5:15 PM - 7:15 PM
[MIS16-P06] Numerical modelling of heat and fluid transport in thin fractures in the oceanic crust: Solving slow flow fast, Part 5
Keywords:hydrothermal circulation, numerical modelling, explicit method, fractures and faults
Heat flow on the seafloor is expected to depend on the plate age since the temperature structure of old oceanic plates is constrained by the age. Heat flow data observed at the centre of the plate show the expected trends (Stein and Stein, 1992); however, recent heat flow surveys have revealed that heat flow data near the trench axis of subduction zones deviate from the expected value depending on their age. For example, on the seaward side of the Japan Trench, heat flow data show scatter up to twice that of the expected values (Yamano et al., 2008, 2014). In addition, the data show wavelengths on the order of kilometres (Ray et al., 2015; Sasaki, 2017). This scatter data may be attributed to the fluid flow associated with heat transport within fractures and faults. However, at least at this specific site, there has been no detailed three-dimensional modelling of such processes. This study addresses, as a first step, the efficient modelling of fluid circulation within thin fractures and faults.
Numerical modelling of fluid flow in fractures and faults suffers from small time steps due to the small spatial scale (thin) in the system, especially when solving this system by explicit time integration. Thus, this kind of system is usually solved by the implicit methods involving iterations. However, systems with thin fractures tend to have irregular numerical grids (like trump cards) that suffer from poor convergence in the iteration. We therefore adopt a different strategy, taking into account the following ingredients:
(1) Two-dimensional fractures are included in of the three-dimensional system following the method of Yang et al. (1998). In solving conservation equations for the fractures, thin regions are solved with the surrounding region treated as the external heat fluxes. On the other hand, when solving the surrounding area, the fractures are treated as planar heat fluxes without the volume. The numerical grids are in two dimensional for the fractures and three dimensional for the surrounding area, and these do not suffer from the irregular shape.
(2) Asynchronous time integration is introduced between the fractures and the surrounding area, because the time step is limited to a small value even with Step (1) due to the thin fractures themselves. For each time step in the surrounding area, the fractures are integrated in multiple times, keeping the contribution from the surrounding are the same. The timestep of the multi-timesteps should be smaller than the time scale of the thin fracture. Because the number of grids of the fracture is much smaller than that of the surrounding area, the overall computation may not increase a lot.
With the above strategy, we calculate a problem of heat transport within thin (two-dimensional) fractures in the surrounding (three-dimensional) oceanic crust. The system is heated from below and cooled from above. We assume that the surrounding area is impermeable for simplicity. Two types of calculations are carried out:
(a) Heat transport using a high thermal conductivity proxy for hydrothermal circulation (e.g. Spinelli and Wang, 2008). This model replaces the effects of fluid flow with a very high thermal conductivity. Only the heat conduction equation is solved for this model. Because the fractures are thin and the conductivity is high, the timestep limitation must be severe. The asynchronous time stepping allows this system to be calculated fast and in a stable manner.
(b) Heat transport with fluid flow (Darcy flow) within the fractures. This model solves the heat and fluid transport in the fractures as well as the conductive heat transport in the surrounding area. Normally, implicit time integration may be used to obtain the Darcy flow; the present model uses an explicit time integration with the reduced sound speed technique (Kawada, 2020, JpGU). With the asynchronous time integration between the fractures and the surrounding region, calculations can be performed fast and stably to obtain the temperature and flow fields.
Finally, some remarks are added. The present model is based on the finite difference method with control volumes and the grid system is restricted to rectangular shapes. Extending the model to the finite element method includes irregular shaped and interconnected fractures. Also note that the explicit time integration can be further accelerated using the Runge-Kutta method (Meyer et al., 2014), especially when the system is dominated by diffusion like the present system.
Numerical modelling of fluid flow in fractures and faults suffers from small time steps due to the small spatial scale (thin) in the system, especially when solving this system by explicit time integration. Thus, this kind of system is usually solved by the implicit methods involving iterations. However, systems with thin fractures tend to have irregular numerical grids (like trump cards) that suffer from poor convergence in the iteration. We therefore adopt a different strategy, taking into account the following ingredients:
(1) Two-dimensional fractures are included in of the three-dimensional system following the method of Yang et al. (1998). In solving conservation equations for the fractures, thin regions are solved with the surrounding region treated as the external heat fluxes. On the other hand, when solving the surrounding area, the fractures are treated as planar heat fluxes without the volume. The numerical grids are in two dimensional for the fractures and three dimensional for the surrounding area, and these do not suffer from the irregular shape.
(2) Asynchronous time integration is introduced between the fractures and the surrounding area, because the time step is limited to a small value even with Step (1) due to the thin fractures themselves. For each time step in the surrounding area, the fractures are integrated in multiple times, keeping the contribution from the surrounding are the same. The timestep of the multi-timesteps should be smaller than the time scale of the thin fracture. Because the number of grids of the fracture is much smaller than that of the surrounding area, the overall computation may not increase a lot.
With the above strategy, we calculate a problem of heat transport within thin (two-dimensional) fractures in the surrounding (three-dimensional) oceanic crust. The system is heated from below and cooled from above. We assume that the surrounding area is impermeable for simplicity. Two types of calculations are carried out:
(a) Heat transport using a high thermal conductivity proxy for hydrothermal circulation (e.g. Spinelli and Wang, 2008). This model replaces the effects of fluid flow with a very high thermal conductivity. Only the heat conduction equation is solved for this model. Because the fractures are thin and the conductivity is high, the timestep limitation must be severe. The asynchronous time stepping allows this system to be calculated fast and in a stable manner.
(b) Heat transport with fluid flow (Darcy flow) within the fractures. This model solves the heat and fluid transport in the fractures as well as the conductive heat transport in the surrounding area. Normally, implicit time integration may be used to obtain the Darcy flow; the present model uses an explicit time integration with the reduced sound speed technique (Kawada, 2020, JpGU). With the asynchronous time integration between the fractures and the surrounding region, calculations can be performed fast and stably to obtain the temperature and flow fields.
Finally, some remarks are added. The present model is based on the finite difference method with control volumes and the grid system is restricted to rectangular shapes. Extending the model to the finite element method includes irregular shaped and interconnected fractures. Also note that the explicit time integration can be further accelerated using the Runge-Kutta method (Meyer et al., 2014), especially when the system is dominated by diffusion like the present system.