Heat exchange during laminar flow is studied at the fracture scale on the
basis of the Stokes equation. We used a synthetic aperture model (a self-affine
model) that has been shown to be a realistic geometrical description of the
fracture morphology. We developed a numerical modelling using a finite
difference scheme of the hydrodynamic flow and its coupling with an
advection/conduction description of the fluid heat. As a first step,
temperature within the surrounding rock is supposed to be constant. Influence
of the fracture roughness on the heat flux through the wall, is estimated and a
thermalization length is shown to emerge. Implications for the
Soultz-sous-For\^{e}ts geothermal project are discussed

Neuville, Amélie

Schmittbuhl, Jean

Toussaint, Renaud

English

arXiv

Heat exchange during laminar flow is studied at the fracture scale on the basis of the Stokes equation. We used a synthetic aperture model (a self-affine model) that has been shown to be a realistic geometrical description of the fracture morphology. We developed a numerical modelling using a finite difference scheme of the hydrodynamic flow and its coupling with an advection/conduction description of the fluid heat. As a first step, temperature within the surrounding rock is supposed to be constant. Influence of the fracture roughness on the heat flux through the wall, is estimated and a thermalization length is shown to emerge. Implications for the Soultz-sous-Forêts geothermal project are discussed

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Hydrothermal coupling in a rough fractureAme´lie Neuville, Renaud Toussaint, Jean SchmittbuhlTo cite this version:Ame´lie Neuville, Renaud Toussaint, Jean Schmittbuhl. Hydrothermal coupling in a roughfracture. EDHRA conference 2006, 2006, France. EU, 3 p., 2006. <hal-00110578>HAL Id: hal-00110578https://hal.archives-ouvertes.fr/hal-00110578Submitted on 30 Oct 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.                                                                     Hydro-thermal flow in a rough fracture  EC  Contract SES6-CT-2003-502706PARTICIPANT ORGANIZATION  NAME: CNRSSynthetic  2nd year reportRelated with Work Package…………HYDRO-THERMAL FLOW IN A ROUGH FRACTURE A. Neuville, R. Toussaint, J. Schmittbuhl*Institut de Physique du Globe de Strasbourg, CNRS-UMR7516, 5 rue René Descartes, 67084 Strasbourg Cedexe-mail: Jean.Schmittbuhl@eost.u-strasbg.frABSTRACTHeat exchange during laminar flow is studied at the fracture scale  on  the  basis  of  the  Stokes  equation.  We  used  a synthetic aperture model (a self-affine model) that has been shown to be a realistic geometrical description of the fracture morphology.  We developed a  numerical  modelling  using  a finite  difference  scheme of  the  hydrodynamic  flow  and  its coupling with an advection/conduction description of the fluid heat. As a first step, temperature within the surrounding rock is  supposed  to  be  constant.  Influence  of  the  fracture roughness on the heat flux through the wall, is estimated and a thermalization length is shown to emerge. Implications for the Soultz-sous-Forêts geothermal project are discussed.INTRODUCTIONThe modelling of the fluid transport in low permeable crustal rocks is of central importance for many applications (Neuman, 2005).  Among  them  is  the  monitoring  of  the  geothermal circulation  in  the  project  of  Soulz-sous-Forêts,  France (Bachler et al, 2003). The transit time of the water into the main fractures between injection and extraction wells has to be carefully estimated in order to provide precise estimates of the experiment  duration.  Water  flux  controls  both  the  heat transfer to the fluid and the cooling of  the massif  (Murphy, 1979; Glover et al, 1998; Tournier et al, 2000). As shown from previous studies (Benderitter and Elsass, 1995; Tournier et al, 2000) only a few important fractures are controlling the fluid exchange between wells.  Accordingly the study of  the flow within a single fracture appears of central interest. It is also important  for  large  scale  numerical  simulations  of  the convective  circulation  in  the  massif  (Fontaine  et  al,  2001; Zhao et al,  2003).  