This study started by loading data from a file named Flow Unit 5, a DATformat data file containing thick, porosity and permeability data from about 68 wells in Flow Unit 5 Oil Field.
Thickness
Simple kriging and ordinary kriging of isotropy
Simple kriging Figure 1a shows the gross thickness map generated with isotropic simple kriging. Overall, the map is not that smooth. In the northwest corner, the conditioning data is undersampled with respect to the rest of the grid. The isotropic simple kriging variance map is shown in Figure 1b. The estimation variance is pretty high through the entire map. Small variance only exists in places where there are wells distributed. The histogram, for the simple kriging estimates, is shown in figure. Most of the data are distributed close to the centre, with some extreme values more than 10 ft. The maximum estimate value is 24.61 ft, while the maximum for the data is 27 ft. There is almost no improvement compared to the histogram of the conditioning data. 92 points are compared and plotted in the scatter plot of gross thickness data and isotropic simple kriging estimation data. The correlation coefficient is 2.082, indicating that isotropic SK tends to overestimate the estimates.
Ordinary kriging
Figure 2a shows the gross thickness map generated with isotropic ordinary kriging. Overall, the map is not that smooth. The continuity is not that good even if it is a bit better than the map generated from simple kriging. In the northwest corner, the conditioning data is undersampled with respect to the rest of the grid. The isotropic ordinary kriging variance map is shown in Figure 2b. The estimation variance is pretty high through the entire map. Small variance only exists in places where there are wells distributed. The histogram, for the isotropic ordinary kriging estimates, is shown in figure. Most of the data are distributed close to the centre, with some extreme values more than 10 ft. The maximum estimate value is 25.036 ft, while the maximum for the data is 27 ft. There is almost no big improvement compared to the histogram of the conditioning data. 92 points are compared and plotted in the scatter plot of gross thickness data and isotropic ordinary kriging estimation data. The correlation coefficient is 1.384, indicating that isotropic simple kriging tends to overestimate the estimates.
Simple kriging and ordinary kriging of anisotropy
Simple kriging
Figure 3a shows the gross thickness map generated with simple kriging. Overall, the map has a smooth appearance that is typical of simple kriging. The good spatial continuity from east to west corresponds to the principal direction of the variogram. The north/south trend, observed in a small area in the northwest corner of the map, is a result from the configuration of the conditioning data and the search neighborhood. In the northwest corner, the conditioning data is undersampled with respect to the rest of the grid. The simple kriging variance map is shown in Figure 3b. The estimation variance is small in gridblocks close to the conditioning data, and it becomes large in area far from the data. Near the conditioning data, the kriging variance becomes the nugget effect of the variogram. The histogram, for the simple kriging estimates, is shown in figure. The mean is 5.43, the median is 4.88 and the mode is around 4, all of which are pretty close to the conditioning data. The standard deviation of simple kriging estimates is 2.63 ft, which for conditioning data, it 4.53 ft. So it is likely that has a narrower spread than the conditioning data. In general, simple kriging does not closely reproduce the extreme values of a distribution with a gross thickness greater than 10 ft, observed in conditioning data. The maximum estimate value is 17.57 ft, while the maximum for the data is 27 ft. Simple kriging tends to make the data follow normal distribution, most of the data are distributed close to the mean value. 92 points are compared and plotted in the scatter plot of gross thickness data and simple kriging estimation data. Simple kriging provides good estimates even for values that are much larger than the mean of the data (5.05 ft). All the plotted points fall approximately along the straight line, with correlation coefficient 0.891, indicating high accuracy of the estimates.
Ordinary kriging
Figure 4a shows the gross thickness map generated with ordinary kriging. Overall, the map has a more smooth appearance than that of simple kriging. Transition areas exist in ordinary kriging map, which represent gradual change among the subareas. The good spatial continuity from east to west corresponds to the principal direction of the variogram. The north/south trend, observed in a small area in the northwest corner of the map, is a result from the configuration of the conditioning data and the search neighborhood. In the northwest corner, the conditioning data id undersampled with respect to the rest of the grid. And this area tends to be larger than that of the simple kriging. Generally, the two maps are similar with each other. The ordinary kriging variance map is shown in Figure 4b, which is smoother than that of the simple kriging variance map. The blue area that represents small variance tends to be wider, which stands for a better estimation result compared to simple kriging. The estimation variance is small in gridblocks close to the conditioning data, and it becomes large in area far from the data. Near the conditioning data, the kriging variance becomes the nugget effect of the variogram. The histogram, for the ordinary kriging estimates, is shown in figure. The mean is 5.68, the median is 4.89 and the mode is around 4, all of which are pretty close to the conditioning data. The standard deviation of ordinary kriging estimates is 2.98 ft, while for conditioning data, it is 4.53 ft. So it is likely that has a narrower spread than the conditioning data. In general, simple kriging does not closely reproduce the extreme values of a distribution with a gross thickness greater than 10 ft, observed in conditioning data. The maximum estimate value is 18.08 ft, while the maximum for the data is 27 ft. Ordinary kriging tends to make the data follow normal distribution; most of the data distribute close to the mean value. 92 points are compared and plotted in the scatter plot of gross thickness data and ordinary kriging estimation data. Simple kriging provides good estimates even for values that are much larger than the mean of the data (5.05 ft). All the plotted points fall approximately along the straight line, with correlation coefficient 0.825,though it is slightly smaller than the simple kriging.
