 Research
 Open Access
THM modeling of hydrothermal circulation at Rittershoffen geothermal site, France
 Bérénice Vallier^{1}Email author,
 Vincent Magnenet^{2},
 Jean Schmittbuhl^{1} and
 Christophe Fond^{2}
https://doi.org/10.1186/s4051701801081
© The Author(s) 2018
 Received: 16 April 2018
 Accepted: 17 October 2018
 Published: 2 November 2018
Abstract
Background
The Rittershoffen deep geothermal project located 6 km east from SoultzsousForts EGS site (France) includes a doublet GRT1 and GRT2 to exploit the geothermal resource at the sediments–granite transition where higher temperatures than those of SoultzsousForêts have been measured. Detailed stratigraphic and geophysical data, temperature logs, and tracer surveys have been collected. However, no reservoir model, integrating largescale geophysical measurements, exists for this site.
Methods
We developed a reservoir model in two dimensions (10 km × 5 km) based on a finite element method. It includes thermo–hydro–mechanical (THM) coupling and extended brine properties. A representative elementary volume of 100 m is assumed to homogenize the fault network complexity at small scales. A back analysis is performed to obtain largescale rock properties using GRT1 temperature log and regional stressdepth profiles.
Results
The inverted largescale properties are consistent with their counterparts measured at the laboratory scale. The bottom of the hydraulic cap rock is 1.2 km ± 0.1 km deep. It is shallower than the discontinuity of the thermal gradient. Hydrothermal convection cells are 2.7 km high which is larger than that previously proposed.
Conclusions
A very good fit of the GRT1 temperature log is obtained using our simplified twodimensional THM model with four homogenized units at a 100 m scale. The comparison between Rittershoffen and SoultzsousForêts models highlights many similarities in terms of rock properties, decoupling of hydraulic and thermal cap rocks and temperature spatial variability (about 50 °C). Predictions of the relationship between reservoir temperature and surface thermal gradients are proposed for future explorations.
Keywords
 Deep geothermal reservoir
 Thermal anomaly
 EGS
 Thermo–hydro–mechanical model
 Hydrothermal convection
Background
Introduction
In the Upper Rhine Graben (URG), the geothermal gradient is unusually high at the near surface (more than 100 \(^\circ\)C/km in the first kilometer in depth, whereas the main value in Europe is 30 \(^\circ\)C/km) (Haas and Hoffmann 1929).
The high underground temperatures in the URG make the region the most studied one in Europe for geothermal applications (Genter et al. 2016; Olasolo et al. 2016; Huenges and Ledru 2011). Geothermal projects, such as the wellknown pilot research SoultzsousForêts site, are based on enhanced geothermal system (EGS) technology. The EGS concept consists in increasing the reservoir permeability using hydraulic, thermal and/or chemical stimulations and then forcing a circulation of the natural brines in the deep wells taking advantage of the thermal anomaly related to the largescale hydrothermal system in fractured rocks (Tester et al. 2006; Schindler et al. 2010; Gérard et al. 2006; Schill et al. 2017). After the development (during more 25 years) of the SoultzsousForêts pilot site as an EGS demonstrator, a new industrial project at Rittershoffen was initiated in 2011 and operated in 2016 (Baujard et al. 2015; Genter et al. 2015). The Rittershoffen site is located 6 km east from SoultzsousForêts in Northern Alsace. The project is based on a geothermal doublet, GRT1 and GRT2, drilled around 2.6 km deep to intersect the normal Rittershoffen fault and its associated fracture network at the interface between sediments and granite. Structural and stratigraphic studies (Aichholzer et al. 2016; Hehn et al. 2016; Vidal et al. 2016a), temperature logs (Baujard et al. 2016, 2017) as well as seismic (Gaucher et al. 2013; Maurer et al. 2015; Lengliné et al. 2017) and geochemical surveys (Dezayes et al. 2014; Sanjuan 2016) have been already established to identify permeable zones and the hydraulic connections between GRT1 and GRT2 wells. The knowledge of the Rittershoffen site is also completed by the huge database collected in the nearby SoultzsousForêts site. Five permeable zones have been currently identified in the granite reservoir, but none in the sediments (Vidal et al. 2016b).
As already demonstrated for the SoultzsousForêts research site, numerical modelings can provide significant insights to better understand the hydrothermal circulation or the rock physics of EGS reservoirs (Jain et al. 2015; Kolditz and Clauser 1998; Pruess 1990; Sanyal 2000; Tomac 2017). Numerical reservoir models can typically be classified according to different aspects (WillisRichards and Wallroth 1995): (i) the description of the complex fracture network geometries they integrate via stochastic distribution (Baujard and Bruel 2006; Cacas et al. 1990) or regular grids (Watanabe and Takahashi 1995; Kohl and Mégel 2007; WillisRichards et al. 1996), (ii) the analysis of detailed physical processes such as full thermo–hydro–mechanical–chemical (THMC) couplings they account for (Kohl et al. 1995; Gelet et al. 2012; Diersch and Kolditz 1998; Bachler and Kohl 2005; McDermott et al. 2006a, b). Thermohydromechanical (THM) modeling based on a homogenized description of the reservoir has been recently presented in Magnenet et al. (2014) for the SoultzsousForêts site to describe the natural hydrothermal circulation.
