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COVER
PREFACE
AUTHORS
1. CONTRIBUTORS
1 The Earth Resources Challenge
1. 1.1. WHEN CHALLENGES BRING OPPORTUNITIES
2. 1.2. PRODUCTION PLANNING AND DEVELOPMENT FOR AN OIL FIELD IN LIBYA
3. 1.3. DECISION MAKING UNDER UNCERTAINTY FOR GROUNDWATER MANAGEMENT IN DENMARK
4. 1.4. MONITORING SHALLOW GEOTHERMAL SYSTEMS IN BELGIUM
5. 1.5. DESIGNING STRATEGIES FOR URANIUM REMEDIATION IN THE UNITED STATES
6. 1.6. DEVELOPING SHALE PLAYS IN NORTH AMERICA
7. 1.7. SYNTHESIS: DATA–MODEL–PREDICTION–DECISION
8. REFERENCES
2 Decision Making Under Uncertainty
1. 2.1. INTRODUCTION
2. 2.2. INTRODUCTORY EXAMPLE: THE THUMBTACK GAME
3. 2.3. CHALLENGES IN THE DECISION‐MAKING PROCESS
4. 2.4. DECISION ANALYSIS AS A SCIENCE
5. 2.5. GRAPHICAL TOOLS
6. 2.6. VALUE OF INFORMATION
7. REFERENCES
3 Data Science for Uncertainty Quantification
1. 3.1. INTRODUCTORY EXAMPLE
2. 3.2. BASIC ALGEBRA
3. 3.3. BASICS OF UNIVARIATE AND MULTIVARIATE PROBABILITY THEORY AND STATISTICS
4. 3.4. DECOMPOSITION OF DATA
5. 3.5. ORTHOGONAL COMPONENT ANALYSIS
6. 3.6. FUNCTIONAL DATA ANALYSIS
7. 3.7. REGRESSION AND CLASSIFICATION
8. 3.8. KERNEL METHODS
9. 3.9. CLUSTER ANALYSIS
10. 3.10. MONTE CARLO AND QUASI MONTE CARLO
11. 3.11. SEQUENTIAL MC
12. 3.12. MARKOV CHAIN MC
13. 3.13. THE BOOTSTRAP
14. REFERENCES
4 Sensitivity Analysis
1. 4.1. INTRODUCTION
2. 4.2. NOTATION AND APPLICATION EXAMPLE
3. 4.3. SCREENING TECHNIQUES
4. 4.4. GSA METHODS
5. 4.5. QUANTIFYING IMPACT OF STOCHASTICITY IN MODELS
6. 4.6. SUMMARY
7. REFERENCES
5 Bayesianism
1. 5.1. INTRODUCTION
2. 5.2. A HISTORICAL PERSPECTIVE
3. 5.3. SCIENCE AS KNOWLEDGE DERIVED FROM FACTS, DATA, OR EXPERIENCE
4. 5.4. THE ROLE OF EXPERIMENTS: DATA
5. 5.5. INDUCTION VERSUS DEDUCTION
6. 5.6. FALSIFICATIONISM
7. 5.7. PARADIGMS
8. 5.8. BAYESIANISM
9. 5.9. BAYESIANISM IN GEOLOGICAL SCIENCES
10. REFERENCES
6 Geological Priors and Inversion
1. 6.1. INTRODUCTION
2. 6.2. THE GENERAL DISCRETE INVERSE PROBLEM
3. 6.3. PRIOR MODEL PARAMETERIZATION
4. 6.4. DETERMINISTIC INVERSION
5. 6.5. BAYESIAN INVERSION WITH GEOLOGICAL PRIORS
6. 6.6. GEOLOGICAL PRIORS IN GEOPHYSICAL INVERSION
7. 6.7. GEOLOGICAL PRIORS IN ENSEMBLE FILTERING METHODS
8. REFERENCES
7 Bayesian Evidential Learning
1. 7.1. THE PREDICTION PROBLEM REVISITED
2. 7.2. COMPONENTS OF STATISTICAL LEARNING
3. 7.3. BAYESIAN EVIDENTIAL LEARNING IN PRACTICE
4. REFERENCES
8 Quantifying Uncertainty in Subsurface Systems
1. 8.1. INTRODUCTION
2. 8.2. PRODUCTION PLANNING AND DEVELOPMENT FOR AN OIL FIELD IN LIBYA
3. 8.3. DECISION MAKING UNDER UNCERTAINTY FOR GROUNDWATER MANAGEMENT IN DENMARK
4. 8.4. Monitoring Shallow Geothermal Systems in Belgium
5. 8.5. DESIGNING URANIUM CONTAMINANT REMEDIATION IN THE UNITED STATES
6. 8.6. DEVELOPING SHALE PLAYS IN NORTH AMERICA
7. REFERENCES
9 Software and Implementation
1. 9.1. INTRODUCTION
2. 9.2. MODEL GENERATION
3. 9.3. FORWARD SIMULATION
4. 9.4. POST‐PROCESSING
5. REFERENCES
10 Outlook
1. 10.1. INTRODUCTION
2. 10.2. SEVEN QUESTIONS
INDEX
END USER LICENSE AGREEMENT

List of Tables

Chapter 01
1. Table 1.1 Overview of the various modeling components, fields of study, and data sources for UQ and decision making in conventional oil/gas reservoirs.