Comparison between isotropic and anisotropic variogram modeling
From the thickness estimate of isotropic and anisotropic modeling, as shown in Figure 1 and Figure 2, it is obvious that anisotropic variogram is much smoother than the isotropic one. From the isotropic and anisotropic variance maps, the estimation variance is much lower for the anisotropic variogram than the isotropic one. This can further be demonstrated by the scatter plot of isotropic OK and anisotropic OK variance, where the isotropic variance is much larger than the anisotropic variance. All these illustrate that anisotropic estimation is more close to the conditioning data and is more accurate.
Through the above comparison between the isotropic modeling and anisotropic modeling, we draw a conclusion that anisotropic variogram can achieve better spatial structure capture more interpretable spatial relationship. In the following analysis of porosity and permeability properties, anisotropic variogram is mainly used to generate the spatial relationship of flow unit 5.
Porosity
Simple kriging
Figure 5a shows the porosity map generated via simple kriging. Overall, the map has a smooth appearance. The good spatial continuity from middle to west corresponds to the principal direction of the variogram. The simple kriging variance map is shown in Figure 5b. The estimation variance could be as low as 0.6369 at where gridblocks are close to the sample data. However, in the northeast corner, the estimation variance could be as large as 64.28 due to the scarcity of sample data. For porosity data, the mean and median are 0.41 and 0.97, respectively. They are very close to each other, indicating symmetry in the distribution. The coefficient of variation could be obtained via equation, and its value is 0.2735, indicating a relatively small variation within the sample. The histogram is shown in Figure 5d. The mean is 0.67, the median is 0.39 and the variance is 25.5767. Comparing with the histogram of sample data, the estimated values are generally a little smaller than the sample data. However, the maximum and minimum values are quite close to that of samples respectively, indicating a relatively fine estimation. Moreover, the estimation variance is much smaller than the sample variance, which implies a relatively low variability in the estimated values. Similar conclusion can also be obtain via the coefficient of variation, which is 0.2446.
Ordinary kriging
The porosity map from ordinary kriging is shown in Figure 6a. In general, this map is quite similar to the map generated via simple kriging. However, the appearance is much smoother. The variance map is shown in Figure 6b. The minimum variance is 0.637, which is quite close to that from simple kriging. However, the maximum variance would be as large as 78.21, which is noticeably larger than the maximum variance from simple kriging. The true value vs. the estimated value plot is shown in Figure 6c. All the points regularly spread around the 45° line, indicating that the estimates generated via ordinary kriging can also match the sample data properly. The histogram is shown in Figure 6d. The mean is 0.7169, the median is 0.4647 and the variance is 31.3116. Comparing with the histogram of sample data, the estimated values are also slightly smaller than the sample data. However, the maximum and minimum values are also quite close to that of samples respectively, indicating a relatively fine estimation. Comparing with the histogram of simple kriging’s estimation, there is no significant difference in terms of their mean values. However, the estimation variance from ordinary kriging method is much larger than that from simple kriging method.
Sequential Gaussian simulation of porosity
Next, we generate five realizations of porosity using sequential Gaussian simulation. The sequential Gaussian simulation will use a normal score transform to turn the porosity values at the wells into a set of values that perfectly follow a standard normal distribution (zero mean, unit standard deviation) and will then generate grids of simulated values whose univariate distribution is also standard normal. Simple kriging is to be applied since the spatially constant mean will be assumed to be zero. In addition, we assume that the variogram of the normalscore transformed data would look very similar to the variogram of the raw data scaled to a unit sill.
The simulation map is shown in Figure 7a. As can be seen from this figure, the distribution trend of the simulated porosity is quite similar to that from the simple kriging estimation as well as the ordinary kriging estimation. High porosity locations are spreading from middle to west. Noticeably, in the southwest corner, the simulated values of porosity are higher than either of the estimation results from the two kriging estimation methods. Figure 7b shows the true value vs. the simulated value plot. All the points lie in the 45° line, indicating a perfect match between true values and simulated values at sampled locations.
