The GTS is an underground rock laboratory located in the Aar Massif in the Swiss Alps. The ISC experiments, which serve as the basis for this study, utilized 15 boreholes of 20–50 m depth, including two injection boreholes (Inj1 and Inj2). The other boreholes are used for stress and strain measurement as well as seismic, pressure, and temperature monitoring during the hydraulic stimulation phases . A general overview of the site showing the persistent structures and the boreholes is given in Fig. a. A summary of the experiments conducted during the ISC measurement campaign and their results are given in and .

The crystalline rock in the southern part of the GTS (ISC experiment volume) has been moderately fractured. Ductile (S1) and brittle–ductile (S3) shear zones can be distinguished from the investigated rock volume (Fig. a; ). The shear zones consist of a fault core, a damage zone, and unperturbed host rock . A 4–6 m highly fractured zone with a fracture density (P10) of around 3 m-1 is present between the fault cores of the two S3 shear zones and is displayed in Fig. b.

The fractures can be distinguished in wall damage zones adjacent to the S3 faults and linking damage zones, i.e., fractures connecting both fault cores . Testing campaigns on the connectivity between several intervals of the injection boreholes revealed that the best response occurs between the intervals 3 and 4 of both injection boreholes, which are located in the aforementioned highly fractured zone. Therefore, this is not only a highly fractured zone but also the most permeable region with conductive fractures . For this reason, the characterization of the hydraulic and structural properties of this region (Fig. b) is the target of this study. The geological mapping of the structures intercepted by the boreholes and tunnels provides the basis for the setup of the conceptual model .

The hydraulic tomography tests that are applied in this study are part of the characterization phase of the ISC experiment. We utilize transient pressure signals from constant-rate injection tests in the intervals 3 and 4 of the injection boreholes Inj1 and Inj2. The different intervals are isolated by a multi-packer system. The properties of the packer intervals and the parameters of the injection are summarized in Table .

Between each injection experiment, pressure recovery was possible. The pressure response of the fluid is measured using piezoresistive pressure transducers. The resolution of the pressure response data is approximately 0.5 kPa. The minimum principal stress is of the order of 8 MPa. As the injected fluid pressure is much below the minimum principal stress, the coupling between hydraulic and mechanical effects can be neglected in the forward modeling of the experiment. The fluid pressure is measured with Δt=2 s. In general, we use similar hydraulic tomography experiments as those applied by except for a shorter injection time (Table ), which was chosen for computational reasons.

The pressure signals are shown in Fig.  for each injection interval. Due to the stochastic inversion approach, the noisy pressure response data can be directly utilized for the inversion without the necessity to smooth or filter the signals.

Fractures are modeled as 2D objects with constant properties normal to the fracture midplane in a 3D rock matrix that is assumed to be impermeable. The pressure diffusion in a single fracture is described by 1aρS∂p∂t-∇T⋅aρkfμ∇Tp=aq, where a (m) is the hydraulic aperture, ρ (kg m-3) is the density of the fluid, S (Pa-1) is the specific storage, kf (m2) is the permeability, μ (Pa s) is the dynamic viscosity, and q (kg m-3 s-1) is a source/sink term. The pressure p (Pa) consists of the static pressure and the piezometric pressure. The permeability is related to the aperture by 2kf=a212, and the subscript T of the gradient (∇T) denotes that it is evaluated in the fracture plane . In this study, flow in the shear zones is modeled using the same approach as flow in the DFN, i.e., the shear zones are represented as 2D objects whereby the flow parameters are given by hydraulic aperture and specific storage (Eq. ). The equations are solved numerically using the finite element method (FEM) with a conforming discretization at the intersections of different fractures. The generation of the geometry and the meshing of the fractures and shear zones are implemented using the open-source mesh generator “Gmsh” . The geometry of each structure is created separately by the built-in geometry module of Gmsh. The fractures and the shear zones are connected for a conforming discretization at the intersections of different structures by the Boolean operations implemented in Gmsh.

The implemented boundary conditions are shown in Fig.  along with the S3 faults and the fractures intercepted by the injection boreholes obtained from optical televiewer logs .

The boundary conditions are chosen considering the fact that only a small volume of the ISC experiment is investigated in this study. Therefore, the following boundary conditions are applied:

The AU (Auflockerungszone, i.e., excavation effects) tunnel is represented by a pressure boundary condition – in this case, ambient pressure.

