# Statistical Rock Physics

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1 Statistical - Introduction Book review Min Sun March. 13, 2009

2

3 Outline. What is Statistical. Why we need Statistical. How Statistical works

4 Statistical Rock physics Information theory Statistics reduce Uncertainty

5 Statistical Rock physics - link seismic response and reservoir properties ( well log, geology) - extend the available data to generate training data for the classification system. - seismic response indirect, but spatially exhaustive, lateral and vertical information that are not available from well data. Information theory - rock property estimation - simple yet powerful tools to quantify the attribute information for discriminating the different facies. - Shannon s information entropy (3.4) : get best attributes that most reduce uncertainty in reservoir properties identification.

6 Statistical Statistics - Quantify uncertainty - Classification and estimation - based on computational statistical techniques nonparametric Bayesian classification, bootstrap, and neural networks, etc - quantitatively measure interpretation uncertainty and the misclassification risk at each spatial location. - Geostatistics - add spatial correlation - add small-scale variability which is hard to identify from seismic only because of the limits of resolution.

7 . What is Statistical. Why we need Statistical. How Statistical works

8 Uncertainty - ubiquitous interpretation uncertainty - Subsurface heterogeneities lithology, pore fluids, clay content, porosity, pressure, temperature, etc - Subsurface properties estimated from remote geophysical measurements many inevitable difficulties and ambiguities in data acquisition, processing, and interpretation. - Data have errors - Models are approximate - Our imperfect knowledge

9 Uncertainty - quantified Quantifying uncertainty to - assess risk - integrate data from different sources - estimate value of additional data

10 Uncertainty quantifying methods - purely statistical - Based on multivariate techniques - Used in seismic reservoir characterization - purely deterministic - Based on physical models derived from elasticity theory and laboratory observations. Derived distribution combining deterministic rock physics models with the observed statistical variability to build a more powerful strategy for reservoir prediction

11 Uncertainty quantifying tools - Probability - frequency: long sequence of identical repetitions - Bayesian: degree of belief based on given evidence - Random variable - to model uncertainty - pdf ( probability density functions) - Completely describe the random variables - Joint pdfs for the relations of multiple random variables

12 Uncertainty quantifying tools -pdf estimate - Must from prior knowledge or available training data - The training data need to be extended or enhanced using rock physics models -methods -parametric approach -nonparametric approach -histogram

13 Uncertainty quantifying tools Example. nonparametric approach -Less rigid assumption brine sands -Low dimensions gas sands

14 Uncertainty quantifying tools Example. Histograms and kernel-based pdf estimates - Oldest and simplest - Explore data variability - Kernel or window function e.g. triangular Gaussian Epanechnikov - Discretization vs. Smoothing

15 Uncertainty flaw of averages Example. Ignoring the variability of rock properties can drastically distort decisions Backus average ignoring the variability of sand and shale velocity, density, and impedance Monte Carlo simulation incorporate the variability

16 Uncertainty reduced by additional data Example 1

17 Uncertainty reduced by additional data Example 2 b. best - discriminant analysis and Bayesian classification (3.6 ) C. D. Vp shift might be detectable in time-lapse seismic data

18 . What is Statistical. Why we need Statistical. How Statistical works

19 Workflow facies definition pdf estimation statistical classification Geostatistics The steps modified depending on the stage in the cycle of exploration, development, and production.

20 Workflow 1. Facies definition identifying variables ( facies or categorical groups ) from well log and geology analyses -Facies - Collection of geologically similar rocks that span a range of petrophysical and seismic properties. - Not necessarily only by lithology type, facies <> single rock. - Intrinsic variability of rock properties - One of the biggest challenges of quantitative seismic interpretation : when does an observed attribute change indicate a significant change across facies rather than a minor fluctuation within a facies.