2. Table 1.2 Danish government’s 10‐point program from 1994.
3. Table 1.3 Example of stakeholders in the decision context of contamination remediation.
4. Table 1.4 Overview of some of the parameters involved in designing unconventional shale operations.
Chapter 02
1. Table 2.1 Hypothetical performance score matrix in a binary decision problem.
2. Table 2.2 Calculating swing ranks.
3. Table 2.3 Outcome of the hypothetical score matrix with cleanup being the winning alternative.
Chapter 03
1. Table 3.1 Overview of the various parameters and their uncertainty for hydro case.
2. Table 3.2 Overview of methods of regression and classification and how they score on various criteria.
Chapter 04
1. Table 4.1 Linear regression analysis, without interactions.
2. Table 4.2 Summary table of the SA techniques described in this chapter.
Chapter 06
1. Table 6.1 Overview of geophysical methods, geophysical variable or property, physical variables (or the variable of interest), and their area of application.
2. Table 6.2 Overview of various parameterization to preserve prior geological information in EnKf.
Chapter 07
1. Table 7.1 Example decisions and the associated data/prediction variables in subsurface applications.
Chapter 08
1. Table 8.1 Prior uncertainties on non‐gridded reservoir model variable.
2. Table 8.2 Estimated injection efficiency for “no action” and optimized rate, together with the resulting cumulative oil production.
3. Table 8.3 Summary of parameters and uncertainties.
4. Table 8.4 Overview of the most impacting model variables for each location and for each prediction variable.
5. Table 8.5 Comparison in terms of correlation coefficients between the simulated prediction variables (using MODFLOW) and the tree‐based models.
6. Table 8.6 P50 values of each alternative vs. objective with assignment of the swing rank.
7. Table 8.7 Scoring alternatives vs. objectives, for a risk neutral decision maker.
8. Table 8.8 Modeling parameters for deterministic and stochastic model.
9. Table 8.9 List of uncertain model variables and prior pdfs.

List of Illustrations

Chapter 01
1. Figure 1.1 Idealized vs. real setting in estimating original oil in place.
2. Figure 1.2 Structural elements of Sirte Basin. Schematic, structural cross‐section from the Sarir Trough showing hydrocarbons in the Sarir Sandstone [Ambrose, 2000; Ahlbrandt et al., 2005].
3. Figure 1.3 (a) Differential hydrodynamic trapping mechanism leading to different levels in fluid contact. (b) The perched aquifer explained as the reason. Contact levels depend on the number of faults in the system.
4. Figure 1.4 Three decision scenarios with three decision variables: injector efficiency, quality map, and production decline.
5. Figure 1.5 Location of the decision problem near the city of Kasted. The blue diamonds are the four alternative well locations (A, B, C, and D) in the decision problem. The grey lines are locations with SkyTEM data. Bottom: vertical profile of inverted SkyTEM data showing buried valleys.
6. Figure 1.6 Conceptual geological model (a sketch) of the buried valet deposits. Valleys are with different lithologies. Hence, hydraulic properties cross‐cut each other [Hoyer et al., 2015].
