The hydraulic and mechanical characterization of fractures is crucial for a wide range of pertinent applications, such as geothermal energy
production, hydrocarbon exploration,

Tube waves are interface waves propagating along the borehole wall. They are sometimes also referred to as Stoneley waves, but, as

Various modeling approaches have been proposed to study the properties of tube waves. A number of analytical techniques to calculate the tube-wave velocity

As tube waves propagate along the borehole, no geometrical spreading occurs, and, therefore, tube waves are much less attenuated than body waves and retain high
amplitudes even at large distances from the source. Thus, if vertical seismic profiling (VSP) data are recorded with pressure sensors, such as hydrophones, tube
waves tend to pose a problem as they cover body-wave reflections

Here, we do not aim at suppressing or removing tube waves but rather consider them as signals containing valuable information for characterizing hydraulically
open fractures along the borehole, which is important for a wide variety of applications, such as groundwater management, geothermal energy production,
hydrocarbon exploration,

The above methods do, however, require extensive manual conditioning of the data, like amplitude picking or time-gating of events. Furthermore, they are unable
to provide an estimate of uncertainty and/or to identify multiple solutions that are equally likely. The objective of this work is to alleviate these
limitations by providing an algorithm that considers the entire wave field for characterizing fractures in terms of their hydraulic apertures and mechanical
compliances as well as the associated uncertainties with a minimal amount of human interaction. To this end, we propose a Bayesian full-waveform inversion
approach in combination with a recent semi-analytical approach

The goal of our stochastic inversion approach is to estimate the posterior probability density function (PDF)

We use a novel semi-analytical algorithm for

In the considered forward operator

When a tube wave propagating along the borehole interface encounters a fracture, fluid flow from the borehole into the fracture
is triggered. This leads to reflection and transmission of tube waves. This process is described with the scattering potential

For the forward operator

To improve the estimation of the fracture compliance

In order to fit the observed data, we implemented the forward operator of

Due to the nonlinearity of the problem, we cannot infer the posterior PDF directly; instead we need to infer it by sampling the prior PDF and the likelihood
according to Relation (

The viability and accuracy of the algorithm have been tested and verified in a variety of synthetic case studies, an example of which is shown in the next section. Subsequently, we apply the proposed inversion scheme to hydrophone VSP data acquired at the underground Grimsel Test Site in the central Swiss Alps.

Before applying our inversion algorithm to observed data, we ran tests on synthetic data to ensure that the algorithm performs as expected. As in these
experiments the same forward solver was used for the generation and the inversion of the data, the corresponding results only allow conclusions to be drawn
with regard to the inversion algorithm itself, but not with regard to the information content of the data. The test case shown here features two fractures
at 10 and 19 m depth. The receiver spacing is 1 m. To make this synthetic study more pertinent and challenging, we contaminated the dataset with actual ambient
noise from a corresponding field dataset at the underground Grimsel Test Site in Switzerland. The resulting data are plotted in Fig.

This synthetic test differs from the field-data example shown in the next section in two ways: (1) it uses as a forward solver the algorithm
proposed by

The inversion was run once with three parallel Markov chains. Figure

Estimates of

Simulated data based on a model proposed at the end of the first Markov chain agree very well with the input data (Fig.

The VSP data, considered in the following, were recorded in crystalline rocks at the underground Grimsel Test Site in Switzerland using a
12-receiver hydrophone string with a receiver spacing of 1 m. In the course of the experiment, the hydrophone string was repositioned, such that, the recorded
traces are separated by 0.5 m. The borehole had a diameter of 0.147 m. As a source, a single-handed 2 kg hammer was used at the wellhead, which excited
frequencies between 0.1 and 4 kHz with a peak around 1.5 kHz. In this study, we consider a 20 m long subsection between 17 and 37 m depth, consisting of 41
hydrophone receiver positions. Through visual inspection of the VSP dataset, complemented by evidence from optical and acoustic televiewer data

Preprocessing of the data included a gentle bandpass filter to suppress high-frequency noise, a static shift correction to remove positioning errors, and a
cosine taper to blank out the later-arriving S wave and associated tube waves. The data after preprocessing are shown in Fig.