Up to now, only very simple models for fractures,  like  the  parallel  plates  model,  have  been considered, ignoring the impact of the fracture morphology on the fracture transmitivity (Tournier et al, 2000).In this study, we are interested in the influence of a realistic geometry  of  the  fracture  on  its  hydro-thermal  response. Several  studies  have addressed  the  permeability  of  rough fractures (Brown, 1987; Plouraboué et al, 1995; Hakami and Larsson,  1996;  Adler  and  Thovert,  1999;  Méheust  and Schmittbuhl,  2000,  2001,  2003).  Our  goal  is  to  show that realistic  fracture  morphology  can  be  introduced  as  a perturbation of the parallel  plate model and  have a strong influence on the heat transport properties even in the limit of laminar flow.MODEL DESCRIPTIONAperture modelA  self-affine  surface  is  a  possible  geometrical  model  for roughness  topographies  of  fractures  as  shown  in  several previous studies (Brown and Scholz, 1985; Schmittbuhl et al, 1993; Schmittbuhl et al, 1995, Bouchaud, 1997). Self-affinity is a scaling invariance property that reads as: x ® x, y ® y, z  ® z where   is a scaling factor and   is the roughness exponent.  As  can  be  seen   the  scaling  transformation  is supposed to be isotropic within the mean plane (x,y) since the scaling factors are the same along x and y axes. The scalingfactor, () is not the same along the mean plane (x,y) and the out  of  plane  direction  z, ().  For  the  latter  direction,  the roughness  exponent   describes  how  anisotropic  is  the property. Figure 1 shows an example of a synthetic self-affine aperture field a(x,y). In this case, the aperture defined as the volume between two facing  fracture  surfaces,  is  also  self-affine (Méheust and Schmittbuhl, 2003). Fluid flow modelTo study the viscous flow in a rough fracture, we developed anumerical simulation based on the Stokes equation for low Reynolds number (Méheust and Schmittbuhl, 2001; 2003):∇ P=vwhere  P is the fluid pressure,   the fluid viscosity and v the Eulerian fluid velocity.Figure 1: Sample of a synthetic self-affine aperture field a(x,y) (in  mm): red corresponds to largest apertures and blue to quasi-closed areas. The rms of the aperture field is 0.46mm.                                                                     Hydro-thermal flow in a rough fracture  In  the  limit  of  small  aperture  variations  ( ∥ ∇ a∥≪1 )  ,  the lubrication approximation might hold and the local flux per unit length is then related to the local pressure gradient:u=− a312 ∇ PThis  leads  to  the  Reynolds  equation  which  is  a  two dimensional approximation of the local flux within the fracture aperture:∇ . a3 ∇ P =0We used a finite  difference scheme to solve the Reynolds equation with a conjugated gradient method. Precision was set to 10-15. Figure 2 shows the result of the computation for the aperture field shown in Fig. 1 when the fluid is injected from the left side. Upper and lower boundary are closed. The fracture is supposed to be horizontal.It  is of important to note the strong channelling of the flow which is induced by the fracture roughness. Despite isotropic aperture fluctuations along the mean fracture plane, as shown in Fig. 1, most of the flow occurs in a significant sub-part of the fracture. This localization of the flow stems from the long range  correlations  included  in  the  aperture  fluctuations.  A direct  consequence  is  the  clear  separation  between hydraulically  active  and  inactive  zones  with  a  non-direct mapping between aperture magnitude and local flux. In other terms,  at  a mesoscopic scale,   mechanical  and hydraulical apertures might strongly differ even if the average aperture is well defined and constant over the whole fracture. Heat flow modelWe  include  heat  flow  in  our  modelling  to  describe  the temperature field within the fracture when the fluid is injected at a different temperature from the surrounding rock. T0 is the temperature of the fluid at the inlet of the fracture and Tw is that of the rock. As a first step in our modelling, we assume the temperature of  the rock to be homogeneous.  