In the continuity of this work, the current study aims at proposing a new model of the largescale hydrothermal circulation in the recent Rittershoffen EGS site. The numerical modeling is based on the current local geological and geophysical knowledge from Rittershoffen, but also the closeby SoultzsousForêts site. The 2D reservoir model includes all major THM couplings. The equations governing THM processes are solved by a finite element approach using the Code_Aster software. The reservoir is homogenized at the scale of 100 m. The fluid rheology (e.g., density, viscosity, heat capacity) is considered as dependent on temperature and fluid pressure as shown by laboratory measurements (Zaytsev and Aseyev 1992; Kestin et al. 1981; Rowe and Chou 1970). We include different a priori settings: (i) the main geological structures of the sedimentary cover (Aichholzer et al. 2016) and the basement (Vidal et al. 2016a), (ii) the temperature–depth profiles through the deep boreholes GRT1 and GRT2 (Baujard et al. 2017), (iii) the distribution of the natural radioactivity (Rummel 1992; Pribnow et al. 1999; Pribnow and Schellschmidt 2000), (iv) the regional stressstate (Evans et al. 2009; Cornet et al. 2007; Valley 2007), (v) the rock properties and their upscaling (GeORG13), and (vi) the geochemical data obtained from brine samples (Sanjuan 2016). We proceed to a back analysis to find the reservoir parameters from the temperature and stress logs. We finally compare the results with the insights from the SoultzsousForêts site and discuss the impact of the geological settings and largescale fault on the hydrothermal circulation, the location of the hydraulic and thermal cap rocks, the lateral variability of the reservoir temperature and the link between the geothermal gradient and the reservoir temperature at a depth of 2.0 km.
Presentation of the Rittershoffen site
Geological settings
Core studies show that the top of the granitic basement is divided into three parts (Vidal et al. 2017). From the top to the bottom, it is composed of reddish oxidized granite due to paleoweathering, hydrothermally altered granite and finegrained low altered granite. Here, we assume that after 2.5 km in depth, the granitic basement is the same in Rittershoffen as in SoultzsousForêts. The basement until 3.9 km deep is composed of a porphyritic monzogranite with Kfeldspar megacrysts in SoultzsousForêts (Dezayes et al. 2010, 2005c). Located below this is the first transition at about 3.9 km to a biotite and amphibole enriched granite and the second transition at about 4.6 km to a rather different leucogranite with very finegrained micas.
Ultra borehole images (UBIs) logs and geochemical analyses have been performed to investigate the structural properties of the fracture networks in Rittershoffen (Vidal et al. 2016a; Dezayes et al. 2014; Vidal et al. 2017). As in SoultzsousForêts, two main natural fracture systems have been identified (Dezayes et al. 2014). The first is composed of closely connected mesofractures. The second is a set of large fractures crossing the former system. From structural analysis, the main set of fractures is oriented around N15–20\(^\circ\)E with a dip of 80\(^\circ\)W in GRT1, but more scattered in GRT2. In the sediments, the density of fractures is about 0.33 fractures per meter. Among eight major fracture areas (i.e., with a thickness higher than 1 cm), only one fracture cluster is considered as permeable (Vidal et al. 2016b). It has an orientation of N20\(^\circ\)E with a dip of 85\(^\circ\)W. In contrast, the top of the basement is highly fractured with about 2.51 fractures per meter, even more than in SoultzsousForêts (with 0.65 fractures per meter). Four permeable fracture areas have been observed among 11 major fracture zones. The fracture zones in the granite have the same orientation as in the SoultzsousForêts site, parallel to the regional orientation.
Temperature profiles and hydrothermal circulation
Figure 1b illustrates the temperature–depth profiles in GRT1 and GRT2 (Baujard et al. 2017). From the surface to the top of the Muschelkalk, the geothermal gradient is constant (around 85 \(^\circ\)C km\(^{1}\)) in both wells. The value is slightly lower than in SoultzsousForêts where it is about 110 \(^\circ\)C km\(^{1}\). Below, the geothermal gradient suddenly declines about 30 times, i.e., around 3 \(^\circ\)C km\(^{1}\) in GRT1 and about 18 \(^\circ\)C km\(^{1}\) in GRT2 at the time of the measurement. The difference in geothermal gradient between the two wells is explained by the thermal nonequilibrium of the GRT2 well (Baujard et al. 2016). Some local temperature perturbations have been recorded in the profiles between 1500 and 2700 m in depth. They are commonly considered as evidence of hydrothermal circulation through fracture zones in particular around 1650 and 2350 m deep in GRT1. The temperature evolution with depth is unknown below the bottom of the wells. Temperature logs in SoultzsousForêts suggest that the geothermal gradient tends to the average Central European gradient (around 30 \(^\circ\)C km\(^{1}\)) in the deep granitic basement (Genter et al. 2010; Pribnow et al. 1999). From the surface to the top of the Muschelkalk, the linear temperature trend suggests that the thermal state is purely diffusive.