7. Figure 1.7 (a) An open (ATES) and (b) closed (BTES) shallow geothermal systems [Bonte, 2013].
8. Figure 1.8 The use of electrical resistivity tomography in the design of shallow geothermal systems.
9. Figure 1.9 Example of temperature monitoring during a heat‐tracing experiment from ERT.
10. Figure 1.10 Rifle site with locations of the four uranium bioremediation experiments conducted in the years 2002, 2003, 2007, and 2008. The areal extent of 2007 and 2008 experiments are approximately 20 m × 20 m. The terrain map is obtained from Google Earth. Location of the wells (indicated by the red dots) for different years are referred from Yabusaki et al. [2007].
11. Figure 1.11 The 2007 Winchester experimental setup (a). The aim is to predict the volume and spatial distribution of immobilized uranium. One simulation result for immobilized uranium is shown (b). M1–M12 are wells where tracers or concentrations are monitored.
12. Figure 1.12 Overview of world shale resources (oil or gas).
13. Figure 1.13 Overview of shale HF operations.
Chapter 02
1. Figure 2.1 Decision tree for a simple game of investing and betting. Squares represent decisions nodes and circles represent uncertainty nodes.
2. Figure 2.2 Illustration of the concept of value, utility, and certainty equivalent.
3. Figure 2.3 Example of a hierarchical tree with objectives.
4. Figure 2.4 Hypothetical value functions, turning a score into a common scale.
5. Figure 2.5 Efficient frontier.
6. Figure 2.6 A hypothetical decision tree. The first node has to be a decision node. Uncertainties here are the type of geological system (channel vs. bar), the orientation of geological bodies and the degree of connectivity between them as measured by a probability. The latter is calculated from actual models. The best alternative is “clean.”.
7. Figure 2.7 Example of a hypothetical influence diagram.
8. Figure 2.8 Example of a complex influence diagram for assessing the value of 4D seismic data.
9. Figure 2.9 Example of a decision tree that involves collecting data as an alternative.
Chapter 03
1. Figure 3.1 Setup of our simple illustration case.
2. Figure 3.2 Concentration of DNAPL at the drinking well location with histogram of arrival times.
3. Figure 3.3 A matrix multiplication representing a mapping from one Cartesian axis space to another Cartesian axis space. Eigenvalues and eigenvectors characterize the nature of such transformation.
4. Figure 3.4 Geometric explanation of SVD as a transformation from one orthogonal system to another orthogonal system.
5. Figure 3.5 Creating a Hausdorff distance between two objects that are discretized with points.
6. Figure 3.6 Evaluation of the modified Hausdorff distance for images of numbers 1, 4, and 7. Modified Hassdorf distance (MHD) requires the rasterization of the images into points sets (shown on the right). The smallest MHD is observed between images of 1 and 7, which is consistent with visual inspection. This is not the case for the pixel by pixel Euclidean distance (left).
7. Figure 3.7 The proximity transforms to measure differences in images constituted by discrete objects.
8. Figure 3.8 Two views of the same data matrix.
9. Figure 3.9 Curse of dimensionality, the ratio of hypersphere with hypercube. Space becomes virtually empty after dimension 10.
10. Figure 3.10 Projection of data onto an eigenvector.
11. Figure 3.11 Five spatial models out of a set of 1000.
12. Figure 3.12 Scree plot, average of all models, score plot and 6 out of 1000 PCs, each of dimension 125 × 100.
13. Figure 3.13 Examples of reconstruction of two spatial models with different spatial variability.
14. Figure 3.14 PCA applied to the set of concentration curve of Figure 3.2.
15. Figure 3.15 MDS: turning a distance table between cities into a 2D map.
16. Figure 3.16 (a) MDS projection of 136 binary overhead snapshots of a flume experiment, using the modified Hausdorff distance. (b) Kernel density estimation in MDS projection.
17. Figure 3.17 (a) Scree plot and MDS projection when the distance is the Euclidian distance between hydraulic conductivity realizations. (b) MDS projection when the distance is the Euclidian distance between realizations of concentration in time. The plots are colored by model parameters.
18. Figure 3.18 (a) PCA of seven head measurements and score plot. (b) Correlation between PCs is almost zero. CCA shows correlation of 0.86 and 0.6 between the canonical components of data and prediction.
19. Figure 3.19 Raw data of oil rate decline in 200 wells; basis function and fitting with smoothing; all 200 smoothed data.
20. Figure 3.20 FPCA with harmonics, PC functions, and score plot.
21. Figure 3.21 Example of a regression tree with two variables and four splits.
22. Figure 3.22 Regression tree for the case of Section 3.1, involving six variables and 200 runs of the model. The color is the average arrival time. The color indicates the deviation from the global average (26.3).
