cokriging

Cokriging is a multivariate geostatistical interpolation technique that estimates the value of a primary variable at unsampled locations by simultaneously exploiting the spatial auto-covariance of the primary variable and the cross-covariance between it and one or more spatially correlated secondary variables sampled at higher density, allowing the secondary data to guide interpolation of sparse primary measurements beyond what ordinary kriging of the primary variable alone could achieve; the method applies the same linear unbiased minimum-variance framework as ordinary kriging but adds cross-variogram model terms that quantify how the primary variable (such as well-log porosity, permeability from core plug analysis, or net pay from petrophysical cutoffs) co-varies spatially with the secondary variable (such as acoustic impedance from 3D prestack seismic inversion, seismic amplitude extracted at a mapped horizon, or formation thickness from seismic interpretation), and the resulting cokriging system weights both primary observations and secondary observations at each estimation node to minimize the estimation error variance subject to the unbiasedness constraint on the weighted sum. In the Western Canada Sedimentary Basin, cokriging is applied in tight reservoir characterization workflows where 3D seismic data provides spatially dense secondary information at 12.5 to 25 m inline and crossline spacing across areas where well control exists only at 500 m to 2,000 m spacing: WCSB Montney siltstone cokriging programs at Sunrise, Septimus, and Cutbank Ridge in northeast British Columbia use acoustic impedance from simultaneous prestack inversion as the secondary variable correlated with sonic-density porosity at 40 to 80 well locations per township to produce 25 m resolution porosity cubes that guide horizontal well placement decisions by identifying porous sweet spots between existing wells; WCSB Cardium tight oil operators at Pembina field and Viking tight oil operators at Dodsland and Kindersley apply cokriging of acoustic impedance to net pay estimation, improving EUR prediction accuracy in infill drilling programs by conditioning pay maps to seismic character rather than simple inverse-distance interpolation between wells at 500 m to 1 km spacing; and WCSB Athabasca oil sands thickness mapping uses cokriging of McMurray Formation net bitumen pay (primary variable from well logs) with tidal channel amplitude anomalies (secondary variable from 3D seismic reflection character) to improve SAGD well pair design across areas with 50 to 200 m well spacing in developed sectors and 500 m to 2 km in exploration margins.