Original Oil In Place (OOIP)
OOIP is calculated with gross thickness and porosity maps generated with ordinary kriging OOIP in stock tank barrel (STB) for each gridblock in the map is given by:
\mathit{OOIP}=\frac{A\times h\times \mathit{ng}\times \left(\varphi /100\right)\left(1\mathit{Sw}\right)}{5.615\times \mathit{Bo}}
(1)
A = the surface area of a block (200 × 200 = 40,000 ft2), h = gross thickness(ft), ng = net to gross ratio, Φ = porosity(%), Sw = the water saturation and Bo = formation volume factor (rest bbl/STB). So:
\begin{array}{l}\mathit{OOIP}=\frac{40000\times 5.687\times 0.7\times \left(20.717/100\right)\left(120\%\right)}{5.615\times 1.2}\\ \phantom{\rule{2.7em}{0ex}}\approx 3916.76\end{array}
(2)
Permeability
When considering the permeability, the values of K range from as low as 0.01 to 750 md, the majority of the values are at the lower end of the region. This type of histogram is rarely useful for characterizing a sample because the values are clustered at one end. One way to overcome this problem is to transform the sample data in some way so that some sample characteristics are evident from the histogram plot. The most commonly used approach for permeability values is the log transform. From analysis, the log k distribution is much more symmetric than the permeability distribution. In addition, the log k and porosity histogram are remarkably similar. Both show similar trends with two peaks in the histogram plot, one of which is at the higher end of the values. Although this needs to be validated, such characteristic similarity way indicates a relationship between log k and porosity.
Simple kriging
Figure 8 shows the permeability map generated via simple kriging. Overall, the map has a smooth appearance. The good spatial continuity from middle to west corresponds to the principal direction of the variogram. The simple kriging variance map is shown in Figure 8b variance of permeability. The estimation variance could be as low as 0.6369 at where gridblocks are close to the sample data. However, in the northeast corner, the estimation variance could be as large as 64.28 due to the scarcity of sample data. Then we use the Scatter Plot to compare the value of log k (hard data) with SK value of logk. The plot shows that result of comparison is close to a 45° slant, which means the value of estimation is really match the hard data very well.
Ordinary kriging
The permeability map from ordinary kriging is shown in Figure 9a. In general, this map is quite similar to the map generated via simple kriging. However, the appearance is much smoother. The variance map is shown in Figure 9b. The minimum variance is 0.637, which is quite close to that from simple kriging. However, the maximum variance would be as large as 78.21, which is noticeably larger than the maximum variance from simple kriging.
Simple cokriging
Figure 10a shows the gross thickness map generated with simple cokriging. Overall, the map has a smooth appearance. The estimated distribution matches the original distribution well. The yellow and red areas represent the areas of which the permeability is relatively higher, while the blue areas stand for where the permeability is relatively lower. The good spatial continuity corresponds to the principal direction of the variogram. The small areas in the corner and bound of the map, is a result from the configuration of the conditioning data and the search neighborhood. In the northwest corner, the conditioning data is undersampled with respect to the rest of the grid. The simple cokriging variance map is shown in Figure 10b. The estimation variance is small in gridblocks close to the conditioning data, and it becomes large in area far from the data. Near the conditioning data, the kriging variance becomes the nugget effect of the variogram. The histogram, for the simple kriging estimates, is shown in figure. The mean is 1.90, the median is 1.95, both of which are a bit smaller than the conditioning data. The standard deviation of simple cokriging estimates is 0.78; while for conditioning data, it is 0.941. So it is likely that has a narrower spread than the conditioning data. In general, Cosimple kriging tends to gather the data to the center, making the data follow normal distribution. That is, most of the data distributes close to the mean value. 55 points are compared and plotted in the scatter plot of gross log k data and simple cokriging estimation data. Simple kriging provides good estimates even for values that are a bit smaller than the mean of the data (1.90 to 2.13). All the plotted points fall approximately along the straight line, indicating high accuracy of the estimates.
