# Comparison of Hydraulic and Tracer Tomography for Discrete Fracture Network Inversion

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. DFN Case Study

#### 2.2. Transdimensional Inversion

#### 2.3. Estimation of the Noise Variance

## 3. Results

#### 3.1. Results of the DFN Inversion

#### 3.2. Results Estimating the Noise Variance

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Synthetic discrete fracture network with three injection points (number one to three) and three receiver points (number four to six) based on an outcrop in Switzerland. It contains two fracture sets with two different fracture angles and fracture apertures.

**Figure 3.**Scheme to explain the procedure of drawing samples from a variance posterior probability density function (pdf).

**Figure 4.**(

**a**) Synthetic discrete fracture networks (DFN) and (

**b**) the results using the pressure signals for the inversion illustrated as fracture probability map.

**Figure 5.**(

**a**) Synthetic DFN and (

**b**) the results using the tracer breakthrough curves for the inversion illustrated as fracture probability map.

**Figure 6.**Noisy tracer breakthrough curves (${\sigma}_{noise}=3$) for different combinations of source and receiver points: (

**a**) tracer injection at source 1; (

**b**) tracer injection at source 2; and, (

**c**) tracer injection at source 3. The colors of the graphs accord with the colors of the receiver points in Figure 1.

**Figure 7.**Histogram of variance samples during the inversion and posterior pdf of ${\mathsf{\sigma}}_{noise}^{2}$ $p({\sigma}_{noise}^{2}|{\xi}_{obs},{\theta}_{-{\sigma}_{noise}^{2}})$ in the last iteration of the rjMCMC loop (

**a**) inversion of ${\sigma}_{noise}^{2}$ using the inverse gamma prior; and, (

**b**) inversion of ${\sigma}_{noise}^{2}$ using the uniform prior.

**Figure 8.**Noisy pressure signals (${\sigma}_{noise}=3000$) for different combinations of source and receiver points: (

**a**) water injection at source 1; (

**b**) water injection at source 2; (

**c**) water injection at source 3. The colors of the graphs accord with the colors of the receiver points in Figure 1.

**Figure 9.**Histogram of variance samples and posterior pdf of ${\mathsf{\sigma}}_{noise}^{2}$ in the last step of the inversion.

**Table 1.**List of geometric constraints, experimental and inversion parameter settings (FLD: fracture length distribution).

Category | Parameter | Fracture Set #1 | Fracture Set #2 |
---|---|---|---|

Geometric constraints | Fracture inclination (°) | −19.48 | 74.48 |

Fracture aperture (mm) | 1.5 | 1 | |

FLD-mean (m) | 9.9 | ||

FLD-variance m^{2} | 8.5 | ||

Experimental parameter settings | Injection pressure (Pa) | 3 × 10^{5} | |

Injection concentration (mg/l) | 40 | ||

Inversion parameter settings | Discretization length (m) | 1 | |

${p}_{add}/{p}_{del}/{p}_{shift}$ | 0.4/0.4/0.2 | ||

Number of rjMCMC iterations | 100,000 |

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**MDPI and ACS Style**

Ringel, L.M.; Somogyvári, M.; Jalali, M.; Bayer, P.
Comparison of Hydraulic and Tracer Tomography for Discrete Fracture Network Inversion. *Geosciences* **2019**, *9*, 274.
https://doi.org/10.3390/geosciences9060274

**AMA Style**

Ringel LM, Somogyvári M, Jalali M, Bayer P.
Comparison of Hydraulic and Tracer Tomography for Discrete Fracture Network Inversion. *Geosciences*. 2019; 9(6):274.
https://doi.org/10.3390/geosciences9060274

**Chicago/Turabian Style**

Ringel, Lisa Maria, Márk Somogyvári, Mohammadreza Jalali, and Peter Bayer.
2019. "Comparison of Hydraulic and Tracer Tomography for Discrete Fracture Network Inversion" *Geosciences* 9, no. 6: 274.
https://doi.org/10.3390/geosciences9060274