Keywords

1 Introduction

The objectives of this contribution are to demonstrate an important limitation in object- and pixel-based facies modelling with respect to facies connectivity, and to highlight the main features of the compression-based modelling method developed to overcome this limitation. Further details of the method can be found elsewhere [1,2,3].

2 Connectivity in Facies Models and Natural Systems

Two different ways of considering facies connectivity are important to this work (Fig. 1). Global connectivity refers to characteristics of the largest connected cluster of objects and have been examined for object-based models (OBM) in the context of percolation theory using the net:gross ratio (NTG). The simplest models consisting of aligned cuboids have a well-defined connectivity threshold at NTGC ≈ 28% [4]. Connectivity thresholds for OBM containing geometrically representative, stationary, three-dimensional systems of more geometrically diverse [5] or geologically realistic [1, 6] elements show similar or lower thresholds (Fig. 2a), with the lowest NTGC for systems of more anisotropic and misaligned objects. Pixel-based models (PBM) built using the sequential indicator (SIS), or truncated Gaussian (TGS) methods have similar thresholds as OBMs, with NTGC ≤ 28%, as do models built using the pixel-based SNESIM multiple point (MPS) method even if the training images used to create the models have much lower connectivity (Fig. 2b, [7]). The inability of the SNESIM MPS method to honour the connectivity of the training image is seldom acknowledged but is a recognised restriction of the method [3, 8,9,10].

Fig. 1
figure 1

Local and global connectivity. a Global connectivity is determined as a function of the properties of the largest connected cluster of objects. Here, that cluster occupies ca. 60% of the total volume of objects, and spans entirely across the model. b The simplest definition of local connectivity is amalgamation ratio (AR) measured in a 1D vertical sample. In this example, one of the four object bases sampled is amalgamated with a lower object, and hence AR = 0.25

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Fig. 2
figure 2

Global connectivity as a function of NTG or AR for: a object-based models. Red line: trend line for high resolution models of aligned cuboids [4]. Black points: individual sinuous channels models [6]. b Pixel-based models generated with different methods and assumptions. Black: SIS. Red: TGS. Blue: SNESIM [7]. c Rule-based models. Gross trend and individual points for lobate models generated with different erosion rules. All these models have NTG ≈0.8 [14]. d Compression-based models with different compression factors (cF) [1]

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These consistent thresholds at NTGC ≤ 28% in OBM or PBM have led some to conclude that they are transportable to natural geological systems [6]. It is hard to test this generalisation directly since it is impossible to estimate global 3D connectivity from limited outcrop data, and very difficult to do so from subsurface data. Therefore, local measures of connectivity such as the amalgamation ratio (AR, Fig. 1b) are useful. AR can be measured in OBM or natural systems but not in PBM. Cross-plots of AR versus NTG for numerous natural depositional systems show that AR ≪ NTG (Fig. 3a, [1, 11, 12]). This contrasts with object-based models, for which AR = NTG if all objects are of constant thickness, or is slightly lower for variable sized objects ([1, 2, 7], Fig. 3b). Hence, the local connectivity behaviour of OBM and natural systems are not the same, and so it is unlikely that their global connectivity behaviour is.

Fig. 3
figure 3

Local connectivity (AR) as a function of NTG. a Compilations of non-hierarchical (red, [1]) and hierarchical lobe (black, [11]) and channel (blue, [12]) systems. b Multi-facies hierarchical OBM. The points show effective values for the 2D cross-sectional model shown, using the same colours as the model. c 1D measurements through two different hierarchical rule-based models generated with different erosion rules. The curves show representative model cF values [15]. d Multi-facies hierarchical compression-based OBM. Symbols as b. The curves show cF values representative of the different facies

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Rule-based facies models (RMB) are created by stacking objects in stratigraphic order using geometrical rules that mimic the depositional processes, and are recognised from a qualitative perspective as being more geologically realistic than object- or pixel-based models [13]. Both local and global connectivity can be measured in rule-based models. Different rules governing erosion and aggradation of the depositional elements provide models with a diversity of local connectivity representative of natural systems ([14], Fig. 3c). Like OBMs and PBMs, RBMs have a well-defined global connectivity threshold. However, this occurs at a critical amalgamation ratio (ARC ≈ 28% [15], Fig. 3c) rather than at a critical NTG, since RBMs can remain at low connectivity to very high NTG. The RBMs examined [14, 15] had approximately circular bodies, but by analogy with OBM, it is likely that ARC for channelized RBMs will be lower.

