Cover
Title Page
Copyright Page
Contents
1 Introduction
1.1 Introductory Concepts
1.1.1 Stochastic Reservoir Modeling
1.1.2 Flow Modeling in Highly Heterogeneous Reservoirs
1.1.3 Linking Geostatistical Modeling to Flow Modeling
1.2 Reference
2 Spatial Correlation
2.1 Spatial Covariance
2.2 Semi-Variogram
2.2.1 Variogram Inference
2.2.2 Variogram Characteristics
2.2.3 Pure Nugget Effect
2.2.4 Behavior Next to the Origin
2.2.5 Outliers and Semi-Variogram Inference
2.2.6 Proportion Effect
2.3 Variogram Modeling
2.3.1 Positive Definiteness
2.3.2 Allowable Linear Combinations
2.3.3 Legitimate Structural Models
2.3.4 Positive Combinations
2.3.5 Geometric Anisotropy
2.3.6 Coordinate Transform in 2D
2.3.7 Modeling Anisotropy in 3D
2.3.8 Semi-Variogram Computation in 3D
2.3.9 Zonal Anisotropy
2.3.10 Modeling Strategy
2.3.11 3D Model
2.4 References
3 Spatial Estimation
3.1 Linear Least Squares Estimation or Interpolation
3.2 Linear Regression
3.2.1 Linear Least Squares—An Interpretation
3.2.2 Application to Spatial Estimation
3.3 Estimation in General
3.3.1 Loss Function—Some Analytical Results
3.4 Kriging
3.4.1 Expected Value of the Error Distribution
3.4.2 Error Variance
3.4.3 An Example
3.5 Universal Kriging
3.5.1 Kriging with a Trend Function
3.5.2 Ordinary Kriging
3.5.3 Universal Kriging Estimate for Trend
3.6 Kriging with an External Drift
3.7 Indicator Kriging
3.7.1 Non-Parametric Approach to Modeling Distributions
3.7.2 Kriging in Terms of Projections
3.7.3 Indicator Basis Function
3.7.4 Indicator Kriging
3.8 Data Integration in Kriging
3.8.1 Simple Co-Kriging Estimator
3.8.2 Simplified Models for Data Integration
3.8.3 Linear Model of Coregionalization
3.9 References
4 Spatial Simulation
4.1 Introduction
4.2 Kriging—Limitations
4.3 Stochastic Simulation
4.3.1 Lower-Upper (LU) Simulation
4.3.2 Sequential Simulation
4.4 Non-Parametric Sequential Simulation
4.4.1 Interpolating within the Range of Thresholds Specified
4.4.2 Tail Extrapolation
4.4.3 Data Integration within the Indicator Framework
4.4.4 Markov−Bayes Approach
4.5 Data Integration Using the Permanence of Ratio Hypothesis
4.5.1 The Tau Model
4.5.2 The Nu Model
4.6 References
5 Geostatistical Simulation Constrained to Higher-Order Statistics
5.1 Indicator Basis Function
5.1.1 Establishing the Basis Function—Projection Theorem
5.1.2 Single Extended Normal Equation
5.1.3 Single Normal Equation Simulation
5.1.4 Returning to the Full Indicator Basis Function
5.2 References
6 Numerical Schemes for Flow Simulation
6.1 Governing Equations
6.1.1 Conservation of Mass
6.1.2 Conservation of Momentum
6.2 Single-Phase Flow
6.2.1 Simulation Equations
6.2.2 External Boundary Conditions
6.2.3 Initialization
6.2.4 Well Models
6.2.5 Linearization
6.2.6 Solution Methods
6.3 Multi-Phase Flow
6.3.1 Simulation Equations
6.3.2 External Boundary Conditions
6.3.3 Initialization
6.3.4 Well Models
6.3.5 Linearization and Solution Methods
6.4 Finite Element Formulation
6.5 Solution of Linear System of Equations
6.6 References
7 Gridding Schemes for Flow Simulation
7.1 Gridding Schemes
7.1.1 Overview
7.1.2 Cartesian Grid
7.1.3 Corner-Point Grid
7.1.4 Perpendicular Bisector Grid
7.1.5 General Unstructured Grid
7.1.6 Other Specialized Gridding Options
7.2 Consistency, Stability, and Convergence
7.2.1 Consistency
7.2.2 Stability
7.2.3 Convergence
7.3 Advanced Numerical Schemes for Unstructured Grids
7.3.1 Generalized CVFD-TPFA Formulation
7.3.2 CVFD-MPFA Formulation
7.3.3 Control Volume Finite Element Formulation
7.3.4 Mixed Finite Element Formulation
7.4 Dual Media Models
7.4.1 Dual-Permeability Formulation
7.4.2 Dual-Porosity Formulation
7.4.3 Embedded Discrete Fracture Model
7.5 References
8 Upscaling of Reservoir Models
8.1 Statistical Upscaling
8.1.1 Power Average
8.1.2 Statistical Re-Normalization
8.1.3 Facies Upscaling
8.2 Flow-Based Upscaling
8.2.1 Effective Medium Approximations
8.2.2 Single-Phase Flow
8.2.3 Two-Phase Flow (Relative Permeability)
8.2.4 Compositional Flow Simulation
8.3 Scale-Up
8.3.1 General Concepts
8.3.2 Scale-Up of Linearly Averaged Attributes
8.3.3 Scale-Up of Non-Linearly Averaged Attributes
8.4 Scale-Up of Flow and Transport Equations
8.4.1 Dimensionless Scaling Groups
8.4.2 Stochastic Perturbation Methods
8.4.3 Volume Averaging Methods
8.5 Final Remarks
8.6 References
9 History Matching—Dynamic Data Integration
9.1 History Matching as an Inverse Problem
9.2 Optimization Schemes
9.2.1 Gradient-Based Methods
9.2.2 Global Methods
9.3 Probabilistic Schemes
9.3.1 Optimization-Based Bayesian Methods
9.3.2 Sampling Algorithms
9.4 Ensemble-Based Schemes
9.4.1 Ensemble Kalman Filters
9.4.2 Ensemble Pattern Search and Model Selection
9.5 References
A Quantile Variograms
B Some Details about the Markov–Bayes Model
Index