ow,calc

Pc,,ow,calc

Pc,ow,spe

Kr,ow

Kr,ow upsc

Kr,ow upsc

Kr,ow spe

Kr,og

Kr,og upsc

Kr,og upsc

Kr,og spe

K

Kmodel

Kmodel

Kspe

Kh/Kv (eff)

100

5.6

100

5.6

100

5.6

Tlayer

Variable

Variable

Variable

Gridblock #

Pointbar: 160

Pointbar: 152

600

Floodplain: 375

Floodplain: 384

Table 3.5. Initial values and production results for the different simulation scenarios. STB stands for stock tank barrel, SCF stands for stock cubic feet. SPE stands for the model as in the ninth SPE comparative study (Killough, 1995). N1 is our river model with the channel belt along strike. N3 is the same model with the channel belt along dip (Figure 3.9). SPE, N1, N3 have Kv/Kh = 100, SPEv, N1v, N3v have Kv/Kh = 5.6. SPE SPEv N1 N1v N3 N3v Total Pore volume (106 STB)

453

1324

1324

Initial Oil in place (10 STB)

186

234

261

Mobile Oil volume (106 STB)

156

82

92

Total solution Gas volume (10 SCF)

259

325

327

Free Gas volume (109 SCF)

0

0

0

246

1061

1063

6

9

6

Water volume (10 STB) 6

Cumulative Oil produced (10 STB)

17.5 25.2 25.9 27.1 25.5 27.2

Cumulative Gas produced (109 SCF) 71.1 101 29.2 31.2 28.4 30.9 Cumulative Water produced (106 STB) 2.1

1.6

18.2 23.7 18.9 24.6

Cumulative Water Injected (106 STB) 1.5

2.5

2.6

82

3.9

2.7

4.0

Figure 3.10. Capillary-pressure- and relative-permeability curves for the SPE model (Killough, 1995). Krw is relative permeability to water, Krow is relative permeability to oil in the presence of water, Pcow is capillary pressure between oil and water. Krg is relative permeability to gas, Krog is relative permeability to oil in presence of gas, Pcog is capillary pressure between oil and gas.

Figure 3.11. Oil, Water and Gas Simulation Production Results by IMEX. STB stands for stock tank barrel, SCF stands for stock cubic feet. SPE stands for the model as in the 9th SPE comparative study (Killough, 1995). N1 is our river model with the channel belt along strike. N3 is the same model with the channel belt along dip. SPE, N1, N3 have Kh/Kv = 100, SPEv, N1v, N3v have Kh/Kv = 5.6.

3.3 Discussion/conclusions To justify our simplifications with regard to permeability and geometry of crossbeds, we argue that these simplifications do not yield the largest error in the upscaling procedure. We state, that the largest errors occur in the facies interpretation (uncored facies, misinterpretation), bottomset characteristics (bottomset permeability and continuity), relative permeability curves (residual oil saturation and connate water saturation), capillary pressure curves (entry pressure, displacement pressure), and investigated area (covering ~ 1·10-8 of the reservoir volume). This is why we deliberately choose to incorporate all aspects in one procedure, instead of detailing on few elements of the procedure, disregarding others. The strength of the procedure is the calculation of relative permeabilities and capillary pressures for the laminae and in the physical incorporation of laminae into the upscaling procedure. The other strong point is its simplicity that allows us to study the effect of variations in parameter values. Our upscaling procedure is an attempt for a better synthesis of all aspects in upscaling. Ideally, we would have tested it on a reservoir that has been producing for a long number of years. 83