We also consider a stationary condition.Conduction describes the heat flow between two regions of different temperature and fulfills the Fourier law:  q=− ∇ Tthat  relates  linearly  the  heat  flux  q and  the  temperature gradient  by  the  mean  of  the  thermal  conductivity  l. Convection is due to fluid motion and it is described by the estimate of the advected heat. From the energy conservation law (Landau and Lifchitz, 1994),  conduction effects balance convective effects and the local temperature within the fluid fulfills:T=v . ∇Twhere v is the fluid velocity and c the thermal diffusivity. For the fluid flow modelling we based our approach on the lubrication  approximation  which  relies  on  an  averaging procedure over the thickness of  the aperture.  Similarly,  we propose a “thermal lubrication” approximation which is based on  the  average  of  the  temperature   over  the  fracture thickness, weighted by the local fluid velocity:T  x , y =∫a v  x , y , z T  x , y , z dz∫h v  x , y , z dzWe also introduce the Nusselt  number  Nu=− jw /qref which compares, when no convection holds, two estimates of the conductive flux: the local flux at the boundary of the fracture jw=−∂T∂ z z=a / 2 and   the  mesoscopic  flux  at  the  fracture thickness scale qref=Tw−Ta .Finally, assuming that the horizontal conduction is negligible when compared to the vertical conduction and that the vertical convection is limited,  the local average temperature T  x , y is obtained from the following equation:a v . ∇ T2 a NuT−Tw=0We  discretized  this  equation  using  a  first  order  finite difference  scheme  and  solved  it  using  also  a  conjugated gradient method. Fig. 3 shows the fluid temperature field for the  fracture  aperture  described in  Fig.  1  and in  horizontal position. The cold fluid is injected from the left side within the fracture which is assumed to be embedded in hot rock at a homogeneous temperature. The  channelling  effect  of  the  temperature  field  is  very significant and directly related to the channelling of the fluid flow  (for  a  convection  dominated  regime).  Accordingly  the temperature is far from homogeneous in the fracture though its aperture is constant on average. It therefore emphasizes the importance of aperture fluctuations on the heat exchange between the cold fluid and the hot rock.Influence of the fracture roughness can be monitored from the comparison  with  the  parallel  plate  case  with  the  same average  aperture,  i.e. same  mechanical  aperture.  The temperature profile along the fluid flow direction (x direction) can  be  analytically  computed in  the  case of  parallel  plate boundary conditions. It reads as:T−T w=T 0−T wexp −x / l ref  where l ref=PeNu .Figure  2: Map of the normalized local flux for the aperture field  shown in Fig. 1. Red areas correspond to maximum fluxes and blue regions to immobile zones.  Figure 3: Map of the normalized temperature within the fracture shown in Fig. 1. The surrounding rock is hotter than the injected fluid. Low temperature zones are in blue and hot zones in red.                                                                      Hydro-thermal flow in a rough fracture  The  Peclet  number  is  defined  as:  Pe=v a2 .  Therefore  a characteristic length exists in the problem:  l ref . It describes the  typical  length  scale  for  the  fluid  to  be  in  thermal equilibrium with the surrounding rock.  Accordingly  this is  a thermalization  length. Figure 4 shows the temperature profile for the parallel plate model and its comparison with profiles for rough fractures. It is of interest to note that rough fractures still show  a  similar  trend  but  the  thermalization  length  is significantly larger (smaller increase of the temperature within the fracture).  In other words, cold fluid migrates further into the fracture in the rough case compared to the parallel plate case owing to the channeling effect. At large scales, numerical  oscillations emerge which stems from a too loose spatial discretization of  the thermalization length scale.CONCLUSIONSWe propose a numerical model to estimate the heat exchange at the fracture scale between a cold fluid and hot surrounding rock.  