## 1 Introduction

[2] Conductive and convective transport of heat or chemical species, is omnipresent in Earth sciences [*Stephansson et al.*, 2004; *Steefel et al.*, 2005], within porous or fractured media. Some technologies related to chemical transport, like radioactive storage [*Cvetkovic et al.*, 2004; *Amaziane et al.*, 2008; *Halecky et al.*, 2011; *Hoteit et al.*, 2004] or well acidizing requires a good resolution of advection-diffusion of chemical concentration [*Szymczak and Ladd*, 2009; *Cardenas et al.*, 2007]. The transport is influenced by the temperature which may play a role by modifying (1) the fluid transport—notably by natural convection, (2) the chemical constants of the reactions, or (3) the geometry of the porous medium. For instance, some chemical reactions or rheological rock transformations only occur in given ranges of temperature (e.g., decarbonation, dehydration of sediments, decomposition of kerogen [e.g., *Mollo et al.*, 2011; *Petersen et al.*, 2010]). Thermal fracturing can result from chemical reactions generating gas [*Kobchenko et al.*, 2011], from thermal stress [*Lan et al.*, 2013; *Bergbauer et al.*, 1998], or from hydro-thermal stress, when injecting hot or cold fluid into a rock. Temperature monitoring is, for example, necessary to prevent potential damages of well installations. Apart from fracturing processes, other changes of geometry of the porous medium can also occur during injection of cold or warm fluid in Enhanced Geothermal Systems (EGS) because of poroelastic effects [*Gelet et al.*, 2012] and also because of chemical reactions like acidizing. Exploitation of EGS requires also the heat exchange to be efficient and durable. An important step is, therefore, to understand the hydro-thermal coupling between fluid and rock; this is the aim of our present modeling.

[3] Hydraulic transport mostly occurs in fractures. It was shown under lubrication approximations and steady state conditions, that the complexity of the fracture topography influences the hydrothermal exchange when a cold fluid is injected into a hot fractured bedrock [*Neuville et al.*, 2010]. More specifically, the lubrication approximations state that the aperture and its wall topographies vary smoothly so that the velocity field is parallel to the main plane of the fracture (the component of the hydraulic flow perpendicular to the main fracture plane is neglected), and that the thermal diffusion only occurs in the directions perpendicular to the main plane of the fracture. For this modeling, the rock temperature was also supposed to be constant. In other models, as, e.g., in the study of *Natarajan and Kumar* [2010], the heat diffusion in the rock is considered, but with a simplified fluid flow. Some features which are observed in nature, like fluid recirculation and time-dependent temperature at the pumping well, can, however, not be explained with this model. We, therefore, wish to go beyond this lubrication assumption and be especially able to observe effects due to highly variable morphology of the fluid-rock interface. Indeed, even if many fluid-rock interface topographies or fracture apertures are statistically self-affine (multi-scale property) [*Brown and Scholz*, 1985; *Bouchaud*, 1997; *Neuville et al.*, 2012; *Candela et al.*, 2009], it is very often possible to observe some isolated asperities with sharp variations of the topography, for instance, along cleavage planes [*Neuville et al.*, 2012] or due to particle detachment, or along stylolite teeth [*Renard et al.*, 2004; *Ebner et al.*, 2010; *Koehn et al.*, 2012; *Laronne Ben-Itzhak et al.*, 2012; *Rolland et al.*, 2012]. The roughness of the fracture can also be perturbed by intersections with other fractures [*Nenna and Aydin*, 2011] or, for microfractures, by the matrix porosity [*Renard et al.*, 2009].

[4] Investigations on the validity of the lubrication approximation, based on the study of some geometrical parameters have been performed, e.g., by *Zimmerman and Bodvarsson* [1996]; *Oron and Berkowitz* [1998]; and *Nicholl et al.* [1999]. Without the lubrication approximation, i.e., with full solving of the Navier-Stokes equation, the hydraulic behavior in channels or fracture with sinusoidal walls were studied, e.g., by *Brown et al.* [1995]; *Waite et al.* [1998]; and *Bernabé and Olson* [2000] with lattice gas methods. It was shown that for a sinusoid with a short wavelength and large amplitude compared to the mean aperture, the hydraulic aperture is smaller than that expected with the lubrication approximation. Errors on the hydraulic aperture, computed with a lubrication approximation, have been analytically obtained by *Zimmerman and Bodvarsson* [1996] and experimentally by *Oron and Berkowitz* [1998]; and *Nicholl et al.* [1999]. In these studies, the fluid flow was reported to be important in the middle of the channel, while it is quasi-stagnant in the sharp hollows of the walls. Eddies were numerically observed in this quasi-stagnant zones [*Brown et al.*, 1995; *Brush and Thomson*, 2003; *Boutt et al.*, 2006; *Cardenas et al.*, 2007; *Andrade Jr. et al.*, 2004] in various fracture geometries, including realistic fracture geometries. These eddies are similar to those analytically predicted by *Moffatt* [1964], who studied eddies formation in a corner between two intersecting planes, when the flow is imposed to be tangential to the planes at an infinite distance from the corner. Microfluidics has also been investigated using lattice Boltzmann methods (LBM) [e.g., *Harting et al.*2010].

