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Reservoir characterization, which could be broadly defined as the process describing various reservoir characters based on the all available data. The data may come from diverse source, including the core sample experiment result, the well log data, the well test data, tracer and production data, 2D, 3D and vertical seismic data, well bore tomography, out crop analogs, etc. Ideally, if most of those data are analyzed and included in the characterization, the reservoir description would be better. However, not all the data are available at the same time. This paper provides a reservoir characterization analysis in the early-stage application, which means before the production, based on real oil filed data by using the SGeMS (Stanford Geostatistical Modeling Software).

The properties involved in the reservoir description may include the permeability, porosity, saturation, thickness, faults and fractures, rock facies and rock characteristics [

A large amount of work has been done by the scholars to join the simulation approached and solve the upscaling description and multivariable attributes problem. Chilès and Delfiner [

The data used for field examples comes from Burbank Oil Field in Oklahoma, which is a sandstone reservoir that consists of several flow units. The data contains 79 wells in the field and all of them are vertical with constant coordinates through the flow units. According to the well information, ten flow units that capture changes in geologic description and variation of petrophysical properties. The flow units data was described at different well locations by analyzing several geological cross sections throughout the field, including top and thickness, porosity, permeability (in log form) from cores and logs. There are totally ten flow unit data which are ordered from the top to bottom of the reservoir and each of them has overall gross thickness ranging from 46 - 85 ft with the mean of 65 ft. Flow units one, two and eight to ten are located at the top and bottom of the reservoir which are less productive, while flow unit three to seven are in the middle of the reservoir, which are most productive and provide best petrophysical properties [

Figures 1(a)-(c) show the overview plot of the data (i.e., see original data source at Burbank Oil Fiel in Oklahoma, flow unit 5, 79 wells), and as it can be seen, under the same coordinate system, the porosity and thickness data is consistent with the well locations, indicating that they are profile of all well data, while permeability data is clustered around one well, suggesting that the permeability data may come from only one or two wells. The permeability data file only offers depth data without any X or Y coordinate which makes it one dimension data. This is a typical case shows that geostatistic data may have different scales and this usually cause difficulty in geoscience work.

The flow unit data contains thickness data, porosity data and permeability data

(in log form) at different X and Y coordinates. At the first step of the project, the data distribution should be viewed.

From the above graphs, the distributions of thickness, porosity and permeability can be observed. The coefficient of variance can be calculated as [

Thus, the coefficient of variance for thickness, porosity and log permeability is

0.6296, 0.3146 and 0.8238 respectively. As coefficient of variance provides a relative spread of a sample and high coefficient of variance reflects the existence of extreme values. All these three are low, suggesting that there are no unacceptable outliners.

However, it is difficult to determine the exact distribution of each variable simply based on eyes judgments. Especially the distribution may look differently when different bin number is chosen. Thus, it is necessary to draw the Q-Q plots [

Figures 3(a)-(c) shows the Q-Q plot of the distribution of the variables, and from the Q-Q normal plot (a), it is very clear that there are many replicate integer numbers for thickness data and the normal distribution may not suit well

for thickness data. However, for Q-Q normal plot (b), it could be observed that the data points fit well for the 45˚ line, except when the theoretical quantile is larger than 1, the fitting becomes poor and this is common when a realistic sample fits a standard normal distribution and this is the tail effect which could be tread as the outliner effect. Thus, for the porosity, except those extreme points at the two ends, the normal distribution seems well for porosity. For the permeability, since the data is log transformed and the normal distribution fits well, it indicates that the original permeability data fits well for the log-normal transformation. Thus, from the Q-Q normal plots, the distribution of the data can be known. Meanwhile, it could be used as an indication for Kriging estimation methods for later use. The thickness data, as is shown, has many replicate integer numbers and this indicates an obvious property of discrete. In this way, the multi Gaussian transformation method, which could transform the data to continuous normal form, is worthy trying. While the permeability data seems to be clustered together, which is described previously in

After the distribution for the variables are analyzed, the relationship between variables should also be checked.

The slope of the above equation equals

covariance between

Variogram is used as a technique to describe the spatial relationships for geoscience data. Traditionally, there are four commonly-used basic variogram-with- still models, which is briefly shown as follows.

Nugget-Effect Model

Spherical Model

where

Exponential Model

Gaussian Model

To make sure the pair of data is sufficient in a variogram, lag numbers, tolerance and lag separation should be carefully selected. At the beginning, the course example of thickness variograms for flow unit 5 is followed.

The overall results for variograms of all the 19 directions could be shown in the software, then the next step is to model these variograms individually based on the four basic still models in the textbook, including Nuggget-effect, Spherical, Exponential and Gaussian, and usually Spherical model and Exponential model are used.

