The onset of natural convection in vertical fault planes: Consequences for the thermal regime in crystalline basements and for heat recovery experiments

Abstract

Very few results are available on free convection in fractured zones, although this is a major mechanism for heat and mass transfer in crystalline rocks. Murphy (1979) has shown, using analytical stability analysis, that the critical Rayleigh number for the onset of free convection in a fracture greatly exceeds the value of 4π2, which is the value for an infinite porous medium, and, even for a subcritical Rayleigh number, convection may occur after a time delay. Murphy proposed that this delayed convection results from a blanketing effect of the fracture induced by the progressive development of a thermal skin inside the fracture walls. The present paper extends Murphy's results by means of numerical modelling. Our numerical method involves a 2-D computation of convection in the fracture plane, and a 3-D solution of the conduction problem inside the fracture wall. The coupling of the codes is achieved by imposing a common temperature at the mid-fracture plane, together with the conservation of energy at the fracture-wall interface. We use two kinds of initial perturbation, which are assumed to constitute end-members for natural or application cases. For an A-type initial condition the thermal field is disrupted in the fracture only, while for a B-type initial condition the perturbation is introduced in the fracture and in the walls. For a given perturbation wavenumber, three distinct domains can be defined according to the Rayleigh number (R). In the first domain, convection takes place immediately; in the second one, convection starts after a delay; and in the third one, convection is damped. These three domains are therefore termed the instantaneous convection (R > R(s)), delayed convection (R(d) < R < R(s)), and conduction (R < R(d)) domains, respectively. It is noteworthy that these three domains are bounded by the same values of the Rayleigh number for both A-type and B-type perturbations. Except for R close to R(d), the time lapse required for the onset of delayed convection is less than 10 conductive times of the fracture width. This time lapse is therefore negligible compared with geological timescales, and R(d) can be considered as the critical Rayleigh number for the onset of free convection in a fracture zone. However, the characteristic conductive time of a fracture, which amounts, for example, to 3 years for a 10-m thick fracture, must be considered for thermal recovery experiments. It is suggested that convection may start during exploitation and will be superimposed on the forced flow. We expect that this will enhance the efficiency of thermal recovery.