Two-dimensional THM modelling of the large scale natural hydrothermal circulation at Soultz-sous-Forêts
- Vincent Magnenet^{1}Email author,
- Christophe Fond^{1},
- Albert Genter^{2} and
- Jean Schmittbuhl^{3}
DOI: 10.1186/s40517-014-0017-x
© Magnenet et al.; licensee Springer. 2014
Received: 14 April 2014
Accepted: 20 November 2014
Published: 17 December 2014
Abstract
Background
A two-dimensional numerical model is developed for the Soultz-sous-Forêts reservoir from an idealized cross-section containing six homogeneous horizontal layers. The considered constitutive equations are those of homogenized saturated porous media involving Thermo-Hydro-Mechanical (THM) couplings, and most of materials properties (for brine and rocks) are taken dependent on temperature and pressure.
Methods
The constitutive equations are solved in transient regime with the finite element software Code_Aster to reach a stationary state of the reservoir.
Results
We show that a large scale natural convection is compatible with present boundary conditions if the permeability of the reservoir is of the order of 1.0×10^{−14}m^{2}. Convection cells are of the order of 1.3 km in width and we analyze several vertical profiles and maps of physical properties.
Conclusions
A stationary convective solution at large scale is highlighted.
Keywords
Geothermal reservoir Finite element method Thermo-hydro-mechanical model Natural convectionBackground
The interest of obtaining a numerical and coupled model of a given geothermal reservoir is fourfold. First, it allows the physical integration of laboratory measurements (rock physics), such as well logging, well head parameters, geological description, and geophysics field measurements. It shows how data are precious input parameters of the model and gives them an utility of great importance. Furthermore, numerical simulations can not only help to interpret and understand physical processes measured with laboratory and field experiments but also address and quantify the relevant processes occurring in a reservoir. Secondly, it provides a direct model based on geophysical inversion of field measurements: fluid flow, fluid pressure, temperature profile, seismicity monitoring, deformation of the ground surface (INSAR/GPS) related to reservoir modification, and gravity or electromagnetic geophysical measurements. Another advantage of simulating the reservoir behaviour is the possibility to analyse the sensitivities of parameters involved in the hydrothermal circulation (or in other physical processes). This analysis can lead to the identification of relevant processes occurring in the reservoir and material properties having the greatest influence on the model outputs, thus providing useful informations for the planification of new experimental investigations. Finally, the model can also be used as a decision tool for drilling and implacement planning, stimulation and exploitation.
Several models have been developed in literature to reproduce field measurements or to predict the value of physical properties in reservoirs. The most simplified approaches are unidimensional. They couple hydraulic with thermics as in (Pasquale et al. [2011b]), but more complex geometries have been considered in two dimensions as in ([Guillou-Frottier et al. 2013]; [Kohl et al. 2000]; [Cerminara and Fasano 2012]; [Magri et al. 2010]) or in three dimensions ([Bächler et al. 2003]; [Kohl and Mégel 2007]). Most of these models aim at reproducing or predicting the temperature profile measured in wells and/or hydraulic data obtained during injection and production phases (water flow, water pressure and temperature vs. time). Some numerical models have also been developed to account for mechanical, thermal and hydraulic couplings at the same time, with a simplified geometry of the fault network, see, e.g. [Kohl et al. (1995]) or [Gelet et al. (2012]). In addition, other thermodynamical aspects have been addressed like chemical couplings in [Bächler and Kohl (2005]).