Also, the above obtained vairograms show different spatial continuity in different directions. Noticed that it is needed to check the anisotropic effect, the basic requirements for anisotropic models should also be applied. First, the parameters used for the variograms should be as low as possible. Second, the condition of positive definiteness should be satisfied. Third, it is assumed that directions of maximum and minimum continuity are perpendicular to each other.

Between the two basic anisotropic models, Geometric model and Zonal model, it should be realized that both these two models indicates that the structures of all variograms at different direction should be the same, the difference is that the Geometric model has similar shape and still value with different ranges, while Zonal model has different still values as well as different ranges. Thus, after simply check Geometric model is not suitable in this case because of the same still value.

The summary of model, range and still values at different directions are showed in the

From the summary table, a half of a rose diagram [

Direction (Angle) | Model | Still Value | Range |
---|---|---|---|

0˚ (Maximum) | Spherical Model | 9 | 6768 |

10˚ | Spherical Model | 7 | 4464 |

20˚ | Spherical Model | 6 | 4032 |

30˚ | Spherical Model | 6 | 4608 |

40˚ | Spherical Model | 5.5 | 4896 |

50˚ | Spherical Model | 4 | 2880 |

60˚ | Spherical Model | 3.3 | 2160 |

70˚ | Spherical Model | 3.4 | 2592 |

80˚ | Spherical Model | 3 | 2592 |

90˚ (Minimum) | Spherical Model | 3 | 1440 |

100˚ | Spherical Model | 4 | 3600 |

110˚ | Spherical Model | 4.5 | 4032 |

120˚ | Spherical Model | 4 | 4464 |

130˚ | Spherical Model | 5 | 3456 |

140˚ | Spherical Model | 4 | 3312 |

150˚ | Spherical Model | 5 | 1440 |

160˚ | Spherical Model | 4.5 | 3312 |

170˚ | Spherical Model | 4.5 | 3168 |

180˚ | Spherical Model | 6 | 6768 |

Similar process is done to porosity and log permeability as well and the results are shown as follows.

As is shown in

Direction (Angle) | Model | Still Value | Range |
---|---|---|---|

0˚ | Spherical Model | 57 | 2592 |

10˚ | Spherical Model | 55 | 2448 |

20˚ (Minimum) | Spherical Model | 40 | 2304 |

30˚ | Spherical Model | 50 | 4464 |

40˚ | Spherical Model | 50 | 5040 |

50˚ | Spherical Model | 45 | 1008 |

60˚ | Spherical Model | 30 | 2448 |

70˚ | Spherical Model | 36 | 4752 |

80˚ | Spherical Model | 36 | 3888 |

90˚ | Spherical Model | 28 | 2304 |

100˚ | Spherical Model | 28 | 2736 |

110˚ (Maximum) | Spherical Model | 28 | 5904 |

120˚ | Spherical Model | 38 | 1296 |

130˚ | Spherical Model | 35 | 2592 |

140˚ | Spherical Model | 40 | 1296 |

150˚ | Spherical Model | 45 | 4032 |

160˚ | Spherical Model | 50 | 3888 |

170˚ | Spherical Model | 55 | 4752 |

180˚ | Spherical Model | 57 | 2592 |

After the variograms are modeled, the next step is to build up the grid estimation map based on the sample data. The aim of Kriging estimation is to utilize the models before to estimate values at unsampled locations with minimized variance condition. First, the data is loaded with grids divided.

values in the “4 h-point” point set data. The size of the searching neighborhood is determined by the maximum range of the variograms, which could be recalled in the previous section. The searching neighborhood is set in “search ellipsoid” with searching radius of 10,000 ft, requiring a minimum of 2 data points to be found within 10,000 ft of each grid point and using no more than 12 nearest neighboring points in the estimation for each grid point.

The “variogram” tab is filled with the information on the variogram modeling of thickness which is developed before. Both range and direction of minimum continuity and maximum continuity are tried at first. The “raw” sill is set at 0.8 instead of 1.0 is to show that the raw data is applied for Kriging estimation and the structure of variogram is not exactly matched for the raw data. However, this makes no difference of 0.8 - 1.0 after carefully checking, excepting that the magnitude of the estimated Kriging variance at each grid is changed.

After simple Kriging is applied, the “Simple Kriging (SK)” bar is then changed to “Ordinary Kringing (OK)”bar, which is as shown.

Figures 11-13 give the comparative result of simple Kriging with maximum range, a medium range of 3600, and minimum range respectively. As it can be seen, the less the range, more isolated points are shown in the map.

