## Abstract

In the early stages of field development, there are many uncertainties and the current trend is to generate coarse-scale models so that many simulations can be run rapidly in order to determine the main sensitivities in a model. However, at a later stage, more data are available and there is less uncertainty in the model, so a more detailed modelling approach is desirable. In this paper, we discuss two modelling procedures which are suitable for different stages of the life of a field, and address the problem of upscaling at each stage. First, we consider a novel approach for generating coarse-scale models for evaluating uncertainty. When there is a large amount of uncertainty, the generation of fine-scale models is too time-consuming, and upscaling them by large factors may produce large errors. We demonstrate an alternative approach for modelling at a coarse scale, while preserving the heterogeneity of the fine-scale distribution, in such a way as to reproduce the fine-scale flow results. Secondly, we focus on more detailed geological models, generated at a later stage of field development. These models may comprise millions of grid cells, and may be highly heterogeneous. Such models require upscaling, but traditional methods may be very inaccurate. We have developed a method for upscaling using well-drive boundary conditions. Tests of this method show that it can reliably reproduce the fine-scale recovery in a range of models.

The conventional approach for estimating hydrocarbon recovery is to generate complex geological models with multimillion grid cells. Such models are time-consuming to construct, so that relatively few versions are generated, despite the fact that there is much uncertainty in reservoir structure. Large models generally require upscaling to reduce the number of cells for flow simulation, and much fine-scale detail may be lost during this stage. In addition, engineers may significantly alter the model permeabilities during history-matching, so the final model may be quite different from the original one. To overcome these problems, ‘top-down’ and ‘scenario-based’ approaches are now being developed (e.g. Williams *et al.* 2004; Bentley & Woodhead 1998). In cases where there is much uncertainty, for example at the start of field development, large numbers of coarse-scale models are generated so that the effects of uncertainty can be fully investigated. Later in the life of a field, the data may be more certain, and more detailed models may be required to assess issues such as infill drilling and EOR strategies. At this stage, more careful simulation and upscaling may be necessary.

The aim of this paper is to consider upscaling and its applicability during early and late stages of field development. We start with a brief discussion on conventional upscaling methods, and then introduce the SPE 10 model which was used in this study. Then we investigate the nature of coarse-scale permeabilities, using history-matched values which reproduce the fine-scale results. This helps us to explain how errors in conventional upscaling arise, and leads to a suggestion for an alternative modelling approach for the early stages of field development. We then consider cases where more detailed modelling is required (later stages of field development), and put forward an improved method for upscaling, suitable for highly heterogeneous models with complex geological structures.

## Upscaling

In this paper, we are focusing on upscaling as a means of reducing the number of cells in the geological model, so that it can be used for full field simulation. At this level, usually only single-phase upscaling is performed, because two-phase upscaling is time-consuming and difficult to apply (e.g. Barker & Thibeau 1997). A number of reviews of single-phase upscaling methods have been written (e.g. Renard & Marsily 1997), so we do not provide a full review here. Instead, we concentrate on pressure solution methods. Figure 1 shows a schematic diagram of this approach. In this figure, we show a particular form of boundary conditions (step 1) – i.e. fixed pressure at the edges and no flow through the sides. Although other boundary conditions may be applied, these are most commonly used, and have been employed in this study. (In the SPE 10 study (Christie & Blunt 2001), these boundary conditions gave the most accurate results.) Also, in this version of the method, the boundary conditions are applied to each coarse cell, and this is referred to as the local upscaling method.

## The SPE 10 model

In this study, we have used the geological model (model 2) from the 10th SPE Comparative Solution Project on Upscaling (Christie & Blunt 2001), and we present an overview of the main features of the model, before describing the tests. This model represents part of a Brent sequence, and consists of 60×220×85 cells, each of 20×10×2 ft. The top 35 layers represent the Tarbert formation (a prograding nearshore environment) and the bottom 50 layers represent the Upper Ness formation (fluvial); see Figure 2. The k_{v}/k_{h} ratio in the model was 0.3 in the channels and 10^{−3} in the background. There are five wells: a water injector and four oil producers, arranged in a five-spot pattern. The relative permeabilities are similar for oil and water: a power-law with an exponent of 2. The viscosity of water is 0.3 and that for oil is 3.0, making the flood unstable.

## Calculation of coarse-scale permeabilities using history-matching

A comparison of methods for upscaling the SPE 10 model is presented in Christie & Blunt (2001) and additional upscaling tests are presented in Pickup *et al.* (2004). Usually, upscaling tests involve comparing fine- and coarse-scale results. However, in this paper, we turn the problem round, and calculate the coarse-scale permeabilities using history-matching. This means that we adjust the coarse-scale permeabilities until we obtain the same recovery rates as in the fine-scale simulation. In this way, we investigate the type of permeability distribution which gives the ‘correct answer’. (Note that, in this example, it was sufficient just to use the oil rate for history-matching. In more complex examples, you might need to use additional quantities, such as pressure, watercut or GOR.)

