5. Results
5.3 The standard approach in fracture modeling
5.3.2 Evaluation of the models
Since the permeability distribution show similarities in all three directions (i.e. I, J K), the models are further evaluated based on the upscaled permeability in the vertical direction (Kk). Figure 33 displays the permeability of the same layer and the six models discussed above. Analyzing the figures one can observe that the permeability pattern is consistent with the fracture intensity. Therefore, for the shown layer, high permeability values are absent for the models based on kriging shown in Figure 33a-c.
48 Figure 33. Permeability (Kk) layer of the models guided by the interpolation algorithms for the three discussed variogram cases (1-3): (a-c) Kriging and (d-f) Gauss simulation.
Figure 33d-f show the heterogeneous permeability distribution based on the Gauss simulation. The maps display high and low permeability zones; however, the location of these zones cannot be justified. In fact, each simulation will result in a different pattern of these zones. In addition, Figure 33d-f illustrate that the variogram range controls the size of the zones of high and low permeability.
Figures 34 and 35 compare the models via their histograms, cumulative distributions, P-values and cross sections along the pseudo wells. The histogram (Figure 34a) illustrates the lognormal distribution of the permeability and emphasizes the variations between the models. Models conditioned by Kriging show more dissimilarities for the three variogram ranges. The large variogram range delivers a broader permeability
49 distribution by means of lower P10 and higher P90 value (dark blue) compared to a small range. Due to this wide distribution, the model displays a more heterogeneous pattern on the cross section (Figure 35c).
The models based on the Gaussian simulation show similar distributions (Figure 34 a, c), and consequently similar P-values. However, the cross sections reveal a more heterogeneous pattern in case of small variogram ranges (Figure 35a, c). It is worthwhile mentioning that both algorithms shows very similar P90 values for the large variogram range (case 3) while P10 and P50 are quite different. In addition, the Gauss based model is characterized by a larger spread of the distribution. This is because Kriging assigns the mean value to the grid cells outside the radius of influence of the data points.
A further model comparison can be performed based on the connected volumes generated for the pseudo wells. The pseudo wells are located along the anticline eastern limb in close vicinity of the NE-SW striking main faults (Figure 24a, section 4.3.1). The connected volumes incorporate only grid cells with permeability larger than the P90 value derived from the measured fracture data.
The highest dissimilarities can be observed for the wells 4 and 5 (Figure 36). The models guided by Kriging demonstrate a strong dependency on the applied variogram ranges (Figure 36a-f). An increasing variogram range yields larger connected volumes (Figure 36a-f). Also the models guided by Gauss simulation show a dependence of the variogram parameters (Figure 36g-l). A model with a small variogram range displays a far greater degree of fracture connectivity, compared to a large variogram range (Figure 36g-l).
50 Figure 34. The permeability distribution (Kk) for the models guided by Kriging and Gauss simulation: a) histogram of log permeability (Kk) for the generated models, b) Cumulative distribution of the permeability for the models conditioned by Kriging and c) Cumulative distribution of the permeability for the models conditioned by Gauss simulation.
Case 1 Case 2 Case 3
Case 1 Case 2 Case 3
c) b)
a)
Kriging Gauss simulation
51 Figure 35. The cross sections along five pseudo wells shown in figure 24 for the models guided by interpolation algorithms for the three different variogram cases: (a-c) Kriging and (d-f) Gauss simulation.
52 Figure 36. Connected volumes for the pseudo wells 4 and 5, for the models based on Kriging (a-f) and Gauss simulation (g-l). For each well, the connected volumes are shown for three variogram cases (Table 5).
53 5.4 Seismic attributes based approach
This section describes the usage of seismic attributes for guiding the fracture intensity distribution.
5.4.1 Variogram parameters derived from seismic attributes
The well data do not allow a correct estimation of the horizontal variogram range. Only the vertical variogram range can be inferred reliably because of the densely sampled fracture intensity logs. On the other hand, the seismic attributes provide additional data points and allow a reliable estimation of the horizontal variogram range. Assuming a close relationship between the fracture intensity and the seismic attributes, the horizontal variogram range is derived from the seismic attributes sampled into 3D grid.
This variogram range is used in collocated co-kriging to interpolate the fracture intensity using a seismic attribute as secondary property (see the chapter Methodology, section 4.2.4). In order to maintain the pattern displayed by the seismic attributes a large correlation coefficient between the measured fracture intensity and seismic attribute is chosen in collocated co-kriging. The used variogram ranges are given in Table 6. The results are discussed in the following sections.
Table 6. The variogram parameters used in collocated co-kriging process.
