A B‐Spline Framework for Smooth Derivative Computation in Well Test Analysis Using Diagnostic Plots

Abstract

In the oil and gas industry, well test analysis using derivative plots, has been the core technique in examining reservoir and well behavior over the last three decades. Recently, diagnostics plots have gained recognition in the field of hydrogeology; however, this tool is still underused by groundwater professionals. The foremost drawback is that the derivative function must be computed from noisy field measurements, usually based on finite-difference schemes, which complicates the analysis. We propose a B-spline framework for smooth derivative computation, referred to as Constrained Quartic B-Splines with Free Knots. The approach presents the following novelties in relation to methodological precedents: (1) the use of automatic equality derivative constraints, (2) a knot removal strategy and (3) the introduction of a Boolean shape parameter that defines the number of initial knots to choose. These can lead to evaluate both simple (manually recorded drawdown measurements) and complex (transducer measured records) datasets. Furthermore, we propose an additional shape preserving smoothing preprocess procedure, as a simple, fast and robust method to deal with extremely noisy signals. Our framework was tested in four pumping tests by comparing the spline derivative with regards to the Bourdet algorithm, and we found that the latter is rather noisy (even for large differentiation intervals) and the second derivative response is basically unreadable. In contrast, the spline first and second derivative led to smoother responses, which are more suitable for interpretation. We concluded that the proposed framework is a welcome contribution to evaluate reliable aquifer tests using derivative-diagnostic plots.

Improving hydrogeological understanding through well-test interpretation by diagnostic plot and modelling: a case study in an alluvial aquifer in France

“…Bourdet's algorithm was adopted in this study to smooth the logderivative signal although studies have assessed its limitations regarding noisy signal (Lane et al 1991;Escobar et al 2004). The B-spline algorithm presents valuable improvements on the smoothing of the diagnostic plot and the use of the second-derivative response might assess the nonuniqueness problem of the diagnostic plot (Tago and Hernández-Espriú 2018). Here, the nonuniqueness issue is tackled down using four plausible analytical solutions, and the main argument to choose one over the others is a homogeneous and plausible set of hydraulic parameters.…”