Thermal convection in three-dimensional fractured porous media

Thermal convection is numerically computed in three-dimensional (3D) fluid saturated isotropically fractured porous media. Fractures are randomly inserted as two-dimensional (2D) convex polygons. Flow is governed by Darcy's 2D and 3D laws in the fractures and in the porous medium, respectively; exchanges take place between these two structures. Results for unfractured porous media are in agreement with known theoretical predictions. The influence of parameters such as the fracture aperture (or fracture transmissivity) and the fracture density on the heat released by the whole system is studied for Rayleigh numbers up to 150 in cubic boxes with closed-top conditions. Then, fractured media are compared to homogeneous porous media with the same macroscopic properties. Three major results could be derived from this study. The behavior of the system, in terms of heat release, is determined as a function of fracture density and fracture transmissivity. First, the increase in the output flux with fracture density is linear over the range of fracture density tested. Second, the increase in output flux as a function of fracture transmissivity shows the importance of percolation. Third, results show that the effective approach is not always valid, and that the mismatch between the full calculations and the effective medium approach depends on the fracture density in a crucial way.

Figure 1

An isotropic 3D fracture network composed of hexagonal fractures. The colors are only used to distinguish the various hexagons.

Figure 2

Evolution of the temperature field in a vertical cross section of a pseudo-two-dimensional box ($1\times 1\times 0.025$). $\text{Ra}=50$, ${\delta }_M^{\prime }=0.0167$. (a) The simulation starts from a uniform temperature field. (b) A conductive temperature field is established. (c) Because of the overcritical Ra, the conductive temperature field is destabilized. (d) Steady-state 2D convection for $\text{Ra}=50$.

Figure 3

The Nusselt number as a function of time for $\text{Ra}=50$ in a square unit box. Data for ${\delta }_M^{\prime }=0.033$ (solid red line), 0.025 (dashed blue line), and 0.017 (dotted green line).

Figure 4

(a) The Nusselt number as a function of the dimensionless mesh size ${\delta }_M^{\prime }$. $\text{Ra}=50$. Black dots correspond to simulations; the red star is the extrapolated value ${\mathrm{Nu}}_{\mathrm{ref}}$, for ${\delta }_M^{\prime }=0$. (b) The deviation $\mathrm{\Delta }\mathrm{Nu}=|\text{Nu}-{\mathrm{Nu}}_{\mathrm{ref}}|/{\mathrm{Nu}}_{\mathrm{ref}}$ between our simulations for various ${\delta }_M^{\prime }$, and the extrapolated ${\mathrm{Nu}}_{\mathrm{ref}}$. Data for ${\delta }_M^{\prime }=0.05$ (red dots), 0.033 (blue squares), 0.025 (black triangles), and 0.017 (green diamonds). The grey area corresponds to errors smaller than 2%.

Figure 5

The Nusselt number as a function of the Rayleigh number Ra. Red squares and blue dots are for simulations of two-dimensional and three-dimensional convection in homogeneous porous media, respectively. Red squares denote two-dimensional convection [Eq. (24a)], and blue circles denote three-dimensional convection [Eq. (24b)].

Figure 6

The Nusselt number as a function of the matrix Rayleigh number ${\mathrm{Ra}}_m$. The solid line is the interpolation (24b). Data for ${\rho }^{\prime }=2{\sigma }^{\prime }=1$ (purple triangles), ${\rho }^{\prime }=3$ and ${\sigma }^{\prime }=5$ (green stars), ${\rho }^{\prime }=4$ and ${\sigma }^{\prime }=1$ (red dots), and ${\rho }^{\prime }=7$ and ${\sigma }^{\prime }=2$ (blue squares).

Figure 7

The temperature field in the two orthogonal vertical midplanes at steady state for three different dimensionless fracture transmissivities: (a), (d) ${\sigma }^{\prime }=0.1$; (b), (e) 1; and (c), (f) 5. Data for ${\rho }^{\prime }=4$ and ${\mathrm{Ra}}_m=41.5$.

Figure 8

The Nusselt number as a function of the dimensionless fracture transmissivity ${\sigma }^{\prime }$. Data for ${\rho }^{\prime }=8$ (red circles), 4 (blue diamonds and green triangles), and 1 (black squares); ${\mathrm{Ra}}_m=41.5$. Solid and dashed lines correspond to small and large Nu increments, respectively. The error bars correspond to $\pm 2{\sigma }_d$, ${\sigma }_d$ being the standard deviation observed over a set of 18 realizations (Fig. 10).