Attributes Amplitude
contrast Chaos Curvature Consistent dip Major
direction (ft) 4693 5615 5590 14000
Minor
direction (ft) 2200 1555 3908 5500
5.4.2 Fracture intensity
The fracture intensity models are displayed for a layer of the 3D grid in Figure 37. The four maps correspond to the seismic attributes amplitude contrast, chaos, curvature and consistent dip. The maps show quite different intensity patterns. In the southern area, the maps a and b show a significantly different pattern compared to the maps c and d. In addition, the models guided by chaos and curvature (Figure 37b, c) deliver a high fracture intensity along the fold hinge. This pattern is not observed in the other
54 two models. In addition, the models in Figure 37b and c display a low fracture intensity in the northern area. The models guided by amplitude contrast and consistent dip (Figure 37a, d) deliver a high fracture intensity in this area. However, all maps share some similarities. In general, they display higher intensities in the vicinity of the faults along the eastern limb. Also, in the central part the maps display a similar fault pattern (black squares). The fracture intensity varies within the area from medium to high depending on the model.
Figure 37. The fracture intensity maps for the models based on different seismic attributes:
a) amplitude contrast, b) chaos, c) curvature and d) consistent dip.
55 5.4.3 Evaluation of the models
The models are further evaluated using the upscaled permeability. Figure 38 displays the permeability properties for one layer of the models conditioned to the seismic attributes shown in Figure 37.
Figure 38. The permeability (Kk) maps for the models based on different seismic attributes:
a) amplitude contrast, b) chaos, c) curvature and d) consistent dip.
A comparison of figures 37 and 38 shows that high permeability zones are in accordance with high intensity zones and vice versa. In general, higher permeability
56 zones are closely related to the main structures of the area. Also, similar to the intensity models, only the chaos and curvature based models reveal higher permeability zones along the fold hinge.
Figure 39. The permeability (Kk) distribution for the models based on different seismic attributes: a) histogram of log permeability (Kk) for the generated models, b) Cumulative distribution of the permeability of the models conditioned to the seismic attributes.
Figure 39 provides insight into the permeability distribution through the histograms, cumulative distributions and characteristic P-values. The histograms display similar permeability distributions for all models conditioned to the different seismic attributes (Figure 39a). Also the cumulative probability distribution confirms the similarities between the models. Consequently, the P-values (P10-P50-P90) do not deviate between the four models (Figure 39b). A further comparison of the models is provided by cross sections. Figures 40 and 41 show the cross sections along the pseudo wells and the provided wells, respectively. The cross sections show patches of high permeability above 100 mD. They are linked by zones of lower permeability (1-50 mD).
In addition, the model based on consistent dip (Figure 40d) shows a very low permeability zone in the southern part.
a) b)
57 Figure 40. The cross sections along five pseudo wells shown in figure 24 for the models guided by seismic attributes: a) amplitude contrast, b) chaos, c) curvature and d) consistent dip.
58 Figure 41. The cross sections along the provided wells shown in figure 24 for the models guided by the seismic attributes: a) amplitude contrast, b) chaos, c) curvature and d) consistent dip.
59 The cross sections based on the chaos and curvature attributes (Figure 41b, c) show extended high permeability zones mainly between the wells 71-1-x-4 and 67-1-x-10 along the fold hinge. The models displayed in Figure 41a and d, deliver local zones of the high permeability. The connected volumes are examined for the wells 3, 4 and 5 (Figures 42-44), since they are located in areas that highlight similarities and dissimilarities of the permeability between the models. Again, the connected volumes are based on grid cells with a permeability higher than the P90 value of the fracture data.
Figure 42. Connected volumes for the pseudo well 3, for the models based on the seismic attributes: a) curvature, b) consistent dip, c) chaos and d) amplitude contrast.
Figure 43. Connected volumes for the pseudo well 4, for the models based on the seismic attributes: a) curvature, b) consistent dip, c) chaos and d) amplitude contrast.
60 Figure 44. Connected volumes for the pseudo well 5, for the models based on the seismic attributes: a) curvature, b) consistent dip, c) chaos and d) amplitude contrast.
The similarities are mainly noticeable for the wells 3 and 4 of the models guided by chaos and amplitude contrast (Figures 42-43, c and d). The other two models (Figures 42-43, a and b) provide a surprisingly large difference between the two closely positioning wells 3 and 4. In addition, all models do not show an extension of the connectivity to the south of the well 4. While the models based on chaos and amplitude contrast deliver a considerable large area of the connected volumes linked to the well 5 (Figure 44, c and d), the models guided by the curvature and consistent dip do not show connected volumes of high permeability (Figure 44, a and b).
5.5 Discussion
5.5.1 Fracture intensity
The limited well data does not allow deriving a reliable fracture intensity property by Kriging or Gauss simulation. The variogram range used by the two interpolation techniques determines the radius of influence of the data points. Unfortunately, the horizontal variogram range is not known because of lack of well data. This underlines the uncertainty linked to the modeled fracture intensity and consequently to the resulting fracture model. The advantage of Gauss simulation lies in its ability to derive several equi-probable models that can be used for capturing the uncertainty of the fracture intensity distribution. The results seem to be more reliable compared to the Kriging in the sense that the distribution of the input data is honored. However, due to
61 the limited number of well data, the simulated fracture intensity cannot provide accurate information about the locations of high and low fracture density.