However, we were not so fortunate to gain access to production data or to fresh cores from a hydrocarbon reservoir. Those who have access can easily do testing themselves. Those that critic simplification at any point of the procedure, can simply add sophistication at will. Where the statistical validity of quantitative data is missed, we refer to the many data that have been published viz.: (1) sampling of lamina permeabilities with instruments that were visibly not well calibrated and (2) distributions of lamina permeabilities that were visibly incorrectly sampled. We prefer a correct theory with few samples instead of no or an incorrect theory based on many incorrect samples. Quantitative geometrical data are best acquired from the reservoir from wells, cores, and seismic lines. If not available an estimate of minimum and maximum values of parameters is to be made. In the modelling we then take minimum and maximum values for all parameters and see their effect on the result. Nonetheless, statistically significant quantitative geometrical data are difficult to acquire. Seismic lines might resolve parasequences, but not facies. Stratigraphic sections supply (cross) bed thickness, but not width and length of individual (cross) beds, although suggested by many authors. Well logs and cores supply bed thickness, but not maximum thickness for a continuous set of trough crossbeds. Well logs and cores supply lamina thickness and dips. Cores also supply lamina permeabilities, but yield very poor lateral coverage. We propose parasequence boundaries to be barriers and facies boundaries to be conductors to flow. We argue that a parasequence boundary inherently represents some time of non-deposition and/or reworking enough to consolidate the surface and render it significantly less permeable than the underlying facies. We argue that a facies boundary within a parasequence is a gradual transition from one lithology to another that doesn‟t limit flow. These assumptions would require testing for every case, but it‟s difficult to obtain permeability values from such a boundary. In this work we have chosen all flow cells within a facies to have the same orientation, which is obviously naive. However, we believe it appropriate to yield regional flow paths. We can easily add orientated flow cells in a later stage using a tri-dimensional flow cell model that yields full-tensor permeability and use a reservoir flow simulator able to handle tensors. Rate dependence is yet another important issue that we didn‟t incorporate in our flow-cell models and that should be addressed in future studies. Our approach can be applied to a reservoir at any stage of exploration or production. One can use it with relatively few data and simple methods and subsequently add data and more sophisticated methods to availability of data, will, and time. The idea of this approach is to create a straightforward procedure that works for any depositional system and for any level of sophistication. For now we only tested our approach on the SPE 9th comparative program (Killough, 1995). It shows that when using equal average values, incorporating the effect of small-scale heterogeneities yields a significantly different outcome from a stochastic distribution of values. It also shows that the orientation of the sedimentary system and hence the distribution of facies controls the outcome. These findings, although trivial they might seem, indicate that this approach is worth pursuing.

84

Our study renders the following conclusions: A six scale hierarchy with crossbed and facies as key elements, i.e. lamina → bed/flow cell → facies/flow unit → facies association/parasequence → systems tract/parasequence set → sequence, is appropriate to construct a reservoir model based on geological structures. It is possible to simplify the reservoir model such that it becomes amenable for upscaling and reservoir simulation using state of the art technology whilst preserving the effect of small-scale heterogeneities/structures. The lamina/cross bed structure naturally incorporates the small-scale heterogeneity and anisotropy via a two-step numerical flow simulation. Simulations show the importance of the total anisotropy i.e. the product of the one-phase permeability and the additional anisotropy originating from two-phase flow effects. Simulations using upscaled lamina permeability and capillary pressure yield significantly different effective values from stochastically assigned values. Further research is necessary to quantify the effect of the distribution and shape of facies and other parameters used in the model. This procedure is a quick and straightforward method to model a reservoir at any stage of its exploration, development, or production stage. The more data the better, but little data suffice. The method enables routine use in reservoir modelling as well as quick sensitivity analysis of parameters.

3.4 Nomenclature = permeability, L2, m2 = relative permeability = saturation, fraction = capillary pressure, m/Lt2, Pa = interfacial tension, m/t2, N/m = empirical constant = sorting factor = porosity, fraction Subscripts o = oil w = water wc = connate water we = normalised water (saturation) g = gas k kr S Pc σ γ λ φ

85

Part II Permeability sampling 4 Sampling Summary Sampling permeability from individual laminae in cross-bedded rocks is very well possible by making use of a so called probe permeameter. The methodology and calculation of permeabilities from the output of this instrument are fairly well-established. However, what is hardly established is a sampling strategy that guarantees adequate representation of permeability contrast between laminae. Especially so, because finer foreset and bottomset are difficult to sample, yet their permeabilities are critical to fluid flow through a crossbed. Therefore we propose to systematically sample laminae. To this end we suggest a sampling strategy that is adapted to the heterogeneity under study, namely to the thickness of the finer foreset.