Figure 9

The temperature field in two perpendicular vertical midplanes at steady state for (a), (d) ${\rho }^{\prime }=2$, (b), (e) ${\rho }^{\prime }=6$, and (c), (f) ${\rho }^{\prime }=10$. Data for ${\sigma }^{\prime }=1$ and ${\mathrm{Ra}}_m=41.5$.

Figure 10

The average Nusselt number as a function of the dimensionless fracture density ${\rho }^{\prime }$ for ${\sigma }^{\prime }=1$ (blue circles), ${\sigma }^{\prime }=2$ (green squares), and ${\mathrm{Ra}}_m=41.5$ (the red star represents ${\mathrm{Nu}}_{\mathrm{mat}}$). Solid and broken lines are the linear fits [Eqs. (25a) and (25b)].

Figure 11

The Nusselt number as a function of ${\mathrm{Ra}}_{\mathrm{eff}}$. Black dots represent Nu with the FPM approach for $0.1\le {\sigma }^{\prime }\le 10$, $0.5\le {\rho }^{\prime }\le 10$, and $7\le {\text{Ra}}_m\le 139$. The dotted red line and solid blue line stand for two- and three-dimensional convection in a HPM.

Figure 12

The Nusselt number ${\mathrm{Nu}}_{\text{HPM}}$ computed in a homogeneous porous medium vs the Nusselt number ${\mathrm{Nu}}_{\text{FPM}}$ computed in the discrete fracture medium, both presenting the same effective permeability. Data for two-dimensional convection (red dots) and three-dimensional convection (blue squares); one dot represents one realization, and the first bisector is shown. Additional data: $7<{\mathrm{Ra}}_m<139$, $0.5<{\rho }^{\prime }<10$, and $0.1<{\sigma }^{\prime }<10$.

Figure 13

The Nusselt number ${\mathrm{Nu}}_{\text{HPM}}$ computed in a homogeneous porous medium as a function of the Nusselt number ${\mathrm{Nu}}_{\text{FPM}}$ computed in the discrete fracture medium with the same effective permeability. One dot represents one realization, and the first bisector is added for clarity. Data for (a) ${\rho }^{\prime }=1$ and 2, (b) ${\rho }^{\prime }=3$ and 4, (c) ${\rho }^{\prime }=5$, 6, and 7, and (d) ${\rho }^{\prime }=8$ and 10.

Figure 14

The Nusselt number as a function of the dimensionless density of fractures, ${\rho }^{\prime }$, for ${\sigma }^{\prime }=1$ (red dots) and ${\mathrm{Ra}}_m=41.5$. Dots represent the statistical average over 18 different networks with the same ${\rho }^{\prime }$. Error bars represent the standard deviations. Solid green and dotted blue curves are the predictions for Nu deduced from Eqs. (24a) and (26a) and Eqs. (24b) and (26b), respectively, with solid and dotted lines in their range of validity.

Figure 15

The deviation $\mathrm{\Delta }\mathrm{Nu}=|N{u}_{\text{FPM}}-N{u}_{\text{HPM}}|/N{u}_{\text{FPM}}$ between fractured and homogeneous porous media with the same effective permeability as a function of the dimensionless transmissivity of fractures ${\sigma }^{\prime }$ for a density ${\rho }^{\prime }=4$ and $12<{\mathrm{Ra}}_m<138$. Each data point corresponds to one realization.

Figure 16

The equivalent Rayleigh number ${\mathrm{Ra}}_{\mathrm{eq}}$ as a function of the effective Rayleigh number ${\mathrm{Ra}}_{\mathrm{eff}}$ for four individual fracture networks. Data for ${\rho }^{\prime }=1$, ${\sigma }^{\prime }=5$ (red circles); ${\rho }^{\prime }=3$, ${\sigma }^{\prime }=5$ (blue triangles); ${\rho }^{\prime }=4$, ${\sigma }^{\prime }=0.1$ (blue squares); and ${\rho }^{\prime }=8$, ${\sigma }^{\prime }=1$ (blue stars). Solid and broken lines correspond to Eq. (28).

Figure 17

The linear relation (29): $a$ as a function of (a) $B$, (b) ${\rho }^{\prime }$, (c) $b^{\prime }$, and (d) $K_z^{\prime }=2K_z/(K_x+K_y)$. The solid line in (a) is $a=1-b/100$ and that in (d) is $a=3.4-2.5K_z^{\prime }$. Red dots and blue squares represent nonpercolating and percolating networks, respectively.