A stable computation of log-derivatives from noisy drawdown data

Pumping tests interpretation is an art that involves dealing with noise coming from multiple sources and conceptual model uncertainty. Interpretation is greatly helped by diagnostic plots, which include drawdown data and their derivative with respect to log-time, called log-derivative. Log-derivatives are especially useful to complement geological understanding in helping to identify the underlying model of fluid flow because they are sensitive to subtle variations in the response to pumping of aquifers and oil reservoirs. The main problem with their use lies in the calculation of the log-derivatives themselves, which may display fluctuations when data are noisy. To overcome this difficulty, we propose a variational regularization approach based on the minimization of a functional consisting of two terms: one ensuring that the computed log-derivatives honor measurements and one that penalizes fluctuations. The minimization leads to a diffusion-like differential equation in the log-derivatives, and boundary conditions that are appropriate for well hydraulics (i.e., radial flow, wellbore storage, fractal behavior, etc.). We have solved this equation by finite differences. We tested the methodology on two synthetic examples showing that a robust solution is obtained. We also report the resulting log-derivative for a real case.