23. Figure 3.23 Application of a single tree and boosted tree to a test set, shown are observed and predicted arrival time.
24. Figure 3.24 (a) Example calculation of variable importance (sensitivity) for the case of Figure 3.21 using regression trees. (b) Relative importance (sensitivity) using boosted tree model for the arrival time response. Relative importance is expressed in percent.
25. Figure 3.25 Five realizations of Gaussian processes for different covariance models.
26. Figure 3.26 Example of approximate realizations of a Gaussian process generated using Karhunen–Loeve expansion (compare with the left figure in Figure 3.25).
27. Figure 3.27 Five realizations of Gaussian processes for different covariance models that are conditioned to available data (represented as black dots). The black line represents the mean value and the shaded area represents the 95th confidence interval. This illustrates that Gaussian process regression (kriging) is an exact interpolator.
28. Figure 3.28 A simple classification problem with a nontrivial solution. Linear classification is not possible in this XOR problem (exclusive or). Linear discriminant functions are desirable, in particular when going to higher dimension. The “trick” is to transform the distances to obtain a higher‐dimensional space.
29. Figure 3.29 Embedding a nonlinear 2D function in a 3D plane, such that projection retrieves the nonlinear function.
30. Figure 3.30 Changing dot‐product to render linear modeling more appropriate. The space on the right is also termed the “feature space” in machine learning, features being quantified by the kernel function. Features in statistics are termed simply “predictors.”
31. Figure 3.31 Comparison of classical MDS with MDS after kernel transformation. Notice, like in Figure 3.28, the impact of an increase in dimension.
32. Figure 3.32 An example of partitioning.
33. Figure 3.33 Choice of the optimal number of clusters using the silhouette index and Davies–Bouldin index. Five clusters are created using k‐medoid, applied on the set of 136 overhead snapshot of a flume experiment. Note that the clustering is applied in high dimension (and not 2D).
34. Figure 3.34 Sampling from a complex pdf by sampling from a uniform distribution and then accepting the red dots and rejecting the blue dots.
35. Figure 3.35 Sampling from a uniform distribution with seven samples per stratum. Sampling from a Gaussian distribution with four samples per stratum constructed from concentric circles.
36. Figure 3.36 Three Latin hypercube samples in two dimensions.
37. Figure 3.37 Using MC to estimate an exceedance probability from the blue pdf. Thousand samples provide basically no accuracy, while high accuracy can be obtained by using the green proposal distribution.
38. Figure 3.38 Summary of the bootstrap procedure.
39. Figure 3.39 The one‐sample bootstrap.
Chapter 04
1. Figure 4.1 Tornado chart for the simplified hydrological case from Chapter 3 on the pollutant arrival time. Blue bars represent changes when a parameter is decreased by two standard deviations. Red bars represent changes when a parameter is increased by the same degree.
2. Figure 4.2 Morris method applied to the simplified hydrological case.
3. Figure 4.3 Scatterplots of parameters values versus the pollutant arrival time.
4. Figure 4.4 Pareto plots displaying the t‐statistics for (left) regression model without interactions (R² = 0.98) and (right) regression model including interactions (R² = 0.99). Red bars show influential parameters. The black line represent the significance level of statistical test for α = 0.05.
5. Figure 4.5 Sobol’ indices: (a) first‐order effect and (b) total effect (including interactions) for a modified version of the Libyan case described in Chapter 1.
6. Figure 4.6 Schematic illustration of the KS test.
7. Figure 4.7 Histograms of L1‐norm obtained by the resampling procedure. The observed L1‐norm and alpha‐percentile are shown as well.
8. Figure 4.8 Pareto plots illustrating the sensitivity of each parameter on the pollutant concentration curves. (a) Standardized L1‐norm per class (three classes used in this example). (b) Average standardized L1‐norm.
9. Figure 4.9 ‐based sensitivity values for each parameter (in percent) using α = 95%.
10. Figure 4.10 Table showing the ‐based sensitivity values for interactions (in percent).
11. Figure 4.11 Application of CART‐based SA on the simplified DNAPL example, using the cost of Eq. (4.31).
12. Figure 4.12 Uncertainty in contaminant concentration (P10, P50, and P90) with (solid blue lines) and without (dashed red lines) stochastic uncertainty in the construction of the hydraulic conductivities and boundary conditions.
