# Physics-Based Relationship for Pore Pressure and Vertical Stress Monitoring Using Seismic Velocity Variations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Velocity Change Due to Induced Stress

#### 2.2. Velocity Change Due to Surface Load and Pore Pressure

## 3. Model Validation

#### 3.1. Static Model

#### 3.2. Stress Model

^{3}and $g=9.8$ m/s

^{2}.

#### 3.3. Shear-Wave Velocity Change

#### 3.4. Surface-Wave Dispersion Forward Modeling

#### 3.5. Passive Image Interferometry

#### 3.6. Model Validation

## 4. Discussion

^{2}/s, would be limited to 0.1 °C. The surface waves, with frequencies we used, have no sensitivity to changes at depths shallower than 20 m (Figure 5). Furthermore, surface waves are sensitive to changes over a large range of depths, while the wavelength of the thermal “wave” is approximately 25 m, resulting in temperature contributions of a range of seasons. This would not be detectable using surface waves at frequencies lower than 2 Hz.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Stress-Induced Compressional-Wave Velocity Change

## Appendix B. Rotation Approximation

**Figure A1.**Comparison of applying rotation before or after cross-correlation using spectral normalization for different component combinations and for receiver pair G374–G434.

## References

- Rawlinson, N.; Fichtner, A.; Sambridge, M.; Young, M.K. Chapter One—Seismic Tomography and the Assessment of Uncertainty. Adv. Geophys.
**2014**, 55, 1–76. [Google Scholar] [CrossRef] - Sens-Schönfelder, C.; Wegler, U. Passive image interferometry and seasonal variations of seismic velocities at Merapi Volcano, Indonesia. Geophys. Res. Lett.
**2006**, 33. [Google Scholar] [CrossRef] - Brenguier, F.; Shapiro, N.M.; Campillo, M.; Ferrazzini, V.; Duputel, Z.; Coutant, O.; Nercessian, A. Towards forecasting volcanic eruptions using seismic noise. Nat. Geosci.
**2008**, 1, 126. [Google Scholar] [CrossRef] [Green Version] - Wegler, U.; Nakahara, H.; Sens-Schönfelder, C.; Korn, M.; Shiomi, K. Sudden drop of seismic velocity after the 2004 Mw 6.6 mid-Niigata earthquake, Japan, observed with Passive Image Interferometry. J. Geophys. Res. Solid Earth
**2009**, 114. [Google Scholar] [CrossRef] - Salvermoser, J.; Hadziioannou, C.; Stähler, S.C. Structural monitoring of a highway bridge using passive noise recordings from street traffic. J. Acoust. Soc. Am.
**2015**, 138, 3864–3872. [Google Scholar] [CrossRef] [PubMed] - Voisin, C.; Guzmán, M.A.R.; Réfloch, A.; Taruselli, M.; Garambois, S. Groundwater Monitoring with Passive Seismic Interferometry. J. Water Resour. Prot.
**2017**, 9, 1414–1427. [Google Scholar] [CrossRef] [Green Version] - Clements, T.; Denolle, M.A. Tracking groundwater levels using the ambient seismic field. Geophys. Res. Lett.
**2018**, 45, 6459–6465. [Google Scholar] [CrossRef] [Green Version] - Nakata, N.; Snieder, R. Estimating near-surface shear wave velocities in Japan by applying seismic interferometry to KiK-net data. J. Geophys. Res. Solid Earth
**2012**, 117. [Google Scholar] [CrossRef] [Green Version] - Rivet, D.; Brenguier, F.; Cappa, F. Improved detection of preeruptive seismic velocity drops at the Piton de La Fournaise volcano. Geophys. Res. Lett.
**2015**, 42, 6332–6339. [Google Scholar] [CrossRef] [Green Version] - Wang, Q.Y.; Brenguier, F.; Campillo, M.; Lecointre, A.; Takeda, T.; Aoki, Y. Seasonal crustal seismic velocity changes throughout Japan. J. Geophys. Res. Solid Earth
**2017**, 122, 7987–8002. [Google Scholar] [CrossRef] - Liu, C.; Aslam, K.; Daub, E. Seismic Velocity Changes Caused by Water Table Fluctuation in the New Madrid Seismic Zone and Mississippi Embayment. J. Geophys. Res. Solid Earth
**2020**, 125, e2020JB019524. [Google Scholar] [CrossRef] - Andajani, R.D.; Tsuji, T.; Snieder, R.; Ikeda, T. Spatial and temporal influence of rainfall on crustal pore pressure based on seismic velocity monitoring. Earth Planets Space
**2020**, 72, 1–17. [Google Scholar] [CrossRef] - Dost, B.; Ruigrok, E.; Spetzler, J. Development of seismicity and probabilistic hazard assessment for the Groningen gas field. Neth. J. Geosci.
**2017**, 96, s235–s245. [Google Scholar] [CrossRef] [Green Version] - Van Eijs, R.M.; van der Wal, O. Field-wide reservoir compressibility estimation through inversion of subsidence data above the Groningen gas field. Neth. J. Geosci.
