Upscaling Two-Dimensional and Three-Dimensional Porosity Bounds by Using High-Resolution Micro-CT Images in Andesitic Rock Samples

Abstract The pore structure of volcanic rocks such as andesites typically shows a rather heterogeneous structure due to the origin of such rocks. The porosity exhibits a random distribution and fluctuates as the size of the sample increases, complicating its estimation at scales of interest. In this work, we examined the behavior of porosity fluctuations for five andesitic rock samples as a function of scale, and for the first time, two differential equations with initial conditions were formulated and solved analytically. With these equations we derive the mathematical expressions of the upper and lower bounds that describe the maximum and minimum limits within porosity fluctuations as a function of the scale. The equations only require a few input parameters, such as the maximum pore size, maximum solid phase size, and porosity of the characterized sample, along with two porosity measures (maximum and minimum) at a specific scale. The scope of this investigation is focused to the scales accessible throughout micro-CT imaging while providing potential implications for broader-scale applications. It is noteworthy that no previous works have provided mathematical expressions for these bounds. We performed a detailed porosity analysis by dividing binary multi-resolution and multi-scale micro-CT images into smaller images in order to extract an extensive dataset comprising hundreds of thousands of porosity data points. Modeling results show that rock heterogeneities directly affect the behavior of the porosity with the scale, showing more disperse fluctuations in the more heterogeneous samples but stabilizing and approaching a constant value assumed as the representative elementary volume (REV). Although it is not always reached within the investigated scales, all porosity fluctuations can be enveloped by two upscaling porosity bounds (upper and lower). This trend continues until a scale is reached at which either a REV may be identified, or the porosity progressively converges toward a stable value, at least for a range of scales where porosity variations become negligible.