### Methodology

We employed the pilot-point approach for adjusting the permeability distribution during history-matching (e.g. Cuypers *et al.* 1998). Experience from benchmarking and analogue case studies could be used to determine the locations of the pilot points. A set of points (pilot points) is selected, and during the history-matching process, the absolute permeability is altered at these points. The permeabilities in the rest of the model are calculated using Sequential Gaussian Simulation (Deutsch & Journel 1998). The Neighbourhood Approximation (NA) algorithm (e.g. Christie *et al.* 2002) was used for history-matching. This is a stochastic algorithm which identifies regions of parameter space that give good history matches, and then preferentially samples in these regions to obtain better matches.

We used a coarse-scale grid of 5×11×6 (upscaling factor of 3400), with 11 pilot points distributed randomly throughout the model. The procedure is shown schematically in Figure 3, and is described in more detail below.

The spatial structure of the fine-scale model was characterized using semivariograms. The average semivariogram in each of the six coarse layers was calculated and fitted to a spherical or exponential model. These semivariograms were used in Sequential Gaussian Simulation (Deutsch & Journel 1998), along with the fine grid permeability pdf (probability density function) to produce coarse-scale permeabilities.

The starting point for the history match was a coarse-scale model which had been obtained by upscaling using the local pressure solve method described above. During the history-matching procedure, permeability multipliers were applied to the pilot points to adjust the horizontal permeability (k_{x} and k_{y}). The permeability in the z-direction was fixed, as were the permeabilities at the wells. The same permeability multiplier was applied to each of the six layers.

The NA algorithm was run on the resulting coarse models. A waterflood was simulated in each model, and the oil rates for each well were compared with the fine-scale values, using the following misfit function: where WOPR is the well oil production rate; f=fine, c=coarse; i=1 … n is the number of time steps; and j=1 … m is the number of production wells (four in this case). The results from history-matching are never unique, and the NA algorithm is frequently used to generate a range of matched models. However, in this case, we present a single model with the smallest misfit.

### Results

A comparison of the cumulative oil recovery for the fine-scale model, the upscaled model and the best history-matched model is shown in Figure 4. It can be seen that the history-matched model reproduces the fine-scale model very well, but the upscaled model overestimates the recovery. Figure 5 shows a comparison of the probability density function (pdf) for the fine, the upscaled and the history-matched models. The graph is plotted in terms of the natural logarithm of the permeability. The fine-scale distribution is bimodal. However, this has been lost in the upscaling process, and the upscaled pdf is much narrower than the fine-scale one. The pdf from the history-matched model is similar to that of the fine-scale.

### Discussion

The graph of the probability density functions (Fig. 5) can be used to explain why the upscaled model gives a poor result for the cumulative recovery (Fig. 4). Upscaling reduces the variability in the permeability distribution, and this reduces the amount of physical dispersion of the flood front (e.g. Zhang & Tchelepi 1999). Therefore, in the upscaled model, water breaks through later than in the fine-scale model, and the recovery is higher. In the case of the history-matched model, the permeabilities are adjusted to reproduce the fine-scale recovery. This means that the amount of physical dispersion is preserved, and the pdf is similar to that of the fine-scale model.

Instead of upscaling, it might have been more accurate to obtain the coarse-scale permeability values by sampling from the fine-scale values. Figure 6 shows a comparison of random sampling of the fine-scale grid compared with upscaling. Although there is considerable error in the average recovery from 1000 realizations of the random sampling method, the results are slightly more accurate than local upscaling. (Note that these results apparently contradict Durlofsky (1992), who concluded that, unless the scale-up factor was small compared to the correlation length, it was better to upscale rather than to sample. However, Durlofsky (1992) was considering single-phase steady-state flow. In the current study, we are using two-phase flow, and the dispersion in the flood front has to be maintained at the coarse scale.)

This study shows that upscaling by a large factor (3400 in this case) can give rise to large errors in the predicted recovery. Therefore, if many models are to be simulated to take account of uncertainty, it is unwise to generate lots of fine-scale models and upscale them. (Also, the time taken to generate many fine-scale geological models is prohibitive.) In order to produce unbiased results in coarse-scale models, we need to generate an appropriate amount of variability in the model. This suggests that it is better to create coarse-scale models by sampling the fine-scale structure or using fine-scale geostatistics, than to generate fine-scale models and upscale them. This procedure is suitable for reservoirs where there is much uncertainty, such as in the early stages of field development, when a range of models must be evaluated rapidly.