Typical rock properties according to: (1) Magnenet et al. (2014); (2) Kohl (2000); (3) GeORG); (4) Bar (2012); (5) Rummel (1992); (6) Haenel (1983); (7) Freymark et al. (2017); (8) Kirk and Williamson (2012); (9) Sausse (2002); (10) Heap et al. (2017); (11) Griffiths et al. (2016); (12) Hettkamp et al. (1999)
Property (unit)  Upper sediments  Lower sediments  Upper granites  Lower granites 

Porosity, \(\phi _o\) (\(\%\))  3.0\(^{[1]}\)–35.0\(^{[1]}\)  2.9\(^{[10]}\)–20.7\(^{[11]}\)  0.13\(^{[3]}\)–25.55\(^{[3]}\)  0.13\(^{[3]}\)–0.8\(^{[1]}\) 
Total specific mass, \(r_0\) (kg m\(^{3}\))  2300\(^{[1]}\)–2600\(^{[1]}\)  2180\(^{[4]}\)–2660\(^{[7]}\)  2500\(^{[1]}\)–2800\(^{[1]}\)  2650\(^{[6]}\)–2800\(^{[6]}\) 
Young’s modulus, E (GPa)  10.0\(^{[1]}\)–90.0\(^{[1]}\)  8.0\(^{[1]}\)–39.0\(^{[5]}\)  25.0\(^{[9]}\)–80.0\(^{[5]}\)  25.0\(^{[9]}\)–80.0\(^{[5]}\) 
Poisson’s ratio, \(\nu\) (–)  0.1\(^{[9]}\)–0.33\(^{[1]}\)  0.06\(^{[1]}\)–0.46\(^{[1]}\)  0.1\(^{[9]}\)–0.38\(^{[5]}\)  0.1\(^{[9]}\)–0.38\(^{[5]}\) 
Biot coefficient, b (–)  0.65\(^{[1]}\)–0.8\(^{[1]}\)  0.8\(^{[1]}\)–1.0\(^{[1]}\)  0.27\(^{[1]}\)–0.45\(^{[1]}\)  0.27\(^{[1]}\)–0.45\(^{[1]}\) 
Specific heat, \(c_s\) (J kg\(^{1}\) K\(^{1}\))  800\(^{[1]}\)  800\(^{[1]}\)  800\(^{[1]}\)  800\(^{[1]}\) 
Thermal conductivity, \(\lambda _{d}\) (W m\(^{1}\) K\(^{1}\))  1.1\(^{[3]}\)–5.9\(^{[3]}\)  1.2\(^{[3]}\)–4.2\(^{[3]}\)  2.3\(^{[3]}\)–4.3\(^{[3]}\)  2.3\(^{[3]}\)–4.3\(^{[3]}\) 
Thermal dilation, \(\alpha _0\) (10\(^{5}\) K\(^{1}\))  1.3\(^{[8]}\)–1.5\(^{[8]}\)  1.3\(^{[8]}\)–1.5\(^{[8]}\)  1.4\(^{,[1]}\)  1.4\(^{[1]}\) 
Heat source production, \(\theta _{\text{rad}}\) (\(\upmu\)W m\(^{3}\))  0.1\(^{[2]}\)–1.0\(^{[3]}\)  0.5\(^{[1]}\)–1.0\(^{[3]}\)  1.0\(^{[6]}\)–6.2\(^{[5]}\)  1.0\(^{[6]}\)–6.2\(^{[5]}\) 
Permeability, \(K_{\text{int}}\) (m\(^2\))  \(1.0\times 10^{18^{[4]}}\)–\(3.2\times 10^{14^{[4]}}\)  \(1.0\times 10^{18^{[11]}}\)–\(1.0\times 10^{13^{[10]}}\)  \(1.0\times 10^{20^{[12]}}\)–\(3.0\times 10^{14^{[12]}}\)  \(1.0\times 10^{20^{[12]}}\)–\(1.8\times 10^{15^{[3]}}\) 
The contribution of natural granite radioactivity to the origin of the thermal anomaly in the Upper Rhine Graben has been studied from core analyses at the SoultzsousForêts site (Rummel 1992; Pribnow et al. 1999; Pribnow and Schellschmidt 2000; Baillieux et al. 2013). The production rates are typically of the order of 0.1, 1.0, 5.0 \(\upmu\)W m\(^{3},\) respectively, for the upper sediments, the Buntsandstein sandstone and the two granites (Kohl 2000). We will assume that they are similar at Rittershoffen.