21 Workflow 1. Facies definition - identifying facies - based on well information cores, thin sections, geology, logs, production data, etc. - cluster separation common practices using cross-plots - algorithms supervised learning - on the basis of expert knowledge

22 Workflow 2. pdf estimation Rock physics modeling, Monte Carlo simulation and pdf estimation -rock physics modeling ( chapter 1, 2 ) - Basic rock physics relations are defined for the facies: Vp, Vs, etc - What ifs : extend the training data to simulate different physical conditions - translate production or geologic information into elastic properties that condition the seismic response

23 Workflow 2. pdf estimation -Monte Carlo simulation ( 3.5) - assumption for correlated Monte Carlo simulation the well-log data extended by rock physics modeling will be statistically representative of all the possible values of Vp, Vs, and density that might be encountered in the study area. - Monte Carlo realizations - drawn from the distributions of each facies - used in models to calculate seismic observables and attributes - an attribute is any characteristic that can be extracted from the seismic data: AVO intercept, gradient, P- and S- wave impedance, etc - extend the pdfs situations that are of interest but not encountered in the wells

24 Workflow 2. pdf estimation -pdf estimation - select attribute - the easiest way is by color-coded comparative histogram plots or cross-plots of attributes. - union or division of attributes may be used to separate priori defined facies : cluster analysis - Information entropy ( 3.4 ) - estimate pdfs for each defined facies - discretization and smoothing Feasibility check - Compute seismic attributes and estimate their pdfs - Guide for designing surveys suitable for extracting the most promising attributes - Decide which attributes should be extracted from the field seismic data.

25 Workflow 2. pdf estimation B. A cross-plot of AI vs. EI calculated with well logs. C. Pdf contours extended by Monte Carlo simulation

26 Workflow 2. pdf estimation

27 Workflow 2. pdf estimation

28 Workflow 3. statistical classification Seismic inversion, calibration to well pdfs, and statistical classification - get seismic attributes a. derived from seismic data by different processing, analysis, or inversion. b. responded to interface properties: e.g. reflectivity, AVO c. responded to interval properties: e.g. acoustic impedance, elastic impedance

29 Workflow 3. statistical classification

30 Workflow 3. statistical classification -calibration to well pdfs -Differences between the computed and extracted attributes caused by - simplifications of the models - imperfections in the data processing - arbitrary scaling of the field amplitudes - different measurement scales - noisy data -statistical classification or pattern recognition (3.6) - Discriminant analysis - K-nearest-neighbor classification - Neural networks - Classification trees - Bayes classification

31 Workflow 3. statistical classification (a) The result of the Bayesian classification using P-wave AVO attributes R and G.

32 Workflow 3. statistical classification (b) Isoprobability surfaces resulting from applying a statistical classification process (nonparametric Bayesian) using the acoustic and elastic impedance

33 Workflow 4. Geostatistics Geostatistical simulations incorporating spatial correlation and fine-scale heterogeneity - Spatial correlation incorporated by - variograms - multiple-point spatial statistics - Small-scale variability - not captured in seismic data because of their limited resolution - One of the main benefits - estimate joint spatial uncertainty - One of the pitfalls - black-box mode without understanding the underlying physics and spatial models.

34 Workflow 4. Geostatistics - A powerful tools for spatial data integration and an important role at various stages of reservoir exploration and development a. early stages - delineate reservoir architecture geostatistically combine seismic traveltime data and sparse well horizon markers. - obtain geostatistical simulation of reservoir properties e.g. lithofacies, porosity, and permeability - impart the appropriate spatial correlation structure to the seismic impedance inversions. b. late stages - incorporate production data into the analysis.

35 Workflow 4. Geostatistics Example : indicator simulation indicator: 1 facies present 0 otherwise ( top ): vertical section of the multiple equiprobable volumes generated using indicator simulation ( bottom ): prior pdfs - given the seismic attributes posterior pdfs - given seismic attributes, the spatial correlation, the facies indicator data from the wells. Updated by the Markov-Bayes indicator formalism ( Deutsch and Journel, 1996 ).

36 Our most precise description of nature must be in terms of probabilities. - Richard Phillips Feynman Statistical Combining deterministic physical models with statistical techniques leads to new methods to describe reservoir rock properties more precisely.

37 Thanks

38 Notes. Variograms key function in geostatistics as it will be used to fit a model of the temporal/spatial correlation of the observed phenomenon. This is an example of a variogram produced using ArcGIS's Geostatistical Analyst.

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