Observed hydrophone VSP data considered in this study. (1) denotes the downgoing P wave, (2) the upgoing tube wave due to the fractures at 23.5 and 23.9 m depth, and (3) the up- and downgoing tube wave due to the fracture at 25 m depth. Note the amplitude decay associated with the P wave.

For this problem with three fractures, we have 15 unknowns, which are specified in Table

Unknowns of the inverse problem and their prior ranges subdivided by horizontal lines into three groups. The first group from the top comprises the background medium parameters, the second group the fracture parameters, and the third group algorithmic “tuning” parameters.

We ran our algorithm three times to ensure that it successfully locates the posterior PDF and does not get stuck in a local minimum. Each time, three parallel
Markov chains were used to explore the parameter space. More chains would have allowed for a more comprehensive exploration of the solution space, but would
also have required more computational resources. Three chains are in our experience sufficient to exhaustively explore a 15-dimensional solution space well,
such that the posterior PDF is found in most of the runs. The development of the root mean square error (RMSE)
is
plotted in Fig.

RMSE weighted by the standard deviation of the data error for the three inversion runs of the observed VSP data shown in Fig.

In order to assess whether the Markov chains have converged sufficiently to allow for a reliable estimation of the posterior PDF, we calculate the so-called
potential scale reduction factor

The acceptance rate specifies how many of the tested models are accepted. A too-high acceptance rate generally implies that only models in the immediate neighborhood of the current model are explored, while a too-low acceptance rate means that computational resources are wasted by testing unrealistic models. Ideally, the acceptance rate ranges between 10 and 30 %. In our case, it lies between 10 and 20 % for runs one and two and around 5 % for run three.

Development of the most relevant unknowns for the three MCMC inversion runs of the observed VSP data shown in Fig.

The most interesting inferred parameters are the apertures and compliances of the fractures, and to a lesser extent the background rock
properties. The development of these unknowns as a function of the number of iterations is plotted for all three runs in Fig.

The vertical axis of the plots in Fig.

The posterior PDF for the estimates of the aperture of the third fracture is unimodal (Fig.

For the bulk and shear modulus of the background (Figs.

Comparison between simulated (colored) and observed (black) data:

As the RMSE in Fig.

Based on the interpretation of the optical televiewer data by

A bit puzzling is the remarkably low estimate of the tube-wave shear modulus of only about 6 GPa (Fig.

From an inversion perspective, the most interesting point of these results is that two modes of the posterior PDF were identified. This showed that having the first fracture with a large aperture, while the second fracture is thin, is similarly probable to the opposite scenario. Note that a deterministic approach would have provided only one result without any indication that there is another mode that can explain the data equally well, whereas our Bayesian approach clearly supplied us with both options. This nicely demonstrates the value of stochastic inversion approaches.

A downside of our Bayesian approach is its enormous computational cost. Most of it is spent in the forward steps to simulate VSP data for the
proposed model. We have optimized the forward simulation by parallelizing over frequencies. Still, one inversion run with three parallel Markov chains and
60 000 forward steps per chain took approximately 14 d to complete using one node (48 AMD Opteron 6174 processors at 2.2 GHz) of our cluster.
However, the inversion would run 3 times faster if each of the three Markov chains were run on a different node. We did not do this due to limited
availability of resources. In any case, we argue that the computation time is well spent, since the results obtained are much more comprehensive than those that
would be obtained through a deterministic inversion, as they allow, as explained above, multiple modes of the posterior PDF to be discovered. Furthermore,
stochastic inversion approaches do not really depend on the starting model. This is in stark contrast to deterministic full-waveform inversion approaches, which
require starting models whose forward response deviates from the forward response of the true model by less than half a wavelength

For the real-data example, we have decided not to estimate the source wavelet during the inversion process, although the corresponding algorithm was developed and successfully applied for synthetic test cases as demonstrated in the first results section. The reason is that the source wavelet of the observed data includes extensive reverberations and is, thus, extremely long and complicated. Estimating it as part of the inversion procedure would have required more than doubling the amount of unknowns, which would have rendered the problem unnecessarily complex.

