Analysts commonly use the pressure derivative to identify flow regimes and well-test-interpretation models in pressure-transient test analysis. Analysts and commercial analysis software commonly use Bourdet's approach to calculate derivatives from measured pressure data. Bourdet's algorithm includes a weighted central-difference approximation with a certain "window" size based on an increment in the logarithm of time, represented by the symbol L. Although test analysts commonly select L in the range of 0.1 to 0.3 log cycle, no criterion is available for choosing the optimal value of L.An unresolved issue in pressure transient test analysis is how to determine the optimal L based on the data for each individual data set such that the data are smoothed sufficiently to remove noise that obscures the signal but not smoothed to the extent that the signal itself is changed.
This paper presents a new approach to calculate the pressure derivative, and improves the results compared to those we achieved in a previous paper, SPE 84471. This approach determines the optimal window size to use with Bourdet's algorithm by employing the fast Fourier transform, Gaussian filtering, and frequency-domain constraints. Our approach denoises the data in the frequency domain, determines the optimal window size, and, when coupled with Bourdet's algorithm in the time domain using the optimal window size, provides an improved pressure derivative. We also developed a novel adaptive smoothing algorithm by recursive differentiation-integration to further improve pressure derivative calculation. Our method can efficiently suppress measurement errors and produce smooth pressure derivatives from well-test data. Equally important, it can prevent over-smoothing of the data by inappropriate use of large window size, and it can preserve the characteristic behavior of the pressure derivative.
We validate our approach with a synthetic example and demonstrate its applicability to actual field examples.