The  numerical  model  is  based  on  a  lubrication approximation  for  the  fluid  flow  and  a  “thermalization lubrication”  approximation  for  the  temperature  evolution. Gravity  is  neglected  since  we  are  dealing  with  horizontal fractures  but  could  be  included  from  a  redefinition  of  the pressure to include hydrostatic contributions.Our model shows that fracture roughness is responsible for channelling  effects.  Fluid  flow  is  dominant  in  a  significant subpart  of  the fracture where head advection is important. Accordingly,  temperature distribution is strongly affected by small fluctuations of the fracture aperture.On-going  work  addresses  implications  for  the  Soultz-sous-Forêts HDR project where large sub-horizontal fractures are controlling the fluid transfer from the injection well (GPK3) and the extraction wells (GPK2 and GPK4). For this purpose we link our fracture model to a large scale dipolar model of the fluid flow. REFERENCESAdler,  P.,  Thovert,  J.,  (1999).  Fractures  and  Fracture Networks. Kluwer academic publishers, The Netherlands.Bachler, D., Kohl, T., Rybach, L., (2003). Impact of graben-parallel faults on hydrothermal convection - rhine graben case study. Phys. Chem. Earth 28, 431-441.Benderitter, Y., Elsass, P., (1995). Structural control of deep fluid  circulation  at  the  soultz  hdr  site,  france:  a  review. Geotherm. Sci. Technol. 4 (4), 227-237.Bouchaud, E., (1997). Scaling properties of cracks. J. Phys.: Cond. Matter 9, 4319-4344.Brown, S.R., (1987). Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. 92, 1337-1347.Brown, S.R., Scholz, C.H., (1985). Broad bandwidth study of the topography of natural rock surfaces. J. Geophys. Res. 90, 12575-12582.Fontaine,  F.,  Rabinowicz,  M.,  Boulègue,  J.,  (2001). Permeability changes due to mineral diagenesis in fractured crust: implications for hydrothermal circulation at mid-ocean ridges. Earth Planet. Sci. Lett. 184, 407-425.Glover, P.W.J., Matsuki, K., Hikima, R., Hayashi, K., (1998). Fluid flow in synthetic rough fractures and application to the hachimantai geomthermal hot rock test site. J. Geophys. Res. 103 (B5), 9621-9635.Hakami, E., Larsson, E., (1996). Aperture mearurements and flow experiemnts on a single fracture. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 33 (4), 395-404.L. Landau E. Lifchitz, (1994). Physique théorique mécanique des fluides, 3rd ed.Méheust, Y., Schmittbuhl, J., (2000). Flow enhancement of a rough fracture. Geophys. Res. Lett. 27, 2989-2992.Méheust,  Y.,  Schmittbuhl,  J.,  (2001).  Geometrical heterogeneities  and  permeability  anisotropy  of  rough fractures. J. Geophys. Res. 106 (B2), 2089-2102.Méheust, Y., Schmittbuhl, J., (2003). Scale effects related to flow in rough fractures. PAGEOPH 160 (5-6), 1023-1050.Murphy, H., (1979). Convective instabilites in vertical fractures and faults. J. Geophys. Res. 84 (B11), 6121-6130.Neuman,  S.,  2005.  Trends,  prospects  and  challenges  in quantifying  flow  and  transport  through  fractured  rocks. Hydrogeol. J. 13, 124-147.Plouraboué, F., Kurowski, P., Hulin, J., Roux, S., Schmittbuhl, J., (1995). Aperture of rough crack. Phys. Rev. E 51, 1675.Schmittbuhl,  J.,  Gentier,  S.,  Roux,  S.,  (1993).  Field measurements of the roughness of fault surfaces.  Geophys. Res. Lett. 20, 639-641.Tournier, C., Genthon, P., Rabinowicz, M., (2000). The onset of natural convection in vertical fault  planes: consequences for the thermal regime in cristalline basements and for heat recovery experiments. Geophys. J. Int. 140, 500-508.Zhao, C., Hobbs, B., Mühlhaus, H., Ord, A., Lin, G., (2003). Convective  instability  of  3-d  fluid  saturated  geological  fault zones heated from below, Geophys. J. Int. 155, 213-220.Figure 4: Temperature profiles (T(x)) for the parallel  plate  case  (exponential  curve  in  black)  and  two different synthetic fractures (blue and red curves).

Hydrothermal coupling in a rough fracture

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