Summary
Over 75,000 permeability measurements were collected from a meter-scale block of Massillon sandstone, characterized by conspicuous crossbedding that forms two distinct nested scales of heterogeneity. With the aid of a gas minipermeameter, spatially exhaustive fields of permeability data were acquired at each of five different sample supports (i.e., sample volumes) from each block face. These data provide a unique opportunity to physically investigate the relationship between the multiscale cross-stratified attributes of the sandstone and the corresponding statistical characteristics of the permeability. These data also provide quantitative physical information concerning the permeability upscaling of a complex heterogeneous medium. Here, a portion of the data taken from a single block face cut normal to stratification is analyzed. The results indicate a strong relationship between the calculated summary statistics and the cross-stratified structural features visibly evident in the sandstone sample. Specifically, the permeability fields and semivariograms are characterized by two nested scales of heterogeneity, including a large-scale structure defined by the cross-stratified sets (delineated by distinct bounding surfaces) and a small-scale structure defined by the low-angle cross-stratification within each set. The permeability data also provide clear evidence of upscaling. That is, each calculated summary statistic exhibits distinct and consistent trends with increasing sample support. Among these trends are an increasing mean, decreasing variance, and an increasing semivariogram range. The results also clearly indicate that the different scales of heterogeneity upscale differently, with the small-scale structure being preferentially filtered from the data while the large-scale structure is preserved. Finally, the statistical and upscaling characteristics of individual cross-stratified sets were found to be very similar because of their shared depositional environment; however, some differences were noted that are likely the result of minor variations in the sediment load and/or flow conditions between depositional events.
Introduction
Geologic materials are inherently heterogeneous because of the depositional and diagenetic processes responsible for their formation. These heterogeneities often impose considerable influence on the performance of hydrocarbon bearing reservoirs. Unfortunately, quantitative characterization and integration of reservoir heterogeneity into predictive models are complicated by two challenging problems. First, the quantity of porous media observed and/or sampled is generally a minute faction of the reservoir under investigation. This gives rise to the need for models to predict material characteristics at unsampled locations. The second problem stems from technological constraints that often limit the measurement of material properties to sample supports (i.e., sample volumes) much smaller than can be accommodated in current predictive models. This disparity in support requires measured data be averaged or upscaled to yield effective properties at the desired scale of analysis.
The concept of using "soft" geologic information to supplement often sparse "hard" physical data has received considerable attention.1,2 Successful application of this approach requires that some relationship be established between the difficult to measure material property (e.g., permeability) and that of a more easily observable feature of the geologic material. For example, Davis et al.3 correlated architectural-element mapping with the geostatistical characteristics of a fluvial/interfluvial formation in central New Mexico; Jordan and Pryor4 related permeability controls and reservoir productivity to six hierarchical levels of sand heterogeneity in a fluvial meander belt system; while Istok et al.5 found a strong correlation between hydraulic property measurements and visual trends in the degree of welding of ash flow tuffs at Yucca Mountain, Nevada. Phillips and Wilson6 mapped regions where the permeability exceeds some specified cutoff value and related their dimensions to the correlation length scale by means of threshold-crossing theory. Also, Journel and Alabert7 proposed a spatial connectivity model based on an indicator formalism and conditioned on geologic maps of observable, spatially connected, high-permeability features.
The description and quantification of heterogeneity is necessarily related to the issue of scale. It is often assumed that geologic heterogeneity is structured according to a discrete and disparate hierarchy of scales. For example, the hierarchical models proposed by Dagan8 and by Haldorsen9 conveniently classify heterogeneities according to the pore, laboratory, formation, and regional scales. This assumed disparity in scales allows parameter variations occurring at scales smaller than the modeled flow/transport process to be spatially averaged to form effective media properties,10–14 while large-scale variations are treated as a simple deterministic trend.2,15 However, natural media are not always characterized by a large disparity in scales as assumed above;16 but rather, an infinite number of scales may coexist,17–20 leading to a fractal geometry or continuous hierarchy of scales.21
Cosmological discordances. II. Hubble constant, Planck and large-scale-structure data sets
