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## Gustafson-Kessel clustering technique

Neither KM nor FCM are suitable algorithms for oriented or multi-sized clusters. In the FCM algorithm, if the Euclidian distance (Relation 2-17) is replaced by the Mahalanobis distance (Relation 2-18) the new algorithm, Gustafson-Kessel (GK), would be capable of detecting uni-sized (in terms of number of data) but oriented (linear at extreme mode) clusters too (Gustafson and Kessel, 1978; Babuka et al., 2002).

**Gath-Geva clustering technique**

Gath-Geva (GG) algorithm solved clustering problems of (i) multi-sized clusters, and (ii) number of clusters. Fuzzy Maximum Likelihood Estimation (FMLE) or GG is a development on the FCM algorithm (Gath and Geva, 1989):

(i) Primary definitions and cluster centers calculation ( ( ) in Relation 2-12): the same as steps (i) and (ii) in FCM but c is the maximum number of clusters.

(iii) Compute performance measures that is based on three criteria: (iii-i) Separation of clusters (low fuzzy hypervolume, Relation 2-22), (iii-ii) Minimal volume of the clusters, but (iii-iii) Maximal number of data points, concentrated in the vicinity of the cluster centroid (high average density partition, Relation 2-23 or high density partition, Relation 2-24).

If the kth data point is within a hyperellipsoid, centered at the cluster prototype of the ith cluster with radii of one standard deviation of each feature, then , is the distance between the kth data point and the ith cluster. Otherwise, it is zero to exclude far data points from calculations of (average) density partition.

(iv) If number of clusters is lower than the predefined maximum, increase the number of clusters, and go to (iii). Otherwise stop the algorithm, then choose the best cluster number due to performance measures.

### Empirical relations in petrophysics

Using empirical relations in reservoir characterization is very common. All the well-log interpretation software applications contain some of these relations for estimating reservoir properties: porosity, permeability, saturation, rock typing, etc. The focus of the thesis is on the porosity, then permeability. In the next step, results of irreducible water are provided in some cases. Since there is not a measurement for evaluating saturation results, it is not discussed as porosity and permeability properties.

**Porosity study by well-logs**

Porosity could be studied by three well-logs: neutron porosity (NPHI), bulk density (RHOB) and sonic (DT). NPHI is a measure of hydrogen content. Since this element exists in the water and hydrocarbons (not in solid phase of formations), so NPHI is a measure of whole the porous media, i.e. total porosity either effective or non-effective (Johnson and Pile, 2002). RHOB has negative correlation with NPHI since higher the porosity, lower the density. Despite the former logs, DT represents only secondary porosity, like fractures or vuggy porosity. In fact, DT is sensitive to the discontinuities that result in dispersion of sonic waves (Johnson and Pile, 2002). Therefore peaks in DT could be interpreted as fractured or porous zones.

**One-log porosity methods: neutron- or density-based**

The response of NPHI and RHOB could be modelled by Relation 2-25. The acquired log value is summation of matrix, shale, hydrocarbon and water parts.

where and ℎ are effective porosity and volume of shale, both in fraction. , ℎ, ℎ and are log readings (either NPHI or RHOB) in 100% water, hydrocarbon, shale and the ith component of matrix rock, respectively. is water saturation in invaded zone (fractional). Finally, is the volume of ith component of matrix rock.

In this thesis, since the dataset belongs to a gas-free reservoir, the difference between the log reading in water and hydrocarbon horizons is neglected, = ℎ. Also, the invaded zone is considered completely full of drilling water = 1 (Crain, 2000). It means that water and hydrocarbon terms are integrated into a single term.

where ( ) is scaled log value. and are log responses in 100% water saturation and rock matrix, respectively (lithology correction). is the final porosity value from NPHI or RHOB logs. In case of occurrence of gas, applying neutron gas correction factor is necessary (Crain, 2000).

**Two-log porosity methods**

#### Density-neutron (shaly sand) cross-plot

Although this method is widely used, it is not recommended. It is used here since still is found in log analysis software applications. Complex lithology models are recommended instead of shaly sand models, because most of sandy reservoirs contain other minerals than quartz and clay minerals. The output of Relation 2-26 for the logs NPHI and RHOB, i.e. ( ) and ( ), are inputs of this method. The shale correction will be applied through the algorithm (Crain, 2000):

There is a quick-look (Relation 2-30) for density-neutron porosity method which uses the output of Relation 2-27. In the version of quick-look, the gas crossover is not accounted since shale corrections result in apparent gas crossover, which is an artifact. In addition, in the shale zones, there are some recommendations: (i) for ℎ( ), replace zero for the range of -0.03 to 0.12; and (ii) for ℎ( ), replace 0.30 for the range of 0.10 to 0.40 (Crain, 2000).

This model is developed to be used in complex lithological situations, and it is the best available empirical relation for the porosity estimation. If ≥ , i.e. no gas crossover, and really the gas does not exist (Crain, 2000):

But if there is gas crossover after shale correction and we are sure that gas is present:

Finally, if there is no gas crossover after shale correction but we are sure about the presence of gas, a correction based on the photoelectric log is necessary (Crain, 2000), which is out of the scope here. All the above-mentioned relations are applied through a spreadsheet (Figure 2-6) to the five wells, under study.

