Improved Identification of Reservoir Heterogeneity with Advanced Derivative Profiles in Relation to Radii of Investigation
Authors: Rahman N M Anisur , Nooruddin Hasan A , Sarac Sukru 
Published: September 2022

Abstract
The identification of reservoir heterogeneity is normally attempted by constructing profiles of conventional well-test derivatives with transient-pressure data. Second-order derivatives are more sensitive than the first-order derivatives in the respective profiles with heterogeneity. Under actual operating conditions, the identification of heterogeneity is more complex than under ideal conditions because of the presence of noise. This study investigates how the degree of heterogeneity and the level of noise in the data strike a balance in identifying heterogeneity.
In this study, different levels of premeditated, random noise are applied to the ideal pressure data to examine their impacts on the profiles of the first- and second-order derivatives. The resulting distortions are analyzed for clarity in identifying the known heterogeneity located at the predefined distances. Ideal pressure data is generated from well-behaved analytical solutions for flow through porous media with various levels of heterogeneity. Comparisons are made between the ideal and the distorted derivative profiles to assess the impact of noise in identifying heterogeneity. This also allows the determination of the radius of investigation with respect to the observed derivative profiles.
The second-order derivatives enable a high-sensitivity analysis by offering features of variation in the derivative profiles as a function of time, even at low levels of reservoir heterogeneity. These features, however, may get smoothened out in the corresponding derivative profile of the actual data due to the presence of external noise in the raw pressure data. The impact of the noise in the raw data, and the subsequent, unintentional smoothening of the derivative profile depend on the level of reservoir heterogeneity that is being sought. This is also illustrated in the profiles of the first-order (e.g., primary pressure, or well-test or Bourdet) derivatives. Such a smoothening tendency limits the analyst's ability to identify small scales of reservoir heterogeneity in the corresponding derivative profiles. The second-order derivative profiles demonstrate more sensitivity than those of the first-order derivatives, when encountering reservoir heterogeneity. By identifying the heterogeneity at a known distance in the derivative profiles, a relationship with the radius of investigation with time has been established. Applying a de-noising technique to the noisy data restores some of the features in the derivative profile and helps estimate a more reliable magnitude of the radius of investigation.
This study offers opportunities to analyze noisy responses in heterogeneous reservoirs by providing a relationship between the second-order derivative profile and the quantity of noise. As a result of the premeditated noise in the pressure data, the magnitudes of the estimated radius of investigation also get impacted. We have been able to quantify this impact by taking advantage of the known quantity of noise in a given case.