[5] On the one hand, many studies exist about the coupling between the fully solved hydraulic behavior—solving of the Navier-Stokes equation—and solute or particle transport, or dispersion in general [*Boutt et al.*, 2006; *Cardenas et al.*, 2007; *Drazer and Koplik*, 2001, 2002; *Flekkøy*, 1993; *Yeo*, 2001; *Johnsen et al.*, 2006; *Niebling et al.*, 2010; *Vinningland et al.*, 2012].

[6] On the other hand, few studies take into account the fully solved hydraulic flow with the heat transfer, when a cold fluid is injected into a fracture embedded in a hot solid. In the absence of flow, the stationary problem of heat transport across a fractal interface was studied, e.g., by *Grebenkov et al.* [2007]. Even if both heat and solute transport are described with advection-diffusion equation, many differences exist due to different boundary conditions and different range of parameters. *Andrade Jr. et al.* [2004] solved the temperature field when a warm fluid is injected within a two-dimensional (2-D) channel delimited by walls whose topographies follow Koch fractals, using the “Semi-Implicit Method for Pressure Linked Equations” algorithm developed by *Patankar* [1980]. In his study, the walls of the channel were, however, set at a constant temperature. Other studies regarding cooling issues of electrical devices have also been done [e.g., *Young and Vafai*, 1998 and references therein], but in contrast to the current study, the thermal boundary conditions used in these works are less relevant for natural problems (for instance, insulated walls are used). Another branch of algorithms for heat solving are based on lattice Boltzmann methods (LBM). The LBM [e.g., *Wolf-Gladrow*, 2005] are very suitable to model the complexity of hydrothermal transport in a rough fracture morphology, whatever the slopes of the morphology (i.e., without any conditions on the smoothness of the morphology). As the algorithms require only local operations (while other numerical methods requires, e.g., inversion of matrices depending on the geometry of the entire porous medium), they handle complex boundaries very well. Several methods have been proposed: multispeed scheme (a density distribution is used, with additional speeds and high order velocity terms in the equilibrium distribution), hybrid method (hydraulic flow is solved with LBM, and heat transport is solved with another method), and double distribution. A review of these methods can be found in *Lallemand and Luo* [2003]; and *Luan et al.* [2012]. Most of the problems solved so far with LBM deal with benchmarks that consist in close systems (square cavity with impermeable walls). In this paper we will examine the limits of the lubrication approximation and the model developed in *Neuville et al.* [2010]. After briefly recalling the fundamentals of this model, we describe the numerical methods used for our modeling outside the lubrication approximations. We chose a LBM with a double distribution method, where the hydraulic flow is independent of the temperature. The chosen lattices for the flow and temperature variables are respectively hypercubic and cubic, with a single lattice speed for each lattice. This choice is seldom used in literature, but it is suitable for three-dimensional (3-D) mass and heat transport modeling [*d'Humières et al.*, 1986; *Wolf-Gladrow*, 2005; *Hiorth et al.*, 2008]. Despite the methods being implemented in 3-D, for simplicity reasons, the parameter exploration is done in two dimensions, translational invariance being assumed along the third dimension. For a self-affine aperture, the contribution of each asperity on the total hydraulic and thermal exchanges is difficult to single out. For this reason, here we chose as a simpler situation to focus on one single asperity. We can thus lead a precise quantitative study of the flow organization and properties in space and time as function of flow speed, asperity size and shape, and heat transport properties.

[7] We will thus explore the behavior of the flow in a fracture with a triangular asperity, as function of the asperity size. We will compare the results directly to the lubrication approximation results and establish when this one fails to model correctly the mass and heat transport.