Calculating irreducible water is a prerequisite for permeability modelling by well-logs and estimating water-cut. In core testing, it is measurable by increasing capillary pressure. Before perturbing the initial equilibrium of the reservoir fluids by production, above the water contact,the irreducible water is equal to the actual (initial) water saturation ( = ). But in transition, water and depleted zones, it is not: ≤ . So we need to calculate it. The difference between and , and relative permeability are controlling factors of water-cut. The used algorithm for irreducible water calculation has two steps (Crain, 2000):

**Table of contents :**

**1 Introduction **

Highlights of the Chapter 1

1.1 Uncertainty resources in well-logging

1.2 The thesis questions and objectives

1.2.1 Question I: vertical resolution of well-logs

1.2.2 Question II: possibilistic uncertainty range of petrophysical parameters

1.3 The importance of the thesis

1.3.1 Fundamental and scientific importance

1.3.2 Application importance

1.3.3 Economic and management importance

1.4 Literature review

1.4.1 Uncertainty in sciences

1.4.2 Uncertainty in geosciences and petroleum exploration

1.5 Introducing datasets

1.5.1 Basic definitions

1.5.2 Synthetic data

1.5.3 Real data

**2 Theories **

Highlights of Chapter 2

2.1 Dempster-Shafer Theory of evidences

2.1.1 Body Of Evidences

2.1.2 Belief and plausibility functions

2.1.3 Consistency of uncertainty assessment theories

2.2 Fuzzy arithmetic

2.2.1 Fuzzy number

2.2.2 Arithmetic operations on intervals

2.2.3 Arithmetic operations on fuzzy numbers

2.3 Cluster analysis

2.3.1 k-means and fuzzy c-means algorithms

2.3.2 Gustafson-Kessel clustering technique

2.3.3 Gath-Geva clustering technique

2.4 Empirical relations in petrophysics

2.4.1 Porosity study by well-logs

2.4.2 Irreducible water saturation

2.4.3 Wylie-Rose permeability relation

**3 Modelling vertical resolution **

Highlights of Chapter 3

3.1 Volumetric nature of well-log recordings

3.1.1 Different types of resolution

3.1.2 VRmf > spacing > sampling rate

3.2 Modelling logging mechanism by fuzzy memberships

3.2.1 Recording configuration and well-log

3.2.2 Approximating VRmf

3.2.3 Passive log of GR

3.2.4 Active logs of RHOB and NPHI

3.2.5 Complex membership function of compensated sonic log

3.3 Volumetric Nyquist frequency

3.4 Conclusions of Chapter 3

**4 Thin-bed characterization, geometric method **

Highlights of Chapter 4

4.1 Review of thin-bed studies

4.1.1 VLSA Method

4.2 Theory of geometric thin-bed simulator

4.3 Sensitivity analysis of well-logs to a 30 cm thin-bed

4.4 Deconvolution relations for thin-bed characterization

4.5 Thin-bed characterization, the Sarvak Formation case-study

4.5.1 Multi-well-log thin-bed characterization

4.6 Conclusions of Chapter 4

**5 Enhancing vertical resolution of well-logs **

Highlights of Chapter 5

5.1 Combining adjacent well-log records by Bayesian Theorem

5.1.1 The importance of volumetric Nyquist frequency in up-scaling

5.2 Body Of Evidences (BOE) for well-logs

5.2.1 Focal elements of well-logs

5.2.2 Mass function of focal element of recording

5.3 Belief and plausibility functions for focal element of target

5.3.1 Theoretical functions of belief and plausibility

5.3.2 Geological constraints as an axiomatic structure

5.3.3 Practical functions of belief and plausibility

5.3.4 Compensating shoulder-bed effect by epsilon

5.4 Log simulators

5.4.1 Random simulator

5.4.2 Random-optimization simulator

5.4.3 Recursive simulator

5.4.4 Recursive-optimization simulator

5.4.5 Validation criteria

5.5 The algorithm

5.6 Application check on synthetic cases

5.7 Discussion on results of synthetic cases

5.7.1 Validating constraint-based error by synthetic cases

5.8 Application to real data

5.8.1 Simulator selection

5.8.2 Optimizing factor of shoulder-bed effect

5.8.3 Results of resolution improvement of real well-logs

5.9 Discussions

5.9.1 Comparing DST and geometry-based results in thin-bed characterization

5.9.2 Advantages of DST-based algorithm

5.9.3 Uncertainty conversion using DST

5.10 Conclusions of Chapter 5

**6 Uncertainty projection on reservoir parameters **

Highlights of Chapter 6

6.1 Importance of cluster analysis in well-log interpretation

6.2 Porosity analysis by cluster-based method

6.2.1 Methodology of cluster-based porosity analysis

6.2.2 Results of cluster-based porosity analysis

6.2.3 Discussion of cluster-based porosity analysis

6.3 Permeability analysis by fuzzy arithmetic

6.3.1 Methodology of permeability analysis by fuzzy arithmetic

6.3.2 Validation with core data

6.3.3 Results and discussions of analysis by fuzzy arithmetic

6.4 Conclusions of Chapter 6

**7 Ending **

7.1 Pathway of the thesis

7.1.1 Outlined achievements of the thesis

7.2 Recommendations

7.2.1 Recommendations for industrial applications

7.2.2 Recommendations for further researches (perspectives)