4.1 Introduction Permeability for reservoir studies has traditionally been sampled with a Hassler-sleeve permeameter on core plugs. The probe-permeameter introduced higher resolution and empirical methods on thin sections even higher. Nonetheless, although studies on permeability sampling are numerous, a systematic sampling strategy has not yet been proposed. Grain size analyses show that grain-size frequency distributions of individual coarse and finegrained laminae in cross-bedded rocks differ from the composed distribution where the two laminae are combined (Emery, 1978; Grace, Grothaus, & Ehrlich, 1978). It seems only logical to expect a similar behaviour for the permeability frequency distributions of laminae. Since, at present, published data don‟t pre- nor exclude otherwise, we assume the distribution of intra-bed lamina permeability and -thickness to be trimodal and the intra-lamina permeability and –thickness to be lognormal (Basumallick, 1966, Boersma, 1967, Kortekaas, 1985). One of the requirements for reliable prediction of oil production by flow simulators is that the heterogeneity of the reservoir be adequately represented. The question is how to represent parameters spanning 12 orders of magnitude (lamina volume in the order of dm3 and reservoir volume in the order of km3) into a single reservoir simulator. We propose to do so by way of an upscaling procedure (Figure 4.1) as in Chapter 3 (Mikeš et al., 2006). The procedure builds on a six-scale hierarchy that conforms to naturally occurring levels of heterogeneity, viz. lamina → bed → facies → facies association/parasequence → systems tract/parasequence set → sequence (Table 4.1). We consider the key elements to be “flow cell” (Chapter 1; Mikeš, 2006) and “flow unit” (Chapter 2; Mikeš & Geel, 2006). The flow cell is the Periodic Unit Cell (PUC) of the flow unit and consists of an array of laminae. Hence, to quantify the reservoir‟s heterogeneity, we only require permeability of laminae and facies boundaries. Adding the geometric properties of facies and bedding, the reservoir model is ready.

86

Figure 4.1. Schematic representation of the 4-step upscaling procedure of this work. Step 1 and 2 Reservoir characterisation. Step 3 and 4 Numerical flow simulation. Kc,Kf,Kb, are permeabilities of coarse-, fine foreset, and bottomset, resp. Kro,Krw are relative permeabilities of oil and water, resp. C,F,B stand for coarse, fine foreset and bottomset.

87

Table 4.1. Six-scale hierarchy of heterogeneity levels for facies and reservoir models in this study. Geological key elements are facies and bed. Reservoir key elements are flow unit and flow cell. Reservoir units Geological units Facies example Sequence

River valley

Parasequence set

Alluvial plain

Facies association/Parasequence

River

Flow-Unit

Facies

Meander belt

Flow-Cell

Bed

Trough bed

Lamina

Foreset

Permeability of foreset laminae kc,f is measured, permeability of bottomset kb is estimated, relative permeability and capillary pressure curves of the laminae (KREL,PC)c,f,b are calculated analytically, effective permeability, relative permeability and capillary pressure curves of the crossbed kFC, (KREL,PC)FC are calculated numerically (Mikeš et al., 2006). Effective permeability, relative permeability and capillary pressure curves of the flow unit (KREL,PC)FU are considered identical to the values of the flow cell. Adequate sampling is not trivial. The focus of this work is therefore on the acquisition of lamina heterogeneities compatible with our upscaling procedure (Figure 4.1) or for that matter with any upscaling procedure. In our case we are interested in dimensional values of laminae and their permeabilities. Commonly, sampling according to a regular or random grid is endorsed as statistically correct. Quite contrarily, we pose that sampling should be adjusted to the geological structure and its irregularity. We use a pressure depletion probe-permeameter that is corrected for inertia and gas slip by using 3 distinct gases and calibrated using homogeneous and heterogeneous cuboid samples in 3 orientations with a Hassler-sleeve permeameter (van de Waal et al., 1998). Subsequently, we systematically sample laminae. The strategy guarantees that individual laminae are sampled each with the appropriate number of samples and the flow cell heterogeneity is hence reliably represented. Sampling across laminae yields crossbed heterogeneity and sampling along lamina yields average permeability for each lamina type. We obtain the mean lamina permeability simply via the arithmetic average of intra-lamina values. We started this work with the intention to take an existing steady-state probe-permeameter (Hartkamp-Bakker) into the field and measure on so-called cross-bedded “reservoir-analogues”. We planned to collect a great number of samples in regular sampling grids that we would subsequently upscale. However, along the way, we encountered a number of problems. The first was our probe-permeameter that proved inaccurate. It appeared to measure values on samples that were impermeable to the Hassler-sleeve instrument, which was probably due to surface irregularities. We constructed a new instrument, a pressure-decay probe-permeameter, calibrated it extensively on natural and artificial homogeneous and heterogeneous core plugs and it proved close to perfect (van de Waal et al, 1998). The next problem was that outcrops generally proved altered in some way, which, according to the process, rendered them either more or less permeable than the original rock. Hence, permeability values obtained from outcrops are not very reliable as representatives of reservoirs. 88