Ordinary cokriging
Figure 11a shows the gross thickness map generated with ordinary cokriging. Overall, the map has a more smooth appearance than that of simple cokriging. Transition areas exist in ordinary cokriging map, which represent gradual change among the subareas. The estimated distribution matches the original distribution well. The yellow and red areas represent the areas of which the permeability is relatively higher, while the blue areas stand for where the permeability is relatively lower. The good spatial continuity corresponds to the principal direction of the variogram. The small areas in the corner and bound of the map, is a result from the configuration of the conditioning data and the search neighborhood. In the northwest corner, the conditioning data is undersampled with respect to the rest of the grid. The ordinary cokriging variance map is shown in Figure 11b, which is more smooth than that of the simple cokriging variance map. The blue area that represents small variance tends to be wider, which stands for a better estimation result compared to simple cokriging. The estimation variance is small in gridblocks close to the conditioning data, and it becomes large in area far from the data. Near the conditioning data, the kriging variance becomes the nugget effect of the variogram. The histogram, for the ordinary cokriging estimates, is shown in figure. The mean is 1.79, the median is 1.84, both of which are a bit smaller than the conditioning data. The standard deviation of ordinary cokriging estimates is 0.887; while for conditioning data, it is 0.941. So it is likely that has a narrower spread than the conditioning data. In general, ordinary cokriging tends to gather the data to the center, making the data follow normal distribution. That is, most of the data distributes close to the mean value. 92 points are compared and plotted in the scatter plot of gross log K data and ordinary cokriging estimation data. Ordinary cokriging provides good estimates even for values that are much larger than the mean of the data (1.79 to 2.13). All the plotted points fall approximately along the straight line, indicating high accuracy of the estimates.
MM2 models
Figure 12a shows the gross thickness map generated with MM2 cokriging. Overall, the map has a more smooth appearance. Transition areas exist in the estimate map, which represent gradual change among the subareas. The estimated distribution matches the original distribution well. The yellow and red areas represent the areas of which the permeability is relatively higher, while the blue areas stand for where the permeability is relatively lower. The good spatial continuity corresponds to the principal direction of the variogram. The small areas in the corner and bound of the map, is a result from the configuration of the conditioning data and the search neighborhood. In the northwest corner, the conditioning data is undersampled with respect to the rest of the grid.
The MM2 cokriging variance map is shown in Figure 12b, which is smoother than that of the Cosimple kriging variance map. The blue area that represents small variance tends to be wider. The estimation variance is small in gridblocks close to the conditioning data, and it becomes large in area far from the data. Near the conditioning data, the kriging variance becomes the nugget effect of the variogram. The histogram, for the MM2 cokriging estimates, is shown in figure. The mean is 1.86, the median is 1.99, both of which are a bit smaller than the conditioning data. The standard deviation of MM2 kriging estimates is 0.879; while for conditioning data, it is 0.941. So it is likely that has a narrower spread than the conditioning data. 92 points are compared and plotted in the scatter plot of gross log K data and MM2 estimation data. All the plotted points fall approximately along the straight line, indicating high accuracy of the estimates.
Thickness SK vs OK
To have a thorough understanding of the results, the difference analysis among the three methods is required. The difference in the estimation results from the two methods is shown in Figure 13. We can see that the map is, in general, covered with light orange (less than 0.3 from color bar), indicating the slight difference between the two methods. The estimate mean of OK is a bit larger than that of SK. In the westnorth region, the light blue color implies the estimate of OK mean in this region is less than that of SK. This maybe mainly because this region is undersampled and there is less data available.
Porosity
Simple kriging estimation vs. Ordinary kriging estiamtion
The difference in the estimation results from the two methods is shown in Figure 14a. We can see that the map is, in general, covered with light blue, which implies the less difference between the two methods. After checking the sample data, we can find out that the light blue regions are where the true values exist. It thus well explains why there is less difference between the two methods. However, at where sample data are not available, the estimation difference between the two methods is remarkable. For example, in the northeast corner, the estimated values from simple kriging are significantly higher than that of ordinary kriging. In comparison, the estimated values, in the southwest region, are slightly lower than the values obtained via ordinary kriging. The difference in the estimation variance of the two methods is shown in Figure 14b. In general, the difference is inconsiderable. However, in the northeast corner, simple kriging estimates’ variance is remarkably lower than the variance from ordinary kriging estimation.
Simple & ordinary kriging estimation vs. Simulation
The two sets of estimation results are compared with the simulation results, as shown in Figure 15. Overall, the two maps are quite similar to each other. We can see that, in the north region, estimation results are lower than the simulation results. However, in the south region as well as the northeast corner, the estimation results are much larger than the simulation results.
Permeability
SK vs OK
The difference in the estimation results from the two methods is shown in Figure 16a. We can see that the map is, in general, covered with light blue, which implies the less difference between the two methods. After checking the sample data, we can find out that the light blue regions are where the true values exist. However, at where sample data are not available, the estimation difference between the two methods is remarkable. In the northeast corner, the estimated values from simple kriging are significantly higher than that of ordinary kriging.
