Geostatistical Scaling Laws Applied to Core and Log Data

Reconciling data from different scales is a longstanding problem in reservoir characterization. Data from core plugs, well logs of different types, and seismic data must all be accounted for in the construction of a geostatistical reservoir model. These data are at vastly different scales and it is inappropriate to ignore the scale difference when constructing a geostatistical model. Geostatistical scaling laws were devised in the 1960s and 1970s primarily in the mining industry where the concern was mineral grades in selective mining unit (SMU) blocks of different sizes. These principles can be extended to address problems of core, log and seismic data. The adoption of these classic volume-variance or scaling relationships presents some challenges. Some specific concerns are (1) the ill-defined volume of measurement, (2) uncertainty in the small-scale variogram structure, and (3) non-linear averaging of many responses including acoustic properties and permeability. We demonstrate the application of volume-variance relations for upscaling and downscaling techniques to integrate data of different scales. Practical concerns are addressed with data from a chalk reservoir in the Danish North Sea. A direct sequential simulation algorithm accounting for data at all scales is documented. Introduction Within the petroleum industry and many other fields where geostatistical models are constructed, the treatment of data of different scale is often ignored1. The core and log data may be averaged in the vertical direction to the scale of the modelling cells2; however, this only partially addresses scale difference. In other examples, a fine-scale model is constructed and then numerically averaged to larger scale3-5. This may be applied in a nested fashion due to computational limitations. The direct simulation of gridblock values conditioned to fine-scale data carried out by cosimulation using cross-covariance between fineand coarse-scale values has been described6. Notwithstanding the importance of accounting for data at different scales, the use of geostatistical scaling laws has not seen wide application in petroleum geostatistics. This is due mainly to unfamiliarity with the techniques and scaling laws. Recalling and demonstrating such techniques will address this unfamiliarity. The scaling laws tell us how the variogram changes with volumetric scale7. As scale increases, the range of correlation increases, the variance and variogram sill decrease, and the nugget effect also decreases. After a recall of theory, the application of scaling laws is illustrated with a synthetic example and with real data from a Danish chalk reservoir. Core and well log data are used. These data measure significantly different volumes. The volume of the core measurement is well understood; however, the volume of the interpreted well log derived porosity is less well understood. The statistics of each data type together with analytical volume-variance relationships can be used to quantify the volume of investigation of the well log data. An illustration of the different scales (Fig. 1) shows that the change of scale from core to log measurement volumes is nearly as large as the jump from log volume to that of a geological modelling cell. The ultimate goal of this work is to illustrate how data of different scales may be used simultaneously in the construction of high-resolution geostatistical models. When the different types of data are all “hard,” in the sense that they do not contain significant errors or uncertainties relative to the property being modeled, it is possible to use block kriging. Certain data types such as seismic contain uncertainties related to the great distance of measurement and calibration of the measured acoustic properties to the petrophysical properties of interest. In this case, it is necessary to use block cokriging. Recall of Volume-Variance Scaling Relationships Mining Geostatistics8 is the classic reference for volumevariance scaling relationships. The essential results are recalled below. Details and proofs may be found in the original reference. SPE 56822 Geostatistical Scaling Laws Applied to Core and Log Data P. Frykman, SPE, Geological Survey of Denmark and Greenland (GEUS), and C.V. Deutsch, SPE, University of Alberta 2 P. FRYKMAN, C.V. DEUTSCH SPE 56822 Consider the fitted variogram model at arbitrary scale v, where v often represents the small core scale: ( ) ( ) ∑ = Γ ⋅ + = nst