Hydrothermal coupling in a self-affine rough fracture

The influence of the multiscale fracture roughness on the heat exchange when a cold fluid enters a fractured hot solid is studied numerically on the basis of the Stokes equation and in the limit of both hydrolubrication and thermolubrication. The geometrical complexity of the fracture aperture is modeled by small self-affine perturbations added to a uniform aperture field. Thermal and hydraulic properties are characterized via the definition of hydraulic and thermal apertures both at microscopic and macroscopic scales and obtained by comparing the fluxes to the ones of flat fractures. Statistics over a large number of fracture configurations provide an estimate of the average behavior and its variability. We show that the long-range correlations of the fracture roughness induces strong channeling effects that significantly influence the hydraulic and thermal properties. An important parameter is the aspect ratio (length over width) of the fracture: we show, for example, that a downstream elongated rough fracture is more likely to inhibit the hydraulic flow and subsequently to enhance the thermal exchange. Fracture roughness might, in the opposite configuration, favor strong channeling which inhibits heating of the fluid. The thermal behavior is in general shown to be mainly dependent on the hydraulic one, which is expressed through a simple law.

Figure 1

2D sketch of the fracture model with parameter definitions. $x$ axis is along the mean hydraulic flow, $y$ axis is along the mean fracture plane but perpendicular to the main hydraulic flow, and $z$ axis denotes the out-of-mean fracture plane direction. $z=z_1$ and $z=z_2$ are the average positions of the facing fracture surfaces. $a\left(x,y\right)$ is the fracture aperture. $T_r$ is the temperature of the solid, supposed to be homogeneous and constant, $T_0$ is the fluid temperature at the inlet. Fluid properties are $\rho$, $c$, $\chi$, and $\eta$, which are, respectively, density, heat capacity, thermal diffusivity, and dynamic viscosity.

Figure 2

(Color online) Local velocity quadratic profile (short dashed line) and temperature quartic profile (long dashed line) inside a fracture [with coefficients from Eqs. (9, 4)]; arbitrary abscissa units. Along the contact with the fracture, $\mathit{v}=\mathbf{0}$ and $T=T_r$.

Figure 3

(Color online) Fracture model with pressure and temperature boundary conditions.

Figure 4

(Color online) (a) Self-affine aperture with $\sigma /A=0.25$. (b) Dimensionless hydraulic flow norm computed with the aperture of (a), having for dimensionless hydraulic aperture $H^{\ast }=0.94$. (c) Microscopic hydraulic apertures, computed from the third root of the hydraulic flow shown in (b).

Figure 5

(Color online) 2D histogram of the link between the microscopic hydraulic aperture and the microscopic mechanical aperture for the fracture shown in Fig. 4 (the scale indicates the probability in percent); the cross has for coordinates $\left(H/A,⟨a⟩/A\right)=\left(0.94,1\right)$. The straight line is $h=a$, which is the equality given by a local Poiseuille law.

Figure 6

(Color online) Macroscopic hydraulic aperture $H/A$ versus $\sigma /A$ for fractures with aspect ratio $l_x/l_y=2$; crosses, variation of the hydraulic aperture by increasing the roughness amplitude $\sigma /A$ for the aperture shown in Fig. 4; dots, cloud of computed data (about $20000$ aperture realizations); squares, average hydraulic behavior with variability bars. On average, $H/A<1$: the permeability is smaller than expected from the Poiseuille law in parallel plate apertures.

Figure 7

(Color online) Macroscopic hydraulic aperture versus $\sigma /A$, for three aspect ratios $J=l_x/l_y$. Averages computed from data are shown with symbols, with error bars, corresponding to plus or minus the standard deviation (see how the average is computed in Sec. 4C). $J=l_x/l_y=2$ shows an enhanced flow (same data as presented in Fig. 6); $J=1$ shows on average a slightly inhibited flow, i.e., $H\le A$ (computed from a cloud of about 1300 points); for $J=0.5$, on average, higher permeability is observed (computed from a cloud of about 900 points). Continuous curves are fitting models (1) $H/A=1+\alpha {\left(\frac{\sigma }{A}\right)}^{\kappa }$, with parameters $\left(\kappa ,\alpha \right)$ equal to $\left(2.05,-1.46\right)$, $\left(1.57,-0.31\right)$, (2.69,0.67), respectively, for $J$ equal to 2, 1, and 0.5. Dotted curves are obtained with fitting models (2) $H/A=1-\mu \left[{\text{log}}_2\left(J\right)+\delta \right]{\left(\frac{\sigma }{A}\right)}^{\kappa }$, with $\left(\mu ,\delta ,\kappa \right)=\left(0.98,0.59,2.16\right)$, for the three curves.