Rock physics
No direct laboratory measurements of the sediments and granite properties are available for the Rittershoffen site. This is why the rock properties at Rittershoffen are assumed to be the same as at the vicinity of the SoultzsousForêts site. Table 1 presents a synthetic review of the relevant rock properties obtained either from laboratory measurements on core samples or from geophysical investigations. Thermal conductivities vary between 1.1 and 5.9 W m\(^{1}\) K\(^{1}\) (GeORG 2013) and permeabilities between 1.0 \(\times\) 10\(^{20}\) and 3.2 \(\times\) 10\(^{12}\) m\(^2\) with a large variability which will be used as prior distribution for the back analysis (Hettkamp et al. 1999; Kohl 2000; Bar 2012; GeORG 2013; Magnenet et al. 2014; Griffiths et al. 2016; Heap et al. 2017).
Toward a largescale reservoir model at Rittershoffen
The goal of the study is to build the simplest THM numerical model that is consistent with the main characteristics of the Rittershoffen site. The model does not aim to describe all the complexity of the geology or the deterministic details of the faults networks (Fig. 1a). As sketched in Fig. 1b, the whole sedimentary cover is split into two horizontal homogenized units: the upper sediments and the lower sediments. The depth of the transition between the two units is taken as a parameter to be adjusted (named \(e_{1}\)) during the parameter back analysis. The basement is also split into two units: the upper granites and the lower granites. Due to the lack of direct knowledge on the deep granitic basement in Rittershoffen, the transition between the two units is set at a depth \(e_1+e_2+e_3=3.9\,\) km as in SoultzsousForêts site. The transition between the sediments and the granite is also set at \(e_1+e_2=2.2\,\) km in depth. We assume that the radiogenic sources are set at 0.1, 1.0, 5.0 \(\upmu\)W m\(^{3},\) respectively, for the upper sediments, the lower sediments and the two granites (Kohl 2000).
Methods
Governing equations of the THM model

A small perturbation assumption is made and solid grains are considered to remain in the thermoelastic regime;

The Cauchy stress tensor \(\mathbf {\sigma }\) is split into two contributions: an effective stress \(\mathbf {\sigma }'\) and a hydraulic stress \(\sigma _{p}\mathbf {1}\) (\(\mathbf {1}\) being the unit tensor);

The thermodynamic flows (effective Cauchy stress \(\mathbf {\sigma }'\), water surface mass flow \(\mathbf {M}_{w}\), heat flow \({\mathbf {q}}\)) are lineary related to thermodynamic forces (linearized strain \(\mathbf {\epsilon }\), gradient of pore pressure \(\nabla p_{w}\), gradient of temperature \(\nabla T\)), but with coefficients that may depend on temperature, porosity (denoted here \(\phi\)) or pore pressure. Hence, most of the homogenized properties (such as the specific heat at constant stress and the thermal conductivity) appearing in Hooke’s law, Darcy’s law, and Fourier’s law of the porous materials are considered as functions of the form \(f(\phi ,p_{w},T)\) by using classical mixing laws;

Following the approach of Magnenet et al. (2014) and Vallier et al. (2016), we consider a rheology of brine that is extrapolated from experimental results for artificial brines at different salinities (NaCl) (Zaytsev and Aseyev 1992; Kestin et al. 1981; Rowe and Chou 1970). More specifically, we assume that the natural brine is equivalent to a pure NaCl solution with a mean specific mass content of 100 g L\(^{1}\). The retained mathematical expressions of the brine properties (density, dynamic viscosity, thermal dilatation, thermal conductivity) are given in Table 2.