The parameters of the DFN θ are treated as unknowns characterized by probability density functions. Based on the Bayesian equation, the posterior density function p(θ|d) of the parameters given the measured data d is proportional to the likelihood function 3log⁡L(d|θ)∝-∑i=1Ndatadi-f(θ)i22σi2 and the prior distributions p(θ) . The term f(θ) refers to the forward simulation of the hydraulic tomography experiment for the DFN realization defined by the parameters θ. The posterior density function is evaluated by sampling from it according to the Markov chain Monte Carlo (MCMC) method. This is an iterative procedure whereby new samples θ′ are proposed by a proposal function and accepted (θi=θ′) with probability 4α=min1,p(θ′|d)qθi-1|θ′pθi-1|dqθ′|θi-1|J| or rejected (θi=θi-1) . The so-called reversible jump MCMC algorithm allows one to change the number of parameters . In this study, the number of parameters is adjusted by deleting or inserting a fracture within the prior bounds. The determinant of the Jacobian matrix |J| has to be considered for transdimensional updates. It equals 1 for parameters sampled from the prior without linking its value to preexisting parameters . The parameters of the inversion problem are adjusted by proposing a new value from a normal distribution whereby the mean of the normal distribution is given by the current value.

The variance σ2 in the likelihood function (Eq. ) accounts for different sources of uncertainties, such as measurement errors, modeling errors, and errors of the conceptual model. Therefore, the value of the variance is estimated separately for each pressure signal. This is implemented as part of the inversion algorithm after the update of the parameters of the DFN. The measured data are assumed to consist of a mean and a normally distributed error d=d‾+N(0,σ2). With this assumption, the variance can be estimated by sampling from an inverse gamma distribution 5σ2|d,θ∼IGNdata2,∑i=1Ndatadi-f(θ)i22, as introduced by and implemented by authors including and . For this reason, the noisy measured data can be directly utilized for the inversion without filtering or smoothing the signals.

In practice, one iteration of the inversion algorithm operates as follows: assuming that the insertion of a fracture is chosen in the MCMC algorithm, the parameters (position; length; fracture set, i.e., orientation; and hydraulic aperture) of the fracture are generated from the prior functions. The chosen parameters are evaluated by simulating the hydraulic tomography experiment with the proposed parameter set θ (i.e., including the new fracture). The outcome of the simulation is compared to the measured pressure signals. If the error is smaller (the likelihood, Eq. , is higher) or similar to the previous step (without the fracture), the acceptance probability (Eq. ) is high . After accepting or rejecting the proposed parameters, the variance is updated according to Eq. ().

The overall inversion procedure relies on several simplifications concerning parameters with less importance for our research target. For instance, the parameters specifying the properties of the shear zones have to be fixed. In general, our aim is an optimal balance between the accuracy of the generated results and the computational cost of the inversion procedure.

The underlying conceptual model comprises simplifications of the properties of single fractures that serve as inversion constraints. We assume plane ellipses as the fracture shape, and the length of the minor axis equals half of the length of the major axis (i.e., the length ratio is fixed). The assumption of reducing the fracture shape to a 2D plane is a common assumption and is justified by the derivation of the cubic law and the large ratio between the fracture extensions and the fracture aperture . The assumption of the fracture shape as an ellipse is reasonable because the flow is dominated by the path between the intersections of different fractures; therefore, no sharp edges are considered for the simulation of the flow in the DFN. The hydraulic aperture is assumed to be constant over the fracture plane. Two fracture sets are defined with fixed orientations based on the orientations of the structures intercepted by the two injection boreholes. Thus, the fracture set is chosen by the inversion algorithm for the fractures between the boreholes; however, the orientation assigned to the fracture sets is a default. Figure  shows the orientation of the structures between the S3 shear zones intercepted by the two injection boreholes and the orientations defined for the two fracture sets. The appearance and distribution of the fractures dominate the flow. Accordingly, the surrounding rock matrix is assumed to be impermeable.

The investigated volume is limited to the volume between the two S3 shear zones (Fig. ). The shear zones consist of a fault core and a damage zone. The permeability increases with distance from the fault core, where the cores are almost impermeable . As the properties of the shear zones are not the target of this study, the shape is simplified and the associated hydraulic parameters are fixed. The shape of the shear zones is simplified to a plane rectangle (i.e., a linear interpolation between the shear zones' traces at the injection boreholes). A constant hydraulic aperture of aSZ=1×10-5 m is assigned. This small value is chosen based on preliminary in situ tests and the knowledge that the cores of the shear zones are impermeable at their tunnel intersection. A higher permeability of the shear zone at specific locations can be covered by placing fractures in the respective area that also accounts for the spatial variability in the permeability of the shear zone. Moreover, the specific storage value is fixed at SSZ=1×10-5 Pa-1. This high value is prescribed considering the results from cross-borehole tests . Fractures of fracture set 1 are approximately parallel to the S3 faults. Hence, a position close to an S3 fault also accounts for spatial changes in the permeability and specific storage of the S3 faults.