Figure 8

(Color online) $-\text{ln}\left({\overline{T}}^{\ast }\right)$, opposite of the logarithm of the temperature field ${\overline{T}}^{\ast }$ computed from the aperture morphology pattern shown in Fig. 4 with a very low roughness amplitude: $\sigma /A=0.05$. The hydraulic aperture of this fracture is $H/A=0.99$. The temperature field exhibits a normalized thermal length equal to $R^{\ast }=0.97$ and a thermal aperture of $\Gamma /A=0.99$.

Figure 9

(Color online) (a) $-\text{ln}\left({\overline{T}}^{\ast }\right)$, opposite of the logarithm of the 2D temperature field, computed from the apertures in Fig. 4a $\left(\sigma /A=0.25\right)$. (b) Normalized local thermal aperture $\gamma /A$ associated with the temperature field shown in (a).

Figure 10

(Color online) 2D histogram in percent of the fracture shown in Fig. 4 as a function of the local thermal aperture $\gamma$ and local aperture $a$ (the shading indicates the probability density). The straight line is $\gamma =a$. The dispersivity of the cloud around the line shows that there is no simple link between the local aperture and the thermal one.

Figure 11

(Color online) 2D Histogram in percent of the fracture shown in Fig. 4 as a function of the local thermal aperture $\gamma$ and local hydraulic aperture $h$ (the scale indicates the probability in percent). The straight line is $\gamma =h$; the localization of the cloud around the line shows a good correlation between $\gamma$ and $h$.

Figure 12

(Color online) Continuous curve, $-\text{ln}\left({\overline{\overline{T}}}^{\ast }\right)$, opposite of the logarithm of the temperature field computed from the temperature field $\overline{T}$ shown in Fig. 9. Dashed-dotted curve, linear fit of curve A (from $x/d=0$ to $x/d=772$), which provides the thermal length: $-\text{ln}\left({\overline{\overline{T}}}^{\ast }\right)=x/1.09+0.6$, i.e., $R^{\ast }=1.09$. Dashed curve, $-\text{ln}\left({\overline{\overline{T}}}_{\parallel }^{\ast }\right)$, opposite of the logarithm of the temperature law for the same fracture modeled without self-affinity perturbation (i.e., parallel plates), which has for thermal length $R_{\parallel }^{\ast }=1$.

Figure 13

(Color online) Crosses, variation of the thermal aperture $\Gamma /A$ by increasing the roughness amplitude $\sigma /A$ for the aperture pattern shown in Fig. 4; dots, cloud of computed data (about $20000$ points) for fractures with aspect ratio $l_x/l_y=2$; triangles, average thermal behavior with variability bars of the cloud; squares, average hydraulic aperture $H/A$ versus $\sigma /A$, recalled here for comparison.

Figure 14

(Color online) Normalized thermal aperture $\Gamma /A$ versus $H/A$ for fractures with aspect ratio $l_x/l_y=2$. Crosses, variation of the thermal aperture by increasing the roughness amplitude for the aperture pattern shown in Fig. 4a versus $H/A$; dots, cloud of computed data (about $20000$ points); squares, average thermal behavior with variability bars. Continuous curve, $\Gamma /A=H/A$, which holds for parallel plates separated by $a\left(x,y\right)=H$.

Figure 15

(Color online) Averages of the normalized thermal aperture $\Gamma /A$ and their deviation bars versus $\sigma /A$ for various aspect ratios $J=l_x/l_y$, as indicated by the labels. See how the average is computed in Sec. 4C.

Figure 16

(Color online) Averages of the normalized thermal aperture $\Gamma /A$ and their deviation bars versus $H/A$ for various aspect ratios $J=l_x/l_y$, as indicated by the labels (see how the average is computed in Sec. 4C). Models lines are $\Gamma =0.9H+0.2A$ for $H<A$ and $\Gamma =3.5H-2.4A$ for $H\ge A$; no continuity condition between both lines is imposed.