Smooth_log_derivative
Smooth-log derivative code allow to smooth the hydraulic test measurements to draw diagnostic plots.
Description
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.
Verification
We tested the methodology on two synthetic examples showing that a robust solution is obtained (Ramos et al 2017). It has been tested too in the hydraulic characterization of three real sites:
- Trabucchi et al (2018) in a salt flat,
- Del Val et al (2020, in review) in a coastal aquifer,
- Martinez-Landa et al. (2020 in review) in a deep aquifer. Aditional data are availeble for HT1-test and HT2-test.
