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## Uncertainty in geosciences and petroleum exploration

Intrinsic randomness of nature is perhaps the first source of uncertainty that is mentioned in geological texts. However, retrodiction is never as uncertain as prediction (Kitts, 1976). The word “retrodiction” could be of interest of geologists since it is composed of the word “prediction” when the prefix “pre” is replaced by “retro”.

Another pioneer work on uncertainty assessment in core orientation could be found in Nelson et al. (1987). In quantifying uncertainty in petroleum volume estimation, fuzzy arithmetic (Theory of Possibility) is declared to provide better results, comparing to Monte-Carlo method (Theory of Probability) (Fang and Chen, 1990). Monte-Carlo simulation is a traditional technique for calculating uncertainty of estimated hydrocarbon volume. For further study about probabilistic Monte-Carlo method, refer to Hurst et al. (2000) and Masoudi et al. (2011).

The uncertainty is categorized into two types: vagueness (equivalent to fuzziness, haziness, cloudiness, unclearness and sharplessness) and ambiguity (non-specificity, diversity, divergence, generality, variety and one-to-many). Three sources of uncertainty are introduced: lack of information, intrinsic nature and ignorance (Fang and Chen, 1990).

In a comprehensive study, fuzzy aggregation is used for reservoir appraisal (Chen and Fang, 1993). In this work, four trap properties (type, size, closure and timing), five reservoir properties (porosity, permeability, net thickness, depth and saturation), five source rock properties (richness, organic matter, maturity, thickness and area), migration distance and seal integrity, i.e. 16 geologic variables in overall, are fused to assess prospects. In brief, an appraisal method is introduced in this paper, which uses fuzzy aggregation methodology to easily integrate different reservoir properties.

Foley et al. (1997) classified uncertainty into three categories: fuzziness, randomness and incompleteness. Incompleteness is equivalent to ignorance, which is one of the sources of uncertainty, introduced in the work. Some authors also classified incompleteness into four categories: (i) what we know but have not included in the model; (ii) what we know that we do not know; (iii) what we do not know that we are unaware of it; and finally (iv) what is difficult to understand.

The concept “geoscore” was introduced by Dromgoole and Speers (1997) to measure the complexity of petroleum reserves by quantifying nine categories:

Structural complexity: (i) overburden complexity; (ii) fault complexity;

Reservoir quality and architecture: (iii) reservoir layering; (iv) reservoir continuity; (v) permeability channels; (vi) barrier continuity; (vii) fault transmissibility; (viii) fractures and (ix) diagenesis.

The key achievement of this study is: the higher the geoscore, the less hydrocarbon recovery and the more overestimation of reservoir volume. It was also advised at the end of the paper that for accurate reserve estimation during appraisal, we need to: (i) recognize the key uncertainties; (ii) quantify the relative importance of uncertainties; and (iii) collecting data to reduce uncertainties or being prepared to handle potential problems may arise during development or production. This paper (Dromgoole and Speers, 1997), and the paper of Yeh et al. (2014) have well presented the importance of uncertainty assessment in economic evaluation and production forecast, respectively.

Hurst et al. (1999) tried to link between the concepts of sequence stratigraphy and the concept of uncertainty. In their paper, there is a discussion about characterizing sandy pinch-out genesis, i.e. infill or onlap types, and correlating sand bodies between wells. Although mathematical basis of this paper is not in-detail, there is a robust geological debate on how to differentiate infill and onlap pinch-outs based on petrophysical and rock sedimentary properties.

Application of probabilistic and fuzzy partitioning on interpreting satellite images is addressed successfully by Matsakis et al. (2000). In this paper, image classification is distributed into classes of lagoon, conglomerate, vegetation, coral rubble, deep water, etc. with the precisions higher than 75%. For mathematical comparison between fuzzy, probabilistic and possibilistic partitioning, refer to Anderson et al. (2010).

Another application of the concept of uncertainty in production problems is presented in Zheng et al. (2000). In this paper, the uncertainty of permeability estimation is lowered by combining results of two measurements of permeability: well tests (meso-scale) and core tests (micro-scale). Calibrating both types of data has confined estimation of permeability to smaller range. It seems that this methodology is more certain comparing to conventional approach, since (i) training is constrained to well-test, and (ii) well-test is a measure of permeability of the reservoir, whereas, core permeability is only indicator of permeability of intact rock.

Fuzzy logic is inherently a suitable tool to characterize vague and imperfectly defined situations, like in geological datasets. Therefore, Saggaf and Nebrija (2003) proposed using fuzzy logic in lithological and depositional facies predictions. In their work, accuracy of fuzzy logic in facies prediction is stated to be more than 90% in every run.

For probabilistic hydrocarbon pore-thickness evaluation in intervals (47.6 ft = 14.5 m) with beds thinner than 1 ft (30 cm), Volumetric Laminated Sand Analysis (VLSA) method is developed. The method uses Monte-Carlo simulation for generating realizations, and providing PDF of output, i.e. petrophysical parameters. 400% improvement in accuracy of hydrocarbon pore-thickness estimation is reported by this method (Passey et al., 2004). In this thesis, VLSA is used as a base method for comparing the outputs with.

Geological risk mapping in play level is produced by multivariate and Bayesian methodology (Chen and Osadetz, 2006). In this work, risk analysis problem is defined as “equivalent to classification with uncertainty in a multivariate space”.

