We have validated our choice of cell size. The well data are blocked in the 3D geological grid and we are even comforted in our workflow. It is time now to distribute the petrophysical properties… by facies of course.

The previous two chapters talked about the geostatistical algorithms in general terms and about some specifics for modeling discrete properties. Much of what was covered previously also applies to modeling continuous properties such as VSH or porosity. The main algorithms are again kriging for interpolation and Gaussian simulation for generating multiple realizations. Both use variograms and some statistical parameters as input (mean value for the kriging, distributions for the simulation algorithms). Both can also take into account different types of secondary variables. Vertical Trend Curves (VTC) are the equivalent of the Vertical Proportion Curves (VPC) for discrete properties. They capture how the mean of a given continuous property varies with depth (see Figure 2 and Figure 8 for some examples). By depth, we mean by horizontal layer in the 3D geological grid. For example, VTCs would spot a decrease of porosity with depth due to compaction. Sometimes, continuous properties also show horizontal trends. Maybe the VSH increases from the North East corner to the South West for example. Such information can be captured as a map which can be taken into account by the geostatistical algorithms, in the way facies proportion maps are used to guide the modeling of facies. VTCs and trend maps capture how a property should vary spatially. It is also recommended to look at how properties are correlated one with the other. For example, if a cross-plot porosity versus VSH shows that the two properties are highly correlated, we can model first the VSH and then use the 3D distribution of VSH as a guide (a secondary variable) to model the porosity in 3D.

Ultimately, the key is to understand how each property varies spatially, how the different properties are connected one to the other and to convert all this knowledge into secondary data for the geostastistical algorithms. The geologist can help interpreting spatial trends while the correlations between properties are of course deeply rooted in the petrophysical analysis. While a geomodeler might guess them all properly, it is much more efficient to identify the meaningful correlations with the team’s petrophysicist.

We illustrate the whole workflow hereafter with a real dataset. Carbonate or sandstone, conventional or unconventional, Canadian or international, none of this matters here as the general methodology can be easily adapted to any reservoir. For that reason, we won’t detail any specifics. Similarly, it is nearly impossible to read the values on the axes of many of the associated pictures. This is done on purpose as the key is to focus on how one graph compares to another globally. Specific numbers won’t add anything. You will be able to adapt it to your project by working with your team to identify what petrophysical properties should be modeled. In fact, we might have as well called our logs A, B, C and D instead of VSH, porosity, SW and PERM for this example.
Looking at a geomodel in 3D views, in cross-section and in map view is important of course. Such views must be used to validate the 3D petrophysical distributions of course. But sometimes, we conclude that because it looks good in 3D (or 2D), it is correct. This conclusion can be misleading when dealing with petrophysical modeling. In the example below, consider that every distribution we created did “look good” in 3D. Such displays did not help us improve our workflow. For that, we had to rely on three other types of views, which we believe geomodelers do not always use as much as we should: histograms (Figure 1), VTCs (Figure 2) and cross-plots between the different petrophysical properties (Figure 3, Figure 4 and Figure 5). On each of these three displays, the black colour is used for the original log data, while the blue and the red are used respectively for the blocked data and the 3D model. For each type of display, we should look at how the display from the original log data compares with the display of the blocked values and then how both compare with the display of the 3D distributed values. The first comparison relates to our discussion in the section about upscaling the well logs: a good well blocking will respect the original statistics of the different properties. Similarly, a good 3D petrophysical model should also respect the input statistics. By lack of space we did not include figures nor any discussion about trend maps, but those should be looked at as well, at the least to check if such trends exist or not, and if some do, to use it/them.

In this example, an initial modeling workflow is defined, run and the computed model is analyzed (Figure 1 to Figure 5). A few problems are found which lead us to modify the workflow. Figure 6 to Figure 9 illustrate the improvements we gained in doing so. This loop should be repeated until no more fine-tuning is needed and the team considers that the geomodel does properly capture the characteristics of the petrophysics in this reservoir.

Firstly, the well data were blocked. The distributions of the blocked VSH and PERM are very similar to the distributions of the original well logs (Figure 1, first two rows). The blocked porosity and SW are close as well, but they show some suspicious bars at (porosity=0%) and at (SW=100%). This is the kind of thing to look for; discrepancies between original and blocked statistics. In this case, it is normal in fact: a post-process was applied to assign these values in some specific K layers. As it relates to something specific to this dataset and it does not add anything to the overall workflow, no more details will be given here. The VTCs (Figure 2) and the cross-plots (compare Figure 3 and Figure 4) confirm that the upscaling was done correctly as the plots on the blocked cells match the plots of the original logs. The cross-plots of porosity-permeability show that the permeability is obviously a mathematical function of porosity. At this time though we will ignore that fact and we treat the permeability like the other three properties. It can be spotted that something is a little suspicious already though: the cross-plot of the blocked values show that the mathematical relationship is not respected at 100% anymore (Figure 4, zoomed in area). But as it concerns only a few points, we decide to ignore this at first.