Review of the constitutive equations of the brine properties and values of empirical coefficients
Parameter  Expression  Coefficients 

Density, \(\rho ^0_{w}\) (kg m\(^{3}\))  1070  – 
Bulk modulus, \(K_{w}\) (GPa)  2.2  – 
Dynamic viscosity, \(\mu _{w}\) (Pa s)  \(\mu _w^\infty + \Delta \upmu _w^\infty \exp (\beta (T  T_{\text{ref}}))\)  \(\mu _w^\infty = 1.9\times 10^{4}\) (Pa s) 
\(\Delta \mu _w^\infty = 6.2\times 10^{6}\) (Pa s)  
\(\beta = \,0.02\,{\text {K}}^{1}\)  
\(\hbox {T}_{ref} = 406.4\,{\text {K}}^{1}\)  
Heat capacity, \(c^p_{w}\) (\(\hbox {J kg}^{1}{\text {K}}^{1}\))  \(a_{c^{p}_{w}} + b_{c^{p}_{w}}(T  T^1) + c_{c^{p}_{w}}(T  T^1)^2\)  \(a_{c^{p}_{w}}\) = 3.7 (\(\hbox {J kg}^{1}{\text {K}}^{1}\)) 
\(b_{c^{p}_{w}}\) = 0.4 (\(\hbox {J kg}^{1}{\text {K}}^{2}\))  
\(c_{c^{p}_{w}} = 4.6\times 10^{3}\) (\(\hbox {J kg}^{1}K^{3}\))  
\(T^1\) = 273.15 K  
Thermal dilation, \(\alpha _{w}\) (\(K^1\))  \(a_{\alpha _w} + 2b_{\alpha _w}(T  T^0) + 3c_{\alpha _w}(T  T^0)^2\)  \(a_{\alpha _w}=1.3\times 10^{4}\) \({\text {K}}^{1}\) 
\(b_{\alpha _w}=4.3\times 10^{7}\) \({\text {K}}^{2}\)  
\(c_{\alpha _w}=2.5\times 10^{10}\) \({\text {K}}^{3}\)  
T\(^\circ\) = 293.0 K  
Thermal conductivity, \(\lambda _{w}\) (\(W m^{1}K^{1}\))  \(a_{\lambda _w}\left[ 1  b_{\lambda _w} \exp (c_{\lambda _w}(T  T^1)\right]\)  \(a_{\lambda _w}\) = 0.7 (\(\text{W}\,\text{m}^{1}\text{K}^{1}\)) 
\(b_{\lambda _w} = 0.2\)  
\(c_{\lambda _w} = 0.02\,\text{K}^{1}\) 
The whole set of notations as well as a detailed presentation of the governing equations is presented in Appendix 1.
The finiteelement model

Temperatures are, respectively, maintained at 10.0 and 213.0 \(^\circ\)C on the upper and lower boundaries. The lateral boundaries are taken as adiabatic.

A fluid pressure of 0.1 MPa (i.e., the value of atmospheric pressure) is imposed on the upper boundary. The other boundaries are assumed to be impermeable.

The normal displacement is nil on the lower and lateral boundaries. The upper boundary is stress free.
For the initial conditions, a constant and uniform temperature of 10.0 \(^\circ\)C is assumed. The fluid pressure field is also assumed to be constant at 0.1 MPa. To ensure the convergence of the process, the computation has been divided into three steps (Magnenet et al. 2014): (i) during a short time period of 1000 years, the boundary conditions and gravity are progressively applied ; (ii) next, during 100,000 years, the system freely evolves along constant boundary conditions ; (iii) in one last increment, the system reaches a steady state by cancelling the nonstationary terms from the constitutive equations.
Inverse method
In this study, some geometrical and rheological parameters are estimated by back analysis (see next section). To do it, Code_Aster has been coupled to the Parameter ESTimation (PEST) software (Doherty 2005).
Numerous inverse methods have been established to carry out back analysis of the rock properties: MonteCarlo methods, Bayesian approaches which associate probability distribution for each parameter (Vogt 2012; Kosack et al. 2011; Tarantola 2004), the neighborhood algorithm based on random generation of new parameters (Sambridge 1999) or the genetic algorithm (PérezFlores and Schultz 2002).
Here, the PEST back analysis software is an implementation of the socalled Levenberg–Marquardt algorithm which minimizes an “error function”—typically the \(L_2\)norm of the difference between model and observations—with respect to a chosen set of parameters. Each parameter p is taken from a uniform a priori distribution called prior distribution in the range \([p_{\text {min}},p_{\text {max}}]\) chosen to be wider than experimental values (see Table 1).
The main benefit of our back analysis is that the inversion procedure based on deterministic method is less numerically intensive than the stochastic methods. However, the inversion procedure is sensitive to the initial conditions of the back analysis such as the prior distributions of the rock properties. Nonetheless, the prior distributions are well constrained thanks to databases from Rittershoffen and SoultzsousForêts sites.
Ranges of tested values during the back analysis called “prior distributions”
Property (unit)  Upper sediments  Lower sediments  Upper granite  Lower granite 

Permeability \(K_{\text{int}}\) (m\(^2\))  10\(^{21}\)–10\(^{15}\)  10\(^{21}\)–10\(^{15}\)  10\(^{21}\)–10\(^{11}\)  10\(^{21}\)–10\(^{11}\) 
Thermal conductivity \(\lambda _d\) (W m\(^{1}\) K\(^{1}\))  0.4–6.5  0.4–6.5  0.4–6.5  0.4–6.5 
Young’s modulus E (GPa)  5.0–95.0  5.0–95.0  5.0–95.0  5.0–95.0 
Poisson’s ratio \(\nu\) (–)  0.05–0.49  0.05–0.49  0.05–0.49  0.05–0.49 
Values of the rock properties fixed during the back analysis
Property (unit)  Upper sediments  Lower sediments  Upper granite  Lower granite 

Porosity \(\phi _o\) (\(\%\))  9.0  9.0  3.0  0.3 
Total specific mass \(r_0\) (kg m\(^{3}\))  2390  2390  2690  2690 
Biot coefficient b (–)  0.73  0.90  0.36  0.36 
Specific heat \(c_s\) (J kg\(^{1}\) K\(^{1}\))  800.0  800.0  800.0  800.0 
Thermal dilation \(\alpha _0\) (10\(^{5}\) K\(^{1}\))  1.4  1.4  1.4  1.4 
Heat source production \(\theta _{\text{rad}}\) (\(\upmu\)W m\(^{3}\))  0.1  1.0  5.0  5.0 

The depth of the granitic basement is taken as the same as in SoultzsousForêts. More precisely, the interface between the upper and lower granites is set at the depth \(e_{1}+e_{2}+e_{3}=3.9\,\) km. We also set the interface between sediments and granite at \(e_{1}+e_{2}=2.2\,\) km.