**2017**, 96, s117–s129. [Google Scholar] [CrossRef] [Green Version] - Van Ginkel, J.; Ruigrok, E.; Herber, R. Using horizontal-to-vertical spectral ratios to construct shear-wave velocity profiles. Solid Earth
**2020**, 11, 2015–2030. [Google Scholar] [CrossRef] - Fokker, E.B.; Ruigrok, E.N. Quality parameters for passive image interferometry tested at the Groningen network. Geophys. J. Int.
**2019**, 218, 1367–1378. [Google Scholar] [CrossRef] - Zhou, W.; Paulssen, H. Compaction of the Groningen gas reservoir investigated with train noise. Geophys. J. Int.
**2020**, 223, 1327–1337. [Google Scholar] [CrossRef] - Brenguier, F.; Courbis, R.; Mordret, A.; Campman, X.; Boué, P.; Chmiel, M.; Takano, T.; Lecocq, T.; Van der Veen, W.; Postif, S.; et al. Noise-based ballistic wave passive seismic monitoring. Part 1: Body waves. Geophys. J. Int.
**2020**, 221, 683–691. [Google Scholar] [CrossRef] [Green Version] - Mordret, A.; Courbis, R.; Brenguier, F.; Chmiel, M.; Garambois, S.; Mao, S.; Boué, P.; Campman, X.; Lecocq, T.; Van der Veen, W.; et al. Noise-based ballistic wave passive seismic monitoring–Part 2: Surface waves. Geophys. J. Int.
**2020**, 221, 692–705. [Google Scholar] [CrossRef] - Tromp, J.; Trampert, J. Effects of induced stress on seismic forward modelling and inversion. Geophys. J. Int.
**2018**, 213, 851–867. [Google Scholar] [CrossRef] - Fjar, E.; Holt, R.M.; Raaen, A.; Horsrud, P. Petroleum Related Rock Mechanics; Elsevier: Amsterdam, The Netherlands, 2008. [Google Scholar]
- Dinoloket. Groundwater Research; Borehole Identification B08C0952. 2020. Available online: https://www.dinoloket.nl/ondergrondgegevens (accessed on 5 June 2020).
- KNMI. Netherlands Seismic and Acoustic Network. Royal Netherlands Meteorological Institute (KNMI), Other/Seismic Network, 1993. Available online: http://rdsa.knmi.nl/network/NL/ (accessed on 16 May 2020). [CrossRef]
- Kruiver, P.P.; van Dedem, E.; Romijn, R.; de Lange, G.; Korff, M.; Stafleu, J.; Gunnink, J.L.; Rodriguez-Marek, A.; Bommer, J.J.; van Elk, J.; et al. An integrated shear-wave velocity model for the Groningen gas field, The Netherlands. Bull. Earthq. Eng.
**2017**, 15, 3555–3580. [Google Scholar] [CrossRef] [Green Version] - Romijn, R. Groningen Velocity Model 2017; Technical Report; Nederlandse Aardolie Maatschappij: Assen, The Netherlands, 2017. [Google Scholar]
- Hawkins, R. A spectral element method for surface wave dispersion and adjoints. Geophys. J. Int.
**2018**, 215, 267–302. [Google Scholar] [CrossRef] - Park, C.B.; Miller, R.D.; Xia, J. Imaging dispersion curves of surface waves on multi-channel record. In Proceedings of the 1998 SEG Annual Meeting; Society of Exploration Geophysicists: Salt Lake City, UT, USA, 1998. [Google Scholar] [CrossRef]
- Wapenaar, K.; Slob, E.; Snieder, R.; Curtis, A. Tutorial on seismic interferometry: Part 2—Underlying theory and new advances. Geophysics
**2010**, 75, 75A211–75A227. [Google Scholar] [CrossRef] [Green Version] - Lobkis, O.I.; Weaver, R.L. Coda-wave interferometry in finite solids: Recovery of P-to-S conversion rates in an elastodynamic billiard. Phys. Rev. Lett.
**2003**, 90, 254302. [Google Scholar] [CrossRef] - Zhan, Z.; Tsai, V.C.; Clayton, R.W. Spurious velocity changes caused by temporal variations in ambient noise frequency content. Geophys. J. Int.
**2013**, 194, 1574–1581. [Google Scholar] [CrossRef] [Green Version] - Tsai, V.C. A model for seasonal changes in GPS positions and seismic wave speeds due to thermoelastic and hydrologic variations. J. Geophys. Res. Solid Earth
**2011**, 116. [Google Scholar] [CrossRef] [Green Version] - Hillers, G.; Campillo, M.; Ma, K.F. Seismic velocity variations at TCDP are controlled by MJO driven precipitation pattern and high fluid discharge properties. Earth Planet. Sci. Lett.
**2014**, 391, 121–127. [Google Scholar] [CrossRef] - Mao, S.; Campillo, M.; van der Hilst, R.D.; Brenguier, F.; Stehly, L.; Hillers, G. High Temporal Resolution Monitoring of Small Variations in Crustal Strain by Dense Seismic Arrays. Geophys. Res. Lett.
**2019**, 46, 128–137. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Map view of the locations of the measurement equipment employed in this study. The location of the piezometer B08C0952 [22] is plotted as a blue point, around which the blue circle indicates a 10 km radius. All geophones within this radius at 200 m depth [23] are shown as black triangles. The outline of the Netherlands and the Groningen gas field are shown as black and red lines.