Overall, the application of constraints and assumptions about the fracture shape limit an exact reproduction of the structural properties of the tested rock mass. However, those parameters that have a major influence on the flow in the DFN are adjusted by the inversion algorithm within prescribed bounds. These are, in particular, the position and the hydraulic aperture of fractures. In contrast, parameters with minor effects on the flow behavior are fixed (e.g., the exact fracture orientation or the length ratio).

The parameters to be inferred are the number of fractures, the position of the fractures, the fracture lengths, the respective hydraulic aperture for each fracture, and the specific storage coefficient that applies to the whole DFN. The specific storage S (Eq. ) is given by the compressibility of water in theory . However, some fractures are only partially open; thus, due to the roughness of the surface, the specific storage can be increased compared with the theoretical value . Moreover, the hydraulic aperture is generally smaller than the actual aperture . The specific storage is assumed to be valid for the whole DFN because two variable hydraulic parameters for each fracture are not feasible for the inversion algorithm. Accordingly, five different update types are implemented to be applied sequentially: the transdimensional update changes the number of parameters by either inserting a new fracture or deleting a fracture; the other update types keep the number of parameters constant but adjust position, length, hydraulic aperture, or the specific storage. For the update of the position, length, and hydraulic aperture, one fracture is chosen randomly, and a new value is proposed by a random perturbation of the current value.

Uniform prior distributions are applied (i.e., a parameter is specified by a constant probability between minimum and maximum possible values that are given in Table ).

The spatial priors are derived in general from the position of fractures intersecting the injection boreholes. The maximum value in the x direction corresponds to the distance to the AU tunnel to apply the boundary condition. The prior for the north direction is given such that the fractures are located between the cores of the S3 shear zones. The elevation of fractures is expected to have a minor influence on the flow between the two boreholes, and a broader possible range for the elevation would be less resolved. In the following, x refers to easting+667400 m, y refers to northing+158800 m, and z refers to height+1700 m (Fig. ). The minimum value for the fracture length is given by the borehole diameter, and the maximum possible value corresponds to the distance between the shear zones. Fractures proposed during iterative inversion which intersect with the fault cores of the shear zones are reduced to the part of the fracture within the investigated volume (Figs. b, ).

The prior range for the aperture is approximated from the results of single- and cross-borehole tests . The minimum specific storage value is given by the compressibility of water , whereas the maximum value is based on cross-borehole injection tests . Both prior distributions for the hydraulic parameters cause preferential flow in the DFN rather than in the shear zones, due to a smaller specific storage and a larger hydraulic aperture of the fractures.

Overall, 27 000 DFN realizations are considered to be posterior DFN realizations because they minimize the error and fulfill the prior conditions. DFN realizations from the initial 500 iterations are discarded as so-called “burn-in” iterations due to a higher error. The computation of the inversion was executed by an Intel Core i9 workstation with 10 cores and 128 GB RAM and took about 1 week. The posterior realizations are approximately equally likely. They reflect the uncertainty of the inversion results in contrast to a unique solution that would be obtained by a deterministic approach. To reduce the autocorrelation of the results, we keep every 100th realization for further processing, which is called “thinning” . Using the stochastic approach applied here, the fit between the measured and simulated pressure signals of the hydraulic tomography experiment is evaluated by the posterior and prediction uncertainty. The posterior limits are calculated based on the simulated pressure signals of the posterior DFN realizations which correspond to the uncertainty of the inversion method. The uncertainty related to predicting new observations is a measure of the overall error as well as of conceptual simplifications, as it also considers the estimated variance (Eq. ). The DFN realizations are evaluated using a fracture probability map (FPM) over the investigated volume. For this, the inspected rock volume is divided into raster elements. Each element records whether the element is part of a fracture. By taking the element-wise mean over all of the posterior DFN realizations, the probability that each raster element is part of a fracture is derived. The evaluation of the FPM summarizes the estimated position and length of the fractures (i.e., those parameters with major influence on the flow). The hydraulic aperture is evaluated on the same raster elements. If a raster element is part of a DFN realization, the respective aperture is taken from the DFN. Thus, the mean hydraulic aperture can be evaluated for each element.

In the first step, the measured and simulated pressure signals are compared to assess the quality of the posterior realizations. Figure  shows the median fit and the 95 % limits of the forward simulation of the posterior DFN realizations and the 95 % limits of the prediction uncertainty along with the observed data.

