- Express Letter
- Open Access

# Topographic effects on crustal stress around the Atera Fault, central Japan

- Koki Kumagai
^{1}Email author and - Takeshi Sagiya
^{1, 2}

**Received:**30 August 2018**Accepted:**21 November 2018**Published:**28 November 2018

## Abstract

## Keywords

- Atera fault
- Boussinesq problem
- Topographic stress
- Hydraulic fracturing test

## Introduction

Yamashita et al. (2010) investigated the crustal stress condition around the Atera fault by means of hydraulic fracturing of boreholes. They reported that the maximum principal stress axes are oriented in the N–S direction, favoring a right-lateral motion of the Atera fault. They also presented GPS velocity data to illustrate a right-lateral motion around the fault. Based on these observations, Yamashita et al. (2010) concluded that the Atera fault recently underwent a right-lateral motion. This conclusion contradicts the seismologically estimated stress field (Terakawa and Matsu’ura 2010) as well as long-term geologic features of the Atera fault. Therefore, their conclusion needs to be reconsidered carefully if there are other factors affecting the stress condition. As one of such possibilities, topographical effects in the crustal stress has been studied (e.g., Liu and Zoback 1992; Luttrell et al. 2011) and proven to be dominant at a shallow depth (Liu and Zoback 1992). Since the Atera fault is situated in a mountainous region with a complex topography, the topographical effects on the crustal stress are worth considering.

In this study, we demonstrate that the N–S compressional stress obtained from a hydraulic fracturing mainly reflects a topographic effect. We conduct a numeral computation of topographic stress based on a formulation of the Boussinesq problem (Boussinesq 1885; Harding and Sneddon 1945). Our calculation shows that the topographic effect is effective in a shallow part of the crust, significantly changing the stress condition from the tectonic stress field. At the same time, the topographic effect is considered to be negligible at the seismogenic depth below a few kilometers.

## Method

Physical constants for model calculation

Density (kg/m | Poisson’s ratio | Gravitational acceleration (m/s |
---|---|---|

2.7 × 10 | 0.25 | 9.8 |

The area of stress evaluation is denoted as a smaller blue rectangle shown in Fig. 1. This rectangular area includes three sites for hydraulic fracturing by Yamashita et al. (2010). Within this area, the stress is evaluated at a horizontal regular grid with 200 m intervals. The vertical grid intervals are 100 m and 1,000 m above and below the depth of 1300 m, respectively. We also evaluate the topographic stress at the in situ hydraulic fracturing sites by Yamashita et al. (2010). We calculate three horizontal stress components \(\tau_{xx} ,\tau_{yy} ,\tau_{xy}\) and convert them into principal stress axes as well as differential stress \(\tau_{{{\mathrm{diff}}}} = \tau_{{H_{{\mathrm{max} }} }} - \tau_{{H_{{\mathrm{min} }} }}\) at each calculation point.

## Results

Calculation results at depth of HFT carried out and Yamashita et al.’s (2010) results

Depth from ground surface (m) | Altitude (m) | \(\tau_{{H_{{\mathrm{max}}} }}\) (MPa) | \(\tau_{{H_{{\mathrm{min} }} }}\) (MPa) | Diff. (MPa) |
| |
---|---|---|---|---|---|---|

UEN | ||||||

HFT | 347 | 189 | 15.1 | 9.2 | 5.9 | N3.5W |

Calculation | 189 | 8.8 | 5.6 | 3.2 | N1.2W | |

FKO | ||||||

HFT | 243–276 | 106–139 | 12–16.2 | 8.9–10.0 | 2.9–6.2 | N22.5W–N17.3E |

Calculation | 119 | 8.0 | 6.4 | 1.6 | N24.6W | |

HTJ | ||||||

HFT | 249–266.3 | 61–78 | 19.4–24.5 | 11.8–14.2 | 7.6–10.3 | N20.5W–N11.7E |

Calculation | 70 | 9.0 | 7.9 | 1.1 | N36.5W |

The calculated stress orientations are mostly concordant with the hydraulic fracturing measurements. On the other hand, the calculated stress values are less than the measurement by about 40% on average. We discuss a possible cause of the discrepancy later.