Taken together, these observations of global and local connectivity in natural systems and in OBMs, PBMs and RBMs suggest that they all have a connectivity threshold at ARC = 28% (flat lying circular elements) or lower (variably oriented elongate elements). In OBM, AR ≈ NTG, so object-based models have NTGC ≈ ARC. Rule-based models have more degrees of freedom than object- or pixel-based models and no link between AR and NTG. Therefore, RBM have no intrinsic value of NTGC, and in this respect natural systems are likely to behave similarly to RBM.

3 Compression-Based Facies Modelling

The considerations above imply that object- and pixel- methods (including pixel-based MPS) are incapable of creating models with realistically diverse relationships between connectivity and NTG. The compression method was developed to overcome this by providing a means of modifying object-based models so that can have low connectivity at high NTG ([1, 2], Fig. 2d). Compression-based models can be created with user-defined trends of local connectivity representative of natural systems (Fig. 3d). The compression-based geometrical transformation can be applied to pixel-based as well as object-based models, implying that it can be used to create facies models which are both conditioned to well data, and constrained by user-defined facies connectivity [3, 7, 10].

Compression-based facies modelling is a two-step process (Fig. 4, [1, 7]). In Step 1, a conventional object- or pixel-based model is created with a net:gross value equal to the target AR value (Fig. 4a, b). In Step 2 the thickness of cells of the different facies are expanded or compressed vertically by particular factors. This grid transformation modifies the facies proportions but does not alter the grid topology, and therefore the facies connectivity (e.g. AR) is unchanged (Fig. 4a). Compression-based models can have higher or lower connectivity than would be present in a conventional model at the same NTG (Fig. 4c). The extent to which connectivity and NTG differ can be expressed by a facies-specific compression factor (cF, Figs. 3d, 4a), which can be estimated by reference to natural system analogues (Fig. 3a).

Fig. 4
figure 4

The compression algorithm for a two facies OBM (foreground facies yellow, background facies grey). a The Step 1 models (blue squares) have AR = NTG (cF = 1). These are transformed to Step 2 models (red circles) with lower (cF < 1, e.g. system B) or higher (cF < 1, e.g. system D) connectivity than the original Step 1 models. Example Step 1 (b) and Step 2 (c) models

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The two simple systems generated in this example (Fig. 4) are shown alongside photos of the geological systems that inspired them in Fig. 5. Depositional systems (particularly deep marine ones) often consist of laterally extensive sand bodies often entirely enclosed by shale (Fig. 5a, b). They are characterised by AR ≪ NTG (Fig. 2a) and are modelled with cF < 1. (Fig. 4a). Other geological systems such as fluid driven injectite or diagenetic vein system can be more connected than a random system at the same NTG value (Fig. 5c, d), and must be modelled with cF > 1 (Fig. 4a).

Fig. 5
figure 5

2D cross-sectional compression-based models and the geological systems that inspired them. a Model of poorly connected sands (yellow) in shale. b Deepwater lobe deposits of the Ross Formation, Loop Head, Ireland. c Model of a network of well-connected veins (yellow) in mudstone. d Diagenetic gypsum veins in the Mercia Mudstone Formation, Watchet, England. The red rectangles in a, c are representative of the system sizes photographed in b, d

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4 Conclusions

The amalgamation ratio in object-based models is an unconstrained output property that is approximately equal to the model net:gross ratio, and object- and pixel-based facies models have connectivity thresholds at NTGC = ARC ≤ 28%. In natural geological systems and rule-based facies models NTG ≠ AR, but ARC takes similar values and has similar sensitivities as it does in object-based models. The compression algorithm is a geometrical grid transformation which exploits these relationships to provide object- and pixel-based models with user-defined connectivity that is independent of NTG.