As a field is developed, more information, such as logs or cores from additional wells and more production history, becomes available so more detailed models may be created. For example, a second generation of models may be constructed which have several 10s of 1000s of cells. If fewer realizations are constructed (due to less uncertainty), such models can easily be simulated without the need for upscaling. Later in the life of a field though, multimillion cell models may be required for investigating EOR strategies or placement of additional wells. In this case, upscaling will be necessary, and it is important to use an accurate method, otherwise the detail in the model will be lost.

## A more accurate upscaling method

In the study described above, we used the pressure solution method with locally applied no-flow boundary conditions. This method is frequently used in industry because it is simple, but more accurate results may be obtained with a little more effort. For example, frequently non-uniform coarse grids are constructed to maintain the permeability heterogeneity whilst reducing the number of grid cells. (See, for example, Garcia *et al.* 1992; and Durlofsky *et al.* 1996) With this approach, the amount of physical dispersion in the coarse-scale model is closer to that of the fine-scale model, and so the watercut and recovery are more accurately reproduced. However, the use of single-phase upscaling with non-uniform coarsening is likely to break down if the upscaling factor is so large that high and low permeability channels are merged. In that case, the relative permeabilities must also be upscaled to give the correct flux (Wallstrom *et al.* 2002).

Another source of error in upscaling is the effect of boundary conditions. In the study above, we applied local, no-flow, boundary conditions, i.e. we fixed the pressures at each edge of a coarse cell and applied sealed boundaries to the edges. These boundary conditions are unlikely to reproduce the actual pressures and flows within the fine grid. The effect of boundary conditions may be reduced by using a ‘flow-jacket’ or ‘skin’ around each coarse cell. The boundary conditions are then applied to this larger region. This method is sometimes referred to as the extended local method (Chen *et al.* 2003). A more accurate approach is to perform a single-phase flow simulation on the whole fine-scale model. This is referred to as a global approach (Holden & Nielsen 2000). Performing a single pressure solve over a fine grid is feasible, even for very large multimillion cell models. (The problem with large models is simulating *multiphase* flow, because the pressure equations may need to be solved thousands of times.)

### A new approach

A global upscaling approach has been selected here. We refer to the method as the Well Drive Upscaling (WDU) method, because we set the pressures at the wells, in the single-phase flow simulation (Fig. 7). From the results of the global single-phase simulation, we calculate upscaled transmissibilities, where transmissibility, T_{x}, in the x-direction is defined as:
where k_{x} is the x-direction permeability, Δx is the distance between the centres of adjacent grid cells and A is the area perpendicular to flow (=ΔyΔz). The transmissibilities in the y- and z-directions are defined in a similar manner. The upscaled transmissibilities are calculated as follows (Fig. 8):

Sum the fine-scale flows;

Average the pressure, using pore volume weighting; and

Apply Darcy's law.

By calculating the coarse-scale transmissibilities directly, we save time in the coarse-scale simulation, and avoid errors arising from calculating the coarse-scale transmissibilities from the effective permeabilities. Zhang *et al.* (2005) and Zhang (2006) give full details of the method, along with the results of tests on a variety of heterogeneous models, such as sand/shale models and channel models. A technique for estimating the optimum coarse grid size is described in Zhang *et al.* (2007).

The WDU method, as described above, can be applied to problems where a single relative permeability curve is used for the whole geological model. However, when there are multiple relative permeability curves (e.g. one for each rock type), a decision has to be made as to which curve to use in a coarse cell (which may contain a number of different rock types). Often, coarse-scale curves are chosen according to the ‘majority vote’ (the relative permeability curve belonging to the majority of the fine cells). However, this can give rise to errors. We have extended the WDU method to include an analytical calculation of the coarse-scale relative permeabilities: we average the fine-scale relative permeabilities using transmissibility weighting. This approach does not require a two-phase flow simulation, and is therefore quick and feasible for large models. Again, more details may be found in Zhang *et al.* (2005).

### Comparison between the WDU method and dynamic two-phase upscaling

The new WDU method is essentially a single-phase upscaling approach. A number of two-phase dynamic upscaling methods have been developed, but they are not robust (Barker & Thibeau 1997), and they are time-consuming because they require two-phase flow simulations. However, we mention one procedure here, the Pore Volume Weighted (PVW) method (Schlumberger 2004) because we have compared it with the WDU method. Both methods use pore volume weighting to average the pressure in a coarse block. In the PVW method, this calculation is performed for each phase in order to calculate the upscaled relative permeability, while in the WDU method it is only applied to single-phase flow to calculate upscaled transmissibility. (In the PVW method, as described in the Eclipse PSEUDO manual, the upscaled absolute transmissibility is estimated by averaging the fine-scale permeabilities.) Because of the time involved in performing the two-phase flow calculations, the PVW method is not feasible for multimillion cell models. On the other hand, the WDU method involves only a single pressure solve so it is a viable approach.

