In this study we use the transdimensional DFN inversion introduced by (Somogyvári et al., 2017), based on the reversible-jump Markov chain Monte Carlo (rjMCMC) algorithm (Green, 1995). This iterative technique uses subsequent random geometry perturbations to generate and test possible model configurations (Fig. 2). Every iteration consists of two phases: an update and an evaluation phase.

In the update phase, the current DFN realization (θn) is altered by fracture addition, deletion or movement. The type of the update and its properties are chosen randomly, with a predefined probability (qθn→θ∗). Fractures could only be inserted to specific discrete points along the existing DFN. This is an important condition to preserve the reversibility of the chain (see Somogyvári et al., 2017).

In the evaluation phase the temperature distribution in the aquifer is modelled on the updated DFN realization (θ∗) using the forward model (details in Sect. 3.2). The model evaluation is done using the Metropolis-Hastings-Green acceptance criterion (Green, 1995): 1αθ∗|θn=min⁡1,pθ∗pθnLξ|θ∗Lξ|θnqθ∗→θnqθn→θ∗J. The likelihood-ratio (Lξ|θ∗/Lξ|θn) is calculated from the improvement of the RMS error of the simulated observations (ξ|θ). We consider the observations (ξ) normally distributed, hence the L functions have a gaussian shape. The proposal ratio (qθ∗→θn/qθn→θ∗) is calculated as the ratio of the probabilities of the reverse and the forward update. Because these updates are discrete, the calculation of these probabilities are straightforward. For example, the probability of a fracture deletion is one over the total number of fractures. A detailed description on the update probabilities can be found in (Somogyvári et al., 2017). The so-called Jacobian (J) is a consequence of the transdimensional updates, and it is required to maintain the stability of the rjMCMC. With discrete model updates however, its value is always one (Denison et al., 2002).

If a proposal gets accepted the new iteration starts with it, if it gets rejected the chain continues with the previous realization. The final product of rjMCMC inversion is not one calibrated model, but a set of realizations often referred to as ensemble. The ensemble is suitable for further statistical investigations and for uncertainty analysis (Somogyvári et al., 2017). To eliminate the effect of the arbitrary chosen initial model, the first part of the chain is discarded and not used in the ensemble (Somogyvári and Reich, 2019).

To simulate the steady state temperature distribution in the DFN model, the tracer transport simulator developed by Jalali (2013) was used. This efficient algorithm could simulate temperature distributions in two dimensional DFN models within a few seconds, making it suitable for MCMC applications.

The forward model takes the DFN geometry and hydraulic and temperature boundary conditions as an input and returns the steady state temperature distribution in the fracture network. The simulator is transient, both the pressure and temperature simulations start from an initial field with perturbations by the boundary conditions, then the simulation is run until a steady pressure and temperature field is obtained.

First the DFN model is discretized to 1-D segments and nodes. Pressure distribution is simulated by using mass conservation, with an implicit finite difference method. Flow in the fractures are calculated by Darcy's law from the pressure gradient. For thermal transport, the advection-dispersion equation is solved considering a steady-state flow field, with an implicit upwind finite difference solver (Jalali, 2013; Ringel et al., 2019).

In this simple model no temperature transport is simulated within the impermeable rock matrix, only the temperature loss from the fracture space towards the rock matrix is simulated. Hence, cross-fracture heat transport (between non-connected near fractures) is not possible. The simulation is limited to 2-D and no complex hydraulic and thermohaline boundary conditions are considered.

The temperature field inside the non-fractured aquifer matrix is calculated via linear interpolation with triangulation. The interpolated field is then intersected with the boreholes, and the extracted 1-D temperature profiles were used as the simulated observations. In theory it would be possible to compare the interpolated 2-D temperature field, with the interpolated field from the site (see Fig. 1b), but we wanted to demonstrate the robustness of the method with only using temperature logs from a limited number of boreholes. The complete simulation could be done faster than a second on a laptop, thus this simple DFN transport model is efficient and fast to be used in an MCMC framework.

The methodology introduced in (Somogyvári et al., 2017) needed to be modified to work on reservoir scale scenarios. The main modification is a simplification to mitigate the limited amount of observations available. Three fracture sets are defined: two for the major fault directions and one for the stratification. In this modified setting, fractures are completely intersecting the domain. This simplification is done based on the geological model of the aquifer (Fig. 1a). Hence, fracture lengths are not a free parameter of the inversion as they are calculated from the fracture position within the domain. In addition, fracture centers are aligned to the vertical or horizontal centerline of the domain (depending on the fracture set) – basically reducing the free parameters to one per fracture.

Some further modifications were also done. Fracture addition and deletion is only used for the low-inclination fracture set (stratification), and fracture movement only for the high inclination sets (the exact location of these faults is not known).