The ranges of values of rock properties are assumed to be the same as in SoultzsousForêts (see Table 1).

The observed temperature profile is assumed to be at the location of the surface maximum heat flux. This assumption is consistent with surface temperature maps in the Rittershoffen area (Haas and Hoffmann 1929). In practice, the numerical temperature profile has then been calculated at the side of a convective cell where the Darcy's velocity is purely ascending. The stress–depth profiles are taken from the same position.
Results of the back analysis
The goal of the study is to better understand the physics of the Rittershoffen reservoir. To address this issue, a back analysis confronting our THM model with the observed temperature and stress depth profiles is carried out. To reproduce the GRT1 temperature–depth profile, nine parameters are estimated: the thickness of the first geological layer \(e_{1}\), and the permeabilities \(K_{\text {int},i}\) and thermal conductivities \(\lambda _{i}\) of the four layers \(i=1\ldots 4\).
Two cap rocks can be identified from the estimated vertical profiles of \(K_{\text {int}}\) and \(\lambda\). The hydraulical cap rock is associated at its base to the high contrast of permeability and the thermal cap rock to the discrepancy of thermal conductivity. Here, the bottom of the hydraulical cap rock (i.e., the top of the convection cells, see Fig. 3) is identified at the interface between the upper and lower sediments. The contrast of permeability is associated here with the high fracture density in the lower sediments and the granite compared to the upper sediments (Vidal et al. 2016b, 2017). The change of the rock property does not correspond to a contrast in terms of lithology. On the contrary, the contrast of thermal conductivity is located at the interface between the sediments and the granitic basement. The whole sedimentary cover associated with a lower thermal conductivity than in the basement contributes to a thermal blanketing of the insulating sediments. The effect has been already identified as a key factor to explain the higher geothermal gradient at depth than the average European one (Freymark et al. 2017; ScheckWenderoth et al. 2014). The discrepancy in terms of depths between the permeability and thermal conductivity contrast highlights a decoupling of the cap rocks: the whole sedimentary cover corresponds to the thermal cap rock, whereas only the upper sediments behave as a hydraulic cap rock.
Discussion
Influence of the largescale fault in the THM model
In the present study, the hydrothermal circulation is assumed to be mainly driven by fractures and faults networks having a dimension less than the representative elementary volume (REV) with a size of 100 m. The hypothesis is different from many SoultzsousForêts modeling studies, which consider that largescale faults (i.e., larger than 100 m) contribute dominantly to the thermal state and accordingly have to be included in the reservoir model (Baujard and Bruel 2006; Kohl and Mégel 2007; Kohl 2000).
To support our assumption, a largescale fault has been included in our reservoir model. It corresponds to the Rittershoffen fault, a major fault zone with a north–south strike. It extends from the surface to 3.5 km deep (GeORG 2013), with a thickness of about 40 m (Baujard et al. 2016). The fault permeability has been estimated to be 5.34 \(\times\) 10\(^{14}\) m\(^2\) (Baujard et al. 2017). Several values of fault dip have been estimated from different approaches: (i) 45\(^{\circ }\) from 3D geological model at reservoir scale based on seismic and log data (Baujard et al. 2017; ii) 74\(^{\circ }\) in a regional scale best fitting plane of the seismic cloud from induced seismicity (Lengliné et al. 2017; iii) 83\(^{\circ }\) from smallscale acoustic image logs (Vidal et al. 2016a). The difference between the estimates may be linked to the scale discrepancy between the three approaches: the structural model is built at the 5 km scale, whereas the acoustic logs are performed at the scale of 0.2 m. Simulations have been carried out with the different estimated dips: 45\(^{\circ }\), 74\(^{\circ }\) and 83\(^{\circ }\) to the west.
Influence of the different couplings in the THM model
To study the effect of the different couplings on the thermal state, simulations have been carried out after canceling them. Figure 10 shows the temperature–depth profiles and the associated maps of temperatures for these simulations. We can observe that the cancellation of the dependence of the thermics on the mechanical and hydraulic processes (HM) or of the hydraulics on the mechanics and thermics (TM) leads both to a largescale diffusive case. The deactivation of the mechanical effect on the thermal and hydraulic processes (TH) highlights a convection system, but different from the full THM model. It includes only two convective cells larger than the ones from the THM model and the reservoir temperature is slightly higher than in the previous model. To conclude, cancelling the mechanical or hydraulic parts of the THM model does have a clear impact on the thermal regime and the hydrothermal circulation.