We then realised that the only valid permeability values are those obtained on a fresh core from a true and producing hydrocarbon reservoir. We changed strategy and looked for cores when we faced the final and biggest problem of all, oil-companies and geological surveys. None of them proved willing to supply us with a fresh reservoir core. Although we performed measurements on all our outcrop samples and an old core, we couldn‟t use them for a reservoir. We decided to change strategy again and concentrate on sampling instead. We decided to construct a rigid sampling strategy that would guarantee systematic sampling of lamina permeability of beds. This does not only guarantee adequate representation of heterogeneities, it is also quicker since it requires only few data and simple arithmetic.

4.2 Method 4.2.1 Heterogeneity characterisation The lack of a systematic sampling strategy and the use of an inaccurate instrument have produced erroneous sampling, which in turn yielded erroneous data analysis, one of which was the use of semivariograms. One way to state our point would be to present all errors committed in the past and then present our alternative way of sampling. However, we find this unjust since all these errors have contributed to the solution of these problems. Instead, we prefer to logically deduce what sampling is appropriate and what is not. Hence, the arguments will speak for themselves. Although the semivariogram has been commonly used as means of characterising heterogeneity, in the case of geological structures its use is very doubtful, since a number of problems impose: (1) If several facies or heterogeneity scales are sampled as one population, distinct heterogeneities are mixed. (2) If samples of coarse and fine laminae are sampled as one population, lamina heterogeneity is lost. (3) If sample spacing is larger than the lamina thickness or the sample volume exceeds the lamina, variability is lost, underestimating value and volume of fine laminae. This is particularly the case if the fine lamina is thinner than the coarse lamina or if cyclicity is not perfectly regular. In all these cases, the semivariogram will flatten out, instead of showing clean cyclicity. Summarising, we can say that it is difficult, if not impossible, to yield from a semivariogram the reliability and detail that can be obtained from a cross-bedded core. It might be feasible if sampling instrument and grid are perfectly adjusted to the structure. However, the only way to achieve this is via visual inspection and “a priori” sampling. Only then the heterogeneity might show up correctly in the semivariogram. But at that moment the semivariogram has become obsolete, since a ruler and lamina sampling will suffice. In all other cases the method will fail. Why then bother to go through measurement and analysis of a vast number of samples, if few samples will do? However, there are some cases in which semivariograms do serve: (1) to find lateral trends in facies, (2) to discern heterogeneities in rocks that appear homogeneous, or (3) to discern heterogeneities in rocks that we can‟t see, as in boreholes. In the last case it would however be recommendable to run a wall-imaging tool first. For all visible heterogeneities average layer thickness should better be observed with the eye and lamina permeability be sampled systematically 89

