# kitagawa-package: Spectral Response of Water Wells to Harmonic Strain and... In kitagawa: Spectral Response of Water Wells to Harmonic Strain and Pressure Signals

## Description

Provides tools to calculate the theoretical hydrodynamic response of an aquifer undergoing harmonic straining or pressurization. There are two classes of models here: (1) for sealed wells, based on the model of Kitagawa et al (2011), and (2) for open wells, based on the models of Cooper et al (1965), Hsieh et al (1987), Rojstaczer (1988), Liu et al (1989), and Wang et al (2018). These models treat strain (or aquifer head) as an input to the physical system, and fluid-pressure (or water height) as the output. The applicable frequency band of these models is characteristic of seismic waves, atmospheric pressure fluctuations, and solid earth tides. The Wang et al (2018) can explicitly model vertical leakage.

## Details

The following functions provide the primary features of the package:

well_response and open_well_response, which take in arguments for well- and aquifer-parameters, and the frequencies at which to calculate the response functions. They both access the constants-calculation routines as necessary, meaning the user need not worry about those functions (e.g., alpha_constants).

Helper functions:

sensing_volume

can be used to compute the sensing volume of fluid, for the specified well dimensions.

## Scientific background

The underlying model is based upon the assumption that fluid flows radially through an homogeneous, isotropic, confined aquifer.

The underlying principle is as follows. When a harmonic wave induces strain in a confined aquifer (one having aquitards above and below it), fluid flows radially into, and out of a well penetrating the aquifer. The flow-induced drawdown, s, is governed by the following partial differential equation, expressed in radial coordinates(r):

\frac{\partial^2 s}{\partial r^2} + \frac{1}{r} \frac{\partial s}{ \partial r} - \frac{S}{T}\frac{\partial s}{\partial t} = 0

where S, T are the aquifer storativity and transmissivity respectively.

The solution to this PDE, with periodic discharge boundary conditions, gives the amplitude and phase response we wish to calculate. The solution for an open well was presented by Cooper et al (1965), and subsequently modified by Liu et al (1989). Wang et al (2018) modified this solution for the leaky aquifer case. Kitagawa et al (2011) adapted the solution of Hsieh et al (1987) for the case of a sealed well. When there is no leakage, Wang et al (2018) is equivalent to Hsieh et al (1987).

These models are applicable to any quasi-static process involving harmonic, volumetric strain of an aquifer (e.g. passing Rayleigh waves, or changes in the Earth's tidal potential). In practice, however, the presence of permeable fractures can violate the assumption of isotropic permeability, which may substantially alter the response by introducing shear-strain coupling. But these complications are beyond the scope of this model.

## Author(s)

Andrew J. Barbour <andy.barbour@gmail.com> with contributions from Jonathan Kennel

## References

Abramowitz, M. and Stegun, I. A. (Eds.). "Kelvin Functions." \S 9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972.

Cooper, H. H., Bredehoeft, J. D., Papadopulos, I. S., and Bennett, R. R. (1965), The response of well-aquifer systems to seismic waves, J. Geophys. Res., 70 (16)

Hsieh, P. A., J. D. Bredehoeft, and J. M. Farr (1987), Determination of aquifer transmissivity from Earth tide analysis, Water Resour. Res., 23 (10)

Kitagawa, Y., S. Itaba, N. Matsumoto, and N. Koisumi (2011), Frequency characteristics of the response of water pressure in a closed well to volumetric strain in the high-frequency domain, J. Geophys. Res., 116, B08301

Liu, L.-B., Roeloffs, E., and Zheng, X.-Y. (1989), Seismically Induced Water Level Fluctuations in the Wali Well, Beijing, China, J. Geophys. Res., 94 (B7)

Roeloffs, E. (1996), Poroelastic techniques in the study of earthquake-related hydrologic phenomena, Advances in Geophysics, 37

Wang C.-Y., Doan, M.-L., Xue, L., Barbour, A. (2018), Tidal Response of Groundwater in a Leaky Aquifer—Application to Oklahoma, Water Resour. Res., 54 (10)

open_well_response, well_response, sensing_volume, wrsp-methods