### Examples of the WDU method

We have tested the WDU method on a number of heterogeneous models (Zhang *et al.* 2005). Here, we show two sets of results using the SPE 10 model.

#### Case 1

In the first example, we use a single layer (layer 59) of the SPE 10 model (Fig. 9). This layer is very heterogeneous, with a complex channel structure, so provides a rigorous test for the method. Note that the permeability range covers eight orders of magnitude. Also, since this is only a 2D model, we were able to perform a full two-phase flow simulation. The size of the fine grid is 60×220 cells, and this was upscaled to 10×22 cells (an upscaling factor of 60).

For this test, we modified the well locations and the relative permeabilities. Rather than arbitrarily placing the injector well in the middle of the model and the producers in the corners, we shifted the wells to high permeability channels. Since the permeability distribution is bimodal (channel and background facies), we used two relative permeability curves (Fig. 10a). After performing the single-phase upscaling, we calculated the average relative permeability curves for each coarse cell, using the method described above. The results are shown in Figure 10b. The number of relative permeability curves could be reduced by grouping similar curves together, but in this study we used all the curves.

The results for the WDU method were compared with the results of the fine-scale model, the local upscaling method with ‘majority vote’ relative permeabilities, and the PVW upscaling method (Schlumberger 2004). (In the case of the PVW method, we used a full fine-scale two-phase flow simulation, which would not be feasible in a real field model.) The oil saturation distributions after the injection of one pore volume of water are shown in Figure 11. It can be seen that the local upscaling method gave a poor reproduction of the saturation distribution, but the WDU and PVW methods produced good results. Figure 12 shows the cumulative oil production. Again, the WDU and PVW methods reproduced the fine-scale results much better than the local upscaling method. Since the PVW method involved two-phase simulation of the whole fine-scale grid, this method should reproduce the fine-scale results with reasonable accuracy. The WDU method only used a single-phase flow simulation but, by using appropriate boundary conditions, the main flow paths through the model were maintained. The results are as good as the PVW method, but with much less time and effort.

#### Case 2

In this test of the WDU method, we upscaled the full SPE 10 model, using the original specifications for well locations and relative permeabilities (Christie & Blunt 2001). The 60×220×85 cell model was upscaled to 10×22×17 (scale-up factor of 300). A comparison of the oil production rate for well P1 is shown in Figure 13. The fine-scale results used here are those supplied in the SPE 10 study (Christie & Blunt 2001). Note that we could not run the PWV method in this case because there were too many cells in the full 3D SPE model. It can be seen that the WDU method gives much better results that the local upscaling method.

### Discussion

The WDU method is more time-consuming than the local upscaling method, but it is still feasible to use this method for multimillion cell models. We have shown that, in highly heterogeneous models with complex structures, it is more accurate than the local pressure solution upscaling method which is commonly used. However, in models with a low level of heterogeneity, the local method is often adequate, so there is no advantage in applying the WDU method. We suggest that this method is appropriate for highly heterogeneous models, where careful simulation is required, e.g. in mature fields for planning infill drilling or IOR schemes. At this stage, there is likely to be less uncertainty, and it is worthwhile spending more time on modelling and flow simulation.

## Summary

In this paper, we suggest two alternative procedures for modelling and simulating flow in hydrocarbon reservoirs. In cases where there is much uncertainty, such as during the early stages of field life, building detailed reservoir models with millions of grid cells is not worthwhile. Instead, effort should be concentrated on evaluating many (thousands or tens of thousands) coarse-scale models, in order to evaluate the uncertainty. However, flow simulation in coarse-scale models may produce erroneous results. In particular, we show that if the level of heterogeneity is underestimated, the recovery may be overestimated. We propose that, at this stage of reservoir modelling, it may be more accurate to generate coarse-scale models using fine-scale geostatistics, rather than generating fine models and upscaling. Our tests show that, although this method is not very accurate, it is on average slightly more accurate than upscaling.

On the other hand, there are times when more precise simulation is required, such as later in field development, or for simulating parts of a reservoir in more detail. In this case, we suggest that a global single-phase simulation is carried out to reduce the errors in upscaling. This provides more accurate single-phase upscaling, which increases the accuracy of coarse-scale two-phase flow simulation. Although this method is more time-consuming than conventional approaches (such as local upscaling methods), it is feasible for multimillion cells models, and is worth the extra effort to achieve more accuracy, especially in highly complex models.

## Acknowledgments

This work was part of the Uncertainty and Upscaling Project at Heriot-Watt. We acknowledge the support of the following companies: Anadarko, BG, BP, JOGMEC, Petronas and the UK DTI. Hashem Monfared was sponsored by NIOC. We should also like to thank Schlumberger for the use of the Eclipse reservoir simulation package.

- © The Geological Society of London 2008