Gaussian Simulation was used for modeling VSH, porosity and SW, both in the original workflow and the modified workflow. What changed is which secondary variables we used. In the original workflow, VSH is modeled without any secondary variable, while their respective VTCs were used as input for modeling both the porosity and SW. The initial workflow did not use any of the cross-plots as input. In the initial workflow, permeability is also modeled by Gaussian Simulation, without any secondary variable (one might have argued that using porosity could have made sense).

The statistics of the resulting model of VSH, porosity, SW and PERM are then compared to the statistics from the blocked values. The histograms are well preserved (Figure 1) expect for the VSH. The VTCs for the porosity, SW and the PERM are also well respected (Figure 2), which is interesting for the permeability, considering that this VTC was not used as input. This illustrates something important about geostatistics: always run your workflows without secondary variables first; then add the secondary variables and see their impact on the model. In some instances, the well data are all you need to get it right. In others, the statistics of the model without secondary data will contradict the secondary data you intended to use. Which one is correct then? It depends on the reservoir. At the least, you should look into it instead of just adding the secondary variable without giving it any thought. In our example, the VSH model looks problematic again as its VTC really does not match the vertical trend from the data. At last, we look at the different cross-plots (Figure 5, the blue dots are the points from the blocked data; they are displayed here to help visualize the differences with the cross-plot from the 3D modeled distributions). Most of the cross-plots do not match the relationship seen on the input blocked data. This seems normal as we did not use any cross-plot correlation as secondary variable. In the first path at editing our workflow, we will focus on the problem with two cross-plots: porosity versus SW and porosity versus PERM.

For the first cross-plot, the blocked data showed a clear negative correlation while the modeled properties show a positive correlation. This means that we have cells with values of porosity and SW that are not supposed to be associated together in this reservoir. We need to fix this incorrect association. We decide to ignore the SW VTC and instead to use the porosity model as a secondary variable. The porosity is modeled as before, with the porosity VTC as a secondary variable.

For the second cross-plot, the blocked data were showing that the permeability was defined as a function of porosity (as confirmed by our petrophysicist). We can still recognize the relationship in the cross-plot from the 3D model. But many cells do not respect it at all. Of course, this might be fine as the relationship itself has some uncertainty in it. So the current cross-plot could be the sign that we have included some uncertainty in the model. This is true, but not what we intended, and clearly we did not control it. If uncertainty might be needed, maybe it is better to apply a range of mathematical relationships. As stated earlier, using the porosity model as a secondary variable would improve the permeability model, but not as well than if we simply apply directly the mathematical relationship PERM=function(porosity) defined by our petrophysicist. Once we have modeled the porosity in 3D, we apply the function to get permeability everywhere in the 3D grid. This approach is applied in the modified workflow.

In this path, we do not edit how the VSH is modeled. It would still need to be taken care of in another loop.

Analyzing the edited SW and PERM models shows that the problematic cross-plots are now cleaned (Figure 6 for porosity-SW, Figure 7 for porosity-PERM). Interestingly, the new SW model still respects well, and in fact slightly better, the vertical trend captured by the VTC (Figure 8) than the original SW model did, even while this VTC is not used anymore to model the SW. When looking into it, it does make sense. Porosity and SW are negatively correlated: in average, the higher the water saturation, the lower the porosity. The porosity and the SW VTCs show similar relationships (Figure 2): if a K layer has a high average water saturation value, it tends to have a low average porosity value. Our modified workflow still respects the porosity VTC and it now uses the porosity-SW correlation. This modification turned out to be enough to see the vertical trend from the SW VTC respected too.

Similarly, it is interesting to see that the cross-plot of SW-permeability has improved too, even if it was not used as input. SW and PERM have now better distributions and a side-effect was to fix that cross-plot as well.

As mentioned earlier, the work on this specific workflow should be continued and maybe looped a few more times to see if certain displays can be improved even more. Fixing the VSH would be a priority. This work is beyond the scope of this book as it would mean applying the same approach already described here.