Based on the assumption that a strong relation exists between fractures and faults seismic attributes are used for deriving the fault pattern. This information is further used for calculating the fracture intensity models. Unfortunately, the seismic attributes analyzed in this study deliver fault patterns that deviate from each other.
Consequently, the resultant fracture models are again subject to large uncertainty.
However, the uncertainty in the horizontal variogram range can be addressed through deriving the variogram model from the seismic attributes. This workflow is justified under the assumption that the fractures and the faults are related to each other. Note that the vertical variogram range cannot be derived from the seismic attributes because of their low vertical resolution. Comparing the variogram models of the four seismic attributes, one can observe that consistent dip delivers a much larger horizontal variogram range compared to the other three attributes. This support the observation that the fracture intensity derived from the consistent dip attribute deviates considerably from the other three fracture intensity models. Consequently, the fracture model based on consistent dip should be regarded as an extreme case model.
5.5.2 Model parameters
The limited well data prevents an accurate modeling of the fracture properties.
Therefore, in order to estimate and model the fracture properties, analogue studies are used. Statistical laws can be used to approximate the data from analogues.
Numerous studies performed on different scales (from micro m to km) illustrate that the fracture length often follows a power law distribution. This is confirmed for the study area of this thesis. Therefore, all models incorporate the fracture length modeled by the power law with a shape factor of 2.8, a scale factor of 40 ft (13 m) and a maximum fracture length of 500 ft (150 m).
The differences of the dominant fracture orientation with respect to the different stratigraphic layers were discussed previously (see chapter 2, Geologic setting, section 2.3). The dominant fracture set at reservoir level is represented by the oblique striking fractures, but in addition, all models incorporate the fractures normal and parallel fractures to the hinge. The sensitivity analyses demonstrate that the key
62 parameter influencing the permeability is the fracture aperture (Figure 30). The aperture is related to the fracture length by the square root law (Figure 31). The calculation of the permeability from the aperture based on the cubic law delivers values that are in conflict with published data for the study area. In order to match the published permeability, a scaling factor is derived for the base case aperture. This scaling factor is also applied to the high case and low case in order to get an average permeability that is not in conflict with published data. The sensitivity analysis demonstrates that fracture length and orientation affect the upscaled permeability to a smaller degree than the aperture.
5.5.3 Comparison of the models
The comparison of the fracture models is challenging. The large number of fractures suggest analyzing the upscaled permeability derived from the fractures, because it is most influential on the reservoir performance. The simple comparison of statistical parameters turns out to be of limited help, because it does not take into account the size of high permeability zones or areas of "low flow". Therefore, the connected volumes defined by high permeability are chosen for comparing the derived models.
Fracture models based on Kriging deliver very smooth permeability models. In contrast, models based on Gauss simulation give highly heterogeneous permeability models. Consequently, both types of models should show very different dynamic behavior, which could be studied for instance using flow line simulations. The models based on seismic attributes deliver a permeability pattern similar to the fault pattern because of the chosen high correlation coefficient between the fracture intensity and the seismic attributes. As the seismic attributes display fault patterns that differ on the local scale, they provide fracture models, which show very different connected high permeability bodies for the same wells in some areas. It can be expected that seismic attributes will deliver more consistent results when applied to modern high-resolution 3D seismic datasets.
63 Figure 45. Connected volumes for the pseudo well 4, for the models based on the different methods: a) Kriging, b) Gauss simulation, c) chaos attribute and d) consistent dip.
Figure 45 summarizes the large uncertainty in fracture modeling for this dataset. It shows the connected volumes for the models based on Kriging, Gauss simulation and two seismic attributes. The significantly different pattern of the connected volumes linked to well 4 underlines the necessity of broad-band seismic with superior signal-noise ratio to guide the fracture modeling process.
64
6. Conclusion
The fracture modeling based on the dataset of the Teapot Dome reveals several conclusions:
The main controlling factors of the fracture models are fracture intensity and aperture. While the fracture intensity controls the spatial permeability distribution, the fracture aperture represents the parameter that has the greatest influence on the absolute value of the permeability.
The models based on Kriging and Gauss simulation do not deliver results and spatial permeability distribution in accordance with the observed fault and fold structures. Even though the algorithms deliver a permeability distribution comparable to the models based on seismic attributes, the spatial distribution and connectivity of high permeability zones are subject to high uncertainty.
The models guided by the seismic attributes deliver a more reliable spatial permeability distribution illustrated by the zones of higher permeability located along the anticline and in the vicinity of the faults.
Although the models deliver a similar permeability distribution, they show differences associated with the spatial distribution of the high permeability zones. Hence, the seismic guided fracture modeling requires special vigilance and control, since a model based on only one attribute can result in a questionable spatial permeability distribution.
Future work might consider the application of an alternative upscaling method (i.e. flow based method) with the aim of deriving the permeability models more accurately from the simulated fractures. Most importantly, production data would greatly enhance the reliability of fracture models. History matching is an excellent approach for the additional evaluation and uncertainty assessment of the models. Finally, the proposed methodology should be applied to modern high-resolution datasets in order to reduce the fracture model differences introduced by seismic attributes.
65
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