Aquifer Characterization and Uncertainty in Multi‐Frequency Oscillatory Flow Tests: Approach and Insights

Abstract

Characterizing aquifer properties and their associated uncertainty remains a fundamental challenge in hydrogeology. Recent studies demonstrate the use of oscillatory flow interference testing to characterize effective aquifer flow properties. These characterization efforts relate the relative amplitude and phase of an observation signal with a single frequency component to aquifer diffusivity and transmissivity. Here, we present a generalized workflow that relates extracted Fourier coefficients for observation signals with single and multiple frequency components to aquifer flow properties and their associated uncertainty. Through synthetic analytical modeling we show that multi-frequency oscillatory flow interference testing adds information that improves inversion performance and decreases parameter uncertainty. We show increased observation signal length, sampling frequency, and pressure sensor accuracy all produce decreased parameter uncertainty. This work represents the first attempt we are aware of to quantify effective aquifer parameters and their associated uncertainty using multi-frequency oscillatory flow interference testing.

Do Simple Analytical Models Capture Complex Fractured Bedrock Hydraulics? Oscillatory Flow Tests Suggest Not

“…Furthermore, we collected a time series that is at least 10 full cycles in length for each oscillatory flow test, which generated at least five full cycles available for analysis and avoided any transience associated with testing "ramp-up" or "ramp-down." This approach balanced the trade-off in minimizing uncertainty in the estimated parameters and duration of the field experiments (Bakhos et al 2014;Patterson and Cardiff 2022).…”

“…In this section we describe our largely automated data processing and analysis workflow used to analyze the head change signals collected during an individual oscillatory flow test to generate estimated fracture flow parameters with uncertainty. To maintain clarity throughout the manuscript we adopt the language presented by Patterson and Cardiff (2022), where "observation signal" refers to the timeseries of recorded head changes with noise at an observation well, "data" refers to the extracted Fourier coefficients estimated through linear signal processing techniques, "error" refers to the uncertainty in the estimated Fourier coefficients as a result of noise in the observational signal, and "uncertainty" refers to the 95% confidence interval associated with fracture flow parameter estimates determined through linearized error propagation. We applied the gradient-based inversion approach described below to optimize parameters associated with all conceptual models described later: the ideal fracture analysis, the leaky fracture analysis, and borehole storage-impacted analysis.…”

Stiff, Smooth, and Solid? Complex Fracture Hydraulics' Imprint on Oscillatory Hydraulic Testing

“…We applied the numerical gradient-based inversion strategy described by Patterson and Cardiff (2022) that best fit the simulated head phasors. This inversion strategy employs a Levenberg-Marquardt algorithm under a Bayesian framework to find the parameters that minimize the objective function given by:…”

“…Across all oscillation periods, the numerically simulated head amplitudes matched the analytical head amplitudes within 1 mm or less. This level of numerical modeling error cannot be differentiated from data measurement error of commonly employed head change sensors (Leven & Barrash, 2022; Patterson & Cardiff, 2022).…”

“…We applied the numerical gradient‐based inversion strategy described by Patterson and Cardiff (2022) to estimate the effective fracture transmissivity $$\left({T}_{\mathfrak{f}}\right)$$ and storativity $$\left({S}_{\mathfrak{f}}\right)$$ that best fit the simulated head phasors. This inversion strategy employs a Levenberg‐Marquardt algorithm under a Bayesian framework to find the parameters that minimize the objective function given by: $$\begin{align*}\underset{\mathrm{s}}{\min }\frac{1}{2}{({\Phi }-h(\mathbf{s}))}^{T}{\mathbf{R}}^{-1}({\Phi }-h(\mathbf{s}))\end{align*}$$ where Φ is the observed data (i.e., head phasor), h ( s ) is the forward model that takes flow parameters as inputs and returns the steady‐periodic head phasor, and R = σ 2 I n is the data error covariance matrix, where I n is the identity matrix and n is the number of phasor coefficients (i.e., data points) used during inversion.…”