Simple cokriging vs ordinary cokriging
The difference in the estimation of the two methods is shown in Figure 16b. In general, the difference is inconsiderable. However, in the northeast, southwest and southeast corner, simple kriging estimate is remarkably higher than the that of ordinary kriging estimation, where sample data is undersampled.
OK vs ordinary cokriging
The difference in the estimation results from the two methods is shown in Figure 16c. The map is, in general, saturated with light blue, which implies less difference between the two methods. However, in the places where the color is dark blue and red, the estimation difference between these two methods are remarkable. This happens mainly because there is no enough data collected in these regions.
Ordinary cokriging vs MM2
The difference in the estimation results from the two methods is shown in Figure 16d. We can see that the map is, in general, covered with orange (with the value of 0.05 from colorbar), indicating the slight difference between the two methods. And the estimate mean of ordinary kriging is a bit larger than that of MM2. Dark blue only exists in a few regions. These are where the estimate of MM2 is remarkably large than ordinary cokriging. This happens because there is not enough data collected.
Thickness comparison between SK and OK

a.
The scatter plot of the simple kringing estimate and the ordinary kriging estimate is shown in Figure 17. The correlation coefficient is 0.973, indicating a high similarity between these two estimates. Most differences occur for gross thickness between 4 and 7 ft. And the ordinary kriging estimate tends to be a little larger than the simple kriging estimate. b. The scatter plot of the simple kringing variance and the ordinary kriging variance is shown in Figure 17. Ordinary kriging tends overestimate thickness. Figure 17 shows that the simple kriging variance is smaller than the error variance estimated with the ordinary kriging. In view of the estimation mean and variance, although SK estimate tends to be more close to the conditioning data for this case, OK and SK both can generate good estimation result.
Porosity comparison between SK and OK

a.
The scatter plot of the simple kringing estimate and the ordinary kriging estimate is shown in Figure 18. The correlation coefficient is 0.984, indicating a high similarity between these two estimates. Most differences occur for gross thickness between 12 and 19. The ordinary kriging estimate tends to be a little larger than the simple kriging estimate. b. The scatter plot of the simple kringing variance and the ordinary kriging variance is shown in Figure 18.The simple kriging variance is only slightly smaller than the error variance estimated with the ordinary kriging.
Log K comparison between SK and OK

a.
The scatter plot of the simple kringing estimate and the ordinary kriging estimate is shown in Figure 19. The correlation coefficient is 0.982, indicating a high similarity between these two estimates. Most differences occur for gross thickness between 0.5 and 1. And the ordinary kriging estimate tends to be a little larger than the simple kriging estimate. b. The scatter plot of the simple kringing variance and the ordinary kriging variance is shown in Figure 19. The correlation coefficient is 0.998, indicating there is almost no difference between the estimation variance of the two.
Log K comparison between simple cokriging and ordinary cokriging

a.
The scatter plot of the simple cokringing estimate and the ordinary cokriging estimate is shown in Figure 20. The correlation coefficient is 0.972, indicating a high similarity between these two estimates. Most differences occur for log K between 1 and 3. And the ordinary cokriging estimate tends to be a little larger than the simple cokriging estimate. b. The scatter plot of the simple cokringing variance and the ordinary cokriging variance is shown in Figure 20. Figure 20 shows that the simple kriging variance is smaller than the error variance estimated with the ordinary kriging. In view of the estimation mean and variance, although simple kriging estimate tends to be more close to the conditioning data for this case, ordinary and simple kriging both can generate good estimation result.
Log K comparison between ordinary kriging and ordinary cokriging

a.
The spatial distributions of log K are significantly different for ordinary cokriging and ordinary kriging methods. The map, generated with ordinary cokriging is not as smooth as ordinary kriging. The influence of porosity adds some unique features to the log k data map that ordinary kriging cannot capture with a single variogram model. The ordinary cokriging and ordinary kriging estimates differ significantly, as shown in Figure 21b. Ordinary kriging tends to overestimate log K. Figure 21b shows that the cokriging variance was smaller than the error variance estimated with the ordinary kriging. This shows that additional information used in cokriging reduces the error variance in estimates.
Log K comparison between ordinary cokriging and MM2

a.
The spatial distributions of log K are significantly different for MM2 and ordinary cokriging methods. From the scatter plot of both, the MM2 estimates tend to be larger than the ordinary full cokriging estimates, as shown in Figure 22b. MM2 cokriging tends overestimate log K. Figure 22b shows that the full cokriging variance is smaller than the error variance estimated with the MM2. This shows that additional information used in ordinary cokriging reduces the error variance in estimates.