13. Figure 4.13 OAT results using three different random seeds.
14. Figure 4.14 Morris method results using three different seeds.
Chapter 05
1. Figure 5.1 How many circles do you see?
2. Figure 5.2 What do you see?
3. Figure 5.3 No art, just a geophysical image.
4. Figure 5.4 The harvest moon appearing gigantic as compared to the moon in the high sky.
5. Figure 5.5 A reservoir modeled developed to reflect super permeability channels. Note the legend with permeability values [Valle et al., 1993].
6. Figure 5.6 Example of t‐norms for conjunction operations.
7. Figure 5.7 Bayes’ billiard table: “to be so made and leveled that if either of the ball O and W thrown upon it, there shall be the same probability that it rests upon any one equal part of the plane as another” [Bayes and Price, 1763].
8. Figure 5.8 Flume experiment of a delta with low Froude number performed by John Martin, Ben Sheets, Chris Paola and Michael Kelberer.
9. Figure 5.9 Examples of a few photographs images of the flume experiment for different time. Flow is from top to bottom. (a) Photographs of the experiments. The blue pixels indicate locations where flow moves over the surface. The black sediment is coal which is the mobile fraction of the sediment mixture, and the tan sediment is sand. (b) Binary representation of the photographs. Black represents wet (flow) pixels and white represents dry (no flow) pixels.
10. Figure 5.10 Selected images by clustering based on the modified Hausdorff distance. The value at the top of the image represents the time in minutes of the experiment.
11. Figure 5.11 (a) QQ‐plot of the MHD distances between the 136 images from the experiment and 136 images generated using direct sampling (DS). (b) Comparison of the variability, as defined by MHD, between generated realizations per each training image (TI) (red) and the images from the experiment (blue) closest (in MHD) to the selected TI.
12. Figure 5.12 Example of a FLUMY model with several realizations of the prior generated from FLUMY with uncertain input parameters.
Chapter 06
1. Figure 6.1 (a) Prior densities depend on the parameterization. (b) Prior densities are invariant to the parameterization.
2. Figure 6.2 (a) Metric density of which the integral need not exist. (b) The integral always exists.
3. Figure 6.3 (a) Generation of point sets interpreted from a geophysical image of the subsurface, (b) initial fault surface, (c) generating proper intersection, (d) creation of the point set for layering, (e) construction of layer surfaces, and (f) putting it all together [Caers, 2011].
4. Figure 6.4 Assigning properties in a stratigraphic grid by creating a “depositional domain” [Caers, 2011].
5. Figure 6.5 MPS: generating realizations from training images, constrained to hard data. Here three “snapshots” of the prior movie are shown.
6. Figure 6.6 (a) A 2D GPR section with interpreted scour features. (b) A graphical presentation of a 2D GPR. (c) A 3D object model with 3D scour objects constrained to the interpreted section (red plane).
7. Figure 6.7 Intersection of two surfaces as constructed by a Boolean operation on two level‐set functions.
8. Figure 6.8 Discrete model realization and expansion of the sampling space using various dimension reduction methods and their regularizations to generate discreteness in the reconstruction [Vo and Durlofsky, 2016].
9. Figure 6.9 Cases of inverse problems. The ellipses are multivariate Gaussian distributions. (a) The linear case, (b) approximation of the linear case by expansion in the mean, (c) expansion in the maximum likelihood point, (d) a nonlinear problem with multiple solutions, and (e) nonlinear and non‐Gaussian case.
10. Figure 6.10 Well‐log of P12 with measured resistivity (Em39), plus the various inverse solutions at that location. Inversion (a) with smoothness constraints, (b) using a homogenous reference model, (c) using a heavily weighted homogenous reference model, (d) using a three‐layer reference model, and (e) using a heavily weighted three‐layer reference model. (f) Cumulative sensitivity [Hermans et al., 2012].
11. Figure 6.11 Comparison between structural inversion and a homogeneous reference inversion.
12. Figure 6.12 Comparison with the well‐log: (a) Smoothness based on inversion, (b) imposing a spatial covariance, (c) multiplying the horizontal range by two, and (d) multiplying the vertical range by two.
