Estimating Fractal Dimension as a Spatially Correlated Pore Structure Heterogeneity Measure from Rate-Controlled Capillary Pressure Curves
Authors: Daniels Jeffrey K. , Myers Michael T. , Hathon Lori A. 
Published: October 2023

Abstract
Pore structure heterogeneity is present in reservoir rocks at multiple length scales, which makes it a challenge to optimally assess and integrate into digital rock and pore-scale models, especially for complex reservoir rocks. The fractal nature of reservoir rocks causes variation in their physical properties over multiple length scales. The fractal dimension governs the power law scaling of fractals and has been estimated from experimental measurements and rock images of the pore space to quantify pore structure heterogeneity. Each experimental technique and imaging modality has limitations on the pore structure characteristics and the level of detail it can provide, necessitating combining them for comprehensive pore structure characterization. However, challenges in spatially correlating pore structure heterogeneity at multiple length scales remain.
An Apparatus for Pore Examination (APEX), with the highest known pressure and volumetric resolutions (5E-6 psi and 1.3E-10 cc), is proposed to make high-resolution rate-controlled capillary pressure measurements, which reflect comprehensive pore structure and fractal characteristics of the rock. Detrended fluctuation analysis (DFA) of the APEX capillary pressure curve estimates a fractal dimension to describe the spatial correlation in pore structure heterogeneity quantitatively. The rock samples analyzed were approximately 0.5-inch in diameter and 0.5-inch long right circular cylindrical core plugs of the Berea sandstone and Indiana limestone.
Amplitude spectra of the APEX capillary pressure curves indicated they were "1/fβ" scaling signals (fractional noises) with self- affine fractal properties and power law correlated statistics. Fractal dimension estimates for the pore structure of both rock samples from the APEX capillary pressure curves and thin section images showed agreement, with lesser than 10% relative differences. Additionally, the fractal dimension estimates agreed (within a 10 % relative difference) with published Berea sandstone and Indiana limestone results from SEM and thin section images. Detrended fluctuation analysis (DFA) of the APEX capillary pressure curves showed that the Berea sandstone had a single pore system with short-range power-law correlated pore structure statistics, indicated by a fractal dimension, D = 2.533. The fractal dimension and amplitude spectrum showed a relatively well-connected pore space with mild pore structure heterogeneity at the pore scale. The Indiana limestone had two pore systems with short-range power-law correlated pore structure statistics indicated by two fractal dimensions, D= 2.735 and D = 2.911. The fractal dimension and amplitude spectrum indicated a poorly connected pore space with smaller pores connecting the larger pores. The results presented in this study showed that high-resolution APEX capillary pressure measurements reflect the fractal characteristics of a reservoir rock's pore structure. In this context, fractal dimensions can be estimated from high-resolution APEX capillary pressure measurements to describe spatial correlation in pore structure heterogeneity quantitatively. The stochastic fractal functions, fractional Brownian motion (fBm) and Lévy Flights can describe the spatial correlation in pore structure heterogeneity of the Berea sandstone and Indiana limestone, respectively.
The results can be used to integrate spatially correlated pore structure heterogeneity at the pore and core scales in computational rock models to enhance their predictive capabilities. They can also complement the results from techniques of quantifying heterogeneity in reservoir properties with significant pore structure dependencies, which do not account for their spatial correlation.