The hydraulic and thermal fracture properties were chosen after available borehole information. The two fault sets are simulated with an aperture of 0.5 mm. The stratification aperture is set to 0.8 mm. These values are chosen after (Kühn et al., 2016) using the cubic law to convert transmissivities to apertures. The values were further tuned by preliminary modelling trying to obtain similar extent of the temperature anomaly as the observed one. These tests also showed that the anisotropy has a stronger effect than the exact aperture values. The hydraulic conductivities of the fractures are calculated from the apertures using the cubic law. Note that both flow and heat transport is limited to 2-D in the model, and no 3-D effects are considered.

Boundary conditions are defined on the top and bottom borders (Fig. 3). In the DFN forward models, they appear when a fracture intersects with these borders. Note that the used pressure values are not absolute and were selected after the hydraulic gradient. For the source of the geothermal water, a 100 m long source region with increased pressure and temperature is defined in the bottom center of the model. The observed maximum temperature of 50 ∘C was assigned. Pressure of the source region was arbitrarily selected after the parameter tests with inspecting the extent of the temperature anomaly (Fig. 3).

Note that the used methodology is limited to invert the geometry, and does not estimate the physical fracture properties. Introducing the hydraulic parameters of the fractures as free parameters into the inversion would result in an increase in model freedom, which without involving additional data would lead to different equivalent solutions (Somogyvári et al., 2017).

The method was first tested on a synthetic dataset. The chosen synthetic DFN model is shown in Fig. 4a. The inversion here is limited to the fractures representing the stratification. The high inclination faults are considered known, and not estimated by the algorithm.

The transdimensional DFN inversion was implemented in python and was run on a standard office laptop with an Intel i5-7267U CPU. The chains were run for 1000 iterations which took 5 min on average. Because of the stochastic nature of the used inversion technique, we ran several simulations with the same initial parameters to see the stability of the results. All these simulations led to reconstructed geometries with the same general features, showing the robustness of the algorithm.

The result of the synthetic example is shown in Fig. 4b in the form of a fracture probability map. This plot visualizes the realizations of the rjMCMC chain by rasterizing the DFN models first, then counting the pixels with fractures along the ensemble. This is equivalent of taking the mean of the MCMC chain for visualization (Bodin and Sambridge, 2009). The fracture probability map shows perfect fits with the two fractures in the bottom part of the profile. The three fractures in the top are not captured exactly, but the extension of this fractured zone is well resolved. The absence of fractures in the center part is captured perfectly. This result is a proof of concept that the methodology is applicable on the proposed geological problem.

For field application, the three virtual boreholes defined as 1-D slices of the profile shown in Fig. 1b were chosen. One borehole is taken from the center of the anomaly, the other two from the two sides. These virtual boreholes were selected instead of real wells for practical reasons, as all temperature profiles from the site were already compiled and integrated to a 3-D temperature dataset. This removes any local effects of the actual boreholes and also provide room for sensitivity analysis of different borehole locations numbers along the profile, which is to be done in the future.

To improve the estimation, background temperature trends were removed from the virtual boreholes (using the temperature trends from the two sides of the model). With the removal of these trends, we have also mitigated the effect of freshwater-seawater mixing, which is not simulated by the forward model.

Simulations were run for 1000 iterations, and similarly to the synthetic case the first half of the chain was discarded. The simulations were completed within 15 min, a little longer then the synthetic case. The results are presented in Fig. 5.

The reconstructed fracture probability map has the following three distinct features. Strong stratification is present close to the bedrock with high probability. Fractures in this area are close to the source and provide pathways for the heat to spread out horizontally. A non-stratified aquifer body present in the center of the domain. It could not only mean that this part is non-fractured, this could be also explained with the existence of fractures that do not play a role of forming the anomaly. This is supported by the similarity of additional synthetic examples where this feature was usually present. In contrast, there is high probability of stratification in the top. The role of these fractures is to close the anomaly, by allowing mixing with the colder waters. The inversion also placed faults close to the location of two boreholes for similar reasons.

Figure 5b shows the mean of the reconstructed temperature fields of the ensemble. Compared to the observed temperature anomaly, the reconstruction shows similar properties. The two anomalies have a comparable extent horizontally and vertically. The skewedness of the anomaly is reconstructed. Originally this was explained by the mixing of different waters, but our results show that it could be explained by the structure of the aquifer.

Figure 5c shows the fit quality of the borehole temperature profiles. Continuous models from the same site provide better fits for most of the temperature observations (Kühn and Stöfen, 2005) but cannot match some observations which are supposed to be based on the fractured characteristic of the reservoir rock. The main limitations here are the 2-D simulation and the lack of complex geometry options, which restricts the shape of the reconstructed anomaly. This could be addressed by generating and testing more complex fracture patterns, but can only be executed by involving additional data into the inversion.