13. Figure 6.13 Overview of the advances in stochastic modeling of faults. The reader is referred to a history of fault modeling including the following: Priest and Hudson [1976], D. Veneziano [unpublished data, 1978], Andrews [1980], Chilès [1988], Thore et al. [2002], Hollund et al. [2002], Cherpeau et al. [2012] Caumon et al. [2004]; Cherpeau et al. [2010], Lajaunie et al. [1997] Hesthammer and Fossen [2000]. MPP = marked point process.
14. Figure 6.14 Kumano Basin, consisting of two areas with clearly interpretable faulting (normal faulting and Y‐faulting) and a zone of interest with poor seismic imaging.
15. Figure 6.15 Various perturbations in the Metropolis sampling of the Strauss process.
16. Figure 6.16 Data, prior model realizations, initialization with an ad‐hoc matching, and posterior fault networks. Histogram of the size (in percent of total domain size) of the various model sets.
17. Figure 6.17 Six iterations of the sequential Gibbs. At each iteration a 4 × 4 window is re‐simulated [Hansen et al., 2012].
18. Figure 6.18 Extracting summary statistics from DCT‐compressed prior model realizations. These summary statistics are then used as prior model statistics in a multichain sampler [Lochbuhler et al., 2015].
19. Figure 6.19 Deterministic versus stochastic imaging.
20. Figure 6.20 (a) A cross‐section of the salt body with illumination (SEAM Model [Fehler and Larner, 2008]). (b, c) Migrated images of an anticline from the location beneath the salt body structure. (b) is the true image of the structure, while (c) indicates a typical image that can be reasonably obtained from an estimate of the velocity model. The black dotted line indicate the volume V of interest.
21. Figure 6.21 (a) Three realizations of the salt body viewed from the top. The green color refers to the base salt body obtained from deterministic inversion, while the red/blue color indicates perturbation of the body. The perturbations are such that they preserve the body roughness. (b) Uncertainty on reservoir volume due to imaging uncertainty.
22. Figure 6.22 Most common workflow from geophysical data acquisition to subsurface geostatistical models.
23. Figure 6.23 Closing the loop between geophysical inversion and geostatistical modeling.
24. Figure 6.24 A workflow for falsification of the prior, before doing any actual inversion.
Chapter 07
1. Figure 7.1 A Bayesian evidential analysis uses prior samples to construct a statistical relationship (evidential learning) between salary and apartment size. The actual salary is then used to state uncertainty on apartment size, or the mean is used as an estimate.
2. Figure 7.2 Two paradigms for uncertainty quantification. The traditional framework (a) applies causal analysis to match the subsurface models to the data, then use those matched models for predictions. The proposed methodology (b) uses Bayesian evidential learning by which the model is used to construct a statistical relationship between the data and prediction variables.
3. Figure 7.3 Overview of the elements of Bayesian evidential learning and prediction for UQ.
4. Figure 7.4 Steps in performing mixed PCA.
5. Figure 7.5 Example of density estimation when both the data and prediction variables are univariate. (a) Each blue point represents a prior sample and the desired conditional distribution is f(h|d_obs) where d_obs is indicated by the red line. (b) Joint f(h, d) estimated by performing KDE using a Gaussian kernel. (c) The conditional distribution f(h|d_obs) obtained using the joint distribution and Eq. (7.19).
6. Figure 7.6 (a) Prior and observed data in time domain. (b) Performing outlier detection in 2D functional space. The boundary learned by the one‐class SVM is show in orange; observed data falling outside of the boundary suggests that the prior is inconsistent.
7. Figure 7.7 (a and b) Prior realizations of data variables for two of the five existing producers (gray). These responses were obtained by flow simulating each of the 500 prior reservoir models. The observed d_obs production rate is shown in red, as expected very few of the prior production rates match d_obs. (c) Prior realizations of production rates from 10 to 30 years.
8. Figure 7.8 Correlation between d and h in canonical space for first three (of four) components. The canonical projection of d_obs is indicated by the red line. A sufficiently strong linear correlation between data and prediction variables can be observed.
9. Figure 7.9 (a) Prior (gray) versus posterior (blue) samples plotted in the first two dimensions in canonical space. Note that the posterior covariance is smaller than that of the prior: Bayesian evidential learning reduced uncertainty. (b and c) Posterior of prediction variables in the time domain obtained by undoing each of the previously applied transformations (normal score, CCA, FPCA) on the canonical posterior samples. Quantiles of both prior and posterior forecasts are shown in (b).