The present work is in line with these previous approaches and can be viewed as another thermo-hydro-mechanical (THM) model of the Soultz-sous-Forêts geothermal reservoir. Our geomechanical model aims at solving constitutive equations using poroelastic medium theory, completed by reasonable boundary conditions. The goal is first to show the existence of a convective and stationary solution at large scale, considering only the regional stratigraphy and known rheophysic properties of the rock matrix and the saturating brine. This solution, different from the Elder problem ([Elder 1967]), is the starting point to proceed to a stability analysis of the reservoir when it is submitted to some mechanical, thermal or hydraulic perturbation specific of stimulation or production phases. As it will be shown below, the constitutive equations governing the evolution of the reservoir are strongly nonlinear. Consequently, one expects that the time evolution of the reservoir may be highly sensitive to initial conditions and that it can reach several metastable equilibrium states during the simulation. Given the complexity of the stability analysis, we focus in this paper on the existence of a stationary state of the Soultz-sous-Forêts reservoir. We consider it at large scale (about 10 km in width and 5 km in height). Our specificity is to assume that media are homogenized, i.e. at a scale above the representative elementary volume (REV) of the well-known contribution of [Coussy (2004]), as porous materials saturated with a single-phase fluid but including all major THM couplings. Our work is limited to a two-dimensional modelling, as in the recent contribution from [Guillou-Frottier et al. (2013]). The main geological structures retained here are as follows: (i) the main sedimentary beddings of the Rhine Graben and (ii) major petrographic transitions in the granite, which are supposed to be horizontal. No fault is included in the model at this stage. Despite these strong geometrical assumptions, we aim at accounting for the rich rheologies of rocks and brine, which constitute the main contribution of the present work compared to the above mentioned studies. At last, let us notice that a transient simulation has been chosen to ‘slowly’ evolve towards a stationary state starting from an initial (given) state of the reservoir. Indeed, integrating the constitutive equations governing THM processes is strongly constrained by stability issues due to the non-linearity of these equations.
The outline of the paper is the following: first, we present the constitutive equations considered in the model. We then develop numerical aspects and finally discuss the main results of our simulations.
Methods
Governing equations
Set of thermodynamic variables associated with each form of energy
Phenomenon | Generalized displacement | Generalized strain | Generalized stress |
---|---|---|---|
Mechanics | ξ [m] | ε =(∇ ξ+∇^{ T }ξ)/2 | σ [Pa] |
Mechanical displacement | Linear strain | Cauchy stress tensor | |
Hydraulics | p_{ w } [Pa] | ∇p_{ w } [Pa.m^{-1}] | M [kg.m^{-2}.s^{-1}] |
Water pressure | Hydraulic flow | ||
Thermics | T [K] | ∇T [K.m^{-1}] | q [J.m^{-2}.s^{-1}] |
Temperature | Heat flow |
where σ [Pa] is the Cauchy stress tensor, r [kg.m^{-3}] the total homogenized specific mass, and F^{ m } [N.kg^{-1}] the massic force density (gravity in the present paper). In this work, r is decomposed into two contributions, r=r_{0}+m_{ w }, r_{0} [kg.m^{-3}] being the initial total homogenized specific mass and m_{ w } [kg.m^{-3}] the mass content of water, that is the mass of water d m_{ w } that entered or left an elementary volume dV of the porous medium since the initial state, per unit of volume. The vector M_{ w } [kg.m^{-2}.s^{-1}] appearing in Equation (2) is the massic flow of water. Concerning Equation (3), ${h}_{w}^{m}$ [J.kg^{-1}] is the specific enthalpy of water, Q [J.m^{-3}] is the ‘non-convective’ heat (see below) and q [J.m^{-2}.s^{-1}] the heat flow due to conduction. The radioactivity of rocks is taken into account through the heat source term Θ [W.m^{-3}].
with ${\rho}_{w}^{0}$ [kg.m^{-3}] the initial specific mass of water and ϕ_{0} the initial porosity.
Physical properties of constituents
At this stage, the main constitutive equations have been presented, and this section is devoted to the presentation of relationships governing the evolution of physical properties with temperature, porosity, and pressure. The entire set of properties is extracted from the literature dealing with experimental investigations.