Comparison with the back analysis for the closeby SoultzsousForêts site
A twodimensional THM model has been also developed for the SoultzsousForêts site (Vallier et al. 2017). Permeabilities, thermal conductivities, Young’s moduli and Poisson’s ratios have been estimated to reproduce the GPK2 temperature log and the total stress profiles via a similar back analysis. In Soultz, the best model highlights a hydrothermal circulation below a shallow bottom of the hydraulic cap rock at a depth of 100 m, whereas it is at a depth of 1200 m in Rittershoffen. Figure 4 illustrates the estimated permeabilities and thermal conductivities–depth profiles for both Rittershoffen and SoultzsousForêts sites.
In Soultz, the inverted permeabilities are, respectively, \(1.0\times 10^{17}\hbox {m}^{2}\), \(3.5 \times 10^{15}\hbox {m}^{2}\), \(6.0 \times 10^{15}\hbox {m}^2\) and \(2.5 \times 10^{16}\hbox {m}^2\) for the upper sediments, the lower sediments (below 100 m deep), the upper and the lower granites. The permeability increases with depth from the upper sediments to the lower sediments and upper granites. Importantly, the permeability of the lower sediments and the upper granite is very similar suggesting that the lithological transition between the sedimentary cover and the granitic basement is not significant for the hydraulic properties. The permeability decreases after 3.9 km in depth.
For the thermal conductivity in Soultz, the values are 3.1 \(W m^{1} \hbox {K}^{1}\) in the granites and 2.1 \(W m^{1} \hbox {K}^{1}\) in the sediments. The thermal conductivity increases at the interface between sediments and granitic basement. Then, its value remains constant in the whole granitic basement. The thermal property is controlled by the interface between the sediments and granites on the contrary to permeability.
Profiles for the two properties highlight that both geothermal sites share noticeable similarities. Inverted properties of the best models show common general trends and in particular, the decoupling of behaviors between the thermal and hydraulic cap rocks. Both sites show a blanketing effect from the whole sedimentary cover, whereas the top of the convective cells (i.e., the bottom of the hydraulic cap rock) is at the transition between the upper and lower sediments. The other similarity between both geothermal sites concerns the rock properties. The reservoir permeability is similar between both sites, about 1.0 \(\times\) 10\(^{14}\,\)m\(^2\) in the granitic basement and the lower sediments. The permeability is also very close in the deep granites, about 6.0 \(\times\) 10\(^{15}\,\)m\(^2\). The slightly higher permeability for the Rittershoffen reservoir can be explained by the more important fracture density in the granite compared to SoultzsousForêts (Vidal et al. 2016b).
There is a noticeable discrepancy between the two sites: the thicknesses of the hydraulic cap rocks. The difference is more than 1 km and leads to a discrepancy between the permeabilities in the upper sediments. This difference can be explained by a higher fracture density for the sediments in SoultzsousForêts than in Rittershoffen. This is consistent with the recent stratigraphic studies comparing the two geothermal sites (Aichholzer et al. 2016). Indeed, a more intense fault network has been observed for the sediments in SoultzsousForêts (Aichholzer et al. 2016; Vidal et al. 2016a).
Figure 6 also provides a comparison of the estimated moduli for Rittershoffen and Soultz sites. The Young’s modulus and Poisson’s ratio are very similar in particular for the granitic basement between sites.
Comparison with the hydrothermal characterization of the GRT1 and GRT2 wells
The study of (Baujard et al. 2017) analyzes the database obtained after stimulation and circulation testing of GRT1 and GRT2 wells in the Rittershoffen reservoir. In particular, a double porosity model area has been carried out with AQTESOLV software (Baujard et al. 2016) from the hydraulic data of the pumping tests. The reservoir thickness is assumed to be about 500 m and the model includes a fracture area 40 m thick. We aim to compare the results from our modeling approach with their interpretations of production and circulation tests.
Baujard et al. (2017) proposed that the depth of the top of the Muschelkalk formations, i.e., 1.65 km, is the depth of the hydraulic cap rock owing to the transition in the temperature log. In our study, the depth of the hydraulic cap rock has been evaluated at 1.2 km ± 0.1 km after our back analysis, which is significantly above the important change of the temperature gradient. From the interpretation of temperature logs, the height of the convection cells is 1350 m (Baujard et al. 2017). In our model, the simulated convection cells have a height of 2.7 km. They extend shallower into the sediments and much deeper into the granitic basement until 3.9 km in depth assuming that the deep granites in Rittershoffen are similar to the ones from the SoultzsousForêts site.