As it can be seen, the less the range, more isolated points are shown in the map and this is effect especially significant in the variance map. All the map of variance show a contour of variance with bull’s eyes, which indicates no spatial

relationship of error values, and the less the range, the bull’s-eyes effect is more obvious. This phenomenon is easy to understand, according to the definition, the grids out of the range have no spatial relationship. Thus, the less the range is set, the more isolated points in the variance map which is only at the sample location would be. It is known that the most idea case is error estimates should show no spatial correlation as values should be independent of spatial location. The range should be chosen consistent with the minimum range, since the minimum continuity direction is the principal direction and maximum continuity direction is the minor direction according to the textbook. However, since the sample data is not that sufficient, the small range cannot be chosen because the spatial relation is too ideal, which reduced the chance to know the information of other grids based on the known data. Thus, in all case of estimation below, all of the maps are built based on the maximum range of variogram. According to the textbook, the assumption that error variance is independent of surrounding samples and, in turn, of the estimated value, is called assumption of homoscedasticity. This is a typical case where the assumption of homoscedasticity is not satisfied in field data and is rarely required by the user.

Overall, the ordinary Kriging shows similar result as simple Kriging. The map has a smooth appearance with both Kriging methods. The spatial good continuity from north to south could be observed which is corresponded to the principal direction. The east to west trend which corresponds to the minor direction is less observed. The estimation variance is small in grid blocks close to the conditioning data, and it becomes larger in areas far from the data points. The variance maps of two methods has some difference, where the ordinary variance map for seems to be smoother. This is because of the ordinary method, the assumption of first-order stationary may not be strictly valid, where the local mean is dependent on the local location. This leads the estimates is relied on the neighboring girds, which produces a local average effect, making the variance seems to be smoother.

Since the porosity sample data come from the same locations, the similar process is done to porosity data as

After observation, the difference of the results for simple Kriging method and ordinary Kriging method for porosity data seem to be obvious at the area without any data points. This is acceptable because the simple Kringing estimate of these unsampled areas relies more on the global mean, which is 19.62, while for the ordinary Kriging method, the estimate relies more on the surrounding grid

values, which vary a lot.

Finally, it is time to deal with the permeability data. As discussed before, permeability data is first transformed in log form. In this case, simple Kriging method, ordinary Kriging method and cokriging method are used to estimate the permeability data grid which is different from the previous grid system. At first, it is supposed to use cokriging method to estimate the whole grid system of permeability based on the linear relationship of porosity and permeability, which is checked before. However, this idea could not be realized based on the data offered for three reasons.

First, the permeability file is a one dimension data and only has depth data without any X and Y coordinate, which is shown in

Another way to estimate overall permeability is to use the linear relationship to estimate permeability at the all sample point of porosity. According to the definition, the equation is

The final step is to do a simulation with attempt to find the difference between simulation method and estimation method. Sequential Gaussian Simulation is used in this case, which assumes that the normalized local error variance follows a standard normal distribution. The software set for thickness data for the method is shown in

The result for one simulation realization with seed 359,357 for thickness data is shown as

Obviously, from the above two graphs, simulation with different seed numbers will result different map. Simulator will generate different random numbers

each time. They have equal probability, but since each grid is with different value, the overall map the looks quite varied.

Then the simulation results are compared with Kriging estimation, which are shown in

As is discussed, the estimation is to minimize error variance, while the simulation technique is to simulate reality. The sequential Gaussian simulation is a conditional simulation method, which provides the local variability by creating alternate, equiprobable images follows normal distribution instead of defining and estimate. The uncertainty is then characterized by multiple possibilities that exhibit local variance. Since it assumes a continuous normal distribution, as it can be observed, although grid values are different in two maps, the change in the simulation result is not as sharper as in the estimation method. The sequential Gaussian simulation gives a smoother result.

The similar results for porosity are in

In the early stage of geostatistics, which is before the production of the reservoir with the limited amount of information, the reservoir description has more

uncertainty. The reservoir characterization is realizable in this stage by using computers to solve the upscale and heterogeneity problem. The whole map of the properties of the reservoir could be drawn based on the discrete local information of the reservoir. However, the accuracy of these descriptions is very hard to be determined, as the real situation in the reservoir formation is unknown. Different engineers may have different understanding of the data as the analysis process cannot avoid subjective judgments. Thus, the reservoir characterization should be updated in time once new information is available.

2016ZX05060026 (Thirteen-Five National Science and Technology Major Projects: Study on Shale Gas Fracturing and Gas Production Regularity in Wufeng Formation and Longmaxi Formation in Fuling Area), 2016ZX05025001 (Thirteen-Five National Science and Technology Major Projects: Study on the Fine Representation of Residual Oil of Offshore Sandstone Heavy Oil Reservoir Based on the Nonlinear Fluid Flow), 2016ZX05027004 (Thirteen-Five National Science and Technology Major Projects: Study on the Key Techniques for Effective Development of Large-scale Gas Field in Thick Heterogeneity of East China Sea).

Zhang, M.L., Zhang, Y.Z. and Yu, G.M. (2017) Applied Geostatisitcs Analysis for Reservoir Characterization Based on the SGeMS (Stanford Geostatistical Modeling Software). Open Journal of Yangtze Gas and Oil, 2, 45-66. https://doi.org/10.4236/ojogas.2017.21004