Properties of brine retained in this work
Parameter | Unit | Range of values/formula | Comments | Retained value |
---|---|---|---|---|
${\rho}_{w}^{0}$ | kg.m^{-3} | Obtained by fitting | 1,070 | |
Equation (15) with the experimental | ||||
correlation of ([Rowe and Chou 1970]) | ||||
K _{ w } | GPa | 2.0 — 4.0 | 2.2 | |
μ _{ w } | Pa.s | ${\mu}_{w}^{\infty}+\Delta {\mu}_{w}^{\infty}exp\left[\beta (T-{T}_{\mathit{\text{ref}}})\right]$ | Obtained by fitting | ${\mu}_{w}^{\infty}=\phantom{\rule{0.3em}{0ex}}193.4\times $10^{-6} Pa.s |
data found in ([Kestin et al. 1981]) | Δ μ_{ w }= 61.7 ×10^{-6} Pa.s | |||
β= −0.02395 K^{-1} | ||||
T_{ ref }= 406.4 K | ||||
${c}_{w}^{p}$ | J.kg^{-1}.K^{-1} | ${a}_{{c}_{w}^{p}}+{b}_{{c}_{w}^{p}}(T-{T}^{1})+{c}_{{c}_{w}^{p}}{(T-{T}^{1})}^{2}$ | Experimental formula | ${a}_{{c}_{w}^{p}}=3,703.3\phantom{\rule{0.3em}{0ex}}$J.kg^{-1}.K^{-1} |
found in ([Zaytsev and Aseyev 1992]) | ${b}_{{c}_{w}^{p}}=0.395773\phantom{\rule{0.3em}{0ex}}$J.kg^{-1}.K^{-2} | |||
${c}_{{c}_{w}^{p}}=4.64025\times 10$^{-3} J.kg^{-1}.K^{-3} | ||||
T^{1}=273.15 K | ||||
α _{ w } | K^{-1} | ${a}_{{\alpha}_{w}}+2\phantom{\rule{0.3em}{0ex}}{b}_{{\alpha}_{w}}(T-{T}^{0})+3\phantom{\rule{0.3em}{0ex}}{c}_{{\alpha}_{w}}{(T-{T}^{0})}^{2}$ | Obtained by fitting | ${a}_{{\alpha}_{w}}=1.3224\times 1{0}^{-4}\phantom{\rule{0.3em}{0ex}}$K^{-1} |
Equation (15) with the experimental | ${b}_{{\alpha}_{w}}=4.3315\times 1{0}^{-7}\phantom{\rule{0.3em}{0ex}}$K^{-2} | |||
correlation of ([Rowe and Chou 1970]) | ${c}_{{\alpha}_{w}}=2.49962\times 1{0}^{-10}\phantom{\rule{0.3em}{0ex}}$K^{-3} | |||
λ _{ w } | W.m^{-1}.K^{-1} | ${a}_{{\lambda}_{w}}\left(1-{b}_{{\lambda}_{w}}exp\left[-{c}_{{\lambda}_{w}}(T-{T}^{1})\right]\right)$ | Obtained by fitting the | ${a}_{{\lambda}_{w}}=0.691131\phantom{\rule{0.3em}{0ex}}$W.m^{-1}.K^{-1} |
experimental correlation found | ${b}_{{\lambda}_{w}}=0.231942$ | |||
in ([Zaytsev and Aseyev 1992]) | ${c}_{{\lambda}_{w}}=2.22312\times 1{0}^{-2}\phantom{\rule{0.3em}{0ex}}$K^{-1} | |||
T^{1}=273.15 K |
Typical properties of some sedimentary rocks and their dependence with temperature
Property | Unit | Range of values | Comments |
---|---|---|---|
ϕ _{0} | % | 3 to 35 | Mean value of approximately 15 |
ρ _{dry} | kg.m^{-3} | 1,900 to 2,600 | Mean value of approximately 2,300 |
b | 1 | 0.65 to 0.80 | Lavoux limestone |
0.8 to 1.0 | sandstone | ||
${c}_{\text{dry}}^{\sigma}$ | J.kg^{-1}.K^{-1} | Approximately 800 | Increase of approximately 15% |
if T∈ [20; 250]°C | |||
K _{int} | m^{2} | 10^{-16} to 10^{-9} | High variability |
λ _{dry} | W.m^{-1}.K^{-1} | 0.5 to 6 | Mean value of approximately 2; |
decrease of approximately 25% | |||
if T∈ [20;250]°C | |||
α _{0} | 10^{-5} K^{-1} | 1.3 to 1.5 | Approximately constant if T∈ [50; 200]°C |
E | GPa | 10 to 92 | Limestones (mean value of approximately 50) |
2 to 39 | Sandstones (mean value of approximately 15) | ||
8 to 22 | Schists (mean value of approximately 14) | ||
ν | 1 | 0.12 to 0.33 | Limestones (mean value of approximately 0.25) |
0.06 to 0.46 | Sandstones (mean value of approximately 0.24) | ||
0.03 to 0.18 | Schists (mean value approximately 0.08) | ||
Θ | µW.m^{-3} | 0.3 to 1.8 |
Typical properties of granite and their dependence with temperature
Property | Unit | Range of values | Comments |
---|---|---|---|
