Gridding Algorithm

Key Takeaways

• Minimum curvature is the most widely used gridding algorithm in petroleum exploration mapping because it produces visually smooth surfaces that geologically resemble the continuous depositional or structural surfaces being mapped, minimizes artificial oscillations (bull's-eyes) around isolated data points, and is computationally efficient for large grids; minimum curvature works by finding the surface that minimizes the integral of the squared curvature (the second derivative of the surface) subject to the constraint of passing through or near the input data points; the smoothness parameter (also called the tension parameter in some implementations) controls the tradeoff between fitting data exactly (high tension) and producing the smoothest possible surface (low tension), with high-tension minimum curvature producing surfaces that more closely honor isolated high or low values at the expense of creating local bull's-eyes, and low-tension surfaces that are smoother but may underrepresent sharp features like faults or salt contacts; minimum curvature has no statistical basis (it does not characterize spatial uncertainty) and treats all input data as equally reliable, which limits its use for quantitative uncertainty analysis.
• Kriging is the statistically optimal linear interpolation method (minimum variance estimation) that uses the variogram (a spatial statistical model describing how the variance of the property changes with distance and direction between sample pairs) to determine the optimal weights for combining nearby data points into an interpolated estimate at each grid node; kriging has two major advantages over deterministic algorithms: it provides an estimate of the interpolation uncertainty at each grid node (the kriging variance, which is large in areas far from data and small near data points), and it incorporates directional anisotropy from the variogram (so that interpolation in the direction of geological continuity uses longer-range samples than interpolation across the grain of the stratigraphy); ordinary kriging, which is the most common form, assumes a locally stationary mean and produces smooth maps that honor the variogram model; sequential Gaussian simulation (a stochastic extension of kriging) generates multiple equally probable realizations of the spatial distribution, enabling probabilistic reserve estimation and flow simulation uncertainty analysis; the variogram must be fitted to the data before kriging, and variogram model choices (nugget, range, sill, anisotropy ratio) significantly affect the kriging output, requiring geostatistical expertise to produce defensible results.
• Inverse distance weighting (IDW) is a simple, intuitive interpolation algorithm that assigns each grid node the weighted average of all input data points, where the weight of each data point is proportional to the inverse of its distance (or distance raised to a power p) from the grid node; IDW is easy to implement and computationally fast, but it has significant deficiencies for petroleum mapping: it produces bull's-eye patterns around isolated data points (all isolines become concentric circles around wells in areas without other data), it does not extrapolate beyond the range of input data values (it cannot produce interpolated values higher than the maximum or lower than the minimum of the input data), it requires selection of the power parameter p (higher p gives more weight to nearby data and sharper local features; lower p produces smoother, more global averages), and it has no statistical basis so no uncertainty quantification is possible; IDW is occasionally used for rapid reconnaissance mapping and for cases where data are so sparse that variogram fitting is not feasible, but it has been largely replaced by minimum curvature and kriging in professional practice.
• Fault handling in gridding algorithms is one of the most technically challenging aspects of structural mapping because faults are discontinuities in the geological surface that violate the smoothness assumptions built into most interpolation algorithms: a standard minimum curvature or kriging algorithm applied to a faulted surface would interpolate across the fault plane, artificially smoothing the displacement and producing an incorrect representation of the fault geometry and the footwall-hangingwall offset; fault-respecting gridding algorithms work by either explicitly honoring fault polygons as internal boundary conditions that prevent interpolation across the fault (the grid is broken into fault-bounded domains with separate interpolation in each domain), or by incorporating fault proximity into the variogram or distance calculation (reducing the effective correlation range across faults); commercial mapping packages (Petrel, Kingdom, GeoFrame, Paradigm) implement fault-respecting gridding via their structural modeling workflows, but the quality of the fault-respecting output is sensitive to the accuracy of the interpreted fault sticks, fault polygons, and horizon picks on both the footwall and hangingwall.
• Grid resolution (cell size) in petroleum mapping involves a fundamental tradeoff between spatial detail and computational cost: fine grids (small cell size, many nodes) capture short-wavelength geological features such as channel edges, fault throws, and thin reservoir pinch-outs more accurately, but require more memory and computation for both the gridding step and any subsequent simulation or mapping operations; coarse grids lose spatial detail but are faster to generate and manipulate; the optimal grid resolution is the finest grid that can be supported by the density of input data without excessive extrapolation between data points (a rule of thumb is that the grid cell size should be no smaller than one-quarter to one-half of the typical well spacing so that each grid cell contains at most one or two wells), and the finest grid that can be processed in acceptable time for the intended application (static model grids used as input to flow simulation are typically upscaled to coarser simulation grids to reduce the number of cells from millions to hundreds of thousands); grid orientation is also a consideration for anisotropic geological settings -- aligning the grid axes with the principal directions of geological continuity (along strike and dip for layered reservoirs, along the channel axis for fluvial systems) reduces interpolation artifacts compared to a north-south/east-west grid that cuts across the geological grain.