Baujard et al. (2017) has also evaluated the Rayleigh number at Rittershoffen. This calculation aims to confirm that a hydrothermal convection occurs inside the Rittershoffen reservoir. The Rayleigh number is found to be included between 11.1 and 535.7. By using the same definition of the Rayleigh number (Desaive 2002; Turcotte 2014), we found a value of 50. Interestingly our value is included in the range found by Baujard et al. (2017). Knowing that the critical Rayleigh number is about 39.5 (Turcotte 2014), it confirms that a spontaneous convection inside the fractured granite is expected in the Rittershoffen model (Murphy 1979).
Temperature lateral variability
In the prospect of future geothermal exploitation, a precise assessment of the reservoir temperature at depth is required from measurements acquired on the near surface. To address the issue, we aim to bring some insights concerning the link between the lateral variability of the reservoir temperature at 2000 m depth and the one of the geothermal gradient obtained at the near surface.
Figure 11 shows different temperature–depth profiles taken at several horizontal positions (every kilometer). Measured Tlogs GRT1 and GPK2 have been added for comparison. The lateral variability of the reservoir temperature at 2000 m deep is about 40–50 \(^\circ\)C. To be noted, the variability of the temperature–depth profiles in the Rittershoffen largescale model is not enough to reproduce the GPK2 temperature–depth profile observed at the SoultzsousForêts site even if both geothermal sites share similarities in terms of rock properties.
To better understand the link between the lateral variabilities of the reservoir temperature and the geothermal gradient, Fig. 11 illustrates also the variation of geothermal gradient at the near surface (for the first 200 m depth) along the xaxis. The thermal gradient varies between 76 and 91 \(^\circ\)C km\(^{1}\). The same periodicity of 6 km is observed for both the geothermal gradient and the reservoir temperature. Interestingly, this periodicity of 6 km can be compared to the distance between SoultzsousForêts and Rittershoffen sites (around 6.5 km).
Conclusion
By using a back analysis confronting a THM model to the temperature and stress profiles observed at Rittershoffen, an excellent fit of the Tlog has been found as well for the regional stress–depth trends. The bottom of the hydraulic cap rock (i.e., the top of the convection cells where a contrast of permeability is obtained) is identified at a depth of 1.2 km ± 0.1 km. This depth is close to the bottom of the Tertiary formations and does not correspond to the discontinuity of temperature–depth profile observed in the GRT1 Tlog at 1.65 km deep. The computed permeability is 1.6 \(\times\) 10\(^{14}\,\)m\(^2\) for the lower sediments and the upper granite. This highlights that the lithological transition between the sediments and the granitic basement has little influence on the hydraulic property. Contrary to the hydraulic cap rock, the bottom of the thermal cap rock (i.e., the zone of thermal conductivity contrast) occurs at the interface between the sediments and the granite. The thermal conductivity is, respectively, 1.4 and 3.1 W m\(^{1}\) K\(^{1}\) in the sediments and granites. This means that the whole sedimentary cover contributes to a blanketing effect and that the thermal and hydraulical cap rocks are decoupled. The largescale permeabilities, thermal conductivities and elastic moduli are mostly consistent with the values observed at the laboratory scale. They have been compared to the ones obtained from a similar back analysis at the closeby Soultz geothermal site. The permeability, thermal conductivity, Young’s modulus and Poisson’s ratio have the same general trend with depth and similar values at Soultz and Rittershoffen. Both sites highlight the same decoupling of the hydraulic and thermal cap rocks. The lateral variability of the reservoir temperature at 2.0 km deep is similar between Rittershoffen and SoultzsousForêts around 40–50 \(^\circ\)C. The same lateral periodicity of 6 km has been found for the geothermal gradient obtained from the near surface and the reservoir temperature. This might lead potentially to a promising tool to assess future geothermal resources. Moreover, further works are currently being done to investigate the potential influence of major faults in the model (e.g., here the Rittershoffen fault). This will allow us to better understand their influence on both thermal and mechanical behaviors inside the reservoir.
Declarations
Authors’ contributions
BV wrote the first version of the paper. The three other authors contributed to the final version of the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The present work has been done under the framework of the LABEX ANR11LABX0050GEAUTHERMIEPROFONDE and benefits from a state funding managed by the French National Research Agency (ANR) as part of the “Investments for the Future” program. It has also been funded by the EGS Alsace Grant from ADEME. The authors would like to thank Christoph Clauser, Albert Genter, Clément Baujard, Thomas Kohl, Chrystel Dezayes, David Bruhn, Nima Gholizadeh Doonechaly, Bernard Sanjuan, Benoit Valley, Judith Sausse, Philippe Jousset, Dominique Bruel, Eva Schill, Patrick Baud, Mike Heap, Luke Griffiths, Alexandra Kushnir, Olivier Lenglinè, Coralie Aichholzer, Philippe Duringer and François Cornet for very fruitful discussions. We thank also the anonymous reviewers and the editorial team for their comments.
Competing interests
The authors declare that they have no competing interests.
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