ϕ _{0} | % | 0.8 | |
ρ _{dry} | kg.m^{-3} | 2,500 to 2,800 | Mean value of approximately 2,600 |
b | 1 | 0.27 to 0.45 | |
${c}_{\text{dry}}^{\sigma}$ | J.kg^{-1}.K^{-1} | ∼ 800 | Increase of approximately 25% |
if T∈ [20; 250]°C | |||
K _{int} | m^{2} | 10^{-20} to 10^{-18} | Sane |
10^{-16} to 10^{-11} | Fractured | ||
λ _{dry} | W.m^{-1}.K^{-1} | 2.3 to 3.2 | Decrease of approximately 1.2 |
if T∈ [20; 250]°C | |||
α _{0} | 10^{-5} K^{-1} | 1.4 | Approximately constant if T∈ [30; 200]°C |
E | GPa | 26 to 78 | Mean value of approximately 59 |
30 | Approximately constant if T∈ [30; 160]°C | ||
ν | 1 | 0.10 to 0.38 | Mean value of approximately 0.23 |
0.25 | Approximately constant if T∈ [30; 160]°C | ||
Θ | µW.m^{-3} | 0.7 to 7.6 |
Values of properties taken as inputs of the model
Property | Unit | Layer 1 | Layer 2 | Layer 3 |
---|---|---|---|---|
ϕ _{0} | % | 0.09 | 0.09 | 0.09 |
r _{0} | kg.m^{-3} | 2,390 | 2,390 | 2,390 |
E | GPa | 50 | 50 | 15 |
ν | 1 | 0.25 | 0.25 | 0.24 |
b | 1 | 0.73 | 0.73 | 0.9 |
${c}_{\text{dry}}^{\sigma}$ | J.kg^{-1}.K^{-1} | 950+0.5 (T−293.15) | 950+0.5 (T−293.15) | 500+0.5 (T−293.15) |
λ _{dry} | W.m^{-1}.K^{-1} | 2.2−0.0025 (T−293.15) | 2.2−0.0025 (T−293.15) | 2.6−0.0025 (T−293.15) |
α _{0} | 10^{-5} K^{-1} | 1.4 | 1.4 | 1.4 |
Θ | µW.m^{-3} | 0.5 | 0.5 | 0.5 |
K _{int} | m^{2} | 1.0×10^{−14} | 1.0×10^{−14} | 1.0×10^{−14} |
Layer 4 | Layer 5 | Layer 6 | ||
ϕ _{0} | % | 0.03 | 0.03 | 0.03 |
r _{0} | kg.m^{-3} | 2,630 | 2,630 | 2,630 |
E | GPa | 59 | 59 | 59 |
ν | 1 | 0.23 | 0.23 | 0.23 |
b | 1 | 0.36 | 0.36 | 0.36 |
${c}_{\text{dry}}^{\sigma}$ | J.kg^{-1}.K^{-1} | 750+0.5 (T−293.15) | 750+0.5 (T−293.15) | 750+0.5 (T−293.15) |
λ _{dry} | W.m^{-1}.K^{-1} | 2.2−0.0025 (T−293.15) | 2.2−0.0025 (T−293.15) | 2.2−0.0025 (T−293.15) |
α _{0} | 10^{-5} K^{-1} | 1.4 | 1.4 | 1.4 |
Θ | µW.m^{-3} | 3.0 | 3.0 | 3.0 |
K _{int} | m^{2} | 1.0×10^{−14} | 1.0×10^{−14} | 1.0×10^{−18} |
Numerical model
The geological layers considered in this work are slightly adapted from the idealized cross section presented in the paper of [Guillou-Frottier (2011]) and inspired by the contribution of [Genter et al. (1997]) and [Genter and Traineau (1992]). We only consider six layers: the first three layers correspond to the sedimentary cover, i.e. Tertiary to Jurassic marls and clays between 0 and 800 m of depth, Keuper and Muschelkalk formations between 800 and 1,000 m, and Buntsandstein sandstones between 1,000 and 1,400 m. The granitic basement is represented by three other layers corresponding to the different petrographic units of the granite massif. On top, a thin layer between 1,400 and 1,550 m is considered to reproduce the strongly altered granite from the paleo-erosion surface. The second layer, between 1,550 and 3,700 m, corresponds to a more fractured monzonitic granite in which the homogenized permeability is high. The last layer, located between 3,700 and 5,350 m, corresponds to a crystalline unit made of a porphyritic monzogranite and a fine grained granite. The height of the model corresponds to the height of the measured temperature profile obtained in GPK2, see, e.g. [Pribnow and Schellschmidt (2000]).
Once a stationary state is obtained, post-processing operations - such as the calculation of the simulated vertical profiles of temperature and vertical stress - are performed.