**Figure 2.**Static models: (

**a**) Compressional-wave velocity $\alpha $, (

**b**) shear-wave velocity $\beta $, (

**c**) mass density $\rho $, (

**d**) bulk modulus $\kappa =\rho {\alpha}^{2}-\frac{4}{3}\rho {\beta}^{2}$, (

**e**) shear modulus $\mu =\rho {\beta}^{2}$, (

**f**) confining pressure $P={\int}_{0}^{z}\rho \left(z\right)g$ $dz$, with g the gravitational acceleration and z the depth below surface, (

**g**) the shear-modulus pressure derivative ${\mu}^{\prime}=d\mu /dP$, based on the smoothed derivative of the shear modulus with respect to confining pressure.

**Figure 3.**(

**a**) Time-lapse changes in pressure head $dh({z}_{i},t)$ with respect to the average pressure head between 1 January 2017 and 31 December 2019, for ${z}_{i}$ = $7.3$, $27.3$, $105.3$, $132.3$, and $170.8$ m depths. (

**b**) Induced pore pressure ${u}^{0}(z,t)$ = ${\rho}_{w}g\phantom{\rule{2.84544pt}{0ex}}dh(z,t)$ for parameters ${\rho}_{w}$ = 1000 kg/m

^{3}and g = 9.8 m/s

^{2}, obtained from linear interpolation of $dh({z}_{i},t)$. The dashed lines indicate the measurement depths of the pressure head. (

**c**) Estimate of induced vertical compressional stress ${T}_{33}^{0}=-\varphi {\rho}_{w}g\phantom{\rule{2.84544pt}{0ex}}dh(z=7.3$ m$,t)$ for $\varphi \approx 0.25$.

**Figure 5.**Absolute value of the sensitivity kernels for fundamental Rayleigh and Love waves for density, compressional-wave velocity and shear-wave velocity at a frequency of 1 Hz.