• Cross-variogram modeling requirements and isotopic versus heterotopic sampling in WCSB cokriging programs: Cokriging requires a model for three spatial functions: the variogram of the primary variable, the variogram of the secondary variable, and the cross-variogram describing their joint spatial dependence. The cross-variogram is computed from paired primary and secondary observations at the same location (isotopic sampling, where both variables are measured at every well) or estimated from nearby co-located pairs (heterotopic sampling, where some wells have only primary data and seismic nodes have only secondary data). In WCSB Montney cokriging, the cross-variogram between acoustic impedance and porosity is fitted using Hampson-Russell Emerge or Petrel Facies modeling tools that enforce a linear model of coregionalization (LMC), a mathematical requirement that all three variogram models share the same geometric anisotropy and structure proportions so the cokriging system matrix remains positive definite; an LMC that fits the cross-variogram but violates positive definiteness produces negative kriging weights that can introduce non-physical negative porosity estimates at locations where the secondary variable conflicts with neighboring primary data.
• Collocated cokriging as the standard practical approximation for seismic-well integration in WCSB workflows: Full cokriging in 3D seismic-conditioned reservoir studies is computationally impractical because the secondary variable (acoustic impedance at 12.5 m inline-crossline spacing) has millions of nodes, and a full cokriging system including all secondary neighbors would require inverting matrices of order 10,000 to 100,000 for each primary estimation node. Collocated cokriging resolves this by retaining only the single secondary observation co-located at the estimation node and discarding all other secondary neighbors, under the Markov screen assumption that the co-located secondary observation screens the influence of all other secondary observations once the local primary neighbors are included. WCSB Montney cokriging workflows in Petrel or Roxar RMS implement collocated ordinary cokriging (COCOK) or the simulated Bayesian updating variant (Bayesian cokriging) that expresses the collocated cross-correlation as a simple regression slope applied to the seismic residual from its global mean, reducing the full cokriging system to a standard kriging system augmented by a single secondary variable term at each estimation node, enabling 3D cokriging of a full Montney township in 2 to 4 hours on a standard workstation rather than days for full cokriging.
• Correlation coefficient threshold, screening benefit, and the conditions under which cokriging improves on ordinary kriging in WCSB tight reservoir studies: Cokriging improves on ordinary kriging of the primary variable only when the cross-correlation between primary and secondary is statistically significant; the theoretical screening benefit (reduction in estimation error variance relative to ordinary kriging) is proportional to the square of the Pearson correlation coefficient between the two variables at co-located well positions. In WCSB Montney tight siltstone, the correlation between acoustic impedance from seismic inversion and log-derived porosity typically ranges from minus 0.55 to minus 0.75 at well control locations (higher impedance correlates with lower porosity silicified facies), yielding a theoretical error variance reduction of 30 to 56 percent over ordinary kriging; when seismic-to-well tie quality is poor (cross-correlation below absolute value 0.4 from wavelet phase errors or poor seismic vintage), cokriging provides minimal improvement and can introduce seismic-noise artifacts into the porosity map, requiring the geostatistician to fall back to ordinary kriging of the primary variable or Bayesian updating with a reduced secondary correlation weight.
• Cokriging for permeability estimation in WCSB Montney and Duvernay tight reservoirs using multiple secondary variables: Permeability estimation is more challenging than porosity estimation because permeability varies over several orders of magnitude and has a non-linear relationship with most seismic observables; WCSB Montney and Duvernay cokriging workflows for permeability use log-transformed permeability as the primary variable (so that the log-permeability is approximately Gaussian and satisfies cokriging's implicit multi-Gaussian assumption) with two secondary variables: acoustic impedance from seismic inversion correlated with clay content (clay-rich Montney facies have higher impedance and lower permeability) and seismic brittleness (Vp-Vs derived) correlated with quartz content and natural fracture density that contribute to effective permeability. When two secondary variables are used, the cokriging system requires three variogram models and three cross-variogram models (one for each pair of variables), all fitted within a valid LMC structure, and the partial cross-correlations between seismic attributes must be tested to ensure they add independent information rather than introducing multicollinearity that inflates the cokriging solution matrix condition number.
• Cokriging cross-validation and uncertainty quantification in WCSB reservoir property estimation: Cokriging produces not only a best-estimate at each node but also an estimation variance (or standard deviation) that quantifies local uncertainty based on the data configuration and variogram model; these uncertainties are used in WCSB Montney resource assessment workflows to generate probabilistic P10-P50-P90 porosity maps by running sequential Gaussian cosimulation (SGS with seismic as secondary trend) multiple times, each simulation honoring primary well data, conditioning to the seismic secondary variable via collocated cokriging, and reproducing the fitted variogram model. Cross-validation of WCSB Montney cokriging results is performed by jack-knifing (leaving one well out at a time and comparing the cokriging estimate at that well location to the actual log value), with the mean absolute error (MAE) and standardized mean squared error (SMSE) metrics used to verify that the cokriging system is accurate (low MAE) and properly calibrated (SMSE near 1.0 indicates the estimation variances match the actual squared errors); WCSB operators typically require cross-validation SMSE between 0.8 and 1.3 before accepting the cokriging model as the basis for infill well placement or resource booking.

WCSB Montney Cokriging Program Guiding Well Placement at Sunrise Field

A northeast British Columbia Montney operator at Sunrise field applied collocated cokriging to condition porosity interpolation between 52 wells on a 1 km by 1 km spacing grid using acoustic impedance from a simultaneous prestack inversion volume as the secondary variable. The cross-correlation between log porosity and co-located impedance at well locations was minus 0.68, yielding a predicted error variance reduction of 46 percent over ordinary kriging. The cokriging porosity cube identified a 3 km by 2 km northeast-trending high-porosity corridor (average 6.8 percent versus 4.2 percent background Montney porosity) aligned with a low-impedance seismic anomaly between two existing wells. Four infill horizontal wells drilled into the cokriging-identified sweet spot produced 30-day initial production rates of 7 to 12 MMcf/d per well versus 3 to 6 MMcf/d for wells drilled outside the anomaly in the same program, validating the cokriging-guided placement and justifying expansion of the 3D inversion cokriging workflow to three additional Montney townships in the Sunrise development area.

Related Terms

Kriging is the single-variable spatial interpolation method that cokriging generalizes; cokriging reduces estimation variance beyond kriging only when the cross-correlation between primary and secondary variables exceeds approximately 0.4 to 0.5 at WCSB well control locations. Variogram quantifies spatial auto-covariance of a single variable; cokriging requires variogram models for both the primary and secondary variables plus a cross-variogram model, all fitted within a linear model of coregionalization that preserves positive definiteness. Seismic inversion produces the acoustic impedance volumes most commonly used as secondary variables in WCSB Montney and Duvernay cokriging workflows; prestack simultaneous inversion provides Vp, Vs, and density impedance cubes correlated with porosity, clay, and brittleness. Geostatistics is the discipline encompassing kriging, cokriging, sequential Gaussian simulation, and stochastic modeling that forms the mathematical foundation for WCSB reservoir characterization; cokriging is the standard multivariate estimation tool when secondary seismic data is available. Reservoir characterization integrates cokriging porosity and permeability cubes with structural interpretation and fluid contact mapping to define WCSB Montney and Duvernay sweet spots for horizontal well targeting and hydraulic fracture stage design.