Initial state
and no mechanical displacements are considered, ξ^{ini}=0 .
Obviously, both formula are evaluated for T≡T^{ini}(x, y) and ${p}_{w}\equiv {p}_{w}^{\text{ini}}(x,y)$.
Coefficients used in Equation (32)
Component | Notation | a_{ ij }(kPa) | b_{ ij }(MPa) |
---|---|---|---|
σ _{ xx } | σ _{ h } | −14.9 | 5.92 |
σ _{ xy } | σ _{ hv } | 0.0 | 0.0 |
σ _{ yy } | σ _{ v } | −33.6 | 25.3 |
σ _{ zz } | ${\sigma}_{h}^{\perp}$ | −25.5 | 1.31 |
Results and discussion
where the blue nodes are the nodes of the linear mesh (used for water pressure and temperature degrees of freedom) and the red ones are the supplementary nodes of the quadratic mesh (used for mechanical displacements). Based on the results of the most refined meshes, it can be concluded that a suitable mesh refinement was reached and that element type have no significant effect on the simulation. Subsequently, we decided to work arbitrarily with quad elements.
The temperature map of Figure 11 illustrates the fact that the vertical profile of temperature depends on the horizontal coordinate x. To quantify the lateral variation of temperature and other calculated quantities with the horizontal coordinate x, different vertical profiles have been plotted in Figure 7a every 500 m between 0 and 2,500 m. We recall that the value x=0 corresponds to the left of the mesh. A lateral variation of about 50°C is observed in the interval of depth corresponding to the zone of convective flow. At this stage, the numerical profile reproduces the main tendency of the experimental profile, whilst not fitting exactly the data in particular in the upper part of the convective cell (Buntsandstein layer) where the large scale permeability is possibly underestimated. This difference can also be explained by the fact that we used values of input parameters extracted from the literature without any back analysis to refine them.
To support the last argument, we return to the definition of the Eulerian porosity as the ratio between the actual void volume and the total actual volume of an elementary representative piece of saturated porous medium. Since the dilation of solid grains (≈1.4×10^{−5}, quasi independent of temperature) is about ten times smaller than the dilation of water (≥1.5×10^{−4} at 50°C), and since the system is globally unconfined and drained because of the upper stress-free and drained boundary, one may surmise that an increase of temperature will greatly increase the volume of water whilst keeping the volume of solid grains quasi constant, thus explaining a possible increase of porosity due to thermal effects. The vertical profiles presented in Figure 7b indicate that the two mentioned contradictory effects seem to globally balance. Finally, one should keep in mind that no dependences of the bulk moduli (solid framework and water) with temperature and pressure have been taken into account, and that these dependences may significantly affect the variation of porosity.
Conclusions
The two-dimensional numerical model developed in this work lead us to the following main conclusions:
A stationary convective solution at large scale is highlighted. The order of magnitude of the convective cell size in the reservoir is about 1.3 km, a value being independent of the model width. A periodic pair of cells is then 2.6 km wide. We insist on the fact that the unicity of this solution is not guaranteed. Furthermore, the way the initial conditions influence the final result is a difficult task accounting for the strong non-linearity of the constitutive equations. The unicity of this solution, its stability, and the way initial conditions may influence the final results will be analysed in a future work.
The order of magnitude of the maximal vertical Darcy velocity is 20 cm.year^{-1}, a value confirmed by previous works found in the literature.
The field of water pressure keeps globally linear with depth, and the influence of thermo-hydraulic coupling on the vertical stress state of the reservoir is rather low.
The large scale convection is triggered with a permeability in the five upper geological layers of about 1.0×10 to 1.0×10^{−15} m^{2}.
One should keep in mind that these first conclusions have been obtained with a two-dimensional model under the assumption of plane strains. They should consequently be confirmed or infirmed by a three-dimensional model, accounting for the presence of main faults in the reservoir. Works are currently in progress to follow this route and to make our model more realistic.
Declarations
Acknowledgements
The author are grateful to the reviewers for their constructive expertise and the useful hints aimed at improving the content of our manuscript. They also gratefully acknowledge Dr. François Cornet for very helpful discussions. This work has been published under the framework of the LABEX ANR-11-LABX-0050-G-EAU-THERMIE-PROFONDE and benefits from a funding from the state managed by the French National Research Agency as part of the ‘Investments for the Future’ program.
Authors’ Affiliations
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