**Figure 6.**Eigen-mode amplitudes as function of depth for Rayleigh waves (amplitudes of both the horizontal and vertical components normalized to preserve amplitude ratios) and Love waves at 1 Hz.

**Figure 7.**Reference cross-coherence ${H}_{ref}$ (i.e., average cross-coherence between 1 January 2017 and 31 December 2019) for all combinations of receivers shown in Figure 1 as a function of receiver-pair distance for components (

**a**) RR, (

**b**) ZZ and (

**c**) TT. The red lines indicate the arrival times $\left|t\right|=x/300+5$, between which we expect all arrivals of direct surface waves, while the red area indicates the coda time windows $(x/300+5)$ s $<\left|t\right|\le 100$ s, used in Equation (15) to retrieve relative velocity change.

**Figure 8.**Multichannel analysis of surface waves [27] of the reference cross-coherences shown in Figure 7 for components (

**a**) RR, (

**b**) ZZ, and (

**c**) TT, visualized in a power plot. The red lines indicate the fundamental dispersion curves of (

**a**,

**b**) Rayleigh and (

**c**) Love waves, obtained with the adjoint method [26].

**Figure 9.**(

**a**) Modeled surface-wave velocity changes, for example, frequency 1 Hz as a function of date, and (

**b**) for the example date 31 August 2019 as a function of frequency. The individual lines represent velocity change in Love waves (black), Rayleigh waves (red) and their Voigt average ${(dv/v)}_{Voigt}=\frac{2}{3}{(dv/v)}_{Rayleigh}+\frac{1}{3}{(dv/v)}_{Love}$ (blue).

**Figure 10.**Time-lapse seismic velocity change retrieved from seismic ambient noise within frequency bandwidth [0.1 0.2] Hz, using the coda of the cross-coherence of horizontal (RR, RT, TR, TT) components between 78 receiver combinations. The background colors show the probability distribution of 312 estimates from dark blue (low probability) to yellow (high probability), while the black line shows the average velocity change.

**Figure 11.**Time–lapse seismic velocity change retrieved from seismic ambient noise within frequency bandwidth [1.0 1.2] Hz, using the coda of the cross-coherence of individual horizontal component configurations, averaged over 78 receiver combinations.

**Figure 12.**Time–lapse seismic velocity change retrieved from seismic ambient noise within the indicated frequency bandwidths, using the coda of the cross-coherence of vertical (left; ZZ) and horizontal (right; RR, RT, TR, TT) components between 78 receiver combinations. The background colors show the probability distribution of 78 (left) and 312 (right) estimates from dark blue (low probability) to yellow (high probability), while the black line shows the average velocity change. The purple and red lines show the results of the independent forward model for velocity change of Rayleigh waves (

**a**,

**c**,

**e**,

**f**,

**g**,

**h**) and the Voigt average (

**b**,

**d**) as in Figure 9. The match between the low-frequency trends of the modeled and the observed velocity change was quantified with a Pearson correlation after a low-pass filter was applied with a cut-off period of 60 days. The correlation coefficient R is shown in the legend.

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**MDPI and ACS Style**

Fokker, E.; Ruigrok, E.; Hawkins, R.; Trampert, J.
Physics-Based Relationship for Pore Pressure and Vertical Stress Monitoring Using Seismic Velocity Variations. *Remote Sens.* **2021**, *13*, 2684.
https://doi.org/10.3390/rs13142684

**AMA Style**

Fokker E, Ruigrok E, Hawkins R, Trampert J.
Physics-Based Relationship for Pore Pressure and Vertical Stress Monitoring Using Seismic Velocity Variations. *Remote Sensing*. 2021; 13(14):2684.
https://doi.org/10.3390/rs13142684

**Chicago/Turabian Style**

Fokker, Eldert, Elmer Ruigrok, Rhys Hawkins, and Jeannot Trampert.
2021. "Physics-Based Relationship for Pore Pressure and Vertical Stress Monitoring Using Seismic Velocity Variations" *Remote Sensing* 13, no. 14: 2684.
https://doi.org/10.3390/rs13142684