## Discussion

Our calculation of topographic stress is consistent with the hydraulic fracturing results in their orientation, but the calculated stress value is smaller by about 40%. It is rather a common characteristic for in situ stress measurement at a shallow (less than 1 km) depth that the measured horizontal stress is much larger than the lithostatic stress (Brown and Hoek 1978; Sano 2005). According to Sano (2005), such a large horizontal stress can be explained by stress redistribution due to surface erosion (Goodman 1980) or self-gravitation of aspherical shell (McCutchen 1982). So it may not be appropriate to ascribe the measured stress pattern to a single cause like faulting or topography. However, the good correspondence between the observed and calculated stress orientations suggests the topographic effect is considered to be the major controlling factor for the measured stress around the Atera fault at a shallow depth.

If we assume \(D\) = 15 km and \(V\) = 3 mm/yr based on the regional seismogenic depth and geological studies (Headquarters for Earthquake Research Promotion 2004), the estimated shear strain rates at three sites (UEN, FKO and HTJ) are about 3 × 10^{−8} yr^{−1}. Since the last earthquake of the Atera fault is considered to have occurred about 400 years ago (Hirouchi and Yasue 2011), the tectonic left-lateral shear stress on the Atera fault accumulated during 400 years is expected to be around 0.7 MPa, if we assume the crustal shear modulus to be 30 GPa. Therefore, if the last earthquake about 400 years ago completely released the tectonic stress, the topographic stress is considered to be still significant over the tectonic stress at a shallow depth less than a few km, supporting our interpretation about the hydraulic fracturing result.

Our calculation suggests that Yamashita et al.’s (2010) in situ stress measurements mainly reflect a topographic effect. It should be noted that the Atera fault is situated at a special topographic feature so that stress condition can be largely perturbed. The Atera fault is running along a topographic boundary where a large altitude difference of about 800 m exists (Fig. 1). Since the topographic stress is proportional to the topographic load, an altitude contrast less than a few hundred meters does not affect the stress condition so much. On the other hand, at a greater depth, the topographic stress perturbation becomes smaller compared with the regional tectonic stress because of the elastic stiffness of the crust (Fig. 3). The maximum principal stress axes around the Atera fault are in the E–W direction at the depth of 10 km based on the moment tensor inversion (Terakawa and Matsu’ura 2010). So our result is consistent with the seismologically estimated regional stress field, which is also consistent with the long-term activity of the Atera fault, that is, the left-lateral strike slip motion based on geological and geomorphological evidence.

Another evidence of Yamashita et al. (2010) to support right-lateral motion of the Atera fault was the GPS observation data showing a right-lateral movement across the Atera fault. But this observation captures only the short-term surface velocities, which should be strictly distinguished from the absolute strain or stress. Actually, the current study area is not free from elastic deformation due to interplate coupling at the Suruga–Nankai Trough where the Philippine Sea plate is subducting beneath southwest Japan. One supporting evidence of this inference is crustal deformation associated with the Tokai slow slip event during 2000–2005 (e.g., Ozawa et al. 2002; Miyazaki et al. 2006). We can identify southeastward displacement of the southwest part of the Atera fault that releases left-lateral shear strain across the fault. During the interseismic time period at the Suruga–Nankai Trough, right-lateral shear strain accumulation is expected. Such elastic strain should be released when a large earthquake occurs at the Suruga–Nankai Trough and does not accumulate over a longer time scale.

## Conclusion

We evaluate the topographic perturbation on the crustal stress around the Atera fault using the Boussinesq problem with actual topographic information. The calculation suggests that the topographic effect is significant at a shallow depth and greatly affect the crustal stress pattern. The calculated maximum compressional axes at the hydraulic fracturing sites are directed in the N–S direction, consistent with the observation. The results demonstrate importance of topographic effects on crustal stress at a shallow depth. The Atera fault is subject to E–W compressional stress, consistent with its geologically identified left-lateral motion.

## Declarations

### Author’s contributions

KK conducted the numerical calculation and drafted the manuscript. TS supervised KK, conceived of the study and helped to draft the manuscripts. Both authors read and approved the final manuscript.

### Acknowledgements

We used topographic data published by the Geospatial Information Authority of Japan. Earthquake catalogue data were published by Japan Meteorological Agency. The manuscript was improved by comments from two anonymous reviewers.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data materials

The computation code for topographic stress and the calculation result can be provided on request.

### Funding

This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers JP25350427 and JP26109003.

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## Authors’ Affiliations

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