Compatible Scales
Compatible scales, in petroleum reservoir engineering and petrophysics, refers to the concept and practice of ensuring that measurements made at different spatial scales — from core plug measurements (centimeter to decimeter scale), to wireline log measurements (decimeter to meter scale), to well test measurements (10-100 meter scale), to seismic measurements (meter to kilometer scale), to production data (field scale) — are properly upscaled, downscaled, or otherwise reconciled before being combined in a reservoir model or used to calibrate one measurement against another, recognizing that a property measured at the scale of a core plug (say, permeability of a 1-inch diameter plug over 1-2 inches of length) is not numerically equivalent to the effective permeability of the same formation as measured by a well test (which averages the heterogeneous permeability over hundreds of thousands of times the volume of the core plug), and that using plug-scale measurements directly as input to a reservoir simulation grid cell (which may represent millions of times the plug volume) produces a systematic bias in the simulation that cannot be corrected after the fact without proper upscaling; compatible scales is both a petrophysical concept (ensuring that log-derived properties are consistent with core-measured properties at the same measurement scale before the log calibration is used to predict properties throughout the uninstrumented wellbore) and a reservoir modeling concept (ensuring that the grid cell properties in the simulation model correctly represent the volume-averaged effective properties of the heterogeneous rock at the simulation cell scale).
Key Takeaways
- The scale problem in reservoir characterization arises from the fundamental heterogeneity of reservoir rock at all scales — from the pore scale (micrometer), through the grain scale (millimeter), the lamination scale (centimeter), the bed scale (decimeter to meter), the parasequence scale (meter to tens of meters), the facies association scale (tens to hundreds of meters), and the depositional system scale (kilometers) — with different reservoir properties (porosity, permeability, capillary pressure, wettability) varying at each scale by different amounts: porosity is relatively slowly varying with scale (a core plug porosity of 20% from a 1-inch plug is often a reasonable approximation of the bed-average porosity over a 1-meter interval if the plug was representative of the interval), while permeability varies much more rapidly (a single plug permeability in a heterogeneous sandstone can be 1,000 millidarcy in a high-quality lamination and 0.1 millidarcy in an adjacent tight lamination separated by 1 centimeter, while the effective vertical permeability of the layered interval measured by a well test may be the harmonic mean of the two values, much closer to 0.1 millidarcy than to 1,000 millidarcy); the permeability scale dependence means that any reservoir model that uses single-plug permeabilities without upscaling will systematically over-predict vertical flow rates (if the highest-permeability plugs are selected for testing, which is common) and mis-predict the volumetric sweep of waterflooding (if the vertical permeability heterogeneity is not captured at the appropriate scale).
- Core-to-log calibration requires scale-compatible measurements: wireline logs measure a physical property over a vertical resolution of 0.5-2 feet (sonic and density logs) to 2-6 feet (resistivity tools) and a radial depth of investigation of several inches to several feet into the formation, while core plug measurements are made over 1-2 inches of plug length and a volume of less than 20 cubic centimeters; when a core plug permeability of 500 millidarcy is measured from a 1-inch plug taken from a fine laminated sandstone where a 2-mm high-permeability lamina happens to be the entire thickness of the plug, the wireline log over the same 2-foot interval averages both the high-permeability lamina and the adjacent low-permeability mudstone, producing a log response (bulk density, sonic velocity, resistivity) that reflects the combined interval properties rather than the plug's properties alone; calibrating the log to the plug permeability in this case would produce a systematic overestimate of the log-predicted permeability because the log averages over a volume that includes the mudstone, while the plug sampled only the sandstone; scale-compatible core-log calibration requires either averaging the log response over the precise interval sampled by the core plug (using high-resolution logs with sampling intervals close to the plug length), or averaging multiple closely-spaced core plug measurements to represent the log-scale interval, or both; the failure to apply scale-compatible core-log calibration is a common source of systematic bias in permeability prediction from wireline logs in heterogeneous reservoirs.
- Permeability upscaling from the core plug scale to the reservoir simulation grid cell scale is necessary because simulation grid cells in practical reservoir models (10-50 meters in the areal direction, 0.5-5 meters in the vertical direction for a typical geological model) represent rock volumes orders of magnitude larger than the plugs from which the permeability data is derived: the effective permeability of a grid cell containing a heterogeneous sequence of laminations with different plug permeabilities depends on the spatial arrangement of the laminations relative to the flow direction — effective horizontal permeability is approximately the arithmetic mean of the layer permeabilities (weighted by thickness) for flow parallel to the layering, while effective vertical permeability is approximately the harmonic mean of the layer permeabilities for flow perpendicular to the layering; the arithmetic mean overestimates the effective permeability by a factor that depends on the degree of heterogeneity (the coefficient of variation of the permeability distribution), while the harmonic mean underestimates it similarly; in practice, the effective permeability of a heterogeneous grid cell lies between the harmonic and arithmetic means, closer to the geometric mean for isotropic permeability distributions, and is most accurately calculated by numerical flow simulation of the fine-scale model (geological model with lamination-scale cells) to determine the upscaled effective permeability that produces the same flow behavior in the coarse simulation model; this numerical upscaling approach is the standard method for generating simulation-scale permeability from geological model-scale permeability in complex heterogeneous reservoirs.
