Keywords

1 Introduction

Many deep-water lobe depositional sequences are characterized by laterally extensive, but finite, sand beds interbedded with continuous low permeability shales. Non-random geological processes such as compensational stacking result in these systems often having poor sand connectivity at high net:gross ratios (NTG). It is challenging to reproduce this low connectivity in reservoir models, but it is important to do so if the models are to be used to predict reservoir performance.

The advantage of using a pixel-based method to model these sequences is that it reproduces geological patterns while honouring conditioning well data. The connectivity of a number of training image (TI) inputs including object- and surface-based models (OBM and SBM respectively) are investigated in this study and compared to the connectivity of output MPS models built using the SNESIM algorithm [1]. The findings from these models, along with the connectivity characteristics of OBM models, aided in the development of workflow for generating poorly connected MPS models using a simple object-based TI followed by a grid transformation using the compression algorithm.

2 Pixel-Based MPS Modelling with Common Training Images

The dimension of the model is 45 km × 45 km and 20 m thick. The target system to be reproduced contains sand beds which are about 4.5 km wide and 0.5 m at their thickest, NTG equal to about 40% and amalgamation ratio (AR) equal to about 10%. This AR implies that 10% of the total area of sand bed bases should be connected to an underlying sand bed. AR cannot be measured in an MPS model, so the connectivity measure used to compare models is the proportion of total sand that is connected to a vertical well at the centre of the model, which is about 5% in this case (Fig. 1).

Fig. 1
figure 1

Summary of the various TIs (a OBM, b irregularly gridded SBM, c regularly gridded SBM) and the corresponding MPS models. The graphs show the connectivity in each case of the target system , the training image (x) and the MPS model

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An unconditioned OBM is easily generated and works by placing user defined objects within the model volume until the target NTG is reached. A downside of OBM modelling is that are no under-defined constraints on the sand connectivity, and it is clear in Fig. 1a that the object-based TI does not have the target low connectivity since about 95% of the sand beds are connected to the well, either directly or via their connections with different beds. Therefore, although the MPS model has reproduced the connectivity of the TI, it does not have the low connectivity of the target system. The second TI is a SBM generated using a number of depositional rules. These rules control the placement of sand elements and shale interelements in a stratigraphic order, and the probability that an element is capped by an interelement controls the degree of sand amalgamation. An important feature of the TI that impacts the MPS modelling is that the TI contains an irregular grid structure since each sand or shale element is contained within an individual layer which is one grid cell thick. Hence, although the TI represents the target system (Fig. 1b), the MPS algorithm produces very different model characteristics consisting of thin sand and shale layers resulting in complete sand connectivity. The final TI is a SBM constructed in a regular grid (Fig. 1c). This TI is a simple model generated with the same object-dimensions and 90% probability of shale drapes and therefore has low connectivity representative of the target model. However, again the MPS algorithm is unable to reproduce the low connectivity represented in the TI, and all of the sand is connected.

These results indicate that a simple workflow of constructing an accurate TI and applying this directly in MPS modelling is inappropriate for geological systems with non-random connectivity, such as the deep-water deposits considered. The only MPS model which reproduced the connectivity of the TI was the object-based case. The connectivity of MPS models generated by both OBM and SBM models are investigated in much greater detail [2] and it was found that these models follow the well-known percolation threshold seen in OBM whereby models with sand proportion greater than about 27% are well connected.

3 Pixel-Based Modelling with Low Connectivity

There are many modelling methods which allow the user to have control over the output connectivity including surface- or rule-based modelling, process-based and object-based methods. One such method, compression-based modelling, was developed originally in conjunction with OBM [3] and is based on the observation that the NTG and AR are equal in these models if all sand elements are of equal thickness. The compression method is a two-step process where modelling is undertaken with an initial low NTG (equal to the target AR) and cells containing sand and shale are expanded and compressed using the compression algorithm to reach the target NTG but preserved the low connectivity.

Combining the compression method with the MPS workflow requires that a low NTG OBM model is used as the TI input and, for the case considered here, it must equal the target low AR of 10% (Fig. 2a). The resulting MPS model reproduces the low connectivity of the TI (Fig. 2b). Then the compression algorithm is applied to the MPS model to rescale its NTG to the target value of 40% (Fig. 2c).

Fig. 2
figure 2

The compression-based MPS workflow

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The properties of the training image, such as the thickness of the objects, are defined by the compression algorithm. This means that the decompression algorithm [4] is applied to the conditioning data prior to modelling. This is the inverse of the compression algorithm and ensures that the conditioning well data will be honoured when the compression algorithm is applied to the final model (Fig. 2d).

4 Summary

Several commonly used TIs have been investigated to examine both the suitability the TI to honour low connectivity at high NTG, and the ability of the MPS algorithm to reproduce the connectivity represented in the TI (Table 1). Although the object-based TI does not contain low connectivity at NTG values greater than 27%, MPS models based on these TIs reproduce accurately their connectivity. In contrast, the two surface-based TIs have the target low connectivity, but this connectivity is not reproduced in the MPS models. A detailed investigation indicates that MPS models follow the same relationship between NTG and connectivity as object-based models when built using surface- or object-based TIs [2]. This equivalence provides the basis for a new workflow in which MPS models are built using OBM TIs, and then rescaled to the target NTG using the compression algorithm. The conditioning well data is incorporated by using the inverse transformation prior to the MPS modelling step. The final MPS model honours the well data and contains independently user defined NTG and connectivity.

Table 1 Summary of the connectivity of a number of TIs (OBM and SBM) and the resulting MPS models generated with or without the compression algorithm (CA)
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