Determination and geostatistical inversion are compared with each other for the purpose of net-pay determination through seismic data (Sams and Saussus, 2008). Determination method provides higher uncertainties, comparing to geostatistics method. For reservoirs with thinner stratigraphic layers, the magnitude of this difference rises, especially when the beds become thinner than vertical resolution of seismic data.

Grandjean et al. (2007, 2009a,b) performed some researches on the application of fuzzy logic in slope stability, hydrogeology and geo-engineering. Their papers present a systematic algorithm for implementing a fuzzy inference system to fuse multi-source geo-data. The systematic algorithm they used contains four stages: (i) preparing geo-dataset; (ii) creating possibility function (membership function), regarding the purpose; (iii) providing technical hypothesis for fusing variables; and (iv) checking or discussing hypothesis or outputs of fusion (Grandjean et al., 2007, 2009a,b; Hibert et al., 2012).

In order to handle structural uncertainty in petroleum reserves, Thore et al. (2002) considered aggregation of side-effects of all processing and interpreting stages on the final results. Preparation of structural model by seismic studies generally consists of six stages, each one is a source of uncertainty in constructing structural model: acquisition, pre-processing, stacking, migration, time-to-depth conversion and interpretation. In this paper, migration, picking and time-to-depth conversion are introduced as dominant uncertainty resources; and amplitude, direction and correlation length of each are incorporated in calculations. In the article, it is also specified that computation of structural uncertainties has several benefits: (i) providing a distribution of gross rock volume; (ii) defining optimal well trajectories; and (iii) reservoir history matching.

In a recent paper about assessing structural uncertainty, Seiler et al. (2009) proposed an elastic grid to be adjustable and trainable due to history of reservoir production. The methodology is approved by synthetic dataset, and is potentially a new frontier for structural uncertainty handling in petroleum appraisal for the next years.

The most comprehensive text book about uncertainty in geosciences is composed by Caers (2011). Five different sources of uncertainty in earth sciences are introduced in the book: (i) measurement and processing errors; (ii) multiple ways of interpreting processed data; (iii) type of geological setting; (iv) spatial uncertainty, which is related to heterogeneity and scale of the study; and (v) response uncertainty, e.g. solving partial differential equations needs initial and boundary conditions that is sometimes uncertain.

The most important prerequisite to uncertainty assessment is stated to be “purpose of the study”; therefore, fit-for-purpose approach is suggested for quantifying uncertainty in each study (Caers, 2011). The majority of the book is concerned with geostatistical methodologies, and how to assess uncertainty besides geostatistical modelling. However some simpler uncertainty tools as tornado chart is introduced likewise.

A bootstrap-based methodology for uncertainty analysis when predicting effective porosity by seismic attributes is presented by Ortet et al. (2012). The authors have introduced this novel methodology as an alternative to standard geostatistical simulation. They have also stated that in cases the main source of uncertainty is related to the calibration set, bootstrap-based method is well adapted.

For assessing uncertainty of seismic interpretations, picking, four constraints are introduced (Yang et al., 2013): (i) best estimate control point; (ii) best estimate surface; (iii) uncertainty envelope; and (iv) uncertainty envelop surface. Based on this method, more than one realization would be generated that helps interpreters to calculate the probability of each realization.

Uncertainty assessment in pore pressure prediction is carried out by Wessling et al. (2013). The paper discusses that the uncertainty, which is associated with the pore pressure, is contributed by geophysical measurements, geological model and manual processing steps. Measurement-related uncertainties could be quantified (and maybe compensated); whereas, it is difficult to handle the uncertainty of descriptive geologic models or manual processing stages. It is suggested in Wessling et al. (2013) that, in order to quantify and control uncertainty of a manual processing, we can substitute this part with an automation. This change will result in a more transparent output for interpreters, hence, easier to quantify and assess the uncertainty.

Within a recently defended PhD thesis, the uncertainty of static models is projected to dynamic models. So, the quantiles (P10, P50 and P90) of production prediction curve were calculated, considering a set of geostatistical realizations (Bardy, 2015).

**Application points**

Though there are many publications about the concept, definition and categorization of uncertainty in earth-related studies, the application of uncertainty assessment in exploration activities is not well-developed. Two common pragmatic points could be derived from the literature for all the uncertainty assessment applications: (i) what is the aim from uncertainty assessment (Caers, 2011)? (ii) What are the sources of uncertainty in the dataset (Chen and Fang, 1993; Dromgoole and Speers, 1997; Lia et al., 1997; Thore et al., 2002; Wessling et al., 2013)?

To rebuild uncertainty bounds of well-logs and propagate it to the petrophysical outputs, Monte-Carlo simulation is used in industrial software applications Techlog- Schlumberger (Gimbe, 2015) and Geolog Datamin Uncertainty Module Paradigm. The algorithm is the same as VLSA, presented for thin-bed studies by researchers of ExxonMobil (Passey et al., 2004).

**Introducing datasets**

**Basic definitions**

We used four key terms to introduce the datasets: “Well-log”, “real-log”, “synthetic-log” and “ideal-log”. The first two terms describe real data, while the last ones are related to synthetic data.

– A well-log records intrinsic or induced properties of the rocks and their fluids (Gluyas and Swarbrick, 2009). Such records, acquired through a well, are either one dimensional, like gamma ray log, or two dimensional as image logs. Well-log is also known as borehole log since the data are captured through the wellbore.

**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)

**References**