with respect to geological structures, i.e., facies, bedding, and laminae. In contrast to what is generally believed in statistics, sampling of geological heterogeneities should hence be “a priori” and not “a posteriori”, which disqualifies at random or fixed grid sampling. Crossbed lamination is a repetitive and more or less regular heterogeneity. Its variability is attributable to hydrodynamic variations in the course of bedform progradation. Essentially, it is an alternation of coarse and fine foresets with a dip of 20 – 35 ° and most commonly 20 – 30 º (Collinson & Thompson, 1989). Additional variabilities might occur in lamina dip, lamina thickness, grain size and - contrast, which might result in rather complex heterogeneities (Boersma, 1967). As a result of above-mentioned variabilities, the heterogeneity in crossbeds manifests itself at three orders of magnitude: (1) Intra-lamina; laminae more permeable towards their top and/or apex. (2) Intra-crossbed (inter-lamina); permeability contrasts between laminae greater towards the top of the crossbed. (3) Intra-facies (inter-crossbed); crossbeds more permeable towards the top and/or apex of the facies. We can observe all these trends in a vertically oriented core, assuming that lateral and vertical trends correspond. That is to say, an intra-lamina trend for each lamina type (coarse and fine foreset) can be observed through an inter-lamina yet intra-crossbed trend for each lamina type. Similarly, an inter-bed trend can be observed through an intra-facies trend in a core. The way in which to obtain the averaged or upscaled value of the facies is via upscaling (Mikeš et al., 2006).

4.2.2 Statistics and measurement procedure The parameters required to characterise the flow cell are (Figure 4.2): (1) Crossbed dimensions: length L, width W, height H, (2) Foreset dip angle α, (3) Foreset lamina thicknesses: coarse foreset tc f and bottomset tb, (4) Foreset lamina permeabilities kc,f,b: coarse foreset k fine foreset kf and bottomset kb. We propose to characterise the distribution of each flow cell parameter by its mean and standard deviation values. The heterogeneity across laminae can be represented by the coefficient of variation CV. Each parameter can be tested to lognormal, rootnormal or normal distribution via the Kolmogorov-Smirnov (or χ2) test. In the following section top and base refer to top and base of the bed (a ≠ b means that a and b are significantly different, a = b that they are not significantly different).

4.2.2.1 Crossbed dimensions L,W,H Crossbed thickness H is the only dimension of crossbeds observable in a core. Examples of distributions are given in literature (Johnson, 1975, Corbett et al., 1994, Bridge, 1997). Width W and length L are estimated from values obtained at outcrops or from equations (Hartkamp-Bakker, 1993, Weber, 1987 and Mikeš, 2006). We take sedimentary bedding horizontal in the flow cell simulation.

90

Figure 4.2. The flow cell and its properties: β = inclination of flow cell, α = inclination of laminae, L/W/H = length/width/height of flow cell, tc,f,b = thickness of laminae, kc,f,b = permeability of laminae. The orientation of the long axis of the crossbed can be obtained from dipmeter logs or borehole imaging logs. The geometry of a crossbed is hence characterised with 10 parameters and the crossbeds 1-phase permeability with 3 parameters.

4.2.2.2 Lamina dip α Lamina dip α can be measured from an oriented core if the sedimentary bedding plane is known. Values and distributions can be taken from literature (Corbett et al., 1994, Johnson, 1975, Storms, 1999 and Mikeš, 2006) or from borehole imaging tools (Mercadier & Livera, 1993, Williams & Soek, 1993, 1997, Knight 2002).

4.2.2.3 Lamina thickness tc,f,b If lamination is distinct, average lamina thicknesses tc,f are best measured with a ruler on a core. Where lamina transitions are not well marked, colour and grain size help to distinguish lamina boundaries. In essence, there are two important variabilities: ttop vs. tbase and tc vs. tf. This yields four different permutations. If (tc,tf)top=(tc,tf)base, then tc and tf distributions can be taken from the total t population of the crossbed. If (tc,tf)top≠(tc,tf)base, then tc and tf distributions are to be taken from the individual populations for coarse c and fine laminae f of a bed. The thickness contrast is one of two alternatives: tf = tc, tf ≠ tc. There is also the case of lamina foresets being of one type only, the lamina boundaries then take the place of f, and the model collapses to an alternation of c and f, with tf