10. Figure 7.10 Scatterplot comparing the top three mixed PCA components of the data variable with the forecast variable. The projected value of d_obs is shown by the red line. Prior distribution of the prediction variable “injector efficiency” with posterior distribution estimated by means of KDE.
11. Figure 7.11 (a) Setup of a situation involving injecting water into oil and aiming to track what happens at two producers, an existing and a new one. (b) Color indicates water saturation at 5 years. (c) The data variable is a set of seismic maps (4D seismic) over 5 years, which represents the change in oil saturation.
12. Figure 7.12 (a) Two realizations of the data variables. (b) An example of data realizations and its reconstruction using the first 30 eigen‐images. (c) Low‐dimensional projection of prior forecast variable using functional data analysis. First three dimensions are shown.
13. Figure 7.13 Posterior forecasts in the time domain obtained by undoing each of the previously applied transformations (normal score, CCA, FPCA) on the samples drawn in canonical space. Quantiles of both prior and posterior forecasts are shown on the left.
14. Figure 7.14 (a) 3D seismic survey of a reservoir zone in West Africa. (b) Prior model realizations and forward simulated 3D seismic.
15. Figure 7.15 MDS plot based on the distance between wavelet coefficients of any two forward simulated 3D seismic models. The color is the width of the turbidite channels. The cross indicates the actual field seismic (right) Using KDE and Bayes’ rule the uncertainty of the proportion of sand containing oil is significantly reduced, indicating significant potential in this field.
Chapter 08
1. Figure 8.1 A common strategy for UQ in all three decision scenarios.
2. Figure 8.2 Structural framework (a) and stratigraphic (b) grids for the Libyan case. The reservoir model consists of four faults and two deformable horizons.
3. Figure 8.3 (a) A training image with channels of high‐quality sand and diatomite bodies is used to construct facies realizations. (b) Permeability realization shown in depositional domain (left) and gridded physical domain (right).
4. Figure 8.4 Histogram of the prior efficiencies realizations for each of the three existing injectors after 800 days of injection.
5. Figure 8.5 Historical production profiles for each of the 500 prior reservoir models (gray). The observed historical production is indicated by the red line.
6. Figure 8.6 (a) Incremental oil production for an infill well drilled at 49 potential well locations are used as data for interpolation (left). Kriging map of reservoir quality using those data points (right). (b) Two realizations of reservoir quality (blue is low quality) drawn from the prior.
7. Figure 8.7 Scatter plots of the first two components of d^mpca vs. the prior injector efficiencies h_prior for each injector. The red line indicates the locations of .
8. Figure 8.8 Prior (gray) and posterior (blue) distributions obtained from Bayesian evidential learning (by KDE) for each of the three injector efficiencies. The “truth” is indicated by the red line.
9. Figure 8.9 Reconstruction of a quality map using the 10 largest eigen‐images in comparison with the original image.
10. Figure 8.10 (a) True quality map with prior and posterior mean, (b) prior and posterior variance.
11. Figure 8.11 Scatterplots of the first canonical components of the data and prediction variables. Posterior field production rates generated by sampling the posterior distribution in canonical space and undoing each of the transformations. Computed quantiles both for the prior and posterior along with the truth is shown in red. For a minimum economically viable rate of 600 stb/day, the posterior predictions indicate that the field should be abandoned at 7897 days, 7136 days, and 6574 days depending on the level of risk the decision maker is willing to accept.
12. Figure 8.12 Weight functions for different values of the exponent.
13. Figure 8.13 Location of infill well selected according to the predicted posterior quality maps and variance, superimposed over the “true” quality map.
14. Figure 8.14 A strategy to come to a decision in the Danish case.
15. Figure 8.15 Embedding of the local “kasted” model into the regional model. The local model lithologies are stochastically varying, while the regional model is deterministic (with uncertain hydraulic conductivities).
16. Figure 8.16 Workflow for generating multiple 3D realizations of the local model architecture. Two major uncertainties need to be addressed: (i) gap‐filling of the SkyTEM data and (ii) spatial uncertainty related to the cross-cutting of buried valleys of different age.
17. Figure 8.17 (a) Score plot of first two mixed PCs of the head and streamflow data variables. The yellow cross represents the actual field observations. (b) Cumulative variance shows that 25 mixed PCs explain 99% of variation in head and streamflow.