An important limitation of our forward model, and indeed of virtually all fracture-based tube wave models, is that fracture aperture and compliance are
correlated. This means that the inversion algorithm tends to compensate for an overestimation of the fracture aperture by underestimating the fracture compliance.
Therefore, we observe that a large fracture aperture for the first fracture is accompanied by a relatively small fracture compliance (Figs.

Inspecting the difference between the observed and the forward modeled data shows that the largest discrepancies are found at the fracture locations. This indicates that the transmission loss of the P wave across fractures may not be reproduced properly in the synthetic data. However, as this affects only the P wave around the fracture locations, the impact on the RMSE is limited. A possible way to improve this issue might be to define a weighting function that peaks at the fracture locations to force the algorithm to obtain a better data fit at these locations, and thus, find a more accurate transmission coefficient. The downside of this, however, is that the weights are new tuning parameters that need to be adjusted through a time-consuming process, which was not feasible to accomplish in the scope of this study.

Limitations of our implementation of the forward operator are its inability to account for scatterers, impedance contrasts related to lithological changes, and borehole enlargements. If corresponding effects are present in the data, they might need to be filtered out prior to inversion. Similarly, changes in the P-wave velocity are not taken into account. If these are present, the data need to be cut into smaller pieces with constant P-wave velocity. Changes in P-wave velocity above the considered borehole section are taken into account by virtually shifting the source depth. The algorithm is also not able to take S waves and corresponding tube waves into account. In our dataset, events of this kind were indeed present and needed to be muted before applying our inversion algorithm to the dataset.

We have developed a Bayesian MCMC full-waveform inversion algorithm based on a semi-analytical forward solver to simultaneously infer the aperture and compliance of individual fractures from corresponding tube-wave data. We mitigate the correlation between fracture aperture and compliance by constraining the fracture compliance by two independent observables: (1) the tube-wave amplitude relative to the P-wave amplitude and (2) the amplitude loss of the P wave across a fracture. The algorithm was applied to a field dataset acquired in crystalline rock at the underground Grimsel Test Site in Switzerland. The subsection of the VSP dataset considered contained three fractures, of which two are very close together. The algorithm identified two equally probable modes in the posterior PDF: either the first fracture features a large aperture and the second fracture a small one or vice versa. In other words, from the information provided, the algorithm can determine that one fracture is larger than the other, but it cannot determine which one is thick and which one is thin. The identification of these two modes clearly illustrates a major advantage of stochastic inversion algorithms as compared to their deterministic counterparts. The latter would not have identified these two modes and would have provided just one of the two possible solutions. Our case study also shows that in a complex geological environment with multiple, closely spaced fractures the hydraulic apertures of individual fractures cannot be determined. However, the method can still provide an effective fracture aperture distribution of a package of fractures. The inferred apertures in our example are consistent with televiewer data, and the inferred compliances are roughly in the same range as those derived from sonic logs at the same site. The data fit is remarkably good, especially when considering the semi-analytical nature of the forward solver and the inherent assumptions it relies on, as well as the rather complex character of the observed hydrophone VSP data.

The forward solver can be downloaded from

JH developed the inversion, contributed to the data analysis and wrote the majority of the manuscript. AG collected and processed the hydrophone VSP data, and contributed to the scientific discussion. SM contributed to the analysis of the results and the final manuscript. NDB contributed to the development of the forward solver and the analysis of the data and the corresponding scientific discussion. EC contributed to the development of the forward solver, the scientific discussion and the understanding of the dataset. KH acted as project leader and participated in the research effort and the manuscript preparation.

The authors declare that they have no conflict of interest.

This work has been completed within the Swiss Competence Center on Energy Research – Supply of Electricity, with support of Innosuisse and the Swiss National Science Foundation in the framework of the National Research Program 70 “Energy Turnaround”.

This research has been supported by the Swiss National Science Foundation (grant no. 407040_153889).

This paper was edited by Michal Malinowski and reviewed by three anonymous referees.