- Well test permeability and core permeability are frequently different by a factor of 2-10 even in apparently simple reservoirs, and the scale effect (the well test averages over a much larger rock volume than the core plugs) is one of the main reasons for this discrepancy: well test permeability (derived from the pressure transient analysis of a build-up or falloff test) represents the effective permeability of the formation over the drainage area of the test (which may be 50-500 meters in radius from the wellbore), averaging over all the heterogeneity within that radius; core plug permeability represents the local permeability at the specific plug location, which may be systematically biased relative to the test-scale average by sampling bias (plugs are more easily selected from well-consolidated, high-permeability intervals), by fracture contribution (well tests in fractured reservoirs include fracture permeability that is not measured by plugs unless the plug happens to sample a fracture), and by flow direction (plug measurements are typically horizontal, while the well test measures the effective permeability for radial flow which has both horizontal and vertical components); the ratio of well test permeability to average core plug permeability — the core-to-test permeability ratio — is a diagnostic of the heterogeneity type (low ratio in layered systems with high vertical permeability contrast, high ratio in fractured systems where the fractures dominate the well test response but are missed by core plugs), and reconciling this ratio quantitatively is a key step in building a reservoir model that is compatible with both the core data and the dynamic production data.
- Seismic-scale properties (acoustic impedance from seismic inversion, seismically derived porosity from rock physics transforms, seismically interpreted facies) have a resolution of 10-30 meters vertically and 25-100 meters laterally (controlled by the seismic wavelength and the acquisition geometry), and must be compared to and constrained by well log properties averaged to the same scale (not to the original 0.5-foot log sampling interval) before the seismic attributes can be calibrated to reservoir properties: a seismic amplitude or impedance value at a specific gridpoint represents the effective property of the 25x25x20 meter voxel centered on that point, averaged over the seismic wavelet in depth and the Fresnel zone in the horizontal plane; calibrating this seismic value directly to the 0.5-foot log sample at the nearest well location gives a scale-incompatible calibration that will produce incorrect predictions of reservoir properties from seismic data in the interwell areas; the correct procedure is to block (average) the log properties over the depth interval corresponding to the seismic resolution (typically 15-25 meters for the dominant wavelength at the target depth), compute the synthetic seismic response from the blocked logs using a forward convolutional model, compare the synthetic to the actual seismic trace at the well location (the well-to-seismic tie), and use the blocked log properties as the calibration data for the seismic attribute-to-reservoir property transform; this scale-compatible calibration ensures that the seismic-derived reservoir property maps are physically consistent with the log-derived properties at the same scale.
Fast Facts
The formal recognition of the scale problem in reservoir characterization and the development of systematic upscaling methods became a major focus of reservoir engineering research in the 1980s and 1990s, driven by the computational capability of reservoir simulation that allowed engineers to build models of increasing detail but simultaneously confronted them with the fundamental question of what properties to assign to each simulation cell. Akhil Datta-Gupta, Larry Lake, and colleagues at the University of Texas at Austin, and parallel research groups at Stanford's Petroleum Research Institute (now part of the Stanford Center for Earth Resources Forecasting), developed the mathematical framework for permeability upscaling that remains the basis of current commercial reservoir simulation software. The resolution of the scale problem in practice — ensuring that a field-scale reservoir model correctly represents core-scale heterogeneity for flow simulation — remains one of the most technically challenging and practically important problems in applied reservoir characterization.
What Are Compatible Scales?
Compatible scales is the principle that a measurement only calibrates another measurement if both are measuring the same volume of rock. A core plug permeability of 500 millidarcy and a well test permeability of 50 millidarcy are not contradictory measurements of the same property — they are measurements of the same physical property at different scales, where the heterogeneity of the formation produces different effective values at different measurement scales. The plug measured a 1-inch cylinder of the best quality sandstone in the interval. The well test averaged over hundreds of meters of rock in all directions, including the tight laminations, the clay-filled zones, and the vertical barriers to flow that the plug did not encounter. Neither measurement is wrong. Both are correct for their scale. The problem arises when the plug value is used directly as input to a reservoir simulation cell that represents the same volume as the well test drainage area — a scale mismatch that systematically biases the simulation by ignoring the heterogeneity that the well test saw and the plug did not. Compatible scales is the principle that prevents this mismatch: use measurements at each scale to constrain the properties at that scale, and use explicit upscaling or downscaling to transfer information between scales in a physically consistent way.