18. Figure 8.18 (a) Sensitivity analysis of the joint data variables (head and stream). (b) Assessment of how well the tree model predicts the mismatch between simulated data and the observed data.
19. Figure 8.19 Sensitivity analysis for three prediction variables at location A.
20. Figure 8.20 Prior and posterior pdfs of recharge (a) and a DEM‐related model variable (b).
21. Figure 8.21 Prior and posterior distribution of two prediction variables: (a) SFRP (Streamflow) at location A and (b) industry pollution at location C.
22. Figure 8.22 Efficient frontiers for a risk neutral (a) and risk averse decision maker (b).
23. Figure 8.23 Strategy to reach a decision (go ahead with the heat pump) in the shallow geothermal case.
24. Figure 8.24 Deterministic predictions of the change in temperature during the pumping phase. (a) Local sensitivity of natural hydraulic gradient for a fixed hydraulic conductivity. (b) Local sensitivity of hydraulic conductivity in fixed gradient conditions. (c) Prior samples of the prediction, the green curve is the prediction for a calibrated hydraulic conductivity distribution. (d) Global sensitivity analysis (GSA) of the parameters considered in the prior.
25. Figure 8.25 (a) Plan view of the experimental setup for the heat tracing experiment. (b) Vertical cross‐section of the ERT panel and electrode layout.
26. Figure 8.26 (a) Sensitivity analysis of the ERT data on model variables, (b) variation of the PCs, (c) red line corresponds to the field data and black lines correspond to average change in resistance for all data configuration as generated from the prior. The red line and red dot represent the same ERT data.
27. Figure 8.27 (a) Falsification with direct raw and PCA transformed temperature measurements. The red line corresponds to the field data, and the gray lines correspond to the average change in resistance for all data configuration as generated from the prior. (b) The score plot shows that the prior covers the data (red dot).
28. Figure 8.28 First and second dimension of canonical correlation analysis for 5‐day experiment. Prior and posterior realizations and quantiles (10, 50, and 90%).
29. Figure 8.29 Posterior realizations, mean, and standard deviation at three different time‐steps.
30. Figure 8.30 Strategy to assess long‐term remediation efficacy by means of acetate injection.
31. Figure 8.31 Simplified reaction network showing the most important reactions for uranium remediation that takes place when injecting acetate.
32. Figure 8.32 Example of heterogeneous realizations of initial U(VI) concentrations conditioned at the monitoring wells. Models are created using sequential Gaussian simulation.
33. Figure 8.33 Spatial realization of log permeability, porosity, and Fe(OH)₃ content.
34. Figure 8.34 Example of realization of the data variables for well5 (a) and the prediction variable, spatial immobilized uranium (b). The gray lines are prior model realization, and the red line is the observed tracer data.
35. Figure 8.35 Eigen‐image decomposition of the ensemble of 500 realizations of immobilized uranium. (a) The first four eigen‐images. (b) An example on how to reconstruct images from a limited set of eigen‐images.
36. Figure 8.36 Results of a global sensitivity analysis using DGSA on three data variables and one target prediction variable.
37. Figure 8.37 CCA analysis between data and prediction variables reveals high correlation in the first canonical score. As a result, the prior uncertainty is reduced considerably into a set of posterior models. This is shown in the score plot as well as with a few posterior realization of the immobilized uranium.
38. Figure 8.38 Location of wells in the Barnett shale (gas) and the Anadarko dataset (location confidential).
39. Figure 8.39 A generic workflow for data‐based predictions.
40. Figure 8.40 Strategy for data‐based learning in shale systems (CART = classification and regression trees).
41. Figure 8.41 Smoothing of oil decline rate in SC2 using functional data analysis.
42. Figure 8.42 Bagplot for outlier detection. Red lines are outliers decline curves that are not used in the training set.
43. Figure 8.43 Semi‐variogram of a function (residual decline rate) with the number of pairs N (h) used for calculation. Gas production (GPRC) at 0, 12, and 24 months estimated by spatial interpolation of existing producers.
44. Figure 8.44 Variable importance plot (relative sensitivity) See Table 1.3 for explanation of variables.
45. Figure 8.45 An example of tree‐based predictions for two wells. Blue curves: training set; red curves: random forest ensemble; thick black line: true well response; dashed black line: forecast of a single tree.
46. Figure 8.46 The results of the Monte Carlo study showing that 50–60 wells are sufficient for training.

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