Figure  demonstrates that the general shape and trend of the measured signals are reproduced by the simulated pressure curves checking the median fit and the 95 % posterior limits. This is especially the case for the response in interval 4 of both boreholes. The weaker fit of some signals in interval 3 indicates effects not covered by the inversion approach or forward simulations, such as deviations from the assumed fracture shape or fracture orientations. For a given DFN realization, the actual measured pressure signals are predicted. Due to measurement noise and simplifications concerning the DFN model, the 95 % limits of the prediction uncertainty are wider than for the posterior uncertainty.

The FPM and the mean hydraulic aperture are shown for different cross sections (z) in Fig. . The fractures intercepted by the injection intervals and the shear zones are fixed; therefore, they appear with a probability of 100 %. Their orientation, as derived from the optical televiewer logs, is assigned to these fractures; thus, the orientation is in the same range as the orientation defined for the fracture set, but the exact values vary.

Overall, two different connections with different levels of permeability are present. A flow path dominated by fractures with a large hydraulic aperture exists between injection interval 4 of both boreholes (Inj1–Int4 and Inj2–Int4). The fractures providing this connection are visible with a high probability in the cross sections z=16 m and mainly z=17 m. In general, a good respective permeable connection between two intervals is possible with a large hydraulic aperture of the fractures, with long fractures, with a long intersection length between different fractures, or with a correlation of these factors. In contrast, a connection with fractures with smaller hydraulic apertures appears between injection interval 3 of both boreholes (Inj1–Int3 and Inj2–Int3) and Inj2–Int4. This flow path is present with an average probability of approximately 50 % primarily in the cross section z=15 m. Fractures linking both flow paths appear more likely the closer the location is to injection borehole 2 (i.e., further east). The described behavior is also reflected in the measured data. All responses provoked by the injection in interval 4 of both boreholes are more distinct than for the injection in interval 3. Although a maximum hydraulic aperture of 10-3 m is enabled by the prior distribution, only a few fractures with a small probability appear with an aperture close to the maximum possible value, as visible in Fig. , at a depth of z=17 m. The specific storage coefficient converges to a mean value of S=7.4×10-7 Pa-1. Only a few updates were possible that occurred mainly during the burn-in iterations. Therefore, this value is interpreted as the result of an optimization (i.e., as the averaged specific storage to be applied for the whole DFN). The estimated specific storage is greater than the theoretical value that functioned as the minimum value of the prior distribution of the specific storage (Table ). This considers a delay in the response that is not related exclusively to the compressibility of water but also to, for example, the surface roughness or fractures that are only partially open. Multiplied by the maximum possible aperture (Table ), the inferred value is well within the storativity range calculated from cross-borehole injection tests . Several fractures of fracture set 1 appear close to the S3.1 shear zone, indicating either permeable fractures close to the shear zone or a higher permeability of the shear zone in this region than the assigned value. This demonstrates that the prescribed assumptions with respect to the hydraulic properties of the shear zone do not induce crucial conceptual constraints in the inversion, but a locally high permeability of a shear zone is indicated by a locally high fracture probability.

Although the volume east of injection borehole 2 towards the AU tunnel is part of the inversion (i.e., fractures can be inserted or moved in this volume), the resolution of the results is low because various DFN realizations (i.e., fracture positions) are possible. Only the volume between the two injection boreholes can be evaluated with a sufficient resolution.

The inferred flow paths consist of fractures with a high or rather low permeability, which is in accordance with the results of . We also compare our results with the structures intersected by other boreholes drilled after conducting the experiments evaluated in this study. While this inversion approach is capable of identifying fractures that are hydraulically relevant, geophysical methods (such as optical televiewer logs) report all structures intercepted by boreholes independent of their permeability. The boreholes PRP1 and FBS1 are partially located within the prior range defined for this study. The 23–25 m depth interval for PRP1 has been identified as the interval with the highest transmissivity by and . In 95 % of the posterior DFN realizations, at least one fracture is present in this interval. Fractures that intersect with the interval between the S3 faults of the FBS1 borehole are present in about 45 % of the posterior DFN realizations. This supports the fact that crucial hydraulic features of the DFN can be identified by the presented inversion approach. Still, even if such successful local validation is possible, there are no other independent measurements available to confirm the validity of the inverted complete DFN structure and its probability. Geophysical measurements, such as seismic data or ground-penetrating radar , were able to characterize the ISC volume on a decameter scale and identify the persistent structures and the highly fractured zone; however, they could not delineate or specify the properties of single flow paths.