Geostatistical Methods

Key Takeaways

• The variogram (or semivariogram) is the core statistical tool of geostatistics, measuring how the variance of the difference between property values at two locations increases as the distance between them increases, with the variogram shape encoding the spatial continuity characteristics of the property: at zero separation distance (the nugget), the variogram value approaches either zero (perfect repeatability of measurements at the same location) or a small positive value (the nugget effect, representing measurement error or spatial variability at scales smaller than the sample spacing); as separation distance increases from the nugget, the variogram rises (indicating that samples farther apart are less similar) until it reaches the sill (the variance of the data, indicating that samples far apart are essentially uncorrelated); the separation distance at which the variogram reaches the sill is the range (the correlation length, beyond which there is no spatial dependence); the variogram is modeled with analytical functions (spherical, Gaussian, exponential) fitted to the experimental variogram calculated from data pairs at increasing separation distances, and the fitted model is used in kriging and simulation to quantify the spatial correlation at all distance scales; anisotropic variograms (with different ranges in different directions, typically longer in the depositional direction than perpendicular to it) are modeled with ellipsoidal anisotropy ratios that reflect the expected greater continuity of fluvial sands along channels versus perpendicular to them.
• Kriging is the geostatistical interpolation method that produces the minimum-variance unbiased linear estimator of a property at unsampled locations, weighting the nearby measured data values according to the variogram model to account for both the distance from the estimate location to each data point and the spatial redundancy between data points (closely spaced data points carry less independent information than widely spaced points and are downweighted accordingly): ordinary kriging estimates the property value at each unsampled location as a weighted average of nearby data points, with weights determined by solving a system of linear equations that minimizes the kriging variance; the kriging variance (a measure of the uncertainty of the estimate at each location) is not a constant but varies with the data density (higher density leads to lower kriging variance at the estimated location) and the variogram range (longer range means more spatial information from each data point, lower kriging variance everywhere); co-kriging incorporates secondary variables (seismic attributes, well log measurements from a different tool) that are spatially correlated with the primary property of interest to improve the kriging estimate by leveraging the denser spatial sampling of the secondary variable; kriging produces smooth, best-estimate property maps that are appropriate for reservoir simulation when the objective is to match observed production history with a single representative model, but underrepresent the actual spatial variability of the property (because kriging is a smoothing operation), making it less appropriate for uncertainty quantification where the full range of possible property distributions must be represented.
• Sequential Gaussian simulation (SGS) is the most widely used stochastic simulation method for continuously variable properties (porosity, net-to-gross) that generates multiple equiprobable realizations of the property distribution by visiting each simulation node in a random sequence and drawing a simulated value from the conditional distribution at that node (conditioned on all previously simulated values and the original data), ensuring that each realization reproduces the variogram structure of the data and honors the hard data at well locations: the multiple realizations from SGS provide a probabilistic model of the property distribution, with the spread among realizations quantifying the uncertainty arising from the limited sampling of the reservoir by existing wells; when all realizations are run through the reservoir flow simulator and the simulation output is compared to the actual production history, the spread in simulated production responses among realizations provides a measure of the production forecast uncertainty attributable to the interwell property uncertainty; the P10/P50/P90 range of cumulative production or recovery factor across the realization ensemble is a standard format for communicating this uncertainty to management and investors in probabilistic reserve estimates.
• Object-based modeling (also called Boolean simulation) is a geostatistical approach for modeling geobodies with complex shapes (such as meandering river channels, submarine fan lobes, and carbonate reefs) that cannot be adequately characterized by variogram-based pixel methods because the spatial continuity of the body geometry is better described by the shape, size, and orientation parameters of the geobody than by a variogram of the rock property values inside and outside the body: an object-based channel model generates individual channel ribbons (each with specified width, thickness, sinuosity, and orientation drawn from a distribution calibrated to analogues or outcrop data) and places them randomly in the simulation volume until the specified net-to-gross ratio is achieved, with the channels representing sand and the interfluve matrix representing shale; the resulting channel model captures the tortuous, elongated, high-permeability connectivity structure of fluvial systems better than a variogram-based simulation that generates blocky, geometrically unrealistic sand bodies; the limitation of object-based modeling is the difficulty of conditioning the simulated objects to hard data at well locations (forcing the object model to honor the observed facies sequence in wells while maintaining the stochastic variability of the channel architecture between wells) and the computational cost of the iterative conditioning algorithms required for dense well control.
• Multiple-point statistics (MPS) simulation extends geostatistical methods beyond two-point variogram statistics (which can capture only linear spatial patterns) to reproduce complex, non-linear spatial patterns extracted from a training image: a training image is a two-dimensional or three-dimensional conceptual geological model (which may be a hand-drawn geological interpretation, an outcrop photograph, or a process-based sedimentary model) that encodes the geological patterns expected in the reservoir, such as the curvilinear geometry of channels, the levee-overbank relationship, or the hierarchical connectivity of carbonate vuggy pores; the MPS algorithm scans the training image to extract the conditional probability of each rock type at a target location given the rock types observed at multiple surrounding template points, and uses these multi-point conditional distributions to draw simulated values during sequential simulation that reproduce the training image's spatial patterns in the simulated reservoir model; MPS simulation has been applied to model geological objects (channels, lobes, fractures) whose geometry cannot be captured by variograms but can be inferred from outcrop analogues or process-based forward models that serve as training images.